Physique statistique d'objets biologiques et des électrolytes aux

Université Paul Sabatier – Toulouse III

Habilitation à Diriger des Recherches

Spécialité : Physique Théorique

présentée par

Manoel MANGHI

Laboratoire de Physique Théorique(UMR UPS-CNRS 5152)

IRSAMC, Université Paul Sabatier, Toulouse

Physique statistique d’objets biologiqueset des électrolytes aux interfaces

soutenue publiquement le 18 novembre 2010 devant le jury composé de

Alois Würger Professeur à l’Université Bordeaux 1 PrésidentJean-François Joanny Professeur à l’Université Paris 6 RapporteurAnthony Maggs Directeur de recherche CNRS à l’ESPCI RapporteurEmmanuel Trizac Professeur à l’Université Paris Sud RapporteurNicolas Destainville Professeur à l’Université Toulouse 3 ExaminateurDavid Dean Professeur à l’Université Toulouse 3 Directeur

Université Paul Sabatier – Toulouse III

Habilitation à Diriger des Recherches

Spécialité : Physique Théorique

présentée par

Manoel MANGHI

Laboratoire de Physique Théorique(UMR UPS-CNRS 5152)

IRSAMC, Université Paul Sabatier, Toulouse

Physique statistique d’objets biologiqueset des électrolytes aux interfaces

soutenue publiquement le 18 novembre 2010 devant le jury composé de

Alois Würger Professeur à l’Université Bordeaux 1 PrésidentJean-François Joanny Professeur à l’Université Paris 6 RapporteurAnthony Maggs Directeur de recherche CNRS à l’ESPCI RapporteurEmmanuel Trizac Professeur à l’Université Paris Sud RapporteurNicolas Destainville Professeur à l’Université Toulouse 3 ExaminateurDavid Dean Professeur à l’Université Toulouse 3 Directeur

Remerciements

Ils vont tout d’abord à mes collaborateurs du laboratoire de Physique Théorique.Nicolas Destainville a joué un rôle important dans ma venue au laboratoire et y a faci-

lité mon intégration. Je lui en suis très reconnaissant. J’apprécie énormément à la fois sarigueur mathématique et sa passion pour les questions à l’interface physique-biologie.

John Palmeri me fait bénificier avec beaucoup de disponibilité de sa grande culturescientifique qui va de la physique de la matière condensée à la chimie des solutions et jel’en remercie.

David Dean me fait partager ses connaissances en physique statistique. Discuter aveclui s’avère toujours enrichissant.

J’ai eu le plaisir de travailler avec Sahin Buyukdagli pendant deux années très fruc-tueuses.

Roland Netz a bien voulu parfaire ma formation de checheur lors de mon stage post-doctoral. J’ai beaucoup appris de sa grande curiosité et inventivité scientifique et je l’enremercie, lui qui, de près ou de loin, garde un œil sur mes travaux.

J’apprécie les nombreuses conversations très instructives sur la biophysique et la biolo-gie avec Laurence Salomé, Catherine Tardin, Evert Haanappel, Philippe Rousseau : à euxtous merci encore.

Je remercie les rapporteurs Jean-François Joanny, Tony Maggs et Emmanuel Trizacqui ont bien voulu s’intéresser à mon travail. Merci à Alois Würger d’avoir bien vouluprésider le jury.

Enfin, c’est un grand plaisir de participer à la vie scientifique du LPT. Je remercieici tous ses membres, permanents ou non, avec qui je partage de nombreuses discussionsvariées et enrichissantes, toujours dans la bonne humeur.

à mes fils,Arthur et Marin

Table des matières

Préambule 1

1 Modélisation tri-dimensionnelle de l’ADN 31.1 État des lieux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Importance des configurations spatiales . . . . . . . . . . . . . . . . . . . . . 91.3 Couplage entre les degrés de libertés d’Ising et de la chaîne . . . . . . . . . 12

1.3.1 Résumé des principaux résultats . . . . . . . . . . . . . . . . . . . . 141.3.2 Trois articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Application à l’ADN observé par microscopie à force atomique . . . . . . . . 631.4.1 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1.5 Remarques et perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2 Membranes fluctuantes 792.1 Interactions entre domaines lipidiques dans les membranes fluctuantes . . . 80

2.1.1 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.2 Membranes empilées . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.2.1 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3 Ions aux interfaces : effets diélectriques et entropiques 1093.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.2 Théorie des champs pour les électrolytes . . . . . . . . . . . . . . . . . . . . 110

3.2.1 Approche champ moyen et limite de faible couplage . . . . . . . . . . 1123.2.2 Calcul à un boucle : théorie de Debye-Hückel . . . . . . . . . . . . . 1133.2.3 Développement du viriel et limite de fort couplage . . . . . . . . . . 1143.2.4 Images électriques et charges de polarisation . . . . . . . . . . . . . . 115

3.3 Approche variationnelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.4 Application au cas d’une solution d’électrolyte . . . . . . . . . . . . . . . . . 1183.5 Electrolyte proche d’interfaces diélectriques planes . . . . . . . . . . . . . . 119

ix

TABLE DES MATIÈRES

3.5.1 Paroi plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.5.2 Pore en forme de fente . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.5.3 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.6 «Évaporation capillaire» des ions dans les nanopores cylindriques . . . . . . 1463.6.1 Pore neutre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.2 Nanopores faiblement chargés . . . . . . . . . . . . . . . . . . . . . . 1483.6.3 Traces expérimentales de cette transition discontinue . . . . . . . . . 1503.6.4 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.7 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4 Un phénomène d’élasto-hydrodynamique :propulsion par un flagelle élastique 1634.1 Bifurcation élastique et propulsion à petit nombre de Reynolds . . . . . . . 1634.2 Confirmation expérimentale . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.2.1 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Conclusion générale 171

Bibliographie 173

x

Préambule

Dans ce mémoire sont présentés les résultats de mes travaux de recherche en physiquethéorique effectués entre 2004 et 2010. Comme souvent de nos jours, ils sont le fruit decollaborations intenses et très fructueuses que je présente ci-dessous dans un ordre chro-nologique.

Une partie de mes travaux de recherche effectués durant mon stage postdoctoral avecRoland Netz, de l’université de Munich, sont présentés dans le chapitre 4. Il s’agit del’étude numérique de la propulsion dans l’eau à très petits nombres de Reynolds, en déve-loppant un modèle simple de flagelle élastique en rotation. Le domaine de recherche toucheici à la fois à l’étude biophysique de la propulsion de certaines bactéries [186] (commel’Escherichia coli) mais aussi à la nanofluidique où la conception de propulseurs biomimé-tiques à très petite échelle est un des objectifs [104, 84, 63].

Les trois premiers chapitres du mémoire portent sur des travaux effectués au Labora-toire de Physique Théorique (LPT) de Toulouse, dans l’équipe de Physique Statistique desSystèmes Complexes (PhyStat) dont je fais partie depuis février 2005. Dans le chapitre 1,une nouvelle approche théorique de la physique de l’ADN est développée, qui met en lu-mière le couplage entre sa structure interne, les paires de bases formées par des liaisonsHydrogène et qui peuvent être dans les états ouvert ou fermé, et les conformations spatialesde la chaîne qui fluctuent sous l’influence de la témpérature. Ces recherches ont été faitesen collaboration avec Nicolas Destainville et John Palmeri du LPT.

Le chapitre 2 concerne l’étude de deux questions mettant en jeu des membranes lipi-diques fluctuantes : le couplage entre l’hétérogénéité de la membrane et ses fluctuationsspatiales, en collaboration avec David Dean du LPT, et l’influence de l’empilement desdeux membranes sur leur interaction et leur dynamique avec N. Destainville.

Enfin le chapitre 3 est dédié à l’étude des électrolytes près d’interfaces diélectriques.En collaboration avec J. Palmeri et Sahin Buyukdagli, post-doctorant au LPT, nousétudions théoriquement l’exclusion diélectrique dans des pores de taille nanométrique. Cemécanisme physique est utilisé dans le domaine de la nanofiltration, qui est à la fois unsujet de recherche fondamentale [204] et la source d’enjeux industriels (comme par exemplepour le déssalement de l’eau de mer [194]), mais joue également un rôle dans le fonction-

1

TABLE DES MATIÈRES

nement des canaux biologiques situés dans les membranes cellulaires [90].

Mon domaine de recherche est donc à la croisée des chemins entre la physique de lamatière molle et la biophysique. Un point commun à tous ces travaux serait la physiquestatistique de molécules ou assemblages moléculaires (ADN, membrane lipidique, ions etflagelle) qui ont une structure interne complexe ou qui sont en interaction avec le milieuenvironnant (substrat, eau).

Depuis une dizaine d’années, l’équipe PhyStat du laboratoire a débuté une collaborationfructueuse avec l’équipe d’expérimentateurs biophysiciens de l’Institut de Pharmacologieet Biologie Structurale (IPBS) de Toulouse dirigée par Laurence Salomé. Je participedepuis mon arrivée au LPT à cette collaboration qui se concentre sur l’étude de la dy-namique de molécules biologiques (lipides, protéines, ADN) par suivi de particule unique.Ces travaux [130] sont juste évoqués dans ce mémoire.

La modélisation de l’ADN et la théorie des ions aux interfaces sont les deux sujetsmajeurs de mes recherches depuis 2004. C’est pourquoi les parties de ce mémoire qui lesrelatent (chapitres 1 et 3) sont les plus développées. J’encadre d’ailleurs la thèse de doctoratde Anil Kumar Dasanna sur la dynamique de l’ADN qui a débutée en octobre 2010.

Neufs de mes articles publiés durant cette période sont insérés dans le mémoire. Mesperspectives de recherche sont décrites à la fin de chacun des trois premiers chapitres etrésumées dans la conclusion générale.

2

Chapitre 1

Modélisation tri-dimensionnelle del’ADN

1.1 État des lieux

Dès le milieux des années 1950, les premières expériences in vitro sur l’ADN en solutionont montré que lorsque la solution est chauffée, l’ADN double-brin se dénature, c’est-à-direque les deux brins se dissocient complètement. Cette dénaturation thermique se fait autourd’un température de transition Tm qui varie de 50 C à 90 C selon la séquence et la taille dudouble-brin d’ADN et sur un intervalle de température très petit ∆Tm ' 1 K [192, 111, 78].Ces deux paramètres caractérisent les profils de dénaturation comme représenté sur lafigure 1.1. En particulier les paires de bases AT formées par l’Adénine et la Thymine reliéespar deux liaisons Hydrogène (H) se dissocient à des températures plus basses (environ 50-70 C selon la salinité) que les paires de bases GC formées par trois liaisons H (100–110 C)(voir figure 1.2).

Cette «transition» de dénaturation 1 est d’ailleurs utilisée depuis lors pour déterminer laproportion des différentes paires de base dans un ADN inconnu par exemple pour le Projetdu génome Humain 2. Une autre application courante est la méthode d’amplification enchaîne par polymérase (ou PCR) qui permet de multiplier des double-brins d’ADN deséquences et tailles bien définies. La température T est le paramètre de contrôle de laréaction qui comprend trois étapes répétées plusieurs dizaines de fois 3 : la dénaturationde l’ADN à dupliquer, l’hybridation de brins cours, les amorces, puis leur extension. Lesvariations de la température de dénaturation avec la longueur des double-brins et leur

1. Le mot transition est ici utilisé pour des raisons historiques, même si savoir s’il s’agit effectivementd’une vraie transition de phase thermodynamique reste sujet à controverse.

2. Voir les site http://www.ornl.gov/sci/techresources/Human_Genome/home.shtml3. Voir par exemple le site http://www.ens-lyon.fr/RELIE/PCR

3

Chapitre 1. Modélisation tri-dimensionnelle de l’ADN

0.2

0.4

0.6

0.8

1

052.5 53 53.5 54 54.5

temperature (°C)

fract

ion

of o

pen

base

pai

rs

Figure 1.1 – Fraction de paires de bases ouvertes en fonction de la température pour unesolution diluée d’ADN synthétique polydA-polydT de 1815 paires de bases. Les symbolescorrespondent aux données expérimentales et la courbe théorique est ajustée avec seulementdeux paramètres. Les configurations de l’ADN sont schématisées pour quatre températuresdifférentes, montrant les trois «étapes» de la dénaturation. La température de dénaturationest ici Tm ' 53.4 C.

2 nm

1 pas d’hélice10 paires de bases3.4 nm

(a)

squelette carboné(desoxyribose et

phosphate)

Adenine

CytosineGuanine

Thymine

O

O

O _

_ O

P

P

P

P

P

P

P

P

O

OO

O

O

O

O

O

OO

OO

O

OO

O

O

O

O

O

O

O

O

OO

O

OO

O _

O _

O _

O _

_ O

_ O

_ O

_ O

OH

OH

O

O

NN

N

NN

NH 2

NH

H 2 N

O

O

N

N

N

N

N

NH 2

HN

O

O

N

N

N

NN

NH 2

H 2 N

HN

O

O

N

NN

N

NH 2 N

NH

bout 3'

bout 5'bout 3'

bout 5'

(b)

Figure 1.2 – (a) Structure secondaire de l’ADN double-brin, les deux brins formant unedouble hélice. (b) Structure primaire de l’ADN définie par la formation de liaisons Hydro-gène entre les bases azotées aromatiques, celles-ci étant liées de façon covalente au squelettecarboné contenant des ions phosphates chargés négativement.

4

1.1. État des lieux

séquence sont calibrées pour contrôler l’accrochage des amorces.Les physiciens ont commencé à modéliser l’ADN double-brin dans les années 1960,

en ce concentrant sur la physique statistique des degrés de liberté internes spécifiques àl’ADN, l’ouverture et la fermeture des paires de bases azotées. En simplifiant le problèmeet considérant que les paires de bases ne peuvent se trouver que dans deux états, ouvertou fermé, le modèle d’Ising s’est rapidement imposé.

Modèle d’Ising modifié

Dans ce modèle d’Ising développé par plusieurs physiciens (voir l’excellente revue deWartell et Montroll [193]), l’ADN est vu comme une chaîne uni-dimensionnelle de N unités,les paires de bases. À la paire de bases i est associée une variable d’Ising σi qui peut prendredeux valeurs, σi = +1 correspondant à une paire de bases intacte ou fermée et σi = −1 àune paire de bases cassée ou ouverte. L’hamiltonien du système s’écrit alors

HIsing[σ] = −N−1∑

i=1

[Jσi+1σi +

K

2(σi+1 + σi)

]− µ

N∑

i=1

σi (1.1)

et fait apparaître trois paramètres d’Ising (voir la figure 1.3) : 2µ correspond à l’énergie(libre) nécessaire pour casser une paire de bases indépendamment de l’état des paires debase voisines, 2J est le coût énergétique pour créer une séparation entre deux séquencesd’état différents et détermine donc la corrélation entre l’état i et l’état i + 1 et enfin 2Kest la différence d’énergie entre deux paires de bases voisines dans l’état ouvert et dansl’état fermé. En regardant plus précisément la structure de l’ADN à l’échelle de la paire debases, on peut associer ces deux derniers paramètres à l’énergie d’empilement des pairesde bases due au recouvrement des orbitales π des cycles aromatiques. Il se trouve que lavaleur des paramètres d’empilement J , de l’ordre de 2 à 5 kBT , est bien plus grande quecelle de µ ' 1−2 kBT (où kBT est l’énergie thermique à température ambiante T = 300 K,soit environ 4× 10−21J). Ces valeurs sont disparates car elles dépendent de la nature despaires de bases pour µ et des doublets de paires de bases successives (au nombre de 10)pour J et K [78, 107].

J2 J2

=! !1!1 +1 +1+1+1+1

K2K K 00 0

2µ 2µ

000 0

Figure 1.3 – Illustration des différents paramètres d’Ising intervenant dans l’éq. (1.1).

5


Ce modèle d’Ising uni-dimensionnel n’est rien d’autre que le modèle d’Ising pour unechaîne de spins classiques, où le potentiel chimique µ joue le rôle du champ magnétique etJ celui de l’intégrale d’échange, modifié pour tenir compte de l’empilement des paires debases aussi possible dans le cas des ADN simple-brins [2]. Il est exactement soluble analy-tiquement pour les homopolymères et, dans la limite N → ∞, pour les ADN comportantdes séquences aléatoires [193]. La technique utilisée est celle des matrices de transfert etla fraction de paires de bases ouvertes dans le cas d’un ADN composé d’une séquence depaires de bases identiques avec conditions au bord périodiques est

ϕ(T ) =12

1− sinh(L)√

sinh2(L) + e−4J

(1.2)

en introduisant le paramètre L = µ + K. Afin de reproduire et d’ajuster le modèle auxprofils de dénaturation expérimentaux, il a fallu postuler que J soit indépendant de latempérature et que L(T ) soit une énergie libre dépendante de la température [193, 78].Une approximation linéaire conduit à L(T ) = a(T − Tm) où a et Tm sont deux paramètresajustables. On comprend alors à partir de l’éq. (1.2) que Tm est bien la températurede dénaturation puisque ϕ(Tm) = 1/2 et que la pente de la courbe, et donc ∆Tm, estdéterminée par le paramètre J . La détermination de ce dernier paramètre ou du paramètrede coopérativité σ = e−4J est une question délicate, car sa valeur dépend sensiblement dela longueur de l’ADN [25].

Modèle de Poland et Scheraga

Le modèle simple précédent permet de rendre compte qualitativement des profils dedénaturation mais impose par construction une température de dénaturation indépendantede N . Ceci est clairement contraire à l’expérience, et le modèle d’Ising a été modifié par Po-land et Scheraga [155] qui ont introduit un poids de Boltzmann différent pour les segmentsouverts lorsqu’ils sont situés au centre du polymère (on parle de bulle de dénaturation)ou à un bout. L’idée est d’introduire un facteur d’entropie de boucle (LE pour Loop En-tropy) en modélisant la bulle de dénaturation de taille n comme une boucle de polymèreADN simple-brin, donc flexible, de longueur 2 + 2n (soit n paires de bases successives ou-vertes). Cette entropie est donc associée au coût de refermeture de la bulle. Le paramètrede coopérativité est alors modifié selon

σLE =e−4J

(n0 + 2 + 2n)c(1.3)

paramétré par une constante n0 (qui, après ajustement pour rendre compte des expériences,se trouve être de l’ordre 100 et prend donc en compte empiriquement la rigidité des bulles

6

1.1. État des lieux

aux petites échelles) et un exposant c qui est déterminé à l’aide de la physique statistiquedes polymères sur réseau. En énumérant les marches aléatoires qui retournent à l’origine, oncalcule c = dν (où c est la dimension de l’espace et ν est l’exposant de Flory). Selon que l’onconsidère une chaîne fantôme ou une chaîne auto-évitante on a donc en trois dimensionsc = 3/2 ou c ≈ 1.764 [71]. Si de plus on tient compte des interactions de volume exclu entreboucles, c prend la valeur 2.115 [100]. L’éq. (1.3) fait alors intervenir un terme non-localqui dépend de la longueur de la bulle n et, comme cette pénalité entropique n’intervientpas pour les brins qui se séparent à partir d’un bout, introduit un effet de bord : l’ouvertureest plus probable à un bout qu’au milieu de la chaîne et donc Tm(N) devient une fonctioncroissante puisqu’en augmentant N , la fraction de bouts de chaîne diminue.

Ce résultat, essentiel pour les expérimentateurs [22], est probablement à l’origine dusuccès du modèle de Poland et Scheraga. Des logiciels accessibles en ligne tel que MELT-SIM [23] utilisent ce modèle pour prédire non seulement les courbes de dénaturation maisaussi, pour toutes les séquences imaginables, les cartes de dénaturation, c’est-à-dire latempérature de dénaturation locale associée de chaque paire de bases au sein de l’ADN enfonction de la séquence. Ils sont très utilisés par les biologistes notamment dans le décryp-tage du génome humain. Leur efficacité est due à une calibration précise des nombreuxparamètres ajustables sur une grande quantité de données [22].

Modèle de Peyrard et Bishop

Les modèles précédents utilisent des variables à deux états σi et ne permettent a prioripas de décrire proprement la dynamique d’ouverture et de fermeture des bulles de dé-naturation à la manière d’une fermeture éclair, comme attendu pour un phénomène trèscoopératif [135]. Peyrard et Bishop [152] puis Dauxois [53] ont développé un modèle non-linéaire de phonons afin de décrire les modes de respiration de l’ADN et la dynamiquedes bords de domaine entre régions ouvertes et fermées [151]. Ce modèle non-linéaire per-met également de décrire la dénaturation de l’ADN comme un processus de transitionde délocalisation 4 ou de désorption, en utilisant l’analogie décrite par de Gennes avec ladésorption d’une chaîne polymère sur un surface adsorbante [77].

Le modèle consiste à écrire un Hamiltonien qui dépend continûment de la distanceentre les deux brins, xi au niveau de la ieme paire de bases :

H[x] =N∑

i=1

V (xi) +N−1∑

i=1

k

2(xi+1 − xi)2 (1.4)

où le potentiel de Morse V (x) = D(e−ax − 1)2 modélise les liaisons H. Les valeurs des

4. La fonction de Green du problème est analogue au propagateur quantique d’un électron dans unpotentiel asymétrique.

7


paramètres microscopiques, et donc indépendants de T , sont choisies pour ajuster le modèleaux expériences et varient sensiblement dans la littérature [152, 39, 54], D = 1.1−13 kBT ,a−1 = 0.02−0.3 nm et k = 10−100 kBT/nm2. 5 La fonction de partition Z =

∫ ∏i dxi e−βH

peut se réécrire simplement en fonction des valeurs propres εi et des vecteurs propres φi(x)de l’opérateur de transfert

∫ ∞

−∞dx′e−βV (x)−βk

2(x−x′)2

φi(x′) = e−βεiφi(x) (1.5)

qui, dans le cas de conditions aux bords périodiques, s’écrit simplement Z =∑

i e−βNεi .

Tant que T < Tm où la température de dénaturation est donnée par

Tm =2√

2kDakB

(1.6)

l’éq. (1.5) a un spectre fini et un état fondamental confiné, φ0(x), d’énergie

ε0 = kBT

(a

√D

2k− a2kBT

8k

)(1.7)

La séparation moyenne entre les paires de bases est 〈x〉 =∫

dx xφ20(x). Peyrard et Bishop

ont montré que 〈x〉(T ) a une forme sigmoïde passant d’une valeur de l’ordre de 0.2 nm à1 nm sur un intervalle de température de 50 K. De plus, d’après l’éq. (1.6) le paramètre k quin’est pas mesurable directement influence sensiblement Tm. Tout comme le modèle d’Ising,le travail de Peyrard et Bishop a montré l’importance des interactions d’empilement, icimodélisés par le second terme de l’éq. (1.4) où k joue le rôle d’un module de cisaillemententre deux paires de bases consécutives. Le modèle de Peyrard et Bishop a ensuite étédéveloppé selon plusieurs directions en considérant les effets de la discrétisation [54], d’unterme d’empilement anharmonique [53, 52, 67] qui conduit à une transition bien plusrapide, l’influence de la séquence [49, 36], de la géométrie hélicoïdale [11] ou le cas deschaînes courtes [206, 39].

Dans le cas de chaînes finies, il faut évidemment considérer tous les modes dans Zet le calcul numérique devient indispensable. Notons que les paramètres sont choisis enfonction de la valeur expérimentale de Tm reliée aux paramètres selon l’éq. (1.6). Mais ilsdépendent également du choix du seuil x0 pour 〈x〉, permettant de discriminer entre unepaire de bases ouverte ou fermée.

Enfin, d’un point de vue plus académique, puisqu’en pratiqueN est fini, une controverses’est développée autour de la nature de la transition de dénaturation comme transition

5. Dans l’article original [152] les valeurs choisies étaient D = 13 kBT , a−1 = 0.05 nm et k =

11 kBT/nm2.

8

1.2. Importance des configurations spatiales

thermodynamique lorsque N →∞. S’agit-il d’un «crossover», d’une transition du premierou du second ordre ? En effet, les expériences ont montré que la dénaturation thermiquede l’ADN est un processus très coopératif où les profils de dénaturation évoluent trèsrapidement proche de Tm [192, 111, 135]. Comme il ne peut y avoir de transition de phasedans un système uni-dimensionnel avec des interactions à courte portée [108], les modèles detype Ising prédisent un crossover dont la pente est contrôlée par le paramètre d’empilementJ . Comme celui-ci est grand devant kBT , les profils de dénaturation sont abrupts prochede Tm. Dans le cas du modèle de Poland et Scheraga, le terme en c lnn dû à l’entropie deboucles conduit à une interaction à longue portée (via les interactions de volume exclu entrebulles), et une transition de phase continue apparaît pour c > 1, qui devient discontinuepour c > 2 [100]. Enfin, dans le cas du modèle de Peyrard et Bishop, le potentiel de Morsebrise la symétrie dans le hamiltonien (1.4), et une transition de phase est possible, qui peutêtre continue ou discontinue selon les valeurs prises par les paramètres du modèle [187].

1.2 Importance des configurations spatiales

Les modèles d’Ising et de Peyrard-Bishop présentés ci-dessus sont essentiellement desmodèles uni-dimensionnels. La chaîne est supposée infiniment rigide et les degrés de libertésconformationnels ne sont pas considérés ou alors sont factorisés dans la fonction de partitionet ne se couplent pas aux degrés de liberté internes de la chaîne, σi ou xi. Dans le cas dumodèle d’Ising, afin de comparer aux expériences, les paramètres d’Ising sont en réalitéconsidérés comme des énergies libres [193] et le hamiltonien effectif (1.1) tient compted’une entropie configurationnelle. Un article très récent tente de justifier le modèle d’Isingà partir d’une formulation microscopique [8].

Dans le modèle de Peyrard et Bishop [54] en revanche, le découplage entre les variablesexternes, le centre de masse d’une paire de bases, X(s), et la distance entre les bases d’unemême paire, x(s), est explicitement formulé. Les paramètres sont supposés microscopiqueset donc indépendants de la température.

Le modèle de Poland et Scheraga quant à lui considère, via l’entropie de boucle quipénalise la création de boucle, les configurations spatiales des boucles uniquement, les sec-tions double-brin de la chaîne étant supposées infiniment rigides. Une dépendance en T

des paramètres d’Ising s’ajoute à l’entropie de boucles afin de contrecarrer cette péna-lité [193, 100].

Quelques modèles d’ADN incluant les configurations spatiales

Plusieurs travaux incorporent l’entropie associée aux configurations spatiales de lachaîne polymère.

9


Une première approche est d’inclure des interactions entre bases d’une même paire,modélisées soit par un potentiel attractif constant à courte distance [75] soit par un po-tentiel similaire à celui de Peyrard-Bishop [184], et la connexité de la chaîne en ajoutantun hamiltonien quadratique

Hel[t] =2∑

i=1

32`0

∫ L

0t2i (s)ds (1.8)

où t(s) est le vecteur tangent au brin i à l’abscisse curviligne s avec 〈|t(s)|〉 = 1 (et nonplus |t(s)| = 1) et `0 est la longueur de Kuhn [200] définie via la relation entre le carré dela distance moyenne bout à bout et la longueur de la chaîne

〈R2〉 = `0L (1.9)

Ces modèles sont intéressants car ils permettent de relier l’approche de Peyrard-Bishop decelle de Poland-Scheraga. De plus, moyennant quelques hypothèses, l’approche de Garelet al. permet d’inclure les interactions de volume exclu entre les deux brins au niveaud’une même paire de bases [75]. Un inconvénient de ce type d’approche est qu’elle mélangedeux échelles de longueur pourtant bien distinctes : l’échelle de l’Angström au niveau del’interaction entre bases d’une même paire et l’échelle de la centaine de nanomètres corres-pondant à la longueur de Kuhn de l’ADN double-brin. En particulier, le hamiltonien (1.8)n’est valable qu’à une échelle supérieure à `0 = 2`p, où `p est la longueur de persistance,qui vaut environ 50 nm pour l’ADN double-brin. Yan et Marko [201] ont proposé un modèlecouplant la longueur de persistance à l’état de la paire de bases suggérant un mécanisme denucléation de bulle de dénaturation pour expliquer la cyclisation des double-brins d’ADNcourts.

Une seconde approche a été développée à partir d’expériences sur l’étirement de l’ADNsous force. Elles ont suggéré l’apparition d’un phase exotique de l’ADN (nommée S pourStretched) pour des forces d’étirement supérieures à 65 pN [46, 178, 162], caractérisée parun plateau dans la courbe force-extension jusqu’à une extension de 2.1 fois sa longueur.Différentes explications théoriques ont été proposées qui suggèrent soit qu’il s’agit d’unetransition associée au des-empilement progressif des paires de bases tout en conservant lesliaisons H intactes [47, 66], soit à leur dénaturation [167]. La plupart de ces modèles sontbasés sur le modèle d’Ising avec un terme supplémentaire qui couple la force à la taille desmonomères, qui dépend elle-même de l’état de spin [43, 1, 182].

Enfin Benham [14, 73] a le premier considéré le couplage entre l’ADN et ses degrésde liberté de torsion afin de rendre compte de la dénaturation observée dans les ADNsur lesquels est appliquée une contrainte de torsion (ou de sur-enroulement). Là encore cecouplage se fait au niveau du modèle d’Ising avec un terme supplémentaire associé à ladéformation de torsion (la différence d’enlacement, ou «linking», résiduelle).

10

1.2. Importance des configurations spatiales

L’ADN en tant que tige élastique

Les expériences précédentes suggèrent que les déformations élastiques de l’ADN, commeson étirement/compression, sa courbure ou sa torsion, jouent un rôle fondamental nonseulement dans ses propriétés physiques mais également biologiques [15].

Le modèle de référence pour décrire les propriétés élastique de l’ADN est le modèle dever (ou Worm-Like Chain WLC) [105, 200] qui est défini par un paramètre, son modulede courbure κ, et qui modélise la chaîne polymère comme une ligne élastique continue.L’hamiltonien associé est

HWLC[t] =κ

2

∫ L

0

(dtds

)2

ds avec |t(s)| = 1 (1.10)

ce qui conduit à une fonction de corrélation exponentielle

〈t(s) · t(0)〉 = e−s/`p (1.11)

où `p = κ/(kBT ) est la longueur de persistance. Il est possible de traiter la contraintelocale |t(s)| = 1 de manière plus globale en moyennant sur toutes les configurations,〈|t(s)|2〉 = 1. En introduisant dans le hamiltonien (1.10) un paramètre de Lagrange, onretrouve l’éq. (1.11) avec le hamiltonien

H′WLC[t] =34

∫ L

0

[κ

(dtds

)2

+t2(s)β2κ

]ds (1.12)

Un modèle plus élaboré consiste à décrire l’ADN comme une tige élastique matériellequi tient compte de l’énergie de torsion. Le modèle de ver hélicoïdal (HWLC) [200] faitintervenir en plus de l’énergie de courbure (1.10), l’énergie de torsion contrôlée par lemodule de torsion C. Le rapport entre les deux modules élastiques vaut κ/C = (1 + σP)dans le cas d’une section circulaire 6. Cette énergie s’écrit

HHWLC[Ω] =κ

2

∫ [Ω2

1(s) + Ω22(s)

]ds+

C

2

∫Ω2

3(s) ds (1.13)

où Ω3 = Ω · e3 = φ cos θ + ψ est la composante le long de l’axe de la molécule du vecteurde Darboux Ω =Ωiei, qui mesure la rotation du repère matériel le long de la chaîne et(θ, φ, ψ) sont les angles d’Euler à l’indice s que forme le repère matériel par rapport à unrepère fixe (s’il n’y a pas de courbure spontanée, t2 = Ω2

1 + Ω22).

6. Dans le cas d’une tige élastique de section circulaire de rayon R, κ = EI (où E est le module d’Younget I = π

4R4 est le moment d’inertie) et C = π

2R4µ (où µ = E/2(1 + σP) est le module de cisaillement et

σP le coefficient de Poisson du matériau, −1 ≤ σP ≤ 0.5) [109, 174].

11


Ces modèles ont été utilisés pour rendre compte des expériences de micro-mécaniquesur l’ADN double-brin à température ambiante. Que se passe-t-il pour T proche de Tm ?Les modules élastiques de l’ADN dépendent-il de l’état de la paire de bases ? Un certainnombre d’expériences physiques sur l’ADN [178, 35, 164] ont montré que l’ADN double-brin a un module de courbure κds bien plus grand que celui de l’ADN simple-brin, κss.Plusieurs valeurs expérimentales sont données dans la littérature, selon la concentration ensel dans la solution [188, 195] (celle-ci modifie l’écrantage des interactions électrostatiquesle long de l’ADN, qu’il soit double ou simple-brin [12, 126]) ou la technique expérimentaleutilisée : mesure du coefficient de diffusion par Fluorescence Recovery After Photobleaching(FRAP) [188], étirement sous force [178] ou encore par suivi de particule unique (TPMpour Tethered Particle Motion) [31]. Le rapport des longueurs de persistance est doncde l’ordre de `ds

p /`ssp ≈ 50 entre les régions intactes et les bulles de dénaturation au sein

du même ADN. On s’attend donc qu’un ADN partiellement dénaturé à une températureT . Tm ait une longueur de persistance effective plus faible que celle d’un ADN double-brinà température ambiante Ta Tm [195, 96].

De même, le module de torsion C a été mesuré dans des expériences de torsion del’ADN double-brin [15, 29, 32] et est de l’ordre de Cds ' 2.4 − 4.5 · 10−19 J.nm. Commeκ ∝ C, le module de courbure de l’ADN simple-brin peut donc être estimé comme étant50 fois plus faible, Css ' 9 · 10−20 J.nm [24].

1.3 Couplage entre les degrés de libertés d’Ising et de lachaîne

Au vu des remarques précédentes, nous avons modélisé l’ADN en couplant les étatsinternes σi ou xi (paire de bases ouverte ou fermée) aux états externes des conformationsde la chaîne polymère en tenant compte des différentes valeurs des modules élastiques.L’hamiltonien WLC pour les deux chaînes k = 1, 2 s’écrit alors en suivant l’éq. (1.12)

H[r1, r2] =2∑

k=1

34

∫ L

0

[κb(x(s))

(dtids

)2

+ κs(x(s))t2i (s)

]ds+

∫ L

0V (x(s)) ds (1.14)

où x(s) = r2(s)−r1(s) est le vecteur reliant les deux brins à l’indice curviligne s, les modulesélastiques de courbure et d’étirement dépendent de cette coordonnée interne (κb ≈ κss auxgrands |x| et κds aux petits) et le potentiel V (x) tient compte des interactions à courteportée de manière similaire au modèle de Peyrard-Bishop. 7 En introduisant le centre de

7. L’allure exacte des fonctions V et κb n’est pas essentielle pour comprendre les mécanismes phy-siques [146].

12

1.3. Couplage entre les degrés de libertés d’Ising et de la chaîne

-2,5 0 2,5 5 7,5 10 12,5

2,5

5

7,5

10

12,5

VR(|x|)

|x|

Figure 1.4 – Allure du potentiel renormalisé éq. (1.17) (en unités de kBTa/`0) en fonctionde |x| (en Å) pour les temperatures T = 0, T = Ta = 300 K et T = 1.33Ta de bas en haut(D`0/kBTa = 5, a−1 = 0.05 nm avec κ(|x|) = κss + (κds/2− κss)e−a|x|).

masse des deux brins X = (r1 + r2)/2 et en intégrant sur les X à x fixé, on obtient unhamiltonien effectif

Heff [x] =38

∫ L

0ds[κb(x)x2 + κs(x)x2 + V (x)

](1.15)

−kBT ln∫DX exp

[−3β

2

∫ L

0ds(κb(x)X2 + κs(x)X2

)](1.16)

Le dernier terme va renormaliser à la fois le terme en κs(x)x2 et le potentiel V , conduisantà un potentiel renormalisé dépendant de la température :

VR(x) = V (x) +32kBT

`0ln[κb(x)κssb

](1.17)

Comme le montre la figure 1.4, le minimum du potentiel renormalisé augmente de 32kBT ln κds

bκssb,

illustrant le rôle important joué par l’entropie de courbure sachant que κdsb /2κ

ssb ∼ 25. De

plus une barrière entropique apparaît, qui joue très certainement un rôle important dansla dynamique de fermeture d’une bulle de dénaturation.

Une approche plus réaliste consiste à discrétiser le hamiltonien (1.14) afin de prendre encompte les liaisons H et les énergies d’empilement au niveau de la paire de bases. Nous avonsdonc couplé le modèle d’Ising à un hamiltonien discret pour la partie conformationnelle enconsidérant une seule chaîne effective représentant le centre de masse des deux brins. Trèsgénéralement, trois types de déformations élastiques sont considérés :

i) les déformations de courbure, dont l’énergie s’écrit en fonction des vecteurs liens

13


normalisés ti = ti/|ti| comme [version discrète de l’éq. (1.13)]

HDWLC[t, σ] =N−1∑

i=1

κ(σi, σi+1)(1− ti · ti+1) (1.18)

où le module de courbure κ(σi, σi+1) dépend de l’état des paires de bases consécu-tives : κ(1, 1) = κU (pour Unbroken), κ(−1,−1) = κB (pour Broken) et κ(1,−1) =κ(−1, 1) = κUB ;

ii) les déformations d’étirement/compression de la distance entre deux paires de basesconsécutives, d’énergie

Hstretch[t, σ] =12

N∑

i=1

ε(σi)2[|ti|2 − a2(σi)

]2 (1.19)

où les valeurs de la taille du monomère a(σi) et du coefficient de Lamé ε(σi) dépendentde l’état de la paire de bases ;

iii) enfin les déformations de torsion, prises en compte dans une version discrète dumodèle de ver hélicoïdal

HDHWLC[φ, θ, ψ, σ] =N−1∑

i=1

C(σi, σi+1)(cos θi,i+1 − cosλi,i+1) (1.20)

oùcosλi,i+1 =

12

[cos(φi,i+1 + ψi,i+1)(cos θi,i+1 + 1) + cos θi,i+1 − 1] (1.21)

et les angles (φi,i+1, θi,i+1, ψi,i+1) sont les angles d’Euler du repère matériel rattachéà la paire de bases i+ 1 dans le repère de la paire de bases i et le module de torsionC(σi, σi+1) dépend de l’état des paires de bases, de la même manière que le modulede courbure (on suppose le coefficient de Poisson fixé).

L’hamiltonien total du système couplé s’écrit alors

H[φ, θ, ψ, σ] = HIsing +Hstretch +HDWLC +HDHWLC (1.22)

Notons que dans la version discrétisée [éqs. (1.18)-(1.20)], les modules de courbures élas-tiques ont la dimension d’une énergie et sont égaux aux modules des éqs. (1.10)-(1.13)divisés par la longueur d’une paire de bases (ads = 0.34 nm, ass = 0.71 nm).

1.3.1 Résumé des principaux résultats

Dans cette partie sont résumés succinctement les principaux résultats du modèle couplépubliés dans les articles joints. Chronologiquement, le couplage avec les déformations de

14


courbure a été considéré en premier [146, 147], celui avec les autres déformations venantplus tard [127].

1. Dans le cas du couplage Ising/WLC, la fonction de partition ainsi que toutes les cor-rélations entre spins et entre vecteurs tangents sont calculées de façon analytique grâceà la technique de matrices de transfert. Ce calcul est formellement équivalent à celuipour un rotateur rigide quantique dont les états propres de l’opérateur sont des spineurs|Ψl,m;τ 〉 = |l,m〉⊗ |l, τ〉 qui couple les états associés aux orientations spatiales (paramétréspar les nombres l = 0, 1 . . .∞, −l ≤ m ≤ l) aux états liants et anti-liants (paramétrés parτ = ±). Ainsi, la projection des états propres dans la base |σΩ〉 où Ω ≡ (θ, φ) conduità 〈σΩ|Ψl,m;τ 〉 =

√4πYl,m(Ω)〈σ|l, τ〉 où les Yl,m sont les harmoniques sphériques (états

propres du modèle WLC). Les valeurs propres λl,τ , sont reliés aux énergies propres du ro-tateur εl,τ = − lnλl,τ et s’expriment en fonctions des fonctions de Bessel modifiées de 1ere

espèce, les Il+1/2. Notre modèle n’est pas le produit direct des modèles d’Ising et WLC,car les états de symétries de rotation différents (l 6= l′) peuvent conduire à des amplitudesde transition 〈l, τ |l′, τ ′〉 non nulles.

2. Grâce à la symétrie d’invariance par rotation du hamiltonien total HIsing +Hchaîne

(les degrés de liberté de rotation de l’ADN global n’interviennent pas dans le couplage), enintégrant sur les degrés de liberté spatiaux, on s’intéresse alors uniquement à la physiquestatistique d’ouverture/fermeture des paires de bases. On retombe alors sur un hamiltoniend’Ising effectif avec des paramètres d’Ising renormalisés qui dépendent de la température.Ainsi, la fraction de paire de bases ouvertes en fonction de la température est toujoursdonnée par l’éq. (1.2) mais avec des paramètres µ0, K0 et J0 renormalisés par les énergieslibres élastiques d’un simple joint

µ0∼= µ− kBT ln

(aB

aU

√εUεB

)(1.23)

K0 = K − kBT

2[G(κU , CU )−G(κB, CB)] (1.24)

J0 = J − kBT

4[G(κU , CU ) +G(κB, CB)− 2G(κUB, CUB)] (1.25)

où

G(κ, C) = 2κ− ln[∫ 1

0dx I0(Cx) e(2κ−C)x

](1.26)

et κ = βκ. Ainsi notre modèle apporte une justification aux paramètres empiriques dépen-dant de T du modèle d’Ising [193]. Les paramètres µ, J et K sont diminués par l’énergielibre, de nature essentiellement entropique, correspondant à l’augmentation des conforma-tions accessibles pour la chaîne ADN lorsque la paire de bases passe de l’état fermé à l’état

15


ouvert (car aB ≈ 2aU, κU/κB = CU/CB ≈ 25).

3. La température de dénaturation Tm se déduit directement de L0(Tm) = 0 dans le casdes ADN longs (N → ∞) et dépend des paramètres d’Ising et des paramètres élastiquesselon

T∞m =2kB

µ+K

ln(a2

BεUκU√CU

a2UεBκB

√CB

) (1.27)

Elle n’est donc plus un paramètre ajustable puisque les valeurs des constantes élastiquespour l’ADN double et simple-brin ont été mesurées indépendamment (voir plus haut. Onchoisira dans la suite C/κ = 1.6). De même, comme le montre la fig. 1.1, le profil de dé-naturation complet est ajusté aux données expérimentales conduisant à des valeurs tout àfait raisonnables des paramètres d’Ising (µ = K = 4.94 kJ/mol, J = 9.13 kJ/mol [127]).

4. En réalité, la taille finie des ADN modifie l’éq. (1.27) et Tm dépend de N . LorsqueTm(N) est une fonction croissante, l’ouverture de l’ADN par les les bouts de chaîne estplus probable que la nucléation d’une bulle de dénaturation au centre de la chaîne. Commedans notre modèle, Tm(N) émerge naturellement et n’est pas un paramètre d’ajustement,la dépendance en N vient de la différence des paramètres d’Ising renormalisés pour lesbouts de chaînes et les monomères centraux. Le modèle couplé est donc très utile pourcomprendre les variations des observables en fonction de N . De plus, notre modèle d’Isingeffectif permet de calculer les énergies libres de nucléation d’une bulle de taille n, qu’ellesoit au centre de la chaîne ou à son bout.

La comparaison avec les expériences impose une prise en compte précise des condi-tions aux bords. Ainsi, dans les expériences sur des homopolynucléotides, deux cas sontenvisagés :

i) des ADN aux bouts libres pour lesquels Tm(N) est une fonction croissante (Tm(30000)−Tm(500) ' 1 K [22]) ;

ii) des bouts fermés, par exemple dans le cas d’un segment polydA-polydT inséré entredeux régions plus stables très riches en GC [22]. Dans ce cas, en revanche, Tm(N) estune fonction décroissante car la probabilité d’ouverture d’un bout en contact avecun monomère GC est plus faible.

Modéliser ces deux cas revient à associer une énergie µ′ aux bouts de chaîne qui peutprendre différentes valeurs : µ′ = µ0 pour des bouts libres, µ′ →∞ pour des bouts fermés,voire µ′ = L0 dans le cas formel où les bouts de chaînes sont identiques aux monomères in-térieurs. La comparaison avec les données expérimentales sur les inserts AT est bonne dansle cas où µ′ est ajusté (le cas µ′ → ∞ est trop contraignant). Le rôle joué par l’entropiede boucles dans ce cas est mis en lumière dans l’approximation d’une seule séquence [155](voir figure 1.5) qui est très précise dans le cas d’inserts de taille N < 104. Les profils

16


(a) (b) (d)(c)

Figure 1.5 – Schéma des quatre états possibles de l’ADN dans l’approximation d’uneseule séquence : (a) chaîne fermée, (b) dégraffage à un bout, (c) hélice interne et (d) bullede dénaturation interne.

de dénaturation sont alors simplement décalés d’1 K vers les plus hautes températures.Enfin, le cas des bouts libres est considéré dans cette approximation en incluant à la foisl’entropie de boucles et le glissement relatif des brins (pour les homopolynucléotides). Lesprofils de dénaturation ainsi obtenus pour différentes valeurs de N reproduisent de façonsemi-quantitative les deux caractéristiques expérimentales, à savoir que Tm croît avec Net ∆Tm décroît. Ils mettent en avant le rôle important de la rigidité des bulles car le para-mètre n0 dans l’éq. (1.3) vaut plutôt 100 que 1, valeur pourtant largement utilisée par lesexpérimentateurs. Les écarts restants par rapport aux expériences viennent probablementde l’approximation d’une seule chaîne, comme «moyenne» des deux brins.

5. Non seulement le couplage modifie les propriétés d’ouverture et fermeture des bases,mais il modifie également les propriétés statistiques et mécaniques de la chaîne. Ainsila fonction de corrélation des vecteurs tangents 〈ti · ti+r〉 fait apparaître deux longueursde corrélation ξ1,± avec des poids associés 〈0 + |1±〉2 correspondant aux probabilités detransition pour passer de l’état fondamental au 1er état excité, d’état interne τ = ±. Cesderniers varient brutalement à Tm à cause du couplage. Un crossover apparaît pour unedistance curviligne typique r∗(T ) ' ξB ln[ϕB/(1 − ϕB)] où ϕB est la fraction de paires debases ouvertes : pour r < r∗, la fonction de corrélation est dominée par la plus petitelongueur de corrélation ξ1,− et pour r > r∗ elle est dominée par ξ1,+. Enfin, pour T < Tm,ξ1,+ ' ξp,U la longueur de persistance du double-brin, alors que pour T > Tm, ξ1,− ' ξp,B.Ce comportement anormal à petites distances pour T . Tm est dû à la présence de bullesde dénaturation qui, étant plus flexibles, font chuter brutalement la longueur de corrélation(ou de persistance). Nous verrons plus loin une application de ce mécanisme [60].

À cause du couplage, on s’attend à une variation brutale de la distance bout à bout del’ADN pour des températures proches de Tm. Cette dernière est définie par

〈R2〉 =N∑

i,j=1

〈(aiti) · (aj tj)〉 (1.28)

17


325.5 326 326.5 327 327.5

20

40

60

80

100

120

140

160

0325

Temperature (K)

R2

2a2UN

Figure 1.6 – Variance de la distance bout à bout (en unités de paires de bases) en fonctionde la température pour les valeurs de paramètres µ = 4.46 kJ/mol, J = 9.13 kJ/mol etK = 0 (Tm = 326.4 K), aB = 2aU et κU = κUB = 26.7κB. Les courbes en pointilléscorrespondent aux cas d’ADN double-brin (en haut) et simple-brin (en bas).

et fait intervenir des fonctions de corrélations du type 〈σαi ti · tjσβj 〉 où (α, β) ∈ 0, 1. Lerésultat est représenté sur le figure 1.6 et montre un chute brutale de la distance bout àbout proche de Tm, en bon accord avec les résultats de Inman et Baldwin [96] qui ontmesuré les variations de viscosité d’une solution diluée d’ADN double-brin en fonction dela température.

1.3.2 Trois articles

Suivent trois articles :J. Palmeri, M. Manghi et N. Destainville, Thermal denaturation of fluctuating DNA drivenby bending entropy, Physical Review Letters 99 088103 (2007) (4 pages)J. Palmeri, M. Manghi et N. Destainville, Thermal denaturation of fluctuating finite DNAchains : the role of bending rigidity in bubble nucleation, Physical Review E 77 011913(2008) (22 pages)M. Manghi, J. Palmeri et N. Destainville, Coupling between denaturation and chain confor-mations in DNA : stretching, bending, torsion and finite size effects, Journal of Physics :Condensed Matter 21 034104 (2009) (18 pages)

18

Thermal Denaturation of Fluctuating DNA Driven by Bending Entropy

J. Palmeri, M. Manghi, and N. DestainvilleLaboratoire de Physique Theorique, Universite de Toulouse, CNRS, 31062 Toulouse, France

(Received 20 March 2007; published 24 August 2007)

A statistical model of homopolymer DNA, coupling internal base-pair states (unbroken or broken) andexternal thermal chain fluctuations, is exactly solved using transfer kernel techniques. The dependence ontemperature and DNA length of the fraction of denaturation bubbles and their correlation length isdeduced. The thermal denaturation transition emerges naturally when the chain fluctuations are integratedout and is driven by the difference in bending (entropy dominated) free energy between broken andunbroken segments. Conformational properties of DNA, such as persistence length and mean-square-radius, are also explicitly calculated, leading, e.g., to a coherent explanation for the experimentallyobserved thermal viscosity transition.

DOI: 10.1103/PhysRevLett.99.088103 PACS numbers: 87.15.Ya, 82.39.Pj, 87.10.+e

Double-stranded DNA (dsDNA) is made up of twointertwined interacting semiflexible single-strand DNA(ssDNA) chains. Over 50 years ago it was recognizedthat the intracellular unwinding of DNA at physiologicaltemperature has as counterpart the thermally induced de-naturation above physiological temperature of purifiedDNA solutions where dsDNA completely separates intotwo ssDNA. Despite the differences between the twomechanisms, this observation has led to an intensive studyof thermal denaturation [1,2]. The stability of dsDNA atphysiological temperature is due to the self-assembly ofneighboring base pairs within a same strand via base-stacking interactions and of both strands via hydrogenbonds between complementary bases. The bonding energyis, however, on the order of kBT (thermal energy) [3] andthermal fluctuations can lead, even at physiological tem-perature, to local and transitory unzipping of dsDNA [1].The cooperative opening of consecutive base pairs leads todenaturation bubbles and the melting temperature Tm,above which bubbles proliferate, depends on sequence,chain length, and ionic strength. Experiments show, forexample, that there exist a bubble initiation barrier of10kBT and free energy cost of 0:1kBT for breakingan additional base pair in an existing adenine-thymine(A-T) bubble [4]. A detailed understanding of equilibrium[1] and dynamical [5] properties of DNA in solution is stillbeing sought, and a consensus concerning the physicalmechanism behind the denaturation transition has not yetbeen reached.

A variety of mesoscopic models has been proposed toaccount for the thermodynamical properties of denatura-tion bubbles in DNA. They range from (i) simple effectiveIsing-like two-state models [1] to more detailed ones suchas (ii) loop entropy models (with or without chain self-avoidance) [1,2,6,7] and (iii) nonlinear phonon models,where the shape of the interaction potential between basepairs is more precisely taken into account [8,9]. To get atransition in models (i) and (ii), an effective temperaturedependent base-pair chemical potential must be inserted by

hand. For finite chains, type (ii) models simply refine thesharpness of the transition [1], but do not attempt toprovide a deeper explanation of the physical mecha-nism—our aim here. For type (iii) models it has beenshown that there can be a denaturation transition analogousto interface unbinding, due to a gain in translational en-tropy. If, however, physically reasonable values for themodel parameters are used [8–10], Tm appears to bemuch too high and the transition width much too large.

It has been shown experimentally that dsDNA is 2 ordersof magnitude stiffer than ssDNA at normal salt concentra-tion. We show in this Letter that taking into account thisdifference in bending rigidity provides a novel physicalexplanation for the denaturation mechanism and leads torealistic values for transition temperatures and widths.Despite important recent advances in understanding thecrucial role played by DNA bending rigidity in explainingforce-extension [11,12] and cyclization [13] experiments,its importance has not yet been clearly elucidated in thecontext of denaturation. We show, via a well-definedcoupled Ising-Heisenberg statistical model, that an entropydriven denaturation transition emerges by integrating outchain fluctuations, due to the entropic lowering of theenergetic barrier for bubble nucleation. This minimalmodel neglects all other (residual) interactions betweenbases arising from, e.g., electrostatics [14], self-avoidance[6], and helical twist [15].

We begin by considering a wormlike chain (WLC)Hamiltonian Hr1; r2 for two interacting ssDNA homo-polymers of length L:

H 1

2

X2

i1

Z L

0ds

3

2bsr2

i s 3

2ss _r2

i s

Z L

0dsVs; (1)

where s is the curvilinear index, ris is the positionof chain i at base position s, and s r1s r2sis the relative (internal) coordinate ( 1=kBT and

PRL 99, 088103 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending24 AUGUST 2007

0031-9007=07=99(8)=088103(4) 088103-1 © 2007 The American Physical Society

_rs @r=@s). The coefficient b is a bending elasticmodulus that is proportional to the short distance cutoff‘0 0:34 nm (the monomer length) and s 1=2b. In order to account for the enhanced stiffnessof dsDNA, b must depend on s, e.g., b ssb be0=, where b ds

b =2 ssb > 0 and

1 A is the range [10]. The persistence length of ssDNA is‘ssp ss

b 1 nm and that of dsDNA is ‘dsp ds

b

50 nm (at 300 K and physiological ionic strength). Thepotential V accounts for both the effective short rangehydrogen bonding interaction between complementarybases at the same s and part of the stacking interactionbetween neighboring bases; a convenient form is the Morsepotential: V D=‘0v 0= where vx e2x 2ex, leading to a well depth of D=‘0 at 0 [8].

At high temperature, s 0 , and therefore thesystem decouples into two semiflexible noninteractingssDNA chains. After introducing the center-of-mass(external) coordinate, Xs r1s r2s=2, thepartition function can be rewritten as a sum overcenter-of-mass and relative configurations: Q RDDX expfHextX; Hintg, where [16]

Hint 3

8

Z L

0dsb 2 s _2 V (2)

Hext 3

2

Z L

0dsb X2 s _X2: (3)

We integrate over X to obtain an effective model for ,Q

RD expfHeffg, where Heff Hint

F ext with F extkBT lnfRDXexpHextX;

g. The external free energy F ext at frozen can beevaluated by introducing the local center-of-mass tangentvectors t _X and then changing variables to ~t t=

b

p. There are two contributions to F ext: one will

renormalize the second term in Hint and the other willrenormalize the potential to VR. The latter mainly givesrise to a purely entropic barrier favoring bubbles,

VR V 3

2

kBT‘0

lnbssb

; (4)

which lowers the well depth from D to DR D3=2kBT lnds

b =2ssb . The entropic bending contribu-

tion can be extremely important when D is in the com-monly accepted range of 1<D=kBT < 5 at T 350 Kand ds

b =2ssb ’ 25. Scaling arguments then show that the

melting temperature gets reduced by a factor 2 down toexperimental values [8,17].

In order to illustrate the above mechanism in more de-tail, we introduce a discretized, exactly soluble, version ofthe above model, which captures the essential physics. Wemap the external tangent vector ts to ti (s ‘0i) and aninternal variable 1 2s 0 (with the stepfunction) to an Ising variable: i 1 for an unbroken bond(state A) and i 1 for a broken one (B). Each link

vector can be denoted by the solid angle i i; i. Theenergy Hi; ti of a state is

H XN1

i1

~i;i11 ti ti1 HI~J; ~K; ~;

HI XN1

i1

~Jii1

~K2i i1

~

XNi1

i:

(5)

Thus H contains only the dimensionless parametersi;i1 ~i;i1, J ~J, ~, and K ~K. Thistype of model was introduced in the context of helix-coiltransitions in 2D [18] and later used in various forms tostudy DNA force extension and cyclization [11–13]. Thefirst term in H, corresponding to Hext in the continuummodel, is the bending energy of a discrete WLC with alocal rigidity i;i1 A ds

b =‘0 for a nearest-neighbor link of type A-A, B 2ss

b =‘0 for B-B, andAB for A-B. In the Ising part, HI, the first term mimics thegradient terms in Eq. (2) and accounts for the local destack-ing energy [8]. The second term accounts for the differencein stacking energy between a segment of dsDNA and adenaturation bubble. The third term corresponds to theenergy (2 ~) required to break a base pair, contributing toD in the continuum model. There is evidence that ~K ~,which justifies the choice of ~K 0 adopted below [19].The bare parameters of the internal (Ising) system ~J, ~,and ~K are taken to be independent of temperatureand bubble loop entropy is not explicitly included; themelting transition is therefore driven only by the dif-ference in bending rigidity. The partition function Z Pfig

RQidi=4eHi;ti in transfer kernel form is

Z X

fi1g

YNi1

Z di

4hVj1ih1jP1;2j2i

hN1jPN1;NjNihNjVi; (6)

where jVi e=2; e=2 is the end vector, which entersto account for the free chain boundary conditions, andPi;;i1 is the transfer kernel given by

P eAcosi1JK eABcosi1J

eABcosi1J eBcosi1JK

!(7)

with cosi ti ti1. The A and B states form the canoni-cal base, jAi j 1i 1; 0 and jBi j 1i 0; 1.Thanks to the rotational symmetry of the bending energywe can again integrate out the chain, leading to an effectiveIsing model with a free energy HI;0, containing renormal-ized parameters: Z eN10

Pfige

HI;0i whereHI;0 HI~J0; ~K0; ~ with J0 J G0A G0B 2G0AB=4 and K0 K G0A G0B=2 K GAB

0 =2 the renormalized Ising parameters, and 0 G0A G0B 2G0AB=4. These parameters de-pend on chain rigidities through G0 which is the freeenergy of a single joint (two monomer) subsystem with


088103-2

rigidity : G0 lnRd=4 expfcos

1g lnsinh=. The renormalized quantity2L0 2 K0 corresponds to DR in the continuummodel. If the bending free energy gain in opening one link,GAB

0 , is greater than the intrinsic energy cost, 2 K,of opening an interior bond, then L0 becomes negative,signaling a change in stability of the A and B states. In thelimit of high A and B, the entropic contribution domi-nates: G0 S0=kB ln2. The discrete modelthen reduces to the continuum one and GAB

0 lnA=Bcorresponds to the correction appearing in DR. The dif-ference between L0 and when K0 0 creates an end-interior asymmetry that plays an important role in finitesize effects.

The Ising partition and correlation functions are ob-tained using transfer matrix techniques. The eigenvectors,j0;i, and the eigenvalues, 0; eJ00fcoshL0

sinh2L0 e4J01=2g, of the effective Ising transfer ma-

trix allow us to calculate the (dimensionless) free energyper Ising spin of the coupled system, F lnZ=N, whereZ

PhVj0; i

2N10; . The average of the internal state

variable is hci hPNi1 i=Ni @F=@, from which

the fractions of A and B links, ’A 1 hci=2 and ’B 1 ’A, can be derived. The melting temperature Tm isthen defined by ’BTm 1=2. When N ! 1, hci getssimplified to

hci1 @ ln0;

@

sinhL0

sinh2L0 e4J01=2

: (8)

If L0 vanishes at a temperature T1m sufficiently low for thecooperativity, or loop initiation, factor e4J0 in thedenominator to be small, then the system will undergo amelting transition: hci1 will sharply cross over from 1for T < T1m (pure A state) to1 for T > T1m (pure B state).Contrary to previous Ising-type models, the meltingtemperature is not put in by hand but emerges naturallyfrom L0 0. In Eq. (8) e4J0 determines the widthof the transition region: T1m 2j@hci1=@Tj

1T1m

2kBT1m

2= ~ exp2J0T1m . In the limit N, i! 1, the

influence of end monomers disappears and the Ising corre-lation function reduces to hi hci1ir hci1i !1 hci21 expr=I, where I ln10;=0; isthe Ising correlation length, the typical size of B (A)domains below (above) Tm.

The tangent-tangent correlation function hti tiri isobtained using the full transfer kernel method [17], whichrequires solving a spinor eigenvalue problem: Pji ji, where ji 1;1. We find ei-genvalues l; labeled by l 0; . . . ;1 and withthe same form as for l 0 given above, but now where G0

in the renormalized parameters is replaced by Gl lnld=dlsinh= [20]. The eigenspinorsare jl;m;i

4p

Ylmjl; i with Ylm thespherical harmonics. The transfer kernel can be expandedin terms of the eigenspinors P

Pl;m;l;jl;m;ihl;m;j

and then be used to calculate the correlation functions. Inthe limit N; i! 1, the expression for the tangent-tangentcorrelation function simplifies to

hti tiri !N;i!1

X

h1; j0;i2 expr=p1;; (9)

which reveals the importance of the two persistencelengths, p1; ln11;=0; (units of ‘0). It is notpossible to extract from hti tiri one unique length for thewhole range of T because the weights associated with p1;strongly vary with T.

We now compare the discrete model predictions withexperiment [1] by focusing on the melting profile ’BT ofa synthetic nonalternating homopolynucleotide, polydA-polydT. A typical profile for a homopolynucleotide has asigmoidal shape characterized by Tm and Tm. Of the fiveindependent parameters that appear in the theory whenK 0, three are determined experimentally (polymeriza-tion indexN and bending moduli assuming AB A [17])and two ( ~ and ~J) are determined by fitting the model toexperiment; hence, Tm Tm ~; ~J;N; A; B. Figure 1(a)shows ’BT for DNA of molecular weight 1180 kDa [1].From the known persistence lengths we obtain A 147and B 5:54 at 300 K. The solid line corresponds to ourmodel fit with ~ 4:46 kJ=mol and ~J 9:13 kJ=mol,leading to Tm 326:4 K. The fitted value for 2 ~ is closeto the experimental energy of 10:5 kJ=mol needed to breakan A-T link [3]. The value for ~J 2 ~ is also consistentwith the idea that destacking energy makes the dominantcontribution to DNA stability [1]. The renormalized coop-erativity parameter at Tm is ~J0 11:5 kJ=mol> ~J. Themodel fit thus leads to parameter values in accord withexperiment (in reality, the fitted values of ~ and ~J implic-itly compensate for effects like loop entropy explicitly leftout of the model [1,17]). In Fig. 1(a), the curve correspond-ing to N ! 1 is shown for the same parameter values. Inthis case, ’BT is given by Eq. (8) and T1m is obtainedfrom L0 0: kBT1m ’ 2 ~ ~K= ln~A=~B. In this limitthe transition width is nonzero, due to the finite coopera-tivity parameter: T1m / 1

I T1m 2 exp2J0T1m .

Since IT1m 2000, typical helix and bubble domains

are flexible within a small window about T1m . When Ndecreases, the width increases [Fig. 1(a)] roughly asTmN T1m N1. Hence even for a long polymer(N 103), finite size effects are non-negligible, in agree-ment with experiments [21]. Then the nature of end mono-mers becomes important, as confirmed in Fig. 1(a), whichshows how short chains begin to melt by end-unwinding atlower temperatures (the trend that Tm increases with de-creasing N will likely be reversed when loop entropy isincluded [2,17]). Our model predictions for experimentallyaccessible A-T pair quantities are in agreement with ac-cepted values [4,5]: (i) 107 at Tm, and at physiologi-cal temperature (ii) an interior single base-pair openingprobability of 106 with a bubble initiation barrier of


088103-3

17kBT, and (iii) a free energy of 0:18kBT for breaking anadditional base pair in an already existing bubble.

In contrast to purely Ising-type models, included in thepredictions of our theory are mechanical and structuralfeatures of the fluctuating chain, such as persistence lengthor mean-square radius, R hR2i1=2 with R ‘0

Piti.

From the expression for R in the limit N ! 1, wecan define an effective persistence length peff : R2 ’

2‘20N

peff 2‘

20N

Ph0;j1; i

2p1;. Because ofthe coupling between bending and internal states, therespective weights h0;j1;i2 associated with each cor-relation length change abruptly at Tm [cf. Fig. 1(b)]. Sincethe transition is very abrupt, it should also be possible toobserve it experimentally by measuring the radius of gy-ration by tethered particle motion [22], light scattering, orviscosity experiments. For instance, since the relative vis-cosity is proportional to R3, it should clearly exhibit an

abrupt thermal transition. Such a transition has indeed beenobserved for the viscosity of synthetic homopolynucleotidesolutions [23], in qualitative agreement with Fig. 1(b).Incorporating bending rigidity into DNA denaturationmodels thus allows us to make explicit predictions forboth melting profiles and DNA mean-size dependent quan-tities. It will be of great interest to both probe such effectsby carrying out experiments on DNA homopolymers andother biopolymers undergoing helix-coil transitions andextend our theory to heteropolymers, mechanical denatu-ration, and DNA dynamics.

[1] R. M. Wartell and E. W. Montroll, Adv. Chem. Phys. 22,129 (1972); O. Gotoh, Adv. Biophys. 16, iii (1983).

[2] D. Poland and H. R. Scheraga, Theory of Helix CoilTransition in Biopolymers (Academic, New York, 1970).

[3] F. Pincet et al., Phys. Rev. Lett. 73, 2780 (1994).[4] A. Krueger, E. Protozanova, and M. D. Frank-

Kamenetskii, Biophys. J. 90, 3091 (2006).[5] H. C. Fogedby and R. Metzler, Phys. Rev. Lett. 98, 070601

(2007); A. Bar, Y. Kafri, and D. Mukamel, Phys. Rev. Lett.98, 038103 (2007).

[6] Y. Kafri, D. Mukamel, and L. Peliti, Phys. Rev. Lett. 85,4988 (2000).

[7] E. Carlon et al., Phys. Rev. Lett. 88, 198101 (2002).[8] M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755

(1989); T. Dauxois, M. Peyrard, and A. R. Bishop, Phys.Rev. E 47, 684 (1993).

[9] J.-H. Jeon, W. Sung, and F. H. Ree, J. Chem. Phys. 124,164905 (2006).

[10] Y. Gao, K. V. Devi-Prasad, and E. W. Prohovsky, J. Chem.Phys. 80, 6291 (1984).

[11] J. Yan and J. F. Marko, Phys. Rev. Lett. 93, 108108 (2004).[12] C. Storm and P. C. Nelson, Europhys. Lett. 62, 760 (2003).[13] P. Ranjith, P. B. Sunil Kumar, and G. I. Menon, Phys. Rev.

Lett. 94, 138102 (2005).[14] N. Korolev, A. P. Lyubartsev, and L. Nordenskiold,

Biophys. J. 75, 3041 (1998).[15] R. M. Fye and C. J. Benham, Phys. Rev. E 59, 3408

(1999).[16] In reality, s in Eq. (2) should include an enthalpic shear

modulus term accounting for DNA stacking energy [8].[17] J. Palmeri, M. Manghi, and N. Destainville (to be pub-

lished).[18] J. Palmeri and S. Leibler, in Dynamical Phenomena at

Interfaces, Surfaces and Membranes, edited by D.Beysens et al. (Nova Science, New York, 1993).

[19] D. P. Aalberts, J. M. Parman, and N. L. Goddard, Biophys.J. 84, 3212 (2003).

[20] G. S. Joyce, Phys. Rev. 155, 478 (1967).[21] G. Altan-Bonnet, A. Libchaber, and O. Krichevsky, Phys.

Rev. Lett. 90, 138101 (2003).[22] N. Pouget et al., Nucleic Acids Res. 32, e73 (2004).[23] R. B. Inman and R. L. Baldwin, J. Mol. Biol. 8, 452

(1964).[24] All curves cross at T ’ 326:1 K, where effects due to the

difference in renormalized stacking energy (K0) exactlycompensate those due to renormalized destacking (J0)[17].

510

50100

5001000

320 325 330 335 340315

ξI

ξAp

ξeff

p

ξ1,+p

ξ1,-p

ξBp

(b)

T (K)Tm

∞

0.2

0.4

0.6

0.8

1

0325.5 326 326.5 327 327.5

T (K)

ϕB (a)∆Tm

∞∆Tm−

N10 102 103 104 105

10

1

10-1

10-2

Tm∞

10 -1

1

10 -2

10 -3

325 326 327 328

0.5

FIG. 1. (a) Melting curves (fraction of broken base pairs versustemperature) for polydA-polydT (data for 0.1 SSC ( 0:015M NaCl 0:0015M sodium citrate, pH 7.0), N 1815[1]). The solid line represents the theoretical result for ~ 1:64kBTm, ~J 3:35kBTm (Tm 326:4 K). The broken linecorresponds to N ! 1 (same parameter values). Lowerinset: Melting curves for N 100; 500; 1815;1 (in decreasingorder at low temperature, T < 326 K). Upper inset: Model re-sults for the shift in transition width Tm T1m versus polymerlength [24]. (b) Temperature variation (N ! 1) of the Isingcorrelation length I, persistence lengths of the coupled system,peff and p1;, and of the pure chains pA;B (in units of ‘0). At T1m ,the effective persistence length peff rapidly crosses over from pAto pB.


088103-4

Thermal denaturation of fluctuating finite DNA chains:The role of bending rigidity in bubble nucleation

John Palmeri, Manoel Manghi, and Nicolas DestainvilleLaboratoire de Physique Théorique, Université de Toulouse, CNRS, 31062 Toulouse, France

Received 18 September 2007; published 22 January 2008

Statistical DNA models available in the literature are often effective models where the base-pair state onlyunbroken or broken is considered. Because of a decrease by a factor of 30 of the effective bending rigidity ofa sequence of broken bonds, or bubble, compared to the double stranded state, the inclusion of the molecularconformational degrees of freedom in a more general mesoscopic model is needed. In this paper we do so bypresenting a one-dimensional Ising model, which describes the internal base-pair states, coupled to a discretewormlike chain model describing the chain configurations J. Palmeri, M. Manghi, and N. Destainville, Phys.Rev. Lett. 99, 088103 2007. This coupled model is exactly solved using a transfer matrix technique thatpresents an analogy with the path integral treatment of a quantum two-state diatomic molecule. When the chainfluctuations are integrated out, the denaturation transition temperature and width emerge naturally as an explicitfunction of the model parameters of a well defined Hamiltonian, revealing that the transition is driven by thedifference in bending entropy dominated free energy between bubble and double-stranded segments. Thecalculated melting curve fraction of open base pairs is in good agreement with the experimental meltingprofile of polydA-polydT and, by inserting the experimentally known bending rigidities, leads to physicallyreasonable values for the bare Ising model parameters. Among the thermodynamical quantities explicitlycalculated within this model are the internal, structural, and mechanical features of the DNA molecule, such asbubble correlation length and two distinct chain persistence lengths. The predicted variation of the mean-squareradius as a function of temperature leads to a coherent explanation for the experimentally observed thermalviscosity transition. Finally, the influence of the DNA strand length is studied in detail, underlining theimportance of finite size effects, even for DNA made of several thousand base pairs. Simple limiting formulas,useful for analyzing experiments, are given for the fraction of broken base pairs, Ising and chain correlationfunctions, effective persistence lengths, and chain mean-square radius, all as a function of temperature andDNA length.

DOI: 10.1103/PhysRevE.77.011913 PACS numbers: 87.10.e, 87.15.Ya, 82.39.Pj

I. INTRODUCTION

The stability of double-stranded DNA dsDNA at physi-ological temperature is due to the self-assembly of its basepairs: self-assembly within a same strand via base-stackinginteractions between neighboring bases; and self-assembly ofboth strands via hydrogen bonds between pairs of comple-mentary bases. These interactions, however, are on the orderof magnitude of a few kBT thermal energy 1–3 and ther-mal fluctuations can lead, even at physiological temperature,to local and transitory unzipping of the double strand see,e.g., 4 or the reviews 5,6. The cooperative opening of asequence of consecutive base pairs leads to denaturationbubbles, which are likely to play a role from a biologicalperspective, since they may participate in mechanisms suchas replication, transcription, or protein binding. For example,it has been proposed 7 that transcription start and regula-tory sites could be related to DNA regions which have ahigher probability of promoting bubbles. Indeed, the energyneeded to break an adenine-thymine A-T base pair 4kBT,connected by two hydrogen bonds, is smaller than the energyneeded to break a guanine-cytosine G-C one 6kBT threehydrogen bonds 1,2. At the same temperature, A-T richsequences present a priori more bubbles than G-C rich ones,even though sequence effects on the occurrence of bubblesare more complex than a simple examination of local A-Tbase abundance 1,7. In addition to the sequence, the frac-

tion of denaturation bubbles in vitro naturally depends ontemperature, as well as on the ionic strength of the solution5,8. In particular, the melting temperature Tm, above whichbubbles proliferate and the two strands completely separate,depends on both sequence and ionic strength. Another pa-rameter that affects the melting temperature is the length ofthe double strand 9,10. This is not a purely academic de-bate because short DNA strands a few tens of base pairs areinvolved in DNA chip experiments where the hybridizationprocess is precisely affected by temperature in a way de-pending on strand length and sequence see 11, and refer-ences therein.

Although the intracellular unwinding of DNA is due toactive and enzymatic processes by imposing unwinding tor-sional stresses 12, the thermally induced denaturation ofpurified DNA in solution has led to an intensive study ofDNA thermal denaturation 4–6,13,14. Mesoscopic modelshave been proposed to account for the thermodynamicalproperties of denaturation bubbles in DNA. The first modelswere Ising-type two-state models, where the base pairs canbe open or closed see 4,5, and references therein. In thesimple base-pair model, the Ising parameters are the base-pair chemical potential and the so-called cooperativity pa-rameter, which accounts for the energetic cost of a domainwall. This type of one-dimensional Ising model is exactlysoluble for homopolymers and for random sequences 4.More sophisticated effective Ising models, including the ten

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kinds of base-pair doublets, lead to better agreement withexperimental data 5. Poland and Scheraga 14, followingthe work of Zimm 15, included “loop entropy” in thesemodels, i.e., the entropic cost of closing large loops formedby destacked single-stranded DNA ssDNA in bubbles4,14. This Poland-Scheraga model is at the core of theDNA melting simulator MELTSIM algorithm first developedby Blake et al. 16,17. The intraloop and interloop self-avoidance corrects the loop statistics and refines the sharp-ness of the denaturation transition 18,19. To get a denatur-ation transition in these models, an effective temperature-dependent Ising chemical potential must be inserted by hand.Recently, Peyrard et al. developed nonlinear phonon modelswhere the shape of the interaction potential between basepairs is more precisely taken into account see 20–22, andthe review 23. By inserting estimated microscopic param-eters, they have shown that their model can both lead to adenaturation transition, analogous to interface unbinding,and be useful for studying bubble dynamics. The predictedtransition temperature is, however, very sensitive to themodel parameter values, and if physically reasonable valuesare used 21,24,25, Tm appears to be much too high and thetransition width much too large.

More generally, the theoretical study of DNA denaturationcan, in principle, be tackled on a least four levels of investi-gation determined by the amount of detail included, goingfrom 1 quantum ab initio approaches and 2 classical allatom molecular dynamical simulations 26, through 3 ef-fective mesoscopic approaches coupling chain conforma-tional and base-pair degrees of freedom 27, to 4 effectivestatistical models for base pairs alone 4,14,23, and refer-ences therein. In theory, it is possible to move up one step inthis hierarchy by integrating out the subset of the degrees offreedom that do not appear at the higher level, giving rise toan effective free energy at each level of description. Ourpurpose here is to show, via a minimal model for DNA ho-mopolymers, that if one starts at the third level and integratesout the chain degrees of freedom, one arrives at a physicallycoherent level 4 explanation for the DNA melting transi-tion: a bending free energy driven denaturation transitionemerges naturally due to the entropic lowering of the ener-getic barrier for bubble nucleation. Working at level 3 alsohas the added advantage of allowing access to the statisticsof the chain degrees of freedom effective persistence length,mean-square radius, etc., something that obviously is notpossible at level 4.

To this end, we have recently proposed a model 28,which considers not only the internal coordinates in terms ofIsing spin variables describing the open or closed states, butalso external coordinates, the chain tangent vectors, whichdetermine the chain configuration and depend sensitively onchain stiffness. Indeed, ssDNA is two orders of magnitudemore flexible than dsDNA at normal salt concentration. Wehave shown that this difference in bending rigidity providesan explanation for the bubble mechanism formation. Furtherevidence for the importance of this bending heterogeneityincludes phenomena such as cyclization, loop formation, andpackaging of DNA into nucleosomes 29 where denatur-ation bubbles facilitate bending of the otherwise rigid poly-mer DNA in structures where it coils up with curvature radii

down to 10 nm, despite a persistence length of dsDNA equalto 50 nm. Our model incorporates precisely this dependenceof the polymer bending rigidity on the state of neighboringbase pairs. It is the discretized version of a continuous model28, and couples explicitly an Ising model, describing theinternal degrees of freedom open or closed, and a Heisen-berg or discrete wormlike chain DWLC one, accounting forthe rotational degrees of freedom between successive mono-mers of the DNA chain. Its originality lies in the fact that theinternal two-state and external bending degrees of freedomare treated on an equal footing and therefore the renormal-ized Ising parameters obtained by integrating out the chaincan be exactly calculated within the scope of the model. Themelting temperature Tm naturally emerges and, together withthe transition width Tm, are explicitly written as a functionof the bending rigidities and strand length, which are experi-mentally known, and bare Ising parameters. In addition, theeffective Ising properties fraction of broken bases and cor-relation length as well as the chain ones persistence lengthand mean-square radius can be computed, allowing, in prin-ciple, direct comparison with experiments. An important fea-ture of this effective Ising model is that the end monomerssee an effective chemical potential that is higher than theinterior one; this end-interior asymmetry leads in a naturalway to a chain length dependence for Tm, the transitionwidth, and other statistical quantities. A similar model hadbeen previously proposed by one of us in two dimensions30, but its application to the study of denaturation bubblesin dsDNA was not made explicit. Recently, a similar ap-proach has been considered in the context of the dsDNAstretching transition 31. The addition in the energy func-tional of the term corresponding to the external force pre-vents an exact solution of the model in Ref. 31 and anapproximate variational scheme had to be implemented.

Beyond dsDNA or dsRNA, our coupled model can beused to describe the properties of any two-state biopolymer,as soon as the local bending rigidity depends on the localstates. As already mentioned, the transition from the B to theS form of dsDNA in force experiments has been investigatedin this framework 31. The helix-coil transition in polypep-tides can also be described by such a theory because the-helix configuration is much more rigid than the randomone 9,14.

The present paper is a detailed account of the results sum-marized in a Letter 28. In Sec. II, we present the coupledclassical Ising-Heisenberg model, which, as we show here,can be used to describe DNA thermal denaturation and writethe partition function in terms of a transfer matrix, usual inone-dimensional statistical systems. We note in passing thatthe coupled Ising-Heisenberg model presented here displaysa rich array of behavior and therefore may be of interest inother contexts, such as one-dimensional 1D classical spinchains or zero-dimensional 0D quantum rotators describinga diatomic molecule with internal states.

Because the full transfer matrix method for the coupledmodel leads to relatively complex calculations, we first showthat the model can be reduced to two effective Ising models,a path that provides a great deal of physical insight: indeed,these effective models allow the calculation of the free en-ergy, as well as the Ising and chain end-to-end tangent-

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tangent correlation functions in terms of an effective Hamil-tonian with temperature-dependent Ising parameters. Adetailed solution of the two effective models is provided inSec. III. Section IV is devoted to the calculation of the Isingand chain correlation quantities, using the effective Isingmodels, as well as a correlator mixing both Ising and chainvariables. Sections V and VI present in detail the full transfermatrix approach, which leads to the complete calculation ofchain correlations and end-to-end distance; a visual interpre-tation of the expression for the two-point correlation func-tions leads naturally to an analogy with a quantum diatomicmolecule. Our theory is compared to experimental denatur-ation profiles of synthetic DNA in Sec. VII and finite-sizeeffects, which are experimentally relevant, are thoroughlyexamined in Sec. VIII. Finally, our concluding remarks aregiven in Sec. IX, where we also summarize our principaltheoretical results of greatest interest for interpreting experi-ments. The principal symbols used in this work are definedand cataloged in Table I.

II. DISCRETE CHAIN MODEL

We model dsDNA as a discrete chain of N monomerslinks, each monomer can be in one of two different states,

U and B, which denote, respectively, unbroken and brokenbonds. The local chain rigidity depends on the nearby linktypes. A denaturation bubble is thus formed by a consecutivesequence of B type monomers. The chain’s conformationalproperties are determined by the set of N unit link tangentvectors ti : i=1, . . . ,N with ti=1. For simplicity, themonomer length a is taken to be the same for both U and Bfor modeling DNA stretching transitions it is necessary tointroduce different monomer lengths 9,31. The position ofthe end of the ith link in the three-dimensional 3D embed-ding space is Xi=X0+a j=1

i t j, where X0 is an arbitrary start-ing point. The end-to-end vector is R=a j=1

N t j. The linkstates are denoted by the value of an Ising variable i=1U or B associated with each link. These Ising variablesallow us to model a system of thermally activated defectssuch as the broken bonds that proliferate on certain macro-molecules like DNA when the temperature is raised. In thiscase the temperature-dependent concentration of brokenbonds is controlled by an appropriately defined chemical po-tential. Because we are interested here in the new phenomenathat arise due to the coupling between the internal Ising andexternal chain conformational degrees of freedom, we will

TABLE I. Index of the main symbols used throughout the paper, with their mathematical definition, and reference.

Symbol Quantity Mathematical definition Reference

Tm Melting temperature UTm=BTm=1 /2 Secs. I and II

N Chain length Sec. II

U /B Unbroken/broken base pair Sec. II

U,B Fractions of U’s and B’s Eq. 6i Internal degree of freedom Ising variable 1 Sec. II

ti Chain unit tangent vector Heisenberg variable ti=1 Sec. II

i,i+1 Local chain bending rigidity coupling U, B, or UB Sec. II

J Half-energy of a domain wall Sec. II

K Difference in stacking energy between ds and ssDNA Sec. II

Half-energy required to open a base pair Sec. II

,J ,K , , . . . Adimensional energies in units of kBT = , . . . Sec. II

G0 Free energy of a single joint of rigidity −lnsinh / Sec. II

J0 Renormalized effective Ising parameter JJ−

1

4G0U+G0B−2G0UB

Eq. 15

K0 Renormalized effective Ising parameter KK−

1

2G0U−G0B

Eq. 16

L0 Effective chemical potential to open an interior base pair +K0 Sec. II

c Infinite-size Ising “magnetization” sinhL0 / sinh2L0+e−4J01/2 Eq. 52U,, B, Fraction of unbroken U and broken B bonds N→ 1 c /2 Sec. III

B,i Site i bonding opening probability melting map 1− i /2 Eq. 117Tm Infinite size N→ melting temperature L0Tm

=0 Sec. II

Tm Transition width N→ 2c /T−1 at Tm

and Figs. 5 and 6 Eq. 53i+ri Ising correlation function Sec. IV

I Ising correlation length e2J0 /2 at Tm and Fig. 4 Sec. IV

ti+r · ti Chain correlation function Fig. 3b Sec. V

1,+p ,1,−

p Persistence lengths Fig. 4 Eq. 100eff

p Effective persistence length Fig. 4 Sec. VI

R Chain mean-square radius R21/2 Secs. II and VI

T* Crossover temperature for finite chains B /NT*=0 Sec. VIII

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not attempt to take into account at the same time the self-avoidance of the chain.

After presenting the model we explain how to calculatethe partition and correlation functions connected with boththe internal and external degrees of freedom for the coupledmodel. Once we have these quantities for the coupled Ising-chain system, we will be able to compare the results for thecoupled system with those for the uncoupled one. We willsee that the coupling can strongly modify the results, namely,the average properties of the system at a given temperature interms of the average concentration of closed and open bonds,average mean-square chain radius, and two-point correlationfunctions.

Up to an absolute location in space a state of the chain isgiven by the 2N variables i , ti. For a chain in 3D each linkvector can be expressed in spherical coordinates as ti= (sinicos i , sinisin i , cosi) and can therefore bedefined by the azimuthal and polar angles i and i, denotedtogether by the solid angle i= i , i. The energy Hi , tiof a state is taken to be

Hi,ti = i=1

N−1

i+1,i1 − ti+1 · ti

− i=1

N−1 Ji+1i +K

2i+1 + i −

i=1

N

i. 1

The angle i,j between two tangent vectors is given by

cos i,j = ti · t j = sinisin jcos i − j + cosicos j .

2

The first term in H is the bending energy of a DWLC with alocal rigidity i+1,i, having the dimension of energy, that de-pends on the neighboring values of the Ising variables. Wehave taken, without any loss of generality, the minimum ofthe bending energy to be zero independent of the values ofthe Ising variables. The second and third terms make up the

energy of the Ising model 4,5, HIHIJ , K , , illustratedin Fig. 1.

The term in J accounts for the local destacking energy

2J of a domain wall where i passes from one value to

another. The term in K accounts for the difference in stack-ing energy between a segment of dsDNA and of a denatur-ation bubble. The last term gives the energy 2 required tocreate a link in the state i=−1 a B link or broken bond.We write a dimensionless Hamiltonian Hi ,i, where =1 / kBT, which thus contains the dimensionless parameters

i+1,ii+1,i, JJ, KK, and . The local rigid-ity is

i+1,i = U for U-U nearest neighbor

B for B-B nearest neighbor

UB for U-B or B-U nearest neighbor, 3

where “U-U nearest neighbor,” etc., denotes nearest neigh-bor link types. In terms of the Ising field i,

i+1,i =1

4U + B − 2UBi+1i +

1

4U − Bi+1 + i

+1

4U + B + 2UB . 4

We identify the B state with two identical noninteractingsingle DNA strands, ssDNA, each with a local rigidity equalto B /2. From Eqs. 1 and 4, the result of the couplingbetween Ising and tangent variables can already be predicted:

both the destacking and stacking parameters J and K will bemodified by the two first terms in Eq. 4, which have exactlythe same functional form in i, whereas will remain un-changed. Moreover, when all the bending rigidities are equal,U=B=UB, the two first terms in Eq. 4 disappear and theHamiltonian 1 decouples into a pure Ising Hamiltonian anda pure DWLC one isomorphic to a 1D classical Heisenbergmodel for magnetism 32: Hi , ti=HDWLCti+HIi. Inthe language of magnetism the model studied here is a clas-sical coupled Heisenberg-Ising spin chain. Although pure ef-fective Ising models have been used extensively to modelhelix-coil and denaturation melting transitions in macro-molecules, it was necessary to introduce phenomenologicallyan effective temperature-dependent chemical potential orstacking energy to obtain a melting transition. A key featureof the coupled model used here is that a melting transitionwill emerge naturally in the effective Ising model obtainedby integrating out the chain conformational degrees of free-dom.

The quantities that we will use to study the behavior ofthe coupled system are the mean of the internal state variable“magnetization” in spin language

c 1

Ni=1

N

i, 5

the local state average i, correlation functions for the Isingvariables, i+ri, and the chain tangent vectors ti+r · ti, andthe mean-square radius RR21/2. These correlation func-

J2 J2

=σ −1−1 +1 +1+1+1+1

K2K K 00 0

2µ 2µ

000 0

FIG. 1. Color online Illustration of the different Ising param-eters appearing in the Hamiltonian H. Open closed base pairs arecoded by a spin = +1−1. The energies indicate the cost of open-ing base pairs with respect to the ground states where all are setto +1. The first line shows the cost, 2J, of a domain wall. Thesecond line indicates the energy, 2, required to open a base pair.The third line gives the difference in stacking energy between asegment of dsDNA and a denaturated one: dark light blue cigarsindicate stacked states in dsDNA ssDNA and the absence of acigar indicates the destacking of adjacent base pairs, which is al-ready taken into account by the J contribution.


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tions measure the extent of cooperativity exhibited by thecoupled system: e.g., the size of the B U domains belowabove the melting temperature, and the length scale onwhich the chain remains rigid. The concentration of U and Blinks is given by

BN,T =1 − cN,T

2= 1 − UN,T . 6

Once the type of homopolymeric DNA is chosen, the bareIsing parameters and the chain bending rigidities can be con-sidered fixed, and therefore c, U, and B become func-tions of the experimental control parameters, namely, tem-perature T, and chain length N. For a pure U chain U=1 andB=0; for a pure B chain U=0 and B=1. A chain with afinite concentration of bubbles B links will have B0 andthe melting temperature Tm, if it exists, will be definedby UTm=BTm=1 /2. The equilibrium statistical averageof a quantity O=Oi ,i that depends on the fluctuatingdegrees of freedom i ,i, is given by

O 4N

Z i=1

i=1

N di

4Oi,ie−Hi,i, 7

where

Z = 4Ni

i=1

N di

4e−Hi,i 8

is the partition function. The partition and correlation func-tions for the coupled system can be calculated using transfermatrix techniques. For example, the partition function can bewritten as

Z = 4Ni

i=1

N di

4V1

1P1,22 ¯ N−1PN−1,NNNV ,

9

where the transfer kernel that appears N−1 times in Eq. 9,is given by

Pi,i+1 = eUcosi+1,i−1+J+K+ eUBcosi+1,i−1−J

eUBcosi+1,i−1−J eBcosi+1,i−1+J−K− .

10

It is written in the canonical basis U= +1 and B= −1 ofthe U and B states. The end vector

V = e/2U + e−/2B 11

enters in order to take care of the free chain boundary con-ditions. Different boundary conditions could be easilyhandled in a similar way, for instance, for closed openends, V= U B and all the following results not explic-itly using Eq. 11 remain valid.

Before presenting the full transfer matrix method, we firstshow that the partition function and averages of any quanti-ties depending only on the Ising variables can be obtained byexamining the effective Ising model obtained by integrating

over the chain conformational degrees of freedom the linktangent vectors in Eq. 9. The problem reduces to that of aneffective Ising model with an “effective free energy” HI,eff

0

containing renormalized parameters. This method works be-cause, for the coupled Ising-chain system, the rotationalsymmetry is not broken absence of a force term ti · z in theHamiltonian 9,31. Hence the matrix obtained by integrat-

ing the kernel Pi ,i+1 in Eq. 9 is the same for any sitei. We thus are able to carry out the solid angle integrations insequential fashion by using the i+1th tangent vector as thepolar axis for the ith solid angle integration. The solid angleintegrated transfer matrix is

PI,eff0 = di

4Pi,i+1 = e−G0U+J+K+ e−G0UB−J

e−G0UB−J e−G0B+J−K− ,

12

where G0 is the dimensionless Helmholtz free energy ofa single joint two-link subsystem with rigidity eitherU-U, B-B, U-B, or B-U, given by

G0 = − ln d

4exp„cos − 1…

= − ln sinh

, 13

an increasing function of , varying linearly with for 1 high T or uncorrelated tangent regime and as ln2 for1 low T entropy dominated regime where the spin waveapproximation for the chain degrees of freedom is valid.The effective transfer matrix PI,eff

0 can be written in Isingform using the renormalized Ising parameters J0, K0, and theprefactor exp−0, all depending on chain rigidities. It isgiven by

PI,eff0 = e−0e+K0+J0 e−J0

e−J0 e−−K0+J0 , 14

J0 J −1

4G0U + G0B − 2G0UB , 15

K0 K −1

2G0U − G0B , 16

0 1

4G0U + G0B + 2G0UB . 17

In the limit of high temperature, the chain tangents will becompletely decorrelated and the uninteresting renormaliza-tions of J and K arise solely from i+1,i Eq. 4, in agree-ment with Eqs. 15 and 16, as revealed by the linear de-pendence of G0 on in this limit. It is rather in theopposite limit of low temperatures and therefore stronglycorrelated chain tangents that the bending entropy driven de-naturation transition arises.

The full partition function Z=ZI,eff0 , in Ising transfer ma-

trix notation,


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ZI,eff0 = 4N

iV1

1PI,eff0 2 ¯ N−1PI,eff

0 NNV , 18

can be rewritten explicitly in terms of an effective Ising-freeenergy HI,eff

0 as follows:

ZI,eff0 = 4Ne−N−10

ie−HI,eff

0 i, 19

where HI,eff0 =HIJ0 , K0 , is given by

HI,eff0 = − J0

i=1

N−1

i+1i −K0

2 i=1

N−1

i+1 + i − i=1

N

i

= − J0i=1

N−1

i+1i −L0

2 i=1

N−1

i+1 + i −

21 + N ,

20

with

L0 + K0 = + K − G0UB/2, 21

and G0UBG0U−G0B. Because HI,eff

0 depends on thetemperature, it cannot be considered as an Ising-state energy,but rather as an effective free energy obtained by integratingout the chain subsystem cf. discussion concerning levels oftheoretical study in the Introduction.

When two links open the renormalized stacking energy ofthe links is K0, which is smaller than K by the difference inbending free energy between U-B and B-B joints. The sec-ond form of HI,eff

0 in Eq. 20 shows that the effectivechemical potential of one interior base pair is renormalizedto L0. If the gain in the one link bending free energy in goingfrom U to B, G0

UB, is greater than the intrinsic energy

2+ K, needed to break a closed interior bond, then theeffective interior joint chemical potential L0 can becomenegative, signaling a change in stability of U and B states.The end links i=1,N, however, feel a different chemicalpotential +K0 /2, which is larger than L0 in the case ofexperimental interest UB and K=0; see Sec. VII. Thisend-interior asymmetry, along with the extra bubble initia-tion energy due to the second domain wall, are reflected inthe difference between the “effective free energy” needed tocreate an n bubble at a chain end,

Gendn = 2J0 − K0 + 2nL0, 22

and in the chain interior

Gintn = 4J0 + 2nL0, 23

which is higher than for an end link by 2J0+K0. As will beconfirmed in Sec. VIII, this difference in effective free en-ergy suggests that at sufficiently low T bond melting willbegin at the chain ends. Written out in greater detail, Eq. 23leads to

Gintn = 4J0 + 2nkBTL0 = 4J + 2n + K − kBTG0U

+ G0B − 2G0UB − nkBTG0UB. 24

In the uncoupled limit U=B=UB only the first twotemperature-independent terms survive. Although this end-interior asymmetry plays no role in the limit of an infinitechain N→, it will play an essential role in determininghow bond melting varies with temperature and bond locationmelting maps and how the melting temperature varies withchain size. These finite size effects are discussed in detail inSec. VIII.

Because the renormalized Ising parameters depend on thechain parameters, the coupled model does not, in general,have the same behavior as the uncoupled one. Indeed, sinceG0 is an increasing function of and UB for dsDNA,if the difference between U and B is sufficiently large, thenat a certain temperature Tm

, the effective interior bubblechemical potential L0 can vanish. Provided that this tempera-ture be sufficiently low for thermal disorder to be weak andend effects due to the finite size of the chain to be unimpor-tant, the state of the system will flip from nearly pure U forTTm

, where L00, to nearly pure B for TTm, where

L00. Precisely at T=Tm, there will be, on average, as many

closed as open bonds and B,=U,=1 /2, where B,T limN→ BN ,T, etc. This transition strictly speaking acrossover, which can be extremely sharp under some cir-cumstances see below, can be interpreted as a melting ordenaturation transition. From the above analysis, we seeclearly that it is the difference in free energy between U-Uand B-B joints, G0

UB, that drives the melting transition. Wewill see below, moreover, that if U and B are much greaterthan one, then the spin wave approximation is valid, and theentropy term in G0

UB dominates. We will see in Sec. VIIthat this is actually the case for real DNA.

By calculating the average of the product of the cosines ofthe N−1 angles, i cosi+1,i and using the same techniqueused above for ZI,eff

0 , we can define another effective model,but now with partition function

ZI,eff1 = Z

i=1

N−1

ti+1 · ti . 25

This expression can be written in Ising transfer matrix formas

ZI,eff1 = 4N

iV1

1PI,eff1 2 ¯ N−1PI,eff

1 NNV , 26

where

PI,eff1 = di

4cosi+1,iPi,i+1

= e−1e+K1+J1 e−J1

e−J1 e−−K1+J1 , 27


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J1 J −1

4G1U + G1B − 2G1UB , 28

K1 −1

2G1U − G1B , 29

1 1

4G1U + G1B + 2G1UB . 30

The function G1=−ln d4 cosexp(cos−1) is

related to the tangent-tangent correlation function betweentwo adjacent monomers isolated two-link subsystem withrigidity in the following way:

t1 · t2two-link = costwo-link

= exp− G1 + G0

= u = coth − 1/ , 31

which is the Langevin function 33. It increases with ,varying as /3 for 1 and as 1−1 / for 1. Thisasymptotic behavior corresponds to G1−G0 varying asln3 / for 1 and as 1 / for large . For a pure chainU or B costwo-link is equal to the nearest-neighbor tan-gent correlation function ti+1 · ti=exp−1 /p with persis-tence length

p = − 1/lnu = G1 − G0−1. 32

In the high spin wave approximation, we obtain p,as expected.

The partition function can be rewritten explicitly in termsof an effective Ising free energy HI,eff

1 as follows:

ZI,eff1 = 4Ne−N−11

ie−HI,eff

1 i, 33

where

HI,eff1 = −

i=1

N−1 J1i+1i +L1

2i+1 + i −

21 + N .

34

By repeatedly using the vector identity a ·cb ·d= a ·db ·c− ab · cd for a ·b= ti+1 · ti, etc., alongwith the property that averages of cross products ti+1 tiare zero and that ti2=1, the partition function ZI,eff

1 can bewritten as the product of the end-end tangent-tangent corre-lation function t1 · tN, and the effective Ising partition func-tion ZI,eff

0 in the following way:

ZI,eff1 = t1 · tNZI,eff

0 . 35

When U=B=UB=, we recover the pure chain tangent-tangent correlation function, given by:

t1 · tN =ZI,eff

1

ZI,eff0 → exp− G1

exp− G0N−1

= exp− N − 1/p . 36

Coming back to the difference in dimensionless free en-ergy G0

UB, we can show that at room temperature it is domi-nated by its entropic part. Indeed, G0 can be split into anaverage dimensionless energy and average entropy contri-bution G0=E0−S0 /kB, where E0=G0 /=G0 /. Hence, the average energy of a two-link sys-tem can be written in terms of G0 and G1 as follows:

E0 = 1 − exp− G1 + G0

= 1 − exp− 1/p = 1 − u , 37

and therefore E0UBE0U−E0B= U−B− UuU

−BuB. Because the function u tends to 1, E0UB→0,

and therefore for temperatures low enough for the spin waveapproximation to be valid for both U-U and B-B links, wesee that G0

UB−S0UB /kB lnU /B. Indeed in this ap-

proximation, the Hamiltonian becomes Gaussian and equi-

partition of energy occurs: E−1⇒ E=E1. In thiscase the melting transition, if it exists, will be driven over-whelmingly by the difference in entropy between U-U andB-B joints. As done for G0

UB, the difference in dimension-less free energy G1

UB can be split into an average dimen-sionless energy and average entropy contribution using thesame formulas as above. If L1=0 at a certain temperature T1

,then we can expect another type of “melting” transition, nowdriven by the free energy difference G1

UBG1U−G1B. We return to this point below.

It should be noticed that the above process could, in prin-ciple, be carried on with increasing difficulty to calculatehigher order correlation functions. Hence these effectiveIsing models give information on multipoint tangent-tangentcorrelation functions. In the next section we obtain the solu-tions to the two effective Ising models.

III. SOLUTION OF THE TWO ISING MODELS

The effective Ising partition and correlation functions canbe obtained using well-known Ising transfer matrix tech-niques 9. In order to treat in parallel ZI,eff

0 and ZI,eff1 , we

introduce the index l=0,1 and compute the associated parti-tion function ZI,eff

l . This index will be useful in Sec. V wherewe introduce the transfer matrix approach. We need theeigenvectors and eigenvalues of the transfer matrices:

PI,effl l=ll, where the expressions for PI,eff

l are givenin Eqs. 12 and 27. The eigenvalues are

l, = eJl−lcoshLl sinh2Ll + e−4Jl1/2 , 38

and they obey the inequality l,+l,−. The two orthonormaleigenvectors l are

l, + =1

2leJl

alU + al−1B

and

l,− =1

2leJl

al−1U − alB , 39

where


011913-7

l = sinh2Ll + e−4Jl1/2 and al = eJlsinhLl + l1/2.

40

The transfer matrices can be expanded in terms of theeigenvectors

PI,effl =

=l,l,l, . 41

Using the decomposition of the Ising 22 identity matrix

II=l ,l , the orthonormality of the eigenvectors l ,for each value of l i.e., l , l ,=,, and the forms Eqs.18 and 26 for the effective Ising partition functions, wethen find

ZI,effl = 4NVPI,eff

l N−1V = 4N =l,

N−1Vl,2.

42

The full partition is given by Z=ZI,eff0 . The matrix elements

V l ,, entering Eq. 42 can be obtained explicitly fromEqs. 11, 39, and 40.

When U=B=UB, the system decouples, but the pure

Ising model with temperature-independent parameters J, K,and , exhibits neither a second order phase transition at a

finite temperature in zero “field” + K=0, nor a melting

transition at a finite temperature for + K0. Indeed, therecan be no melting transition because the inequality B1 /2 holds over the whole temperature range. At low tem-peratures sinh+Ke−2J the system is ordered with U

1 and B0. As the temperature is raised, denaturationbubbles are thermally excited, with a crossover when

sinh+Ke−2J, or roughly = kBT−1+ K+2J−1.At higher temperatures sinh+Ke−2J, the average“magnetization” monotonously approaches a completelythermally disordered state with c=0 and U=B=1 /2.Note that this regime is not reached for DNA since as we willsee below, T8Tm and the model is certainly no longervalid for such high temperatures. By contrast, the coupledIsing-chain model will exhibit a very different behavior, witha finite temperature melting transition. In the following weimplicitly assume that all temperatures of interest obey TT.

Although the eigenvectors l , are orthogonal in for thesame value of l, this is not necessarily the case for differentvalues of l as we will see below, this is a consequence of thedifference in rotational symmetry. In general, depending onthe values of the temperature-dependent effective Ising pa-rameters Ll and Jl, the eigenvectors are complicated mixturesof the canonical basis states U and B. These eigenvectorsand their corresponding eigenvalues can, however, be sim-plified in two important limits:

1 For sufficiently low or high temperatures, below orabove the transition temperature Tl

at which Ll vanishes,the inequality sinh2Lle−4Jl is obeyed the experimentalmelting temperature for infinite chains is thus Tm

T0. As a

consequence, the off-diagonal domain-wall or “tunneling”terms in PI,eff

l can be neglected and the eigenvectors reduce

asymptotically to the canonical ones, with the mapping de-pending on the sign of Ll: l , + U and l ,−−B forLl0 and l ,−U and l , + B for Ll0. In this limitof strong cooperativity, the eigenvalues reduce to

l, expJl Ll − l for sinh2Ll e−4Jl, 43

and therefore the pure U state is strongly favored if Ll0and the B state if Ll0, because

l,+

l,− exp2Ll for sinh2Ll e−4Jl. 44

By introducing the following eigenvalues for pure U andpure B,

l,U expJ + − GlU , 45

l,B expJ − − GlB , 46

and using the definitions of Jl, Ll, and l, these limitingforms for the eigenvalues can be further simplifiedsinh2Lle−4Jl as follows:

l,+ l,U

l,− l,B for Ll 0

and

l,+ l,B

l,− l,U for Ll 0. 47

2 For temperatures at or very near the transition tem-perature, however, the opposite inequality sinh2Lle−4Jl

holds and Ll can be set to zero in Eqs. 38–40: the eigen-vectors then reduce to symmetric and antisymmetric super-positions of the canonical basis vectors U and B as isgiven by:

l, 12

U B for sinh2Ll e−4Jl, 48

with eigenvalues

l, e−leJl e−Jl for sinh2Ll e−4Jl. 49

In this limit of weak cooperativity the eigenvalue ratio isapproximately l,+ /l,−cothJl and the behavior of thesystem is dominated by the domain walls. The average be-havior for large N, which is governed by the ground sym-metrical state, shows vanishing average for the mean of theIsing state variable or magnetization in spin language forl=0 and N→, UB1 /2, since there is no spontaneoussecond order phase transition for the 1D zero field Isingmodel. The exact results for the eigenvalues and eigenvec-tors interpolate smoothly between the above simplified re-sults obtained far from and close to Tl

Fig. 2.Using Eq. 11, these limiting forms can be used to obtain

simple approximations for the following important matrixelements:

V0, + = e/2, T Tm

2 cosh/2 , T = Tm

e−/2, T Tm , 50


011913-8

V0,− = − e−/2, T Tm

2 sinh/2 , T = Tm

e/2, T Tm . 51

The limiting forms for V 0,− reveal that this matrix ele-ment is negative for low temperature and positive at Tm

andtherefore must pass through zero at a temperature T* lowerthan Tm

. This special temperature will be studied in detail inthe section concerning finite-size effects Sec. VIII.

If a melting transition exists for infinite chains at a finitetemperature Tm

, then L0 will go from a positive value belowTm, through zero at the transition, then to a negative value

above Tm. This temperature dependence for L0 is similar to

the time dependence of the uncoupled energy levels in theLandau-Zener problem, a quantum two-state dipole systemin an electrical field varying linearly with time 34. It is notsurprising, therefore, that the two branches for the “adia-batic” states of the Landau-Zener problem are equivalent, asthe time t varies from − to +, to the “eigenenergies,”l,−lnl,, of the states l , presented above, as thetemperature varies from below the transition to above withthe same type of limiting behavior near and far from thetransition temperature T=Tl

, equivalent to t=0 in the quan-tum Landau-Zener problem; see Fig. 2. As in the quantummechanics of diatomic molecules, eigenenergies possessingthe same rotational symmetry same l cannot cross levelrepulsion, although they do reach a point of closest ap-proach at Tm

. As an illustration we show in Fig. 2 theLandau-Zener diagrams l,T for l=0,1 and observe levelrepulsion near T=Tm

and T=T1 for states with the same l

and the possibility of level crossing for states with differentvalues of l.

From the full partition function, we define the dimen-sionless Helmholtz free energy per Ising variable of thecoupled system F=−N−1 ln Z. The average value of the

Ising-state variable or “magnetization” can then be ob-tained for finite chains using c=−F /, from which U

and B can be deduced. The expression for c simplifies inthe limit N→, because only the largest of the eigenvalues,0,+, entering in the l=0 effective Ising partition functionsurvives.

c →N→

c −f

=

sinhL0sinh2L0 + e−4J01/2 , 52

where f =limN→ F=−ln 0,+. Equation 52 can then be usedto find U, and B,.

If L0 vanishes at a temperature Tm, low enough for the

e−4J0 term in the denominator to be sufficiently small, thenthe system will undergo a sharp melting transition: c willjump sharply from +1 for TTm

pure U state to −1 forTTm

pure B state. The size of the e−4J0 term in Eq. 52will determine the width of the transition region,

Tm 2 c

T

Tm

−1

2kBTm

2

+ Kexp− 2J0Tm

, 53

which is exponentially small in J0Tm when J0Tm

1.In a similar manner, from the effective partition function

ZI,eff1 , we can define the free energy per Ising spin, F1

=−N−1 ln ZI,eff1 . A quantity analogous to the average Ising

“magnetization” c, for this partition function can then beobtained for finite chains using

c1 = −F1

=

ct1 · tNt1 · tN

, 54

from which U1= 1+ c1 /2 and B

1 can be obtained.From the definition of c1, we see that it is a mixed corre-lation function that describes how the average system inter-nal state c is correlated with its external state chain configu-ration via the end-end tangent-tangent correlation function.In the limit of an infinite chain c1 reduces to the followingexpression analogous to Eq. 52

c1 →N→

c1 = −f 1

=

sinhL1sinh2L1 + e−4J11/2 , 55

where f 1=limN→ F1. This limiting formula is similar tothe one obtained for c in Eq. 52, which implies that if L1

vanishes at a temperature T1, low enough for the e−4J1 term

in the denominator to be sufficiently small, then the systemcan undergo a second type of “melting” transition: c1 willjump sharply from +1 for TT1

pure c1 “U” state to −1for TT1

pure c1 “B” state. For TmTT1

, c1 +1 while c−1. This implies that in this temperaturerange ct1 · tN−ct1 · tN. This counterintuitive result isanother manifestation of the nontrivial coupling between in-ternal and external degrees of freedom. The value of the e−4J1

term determines again the width of the transition region

T1 2 c1

T

T1

−1

2kBT1

2

+ Kexp− 2J1T1

.

56

320 330 340 350 360

0.4

0.6

0.8

1

T (K)

ε=−ln(λ)λ0,+

λ0,-

λ1,+

λ1,-

Tm∞

T1∞

FIG. 2. Color online “Energies” l,=−lnl, for l=0,1 inthe quantum formalism related to the Landau-Zener problem vstemperature for parameter values used in Sec. VII to fit experimen-

tal melting data: =4.46 kJ /mol, J=9.13 kJ /mol, and K=0. Weobserve that far from the two transition temperatures, the eigenval-ues reduce to the limiting forms as shown in Eq. 47. The inset isa zoom close to Tm

showing the level repulsion between thebranches 0, . A similar level repulsion occurs near T1

betweenthe branches 1, .


011913-9

IV. ISING STATE VARIABLE—ISING AND CHAINCORRELATION FUNCTIONS

The average value of the local Ising variable i and thetwo-point i+ri correlation function can be calculated bystarting from the expression 9 for the partition function andusing the property that an insertion of a term j in the sum ofproducts is equivalent to the insertion of the Pauli matrix inthe canonical basis,

z = 1 0

0 − 1 , 57

at the jth position in the product of transfer matrices definingthe partition function, Eq. 42. This comes from z=, and the equality

j jPI,eff0 j+1 =

=1 jzPI,eff

0 j+1

= jz · PI,eff0 j+1 . 58

We then find

i =4N

Z VPI,eff0 i−1zPI,eff

0 N−iV , 59

i+ri =4N

Z VPI,eff0 i−1zPI,eff

0 rzPI,eff0 N−r−iV .

60

By the same method used for reducing the partition function,we finally obtain

i =4N

Z 1,2

V0,20,2i−1 0,2z0,10,1

N−i 0,1V ,

61

i+ri =4N

Z 1,2,3

V0,30,3i−1 0,3z0,20,2

r

0,2z0,10,1N−r−i0,1V , 62

with i=. The matrix elements appearing in the above ex-pressions can be found explicitly using Eqs. 11, 39, and40.

The Pauli matrix z, which can be interpreted as a quan-tum mechanical dipole moment operator, is diagonal in thecanonical basis U= +1 and B= −1 see Eq. 57, but

not necessarily in the basis that diagonalizes PI,effl . Indeed, in

the l=0 basis we have

z0 = c 1 − c

2 1/2

1 − c2 1/2 − c

. 63

On the one hand, the transfer matrix PI,effl mixes the canoni-

cal basis states, which explains the complicated representa-tion of the effective Ising partition function, Eqs. 18 and26, state variable average, Eq. 61, and correlation func-tion, Eq. 62 in this basis. On the other hand, in the basis

that diagonalizes the transfer matrix, the propagation be-tween measurements is simple no mixing, but now, in gen-eral, the “dipole measurement process, corresponding to z,mixes such states. By directly summing Eq. 61 over i andusing the matrix elements of Eq. 63, we can calculatecN ,T as follows:

cN,T = c 1 −2RV

2

RV2 + eN−1/I

+2RV

1 − c2 1 − e−N/I

N1 + RV2e−N−1/I1 − e−1/I

, 64

where

RV V0,− V0, +

, 65

and I is the Ising correlation length

I = 1/ln0,+/0,− , 66

the typical size of minority BU domains below above Tm.Although V 0,− can be positive or negative and evenzero, V 0, + is for physical reasons strictly positive, be-cause both V and 0, + are linear combinations of the ca-nonical basis states with strictly positive coefficients of pro-portionality see Eqs. 39, 40, 50, and 51. The ratio RVwhich can therefore be negative, zero, or positive is thusalways well defined.

The above expression, Eq. 62, for i+ri can be inter-preted, using “path integral imagery, as a quantum mechani-cal measurement process over an imaginary time period of Nsteps of duration . This interpretation is based on the 1Dclassical Ising representation of the partition function of a 0Dquantum two-state system 35: the transfer matrix becomes

the quantum propagator, PI,eff0 ↔exp−H /, where H is the

quantum Hamiltonian, and the eigenvalues of the transfermatrix are related to , the energy eigenvalues of the Hamil-tonian via ↔−ln . In general the two states for the quan-tum system are coupled by a non-zero tunneling amplitude,which corresponds to the off-diagonal domain-wall termsof the transfer matrix. For instance, following Eq. 62, thesystem is prepared in the initial state V and evolves N−r− i time steps under the dynamics determined by the propa-

gator PI,eff0 , until a measurement of the dipole moment is

performed, determined by the action of z. The state thatcomes out of the measurement then evolves r time steps untila second measurement of the dipole moment is performed.The state that comes out then evolves i−1 further time steps.The correlation function i+ri is thus the normalized am-plitude that the system returns to the initial state V at theend of this double measurement process.

In the limit N→, the results Eqs. 61 and 62 simplifybecause we keep only the leading order terms largest eigen-values which sets 1=+.

i →N→

c + RV1 − c2 1/2 exp− i − 1/I , 67


011913-10

i+ri →N→

3,2

V0,3V0, +

0,3

0,+i−1

0,3z0,2

0,2

0,+r

0,2z0, + . 68

Using the above results we obtain the following limitingform for cN ,T when N→

cN,T →N→

c +2

N

RV1 − c

2

1 − e−1/I. 69

In the double limit N , i→, meaning that we ignore theinfluence of end monomers, expressions 67 and 68 reduceto the simpler cyclic boundary condition forms as follows:

i →N,i→

c, 70

i+ri →N,i→

c2 + 1 − c

2 exp− r/I . 71

Using the limiting values obtained earlier, Eq. 49, we findthat at the melting temperature Tm

, I=−1 / lncothJ0.When e−2J0Tm

1, ITme2J0Tm

/21. We shall see inSec. VII that I can be extremely large, but finite, at Tm

,where it reaches its maximum value. Moving away from Tm

in both directions, we find that I1 / 2L0 decreases asT−Tm

increases. When ITm1, the width of the transi-

tion, Eq. 53, can be rewritten as TmkBTm

2 / + KITm

. Because the system is translationally invariantin the limit N , i→ and i+r− i+ri− i= i+ri− i2, Eqs. 70 and 71 show that for c1 the Isingcorrelation length measures the range of correlations be-tween spatially separated deviations of the local Ising spinfrom the average value.

Using the same Ising-model transfer matrix techniquesemployed above for the partition function, we can calculatethe chain end-end tangent-tangent correlation functiont1 · tN, which is related to the effective partition functionsZI,eff

l , l=0,1 by

t1 · tN =ZI,eff

1

ZI,eff0 =

1,N−1V1,2

0,N−1V0,2

=V1, + 2 exp− N − 1/1,+

p + V1,− 2 exp− N − 1/1,−p

V0, + 2 + V0,− 2 exp− N − 1/I, 72

where the chain persistence lengths are defined by

1,p = 1/ln0,+/1, . 73

This result indicates clearly that, in general, t1 · tN dependson three distinct characteristic lengths: I and 1,

p .In order to better understand the physical content of Eq.

72 and later results, it is useful to derive simplified lim-iting forms for the matrix elements and persistence lengthsappearing therein. Using the same technique employed toobtain the expressions for V 0, shown in Eqs. 50 and51, simple limiting forms can be derived for V 1, inthe temperature range of experimental interest, TT1

, lead-ing to V 1, + e/2 and V 1,−−e−/2.

Using the limiting values obtained earlier for the eigen-values, Eq. 49, we find the following limiting forms for thetwo chain persistence lengths:

1/1,+p 1/U

p , T Tm

1/I + 1/Up , Tm

T T1

and

1/1,−p 1/I + 1/B

p , T Tm

1/Bp , Tm

T T1. 74

The limiting forms for 1,+p in the range Tm

TT1 and

1,−p in the range TTm

have a simple physical explanation:the effective persistence lengths 1,

p of minority domains B,or , below Tm and U, or , above Tm tend to the typicalminority domain size I, when these domains behave as rigidrods U

pI for minority U domains and BpI for minority

B domains.It is interesting to note that the various correlation

lengths can be identified with differences between theeigenenergies appearing in the Landau-Zener diagram Fig.2: I=1 / 0,−−0,+, 1,+

p =1 / 1,+−0,+, and 1,−p =1 / 1,−

−0,+. Hence we already observe in this diagram that I

reaches its maximum at Tm, which is the point of closest

approach of the branches 0, . In the limit of large N theexpression for t1 · tN substantially simplifies and depends ononly one persistence length 1,+

p as follows:

t1 · tN →N→

V1, + 2

V0, + 2 exp− N − 1/1,+p . 75

The value of N for which the limiting form Eq. 75 startsto be a good approximation to Eq. 72 depends on the tem-perature via the weights V l , + 2 and the characteristiclengths I and 1,

p .


011913-11

V. FULL TRANSFER MATRIX APPROACH

To calculate the general chain tangent-tangent correlationfunction ti · ti+r, for the coupled model, we need to intro-duce the more powerful and more abstract transfer kernelmethod. This method will also shed additional light on theorigin of the effective Ising models obtained above by firstintegrating out the chain degrees of freedom. To calculate thepartition and correlation functions using this method, weneed to solve a spinor eigenvalue problem in order to find the

eigenfunctions and eigenvalues of the transfer kernel: P=, or more explicitly,

=1

d

4P,, = , 76

where

=+1U +−1B . 77

For the pure Ising model the eigenvalues and eigenvectorscan be labeled by the index =. For the pure chain modelwith rigidity and no applied stretching force like the 1Dclassical Heisenberg model in zero field the eigenfunctionslm=4Ylm are proportional to the spherical har-monics Ylm, which are indexed by the integer pair l ,m,with l=0,1 , . . . , + and m=−l , . . . , + l. Furthermore, the ei-genvalues for the pure chain model l are indexed only by l,because they are degenerate in m as follows:

l = e−l 1

d

dl sinh

=

21/2

e−Il+1/2 ,

78

where Il+1/2 is the modified Bessel function of the firstkind. These eigenvalues take on the values 0=e− sinh /=exp−G0 and 1=0u=exp−G1for l=0,1 and are related to the l=0,1 two-link free energiesalready calculated, Eqs. 13 and 31 33.

The rotational symmetry of the coupled model impliesthat in this case the eigenspinors can still be labeled by theindices l ,m ; used for the pure Ising and pure chain mod-els, i.e.,

l,m; = lml, , 79

and the eigenvalues l, by l , degenerate in m. The ei-genvalues and kets l ,, which are independent of the solidangle , must be determined by solving the eigenvalue equa-tion 76. In general, the eigenvalues and eigenvectors for thecoupled system are not, however, simply direct products ofthe corresponding eigenvalues and eigenfunctions of the un-coupled Ising and chain systems. By solving the eigenvalueequation 76, we find that the kets l ,, appearing in the fulleigenspinor, and the eigenvalues l,, have already been in-troduced and obtained for l=0 and 1 Eqs. 38–40 in thecalculation of the effective Ising partition functions, Eq. 42.If we define Gl=−ln l, then the same formulas, Eqs.38–40, apply for the kets l , and the eigenvalues l, inthe general case l=0,1 , . . . , +, =. The eigenspinors areorthonormal as shown by

l,m;l,m; = l,l, d

4lm

* lm

= llmm. 80

Once we have the eigenvalues and orthonormal eigen-functions, we can express the transfer kernel in an abstractoperator notation as

P = l=0

+

m=−l

+l

=l,l,m;l,m; , 81

and then use the orthonormality of the eigenspinors, as wellas the decomposition of unity,

I = d

4 =

l=0

+

m=−l

+l

=

l,m;l,m;

82

to calculate the quantities of interest in a straightforwardway. As a check on the method, we can, for example, recal-culate the partition function using the following expression:

Z = 4N i=1

i=1

N di

4V11

11P22 ¯ N−1N−1PNNNNV ,

83

or in kernel product form

Z = 4NVPN−1V = 4N l,m;

Vl,m;2l,N−1. 84

Because the end vector V contains only the rotational

ground state 0,0; i.e., l=0, m=0, its matrix elementswith the eigenspinors of the transfer kernel simplify to

l,m;V = l0m00,V . 85

Inserting this expression for the matrix element into Eq. 84leads immediately to the result, Eq. 42, obtained previouslyfor l=0.

Z = ZI,eff0 = 4N

=V0,20,

N−1. 86

In order to calculate the correlation function ti · ti+r, wecould use

ti · ti+r =1

3 m=−1

+1

1m* i+r1mi , 87

which can be obtained from Eq. 2 and the definition of thespherical harmonics. Thanks, however, to rotational symme-try, the average value of ti · ti+r simplifies to

ti · ti+r = 3ti,zti+r,z = 10i10i+r , 88

where 10=3 cos. The tangent-tangent correlationfunction can be written in a kernel product form similar tothe expression for the partition function, Eq. 83, with thedifference being that we must now insert the projection or


011913-12

dipole operator along the z axis, Z=cos, related to10 in the j= i and j= i+r positions. This operator, whichis diagonal in the canonical basis , has the followingmatrix elements:

iiZi+1i+1 =1310ii+1 −ii+1i

, 89

from which the following equality can be established:

10iiiPi+1i+1

= 3 d

4iiZPi+1i+1

= 3iiZPi+1i+1 . 90

In operator product form, using Eqs. 83 and 87 the cor-relation function then becomes

ti · ti+r = 34NZ−1VPi−1ZPrZPN−r−iV . 91

In the basis that diagonalizes the transfer kernel P, the op-

erator Z, which is not diagonal, has the following matrixelements:

l,m;Zl,m;

= l,l,mm l,l−1 l2 − m2

4l2 − 11/2

+ l,l+1 l + 12 − m2

4l + 12 − 11/2 , 92

which, aside from the factor l , l ,, is the well knownselection rule for quantum dipole transitions, i.e., l=1and m=0 in, for example, the Stark effect 36. Althoughl , l ,= the matrix element l , l , is not neces-sarily zero for l l and , because in this case the ma-trix element is between states of different rotational symme-try. This result indicates that the measurement of the z-axisdipole moment can also induce a change in the internal state of the system.

Equation 85 shows that in the expression for ti · ti+r,the Z projection operator can only connect a l=0, m=0rotational state with an l=1, m=0 one, or vice versa as inthe Stark effect for the 1s state of the hydrogen atom. Toevaluate ti · ti+r using Eq. 91 we therefore only need onematrix element,

1,0;Z0,0; =13

1,0, . 93

By inserting the decomposition of unity between each matrixfactor in Eq. 91 and using the orthonormality of the eigen-spinors, we obtain

ti · ti+r =4N

Z 1,2,3

V0,30,3i−1 0,31,21,2

r

1,20,10,1N−r−i0,1V . 94

When i=1 and r=N−1, using again the decomposition ofunity in the l , space, we recover our previous result fort1 · tN, Eq. 72.

The above expressions for ti · ti+r, Eqs. 91–94 canalso be interpreted, using the “path integral” representationof a quantum statistical partition function, as a quantum me-chanical measurement process over an imaginary time periodof N steps. This interpretation is based on the 1D classicalIsing representation 1D classical Heisenberg represen-tation of the partition function of a 0D quantum diatomicmolecule, modeled as a two-state rigid rotator. The system isprepared in an initial state V that is in a mixture of theinternal states =, but in the spherically symmetric rota-tional ground state. This initial state evolves N−r− i time

steps under the dynamics determined by the propagator P,until a measurement of the dipole moment along the z axis,

determined by the action of Z, is performed. The projectionsof V onto the rotational ground states l=0, m=0,0, V, evolve in a simple way because these states areassociated with eigenspinors of the transfer kernel each timestep gives rise to an eigenvalue factor 0,. This dipole mea-surement excites the system from the rotational ground stateto the l=1, m=0, nonspherically symmetric, first rotationalexcited state, along with a possible transition in the internalstate of the diatomic molecule. The state that emerges fromthe measurement then evolves r time steps, until a secondmeasurement of the dipole moment is performed. This mea-surement deexcites the system from the l=1, m=0 excitedstate back down to the l=0, m=0 ground state or up to thel=2, m=0 excited state see Eq. 92 again with a possiblechange in internal state. The state that comes out of thissecond measurement then evolves i−1 further time steps.From the representation in Eq. 94 we see that the correla-tion function ti · ti+r is then the normalized amplitude thatthe system returns to the initial state V at the end of thisdouble dipole measurement process because the final stateand the l=2, m=0 excited state are orthogonal, no l=2matrix elements appear in Eq. 94. By resolving the initialand final states, both equal to V, into 0, components, thesum in Eq. 94 is over all possible “time” sequences involv-ing 0,.

In order to study the limiting forms of ti · ti+r, we write

ti · ti+r =

1,2,3

C3,2,10,3

0,1

i−11,2

0,1

r0,1

0,+N−1

0,

0,+N−1

V0,2

,

95

where we have introduced the joint amplitude


011913-13

C3,2,1 = V0,30,31,21,20,10,1V .

96

When all the bending rigidities are equal to , we recover, asexpected, the pure discrete wormlike chain result, ti · ti+r=exp−r /p with p given in Eq. 32.

In the limit N→, we keep only the leading order term0,+

N−1 in Z and the surviving terms in the numerator of thetwo-point correlation function 1= + to find

ti · ti+r →N→

2,3

C3,20,3

0,+i−11,2

0,+r

, 97

where

C3,2 =V0,30,31,21,20, +

0, + V. 98

Using the effective chain persistence lengths introducedpreviously in Eq. 73, we can express Eq. 97 in a physi-cally more transparent form as follows:

ti · ti+r →N→

2=

exp− r/1,2p C+ ,2

+ C− ,2exp− i − 1/I , 99

with 1,−p 1,+

p and I the Ising correlation length alreadyintroduced in Eq. 66.

In the double limit N , i→, the dependence on the chainends disappears again and the expression 99 simplifies to

ti · ti+r →N,i→

2=

1,20, + 2 exp− r/1,2p , 100

which reveals the importance of the two persistence lengths1,

p , and the two “transition probabilities” 1, 0, + 2, forgoing from the ground state 0, to the first rotational ex-cited state 1,, with or without a change in internal state .In the temperature range of experimental interest, TT1

,1, + 0, + 2 and 1,−0, + 2 can, to an excellent approxima-tion, be set equal to U, and B,, respectively. When thislast result is used in conjunction with the limiting forms for1,

p , Eq. 74, the following useful approximation is ob-tained for Eq. 100, valid for TT1

ti · ti+r N,i→

U, exp− r/1,+p + B, exp− r/1,−

p .

101

For N , i→ and short distances r1,−p , we find the limiting

linear behavior in r.

ti · ti+r N,i→

1 − r/eff,CFp , 102

where

1/eff,CFp 1, + 0, + 2/1,+

p + 1,− 0, + 2/1,−p

U,/1,+p + B,/1,−

p 103

is an effective persistence length for the correlation functionCF at short distances that clearly reveals the importance ofthe shortest persistence length 1,−

p under these conditions. In

the triple limit N , i ,r→, only one term survives.

ti · ti+r →N,i,r→

1, + 0, + 2 exp− r/1,+p . 104

Exactly at Tm, the above expression, Eq. 100, simplifies to

ti · ti+rTm

N,i→

1

2exp− r/U

p + exp− r/Bp , 105

which for rBp reduces to

ti · ti+rTm 1 −

r

21/U

p + 1/Bp . 106

Because the inverse persistence lengths enter into Eq. 102,the short distance limiting behavior will tend to be domi-nated by the shortest one, 1,−

p above Tm, where 1,−0, + 2

1, + 0, + 2. Below Tm, however, there will be a competi-

tion between the weights 1,−0, + 2 1, + 0, + 2 and thepersistence lengths 1 /1,−

p 1 /1,+p .

The conditions under which these limiting expressions arevalid approximations depends critically on the weightsappearing in the above expressions. If, for example,1, + 0, + 2U, is sufficiently small compared with1,−0, + 2B, at a certain temperature, then the “sub-dominant” term in Eq. 100, 1,−0, + 2 possessing thesmaller persistence length may actually be dominant over awide range of r values, as shown in Fig. 3.

Indeed, due to the coupling between bending and internalstates of DNA, for realistic parameter values cf. Sec. VII,the respective weights 0, + 1, 2 associated with each cor-relation length change abruptly at Tm

: below Tm, we have

0, + 1, + U U=1 and 0, + 1,−U B=0, thus 1,+p

Up . For Tm

TT1, we find 0, + 1, + B U=0 and

0, + 1,−B B=1, which implies 1,−p B

p . For highertemperatures TT1

, the respective weights get swappedagain, but now 1,+

p 1,−p . These considerations lead us to

introduce a critical distance r*, at which the two terms in Eq.100 are equal.

r* 1

1,−p −

1

1,+p −1

ln 1, + 0, + 2

1,− 0, + 2 Bp lnB,

U, ,

107

where in arriving at the last approximation we have usedlimiting forms that are valid when TT1

and assumed thatB

pUp ,I cf. Fig. 4. When rr*, then the correlation

function Eq. 100 is dominated by the shortest persistencelength 1,−

p , and when rr* the correlation function is domi-nated by the longest one, 1,+

p . For sufficiently long chainsand temperatures close enough to Tm, the inequality Nr*

holds and this crossover should be clearly visible see Fig.3.

VI. MEAN-SQUARE END-TO-END DISTANCE

To calculate the mean-square end-to-end distance of thechain, we use the two-point correlation function obtainedabove to find:


011913-14

R

a2

= i,j=1

N

ti · t j = N + 2i=1

N−1

r=1

N−i

ti · ti+r . 108

When all the bending rigidities are equal to , we recover, asexpected, the following pure discrete wormlike chain result

R2 = a2NWNu ,

where

WNz =1 + z

1 − z−

2z

N

1 − zN

1 − z2 , 109

with u defined in Eq. 31. In the limit N→,

R2 →N→

a2N1 + e−1/p

1 − e−1/p . 110

More specifically, three distinct regimes can be identified asfollows:

R2 →a2N , p 1 N

freely jointed2a2Np , 1 p N

spin wavea2N2, 1 N p

rigid .

111

Because p=p is a decreasing function of tempera-ture, the pure DWLC will go from the rigid to the spin waveto the freely jointed Gaussian regime as the temperature israised.

For the coupled model the double summation in Eq. 108can also be carried out and we find

R

a2

= N

+

1,2,3

C3,2,1SN0,3,1,2

,0,10,1

0,+N−1

1

2 0,

0,+N−1

V0,2

,

112

where

SNx,y,z = Ny

x − y−

y

x

1 − y/xN

1 − y/x2 for x = z

z1−NTNy,z − TNy,xz − x

for x z ,113

with TNx ,y=yNx /y− x /yN / 1−x /y. In the limit N→the above complicated expression for R2 simplifies to an ef-fective Gaussian form

320 340 360 380300

0.2

0.4

0.6

0.8

1

0

T (K)Tm

∞ T1∞

1,±|0,+2

(a)

0.01

0.1

1

0.00120 40 60 80 1000

r

ti · ti+r

(b)

FIG. 3. Color online a Variation of the transition probabilities1, 0, + 2 with temperature red solid line for and dashed blueline for . For TT1

, 1, + 0, + 2U, and 1,−0, + 2B,.One observes that for Tm

TT1, 1,−0, + 2=1, which under-

lines the relevance of 1,−p in this temperature range with parameter

values =4.46 kJ /mol, J=9.13 kJ /mol, and K=0; see Sec. VII.b Tangent-tangent correlation function given by Eq. 100 N , i→ for three different temperatures: just before the transitiondotted blue line, controlled by 1,+

p 1 /Up +1 /I−1U

p ; slightlyabove Tm

solid red line, where the correlation length 1,−p B

p isdominant for rr*20; and after the transition Tm

TT1

where the correlation length 1,+p I disappears in favor of 1,−

p

Bp dashed green line.

510

50100

5001000

320 325 330 335 340315

ξI

ξUp

ξeffp

ξ1,+p

ξ1,-p

ξBp

T (K)Tm

∞

FIG. 4. Color online Variation with temperature of the variouscorrelation lengths appearing in the model results: the Ising corre-lation length I solid red line; persistence lengths of the coupledsystem, eff

p appearing for long chains and 1,p dashed-dotted

blue and green lines, respectively; and of the pure chains, U,Bp

dashed purple lines in units of a. At Tm, the Ising correlation

length is peaked but finite, which is related to the point of closestapproach of the two 0, branches zoom of Fig. 2, and the ef-fective persistence length eff

p rapidly crosses over from Up to B

p

parameter values =4.46 kJ /mol, J=9.13 kJ /mol, and K=0.


011913-15

R2 →N→

2a2Neffp ,

where

effp

1

2

1,0, + 21 + e−1/1,p

1 − e−1/1,p 114

is an effective “long chain” persistence length. This expres-sion can be also obtained by using the simplified N , i→limiting form for ti · ti+r, Eq. 100, in the general formulafor R /a2, Eq. 108. It tends to a further limiting form when1,

p 1:

effp →

1,0, + 21,p

U,Up + B,B

p , T Tm

U,1/Up + 1/I−1 + B,B

p , Tm T T1

.115

At Tm this expression simplifies to eff

p Up +B

p /2 whenIU

p , which is actually the case Fig. 4.For TTm

, the longest persistence length dominates:eff

p 1,+p U

p . Above Tm, however, we see once again that

there may be a competition between the persistence lengths1,

p , and the “transition probabilities” 1, 0, + 2, appear-ing in Eq. 115. This competition occurs now for TTm

,contrary to what was found for the short distance behavior ofthe two-point correlation function, Eq. 102, because thepersistence lengths themselves appear in Eq. 115, and nottheir inverses. For Tm

TT1, depending on the weights,

the smaller persistence length 1,−p , may actually be dominant

over the larger one, 1,+p ; if so, eff

p 1,−p B

p , which is actu-ally the case when we consider standard parameter values fordsDNA and ssDNA Sec. VII, as shown in Fig. 4.

VII. APPLICATION TO SYNTHETIC DNA THERMALDENATURATION

Melting or thermal denaturation profiles are experimen-tally obtained by following the UV absorbance of a DNAsolution while slowly increasing the sample temperature.This method allows one to follow the temperature evolutionof the fraction of base pairs that have been disrupted, BT.A typical profile has a sigmoid shape possibly with bumpsthat could appear depending on the DNA sequence. DifferentIsing-type models have been proposed 4,5,13,14 for mod-eling denaturation curves by focusing on the influence of thebase-pair sequence, but they do not attempt to take into ac-count properly the fluctuations of the DNA chains them-selves. Yet, chain fluctuations increase with T and play acrucial role in determining melting profiles. Moreover, thesefluctuations concern both stiff helical segments and flexiblecoils with different bending rigidities.

In this section, we compare the model developed above,whose key element is to account for internal state fluctua-tions on an equal footing with those of the chain, with a setof experimental data. We focus on the evolution of BT fora synthetic homopolynucleotide polydA-polydT. Six in-

dependent parameters appear in the theory: the polymeriza-tion index N, the three Ising parameters K, J, and definedin Eq. 1 and Fig. 1, and bending moduli U for dsDNA andB for ssDNA. Note that we have also introduced a bendingrigidity UB for domain walls. However, UB appears in thetheory only in the effective cooperativity parameter J0. Thuschanging UB is equivalent to varying the bare J, i.e., theenergetic penalty to create a wall. Without any lost of gen-erality, we choose to fix UB=U. Moreover, we choose freeboundary conditions for the end monomers, which is validunless their state is fixed by the experimental conditions 37although any type of boundary conditions can be treatedusing our model. Of the six parameters, three are deter-mined experimentally: N, U, and B. Moreover, there is evi-dence that stacking interactions in dsDNA and ssDNA are of

the same magnitude, which justifies the choice of K=0adopted below 38.

Figure 5 shows BT for a polydA-polydT DNA ofmolecular weight Mw=1180 kDa in a solution of 0.1 SSC0.015 M NaCl+0.0015 M sodium citrate, pH 7.0 takenfrom 4.

In order to compare the data with our model predictions,we choose the experimental values for the persistencelengths, ds

p 50 nm and ssp 1 nm at 300 K, which lead to

U=dsp /a=147 and B=2ss

p /a=5.54 at T=Tm taking a=0.34 nm for one base-pair size and a factor of 2 for twoflexible segments in parallel per coil segment. The two re-

maining parameters and J are determined by fitting theexperimental data. The solid line in Fig. 5 corresponds to

=1.64kBTm4.46 kJ /mol and J=3.35kBTm9.13 kJ /molleading to Tm=326.4 K. We can then deduce several thermo-dynamical features. Noting that the bare enthalpy for creat-ing one A-T link is, in our model, 2, we find a value veryclose to the experimental value of 10.5 kJ /mol 2. Although

the value of J is more difficult to interpret, our result J2 is consistent with the idea that destacking interactionsmake the dominant contribution to DNA stability 5. Chainfluctuations do not only renormalize the effective free energy

2L0 required to break an interior base pair, but also the co-

operativity parameter J0: the latter varies almost linearly with

0.2

0.4

0.6

0.8

1

0325.5 326 326.5 327 327.5

T (K)

ϕB

Tm∞

FIG. 5. Fraction of broken base pairs for a polydA-polydT vstemperature solution of 0.1 SSC, N=1815. The solid line repre-

sents the theoretical law for =1.64kBTm and J=3.35kBTm, whereTm=326.4 K. The case N→ for the same parameter values corre-sponds to the broken line.


011913-16

T following Eq. 15 contrary to previous theories where Jwas taken as constant and supposed to be purely enthalpic incharacter 4. We have for the total cooperativity parameter

J0=4.17kBTm at T=Tm, which shows that the bending contri-bution is roughly 25% remembering that we have chosenUB=U. The model fit thus leads to parameter values inaccord with experiment. Our model predictions for experi-mentally accessible A-T pair quantities are also in agreementwith accepted values 3,39: i the loop initiation LI factorLIe−4J0 10−7 at Tm; and at physiological temperature,Tph, ii an interior single base pair opening probabilityBTph10−6 with a bubble initiation barrier of 17kBT, andiii a free energy of 0.18kBT for breaking an additional basepair in an already existing bubble. In reality, the fitted values

of and J implicitly compensate for effects like loop en-tropy explicitly left out of the model 4. As shown in Fig. 9of 4, effective Ising models without loop entropy, like ours,can be considered to account implicitly and approximatelyfor loop entropy, provided that one allows for loop entropycontributions to both J and K. This loop entropy renormal-ization of the Ising model parameters will depend on thevalue of the loop entropy exponent k and the chain length Nand could allow for a simple approximate way of accountingfor the influence of loop entropy within the framework of aneffective Ising model cf. 40. This implicit renormalizationprobably explains why our model value for LI is at the lowend of the accepted spectrum.

In Fig. 5, the curve corresponding to the thermodynamiclimit N→ is shown for the same parameter values. In thiscase, BT is given by Eq. 52 and the value of the meltingtemperature is obtained analytically as a function of Tm

usingL0Tm

=0, which is given in the limit of low temperatureBkBT by

kBTm 2

+ K

lnU/B. 116

Hence, the melting temperature is reached when the enthalpyrequired to create a link is perfectly balanced by the differ-ence in entropy dominated free energy between the twotypes of semiflexible chains U or B. Another quantitywhich has an experimental relevance is the width of the tran-sition. In the thermodynamic limit this width is narrow, butnonzero, due to the large but finite cooperativity parameter:Tm

1 /ITm2 exp−2J0Tm

see Eq. 53. Hence, thethermodynamic limit clarifies the role of the two free modelparameters: in conjunction with the experimentally knownbending rigidities, sets the melting temperature and J fixesthe transition width. This is in contrast to previous Ising-typemodels, where three fitting parameters were used J, L0 /T,and Tm

with L0 assumed to by a linear function of T 4.Within the scope of our model the measured transition

width is indicative of a very long Ising correlation length Inear the transition temperature, much larger than the pure Uand B persistence lengths; therefore typical helix U andbubble B domains of size I are flexible within a smalltemperature window near the transition.

Included in the predictions of our theory are mechanicaland structural features of the composed chain, such as per-sistence length or mean-square end-to-end radius R. This dif-fers from purely Ising-type models 4,5 and nonlinear mi-croscopic models 20,21 where only thermodynamicalquantities related to base pairing are available. The variationof the effective persistence length eff

p and thus the radius ofgyration for long chains vs T is shown in Fig. 4. It variesfrom U

p for TTm to B

p for TTm. Since the transition is

very abrupt, we suggest that the denaturation transition canalso be followed experimentally by measuring directly theradius of gyration, for instance, by tethered particle motion41, light scattering, or viscosity experiments. For instance,since the relative viscosity is proportional to cDNAR3 wherecDNA is the DNA concentration, it should clearly exhibit anabrupt thermal transition at a given cDNA and N. Such a tran-sition has indeed been observed for the viscosity of synthetichomopolynucleotide solutions 42, in qualitative agreementwith Fig. 4.

In fitting our model to experiment for chains of lengthN=1815, we have found that finite-size effects play an im-portant role Fig. 5. In the following section, we investigatesuch effects in detail.

VIII. FINITE-SIZE EFFECTS

It has been shown experimentally that DNA thermal de-naturation varies with chain length N 10. In this section, wecarefully study the effect of chain ends on the denaturationtransition. In Fig. 6 are shown denaturation profiles for vari-ous chain lengths from N=100 to N→ and the fixed pa-

rameter values =4.46 kJ /mol, J=9.13 kJ /mol, and K=0used in the previous section to fit the melting data. Withinthe scope of our model one observes that i the melting

10-1

1

10-2

10-3

325 326 327 328

0.5

T (K)

ϕB

T* ∆Tm∞∆Tm −

N10 102 103 104 105

10

1

10-2

10-1

FIG. 6. Color online Linear-log plot of melting curves for N=100 dashed-dotted green line, 500 dotted purple line, 1815solid blue line, and dashed red line in decreasing order atlow temperature, T326 K parameter values =4.46 kJ /mol,

J=9.13 kJ /mol, and K=0. We note that all the curves intersect atT*, as discussed in the text. Tm is defined by B=0.5. Inset: Log-logplot of model results for the shift in transition width Tm−Tm

vspolymer length. Dots correspond to the model results and the solidline is a law in 1 /N.


011913-17

temperature TmN is a decreasing function of N, varying asTmN−Tm

/Tm1 / N−1; ii all the denaturation curves

intersect at a temperature T* at which B0.03; iii thetransition width TmN is a decreasing function of N.

Concerning points i and ii, the observed behavior forthe coupled system with the present parameter values is di-rectly related to the model result that T*Tm

, which is radi-cally different from the behavior found for the simple Isingmodel for which melting curves B are strictly decreasingfunctions of N, because formally, T*= when U=B=UB.The present behavior for melting maps is also very differentfrom the predictions of older empirical Ising-type models ofdenaturation and helix-coil-like transitions 9, for which thechemical potential appearing in the end vector V is in-correctly identified with L0. This identification results in amelting temperature independent of N, i.e., Tm=T*.

Concerning point iii, the transition width roughly fol-lows the law Tm−Tm

1 / N−1, which is a classicalresult for finite-size systems where fluctuations decrease inthe thermodynamic limit. One observes that even for a longpolymer N103, finite-size effects are important. For veryshort chains, e.g., N100, such effects get amplified and wepredict a transition width as large as 50 K for N=10. Thispoint is crucial, since it has been observed experimentallythat for polydA-polydT inserts between more stable G-Crich domains, melting curves are much wider for very shortDNA chains N10 bp 37 with a width that decreaseswith increasing N observed for 60N140 in 10. Insuch experiments, the nature of end monomers clearly be-comes extremely important.

For a given N and T the local site-dependent bubble open-ing probability, or melting map,

B,i =1 − i

2, 117

can be obtained from i given in Eq. 61. In Fig. 7 B,i isplotted for six different temperatures using the same modelparameter values employed in Fig. 5; we observe that belowT* the chain unwinds from the ends, whereas above thistemperature an interior bond has a higher probability of be-

ing open than an end one. Far enough below T* the meltingcurve heals rapidly to a plateau value close to B, on alength scale on the order of IN. At physiological tempera-ture 310 K, the interior bond opening probability is 10−6 inagreement with the experimental value for long runs of A-Tpairs which is an order of magnitude lower than randomlyplaced A-T pairs 3. At this temperature the end bonds haveopening probabilities two orders of magnitude greater thanthe interior ones.

At T* the melting curve is perfectly flat. This result,which is independent of chain length N, indicates that eachIsing variable can be considered to fluctuate independently:i is constant independent of i, despite a two-point corre-lation function that does not factorize ii+r ii+r,cf. Eq. 71 and a large Ising correlation length I1, seeFig. 4. Indeed, the influence of the renormalized stacking

energy K00, which favors bubble formation, exactly

compensates that due to the renormalized destacking J0,which suppresses bubble formation, and therefore B,iT*= 1−tanh / kBT* /20.03, which results from takingN=1 or taking K0=J0=0 for arbitrary N. In some ways thiscompensation leads to an effective noninteracting Ising sys-tem with c /NT*= B /NT*=0, which explains whythe melting curves for different values of N cross at T* inFig. 6. Below T* the combined effects of the renormalizedstacking energy and entropy gain favoring interior bubblesare too small to overcome the destacking energy cost asso-ciated with an extra domain wall and the chain ends unwindfirst. Since K0 becomes more negative with increasing Tfaster than J0 increases, a temperature T* is reached wherethe renormalized stacking and destacking effects just com-pensate. Above T* the situation is reversed and the openingprobability is higher in the chain interior. For arbitrary N andT, the thermodynamic chemical potential, defined by = F /NT can be related to c via Maxwell-type relations,leading to /T,N=Nc /NT− c. At T* this generalrelation simplifies to /T*,N= − cT*= f /T*,N,characteristic of a noninteracting system.

Upon examination of Eqs. 61 and 86, we see that T* isdetermined by the condition that i= c, which is ob-tained when the end vector V is identical to the eigenket0, + and orthogonal to 0,−: V 0, + =1 and V 0,−=0.Physically, this means that the coupled Ising-chain system isin a pure state 0, + , that mixes the canonical states in aspecial way. The temperature T* can be obtained by solvingV 0,−=0. Using Eqs. 11, 39, and 40 this translatesinto e=e−2J0sinhL0+ sinh2L0+e−4J01/2. After somemanipulation using Eq. 52, this can be shown to be identi-cal to cT*= cN=1,T*=tanh / kBT*. Further-more, Eqs. 62 and 94 show that the Ising and chain two-point correlation functions, ii+r and ti · ti+r, also getsimplified at T*: the approximate forms, Eqs. 71 and 100valid, in general, only for N , i→, become exact for arbi-trary N and i at this special temperature. It is clear that at T*

the coupled system behaves as if there are no end effects andfinite chains have the same behavior as an infinite one.

To shed additional light on this mechanism and illustratethe important role of internal bubble entropy for long chains,

0 250 500 750 1000 1250 1500 1750i

B,i

106

1

105

104

103102101

FIG. 7. Average melting maps for different temperatures for N=1815 and the same parameter values used in Fig. 5: plot of thefraction of broken bases, B,i as a function of the base position i for,in increasing order, 310 K, 0.99T*=322.87 K, T*=326.13 K, Tm

=326.24 K, Tm=326.4 K, and 1.01Tm=329.66 K.


011913-18

we now study an infinite chain and compare B,int= limN,i→ B,i with B,end= limN→ B,1. Equation 67shows that I plays here the role of a healing length, overwhich end effects relax see Fig. 7. The ratio of matrixelements appearing in Eq. 67 gets simplified in the follow-ing way for special values of T:

RV =V0,− V0, +

= − e−, T T*

0, T = T*

tanh/2 , T = Tm

e, T Tm . 118

At Tm, Eq. 67 simplifies to

iTm = tanh

2kBTmexp− i − 1/ITm

, 119

which shows that for very long chains NI atTmT*B,end = 1−tanh / 2kBTm

/ 20.16B,Tm

= 1 /2, revealing an internal opening probability more thanthree times higher than an end one. In Figs. 7 and 8, how-ever, we observe that for T=Tm and Tm

, IN=1815 cf.Fig. 4, and therefore end effects do not get damped out nearthe center of this finite chain. Indeed, at Tm

the openingprobability B,i near the middle of a chain of length N=1815 is much less than the value of 1 /2 holding for aninfinite one. For N=1815 we still observe, however, notice-able differences 10–20 % between interior and endopening probabilities near Tm.

The one-sequence approximation has been defined by Po-land and Scheraga 14 and consists in neglecting the manysmall bubbles which eventually collapse and consideringonly one large thermally excited bubble. It is valid for tem-peratures sufficiently far below Tm

. For N→, B,int andB,end can be estimated in this approximation by summingover, respectively, all the interior and end bubbles containingthe fixed site in question. In the interior case we thus have

B,int n=1

n exp− Gintn = e−4J0

n=1

ne−2nL0 =e−4J0

4 sinh2L0.

120

The factor of n in the sum is entropic in nature and equal tothe number of ways of placing a fixed interior site within aninterior n bubble. In the end case, where there is no entropicfactor,

B,end n=1

exp− Gendn = eK0−2J0

n=1

e−2nL0 =e−2J0+

2 sinhL0.

121

It is important to note that for the model parameters em-ployed, the additional dimensionless free energy for breakingan additional base pair in an already existing bubble, 2L00.18 at physiological temperature, is less than one andmuch smaller than the bubble initiation energy cost for anend bubble 2J08 and a fortiori for an interior one,4J0. This implies that even at physiological temperature,where the probability of bond opening is very small, bubblescovering a wide range of sizes contribute to the sums in Eqs.120 and 121: both nint=cothL01 /L06 and nend

=e−L0 / 2 sinhL01 / 2L03 are of the same order ofmagnitude as I, which varies as 1 / 2L0 when L01.Because L0 decreases with T going to 0 at Tm

, both B,intand B,end are increasing functions of temperature. For TT* the bubble initiation energy cost dominates and B,intB,end. Thanks to the entropic factor and the increasinglynegative value of K0, however, B,int increases more rapidlythan B,end and the two curves cross over at T*, an estimateof which can be obtained by equating the above two one-sequence approximations for both these quantities. As acheck on the preceding discussion, the one-sequence ap-proximations for infinite chains obtained above for end andinterior opening probabilities can be shown to be in exactagreement with what one gets from the definition of B,i, Eq.117, and Eq. 67 by using the low temperature approxima-tions for V 0,− / V 0, + −e− Eq. 118 and c Eq.52, i.e., expanding to lowest order in e−4J0, assuming thate−4J0sinh2L0, which is the formal criterion for the valid-ity of the one-sequence approximation.

Using the one-sequence approximation, it is now easy tosee how the cost in loop entropy LE associated with inter-nal bubbles will modify the above results. In the so-calledloop entropy models 4,14,43 the internal bubbles formedby the single strands are visualized as one polymer loop,whose entropic cost has been estimated first by Zimm 15.Hence, although B,end does not change, the interior openingprobability does, becoming

B,intLE

n=1

n/n0 + 2 + 2nk exp− Gintn , 122

where we have adopted a common simplified form for theloop entropy factor associated with n broken base pairs,fLEn= n0+2+2n−k that depends on the loop length 2+2n and is parametrized by a constant n0, which may be as

0 250 500 750 1000 1250 1500 1750i

0.1

0.2

0.3

0.4

0.5

B,i

FIG. 8. Zoom of the melting map shown in Fig. 7 for Tm lower

curve and Tm upper curve.


011913-19

large as 100 10, and an exponent k, usually assumed to bein the range 3 /2 k 2.1, depending on the extent to whichchain self-avoidance is taken into account 18. A large valuefor n0 would severely reduce the importance of loop entropyfor short chains and probably reflects the presence of strongbending rigidity effects an important open question con-cerns how to incorporate bending rigidity into fLE in a physi-cally correct way. Loop entropy clearly lowers the probabil-ity of interior bubble opening and will lead to an increase inT*. The situation is further complicated if strand sliding,which may be important for periodic DNA, is taken intoaccount. For homopolymeric DNA like polydA-polydT,strand sliding leads to a modified loop entropy exponent k=k−1 4,14,43, resulting in a significant decrease in theimportance of loop entropy. By the way, it also minimizesthe importance for homopolymeric DNA of recent claimsthat a true first-order phase transition should occur for infi-nite chains because k appears to be greater than 2 whenself-avoidance is fully taken into account 18. The com-bined effects of loop entropy and strand sliding will lead toincreases in both T* and Tm

. Neither the theoretical nor theexperimental situation concerning TmN is entirely clear forhomopolymeric DNA with free ends and further careful ex-periments are clearly called for. If loop entropy and strandsliding are included in the coupled Ising-chain model, T*

might become higher than Tm, which would imply that TmN

would increase with N 14, unlike what we find to occurwhen these two effects are neglected.

In the future, we intend to examine these questions byincorporating loop entropy and strand sliding directly intoour DNA model. With all other factors being equal, addingloop entropy and strand sliding increases the stability of theclosed state, resulting in sharper melting curves and highervalues of TmN see 40 and Figs. 9 and 10 of 4. Theimportance of this loop entropy contribution would changewith chain length N and may lead to an increase in TmNwith increasing N 14 at least for a certain range of chainsizes.

IX. SUMMARY AND CONCLUSION

This paper presents a theoretical model of DNA denatur-ation, already introduced in 28, which focuses on the cou-pling between the base-pair link state unbroken or brokenand the rotational degrees of freedom of the semiflexiblechain. The Hamiltonian includes local chain bending rigidi-ties whose values depend on neighboring base-pair states:around 5kBT for bubbles and 150kBT for connected base-pairsegments. Because of the rotational symmetry, the model canbe rewritten in terms of an effective Ising Hamiltonian byintegrating out the rotational degrees of freedom of thechain. Hence, our model yields considerable insight into theempirical temperature-dependent parameters used in previ-ous Ising-type models 4. In particular, the melting tempera-ture Tm is no longer a fitting parameter, but emerges naturallyas a function of i experimentally known bending rigiditiesU and B; ii the bare energy required to open a base pair

2; iii the bare energy of a domain wall, or destacking, 2J;iv the difference in bare stacking energy between ss and

dsDNA, 2K; and v the polymerization index N. Moreover,our model allows structural features of the DNA chain, suchas the mean size R, to be calculated as a function of T. Anabrupt transition for R is found at Tm and explains, at leastqualitatively, the thermal transition observed in viscositymeasurements 42.

From an experimental perspective, our results obtainedfrom exactly solving the coupled model can be summarizedas follows. First of all, we propose formulas for chain free-boundary conditions, this information being encoded in theend vector V given in Eq. 11. However, any other bound-ary condition can be treated following the same route, eventhough we shall not detail the calculations here. For example,a polydA-polydT sequence of length N sandwiched be-tween more stable G-C sequences 37 can be seen near itsmelting transition as a DNA of length N, with fixed boundaryconditions, resulting in an end vector V= U.

Once boundary conditions are set, our model predictsmelting profiles, as measured, for example, from UV absor-bance experiments. Even if the model relies upon six micro-scopic parameters, as discussed in Sec. VII, most of them areknown experimentally, including the strand length N, andonly two of them must be extracted from melting profiles: ,the bare half-energy required to break a base pair which can

also be estimated experimentally 2, and J, the cooperativ-ity parameter that indicates the cost of creating a domainwall between unbroken and broken base pairs recent experi-ments on single DNA molecules aim at determining thisquantity; see 44. Melting profiles, giving the fraction ofbroken base pairs B, as a function of the temperature T, aredetermined through the average of the Ising state variablec, because BN ,T= 1− cN ,T /2.

For infinite chains N→, the expression for cT israther simple Eq. 52:

cT =sinhL0

sinh2L0 + e−4J01/2 , 123

where the renormalized parameters J0, K0, and L0=+K0 aregiven in Eqs. 15, 16, and 21. In these latter equations,the function G0 has a simple algebraic expression Eq.13 that reduces to a purely entropic contribution, G0 ln2, in the physically relevant low-T spin wave ap-proximation. From the melting profile B,T, the meltingtemperature Tm

is defined by B,=1 /2, in other words byL0=0. The finite transition width is estimated by Eq. 53:Tm

2kBTm2 exp−2J0Tm

/ + K.For finite length strands N finite, the expression for

cN ,T, Eq. 64, is more complex, because N has an in-fluence on BN ,T and thus Tm. In addition, the interplaywith mechanisms, such as loop entropy, not taken into ac-count in this study is nontrivial, as discussed in detail in Sec.VIII. At the level tackled in the present paper, we obtain asimplified expression for cN ,T when N is large, Eq. 69,which simplifies even further when NI1 Fig. 7.

cN,T c + 2RVI/N1 − c2 1/2, 124

where RV, Eq. 65, which is a ratio of matrix elements per-taining to end effects, simplifies at certain special tempera-


011913-20

tures see Eq. 118. Note that finite-size effects are stillimportant for sizes of several thousands of base pairs seeSecs. VII and VIII and Figs. 7 and 8 and are not a purelyacademic debate.

Three important correlation lengths can be calculated inthe framework of our model. On the one hand, the Isingcorrelation length I gives access to the typical size ofbubbles in the low temperature regime TTm, as well as tothe typical size of unbound regions for TTm. This quantityis calculated in Eq. 66 and assumes a simplified form at Tm

:ITm

exp2J0Tm /21, when J0Tm

1. On the otherhand, one effective chain persistence length eff,CF

p

U, /Up +B, /B

p−1, provides information on the shortdistance behavior of the chain tangent-tangent correlationfunction, Eq. 102, and another, eff

p , provides informationon the typical chain conformations, in particular, its mean-square radius, given by

R2 2a2Neffp 2a2NU,U

p + B,Bp , 125

where a is the monomer length 0.34 nm and the approxi-mation is valid for very long chains large N. Knowing thisquantity is of primary importance when interpreting datafrom tethered particle or tweezer experiments 41,45,46,atomic force microscopy 47, or viscosity measurements42. The results that we have obtained here for the chaintangent-tangent correlation function and mean-square radiusare very different from what one obtains by solving aquenched random rigidity model, where the local joint rigid-ity can take on one of two values, 1 and 2, with probability1 and 2=1−1. In this case there is only one effective

correlation length: ti · ti+r=e−r/ranp

and R22a2Nranp ,

where ranp −1 / ln1e−1/p1+2e−1/p2 and p is

given by Eq. 32.The foregoing analysis makes allowance for neither sol-

vent entropic hydrophobic, nor electrostatic effects, whichmight not only change the actual value of the bare Ising

parameters J, K, and , but also lead to additional entropiccontributions. To go further concerning solvent entropic ef-fects would require molecular dynamics simulations, whichis beyond the scope of the present approach. The electrostaticeffects in DNA melting are twofold: an entropic contributionarising from the difference in electrostatic energy betweenstates U and B and an enthalpic contribution arising fromcounterion release. At equilibrium the two effects partiallycompensate and the remaining contribution is, using a simplethermodynamic approach in the low salt limit 48,49, ap-proximately equal to Gel=kBTbB−UlnI / I0, where I isthe ionic strength I0=1 M, b=e2 / 4!kBT is the Bjerrumlength e is the elementary charge and ! the water dielectricpermittivity, and U and B are the linear charge densities ofthe pure U and B chains, respectively. Although this is anapproximate result and the determination of B−U remainscontroversial, these two contributions should be included in amore refined model. A natural extension emerging from this

study concerns the ionic strength dependence of DNA melt-ing profiles and effective persistence lengths.

In a first simplified approach one can of course use thehomogeneous model developed here for inhomogeneousDNA by using average parameter values, although the valid-ity of the average parameter approach depends on the strandlength 40. To go further the approach we present here caneasily be extended to include, explicitly, sequence heteroge-neities, by first integrating over the chain conformationswhich is always possible if the base-pair sequence isknown and then solving the effective inhomogeneous Isingmodel using known analytical or numerical methods4,13,14,16,53. In order to introduce base-pair heterogene-ities into the model, it is necessary then to define not onlyappropriate sequence dependent on-site and nearest-neighborIsing parameters, as in earlier work, but also generalizedbending rigidities.

It would also be extremely interesting to use our model tostudy the influence of exogenous damaging factors on theDNA polymer. Important such factors include UV light, en-vironmental mutagens, and endogenous metabolic by-products 54,55. On the one hand, if it is still possible toapproximate the fluctuations of base pairs under the influ-ence of these factors in terms of open and closed states, thenthe exogenous influence could be accounted for by locallymodifying the values of our model parameters. On the otherhand, if these factors modify the nature of the local base-pairstates, it would be necessary to replace our Ising model by aPotts one, accounting for more than two possible states. Themethods that we have developed here can also be generalizedto solve the ensuing coupled Potts-Heisenberg model neces-sary to describe such situations.

The current rapid development of force experiments inmagnetic or optical tweezer traps 45, “tethered particle mo-tion” experiments 41, or even more recently, atomic forcemicroscopy ones 47, presents a formidable opportunity toinvestigate directly the elastic properties of DNA strands as afunction of temperature, salt concentration, and length N.Indeed, despite the pioneering work by Blake and Delcourt10, a systematic experimental study of the effect of N whilecontrolling the nature of chain ends, is lacking, especially forhomopolymers. This would be a way to discriminate be-tween the different models and also shed light on the role ofloop entropy, which is neglected in the present model.

The approach developed here can be extended to bubbledynamics. Very recent work 39,50,51 studied the growth ofalready nucleated bubbles using the Fokker-Planck equationapplied to the Poland-Scheraga model i.e., an effective Isingmodel including loop entropy. The agreement with experi-mental results obtained by fluorescence correlation spectros-copy is remarkably good 39. The issue of bubble nucleationis, however, not solved and will be continued to be exploredin the near future. In addition, the mutual influence of ther-mally excited bubbles and chain flexibility should play animportant role in determining global DNA conformationsand strongly influence looping dynamics 46,52.


011913-21

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99, 088103 2007.29 J. Yan and J. F. Marko, Phys. Rev. Lett. 93, 108108 2004.30 J. Palmeri and S. Leibler, Dynamical Phenomena at Interfaces,

Surfaces and Membranes, edited by D. Beysens, N. Boccara,and G. Forgacs Nova Science Publishers, Inc., New York,1993, p. 323.

31 C. Storm and P. C. Nelson, Europhys. Lett. 62, 760 2003.32 M. E. Fisher, Am. J. Phys. 32, 343 1964.33 G. S. Joyce, Phys. Rev. 155, 478 1967.34 M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 1998.35 D. Chandler, Introduction to Modern Statistical Mechanics

Oxford University Press, New York, 1987, Sec. 5.8.36 C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quan-

tique Hermann, Paris, 1973, Vol. 2.37 G. Altan-Bonnet, A. Libchaber, and O. Krichevsky, Phys. Rev.

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84, 3212 2003.39 H. C. Fogedby and R. Metzler, Phys. Rev. Lett. 98, 070601

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011913-22

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 21 (2009) 034104 (18pp) doi:10.1088/0953-8984/21/3/034104

Coupling between denaturation and chainconformations in DNA: stretching,bending, torsion and finite size effectsManoel Manghi, John Palmeri and Nicolas Destainville

Laboratoire de Physique Theorique, Universite de Toulouse, CNRS, 31062 Toulouse, France

E-mail: [email protected], [email protected] [email protected]

Received 30 May 2008, in final form 7 August 2008Published 17 December 2008Online at stacks.iop.org/JPhysCM/21/034104

AbstractWe develop further a statistical model coupling denaturation and chain conformations in DNA(Palmeri et al 2007 Phys. Rev. Lett. 99 088103). Our discrete helical wormlike chain modeltakes explicitly into account the three elastic degrees of freedom, namely stretching, bendingand torsion of the polymer. By integrating out these external variables, the conformationalentropy contributes to bubble nucleation (opening of base-pairs), which sheds light on the DNAmelting mechanism. Because the values of monomer length, bending and torsional modulidiffer significantly in dsDNA and ssDNA, these effects are important. Moreover, we explore inthis context the role of an additional loop entropy and analyze finite size effects in anexperimental context, where polydA–polydT is clamped by two G–C strands, as well as for freepolymers.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The study of the physical properties of DNA is seeingintense activity from both a theoretical [1–18] and anexperimental perspective [19–27]. The first theoretical andexperimental studies were published several decades ago, butthe recent development of experimental techniques enablingone to address DNA properties at the single molecule levelhas brought a significant renewal of interest in the field.These techniques provide not only average properties liketheir former bulk counterparts, but also the statistics offluctuations around the average values. Single moleculesetups range from magnetic and optical tweezers [28, 29]or tethered particle motion apparatus [30–33], to atomicforce microscopy [24, 34]. They give access to hugeamounts of data concerning DNA physical properties such asbending, stretching and twisting elasticities or conformationaldynamics [30, 35–37, 31]. In parallel, the genomic revolutionleads to the elucidation of a large number of biologicalfunctions involving nucleic acids. A pressing demand hasfollowed for reliable and precise physical models, ableto validate the many hypotheses emerging from molecular

biology or microscopy experiments. This constitutes a doublemotivation for theoreticians to refine the existing microscopicDNA models: accounting for the new, accurate physicsexperiments, and validating (or invalidating) the physicalassumptions underlying the proposed biological mechanisms.

Denaturation is one of the intimate DNA physical featuresthat are supposed to be involved in many critical cellularfunctions, such as transcription, replication and proteinbinding, but are not fully understood. Even though DNAunwinding at the cellular level is generally an active processdue to enzymes consuming energy, such as helicases [38],understanding the subtle statistical mechanics of this bio-polymer is an essential first step towards the elucidationof more complex, active mechanisms. Furthermore, thespontaneous opening of base-pairs due to thermal activationis likely to play a direct role in several biological processes.Recently, Yan and Marko [12] have, for example, proposed thatcoupling the DNA elasticity to a minimal model of base-pairmelting can account for the increased cyclization probabilityobserved by Cloutier and Widom [39]: even if it is rare, localdenaturation increases short-range flexibility because singlestrand DNA (ssDNA) is nearly two orders of magnitude more

0953-8984/09/034104+18$30.00 © 2009 IOP Publishing Ltd Printed in the UK1

J. Phys.: Condens. Matter 21 (2009) 034104 M Manghi et al

flexible than double strand DNA (dsDNA). This increasedflexibility should play a role everywhere the polymer must bebent or looped on length scales shorter than its persistencelength (typically equal to 50 nm). In the nucleosome, it istwisted around histones, the diameters of which are about11 nm [40].

In order to get more insight into this coupling betweendenaturation and elasticity, we recently proposed a morerefined coupled, nonlinear model, where the internal states ofbase-pairs (open or closed) are described by a one-dimensionalIsing model, whereas the chain configurations are encodedby a one-dimensional Heisenberg one, taking into accountDNA bending [17, 18]. By solving exactly this model, wedemonstrated that taking into account this coupling betweeninternal and external degrees of freedom enables the predictionof the modifications of elastic properties when increasing thetemperature: Ising parameters are renormalized by temperaturein such a way that DNA denaturation is accompanied by acollapse of the chain persistence length. Following this route,we were able for the first time to write the melting temperatureTm as a function of microscopic parameters only—when itwas a fit parameter in previous models—and to give a newdescription of boundary and finite size effects.

However, our model was minimal in the sense that onlybending was taken into account. Torsion is also known to playa role on elasticity because a strong flexion of an elastic rod isin general accompanied by a torsion [41] which decreases theenergy cost of the deformation. Similarly, stretching of base-pairs ought to be included in a complete elastic model. In thepresent paper, we systematically explore these effects in detail,by proposing an exactly solvable discrete helical wormlikechain (DHWC) model, and predicting how Ising parametersare renormalized in this context (section 2).

In section 3, we investigate the influence of the chainlength (or finite size effects) on melting profiles. At theexperimental level, it has been shown in [1, 42] that they aremeasurable even for DNA made of several thousand base-pairs.These effects are usually measured for polydA–polydT flankedby more stable G–C rich strands. Hence we modify our modelto account for such clamped boundary conditions. In othermodels of denaturation [2, 42, 43], chain configurations arepartially incorporated via a so-called ‘loop entropy’ that takesinto account the entropic cost of closing a denaturation bubblewhen it is not located at a polymer end. We investigate the roleof loop entropy in finite clamped and free DNA chains.

2. Coupling between internal and external DNAdegrees of freedom

In [17, 18], we showed that the denaturation meltingtemperature emerges naturally by taking into account thedifference in bending rigidities of ssDNA sequences (bubbles)and dsDNA ones. Indeed, the ratio of both moduli, κds/κss

is of the order of 50. It is at the origin of an entropicbarrier which stems from the fact that in the ssDNA state, theallowed spatial configurations for unit tangent vectors ti , whichdescribe the chain conformations, are much more numerous,leading to a significant increase in entropy. More precisely,

it has been shown that the free energy (mostly of entropicnature) coming out by integrating the Hamiltonian part thatdepends on the external variables ti , renormalizes the bareIsing parameters, K and J , which are the energy costs ofcreating a domain wall and destacking two adjacent base-pairsrespectively. The third Ising parameter, μ, which correspondsto the energy required to break a base-pair (or ‘magnetic field’in a magnetism analogy), is not renormalized. In particular,the full penalty of breaking one base-pair located in DNA’sinterior, L = μ+ K , becomes

L0 = μ+ K − kBT

2

[G0

(κds

kBT

)− G0

(κss

kBT

)]

μ+ K − kBT

2ln

(κds

κss

)for κ kBT (1)

where kBT is the thermal energy and G0(x) = x − ln( sinh xx ).

The approximation is valid in the temperature range of interestsince κss ≈ 6 kBT .

In the infinitely long chain limit, the melting temperatureTm, defined as the temperature at which half of the base-pairs are broken, is simply given by L0(Tm) = 0. Themelting temperature thus naturally emerges in this modeland is determined by the competition between the enthalpiccost of breaking base-pairs (mostly hydrogen bonds and π -overlap of carbon ring wavefunctions of adjacent nucleotidesbut also charge, dipolar and van der Waals interactions) andthe entropic gain in nucleating bubbles made of very flexiblesingle-stranded DNA chains.

However, other external variables than ti , which alsocharacterize the chain elasticity, may lead to a renormalizationof the parameter L. Clearly, two other external degrees offreedom should also be taken into account:

• many force–extension experiments have shown that themonomer size a is not the same in dsDNA and ssDNA(see the review [44] and references therein). Indeed, themonomer size in the B-form of double-stranded DNA isgenerally defined as the rise along the central axis perbase-pair which is ads = 0.34 nm. The generally acceptedvalue [12, 45] of the monomer size in ssDNA is ass =0.71 nm and we choose in the following ass ≈ 2 ads.1

• the B-form of dsDNA is the famous double helix anda torsional energy has to be taken into account in amore refined model. Indeed, in the continuous helicalwormlike chain model for DNA [47], the elastic energyof the chain has two contributions: a bending term alreadytaken into account in [17, 18] and an energy of torsionaldeformations which in the continuum limit reads

Etwist = C

2

∫2

3(s) ds (2)

where 3 = Ω · e3. The Darboux vector Ω characterizesthe rotation of the material frame, e3 is along themolecular axis and s is the curvilinear index. The twist

1 The rotation per base-pair (or equilibrium twist) is 0 = 0.19π rad [46](the pitch is then p = 3.55 nm) and the radius r = 1.19 nm, so onecan estimate the monomer size in the fully extended ssDNA form as ass =0

√r2 + (p/2π)2 ≈ 0.79 nm which is larger than ads. It should be noticed,

however, that ssDNA is usually not fully extended and has a helical shape too.

2


(or torsional) rigidity modulus C has been measured intorsional experiments on dsDNA [48–50, 37], and is of theorder of Cds 2.4–4.5×10−19 J nm. The twist rigidity ofssDNA is lower because it loses its stiff helical structureand has been evaluated to be Css 9 × 10−20 J nm [51].The ratio Cds/Css is of the same order as κds/κss and willcertainly modify the Ising parameters in a similar way asfor the bending energy.

2.1. Discrete helical wormlike chain model

In the present work, the DNA is modeled as a fluctuatingpolymer chain in a space of three dimensions, characterizedby the external chain variables, the set of N bond vectors ti

and their orientation in space (it is thus implicitly assumedthat the monomer has a three-dimensional structure) and aninternal Ising variable σi = ±1 which models the internalstate of dsDNA, unbroken (U) or broken (B) respectively. Themodeling of the base-pair internal state by an Ising model wasdeveloped in the 1960s by Lehman, Montroll and Vedenov (seereview [52] and references therein).

We focus on the coupling of the internal variables with theexternal variables which is included in the Hamiltonian parttreating the fluctuating chain. A material coordinate frame isdefined for each monomer i , eμ,iμ=1,2,3 = ui , ni , ti , whereti is the unit bond vector ti = Ri+1 − Ri = ti ti and thetwo other unit vectors are in the directions of the principalaxes of inertia. This triad is defined with respect to a fixedreferential x, y, z through a rotation matrix Ai characterizedby Euler angles ωi = (αi , βi , γi). The evolution of thetriad along the molecular chain from monomer i to monomeri + 1 is obtained by a rotation, also defined by Euler angles(φi,i+1, θi,i+1, ψi,i+1)

eμ,i+1 = Λμν(φi,i+1, θi,i+1, ψi,i+1)eν,i (3)

where the rotation matrix Λ is the product of three rotationmatrices associated with each Euler angle, but can also beviewed as the product of two rotations of angles θi,i+1 andφi,i+1 + ψi,i+1 [50]

Λ(φi,i+1, θi,i+1, ψi,i+1)

= R(ti , ψi,i+1)R(ni,i+1, θi,i+1)R(ti , φi,i+1)

= R(ti , φi,i+1 + ψi,i+1)R(R(ti ,−φi,i+1)ni,i+1, θi,i+1). (4)

In the material coordinate frame eμ,i , the bond vector ti+1

is thus defined by its spherical coordinates (θi,i+1, φi,i+1).Moreover, the Euler angles (φi,i+1, θi,i+1, ψi,i+1) which willappear in the Hamiltonian are completely determined by thetwo sets of Euler angles ωi and ωi+1 through Λi,i+1 = Ai+1 ·A−1

i .The configurational part of the Hamiltonian is defined as

the sum of two terms

H[σ, t, ψ] = HIsing[σ ] +Hchain[σ, t, ψ] (5)

where HIsing[σ ] is the usual Ising Hamiltonian already definedin [17, 18] with three parameters (μ, J, K ), andHchain[σ, t, ψ]

is the DHWC Hamiltonian

HIsing[σ ] = −μN∑

i=1

σi −N−1∑i=1

[Jσi+1σi + K

2(σi+1 + σi )

](6)

Hchain[σ, t, ψ] = 1

2

N∑i=1

εi

2

(|ti |2 − a2i

)2

+ 12

N−1∑i=1

[κi,i+1(ti+1 − ti )

2 + 2Ci,i+1(cos θi,i+1

− cosλi,i+1)]. (7)

The first term of (7) is a nonlinear stretching term dictatedby rotational and translational invariances. The values ofthe Lame coefficient εi and the monomer length ai dependon the state of the base-pair ((εU, aU) for σi = +1 and(εB, aB) for σi = −1). The second term corresponds to thebending and torsional energies. The latter can be written asCi (tr Λ(0, θi,i+1, 0)− tr Λ(φi,i+1, θi,i+1, ψi,i+1)) and accountsfor the energy penalty associated with the twist defined by theangle φi,i+1+ψi,i+1. Indeed, the angle λ of the rotation definedin (3) is a function of φ+ψ and θ (indices i, i + 1 are omitted):

cosλ = 12 [cos(φ + ψ)(cos θ + 1)+ cos θ − 1]. (8)

The bending κi,i+1 and torsional Ci,i+1 moduli also vary locallywith the state of nearest-neighbor links ((κU,CU) for typeU–U , (κB,CB) for B–B and (κUB,CUB) for U–B). Weassume in this model that all the parameters appearing in (5)are independent of the nucleotide type. Hence we focuson homopolynucleotides. The case of sequence dependentparameters could be handled numerically.

Equation (7) defines our discrete version of the continuoushelical wormlike chain model first employed by Yamakawafor DNA [47] and extended in several articles in theliterature [53, 50, 54, 55]. First, one observes that if thereis no twist, i.e. no rotation around the tangent vector ti , itimposes φ + ψ = 0 and from (7) and (8), the torsional termvanishes. Hence if there is no twist along the chain (or if theDNA chain is modeled as linear), the DHWC becomes theclassical discrete wormlike chain already developed in [17, 18].Furthermore, the DHWC simplifies in the continuum limit,xi+1 − xi → ∂x

∂s s with s → 0 where s is the curvilinear

index. Indeed it is straightforward to see that∑N−1

i=1 κ(ti+1 −ti)

2 → ∫κ[2

1(s) + 22(s)] ds and with more algebra

that∑N−1

i=1 C[trΛ(0, θi,i+1, 0) − trΛ(φi,i+1, θi,i+1, ψi,i+1)]simplifies into (2) where the Darboux vector is defined byeμ,i+1 − eμ,i → Ω× eμ,i andμ(s) = Ω·eμ(s). Finally, in thelow temperature regime where the spin-wave approximationis valid (φ + ψ 1 and θ 1), bending and torsionalcontributions reduce to quadratic terms

12

N−1∑i=1

[κi,i+1θ

2i,i+1 + Ci,i+1(φi,i+1 + ψi,i+1)

2]+O(θ4, φ4, ψ4).

(9)The discrete model defined by (9) has already been used in thecontext of DNA supercoiling [56].

3


2.2. Stretching contribution to the entropy of bubblenucleation

The first stretching term in (7) is local without any couplingbetween the nearest neighbors. Therefore it can be integratedout easily. The Lame elastic constant ε is very large forDNA molecules: εa3 has been evaluated as 8.4 nN for ssDNAby fitting force–extension experimental curves using ab initiocalculations [45]2, and one can expect the same order ofmagnitude for dsDNA. Therefore, εa3 kBT/a 4 pN andthe saddle-point approximation applied below is valid.

By expanding the first term of (7) and writing |ti | = ai +δi

we have

(|ti |2−a2i )

2 = (|ti |+ai )2(|ti |−ai)

2 ≈ 2a2i (|ti |−ai )

2+O(δ3i ).

(10)The elastic term of the Hamiltonian (5) simplifies into

Hchain[σ, t, ψ] N−1∑i=1

εi a2i

2(|ti | − ai)

2 + κi,i+1(1 − cos θi,i+1)

+ Ci,i+1(cos θi,i+1 − cos λi,i+1). (11)

The configurational part of the partition function is

Z =∑σi

e−βHIsing[σ ]∫ (

N−1∏i=1

d3ti dγi

8π2a30

)e−βHchain[σ,t,ψ] (12)

where γi is the second twist Euler angle of ti with respect tothe reference frame and a0 is a normalization length. By usingthe decomposition of ti in spherical coordinates, (ti , αi , βi ),one has d3ti dγi = t2

i dti sinαi dαi dβi dγi ≡ dti d3ωi and thepartial partition function for the chain is

Zchain[σ ] =N∏

i=1

∫ ∞

0

t2i dtia3

0

e− βεi a2i

2 (ti −ai )2

×∫ N∏

i=1

(d3ωi

8π2

)e−βHangle [σ,ω] (13)

where Hangle[σ,ω] is the bending and torsional Hamiltonian.Using the saddle-point approximation for the stretchingintegral, we get in the large stretching constant limit

N∏i=1

∫ ∞

0

t2i dtia3

0

e− βεi a2i

2 (ti −ai )2 ≈

N∏i=1

√2π

βεi

ai

a30

≡ e−∑Ni lni .

(14)As explained above, we assume that the stretching energy hastwo competitive minima for dsDNA and ssDNA. In our modelit means that the elastic constant εi and the monomer sizeai have two different values whether the monomer is in theunbroken (σi = 1) or broken state (σi = −1). Hence, onceintegrated over the local ti variables, the stretching energy partcan be included in the Ising part of the Hamiltonian to get aneffective Ising Hamiltonian with a renormalized μ. Indeed, bydefining lni = δμ σi + where δμ = ln(U

B) and =

ln(UB), the renormalized temperature dependent chemicalpotential is

μ0 = μ− kBT ln

(aB

aU

√εU

εB

)(15)

2 Classical calculations lead to smaller values by one order of magnitude [37].

where the correction accounts for the entropic gain when themonomer state changes. It has two contributions:

(i) in the broken state, the monomer size is greater, aB ≈ 2aU,which implies a larger volume in the phase space and thusan increase in entropy;

(ii) in the case of different elastic constants, εU = εB, sincethe stretching energy 〈E〉 = 1

2 kBT is independent of theseconstants, the elastic free energy difference is purely ofentropic origin, similarly to the simple Einstein model forsolids.

In the present case, the elastic constants εU and εB areunknown. Although several experimental studies seem toshow that the stretching constant of dsDNA is larger than forssDNA [57, 58], we have not been able to find reliable values.If, for example, we assume them equal, then the chemicalpotential μ is lowered by 0.5–1 kBT , which is non-negligible.

2.3. Bending and torsional contributions

In this section, we focus on the partition function integratedover the angles (d3ωi = sinαi dαi dβi dγi ). The full partitionfunction (12) can be written as

Z =∑σi

e−βHIsing,0[σ ]∫ (

N∏i=1

d3ωi

8π2

)

× e−β∑N−1i=1 κi,i+1 (1−cos θi,i+1 )+Ci,i+1 (cos θi,i+1−cos λi,i+1 ) (16)

where HIsing,0 is the same as (6) with μ replaced by μ0

given in (15). Similarly to the discrete wormlike chainmodel [17, 18], the partition function for the coupled systemcan be calculated using transfer matrix techniques. Forexample, we have

Z =∑σi

N∏i=1

∫d3ωi

8π2〈V |σ1〉〈σ1|P(ω1,ω2)|σ2〉 · · · 〈σN−1|

×P(ωN−1,ωN )|σN 〉〈σN |V 〉, (17)

where the matrix elements of the transfer kernel that appearsN − 1 times in (17), are given by (the tilde means in units ofkBT )

〈+1|P(ωi ,ωi+1)|+1〉= eκU(cos θi,i+1−1)+CU(cos θi,i+1−cos λi,i+1 )+ J+K+μ0 (18a)

〈−1|P(ωi ,ωi+1)|−1〉= eκU(cos θi,i+1−1)+CU(cos θi,i+1−cos λi,i+1 )+ J−K−μ0 (18b)

〈+1|P(ωi ,ωi+1)|−1〉= eκUB(cos θi,i+1−1)+CUB(cos θi,i+1−cos λi,i+1 )− J (18c)

= 〈−1|P(ωi ,ωi+1)|+1〉. (18d)

It is written in the canonical base |U〉 = |+1〉 and |B〉 = |−1〉of the U and B states. The end vector

|V 〉 = eμ0/2|U〉 + e−μ0/2|B〉 (19)

enters in order to take care of the free chain boundaryconditions [18] (see also section 3).

The partition function can be rewritten by examiningthe effective Ising model obtained by integrating over the

4


chain conformational degrees of freedom ωi in (17). Theproblem reduces to that of an effective Ising model withan ‘effective free energy’ HIsing,eff containing renormalizedparameters. This method works because, for the coupled Ising-chain system, the rotational symmetry is not broken. Hence thematrix obtained by integrating the kernel P(ωi ,ωi+1) in (17)is the same for any site i .

We thus are able to carry out the angle integrations insequential fashion by using the triad eμ,i−1 as the referentialfor the i th Euler angle integration. Since this correspondsfor each integration to making a rotational transformation forthe variables with the Jacobian equal to 1, the Euler angleintegrated transfer matrix is

PI,eff =∫

d3ωi

8π2P(ωi ,ωi+1)

=(

e−G(κU,CU)+ J+K+μ0 e−G(κUB,CUB)− J

e−G(κUB,CUB)− J e−G(κB,CB)+ J−K−μ0

)(20)

where G(κ, C) is (in units of kBT ) the free energy of a singlejoint (two-link) subsystem with bending and torsional rigidities(κ,C) (either U–U , B–B , U–B):

G(κ, C)

= − ln

[∫sin θ dθ dφ dψ

8π2eκ (cos θ−1)+C(cos θ−cos λ)

](21a)

= 2κ − ln

[∫ 1

0dx I0(Cx) e(2κ−C)x

], (21b)

where I0 is the modified Bessel function of the first kind3. Twointeresting cases are:

• C = 0 leading to G(κ, 0) = G0(κ) alreadydefined in (1) which is an increasing function of κ(cf figure 1), and (21b) is a generalization of the previousresult (1) [17, 18];

• κ = 0, G(0, C) = C − ln[I0(C) + I1(C)] which is alsoan increasing function of C (cf figure 1).

The function G(κ, C) is plotted in figure 1 which showsthat it is a monotonically increasing function. In the spin-wave approximation, the integral (21a) is computed using thesaddle-point approximation, and the asymptotic behavior of Gis

G(κ, C) −→κ,C1

ln(2κ)+ 1

2ln

(2

√2

πC

). (22)

We observe in figure 1 that the asymptotic limit is a very goodapproximation for κ and C larger than 2, and thus for realDNA.

The Hamiltonian of the model (5) then reduces to aneffective Hamiltonian which is now of Ising-type

HIsing,eff[σ ] = −μ0

N∑i=1

σi−N−1∑i=1

[J0σi+1σi + K0

2(σi+1 + σi)

]

(23)

3 This result does not change if we include in the model the equilibrium twist0 = 0.19π rad for dsDNA in the Hamiltonian (7) by changing cos(φ + ψ)

in cos(φ + ψ −0) for unbroken nearest neighbors.

Figure 1. Plots of the function G(κ, 0) and G(0, C) and theasymptotic expressions (22) as broken lines.

where the bare Ising parameters K and J are renormalizedaccording to

K0 = K − kBT

2[G(κU, CU)− G(κB, CB)] (24a)

J0 = J − kBT

4[G(κU, CU)+ G(κB, CB)− 2G(κUB, CUB)]

(24b)

and μ0 is defined in (15).Usually, it is admitted that the torsional modulus is

proportional to the bending modulus C 1.6 κ [50].Taking the same values as in [17, 18] for a polydA–polydThomopolymer, κU = κUB = 147 and κB = 5.54 at T =Tm = 326 K, we get G(κU, CU) = 9.3 and G(κB, CB) = 4.3which leads to a decrease of K and J by about 2–3 kBT and1–2 kBT respectively in the temperature range of interest. Wehave found in [18] μ = 1.78 kBT , J = 3.64 kBT and K wasset to 0. Hence these entropic contributions are of the sameorder of magnitude as the bare values and must be taken intoaccount.

Moreover, with these values, the spin-wave approximationapplies and we can summarize (15) and (24a) as

L0 = μ0 + K0 ≈ μ+ K − kBT

2ln

(a2

BεUκU√

CU

a2UεBκB

√CB

)(25)

showing that the renormalization of the Ising parameters comesessentially from entropic effects, namely stretching, bendingand torsional entropies.

This twist-induced melting might be important in thecontext of single molecule torque experiments [28, 35–37, 59]and in the context of superhelical stressed circular dsDNA [60].For instance, within this model, applying a torque (or a twist)will locally modify the free energy cost L to nucleate a bubbleand will, in return, influence the mechanical response of thechain.

In the rest of the paper, we will be interested in expectationvalues depending only on the spin variables σi . Hence,everything can be computed using directly the effective IsingHamiltonian (23) with renormalized parameters, μ0, L0 and J0.In principle, the DHWC model could be completely solved bytransfer matrix techniques, thus requiring the diagonalizationof the transfer operator P(ωi ,ωi+1) defined in (17). This isout of the scope of the present work.

5


2.4. End-to-end distance

In this section, we compute the end-to-end distance of adsDNA using the model presented in [17] where we neglectthe torsional term. We show that the difference in monomersizes in the unbroken and broken states modifies the end-to-end distance and should be taken into account. Therefore, wecomplete the findings of [18] where the monomer sizes weresupposed to be equal.

The end-to-end distance of the chain is defined as R =√R2, where

R2 =N∑

i, j=1

〈(ai ti) · (a j t j )〉

=N∑

i, j=1

A2〈σi ti · t jσ j 〉 + AB(〈σi ti · t j 〉 + 〈ti · t jσ j 〉

)

+ B2〈ti · t j 〉. (26)

The monomer size, which depends on the internal variableσi , is written as ai = Aσi + B with A = (aU − aB)/2 andB = (aU + aB)/2. In the thermodynamic limit, N → ∞, thisexpression simplifies to

R2

N−→N→∞

(A2〈σ 2

i 〉 + 2AB〈σi〉 + B2)

+ 2∞∑

r=1

[A2〈σi ti · ti+rσi+r 〉 + AB(〈σi ti · ti+r 〉

+ 〈ti · ti+rσi+r 〉)+ B2〈ti · ti+r 〉]

(27)

which is independent of i . By using the transfer matrixapproach and the results already presented in [18], we find aftersome lengthy calculations (see [18] for detailed definitions)

R2

N−→N→∞

A2 + 2AB〈σz〉 + 2B2ξp

eff + 2∑τ

(A2〈1, τ |σz |0,+〉2

+ 2AB〈0,+|1, τ 〉〈1, τ |σz |0,+〉) e−1/ξτ

1 − e−1/ξτ(28)

where the effective persistence length is defined as

ξp

eff ≡ 1

2

∑τ

〈1, τ |0,+〉2 1 + e−1/ξτ

1 − e−1/ξτ. (29)

The Pauli matrix σz acts only on the second part of the basisthat diagonalizes the transfer matrix operator P : |l,m,τ 〉 =|l,m〉 ⊗ |l, τ 〉 (where (l,m) are the quantum numbersassociated to the spherical harmonics and τ = ± labels theeigenstates of the Ising model). In the basis |0,±〉 we have

σz =( 〈c〉∞ √

1 − 〈c〉∞√1 − 〈c〉∞ −〈c〉∞

)(30)

where 〈c〉∞ is the expectation value of the average spin variable(or ‘magnetization’) in the thermodynamic limit

1

N

N∑1

〈σi 〉 −→N→∞

〈c〉∞ = sinh(L0)

[sinh2(L0)+ e−4J0]1/2. (31)

The parameter L0 is defined in (1) and J0 in (24b), settingC = 0 for the three cases. The two orthonormal eigenvectorsfor a fixed l are defined in [17, 18].

Temperature (K)

325.5 326 326.5 327 327.5325

20

40

60

80

100

120

140

160

0

Figure 2. Normalized mean square end-to-end distance R2/(2a2U N)

(in units of base-pair size) as a function of the temperature T for theparameter values μ = 4.46 kJ mol−1, J = 9.13 kJ mol−1, K = 0,corresponding to Tm = 326.4 K and κU = κUB = 147, κB = 5.5,aB = 2aU. The full calculation (28) (in red) and the interpolationformula (32) (in black) coincide. The upper (blue) and lower (green)broken lines correspond to the bare dsDNA and ssDNA values,respectively.

The result (28) is shown in figure 2 for aB = 2aU. Anaccurate interpolating formula is given by

R2interpol = 2N(ϕUa2

Uξp

U + ϕBa2Bξ

pB ) = (1 − ϕB)R2

ds + ϕBR2ss

(32)thus generalizing a similar result given in [18] for the caseaU = aB (φB = 1 − φU defined in (35)).

3. Finite size effects within the DHWC model

In this section, we study the behavior of the fraction of openbase-pairs, ϕB(N, T ), as a function of both temperature andchain length for homogeneous DNA with free and modifiedboundary conditions (necessary for DNA inserts). Despiteearly recognition [61] that a careful experimental study ofsuch homogeneous DNA polymers of varying length wouldbe of great help in advancing our theoretical understanding ofDNA denaturation, unfortunately such a study has not yet beencarried out. As a consequence, important questions concerningthe competition between end unwinding and internal bubbleformation for finite chains, as well as the correct form of theloop entropy factor (including the effect of chain rigidity) andthe role of chain dissociation, remain open. Our goal here is toshed further light on the role of polymer length in the thermaldenaturation homogeneous DNA (see [62] for a recent studyof finite size effects within the framework of generalizations tothe Peyrard–Bishop model).

The model we use here is a generalization of the onepresented in [17, 18] and has been defined in section 2.The renormalized chemical potential is given by (15) andfor purposes of illustration we use the simpler, but accurate,spin-wave approximations for the two other renormalizedparameters, summarized here:

L0(T ) ≈ L − kBT

2ln

(a2

BεUκU√

CU

a2UεBκB

√CB

)(33)

6


J0(T ) ≈ J − kBT

4ln

(κUκB

κ2UB

√CUCB

CUB

), (34)

where L = μ + K . We also adopt the following physicallyreasonable set of model parameters: κU/κB = 147/5.5 =26.7, κUB = κU, aB/aU = εU/εB = 2, and CB/κB =CU/κU = CUB/κUB = 1.6. When loop entropy is not includedin the model, we use a value for J obtained previously byfitting experimental melting data for a homopolynucleotidepolydA–polydT: J = 9.13 kJ mol−1 [17, 18] (we recall thatthe renormalized value J0 is a key parameter in determiningthe transition width). When the effect of loop entropyon the thermal denaturation of free chains is studied, wewill use a smaller value of J , half of the larger one, asit is well known that loop entropy tends to sharpen thetransition [42, 52, 61]. If the model prediction (without loopentropy) for the melting temperature for polydA–polydT oflength N = 30 000 base-pairs is chosen to agree with theexperimental results in [1] (T expt

m = 338.70 K), then we obtainL = μ + K = 9.87 kJ mol−1, close to the value obtained bysetting L0(T

exptm ) = 0 (which gives the model result without

loop entropy for the infinite chain melting temperature, seeequation (33)). Using J = 9.13 kJ mol−1, we find thatJ0 12.3 kJ mol−1 at T = 339 K, which implies thatthe entropic contribution is greater than 25% near the meltingtemperature ( J0 = β J = 3.23 for β = 1/(kBT expt

m )). UsingJ = 4.57 kJ mol−1, we find that J0 7.70 kJ mol−1 atT = 339 K, which gives an entropic contribution of 41%.

In our previous work [17, 18] we assumed that thedifference in bare stacking energy, K , between the U andB states was zero. This choice was based on evidencethat near room temperature single-stranded polyrA remainsstacked [63]. It seems, however, that near the dsDNA meltingtemperature dT single strands are probably completely and dAones partially unstacked [1], with an unstacking fraction closeto 75% near Tm [1]. We can conclude that the single dT anddA strands in polydA–polydT bubbles may have much lessstacking energy than the helical segments and incorporate thiseffect into the model by introducing a weighting parameter, f ,that measures the contribution of K to L at fixed L: K = f Land μ = (1 − f )L. Although the two unbound single dT anddA strands in a polydA–polydT bubble may not behave exactlylike two free single dT and dA strands, the above discussiondoes suggest that f may be large near the melting temperature.Indeed, if we accept the putative experimental value for thebare enthalpy needed to open one A–T base-pair as a measureof μ, then we find μ 5.25 kJ mol−1 [17, 18, 64]. Using thisresult and the above value for L then yields f 0.5. When fis taken to be zero there is no loss in stacking energy whena bubble opens and we recover the case previously studiedin [17, 18].

An important question is how to incorporate bubble loopentropy into statistical models of fluctuating DNA. This loopcontribution arises from the extra cost in free energy (withrespect to two single unbound end chains) needed to form aclosed loop of bases making up a bubble [42, 61, 65, 66]. Whenloop entropy is neglected Poland–Scheraga (PS) type modelsreduce to effective Ising ones, albeit without the end-interiorasymmetry that naturally arises within our approach from the

difference between L0 and μ0 (see equation (20) of [18]).This can arise both from a dissimilarity between μ and Kand from the renormalizations coming from integrating out theconformational degrees of freedom. If, without justification,we formally set μ0 equal to L0, we recover previous Ising/PStype models without loop entropy.

For finite DNA polymers, end effects may have a stronginfluence on both the thermal denaturation transition and chainconformational properties. As already discussed in [18] thecoupled DNA model that we have developed is extremelyuseful for investigating the dependence of various systemproperties on chain length, N . For DNA homopolymers twotypes of situations can be envisaged: (i) finite homopolymerswith free end boundary conditions, and (ii) finite polydA–polydT inserts between more stable G–C rich domains withmuch higher melting temperatures.

Case (i) has already been extensively studied theoreticallyin [18] when the loop entropy associated with bubbles isneglected. Although for very long chains end effects areunimportant and f plays no role (only the value of L isimportant), for not too long finite chains f has a stronginfluence on the melting curves. Within the scope of ourmodel with f = 0 it was found previously that for finite DNAchains thermal denaturation takes place in an inhomogeneousfashion with the probability of base-pair opening being higherat chain ends for temperatures T < T ∗. At the temperatureT ∗ the fraction of broken base-pairs becomes independent ofchain length and the probability of base-pair opening becomesindependent of position on the chain (see figures 6 and 7of [18]). For f = 0 it was also found that the meltingtemperature obeys T ∗ < T ∞

m < Tm(N) (where T ∞m =

Tm(N → ∞)) and, along with the transition width, decreaseswith increasing N . For T < T ∗ the fraction of open base-pairs, ϕB(N), decreases with increasing N , whereas for T >

T ∗, it increases with increasing N . We further this previoustheoretical study here by investigating the influence of theweighting factor f .

Without loop entropy, previous Ising/PS type modelspredict Tm independent of N and therefore Tm = T ∗.When loop entropy is added to these models Tm(N) becomesan increasing function of N , which appears to agree withexperiment (in [1] it was found that the melting transitionfor a free homopolymer of length N = 30 000 takes placeat a temperature 1 K higher than that of chains of lengthN ≈ 500 and is much sharper). We also examine indetail the validity of the one-sequence approximation for freeboundary conditions and investigate the influence of loopentropy in situations where the accuracy of this approximationcan be gauged [42, 61]. For free boundary conditionsthis approximation involves keeping only the base-pair statesforming one interior bubble or one helix section of variablelength 0 n N . Unfortunately, there do not appear to beany detailed experimental studies of the thermal denaturationof DNA homopolymers with free ends as a function of chainlength (see, however [52, 67]) that can be used to test the modelpredictions and clarify the role and importance of both endeffects and bubble loop entropy.

For the case (ii) of an A–T insert of length N in largermore stable DNA polymers, detailed experiments [1] have

7


already been carried out for 60 < N < 140 and alsointerpreted using both a simple two-state approximation forthe A–T insert and the Poland–Scheraga model [42] (includingloop entropy) for the entire polymer [1]. For inserts, theboundary conditions are fixed mainly by the exterior G–C richdomains and only L enters (and not f , i.e. the individual valuesof μ and K ). For inserts the one-sequence approximationinvolves keeping only the base-pair states forming one bubbleof variable length 0 n N . The two-state approximationaccounts only for the completely closed and the completelyopen chain states in the partition function [42] and is a specialcase of the more general one-sequence approximation. Thevalidity of these types of approximations relies intimately onthe relatively large cost in free energy for creating a bubble (orbase-pair domain walls) compared with the cost of changingthe length of an already existing bubble (i.e. |L0| J0). Theupshot is that a one-bubble state can have a variable length(and in dynamics undergoes breathing) and such states shoulddominate the free energy for not too long chains (and for longerchains, temperatures not too close to the melting one).

We reexamine this problem by analyzing the sameexperimental results [1] using our coupled model for a finitechain with modified boundary conditions, because in suchsituations the nature of end monomers becomes extremelyimportant. In doing so, we study the validity of both the two-state and one-sequence approximations without loop entropyby comparing the predictions of these simplified approachesto those obtained from the exact solution to our model.By incorporating the loop entropy into the one-sequenceapproximation, we also examine the role and importance ofthis effect for homopolymer inserts. In order to compare thepredictions of the model with experiments on A–T inserts wehave fitted the DNA melting data presented in figures 6 and 7of [1] using simple fitting functions, the goal being to get asmooth approximation to the data (see the appendix) that willbe useful in this section.

3.1. Exact results for general chain boundary conditions(without loop entropy)

Using transfer matrix techniques we have shown that it ispossible to obtain a compact expression for the average fractionof open base-pairs in a finite chain of length N for arbitraryboundary conditions [18] (with neither loop entropy, nor chainsliding):

ϕB(N, T ; μ′) = 12

[1 − 〈c〉(N, T ; μ′)

](35)

where 〈c〉(N, T ; μ′) ≡ 1/N∑N

i=1 〈σi 〉 is given by

〈c〉(N, T ; μ′) = 〈c〉∞[

1 − 2R2V

R2V + e(N−1)/ξI

]

+ 2RV

√1 − 〈c〉2∞

(1 − e−N/ξI

)N[1 + R2

V e−(N−1)/ξI] (

1 − e−1/ξI) (36)

ξI is the Ising correlation length, and

RV (μ′) ≡ 〈V ′|0,−〉

〈V ′|0,+〉 (37)

with the normalized end vector

|V ′(μ′)〉 = [2 cosh(μ′)

]−1/2(

eμ′/2|U〉 + e−μ′/2|B〉

)(38)

enforcing the chain boundary conditions. The quantities〈c〉(N, T ; μ′), 〈c〉∞ (as given in (31)), RV (μ

′), and ξI are allfunctions of L0 and J0 [18]. For free ends μ′ = μ0, whereasfor closed (open) ends, |V ′〉 = |U〉 (|B〉), which can be seen bytaking the μ′ → ±∞ limits of (38). When μ′ is formally setequal to L0 there is no longer any end-interior asymmetry andthe model reduces to older Ising/PS type [52] models withoutloop entropy.

A simple expression can be obtained for RV by settingN = 1 in (36) and solving for RV :

RV (μ′) = 〈c〉1 − 〈c〉∞√

1 − 〈c〉2∞ +√

1 − 〈c〉21

(39)

where 〈c〉1 = tanh(μ′) is a function of μ′ and therefore reflectsthe boundary conditions.

DNA chains with free boundary conditions. When μ′ = μ0

(free boundary conditions), RV,free = RV (μ0) gets simplifiedin the following way for special values of T [18]:

RV,free =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−e−μ0 , T < T ∗

0, T = T ∗

tanh(μ0/2), T = T ∞m

eμ0 , T > T ∞m

(40)

which shows that RV,free is a monotonically increasing functionof T and vanishes at T = T ∗.

In figure 3 we present model results (with neither loopentropy, nor chain sliding) based on equation (36) for freechains of different lengths and different values of f . Weobserve that T ∗ increases with increasing f ; for f < 0.7,Tm(N) decreases with increasing N , whereas for f > 0.7,Tm(N) increases with increasing N . When f ≈ 0.7, themelting curves are nearly identical with the results obtainedfrom older Ising/PS type models (μ′ = L0) without loopentropy. When loop entropy is added to the model, the meltingtemperatures for the longer chains will be shifted to the right,amplifying the effect of finite f (see below).

DNA inserts with closed boundary conditions. For an A–T insert of length N in more stable G–C domains a simplestarting approximation is to apply closed boundary conditions(i.e. base-pairs i = 1 and N are considered to be held closeddue to their coupling to the adjacent G–C domains). For closedboundary conditions, μ′ → ∞, leading to

RV,cl =√

1 − 〈c〉∞√1 + 〈c〉∞ (41)

which is non-zero for all T > 0, implying that in this caseT ∗ = 0.

8


Figure 3. Fraction of broken base-pairs (35) versus temperature for free boundary conditions (without loop entropy) and chain lengths ofN = 30 000, 136, 105, 83 and 67 (from left to right, above the temperature of intersection, T ∗) (a) f = 0, (b) 0.4, (c) 0.6, (d) 0.7, (e) 0.8(other model parameters used are listed at the beginning of section 3).

Unfortunately in this case only N − 2 base-pairs canopen. A better approach involves artificially extending theinsert length from N to N + 2 and using closed boundaryconditions on the extended chain. In this case the ‘fictitious’(i = 1 and N + 2) base-pairs are held closed by the boundaryconditions in order to simulate the influence of the adjacentmore stable G–C rich domains and the remaining N base-pairscan fluctuate. Since the i = 2 and N + 1 base-pairs areadjacent to closed base-pairs, their probability of opening willbe lower than that of interior ones. It is clear that in this casemelting will begin near the center of the insert. If ϕcl

B (N, T )is the fraction of open base-pairs for a chain of length N withclosed boundary conditions, then simple counting shows thatthe average fraction of open base-pairs in the extended modelis given by

ϕextB (N, T ) = N + 2

Nϕcl

B (N + 2, T ). (42)

A more sophisticated approach is to keep the physicalinsert length of N and account for the coupling to the morestable G–C rich domains via a mean-field-type approximationby taking μ0 < μ′ < ∞. The approaches presented aboveare obviously valid only when the temperature is sufficiently

far below the melting temperature of the G–C rich domainsso that the experimental ultraviolet (UV) absorbance used tomeasure ϕB(N, T ) comes primarily from the A–T inserts inthe temperature range of interest.

In figure 4 we show how ϕB(N, T, μ′) varies as a functionof μ′ for N = 136. The melting temperature as a function ofμ′ interpolates smoothly between the results for free (μ′ = μ0)and closed boundary conditions over a temperature range of∼5 K and the width of the transition increases slightly withincreasing μ′.

In figures 5 we compare the experimental results forA–T inserts (figure A.2(a)) with the model predictions forϕB(T, N) for f = 0 and three different model boundaryconditions: (i) free boundary conditions, (ii) optimized μ′,(iii) extended model, closed boundary conditions. The valueof L = 9.87 kJ mol−1 is held fixed to reproduce theexperimental melting temperature for N = 30 000 and themodel predictions for the optimized (for Tm(N)) μ′ case arepractically insensitive to changes in f and J . For closedboundary conditions ϕB(N) increases with increasing N atfixed T simply because the end effects become attenuatedfor long chains, as illustrated in figure 5. We conclude thatthe model in its present form can reproduce the qualitative

9


Figure 4. Fraction of broken base-pairs (35) versus temperature forN = 136 and f = 0 (μ = L = 9.87 kJ mol−1) as a function of μ′:from left to right, free boundary conditions (μ′ = μ0); μ′/L = 0.86;1.14, 1.43; 2.00; 8.56; for closed boundary conditions the result issuperimposed on the right-hand curve (μ′/L = +∞).

tendencies, but not the quantitative details, of experimentson short A–T inserts (for such short chains including loopentropy into the model will not lead to better fits, see below).The results presented here do allow us, however, to gaugethe importance of chain boundary conditions on the meltingcurves. One difficulty in applying the present model arisesbecause the simplified approach presented here does notaccount for the increased probability of opening for G–C base-pairs adjacent to the A–T inserts. The complete solution ofour model for the full heterogeneous chain is in principalpossible using known numerical methods, as is discussed inthe conclusion.

The exact result for ϕB(N, T ;μ′) (35) does not revealin a physically transparent way which states contribute themost for a given chain length N and temperature T and,

as already mentioned, includes neither the effects of loopentropy, nor of chain sliding. In order to include such effectsin a straightforward way we now study the one-sequenceapproximation to the exact partition function for our model,an approximation that should be valid for sufficiently shortchains.

3.2. One-sequence approximation

One-sequence approximation for closed boundary conditions:DNA inserts. We start by examining the one-sequenceapproximation for homopolymer inserts of length N for closedboundary conditions without loop entropy. The effective freeenergy of creating an interior n-bubble with two base-pairdomain walls is [18],

βG(n)int = 4 J0 + 2nL0 (43)

and therefore the restricted partition function, Z cl1seq, including

only n-bubbles varying in size between n = 0 (helical insert)and N (bubble insert) is given by:

Z cl1seq = 1 +

N−1∑m=0

(m + 1) exp[−βG(N−m)

int

](44)

where the first term equal to one comes from a completelyclosed chain state and for an n-bubble m = N − n is thenumber of remaining intact base-pairs in the insert. The factorof (m + 1) = N − n + 1 in the sum is entropic in natureand equal to the number of ways of placing an n-bubble insidean insert of length N . We recall that L0 becomes negativefor T > T ∞

m and therefore in the high temperature range ofinterest for inserts the term depending on G(n)

int in (44) favorslarge bubbles. The entropic factor, on the other hand, favors

Figure 5. Fraction of broken base-pairs (35) versus temperature for (from left to right) N = 30 000, 136, 105, 83 and 67: fitted experimentalresults from figure A.2(a) (dashed curves) and (a) model predictions using free boundary conditions, (b) μ′ optimized to fit Tm(N) and(c) closed boundary conditions.

10


small bubbles. The one-sequence approximation incorporatesthe first two terms (of order 0 and 1) in an expansion in powersof the loop initiation factor,

σLI = e−4 J0, (45)

which counts the number of bubbles [42, 61].Within the one-sequence approximation, the average

fraction of broken base-pairs can be obtained from Z cl1seq:

ϕclB,1seq(N) = − 1

2N

∂(

ln Z cl1seq

)∂L0

. (46)

The sums in (44) can be carried out to find the followingcompact expression:

Z cl1seq = 1 + e−4 J0C(e2L0) (47)

whereC(x) ≡ x−N

(x p′(x)+ p(x)

)(48)

with

p(x) ≡ x N − 1

x − 1. (49)

By using (47) the following expression can be obtained forϕcl

B,1seq(N):

ϕclB,1seq(N) = −e−4 J0

N

[(∂C∂x

)x

]x=e2L0

. (50)

For sufficiently short chains the one-sequence approachwithout loop entropy defined above will be an accurateapproximation to the exact result for the extended model(N + 2 base-pairs with closed boundary conditions, ϕext

B (42)).When this approximation is valid, multi-bubble states areextremely rare (the range of validity in N of the one-sequenceapproximation depends on the value of J0 via σLI (45)).

Although it is difficult to incorporate bubble loop entropyinto our model in a general way because of mathematicalcomplications arising from the ‘long-range’ nature of theloop entropy factor, it is easy to do so within the one-sequence approximation. Including the loop entropy lowersthe probability of n-bubble opening. We adopt a commonsimplified form for the loop entropy factor associated with nbroken base-pairs [65, 43, 66],

gLE(n) = (n0 + 2 + 2n)−k (51)

that depends on the bubble loop length, B = 2 + 2n, and isparametrized by a constant n0 and an exponent k. The loopentropy exponent k is thought to be in the range 3/2 k 2.1, depending on the extent to which chain self-avoidance istaken into account [7]. The term n0 accounts for the enhanceddifficulty of forming small closed bubbles arising from DNAchain stiffness. Including the loop entropy leads to a modifiedone-sequence partition function, given by

Z cl,LE1seq = 1 +

N−1∑m=0

(m + 1)gLE(N − m) exp[−βG(N−m)

int

].

(52)

Figure 6. Comparison of the two-state approximation (dashedcurves) without loop entropy (55) with the full result (42) (solidcurves) for closed boundary conditions; from left to right N = 136,105, 83 and 67 (same parameters and colors as figure 3).

The introduction of loop entropy (k > 0) in Z cl,LE1seq can have an

exaggerated effect on the calculated melting curves if the loopinitiation factor, σLI (45), is not readjusted at the same time. Ifwe define D = (n0 + 2)/2 and use J0 → J0 + (k/4) ln(2D)in (52) then Z cl,LE

1seq can be rewritten as

Z cl,LE1seq = 1 +

N−1∑m=0

(m + 1)[1 + (N − m)/D]−k

× exp[−βG(N−m)

int

], (53)

with G int still given by (43) (in the fitting of experimental data,the value of D has been taken to be as large as 96 [1] andeven 450 [65]). The above readjustment of J0 means thatonly long n-bubbles (n = N − m > D) ‘feel’ the effect ofloop entropy (the suppression of short bubble formation dueto increased chain stiffness being incorporated directly into thereadjusted J0). We will compare the predictions of the one-sequence approximation with (k, D > 0) and without (k = 0)loop entropy using (53). Although the sums in (53) apparentlycannot be carried out analytically, once they are performednumerically, the analog of (46) can be used to obtain ϕcl,LE

B,1seq.If in evaluating the one-sequence partition function,

Z cl,LE1seq , we retain only the completely closed (m = N) and

completely open (m = 0) states, we obtain the two-stateapproximation:

ϕcl,LEB,2st = 1

1 +(1 + N/D)−k exp

[−βG(N)

int

]−1. (54)

A more general s-state approximation can be defined byincluding the m = 0, . . . , s−2 terms in the sum (52). Withoutloop entropy (k = 0), (54) simplifies to

ϕclB,2st = 1

2

1 − tanh

[−βG(N)

int /2]. (55)

In figure 6, the two-state approximation without loopentropy is compared to the exact result for the extendedcase (42). We observe a cross-over temperature (at which thetwo-state approximation begins to overestimate ϕcl

B ) roughlygiven by the temperature at which G(N)

int goes from positive

11


Figure 7. Free energy of bubble formation for N = 136 and T = 339, 342, 345, 347 K (from top to bottom): (a) intrinsic free energy,βG(n)

int ; total free energy, βF (n)int : (b) k = 0 (without loop entropy); (c) k = 1.7, D = 100; (d) k = 1.7, D = 1 ( f = 0, other model

parameters as in figure 3).

to negative (signaling a vanishing ‘nucleation barrier’ for thecompletely open insert). Contrary to previous claims [1], in thepresent case the two-state approximation overestimates Tm(N)by more than 2 K and underestimates the transition width.

The form (53) suggests defining an effective total n-bubblefree energy

βF (n)int = βG(n)

int − ln(N − n + 1)+ k ln(1 + n/D) (56)

that accounts for the intrinsic free energy of bubble formation(first term), as well as positional (second term) and loopentropy (third term). Following (43), βG(n)

int decreases withincreasing n for T > T ∞

m (L0 < 0) and increases for T < T ∞m

(L0 > 0). The positional and loop entropy contributionsincrease the effective free energy cost of bubble creation as thebubble size n increases.

In figure 7 we plot bubble free energies for N = 136 andincreasingly important loop entropy effects. The intrinsic part,G(n)

int is a linearly decreasing function of n (T ∞m ≈ 338 K see

figure A.2) and vanishes at n = N for T = 345 K, close to thetemperature at which the two-state approximation becomes anoverestimation (see figure 6). We observe that (i) inclusion ofthe positional entropy alone (figure 7(b)) leads to a minimumin (56) near n = N for sufficiently high temperatures and(ii) the loop entropy rigidity parameter D plays a minor rolewhen it is close to 100 (figure 7(c)) and an important onewhen it is close to 1 (figure 7(d)) (the value commonly usedin the modeling of infinite chains). In the latter case (56)remains positive over the whole temperature range studied andhas a maximum for small n and a minimum near n = N forsufficiently high temperatures (figure 7(d)).

In figure 8, we compare the one-sequence approximationswith and without loop entropy (55) for short inserts obeyingclosed boundary conditions. For J = 9.13 kJ mol−1

we find that for inserts without loop entropy the one-sequence approximation is practically indistinguishable fromthe exact result (42) for N < 10 000. Because loopentropy further reduces the probability of bubbles, wetherefore believe that the one-sequence approximation withloop entropy should be an excellent approximation in mostcases of practical interest (i.e. inserts with lengths less thana few thousand base-pairs). We observe in figure 8 thatfor such inserts and fixed L the net result of includingthe loop entropy is to shift the melting curves to theright by about 5 K for D = 1 and about 1 K forD = 100 without much change in the transition width.It therefore seems as if the addition of loop entropy willnot enable us to improve the fits to experiment shown infigure 5(b).

Although it is possible to work out the details of the one-sequence approximation when the end base-pairs in an insertof length N experience a chemical potential μ′ < ∞, we willnot present these results here.

One-sequence approximation for free boundary conditions.We now examine the one-sequence approximation with andwithout loop entropy for DNA homopolymers of length Nwith free boundary conditions. Because most syntheticDNA homopolymers are less than a few thousand base-pairslong [42, 61, 1, 52], the one-sequence approximation may bea useful and accurate simplified approach in such cases. Forfree boundary conditions, besides single interior bubbles, wemust include the possibility of single helical sequences. Theeffective free energy of creating an interior n-bubble with twobase-pair domain walls is given in (43); the effective freeenergy of creating a single end unzipped sequence of length

12


Figure 8. Comparison of the one-sequence approximations with loopentropy (dashed curves) and without (solid curves) (53), from left toright N = 136, 105, 83 and 67 for closed boundary conditions (sameparameters as in figure 3) and k = 1.7 (a) D = 1, (b) D = 100.

n starting at i = 1 or N (with only one base-pair domain wall)is [18]:

βG(n)end = 2 J0 − K0 + 2nL0. (57)

The effective free energy for creating a single interior helicalsequence of length m = N − m (including neither the i = 1nor N base-pair) with two domain walls is [18]:

βG(m)helix = 4 J0 − 2K0 + 2(N − m)L0. (58)

The effective free energy needed to completely denaturethe DNA chain of length N is βG(N)

open = 2L0 N −2K0. The restricted one-sequence partition function for freeboundary conditions, Z free

1seq, includes contributions from (i) thecompletely closed state (dsDNA), normalized to a weight ofone, (ii) interior n-bubbles inserted in a domain of length N −2varying in size between n = 1 and N − 2,

Z Bint1seq =

N−3∑m=0

(m + 1) exp[−βG(N−2−m)

int

](59)

(iii) one unzipped end sequence of length n, Z end1seq with two-

fold degeneracy

Z end1seq = 2

N−1∑n=1

exp[−βG(n)

end

](60)

(iv) a single interior helical sequence

Z Hint1seq =

N−2∑m=1

(N − 1 − m)α exp[−βG(N−2−m)

helix

], (61)

Figure 9. The four DNA states accounted for in the one-sequenceapproximation for free polymers (aside from the dissociated chains):(a) closed chain, (b) end unwinding, (c) internal helix and (d) internalbubble, corresponding, respectively to the first four terms in (63).

where α = 1 without chain sliding (for heteropolymers usingaverage parameter values) and 2 with (for homopolymers likepolydA–polydT) [42, 61, 52], (v) the completely open state(op),

Z op1seq = exp

[−βG(N)

open

]. (62)

Z free1seq can therefore be written as

Z free1seq = 1 + Z end

1seq + Z Hint1seq + Z Bint

1seq + Z op1seq. (63)

The four DNA states accounted for in the one-sequenceapproximation (aside from the dissociated chains) are shownin figure 9. It is now easy to include loop entropy by insertingthe loop entropy factor gLE into the fourth term of (63):

Z Bint,LE1seq =

N−3∑m=0

(m + 1)[n0 + 2 + 2(N − m)]−k

× exp[−βG(N−2−m)

int

]. (64)

It is not possible now to simply readjust J0 as was donefor inserts, because unzipped end sequences ‘see’ the un-readjusted J0. Unzipped end sequences are composed of twounbound chains joined at one end and therefore there is no loopentropy factor in Z end

1seq or Z Hint1seq (a small correction term for

two such self-avoiding chains, however, has been neglected,see [68]). We can, however, rewrite (64) as

Z Bint,LE1seq =

N−3∑m=0

(m + 1)[1 + (N − m)/D]−k

× exp[−βG(N−2−m)

int

], (65)

βG(n)int is the same as βG(n)

int with J replaced by

J ≡ J + (k/4) ln(2D) > J . (66)

It is then possible to define an effective loop initiation factor,σLI ≡ e−4 J0 < σLI, that controls the probability of bubbleformation in the presence of loop entropy and depends on the

13


Figure 10. Melting curves: comparison of the exact result with the one-sequence approximation (including completely open state) for a freechain (no loop entropy, no sliding): exact results (solid curves), from right to left near the upper part of the curves (T > 339 K), N = 500,2000, 10 000; one-sequence approximation, N = 500 (long dashed curve), 2000 (intermediate dashed curve), 10 000 (short dashed curve),f = 0.5, other parameters as in figure 3 (a) J = 4.57 kJ mol−1; (b) as in (a) but now a linear-log plot; (c) J = 9.13 kJ mol−1; (d) as in (c) butnow a linear-log plot. In (c) and (d) the dashed and solid curves are practically superimposed.

readjusted value J (although it is still σLI that controls theprobability of end unwinding and one internal helical section).

Within the free boundary condition one-sequence approx-imation the average fraction of broken base-pairs can be ob-tained from Z op

1seq via

ϕfreeB,1seq(N) = − 1

2N

∂(

ln Z free1seq

)∂L0

. (67)

When chain dissociation is taken into account thecontribution from the completely open chain, Z op

1seq, is dropped

from Z free1seq, which then becomes the internal partition function

for associated chains:

Z free1seq = 1 + Z end

1seq + Z Hint1seq + Z Bint

1seq (associated chains). (68)

The corresponding ϕfreeB,1seq is the fraction of broken base-

pairs in associated chains (clearly a lower bound for theexperimentally measured total fraction of broken base-pairs,because the contribution of dissociated chains is neglected).In this case the one-sequence approximation incorporates thefirst four terms (of order 0, 1/2 and two terms of order1) in an expansion in powers of the loop initiation factor,σLI (the so-called zipper model neglects the last (bubble)contribution) [42, 61]. The next higher term, neglected in (63)and of order 3/2, accounts for one internal bubble with chainsliding. In most cases of practical interest there is littledifference between using (63) and (68).

The above one-sequence approximation should be validfor sufficiently short chains. After determining its range ofvalidity when loop entropy is neglected, we can then use it with

confidence within this range to examine the influence of loopentropy on DNA denaturation.

In figure 10 we test the validity of the one-sequenceapproximation with neither loop entropy nor chain sliding bycomparing it with the exact result (35) for which the partitionfunction includes the completely open state. From now on wefix the weighting factor f at 0.5, which, as explained earlier, isclose to the one estimated from experiment. We observe thatthe one-sequence approximation is accurate when N 500for J = 4.57 kJ mol−1 (figure 10(a)) and is also accuratebeyond N 10 000 for J = 9.13 kJ mol−1 (figure 10(c)); inboth cases studied the melting temperature is well reproduced,although the transition width is underestimated for J =4.57 kJ mol−1 when N = 500 (with the discrepancy increasingwith increasing N). The one-sequence approximation alsosomewhat overestimates the temperature T ∗ at which themelting curves intersect. We conclude that the limiting valueof N for which the one-sequence approximation is accuratedepends critically on the value of J0 via the loop initiationfactor (45).

Because we are now interested in studying the effects ofloop entropy on thermal denaturation, we employ the smallervalue for J (J = 4.57 kJ mol−1). Despite the smaller valueof J , the inclusion of loop entropy reinforces the validity ofthe one-sequence approximation. For J = 4.57 kJ mol−1,k = 1.7,and D = 100 (n0 = 198), the readjusted value J (66)is greater than 9.13 kJ mol−1, implying that in this case bubblesare even more highly suppressed for J = 4.57 kJ mol−1

with loop entropy than for J = 9.13 kJ mol−1 without loopentropy. In figure 11 we observe that at low temperature thechain-sliding-only model gives the highest melting and the

14


Figure 11. Internal melting curves (associated chains): a comparisonof various one-sequence approximations for a free chain withN = 20 00. (a) Linear plot, (b) linear-log plot: neither loop entropynor sliding (long dashed curve), sliding only (intermediate dashedcurve), loop entropy only (short dashed curve), both loop entropy andsliding (solid curve), (k = 1.7, n0 = 198, f = 0.5, other parametersas in figure 3).

loop entropy one the lowest. At higher temperature the sliding-loop entropy model gives the highest melting. For the caseconsidered in figure 11, we therefore expect the accuracy ofthe one-sequence approximation to be comparable to that seenin figures 10(c) and (d) (and not figures 10(a) and (b)).

In figure 12 we plot the melting curves using the loopentropy-sliding model for free chains of three different lengths(N = 500, 200, 10 000) and compare the results obtainedwithout loop entropy and sliding. We note that due to thecombined effects of sliding and loop entropy the meltingtemperature increases with increasing N and the width ofthe transition decreases (figure 12(b)), in agreement withexperiment [1] (for f = 0.5 the temperature T ∗ at whichthe melting curves intersect is now greater than Tm(N), theopposite of what occurs when loop entropy and sliding areneglected, see figure 12(a)). The model prediction for thedifference between the melting temperatures for N = 500 and10 000 is about 0.5 K (the results for N > 10 000 should bevery close to the N = 10 000 one). When chain dissociationis added to the model, one can reasonably expect that themelting temperature for N = 500 will decrease by about0.5 K [42, 61] and that for N 10 000 it will hardly change.This result suggests that once chain dissociation is incorporatedinto the current model, it should be possible to account for theexperimental results of [1] (Tm(30 000)− Tm(500) 1 K anddecreasing transition width as N increases).

Figure 12. Internal melting curves (associated chains) obtained withand without loop entropy and sliding for free chains of three differentlengths: N = 500 (long dashed curve); 2000 (solid curve); 10 000(short dashed curve) with J = 4.57 kJ mol−1, k = 1.7, n0 = 198,f = 0.5 (other parameters as in figure 3): (a) with neither loopentropy, nor chain sliding; (b) with loop entropy and chain sliding.

4. Concluding remarks

This paper presents an extension of a theoretical model ofDNA denaturation [17, 18] that couples the base-pair states,unbroken or broken, and the chain configurational degrees offreedom. The elastic contributions are taken into account,arising from chain bending, torsional and stretching rigidities,the values of which depend on the neighboring base-pairstates. The difference of bond lengths in ssDNA (0.34 nm)and dsDNA (0.71 nm) is also included in the Hamiltonian.This model, tackled by analytical means, provides new insightinto the dependence of the effective Ising parameters, usedin previous Ising-like models, on microscopic elastic moduli.The main conclusion is that all these features lead to arenormalization of the bare Ising parameters of the same orderof magnitude as the thermal energy. Hence, they cannotbe ignored when relating microscopic properties, extractedfor example from ab initio calculations or experiments onDNA fragments, to the collective properties of the wholechain, measured, for instance, in single DNA moleculeexperiments (atomic force microscopy, optical and magnetictweezers, tethered particle motion). As an illustration, withoutconsidering the effects of stretching elasticity and base-pairlength, the energy cost to open a base-pair, 2μ, would bedirectly related to the same quantity measured with a forceapparatus [64]4. But μ is renormalized by these effects and

4 Even though the (screened) electrostatic repulsion of the phosphate atomsof the sugar DNA backbone is not taken into account in those experiments andis also likely to decrease the real μ [69].

15


is lowered by 0.5–1 kBT when the bare value is close to 2 kBT .The same conclusion holds for the destacking, J , or stacking,K , parameters.

In this work, we also analyze finite size effects. Inparticular the role of closed boundary conditions on meltingcurves for finite lengths is investigated in order to model aclamped polydA–polydT DNA insert. Two approximationsare considered: (i) the one-sequence approximation amountsto neglecting configurations with several bubbles and (ii) thetwo-state one keeps only the contributions from the completelyclosed and open chains [1]. In the range of parameters studied,the agreement with the exact result is excellent in case (i),whereas it is much less satisfactory in case (ii). We alsoundertake the integration of loop entropy in case (i), whichleads to an increase in Tm that is associated with the loopentropy cost and depends on the value of the loop entropychain stiffness parameter D (for N ∼ 100 there is a shiftof 1 K for D = 100 and of 5 K for D = 1). Finally, westudy free polymer chains using exact results with neither loopentropy nor chain sliding and the one-sequence approximationwith loop entropy and chain sliding. Our major conclusion isthat the experimentally observed increase in Tm with increasingchain length for homopolymers can be accounted for byincorporating both loop entropy and chain sliding into ourmodel. The simplicity of our method of incorporating loopentropy into the one-sequence approximation paves the way toa deeper study of the role of chain stiffness in the loop entropyfactor, gLE. We underline that careful experiments on freeand clamped homopolymers of different lengths (in solutionor in single molecule experiments) would be extremely usefulin elucidating the role of DNA finite size effects.

From an experimental perspective, our findings arerelevant for free DNA in dilute solutions, without anyconstraint on chain configurations, nor any applied force ortorque. An ingredient that we have not considered so far is thegain in translational entropy due to strand separation in the caseof dissociation [66]. A correct treatment of this mechanismconsists in writing a chemical equilibrium between completelydenatured single strands and partially bound ones (work inprogress).

The case of constrained DNA is more involved. If a forceor a torque is applied, for instance in tweezer experiments,rotational symmetry is lost in the Hamiltonian, whichprevents an analytical solution of the problem. Numericalor approximate schemes, such as variational principles, maybe used. Another interesting constraint concerns polymerlooping [60]. Circular DNAs appear in the case of transposonsor insertion sequences [40, 31]. Writing down the polymerclosure (e.g. for the determination of the J -factor) is aformidable task because it corresponds to the global constraint∑

ti = 0, formally equivalent to an applied force [12].We can, however, partially take into account looping in ourframework by imposing periodic boundary conditions on thevectors eμ,i and/or on σi , instead of the end condition |V 〉.This can be handled using the transfer matrix method. In thecase of superhelical twist, the polymer winds one or severaltimes around its tangent vectors ti . This condition can alsobe enforced via the boundary conditions, by requiring that

the appropriate combination of Euler angles acquires a phasemultiple of 2π when going from i = N to 1. This topologicalconstraint should lead to an increased fraction of denaturedbase-pairs, in order to release the torsional energy cost, andconsequently to an increased flexibility, thereby facilitatingcyclization. Our predictions for the end-to-end distance canalso be compared to experiment, because R is proportionalto the radius of gyration, which can be measured in viscosityexperiments.

All the results presented in this paper concern ho-mopolynucleotides and the numerical applications focused onpolydA–dT. This work can, however, be generalized to het-eropolymers, although a minimal amount of numerical workis necessary to handle the reduction of the transfer matrices.Nonetheless, a numerical study of heteropolymers would re-quire a knowledge of the microscopic elastic moduli, whichare far from being known with any certainty for any pair of thefour nucleotides A, T, G and C.

Appendix

In this appendix we extract smooth melting curves fromthe experimental data in [1]. For the poly dA–dT DNApolymer with free ends and 30 000 base-pairs we have usedthe temperature derivative of

ϕfit = cf

2

[1 − sinh(−af + βμf)√

e−4β Jf + sinh2(−af + βμf)

], (69)

where cf, af and μf are fitting parameters (simplified N = ∞Ising form); this functional form arises in simple Ising modelsof DNA denaturation [52].

For A–T inserts we have used the temperature derivativeof

ϕfit = cf

2hf1 − tanh[hf(Tf − T )] , (70)

where cf, hf and Tf are fitting parameters; this functionalform arises in a two-state treatment of simple Ising modelsof DNA denaturation [42] (the use of the two-state form hereto extract smooth experimental melting curves does not implythat the two-state approximation is a valid one, see figure 6).As shown in figure A.1 the areas under the fitted dϕB/dTfunctions are not normalized to one. We thus assume thatthe normalized fitted ϕB functions (figure A.2) represent agood approximation to the fraction of open base-pairs for theA–T segments. By examining figure A.1 we see that thisassumption is well borne out for the A–T inserts, but less sofor the N = 30 000 base-pair chain because of difficulties inreading the data off the experimental curve and the asymmetryof this curve. Our choice of fitting functions gives symmetriccurves about the melting temperature and thus cannot accountfor the observed asymmetry for N = 30 000. The observedasymmetry probably cannot be explained by loop entropy andchain sliding because (for infinite chains at least) when theyare included in the model, the melting curves become flatterto the left of the melting temperature and steeper to the right(the opposite of what is observed in figure A.1). For finitechains, however, the combined effects of loop entropy and

16


Figure A.1. Absorbance temperature derivative (unnormalizeddϕB/dT ) versus temperature: experimental data points [1] andunnormalized fitted functions (green non-monotonic curves, lefty-axis); UV absorbance (unnormalized fraction of broken base-pairs,ϕB) versus temperature (red monotonic curves, right y-axis) (fromleft to right: N = 30 000, 136, 105, 83 and 67).

Figure A.2. Normalized functions fitted to the experimental data [1]:(a) fraction of broken base-pairs versus temperature, ϕB; (b) dϕB/dTversus temperature (from left to right, N = 30 000, 136, 105, 83and 67).

chain sliding can be different, see figure 11. Although theN = 30 000 base-pair chain melting temperature ∼339 Kis well reproduced, the width of the transition appears tobe overestimated. The general trend is for both the meltingtemperature and transition width to decrease with increasingN . As the length of the insert increases the melting shouldtend to the infinite free chain result.

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18

1.4. Application à l’ADN observé par microscopie à force atomique

1.4 Application à l’ADN observé par microscopie à force ato-mique

Les techniques d’observation de la molécule unique ADN se sont énormément déve-loppées ces dernières années. Alors que les méthodes habituellement utilisées par les phy-siciens de la matière molle (rayons X, neutrons, rhéologie. . .) conduisent à des moyennesd’ensemble, l’intérêt croissant pour les biologistes d’observer en temps réel l’interaction del’ADN avec d’autres molécules biologiques les poussent à développer des méthodes précisesde manipulation de molécules uniques. Le rêve est d’avoir une idée précise des processus invivo comme la condensation de l’ADN autour d’histones, sa transcription, sa réplication,ou sa réparation [2].

Parmi les différentes techniques de biophysique utilisées pour observer l’ADN, on peutdistinguer, d’une part, les techniques de «spectroscopie» sous force [34, 179, 46, 162],sous couple [32, 35] ou sous champ électrique [121] qui étudient la réponse de la macro-molécule aux contraintes imposées et, d’autre part, les techniques d’«observation» où apriori l’ADN n’est pas ou peu perturbé. Citons la technique de Tethered Particle Motion(TPM) [70, 156] qui observe l’ADN en trois dimensions, ou à deux dimensions, la techniquede fluorescence [172, 17] et la microscopie à force atomique (AFM) [82, 163]. Notons quecette dernière peut être également utilisée pour tirer sur l’ADN [93].

Des observations récentes de la courbure de l’ADN double-brin par AFM par Wigginset al. [196] ont montré des «anomalies» (voir figure 1.7) : la distribution de probabilité p(θ)d’angles de courbure θ, pour deux paires de bases séparées d’une distance r, montre unesur-abondance de grands θ pour des petits r par rapport à ce que prédit le modèle WLC(voir figure 1.8a). Pour les grands r, ils retrouvent la distribution prédite par le modèleWLC. Ces observations sont faites à température ambiante Ta ' 298 K sur de l’ADNadsorbé sur du mica à l’aide de ponts ioniques Mg2+. Elles sont à mettre en parallèleavec les différentes expériences montrant une élasticité «anormale» de l’ADN aux courtesdistances :

i) des mesures faites sur la cyclisation d’ADN courts ont montré que la probabilitéde cyclisation en trois dimensions était plus grande que celle attendue [44, 45]. Unmodèle de «kinks» a été proposé pour expliquer ces observations [201] mais une erreurexpérimentale a été découverte, les expériences ont été refaites et il a été montré quele modèle WLC convenait bien [64].

ii) plus récemment, des mesures de longueur de persistance de l’ADN aux petiteséchelles, par des techniques de Fluorescence Resonance Energy Transfer (FRET) [205]et de TPM [31], prédisent une valeur plus faible que celle attendue par le modèleWLC.

Ces résultats expérimentaux suggèrent qu’aux petites distances, l’ADN se comporte dif-

63


Figure 1.7 – Photographie d’un ADN visualisé par AFM (tiré de Wiggins et al. [196]).

féremment d’une simple tige élastique modélisée par le WLC. Ces expériences sont faitesà température ambiante, ce qui a priori ne favorise par l’apparition des bulles de dénatu-ration, dont la probabilité estimée selon notre modèle pour un polydAT-polydAT (le casle plus favorable) est de l’ordre de 10−6.

De plus dans les expériences d’AFM de Wiggins et al., l’ADN est adsorbé électrosta-tiquement et à l’air. Les énergies des liaisons H et de recouvrement des liaisons π entrecycles aromatiques sont donc probablement modifiées, ce qui doit se répercuter sur les va-leurs de paramètres d’Ising. Des modifications structurales (longueur des paires de bases)ont d’ailleurs été observées par AFM sur des ADN synthétiques [28]. Nous avons appliquénotre modèle modifié pour un ADN en deux dimensions [145] 8 et ajusté les paramètresd’Ising afin de reproduire les données expérimentales de Wiggins et al. [196]. La comparai-son est remarquable (voir figure 1.8a). Les paramètres d’Ising modifiés conduisent à uneprobabilité de nucléation d’une bulle de dénaturation estimée à 10−3 et à une températurede dénaturation inaccessible de l’ordre de Tm ' 600 K.

L’augmentation subite de p(θ) pour r < 25 coïncide, dans notre modèle, avec l’appa-rition d’une bulle de dénaturation (d’environ 3 paires de bases) et l’angle critique θc pourlequel apparaît le comportement non-gaussien est simplement estimé suivant κdsθ2

c/2 '∆GB, c’est-à-dire lorsque l’énergie de courbure devient de l’ordre de l’énergie libre denucléation d’une bulle de dénaturation d’une paire de bases ∆GB = 4J0 + 2K0 + 2µ, soit

θc '√

2∆GB

κdsr (1.29)

ce qui correspond bien aux seuils observés. Avec les paramètres 2D, on a ∆GB ' 6.6 kBT ,ce qui conduit à une bonne supérieure pour rmax ' 120 paires de bases (avec θc ' π). En

8. L’idée générale du couplage variables internes/externes ainsi que certains calculs en 2D dans uncontexte différent ont été publiés par John Palmeri en 1993 [145].

64

1.4. Application à l’ADN observé par microscopie à force atomique

(a)

0 1 2 3θ (rad)

−2

0

2

4

6

8

10

(p[nl−θ

])

exp. data, r=15exp. data, r=29exp. data, r=88theoretical fits

(b)0 1 2 3

θ (rad)0

3

∆MB(

θ)

0 1 2 3

0

5

10

15

Nr

Figure 1.8 – (a) Logarithme de la distribution de probabilité p(θ) pour différentes valeursde r. Les symboles représentent les données expérimentales de Wiggins et al. [196] et lescourbes théoriques sont ajustées avec κUB = 20.97, J = 1.3173 et µ = 1.6685 (en unités dekBTa). (b) Couple Nr = ∂ lnZ(θ)

∂θ (unités de kBT ) et excès d’ouverture de paires de bases∆MB = −kBT

2∂∂µ ln p(θ) en fonction de θ. De gauche à droite r = 5, 15, 24 paires de bases.

L’élasticité est linéaire jusqu’à un seuil θc ∝√r pour lequel l’excès de courbure induit

l’ouverture d’une bulle de deux paires de bases.

calculant le couple moyen en fonction de l’angle de déflexion θ imposé, nous avons observéque l’ouverture des paires de bases coïncidait exactement avec le décrochage du couple(voir figure 1.8b). Le couple reste ensuite constant, puisque l’ouverture d’une petite bulleflexible suffit à la chaîne pour supporter de grands couples.

Une conséquence intéressante de ce travail est que, pour ce qui est de l’ADN en solution(en 3D) les valeurs des paramètres d’Ising conduisent également à l’ouverture de paires debases, comme l’avaient suggéré Yan et Marko [201] 9, mais pour r < 50. Même dans ce cas,la probabilité d’ouverture est extrêmement faible de l’ordre de 10−7 si bien que cela estdifficilement observable. Ainsi on ne peut donc pas tirer de conclusions sur les propriétésde l’ADN en solution et son interaction avec des protéines ou des enzymes directement àpartir d’expériences faites par AFM en deux dimensions.

Ce travail réconcilie plusieurs approches qui ont montré que le modèle WLC convenaitparfaitement pour des ADN en 3D, par exemple sous force, et que seul pour des très petitesdistances des non-linéarités apparaissent à température ambiante. Celles-ci ne nécessitent

9. Notons cependant que dans [201], le paramètre de coopérativité est négligé (J = 0) rendant le calculbien plus simple. Son importance est cependant considérable [193].

65


pas un nouveau modèle, sous-linéaire [196, 198] ou «kinkable» WLC [197], mais sont prisesen compte en couplant ce modèle WLC aux degrés de liberté internes.

1.4.1 Article

Suit l’article :N. Destainville, M. Manghi et J. Palmeri, Microscopic mechanism for experimentally ob-served anomalous elasticity of DNA in two dimensions, Biophysical Journal 96 4464 (2009)(6 pages)Le matériel supplémentaire associé à cet article est accessible sur le web à l’adresse :http://www.cell.com/biophysj/supplemental/S0006-3495(09)00695-X

66

Microscopic Mechanism for Experimentally Observed Anomalous Elasticityof DNA in Two Dimensions

Nicolas Destainville,* Manoel Manghi, and John PalmeriUniversite de Toulouse, Universite Paul Sabatier, Laboratoire de Physique Theorique (Institut de Recherche sur les Systemes Atomiques etMoleculaires Complexes), Toulouse, France; and Centre National de la Recherche Scientifique, Laboratoire de Physique Theorique, Toulouse,France

ABSTRACT By exploring a recent model in which DNA bending elasticity, described by the wormlike chain model, is coupled tobasepair denaturation, we demonstrate that small denaturation bubbles lead to anomalies in the flexibility of DNA at the nano-metric scale, when confined in two dimensions (2D), as reported in atomic-force microscopy experiments. Our model yields verygood fits to experimental data and quantitative predictions that can be tested experimentally. Although such anomalies existwhen DNA fluctuates freely in three dimensions (3D), they are too weak to be detected. Interactions between bases in the helicaldouble-stranded DNA are modified by electrostatic adsorption on a 2D substrate, which facilitates local denaturation. This workreconciles the apparent discrepancy between observed 2D and 3D DNA elastic properties and points out that conclusions aboutthe 3D properties of DNA (and its companion proteins and enzymes) do not directly follow from 2D experiments by atomic-forcemicroscopy.

INTRODUCTION

Whereas traditional bulk experiments provide average

behaviors of dominant subpopulations, new methods exist

that address DNA mechanical properties at the single-mole-

cule level (1–3). Observations by atomic force microscopy

(AFM) of double-stranded DNA (dsDNA) adsorbed on

a two-dimensional (2D) substrate (4,5) have recently allowed

a direct quantification of the distribution, p(q), of bending

angles q (6,7). This led to the unexpected observation of

an overabundance of large q (8), with respect to the wormlike

chain (WLC) model, at very short range (z5 nm, much less

than the persistence length z50 nm). These observations

suggest that, even in the absence of any bending constraints,

nonlinearities, such as kinks where DNA is locally unstacked

(9) or small denaturation bubbles, are excited solely by

thermal fluctuations with a high enough probability to be

observable at room temperature (TR ¼ 298.15 K). These

findings cast some doubt upon the adequacy of the WLC

model traditionally adopted in three dimensions (3D) (10).

In this respect, Cloutier and Widom (11) have observed

that short dsDNA, ~100 basepairs (bp) long, formed looped

complexes in 3D with a much higher probability than

expected, which was attributed to partial denaturation (12).

However, these findings have been questioned by new exper-

iments that pointed out a flaw in the experimental procedure

(13) and showed that short-DNA cyclization data were accu-

rately fitted by the WLC model, without invoking kinks. A

recent study based on flow experiments draws similar

conclusions (14). These converging elements are supported

by all-atom numerical simulations (9,15) suggesting that

kinks are not excited by thermal fluctuations with any

measurable probability in unconstrained DNA fluctuating

freely in solution.

Apart from 2D confinement, what is the difference

between both types of experiments? Fig. 1 shows a sketch

of DNA fluctuating in solution or adsorbed on a mica surface

as in AFM experiments (5–7). These experiments are carried

out in air (the solvent is dried) and DNA is electrostatically

adsorbed using magnesium ions, forming an ionic crystal

with the charged substrate. DNA electrostatics are thus

expected to be strongly affected as compared with DNA in

water, hence hydrogen-bonding energies between two

complementary bps and stacking energies between adjacent

base aromatic rings are substantially modified.

Recently, we have proposed a solvable model where

bending elasticity is intrinsically coupled to bp melting

(16,17) in contrast to older approaches for which bending

is not explicitly included (18,19). Single-stranded DNA

being two orders-of-magnitude more flexible than dsDNA,

this coupling must be taken into account because local dena-

turation strongly increases flexibility. Here, we argue that in

2D the modification of the above denaturation parameters

(bonding and stacking energies), due to adsorption, increases

the probability of bp opening, which lowers, in turn, the

bending rigidity. This explanation reconciles the apparent

discrepancy between 3D and 2D experiments.

THEORY

Model background

We model dsDNA as a chain of N bps i (1 % i % N) possessing two degrees

of freedom (16,17): an Ising variable, si, set to þ1when the bp is unbroken

(U) or set to 1 when the bp is broken (B). In addition to this internal vari-

able, an external one, the unit vector ti, sets the spatial orientation of the

monomer. The Hamiltonian couples explicitly the si and ti:

Submitted January 28, 2009, and accepted for publication March 10, 2009.

*Correspondence: [email protected]

Editor: Laura Finzi.

2009 by the Biophysical Society

0006-3495/09/06/4464/6 $2.00 doi: 10.1016/j.bpj.2009.03.035

4464 Biophysical Journal Volume 96 June 2009 4464–4469

H½si; ti ¼XN1

i¼ 1

kðsi; siþ 1Þð1 tiþ 1 , tiÞ

JXN1

i¼ 1

siþ 1si mXN

i¼ 1

si:

(1)

The bending rigidity of the joint between bps i and i þ 1, k(si, siþ1), takes

different values according to the internal state of the two neighboring bps.

We denote kU h k(1, 1), kB h k(1, 1), and kUB h k(1,1) ¼ k(1, 1).

The Ising parameters J and m have the following physical meanings: J is the

destacking energy (energetic cost to unstack two consecutive aromatic

rings); and 2m is the energy difference per bp between open and closed

states.

This discrete WLC model coupled to an Ising model can be completely

solved using a transfer matrix approach (16,17). Calculating the partition func-

tion amounts to solving a spinor eigenvalue problem (formally equivalent to

a quantum rigid rotator). In 3D, the orthogonal eigenstates, denoted by

jbJl;m;ti, are indexed by three quantum numbers: l ¼ 0, 1, ., N; and

m¼ l, ., l are the usual azimuthal and magnetic quantum numbers associ-

ated with the spatial orientation of ti and t ¼ is related to the bonding and

antibonding bp states (as for the one-dimensional Ising model or the H2þ cova-

lent bond). When projecting the eigenstates onto the real space basis jsUi, with

s a bp state and (q, 4) h U, the two spherical angles defining t, one gets

hsUjbJl;m;ti ¼ jl;mðUÞhsjl; ti. The jl;mðUÞ ¼ffiffiffiffiffiffi4pp

Yl;mðUÞ, proportional to

the spherical harmonics, are the eigenvectors of the pure chain model (i.e.,

when all k are set equal). The eigenvalues ll, t are degenerate in m and can

be expressed in terms of modified Bessel functions of the first kind

In (n ¼ lþ 12) (20) (see (17) for the expressions for the jl, ti). We have hl, t0jl,

ti ¼ dtt0, but hl0, t0jl, ti s dll0dtt0, because if l s l0, the matrix element is

between states of different rotational symmetry. This is why our coupled model

is not the trivial direct product of both the Ising and discrete WLC models.

The previous exact solution can also be found when the chain is confined to

2D, as already stated by one of us in Palmeri and Leibler (21); for example,

when DNA is adsorbed on a substrate at thermodynamical equilibrium (7).

The spherical angles (q, 4) become a single polar angle q ˛ (p, p]; the spher-

ical harmonics jl, m(q, 4) become the simpler jn(q) ¼ einq, with n integer;

the 2D analogs of the eigenvalues are denoted by ln, t and the eigenvectors

by jn, ti (21).

In the model as presented here, we do not take into account additional

DNA degrees of freedom, such as torsion or stretching. Although we have

recently demonstrated that it is possible to do so in the context of thermal

denaturation (22), the additional mathematical complications of taking

them into account in the bending-angle distribution calculation would tend

to obscure the basic physical mechanism leading to the onset of nonlinear

effective bending elasticity, and is therefore not warranted here.

Short-distance chain statistics in 3D and 2D

To compute the probability distribution p(ti$tiþr) of finding the polymer with

a given relative orientation between bps i and i þ r, we introduce the partial

partition function, Z(zi, ziþr), where all degrees of freedom are integrated out

except the projections on the z axis of ti and tiþr, which are set to zi and ziþr

(both ˛ [1, 1]),

Zðzi; ziþ rÞ ¼Xfsj ¼ 1g

YN

j¼ 1

ZdUj

4p

dðcosqi ziÞ

dðcosqiþ r ziþ rÞVs1U1

Y

j

sjUj

bPsjþ 1Ujþ 1

sNUN

V;(2)

where bP is the transfer matrix and jVi the boundary vector (16). The

complete calculation from Eq. 2 of p(s) ¼ 4pZ(1, s)/Z, where s h ti$tiþr

h cos q, q is the bending angle between two monomers separated by

a distance r, and Z is the full partition function, is given in the section B

in Supporting Material. It uses the decomposition of bP on the eigenbasis

jbJl;m;ti. We have checked that boundary effects are negligible at TR as

soon as i is larger than a few unities. We thus give the final result for p(s)

in the limit of long DNA when the internal segment [i, i þ r] is far from

both chain ends (i.e., for N / N and i / N),

pðsÞ ¼XNl¼ 0

2l þ 1

2PlðsÞ

Xt¼

0; þ

l; ti2 er=xp

l;t ; (3)

where Pl(s) is a Legendre polynomial (20). Equation 3 is a sufficient approx-

imation of Eq. S12 (in Supporting Material) for fitting purposes. This expres-

sion reveals the role of infinitely many tangent-tangent correlation lengths,

xpl;t ¼ 1/ln(l0, þ/ll, t). At TR, the persistence length, xp ¼ 150 bp, coincides

with the dominant correlation length xp1, þ (17).

The same calculation holds in 2D. We find the probability distribution

(section C in Supporting Material)

pðqÞ ¼ 1

2pþ 1

p

XNn¼ 1

cosðnqÞXt¼

0; þ

n; ti2 er=xpn;t ; (4)

where xpl;nt ¼ 1/ ln(l0, þ/ln, t) are also the tangent-tangent correlation lengths

associated with 2D eigenmodes jn, tiwith eigenvalues ln, t. For the numerical

calculation of infinite series such as Eq. 3 or Eq. 4, the sum is performed up to

order 100 (a higher cutoff has been checked not to change numerical values).

At room temperature, TR, one observes below (see also Fig. 2, a and c) that,

for q smaller than a threshold qc, p(s) and p(q) coincide with the discrete WLC

model probability distribution, pDWLC, which is the simplified version of Eq. 3

or Eq. 4 when no denaturation bubbles appear (formally all k equal to kU),

pDWLCðsÞ ¼XNl¼ 0

2l þ 1

2PlðsÞ

"Ilþ 1

2ðbkÞ

I12ðbkÞ

#r

; (5)

FIGURE 1 Sketch of a dsDNA segment solvated in

water (left) with its sodium counterion cloud (the phosphate

groups of the DNA backbone are negatively charged); and

in air (right), electrostatically adsorbed on a mica substrate

forming an ionic crystal via magnesium ion bridges

between the DNA and the negatively charged substrate.

Therefore, the parameters associated with the hydrogen

bonding of bps and the stacking of adjacent bases are

significantly modified.

Biophysical Journal 96(11) 4464–4469

Mechanism for Anomalous DNA Elasticity 4465

pDWLCðqÞ ¼1

2p

XNn¼N

cosðnqÞ

InðbkÞI0ðbkÞ

r

; (6)

in 3D and 2D, respectively (dotted lines in Fig. 2, a and c), with b¼ (kBT)1.

In the Gaussian spin-wave approximation, bk >> 1, valid here, the discrete

WLC model leads to a quadratic dependence in q. Indeed, in this case,

½Ilþ12ðbkÞrxIlþ1

2ðbk=rÞ. One ends up with the probability distribution for

a single joint of effective bending modulus k/r, and pDWLCxpGSW ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk=ð2prÞ

pexp½bkq2=ð2rÞ in 2D (section E in Supporting Material).

This implies that the free energy required to bend the polymer by an angle

q is quadratic, F(q, r) ¼ kq2/(2r). In this approximation, the bending rigidity

k and the persistence length xp are related through xp ¼ 2bk in 2D and

xp ¼ bk in 3D (23).

RESULTS

We first examine the distribution p(s) h p(ti$tiþr) in 3D.

Whereas it is dominated at large r by the largest persistence

length xp¼ 150 bp and is well described by the WLC model,

this is not true at short r and large q.

Fig. 2 a displays the probability density p(s), s¼ ti$tiþr hcos q, for realistic parameters (16,17). At TR, for q smaller than

a threshold qc, p(s) coincides with the discrete WLC model

distribution, pWLC(s) (Eq. 5), the simplified version of Eq. 3

when no denaturation bubbles appear. For q > qc, the plot

becomes nonquadratic because of partial DNA denaturation.

The threshold qc is estimated by equating the energetic cost

of bending the polymer by an angle q in its unmelted state,

F(q, r) ¼ kUqc2/(2r), with the free-energy cost of nucleating

a single denaturation bubble (of one bp), denoted by DGB,

which is DGB x 17 kBT in 3D (17). Using this scaling

argument, we find

qcx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 DGB

kU

r

r; (7)

which gives a good estimate of the observed thresholds

(Fig. 2 a). The anomalies (or nonlinearities) appear for larger

and larger values of q when r grows, and are nonexistent in

the plots of p(s) as soon as r > 50 bp, i.e., at length-scales

larger than 15 nm, thus recovering standard Gaussian

behavior. Indeed, setting qc ¼ p in Eq. 7 yields the upper

limit, rmax x 50 bp, as observed in the plots. This also

explains why cyclization experiments with r > 50 bp are

correctly described by the WLC model (13). For r < 50 bp,

this local melting effect is extremely weak, occurring with

a probabilityR p

qcpðcosqÞsinq dqz107 for r R 5.

The situation is very different when DNA is confined in 2D.

It has been demonstrated in experiments that DNA is in 2D

thermodynamical equilibrium (5,7). This is the reason why

our statistical mechanical model applies and in the large Nlimit, the probability distribution p(q) is given by Eq. 4. Plots

are provided in Fig. 2, b and c, for realistic parameter values.

At large enough angles, one also sees deviations from the

WLC behavior, appearing as soon as p(q) z 0.01 rad1,

a now measurable value (7).

We fit 2D experimental data (7) in Fig. 2 b, using Eq. 4 with

kUB, J, and m as fitting parameters (section D in Supporting

Material). The fits are good over the whole q-range. For the

best-fit parameter sets, the fraction of melted bps for uncon-

strained DNA is then >0.1% at TR, two orders-of-magnitude

higher than in 3D (16). The predicted melting temperature,

Tm, and transition width, both ~600 K, are also much higher

b

ca

d

FIGURE 2 Theoretical predictions of DNA elastic prop-

erties in two and three dimensions. (a) Logarithm of the

probability distribution p(cos q) ¼ p(s) in 3D (Eq. 3, solid

lines) for different values of r ¼ 5, 15, and 25 bp (from

left to right) compared with the WLC model (dotted lines).

One bp length is a ¼ 0.34 nm. The Ising and elastic param-

eter values (in units of kBTR) come from fits to earlier exper-

iments (16): kU ¼ kUB ¼ 147; kB ¼ 5.54; m ¼ 1.7977; and

J ¼ 3.6674. The probability distribution ~pðqÞ is given by

~pðqÞ ¼ sinq pðcosqÞ, because ds ¼ sin q dq. (b) Logarithm

of the probability distributions p(q) in 2D. Symbols repre-

sent experimental data taken from Wiggins et al. (7),

whereas the curves are now our best fits, from Eq. 4. The

curvilinear distances between monomers in Wiggins et al.

(7), namely 5, 10, and 30 nm, correspond, respectively, to

r ¼ 15, 29, and 88 bp. The value kB ¼ 5.54 (in units of

kBTR) comes from Palmeri et al. (16) and kU ¼ 160.82

comes from fitting the r ¼ 88 bp set of data by a pure

WLC model, as in Wiggins et al. (7) (because for such

large r, the Gaussian character is restored). The remaining

parameters (kUB, J, and m) are fitted. One possible param-

eter set is (kUB, J, and m) ¼ (20.97,1.3173,and 1.6685)

(section D in Supporting Material). Dotted lines show the

predictions of the WLC model, for comparison. (c) Loga-

rithm of the probability distribution p(q) in 2D. Parameter

values are coming from fits (see panel b), and r ¼ 5, 15, and 25 bp (from top to bottom, solid lines). Dotted line shown the predictions of the WLC model

and dashed lines show the same profiles when kB¼ 0. (d) Average excess chain melting DMB(q) in 2D. Same parameter values as in panel b. From left to right,

r ¼ 5, 15, and 25 bp. The elasticity is linear until a threshold qcfffiffirp

, where excessive bending induces bp melting.


4466 Destainville et al.

than their 3D analogs. Despite the high value for Tm in 2D, the

large transition width leads, with respect to 3D, to nonnegli-

gible bubble nucleation, even at TR. In other words, the loop

initiation factor (18), s ¼ e4J0=kBTR z102 where J0 is the

renormalized destacking parameter (17), is increased by

several orders of magnitude with respect to 3D (24). The

same argument as in 3D leads to rmax x 120 bp in 2D, after

modifying DGB¼ 6.6 kBT according to our fitted parameters.

Furthermore, we display in Fig. 2 d the average excess of

melted bps when ti$tiþr ¼ cos q is fixed, as compared with

an unconstrained DNA (see Appendix). As anticipated, the

deviation from the WLC behavior at qc coincides with the

appearance of melted bps making the polymer more flexible.

DISCUSSION

How can the apparent discrepancy between 2D and 3D

parameter values be explained? Not by the fact that the

DNA used in 2D experiments are heteropolymers, whereas

the values derived in 3D come from poly(dA)-poly(dT)

homopolymers (16). Indeed, even for the most robust pol-

y(dG)-poly(dC), Tm ¼ 360 K in solution. A simple and

straightforward explanation for the discrepancy in parameter

values is related to the change in the DNA electrostatic energy

when it is solvated in water (3D) or adsorbed through magne-

sium (Mg2þ) bridges on the mica in a dry environment.

Indeed, it is known that slightly modifying electrostatic inter-

actions (such as by varying the salt concentration) changes

dramatically the denaturation profile of DNA in solution

(see, e.g., (25)). The energy required to break a bp, 2m, and

the energy to destack consecutive bps, 2J, should also be

sensitive to the change in the direct adsorption energy

between mica and ds or single-stranded DNA. Strong support

for this mechanism comes from the experimental results of

Wiggins et al. themselves (7). In their Fig. S3, they present

the angle distribution and end-to-end distance statistics for

DNA adsorbed on a different-quality mica. Even though the

data match to a good approximation those of their Fig. 3,

a detailed analysis of the plots for r ¼ 5 and 7.5 nm leads to

the conclusion that the two data sets do not coincide, even

taking into account error bars. This is an experimental indica-

tion that the substrate on which DNA molecules are adsorbed

does indeed influence its microscopic parameters. Recent

AFM experiments also testified to a DNA structural modifica-

tion after adsorption on mica and drying (26): poly(dG)-

poly(dC) proves to shorten its contour length, supposedly

by taking an A-DNA conformation, in contrast to poly(dA)-

poly(dT) or plasmid DNA, both of which keep their B-DNA

conformations.

As a result, inferring the parameters m and J from their 3D

analogs is a challenging task. At the time of the writing of this

article, the best strategy is certainly to fit them to experimental

data. The above results are confirmed by recent accurate

all-atom molecular dynamics simulations: Mazur has investi-

gated in detail the short-distance angle distribution of 3D

DNA and did not find any evidence for the strong deviations

from a WLC distribution found experimentally in 2D (15).

Now we discuss in greater detail the role of bubble flexi-

bility, kB, and of cooperativity, J, by comparing our model

with earlier ones. In the kinkable WLC model (27), kinks of

vanishing rigidity can be activated by thermal fluctuations.

This model and ours become physically equivalent in the

kB / 0 limit: a 2-bp denaturation bubble plays the role of

a kink, in the sense of a thermally activated local defect

without rigidity. Our microscopic vision of a kink thus differs

from Lankas et al.’s local unstacking one (9), but yields the

same short-range mechanical properties. When kB ¼ 0, the

interesting behavior of p(q) in the denatured region is

destroyed: p(q) becomes flat (Fig. 2 c), as in Wiggins et al.

(27), and is practically insensitive to r once a kink is nucle-

ated, because a chain segment including a kink has vanishing

rigidity. This is the reason why Wiggins et al. appeal to a

different linear subelastic chain (LSEC) model, with a

phenomenological bending energy ELSEC ¼ Ljqj, which

enables them to satisfactorily fit their experimental data

(7,28). In contrast to this LSEC model, our approach proposes

a microscopic explanation associated with bubble nucleation

for the subharmonic behavior of p(q). Due to excess bubble

formation, our model predicts deviations from WLC (or

Gaussian) behavior as soon as r < rmax with rmax h p2kU/

(2DGB) (from Eq. 7). This expression differs from the

LSEC model one, for which rmax z bkU.

Setting J¼ 0 with kB finite also affects the profiles by soft-

ening the transition and increasing significantly the large angle

probabilities, by a factor >10 (data not shown), which

confirms the importance of cooperativity (when in addition

kUB¼ 0, we find again the model proposed in Yan and Marko

(12) in the context of cyclization). Neglecting J or kUB would

require the use of unphysically large kB values when fitting

experimental data, while worsening the fit quality.

Our model is restricted to homopolymer DNA. However,

a more accurate treatment should incorporate sequence

effects by using bp-dependent model parameters (29).

Considering that the heteropolymer case is difficult to treat

theoretically, and experiments provide only an average

description of bending angle probability distribution, we

limit ourselves here to describing the anomalous behavior

using an averaged approach. If more detailed experimental

results become available, it would be worthwhile to extend

our model to treat the heterogeneous case.

Currently, many AFM experiments explore DNA confor-

mations and complexation between nucleic acids and

proteins (see reviews (4,30,31)). When AFM imaging is

carried out on DNA (6,7,32,33) or DNA/histone complexes

(34) to access their statistical and dynamical properties,

effects of surface interactions on DNA structure are likely

to modify sensibly these properties. More generally, our

work suggests that studying DNA/companion proteins inter-

actions by AFM (35–38) does not provide any quantitative

clue to 3D complexation.



In the cell, packaging involves wrapping DNA around

positively charged histones (39). It has been shown that this

adsorption is mainly driven by electrostatics (40). Our results

suggest that in this case, DNA adsorption on a curved charged

surface (such as the histone) is likely to modify profoundly

local elastic and denaturation properties of dsDNA. Enhanced

flexibility due to denaturation is then likely to facilitate wrap-

ping. This mechanism might also be important for improving

the accessibility of enzymes to the single strands in local

bubbles (41,42) when DNA is wrapped.

One way of validating this model at the experimental level

would be to quantify the effects of temperature, which can be

predicted for both our coupled model and the LSEC one (28)

(Fig. 3; see section E in Supporting Material for LSEC

formula). Our model predicts that increasing temperature

enhances flexibility in a more pronounced manner, thanks

to the opening of bps. We believe that such a deviation

between the predictions of both models would be a credible

experimental test of their respective validities. Additional

tests of the quantitative difference between DNA properties

in 3D and 2D would be to compare cyclization rates by

AFM in both situations for the same dsDNA strands, or to

check that denaturation remains weak in 2D when approach-

ing the 3D melting temperature, as predicted by our results.

APPENDIX: BENDING-INDUCED MELTING IN 2D

Following a calculation as in Wiggins et al. (27), we derive the excess chain

melting DMB as a function of q. It measures the average excess of melted bps

in the bended chain as compared with the free, unconstrained one and is

given by DMBðqÞh kBT2

vvm

lnpðqÞ (section F in Supporting Material).

The comparison of Fig. 2, c and d, confirms that the deviation from the

WLC model corresponds to the appearance of melted bps that make the

polymer more flexible at short range. An interesting feature of these calcu-

lations is the saturation of DMB at a finite value, even when r < rmax

increases. In Fig. 2 d, this value is close to 3, which means that the total

excess number of denatured bps does not exceed 3 on average. In other

words, even if r bps, or more, can in principle be melted to relax the

constraint ti$tiþr ¼ cos(q), only a few of them actually do, since it costs

more energy to melt more bases, whereas, owing to the small value of kB,

a small denaturation bubble suffices to give the whole molecule a very small

resistance to torque.

SUPPORTING MATERIAL

Three figures and thirty-four equations are available at http://www.biophysj.

org/biophysj/supplemental/S0006-3495(09)00695-X.

We thank Roland R. Netz and Catherine Tardin for enlightening discussions.

This work was supported in part by the French Research Program ANR-07-

NANO-055, Project SIMONANOMEM.

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1.5. Remarques et perspectives

1.5 Remarques et perspectives

Chaîne effective ou deux brins en interaction ?

Ce modèle couplé généralise un certain nombre de modèles proposés dans la littéra-ture [201, 182]. Son inconvénient réside dans le fait que l’ADN double-brin ou partiellementdénaturé est modélisé par une seule chaîne dont la rigidité de courbure locale peut varier.

Certains travaux considèrent deux chaînes simple-brin en parallèle [75, 98] mais celaconduit à une rigidité de courbure double-brin simplement double, soit 2κss ce qui necorrespond pas aux résultats expérimentaux. Alors que l’on comprend intuitivement quece facteur 50 entre les deux rigidités de courbure est relié aux interactions d’empilementau sein de l’hélice, il semble indubitable qu’un modèle microscopique plus réaliste doitprendre en compte la géométrie hélicoïdale de l’ADN double brin. Nos simulations préli-minaires de dynamique brownienne [41] montrent la difficulté de modéliser proprement ladouble hélice et surtout de relier ces paramètres microscopiques aux paramètres mesurésexpérimentalement (longueur de persistance, module de torsion. . . ).

La prise en compte des deux brins dans les bulles de dénaturation est possible avecnotre méthode au prix de gros efforts de calculs et pour N →∞ [161]. Nous avons montréque dans la limite d’une séquence, l’utilisation du poids de Poland et Scheraga permettaitde reproduire les données pour les brins relativement courts. En revanche, dans la limitedes brins très court où l’approximation des deux états est valable, la dynamique et ladépendance en N du processus de dénaturation est contrôlé par l’équilibre de dissocia-tion [155, 192]

dsDNA 2 ssDNA (1.30)

faisant intervenir une loi d’action de masse de constante d’équilibre ∆(T ) et donc l’en-tropie de translation des brins. La fraction de paires de bases ouvertes est alors ϕB =∆/4(

√1 + 8/∆ − 1) ce qui conduit à ∆(Tm) = 1. À cause du couplage, les approches

simples [155, 16, 78] où les fonctions de partition associées aux degrés de libertés externeset internes se factorisent, Z = ZextZint, n’est plus valable. Dans ce cas, il est indispen-sable de traiter les deux chaînes en interaction. Le calcul de le température de transitions’avère alors difficile. Nos résultats préliminaires montrent qu’il faut introduire les modesde vibration des monomères [paramétrés par ε, voir l’éq. (1.19)] et faire un certain nombred’approximations. Les résultats proposés par Benight [16], où l’ADN double-brin est mo-délisé par un cylindre solide, tandis que les deux simples brins sont modélisés par dessphères (et la factorisation des fonctions de partition est supposée), conduisent à une loiphénoménologique ∆ ∝ Nα où α est choisi dépendant de T afin d’interpoler entre les deuxgéométries. Il sera intéressant d’étudier se problème en alliant simulations numériques pourles petits N et développements analytiques.

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ADN sous contrainte

Le cas d’un ADN sous contrainte, où les angles de courbures sont imposés par l’opéra-teur, a été abordé dans [60]. Deux directions futures sont envisagées. La première concernel’ADN sous force, ce qui dans notre modèle conduit à un modèle d’Heisenberg sous champ.L’opérateur de transfert n’est alors plus diagonal dans la base des harmoniques sphériquesmais tridiagonal ce qui permet tout de même une résolution semi-analytique [171, 181, 161].La prise en compte du modèle complet [127] où la longueur de la paire de bases dépend del’état interne devrait permettre d’ajuster notre approche aux résultats expérimentaux. Ilest à noter que des expériences très récentes [123, 199] suggèrent qu’effectivement la transi-tion de sur-étirement de l’ADN serait reliée à la nucléation locale de bulles de dénaturationet non au simple desempilement de l’ADN double-brin. Des travaux très récents [161, 131]ont utilisé une approche semblable à la nôtre et l’un d’entre eux reproduit de façon remar-quable certaines courbes expérimentales [131]. Nous espérons ainsi relier deux processusphysiques supposés différents que sont la dénaturation thermique et la transition de sur-étirement.

Le cas de la cyclisation induite par des protéines, les ligases, placées aux extrémités,est également envisageable à l’aide de notre modèle couplé, en utilisant les valeurs deparamètres déterminés dans [147, 127]. Dans [60], nous avons calculé la densité de proba-bilité pour un angle de courbure θ donné entre deux monomères donnés. Nous envisageonsd’étendre ce calcul au cas où l’orientation de plusieurs monomères est fixée, par exemplepar les ligases ou lors de l’enroulement de l’ADN autour de protéines. Ici encore, nos cal-culs préliminaires montrent que le seuil de 50-80 paires de bases semble être celui où laprobabilité de cyclisation devient plus grande que pour le modèle WLC.

Notre modèle peut également se généraliser au cas de l’ADN sous torsion [32, 132,14, 15]. La densité de probabilité pour un angle de courbure et de torsion donné se cal-cule à l’aide des fonctions de Wigner, éléments de matrices des rotations [200], Dmjl (Ω) =〈l,m|D(Ω)|l, j〉 couplées aux valeurs propres Λl,j,τ semblables aux valeurs propres du mo-dèle WLC-Ising mais incluant des fonctions Gl,j(κ,C) à déterminer. Notons que G0,0(κ,C)est en fait déjà donnée par l’éq. (1.26) [127]. De manière semblable à l’ADN sous lacontrainte d’un angle de courbure fixé entre deux monomères, on s’attend à ce que lacontrainte d’un super-enroulement conduise à la nucléation de bulles de dénaturation [14,15, 183].

Dynamique de fermeture des bulles

En 2003, Libchaber et ses collaborateurs [4] ont mesuré par spectroscopie de corrélationde fluorescence (FCS) la dynamique de relaxation d’une bulle de dénaturation de 18 pairesde bases AT insérées entre deux régions riches en GC. Le temps caractéristique mesuré,

74


τ ≈ 50µs, paraît très long pour une petite bulle. Ce temps correspond au temps le pluslong de la fonction d’auto-corrélation de l’intensité de fluorescence et est relié d’un manièreou d’une autre à l’ouverture ou la fermeture de la bulle de dénaturation.

Un certain nombre de théoriciens se sont intéressés à ce résultat. Ils ont tour à tourmis en avant : soit la dynamique de la taille de la bulle n, en développant une équation deFokker-Planck sur n à partir de l’énergie libre de Poland-Scheraga [4, 81, 10, 74, 9] 10 ou del’approximation à deux états [20] ; soit la dynamique de la distance x entre brins en utilisantle modèle de Peyrard et Bishop [3, 102, 59, 112] ou en supposant un potentiel d’interactionquadratique [42]. Ces approches apportent souvent des comparaisons qualitatives mais lesrésultats satisfaisant d’un point de vue quantitatif sont rares. Nous pensons d’une part,qu’appliquer le modèle de Poland et Scheraga aux expériences [4] n’est pas valable puisquela bulle de dénaturation a une longueur n = 28 < n0 ≈ 100 où n0 est le cut-off dansl’éq. (1.3) ; et d’autre part, que le modèle Peyrard et Bishop ne considère pas la dynamiquede la chaîne en trois dimensions. La majeure partie de ces travaux [151, 102, 59, 42, 112] seconcentre sur la dynamique de respiration des bulles, en considérant que la chaîne dans sonensemble est pratiquement gelée et en étudiant uniquement la dynamique de la variableinterne x. En particulier, nos simulations de dynamique brownienne en trois dimensionsd’une chaîne WLC avec un module de courbure κds [41], montrent que pour des chaînesde 10 à 30 paires de bases, le temps de relaxation de la chaîne bout à bout est τ ≈ 16à 140 ns (τ ' N3). Nous estimons τ ' 100 µs pour N = 300, loin des valeurs de 1 ns(pour T Tm) à 1 µs (pour T . Tm) trouvées par Jeon et collaborateurs [98, 102]. Dansleurs travaux, l’espace des phases associé aux conformations globales de la chaîne n’est pastotalement exploré avec des temps de simulations si faibles. Nos simulations préliminairesde dynamique brownienne avec couplage [41] montrent que le processus de fermeture d’unebulle de dénaturation se fait sur des échelles de temps très longues, ici supérieures à 100 µs(voir figure 1.9). Le processus est contrôlé par la diffusion des extrémités double-brin quientourent la bulle. Dans cette optique, la modélisation d’une double hélice fluctuante nousparaît prometteuse.

Expériences de Tethered Particle Motion

Nous avons proposé à nos collaboratrices Laurence Salomé et Catherine Tardin de l’Ins-titut de Pharmacologie et Biologie Structurale de monter une expérience de dénaturationd’un ADN unique par TPM. L’idée consiste à utiliser le dispositif de TPM, qui a conduit àplusieurs résultats interessants sur l’interaction ADN/protéines [70, 156, 157], dans lequel

10. Notons que l’éq. (1.3) introduit un terme non-linéaire (mais logarithmique) en n dans l’énergie libreeffective d’une bulle de taille n. Ce terme est essentiel dans l’étude de la dynamique de prolifération desbulles de dénaturation.

75


Figure 15: Left: samples of configuration during our numerical simulation withN = 30. Initial configuration, relaxed configuration for t < 0 and configurationat t = 7µs from top to bottom. Right: ’AT’ base pair interaction at t < 0 (blackcurve) and t ! 0 (red curve).

5.3 Bubble closure

Our primary objective in this study is to demonstrate the influence of the chaindi!usion on the denaturation bubbles dynamics. For this purpose we proposethe following procedure: We consider a DNA of N bases, the three first and threelast bases are forced to be closed with a strong Morse potential to prevent thedenaturation of the tips of the chain. The other bases are initially coupled onlywith the repulsive part of the Morse potential. When the system has relaxed(at t = 0), we switch on the attractive part of the Morse potential.

Uint =!

i

A(!)i exp("2

!i " !0

")" 2A

(+)i exp("!i " !0

") + 1 (14)

18

Figure 1.9 – Exemples de configurations de l’ADN dans les simulations de dynamiqueBrownienne pour 30 paires de bases : de haut en bas, configuration initiale, relaxée et àt = 7 µs. À t = 0 le potentiel d’interaction entre les brins de la region AT bascule derépulsif à attractif. Une petite bulle de dénaturation de 5 à 6 paires de bases se maintientsur des échelles de temps supérieures à 100 µs.

un ADN est accroché à un bout au substrat et à l’autre à une bille fluorescente observéepar microscopie. Le mouvement brownien de la bille dans le plan parallèle au substrat estenregistré et permet de remonter à la distance bout à bout

√〈R2〉 de l’ADN [éq. (1.28)] :

plus l’amplitude des fluctuations latérales est importante, plus cette longueur est grande.Ainsi le TPM apporte des informations sur les changements conformationnels de l’ADN.L’ADN de 596 paires de bases, synthétisé par Philippe Rousseau du Laboratoire de Micro-biologie et Génétique Moléculaires, consiste en un segment de 50 paires de bases AT inséréentre deux régions riches en GC comme dans l’expérience de Libchaber et al. [4]. Ainsi lecœur AT fond avant le reste, ce qui permettra de garder la bille reliée au substrat. Nousavons estimé numériquement une chute du carré de l’amplitude des fluctuations latéralessupérieure à 20 % lors de la nucléation de la bulle (pour une bille de 20 nm et un ADN deplus de 250 paires de bases), ce qui est bien au-dessus de la résolution du dispositif.

Ces travaux sont en cours. Le dispositif de chauffage est maintenant au point mais ungros travail de synthèse d’une liaison covalente entre l’ADN et le substrat et l’ADN etla la bille est en cours afin qu’elles ne se dénaturent pas à T = 57 C, la température dedénaturation du cœur AT mesurée par absorbance UV. Nous espérons également étudierla dynamique de cette bulle, car le TPM permet de mesurer des événements biologiques(bouclage, complexation. . .) avec une résolution inférieure à la seconde. À l’aide de simula-

76


tions de dynamique brownienne et Monte Carlo, nous avons calibré le TPM d’un point devue dynamique [130]. En effet, l’expérience perturbe la mesure de la dynamique de l’ADNseul à plusieurs niveaux :

i) la dynamique observée est celle de la bille et rend compte de la dynamique ducomplexe ADN-bille et non de l’ADN seul ;

ii) les interactions de volume exclu de l’ADN avec la bille et le substrat modifient saconformation et donc sa dynamique ;

iii) la friction hydrodynamique de la bille et de la chaîne avec la surface ralentit nota-blement la dynamique.

De plus les interactions hydrodynamiques au sein de la chaîne sont modifiées. Nous avonsmontré que la calibration dynamique du TPM dépend fortement du temps de relaxationglobal de la chaîne qui augmente avec la longueur de l’ADN et la taille de la bille. Enparticulier, pour des billes inférieures à 20 nm, la friction hydrodynamique est négligeable,ce qui conduit à une résolution temporelle optimale de 20 ms [130]. La question de savoir sila dynamique de fermeture d’une bulle de dénaturation de 50 paires de bases est observablepar TPM avec cette résolution reste à élucider.

La plupart de ces questions et notamment celles concernant la dynamique de fermeturedes bulles de dénaturation constitueront le sujet de thèse de Anil Kumar Dasanna, quivient de débuter au 1er octobre 2010.

77


78

Chapitre 2

Membranes fluctuantes

La membrane plasmique formée d’une bicouche lipidique est un des constituants majeurdes cellules biologiques, car elle joue le rôle de «frontière» entre le milieu extracellulaireet le milieu intracellulaire, le cytoplasme. Elle permet à la cellule d’avoir une compositiondifférente du milieu dans lequel est évolue. Elle est essentiellement constituée de lipides etprotéines (voir figure 2.1) : les protéines contrôlent le transport des molécules (ions, pro-téines, macromolécules,. . .) et de l’information (potentiel électrique,. . .) grâce, par exemple,à des protéines trans-membranaires qui forment des canaux sélectifs. Les lipides sont desmolécules amphiphiles qui s’auto-organisent en bicouche dans l’eau et forment ainsi lastructure membranaire. Par exemple, le cholestérol est très abondant dans les cellules eu-karyotes et joue un rôle prépondérant dans la formation de micro-phases lipidiques.

Si l’on s’intéresse à la membrane lipidique (sans les protéines), elle peut être vue sché-matiquement comme un fluide de lipides à deux dimensions. Ses propriétés élastiques sontcaractérisées par un module de courbure, κ, de l’ordre de quelques dizaines de kBT , ce quiles rend relativement fluctuantes à température ambiante. Les premiers modèles élastiquesdes membranes sont dus à Canham [40] et Helfrich [88] qui caractérisent les fluctuationsd’une membrane essentiellement par son module de courbure κ et sa courbure sponta-née c0. Ce modèle s’est avéré très puissant et a permis de décrire aussi bien le spectre desfluctuations des membranes lipidiques [173] que la forme d’équilibre des erythrocytes [117].

Nous présentons ici deux travaux, basés sur ce modèle, qui étudient, d’une part, le cou-plage entre les fluctuations thermiques de la membrane et l’hétérogénéité de sa compositioninterne (on considère ici un mélange binaire de lipides) et, d’autre part, deux membranesempilées «adsorbées» sur un substrat plan.

Ce travail s’insère dans une collaboration plus large entre le groupe de Physique Sta-tistique du laboratoire et l’équipe de Laurence Salomé de l’Institut de Pharmacologie etBiologie Structurale (IPBS, Toulouse) qui porte sur l’étude de la dynamique des protéines

79

Chapitre 2. Membranes fluctuantes

Proteine

Fluide extra!cellulaire

CytoplasmeFilaments ducytosquelettetransmembranaire

peripheriqueProteine

Chainessaccharidiques Cholesterol

Bicouchelipidique

Figure 2.1 – Schéma de la membrane plasmique constituée d’une bicouche lipidique(d’épaisseur ' 5 nm) dans laquelle sont insérées les protéines. Le cytosquelette situé dansle cytoplasme est connecté à la membrane par des protéines. D’après [169].

et des lipides dans les membranes biologiques (expériences de suivi de particules uniques).

2.1 Interactions entre domaines lipidiques dans les membranesfluctuantes

Nous avons étudié une membrane modèle composée de deux types de lipides A et Bqui ont tendance naturellement à se séparer en deux phases en l’absence de fluctuations dela membrane. Le hamiltonien gaussien de ce liquide binaire s’écrit classiquement

HI [φ, h] =∫

dx√g

[J

2gij∇iφ∇jφ+ V (φ)

](2.1)

où φ = φA − φB est la différence de fraction volumique des lipides A et B. Le paramètreJ est positif et correspond à l’énergie nécessaire pour créer une séparation entre deuxdomaines distincts et V (φ) est le potentiel qui fixe les valeurs de φ et donc la compositionde la membrane à l’équilibre. Le facteur g = 1 + (∇h)2 est le déterminant de la métriquegij de la surface de la membrane courbée dans les coordonnées de Monge où h(x) est lahauteur de la membrane à la coordonnée x dans le plan projeté.

Par exemple, dans le cas d’une membrane incompressible dont les bords sont tenus surun cadre et en contact avec une solution avec laquelle elle peut échanger des lipides, lavaleur de φ à l’équilibre, φ0, est fixée par le minimum de l’énergie libre. Si la membranene fluctue pas elle est donc fixée par V ′(φ0) = 0. En revanche, lorsqu’elle fluctue, il faut

80

2.1. Interactions entre domaines lipidiques dans les membranes fluctuantes

prendre en compte l’énergie des fluctuations. Or, la membrane est caractérisée par unmodule de courbure, κ(φ), qui varie avec la composition lipidique locale, φ. Le hamiltoniende Helfrich régissant les fluctuations h(x) s’écrit alors

HS [φ, h] =12

∫dx κ(φ)

(∇2h

)2 (2.2)

Nous avons fait un développement perturbatif en cumulants à partir du hamiltonientotal exact H[φ, h] = HI [φ, h] + HS [φ, h] autour du champ moyen φ0, dans la limite oùκ varie faiblement avec φ. Dans ce cas, nous avons montré qu’une interaction, de natureentropique, apparaît entre domaines de lipides semblables, v(r), induite par les fluctuationsde la membrane. Cette interaction peut être attractive ou répulsive selon les caractéristiquesdu système (potentiel effectif, interactions de van der Waals). Ainsi, à grandes distances,c’est-à-dire pour r ξ où

ξ =

√κ(φ0)V (φ0)

(2.3)

est de l’odre de 10 à 100 nm pour les membranes lipidiques, cette interaction est dominéepar le potentiel V et une forme approchée est

v(r) ≈ −kBT4π2

[V ′(φ0)V (φ0)

]2 1r4

(2.4)

qui est toujours attractive. Cette interaction à longue portée est semblable à l’interactionde Casimir, VC = −kBT 6π2(r0/r)4 où r0 est le rayon des inclusions [79, 148], et quis’applique entre deux inclusions dans une membrane sans tension. À plus courte portée,l’interaction est dominée par la courbure et se comporte comme

v(r) ≈ −αkBT8π2

[κ′(φ0)κ(φ0)

]2 1(rξ)2

(2.5)

où α = V ′(φ0)κ(φ0)/[V (φ0)κ′(φ0)] peut être négatif ou positif (conduisant donc à uneinteraction respectivement répulsive ou attractive). La nature de cette interaction dépenddonc de la géométrie des lipides A et B ainsi que de l’énergie surfacique par tête lipidique.Par exemple une bicouche faite de DPMC 1 a un module de courbure de l’ordre de 25kBT etqui est multiplié par 10 lorsqu’on y ajoute du cholestérol dans les mêmes proportions [65].Dans la situation présentée ci-dessus où la membrane est en contact avec un réservoir delipides, on trouve en minimisant l’énergie libre par rapport à φ0, α = −πkBT/(2εa) oùεa est l’énergie surfacique moyenne d’un lipide et l’interaction est répulsive aux distancesinférieures à 100 nm.

1. Dimyristoylphosphatidylcholine.

81


Dans le cas où cette interaction est répulsive à une échelle intermédiaire, bien supé-rieure à la taille d’un lipide, elle peut conduire à la formation de mésophases de taillefinie (de l’ordre de ξ). Il s’agit là d’une explication possible de la formation de domainesmésoscopiques dans certains systèmes, comme par exemple les radeaux lipidiques observésdans les membranes cellulaires.

2.1.1 Article

Suit l’article :D.S. Dean et M. Manghi, Fluctuation-induced interactions between domains in membranes,Physical Review E 74 021916 (2006) (9 pages)

82

Fluctuation-induced interactions between domains in membranes

D. S. Dean and M. ManghiLaboratoire de Physique Théorique, UMR CNRS 5152, IRSAMC, Université Paul Sabatier, 118 route de Narbonne,

31062 Toulouse Cedex 04, FranceReceived 1 December 2005; revised manuscript received 3 July 2006; published 18 August 2006

We study a model lipid bilayer composed of a mixture of two incompatible lipid types which have a naturaltendency to segregate in the absence of membrane fluctuations. The membrane is mechanically characterizedby a local bending rigidity which varies with the average local lipid composition . We show, in the casewhere varies weakly with , that the effective interaction between lipids of the same type either can beeverywhere attractive or can have a repulsive component at intermediate distances greater than the typical lipidsize. When this interaction has a repulsive component, it can prevent macrophase separation and lead toseparation in mesophases with a finite domain size. This effect could be relevant to certain experimental andnumerical observations of mesoscopic domains in such systems.

DOI: 10.1103/PhysRevE.74.021916 PACS numbers: 87.16.Dg, 82.70.Uv

I. INTRODUCTION

At the simplest level biological membranes are modeledby homogeneous flexible bilayers of amphiphilic lipid mol-ecules 1,2. However, in many physical and biological situ-ations, these membranes are inhomogeneous on some micro-scopic scale. Indeed, four major distinct lipid types aretypically present in mammalian cell membranes 3. It isnatural to ask what may be the role of this homogeneity inthe biological context and how it influences the mechanicalproperties of the cell. The interplay between the lipid com-position and membrane fluctuations has been addressed inmany recent studies. The local composition of the membranewill clearly affect its fluctuations and local geometry. Indeed,the coupling between membrane fluctuations and local com-position is at the origin of the budding instability 4,5 seenin certain systems. On the other hand, membrane fluctuationswill also influence its local composition. In this paper wewill examine how the coupling of membrane fluctuations tolocal composition can affect the phase ordering of its com-ponent lipids.

In previous works, the way in which the fluctuation-composition coupling is incorporated into the overall freeenergy of system falls into two main classes. i The mem-brane is composed of a homogeneous lipid background withadded insertions such as trans-membrane proteins and at-tached polymers. ii The membrane is modeled as a multi-component system with several lipid types and where themechanical properties of the system are dependent on therelative local concentrations of the various lipid types.

The insertions considered in models of class i modifythe membrane fluctuations via several different mechanisms.First pointlike inclusions, such as polymers, exert a pressuredistribution on the flexible membrane. This involves a cou-pling of the membrane composition, in this case the densityfield of the inclusions, to the height hx over the projectedarea of the membrane. Another possible coupling is via animposed boundary condition on the height field h at theboundary between the inclusions and the membrane. For ex-ample, the contact angle at the boundary can be taken to befixed in order to minimize the hydrophobic free energy of the

insertion. This is an example of a hard constraint. Alterna-tively, one can introduce a general coupling tensor, related tothe orientational degrees of freedom of the inclusions, to thelocal strain tensor i jh, from which the curvature tensorcan be extracted. This then corresponds to an energetic termwhich induces a preferred local curvature. In the literatureseveral types of inclusions are considered: circular 6–8,elliptic 9, more general 10–13 embedded inclusions, aswell as adsorbed cylinders 14. Besides introducing a ten-dency for a spontaneous local curvature, which breaks theup-down symmetry of the system, inclusions may alsomodify the energy associated with terms quadratic in thecurvature tensor. For example, isotropic inclusions maymodify the local bending and Gaussian rigidities of the mem-brane. In the case of two inclusions one may then explicitlyevaluate their effective interaction. To summarize, the den-sity field of the inclusions in all these cases is coupled via ah in the case of insertions exerting a pressure, b an effectivevectorial ih coupling in the case of imposed boundary con-ditions at the inclusion frontier with the membrane, c atwo-tensor coupling to i jh when there is a locally pre-ferred curvature tensor, and finally d a coupling toi jhklh, when the local bending and Gaussian rigiditiesare modified by the inclusions and also when nonisotropiceffects are present. The above are the most physically rel-evant couplings up to quadratic order and consequently arethe most significant in systems where the height fluctuationsare relatively small.

In this case of models of type ii, the variation of theelastic properties of the membrane is more continuous thanin the case of inclusions. If one neglects the possibility ofnonisotropic effects, the most natural parameters that willvary with local lipid composition are the bending rigidity ,the Gaussian rigidity , and the spontaneous local curvaturec. For instance, a concentration-dependent spontaneous cur-vature is considered in 15–18. Linear perturbations to boththe bending rigidity and the spontaneous curvature are stud-ied in 19,20. In 6, linear perturbations to the bending andGaussian rigidity were considered; the interaction arising inthis case is proportional to 1/r4 and the prefactor is given bythe product of coefficients of the linear deviations from the



average values of and . The induced interaction may thusbe attractive or repulsive depending on the sign of these co-efficients. Of course models of type i, with discrete inclu-sions, can be described by models of type ii when it makessense to take a continuum limit for the inclusions. This limitwill be valid for inclusion sizes that are comparable to themicroscopic length scale of the membrane, that is to say, thelateral lipid size.

All of the studies mentioned above assume zero surfacetension. However, the study of membranes under tensionsheds light upon the physics of biological membranes whichare not truly at equilibrium but under external constraints orperturbations. The surface tension can be due to electrostaticinteractions with the aqueous solvent or due to the presenceof molecular protrusions. Furthermore, the external action oflaser tweezers on a vesicle attracts phospholipids and putsthe membrane under tension. This leads to interesting phe-nomena such as pearling instabilities 21. It has been theo-retically shown that the presence of surface tension can in-duced a repulsive interaction between inclusions of the sametype 22,23. The model used in 22,23 is of type i and isbased on a linear coupling of the inclusion to the height ofthe membrane, for example, to model the local pressure ex-erted by an attached polymer. The interaction is sensitive tothe strength of membrane-inclusion coupling. In this system,the up-down symmetry of the membrane is clearly broken bythe linear coupling. Indeed, in many biological situations theup-down symmetry of the membrane is clearly broken, forinstance, by different compositions in the top and bottomleaves or by the presence of conical trans-membrane inclu-sions. However, it is interesting to ask if the presence ofsurface tension can also lead to repulsive interaction betweendomains, with similar lipid composition, even when the up-down symmetry is conserved.

The physics of phase separation may play an importantrole in biological systems. It has been experimentally shownthat erythrocyte membranes which contain many differentlipid types form immiscible two-dimensional liquids, whichare very close to the miscibility critical point 24. The re-sulting thermodynamic forces can affect the mechanicalproperties of the membrane and in particular its shape. How-ever, in turn the fluctuations will also affect the distributionof the components in the membrane. As an example, a long-range fluctuation mediated repulsion between inclusions,combined with a short-range van der Waals attraction, couldlead to the formation of mesoscopic domains 16,25 of theinclusions. It has also been shown that the presence of asurface tension modifies the effective interactions betweenconical inclusions 26; inclusions of the same type are al-ways repelled but oppositely orientated inclusions interactattractively at long distances and then repel at shorter dis-tances. This is in contrast to the case where there is no ten-sion when all interactions are always repulsive.

In this paper, we consider a two-component bilayer withthe up-down symmetry and, in general, with a nonzero sur-face tension. We show that for certain variations of the bend-ing rigidity and the local surface energy the composition-independent component of which can be interpreted as asurface tension with the local composition in lipids, afluctuation-induced lipid-lipid repulsive interaction can ap-

pear between domains of similar composition. This, togetherwith a short-range van der Waals attraction, can induce theformation of mesophases. In the scheme of previous models,our model falls into the class of type ii above and ourcumulant expansion method is similar to that used in6,19,20. In our study we add a nonzero surface energy, asin 22,23, but where this local surface energy fluctuates withthe local lipid composition.

The paper is organized as follows. In Sec. II we presentour field-theoretical model. In Sec. III using a cumulant ex-pansion for small height fluctuations we calculate the in-duced interaction; this rather technical section may beskipped by a reader interested only in the physical conse-quences of the calculation. In Sec. IV the general physicalproperties and asymptotic behavior of this effective interac-tion are discussed. Section V is devoted to a description ofthe results, which are compared to previous studies. In addi-tion we suggest a possible experiment where the effects pre-dicted here could possibly be seen.

II. FIELD-THEORETICAL FORMULATION

We consider a model membrane with two lipid types Aand B and where the top and bottom leaves have the samelipid composition. In the most frequent case, at least when-ever van der Waals interactions are dominant, it is energeti-cally favorable for lipids of the same type to be adjacent. Inthis case, we can write down a typical attractive energy persite E=AB where A and B are the liquid volume frac-tions of lipids A and B and 0 is a Flory parameter relatedto the electronic polarizabilities of both molecules. We willconsider a coarse-grained model for a field related to thelocal surface fraction of the two lipid types, i.e., =A−B, which in the absence of surface fluctuations exhibits acontinuous phase transition at sufficiently low temperatures.The theory is then described by the Ginzburg-Landau Hamil-tonian 16

HI = gd2x J

2giji j + V , 1

which is written in a covariant form that ensures the inde-pendence of the energy from the choice of the two-dimensional coordinate system denoted by x. The parameterJ is positive and related to the Flory parameter ; it is aferromagnetic interaction and energetically favors lipids ofthe same type being next to each other. The potential Vfixes the two characteristic values of and the global com-position via chemical-potential-like terms. As the potential Vappears in H simply integrated over the area of the mem-brane, it can be interpreted as a composition dependent-contribution to the surface energy of the membrane. Indeed,the constant part of V which is V0 can be interpreted as asurface tension because it is coupled to the total physical areaof the membrane g d2x. The term V0 can thus be used asa Lagrange multiplier to fix the physical membrane area. Asmentioned above, V will have a dependence as in the usualLandau models for phase-separating systems. As in standardLandau theory, we will assume that V is a single well at high

D. S. DEAN AND M. MANGHI PHYSICAL REVIEW E 74, 021916 2006

021916-2

temperature and a double well at low temperature. Thismeans that the system on a plane will exhibit a continuousphase transition. At the mean-field level, this transition oc-curs when the mass associated with this field theory, givenby M0

2=V0, vanishes where 0 is the homogeneousmean-field solution. This transition exhibits a divergent cor-relation length and corresponds to a macrophase separationwhich occurs at a critical temperature T=Tc.

In the above, the metric of the membrane surface is de-noted by gij and in the Monge gauge it is given by

gij = ij + ih jh 2

where h is the height of the surface above the projectionplane whose area we will denote by A. The term g denotesthe determinant of gij and is given by

g = 1 + h2. 3

Hence the Hamiltonian given by Eq. 1 already implicitlyincludes a coupling between the local composition, as en-coded by , and the membrane fluctuations, as encoded by h.The interface energy or line tension, which corresponds tothe term quadratic in the gradient, is written as to ensurecovariance; gij is the inverse of gij and is given by

gij = ij −ih jh

1 + h2 . 4

Here we note that the fact that one should use the covariantform of the line tension is often forgotten in the literature. Tolowest, i.e., quadratic, order in the fluctuations h, one has

HI,h = d2x J

22 + V + d2x J

42h2

−J

2 · h2 +

1

2h2V . 5

We now take into account the elastic energy of the mem-brane so the total Hamiltonian of the system is given by

H,h = HI,h + HS,h . 6

The Hamiltonian for surface fluctuations will be taken to be

HS,h =1

2 d2x 2h2, 7

which is the simplest Helfrich Hamiltonian for surface fluc-tuations 1 and correspond, strictly speaking, to the firstterm in a 2h expansion of the mean curvature 2. ThisHamiltonian corresponds to a bending energy with localbending rigidity which depends on the local compositioncharacterized by . The two-dimensional membrane systemis assumed to have no spontaneous curvature and thus has anup-down symmetry. More generally, one could also include acomposition dependence on the Gaussian rigidity; the contri-bution coming from this term would then cease to be a to-pological invariant and should strictly be included.

The effective partition function in the presence of mem-brane fluctuations is given by

Z = ddhexp− H 8

where −1=kBT is the thermal energy scale. We recall that Ais the projected area of the membrane; the physical area ofthe membrane is denoted by A+A, where A is oftencalled the excess area. For typical biological membranes,A /A is small, of the order of a few percent, and we willthus legitimately assume, in the rest of the paper, that heightfluctuations are small compared to the typical length scale ofthe system.

III. CALCULATION OF THE FLUCTUATION-INDUCEDINTERACTION

In this section we explicitly calculate the fluctuation-induced interaction to second order in the cumulant expan-sion.

In the high-temperature regime, lipids form a mixed phasecharacterized by a homogeneous and uniform composition0, with fluctuations about 0. In an ensemble where theaverage value of is fixed we write =0+ where 0=A−1d2x x. Consequently in this case, we haved2x x=0. By assuming that and V behave continu-ously around 0, we expand the total Hamiltonian 6 up toO2 in the fluctuations. This leads to

H0,,h = AV0 + H0I 0, + H0

S0,h + H0,,h ,

9

H0I =

1

2 d2xJ2 + V02 , 10

H0Sh =

1

2 d2x02h2 + V0h2 . 11

When the term proportional to V0 is included in the sur-face Hamiltonian H0

S, as we have chosen to do above, V0can be interpreted as an effective elastic energy. However,because it is constant, V0 can be interpreted as an effec-tive surface tension. The part of the Hamiltonian which wewill treat perturbatively is

H0,,h =1

2 d2xJ1

22h2 − · h2

+ V0h2 + 02h2 +1

2V0

h2 + 02h22 . 12

The scheme of the calculation is just slightly different in thecase where the value of 0 is allowed to fluctuate but nothingintrinsically changes.

We perform a cumulant expansion in the partition func-tion 8 as follows

FLUCTUATION-INDUCED INTERACTIONS BETWEEN¼ PHYSICAL REVIEW E 74, 021916 2006

021916-3

Z Z0S exp− AV0 dexp− H0

I 1 − HS

+2

2 H2S 13

Z0S exp− AV0 dexp− Heff0,

14

where Z0S=dhexp−H0

S and OS= Z0S−1dhO exp

−H0S. The cumulant expansion at this order is clearly ex-

act to O2. The effective interaction at this order is thusgiven by

Heff0, = H0I 0, + Hc

S −

2 H2c

S 15

where the subscript c indicates that it is the connected part ofthe correlation function.

Note that only the first term in the cumulant expansioncan lead to a quadratic term in ; however this term can beseen to be zero by the following:

1

22h2 − · h2

c

S

=1

22 h2c

S − i j ih jhcS

=1

22 h2c

S − i jij1

2 h2c

S = 0,

16

where we have appealed to the isotropy of the system. Thereis therefore no renormalization of the coupling J. Also in thefirst term of the cumulant expansion, terms linear in cancelby definition of as they are integrated against a constantby isotropy and the remaining terms yield Hc

S

= 12 d2x M1

22 where the mass M1 is given by

M12 =

1

2V0 h2c

S + 0 2h2cS . 17

Again to quadratic order in , the second-order term in thecumulant expansion yields

−

2 H2c

S =1

2 d2x d2y xUx − yy 18

where

Ux − y = −

420 2hx22hy2c

S

+ 2V00 2hx2hy2cS + V20

hx2hy2cS . 19

The potential Ux−y is nonlocal and characterizes the in-duced interaction mediated by height fluctuations. The vari-ous connected correlation functions above are evaluated as

2hx22hy2cS =

2

2 4Gx − y2,

hx2hy2cS =

2

2ij

i jGx − y2,

2hx2hy2cS =

2

2 2Gx − y2. 20

Here the Green’s function G is given by

G = 04 − V02−1 =1

V0G0 − Gm , 21

where

=1

m=0

V0. 22

The intrinsic length is usually in the range 10–100 nm forbiological membranes. The Green’s function G0 is G0

=− 12 ln x

L L is an arbitrary length and Gm is the Yukawainteraction given by

− 2Gx + m2Gmx = x . 23

In two dimensions one has

Gmx =1

2K0mx , 24

where K0x is the Bessel function of the second kind oforder 0. Using these results we find that

Ux = Bx + vx . 25

The first term of the right-hand side RHS is short rangedwith B=−m4 /20 /020−2m2Gm0 andneeds to be regularized via an ultraviolet cutoff =2 /acorresponding to a microscopic length scale a which wouldbe of the order of the distance between lipid heads. Thesecond term v of the rhs above is the long-range inducedinteraction and is independent of the ultraviolet cutoff. It isgiven by

vr =m4

20

02

u , 26

u = −1

42K02u + 2K0

2u + 2K0u −1

u22

+1

u2K0u +1

u2 27

where u=mx=mr and

=V0V0

00

. 28


021916-4

IV. PROPERTIES OF THE FLUCTUATION-INDUCEDINTERACTION

Here we discuss the features of the interaction derived inSec. III. First, the strength of this interaction is polynomial inm and therefore reducing the local surface or elastic energyand increasing the local bending rigidity reduces the interac-tion energy considerably. This same reduction, however, alsoincreases the range of the interaction. Second, the interactionstrength is set by kBT, which means that this interaction hasan entropic origin.

If is positive, we see that the interaction between areashaving the same lipid type with the same sign of is al-ways attractive. To see this, we note that in the expression forvu, Eq. 26, the coefficients of 0, , and 2 are functionsof u that are positive and monotonically decreasing, henceyielding an attractive interaction at all distances. In a ferro-magnetic analogy where the lipid types are characterized bya field having the values concentrated about ±1, this cor-responds to a long-range ferromagnetic interaction which en-hances the short-range one already present. However, when is negative, v now has a repulsive component, correspond-ing to the coefficient of , which could prevent macrophaseseparation.

For large u we have 27

K0u

2uexp− u 29

and thus at large distances v behaves as

vr −kBT

42V0V0 2 1

r4 . 30

This 1 /r4 interaction, which is always attractive, is typicallyseen between inclusions of the same type in membraneswithout surface tension and is found in many of the studiesdiscussed in the Introduction. However, this 1 /r4 attractiondoes not have the same physical origin as in previous studiesbecause, as can be seen by examining the prefactor, it isgenerated solely by the fluctuations of the surface energy.

For small r we find

vr − kBT

8200

2m2

r2 + Oln2 r

−kBT

42

0V020

1

r2 + Oln2 r 31

which is again attractive if 0 but is repulsive when 0 or 0V00. In this last case, the overall inter-action is somewhat frustrated: it is attractive at very shortlength scales of the order of the microscopic length scaledue to van der Waals interactions in our model representedby the local ferromagnetic interaction, together with alonger-range membrane-mediated repulsion over intermedi-ate length scales, before becoming attractive at longer lengthscales. One can suppose that the occurrence of these attrac-tive and repulsive interactions can prevent macroscopicphase separation and lead to mesoscopic domains.

V. MICROPHASE SEPARATION

The final effective quadratic Hamiltonian can now bewritten as

Heff0, =1

2 d2xJ2 + M2

22

+1

2 d2x d2y xvx − yy , 32

where the corrected mass is given by M22=M0

2+M12+B. It is

important to note that this mass depends on the microscopiccutoff a since M1

2 includes the expectation values 2h2S

and h2S which diverge and must thus be regularized,and a similar regularization is needed to evaluate B. As al-ready explained in the Introduction, we consider systemssuch that, in the absence of height fluctuations, when M0→0 the system exhibits a second-order phase transition withdiverging correlation length 0=J /M0. In Fourier space, wefind

Heff0, =1

222 d2qJq2 + Me2 + wqq− q ,

33

where q= q and the Fourier transform and its inverse aredefined by

fx = d2q

22 fqexpiq · x , 34

fq = d2x fxexp− iq · x . 35

In the Fourier representation the nonlocal part of the interac-tion is given by wq= vq− v0 and Me

2=M22+ v0 thus

gives the effective mass for the theory.The stability of the homogeneous solution against phase

separation is determined by the lipid-lipid correlation func-

tion in Fourier space qq= 22q+qSq wherethe structure factor is

Sq =kBT

Jq2 + Me2 + wq

. 36

Defining

= 00

2

, 37

we find three dimensionless parameters in the structure factor

Sq =Sq22 =

1

2J/e2 + q2 + Wq38

where q=q, which are , J, and e2=J /Me

2. The Fouriertransform of the dimensionless potential is


021916-5

Wq = 20

u duJ0qu − 1 u , 39

where J0x is the Bessel function of the first kind of order 0.A divergence of the structure factor Eq. 38 at q=0, while Sremains finite for q0, signals a macrophase separation. Inthis case, since we find S0=e

2 / J, the phase separationoccurs when the correlation length e→ and this corre-sponds to the case where the induced interaction does notchange the nature of the transition but only changes where it

occurs. We show the behavior of Wq and Sq for differ-ent values of in Fig. 1. When 0, Wq and its firstderivative are always positive, and the only maximum of thestructure factor occurs at q=0.

However, when 0 we see that for particular values of

J and /e, Sq reaches a maximum at an intermediatevalues of q and this maximum can even diverge at a nonzerowave vector q*. In this case, the homogeneous solution be-comes unstable before e→ which leads to the formationof mesophases mesoscopic phase separation with a finitecharacteristic length scale given by / q*. Note that at largevalues of q, we have Wq ln q but this short-rangecomponent of the induced interaction is dominated by theshort-range van der Waals interaction term whose strength iscontrolled by J, and Sq ultimately decreases as 1/ q2 forlarge q. The maximum of the structure factor diverges for anintermediate wave vector q* which is implicitly defined bythe two following equalities:

Wq* + 4Jq* = 0, 40

Wq* + 2Jq*2 + /e2 = 0, 41

i.e., when the parabola −2Jq2+ /e2 is tangent toWq. Given the number of parameters in our theory theevaluation of a complete phase diagram is not feasible; how-ever, the fundamental question we wish to address is whetherthere is a macrophase or microphase separation. To do thiswe can examine the structure factor at the point where theq=0 mode becomes unstable, that is to say, where the effec-tive mass Me=0. This is thus equivalent to examining tem-

peratures Tc which are critical in the true sense. If the modesq0 are stable at Tc then we expect to see the macrophaseseparation. However, if at Tc there is already a mode q0which is unstable then a microphase separation must havealready occurred at a temperature TTc. Thus, without hav-ing to specify the full theory, we can identify when a mac-rophase separation is converted to a microphase one due tocoupling between membrane fluctuations and its composi-tion.

Since we are interested in the behavior of the structurefactor when approaching the macrophase transition e

→; we calculate the onset of the microphase separationgiven by Eqs. 40 and 41 for /e=0, i.e., at the criticaltemperature Tc. The result is shown in Fig. 2: the gray regioncorresponds to the region of the phase diagram J ,where mesophases appear whereas the white region corre-sponds to macrophase separation. The solid line correspondsto the solution of Eqs. 40 and 41 the dots are the exactsolutions. This “phase diagram” is plotted at a fixed lipidcomposition 0 and fixed temperature T=Tc correspondingto the critical point in the ,T space. It is important to notethat when the temperature deviates from the critical tempera-ture, the correlation length e becomes finite but very large,and the gray region delimited by the solid line shrinks. How-ever, in the case where, for given lipid types, the parametersJ and lie in the gray region, mesophases appear beforethe macrophase separation at a temperature TTc. In theextremal situation where we are far from the macrophaseseparation the region of the phase diagram corresponding tomesophases disappears completely.

For parameter values belonging to the gray region of the

diagram, the structure factor Sq diverges before q=0 at afinite value q*. When moving along the solid line starting atthe origin, the value of q* decreases until we reach the pointJ=3.810−3, =−0.557 where q*=2.095, which is thesmallest value and corresponds to a characteristic lengthscale of 5–50 nm for the mesophases. Then q* increasesagain when increases.

FIG. 1. Plot of Wq for various values of : 0.05 dotted line,−0.1 broken line, and −0.15 solid line.

FIG. 2. Theoretical phase diagram of the bilipidic fluctuatingmembrane in the plane J , at the onset of the macrophaseseparation, i.e., at the critical temperature Tc. The gray region cor-responds to the parameter range for which a phase separation occursat a nonzero wave vector q* leading to the formation of mesophasesinstead of a macrophase separation.


021916-6

On the experimental side, the parameters and J arenot easy to determine. The phenomenological parameter Jcoming from the Landau-Ginzburg theory is somehow re-lated to the van der Waals attractions between lipids andvaries roughly as 1/T such that J is fixed for given lipidtypes and independent on temperature. Although we do notknow exactly the value of this parameter, we can assumeJ1. For lipidic vesicles made of a mixture of two verydifferent lipids such as dimyristoylphosphatidylcholineDMPC/cholesterol a long lipid and a short one, the cur-vature modulus has been experimentally measured 050kBT 26kBT with DMPC alone and increases when theproportion of cholesterol increases up to 250kBT with 50% ofcholesterol 28,29. Hence we find J0.01, which meansthat the mesophases region of the phase diagram can be ex-perimentally reached in such systems.

Finally, in the mesophase region, the denominator of Sqgiven by Eq. 36 becomes negative and the calculation ofthe preceding sections, based on quadratic fluctuations of is no longer valid the modes qq* are unstable. The natureof the resulting stable mesophase requires further analysis todetermine it and this is beyond the scope of the present pa-per.

VI. DISCUSSION AND CONCLUSIONS

In this paper we have shown that the coupling of mem-brane composition via a composition dependence on the lo-cal surface energy and bending rigidity can alter the phasediagram of a membrane composed of a mixture of differentlipids. Indeed, depending on the physico-chemical propertiesof the lipids for instance by modifying the length of theirhydrophobic tail, the membrane can exhibits a microphaseseparation leading to the formation of so-called mesophasesat a temperature TTc, i.e., before an eventual macrophaseseparation. In the previous works where surface tension wasconsidered the composition-fluctuation coupling was linearand the up-down symmetry of the system thus broken.

The long-range part of the interaction given by Eq. 26behaves as −1/r4 for large r. This interaction has the samebehavior as that found between inclusions in several modelswhere surface tension is not present. For instance, in tension-less membranes one finds the effective pairwise interaction7,9

VCr = − kBT62 r0

r4

42

between circular inclusions with the up-down symmetry,where r0 is the radius of the inclusions. The long-range partof the interaction found in Eq. 26–30 is also proportionalto the thermal energy but it is solely due to fluctuations in thesurface or elastic energy.

A few works focused on the effect of the surface tensionon fluctuation-induced interactions 18,22,23,30. Calculat-ing the potential between two circular inclusions which lo-cally apply a pressure on the membrane, Evans et al. foundan interaction that is everywhere repulsive 22 between in-clusions of the same type and is given by

r =12

2K0mr , 43

where i is related to the force distribution of inclusions iacting on the membrane surface and is the surface tension.Here again, this interaction is different from Eq. 26 in ori-gin but has some similar features: it is present for mem-branes under tension and is repulsive with a typical range of30 nm for biological membranes. Our model is very dif-ferent; it does not assume any pressure distribution acting onthe membrane but relies on the behavior of and Vclose to a liquid-liquid immiscibility critical point. Thisproximity to a liquid-liquid immiscibility critical point in areal biological context is supported by beautiful experimentson monolayers made of lipids extracted from erythrocytes24.

In this study we have seen that the induced interactiononly has a repulsive component when 0. Qualitatively,this means that the signs of 0 and V0 are opposite:when a region is locally enriched for instance in lipid A,bending rigidity increased 00 whereas the effectivesurface tension decreases V00. Let us consider for amoment the mean-field theory where one neglects the fluc-tuations about 0. Consider an incompressible membranewhich is constrained to have a constant projected area, forexample a membrane supported by a frame. Also let themembrane exchange lipid species with the bulk solutionaround it 32. The mean-field free energy as a function of 0is given from Eq. 14 as

F0A

= V0 +1

4

0

k dk ln0k4 + V0k2 .

44

This mean-field free energy must be regularized by the ultra-violet cutoff . As the membrane is in a solution containinga reservoir of lipid species, 0 is not fixed but is thermody-namically selected so as to minimize the mean-field free en-ergy. In this case, in our previous treatment we should havethus included a term V0 in the expression for H0

I ; how-ever, this term can be seen to cancel exactly with the firstterm of the cumulant expansion, which in this case is alsonow no longer zero. The part of the free energy F* that varieswith 0 is given by

F*0A

= V0 +1

82 ln0 +

V02

+V00

ln02

V0+ 1 . 45

The calculation carried out in this paper is valid for smallsurface fluctuations; a way of ensuring that the fluctuationsare small is by choosing a very stiff membrane. This can beensured by taking large. The equation minimizingF*0 can be expressed as


021916-7

V01 +1

4

0

k dk1

0k2 + V0+

04

0

k dkk2

0k2 + V0= 0. 46

Now physically we must have V00, as for any ef-fective surface tension it is necessary to have m real; thisresult implies that the system will naturally be in the regionwhere 0. Now it is straightforward to show see, forexample, 31 at the mean-field level used here that the ratioof the excess area to the projected area is given as

A

A=

1

4

0

k dk1

0k2 + V0

=1

80ln02

V0+ 1 . 47

In terms of the ratio of the excess to projected area, Eq. 46can now be written as

V01 +A

A +

00 2

8−

V0A

A = 0.

48

In the limit where A /A is small, using Eq. 28 we obtain

= −2

8V0. 49

Now if we write =2 /a where a is the microscopic lengthscale we find that

= −kBT

2a, 50

where a is the surface energy of a square of the membraneof linear dimension a, i.e., the average surface energy perlipid. Hence we find 0 which suggests a scenario to ob-serve the formation of mesophases experimentally. For in-stance one can use a membrane composed by a mixture ofDPMC and cholesterol and supported by a frame close at atemperature close to Tc.

In a more general context, molecular dynamics 33 andMonte Carlo simulations 34 have shown that the bendingrigidity has a nonmonotonic behavior as a function of theshort-lipid number fraction xs: it first decreases rapidly forsmall xs and then increases slowly, with a minimum aroundxs0.6. These studies suggest that for a two-component bi-layer made of short and long lipids, the gradients of andV could have opposite signs but some tuning may berequired. In this case the effective interaction will have arepulsive component which could induce mesoscopic phaseseparation.

The issue of mesophase formation has been discussed inseveral papers. Taniguchi 16 has shown in a model with alinear coupling of the composition to the mean curvature

that near-spherical vesicles with off-critical compositions ex-hibit circular domains that closely resemble patterns ob-served in red blood cell echinocytosis 24.

A similar study has been carried out in different geom-etries 17 and the same general phenomena are observed.Inspired by the problem of pattern formation of quantumdots at the air-water interface, Sear et al. 25 have studiedthe effects of a short-range attraction on top of a shorter-range hard core and long-range repulsion in Monte Carlosimulations of two-dimensional systems of interacting par-ticles. In their simulations both circular domains and stripeswere observed as is the case in the experiments.

Finally, by adding an attractive short-range interaction tothe potential Eq. 43, Evans et al. have argued that me-sophase formation 22 could be induced. Hence, it couldexplain the formation of caveolae buds from cell membranesand their striped texture. The mechanism proposed in thispaper of course leads to the same phenomenology in the casewhere the effective potential induced by membrane fluctua-tions has an intermediate range repulsive component. How-ever we do not find any repulsion in the situation where 0 which implies some conditions on the membrane com-position which could perhaps be tested experimentally.

The model presented in this paper can be generalized byconsidering lipid distributions without the up-down symme-try, i.e., with different compositions in the top and bottomleaves. In this case, one would introduce a composition-dependent spontaneous curvature c in the Hamiltonian. Ifone assumes that the mixed homogeneous phase has nospontaneous curvature then one takes c0=0 and in thiscase the correction to the long-range interaction is

v*r = −V0

2c02K0mr 51

and the mass is renormalized by a repulsive term. Hencethis correction is attractive and could wipe out the aboverepulsive effect. The two-component membrane could alsocontain trans-membrane proteins. Despite the fact that therepulsive interaction between inclusions described by Evanset al. would appear, it is well known that protein aggregationalso increases the local lipid composition, as observed inerythrocyte membranes where it induces a phospholipid en-richment 35. The inclusion of proteinlike insertions in thistwo-lipid model could thus produce quite rich behavior andis a line worth pursuing.

Our study has predicted that it is possible that a mem-brane whose fluctuations are impeded exhibits a macrophaseseparation whereas if it is allowed to fluctuate freely thistransition becomes a mesophase separation. In a stack ofmembranes the fluctuations are suppressed by Helfrichforces 36 which are of steric origin. Experimentally, there-fore, one could prepare a stack of bilayers at a lipid compo-sition where the bilayers within the stack exhibit a mac-rophase separation. However, according to our predictions, asingle membrane could possibly exhibit a mesophase sepa-ration 37. Another possibility is that one could try to ob-serve the effect predicted here by using charged membranes


021916-8

and then varying their rigidity by changing the bulk solu-tion’s salt content 38.

We emphasize that, in this paper, we have concentrated onan entirely equilibrium mechanism as a possible explanationfor the formation of mesoscopic domains. However, in living

cells, out-of-equilibrium effects are of course important. Re-cently the recycling of lipids between the membrane and cellinterior has been put forward as a nonequilibrium mecha-nism for the formation of raftlike structures in active systems39,40.

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168301 2005.


021916-9


2.2 Membranes empilées

Afin de comprendre plus en détail les mécanismes de confinement dynamique et l’or-ganisation des membranes, il apparaît crucial de travailler sur des membranes modèles.Travailler sur un système modèle permet en effet de s’affranchir de la très grande com-plexité d’une cellule afin d’isoler un mécanisme physique particulier et de comparer plusaisément théorie et expérience. L’équipe de Laurence Salomé à l’IPBS travaille en ce mo-ment sur un tel projet, lequel consiste à empiler deux (voire plusieurs) bicouches lipidiquessupportées pour s’affranchir des interactions, notamment hydrodynamiques, avec le sub-strat. Des protéines membranaires peuvent en outre être incluses dans la dernière bicouche,de manière contrôlée (figure 2.2).

Nous avons, avec Nicolas Destainville, entrepris de modéliser de tels systèmes de bi-couches empilées sur un substrat et en particulier le cas le plus simple de deux bicouches em-pilées. Les interactions entre les bicouches et entre le substrat et la première bicouche sontnon-linéaires afin de pouvoir décrire les transitions de décrochage (ou «unbinding») [118].Nous avons choisi un potentiel de Morse

VMorse(z) = D[e−2α(z−d) − 2e−α(z−d)

](2.6)

qui décrit bien à la fois la répulsion à courte portée associée à la répulsion due au couchesd’hydratation près des membranes, et l’attraction de type van der Waals (éventuellementécrantée) à longue portée. Le hamiltonien du système complet avec la contributions élas-tique d’Helfrich s’écrit alors

H =∫

dr2∑

i=1

12

[σ (∇hi(x))2 + κ

(∇2hi(x)

)2]+ Vi[hi(x)− hi−1(x)]

(2.7)

où κ et σ sont le module de courbure et la tension de surface (supposés identiques pourles deux membranes) et hi la hauteur de la membrane i.

Nous avons étudié, par une approche variationnelle gaussienne (voir l’article [125] inclusci-après), la transition de décrochage des bicouches lorsque la température croît, et mis enévidence des diagrammes de phases très riches. Un de ces diagrammes est représenté surla figure 2.2 et détermine les régimes de paramètres pour lesquels l’empilement des deuxmembranes sur le substrat subsiste. Les distances ainsi que la force d’interaction effectiveentre les membranes ont été calculées en fonction de la température T . Lorsque T augmente,les fluctuations 〈h(x)2〉 augmentent de façon non-linéaire, jusqu’à diverger à la températurede décrochage, Tu. Ainsi, lorsqu’on s’approche de la transition de décrochage de la secondemembrane, celle-ci s’éloigne et s’affranchit de ce fait des interactions hydrodynamiques avecl’autre membrane. Nous montrons comment cette transition est contrôlée par les potentield’adsorption entre le substrat et la première membrane. La connaissance de l’évolution

92

2.2. Membranes empilées

(a)

organisation fonctionnelle de la membrane demeure un des grands defis de la biologie cellulaire.Les di!erentes proteines ayant des a"nites a courtes portees modulees par leurs proprietes

physico-chimiques, elles peuvent etre classees en q familles, une famille regroupant des proteines, pasnecessairement identiques, appelees a etre co-localisees dans les memes nano-domaines. La di!erenced’a"nite entre proteines de la meme famille et proteines de familles di!erentes est mesuree par unparametre d’interaction de Flory, !. J’ai alors montre que pour ! superieur a une valeur critique !c,la competition entre entropie de melange et attraction a courte portee favorise des clusters essentiel-lement monocolores (figure 7). La preuve utilise une theorie champ moyen a la Flory-Huggins, quipresente l’avantage de devenir exacte a la limite des grands q d’interet4.

Mon resultat principal est que ! croıt lentement avec q (! ! ln q), de sorte que meme si plusieursmilliers de familles coexistent, une di!erence d’a"nite de 2 ou 3 kBT su"t a segreguer les famillesdans des clusters distincts. Ce mecanisme est donc realiste du point de vue physique et rend comptede la segregation des proteines membranaires dans des nano-domaines di!erents afin d’y e!ectuerdes fonctions biologiques specifiques.

II-4 Systemes modeles : doubles bicouches supportees

Afin de comprendre plus en detail les mecanismes de confinement dynamique (section II-1) et l’or-ganisation des membranes (sections II-2 et II-3), il apparaıt crucial de travailler sur des membranesmodeles. Travailler sur systeme modele permet en e!et de s’a!ranchir de la tres grande complexited’une cellule afin d’isoler un mecanisme physique particulier et de comparer plus aisement theorieet experience. Laurence Salome travaille en ce moment sur un tel projet, lequel consiste a empilerdeux (voire plusieurs) bicouches lipidiques supportees pour s’a!ranchir des interactions, notammenthydrodynamiques, avec le substrat. Des proteines membranaires peuvent en outre etre incluses dansla derniere bicouche, de maniere controlee (figure 8).

Fig. 8 – Schema de principe de l’elaboration d’une double bicouche supportee (sur verre). Unepremiere bicouche est formee par fusion de vesicules. Puis l’operation est repetee pour deposer laseconde bicouche. Des proteines peuvent etre inserees dans cette derniere pour etudier leur phy-sique statistique. La seconde bicouche est moins liee que la premiere et ses fluctuations sont plusimportantes. Les proteines, qui par ailleurs n’interagissent plus avec le verre, sont dans un milieuqui ressemble a une membrane libre.

Nous avons, avec Manoel Manghi, entrepris de modeliser de tels systemes de bicouches empilees.Notre modele propose une interaction non-lineaire entre bicouches di!erentes (potentiel de Morse) etune energie libre de type Helfrich au sein d’une meme bicouche. Nous avons etudie, par une approchevariationnelle, la transition de decrochage des bicouches lorsque la temperature croıt, et mis en

4Nous avons par ailleurs verifie numeriquement avec Nicolas Meilhac que cet argument reste satisfaisant a petitsq (resultats non publies).

13

(b)50 100 150 200 250 300

1

2

3

4

5

6

00

! =301

!2

T/"

12

3

4

Figure 2.2 – (a) Schéma de principe de l’élaboration d’une double bicouche supportée (surverre). Une première bicouche est formée par fusion de vésicules. Puis l’opération est répétéepour déposer la seconde bicouche. Des protéines peuvent être insérées dans cette dernièrepour étudier leur physique statistique. La seconde bicouche est moins liée que la premièreet ses fluctuations sont plus importantes. Les protéines, qui par ailleurs n’interagissent plusavec le verre, sont alors dans une membrane presque libre. (b) Diagramme de phases dedeux membranes supportées, interagissant avec un potentiel de Morse définit par l’éq. (2.6).L’amplitude du potentiel d’adsorption (adimensionné) de la première membrane est Ξ1 =8α2κD1/σ

2 = 30. Dans le plan température vs force d’adsorption de la seconde membrane,le diagramme est divisé en quatre régions, où l’empilement est lié au substrat (région 1),la membrane supérieure se décroche (2), les deux membranes se décrochent (3) et les deuxmembranes se décrochent tout en restant empilées (4) [125].

des ces paramètres est nécessaire pour l’analyse des spectres de réfléctivité des rayons X.En effet, dans ces expériences [50], l’allure des spectres varie énormément avec la forced’interaction entre membranes : celle-ci est souvent très faible, ce qui suggère que le systèmeest près de la transition de décrochage.

Nous avons également étudié la dynamique de ces membranes supportées à partir del’équation de Langevin qui régit l’évolution temporelle de l’amplitude des oscillations dela hauteur de la membrane i et de vecteur d’onde q, hi(q, t) :

∂hi(q, t)∂t

= −Λij(q)δH0

δhj(−q, t)+ ζi(q, t) (2.8)

oùH0 est le hamiltonien gaussien, ζi(q, t) le bruit thermique et Λij(q) la matrice de frictionqui tient compte des interactions hydrodynamiques. En effet, la présence du substrat etde la seconde membrane, qui imposent une condition de non-glissement sur le champde vitesse, vont modifier les coefficients de friction γi(q) par rapport à une membraneseule (pour laquelle γ(q) = (κq3 + σq)/(4µ) est simplement le quotient de l’énergie des

93


(a) (b)

10

103105

10-1

10-3

0.5 1 5 10 500.1q!

"i~

Figure 2.3 – (a) Schéma de deux membranes supportées sur un substrat montrant l’in-fluence du substrat (et des membranes) sur le champ de vitesse du solvant environnant.(b) Variation des coefficients de friction des deux modes γi(q) (lignes épaisses) en fonctiondu vecteur d’onde (adimensionné par ξ) pour Ξ1 = 30 et Ξ2 = 3. Les distances au substratdes membranes sont d1 = 2 nm and d2 = 3 nm) [125]. Les lignes fines correspondent aucas sans hydrodynamique et la ligne pointillée est le résultat asymptotique (qξ)3.

fluctuations et de la friction visqueuse). Ces coefficients sont définis via la fonction decorrélation temporelle du mode i

〈wi(q, t)wi(−q, 0)〉 = 〈wi(q, 0)2〉 e−γi(q)t (2.9)

En comparant au cas où l’hydrodynamique n’est pas prise en compte, nous trouvonsque les γi(q) sont bien plus faibles aux petits vecteurs d’onde (de 3 à 4 ordres de grandeur)et croissent de façon monotone (voir figure 2.3). De plus le découplage des deux modesest maximal aux vecteurs d’ondes intermédiaires de l’ordre de q−1 ≈ ξ ' 50 nm (oùξ =

√2κ/σ). Enfin, lorsque la seconde membrane s’éloigne de la première lors, par exemple

d’une augmentation de T , le coefficient de friction du mode le plus lent diminue fortement.Ces résultats montrent que les interactions hydrodynamiques jouent un rôle majeur

dans la dynamique des membranes supportées et sont essentielles dans la compréhensionde la dynamique de leurs composants et en particulier de la diffusion des lipides et desprotéines trans-membranaires dans la membrane.

2.2.1 Article

Suit l’article :M. Manghi et N. Destainville, Statistical Mechanics and Dynamics of Two Supported Sta-cked Lipid Bilayers, Langmuir 26 (6) 4057 (2010) (12 pages)

94

DOI: 10.1021/la903504n 4057Langmuir 2010, 26(6), 4057–4068 Published on Web 12/15/2009

pubs.acs.org/Langmuir

© 2009 American Chemical Society

Statistical Mechanics and Dynamics of Two Supported Stacked

Lipid Bilayers

Manoel Manghi* and Nicolas Destainville

Universite de Toulouse, UPS; Laboratoire de Physique Theorique (IRSAMC), F-31062 Toulouse, France,and CNRS, LPT (IRSAMC), F-31062 Toulouse, France

Received September 16, 2009. Revised Manuscript Received October 27, 2009

The statistical physics and dynamics of double supported bilayers are studied theoretically. The main goal indesigning double supported lipid bilayers is to obtain model systems of biomembranes: the upper bilayer is meant to bealmost freely floating, the substrate being screened by the lower bilayer. The fluctuation-induced repulsion betweenmembranes and between the lowermembrane and the wall are explicitly taken into account using aGaussian variationalapproach. It is shown that the variational parameters, the “effective” adsorption strength, and the average distance tothe substrate, depend strongly on temperature and membrane elastic moduli, the bending rigidity, and the microscopicsurface tension, which is a signature of the crucial role played by membrane fluctuations. The range of stability of thesesupported membranes is studied, showing a complex dependence on bare adsorption strengths. In particular, theexperimental conditions of having an upper membrane slightly perturbed by the lower one and still bound to the surfaceare found. Included in the theoretical calculation of the damping rates associated with membrane normal modes arehydrodynamic friction by the wall and hydrodynamic interactions between both membranes.

Introduction

Double supported membranes are composed of two lipidbilayers superimposed on a solid substrate (Figure 1). The lowermembrane (closer to the substrate) is weakly absorbed, the upperone being bound via intermembrane interactions. This system hasa growing interest for physicists and biologists since it allows oneto avoid, in part, the direct influence of the substrate on the uppermembrane (pinning by defects, repulsion, or direct attraction bythe wall) which becomes almost freely floating. Some recentexperimental techniques allow the formation of such systemswhether by Langmuir-Blodgett deposition1-3 or by deposingafter rupture a giant vesicle on a single bilayer.4,5 These doublesupported lipid bilayers can then play the role of model mem-branes, the lipid and protein composition of which can be varied.Indeed, the extreme complexity of cell membranes motivates thedevelopment of artificial ones6,7 that are more easily studied froma physical point of view. In the present case, the planar geometryfacilitates the use of modern spectroscopy techniques to char-acterize these model membranes.1,4,5,8 Furthermore, two stackedmembranes can model cell-cell junctions9,10 where the role oflipids and proteins can be investigated.5,11

Since liquid membranes such as lipid bilayers are highlyfluctuating at room temperature (their bending modulus κ is on

the order of several kBT ),12 a fluctuation-induced repulsionappears at finite temperature, leading to a destabilization of thesystem and eventually, an unbinding from the substrate at acritical temperatureTu. The unbinding transitionofmembranes isa long-standing issue, first studied by Helfrich,13 which has led toa considerable amount of theoretical investigations on homo-geneous stacks of two or more membranes. Lipowsky describes,in a very nice review,14 the various situations where unbindingoccurs, among which are the cases of only steric repulsiveinteractions with applied external pressure or attractive interac-tions between membranes, depending on experimental condi-tions.

These theoretical studies essentially focus on the order of thetransition and the values of the critical exponents, mostly usingMonte Carlo numerical simulations,15,16 or group renormaliza-tion techniques.17-19 Interestingly, an analytical solution can beobtained when considering the unbinding of fluctuating strings(one-dimensional objects) in twodimensions.20 Indeed, this case isexactly mapped onto the delocalization transition of a quantumparticle in an external potential and the unbinding temperatureTu

can been computed exactly by solving the Schr€odinger equationby transfer matrix techniques.

Among these studies one can distinguish among symmetricsystems (two identical membranes or a bunch of identicalmembranes), asymmetric systems where the membranes are notidentical, and eventually the extreme case of one membrane being*To whom correspondence should be addressed. E-mail: manghi@irsamc.

ups-tlse.fr.(1) Daillant, J. et al. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 11639.(2) Lecuyer, S.; Charitat, T. Europhys. Lett. 2006, 75, 652.(3) Charitat, T.; Bellet-Amalric, E.; Fragneto, G.; Graner, F. Eur. Phys. J. B

1999, 8, 583.(4) Wong, A. P.; Groves, J. T. J. Am. Chem. Soc. 2001, 123, 12414.(5) Wong, A. P.; Groves, J. T. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 14147.(6) Sackmann, E. Science 1996, 271, 43.(7) Mouritsen, O. G.; Andersen, O. S. Biol. Skr. Dans. Vidensk. Selsk. 1998, 49,

7.(8) Groves, J. T. Annu. Rev. Phys. Chem. 2007, 58, 697.(9) Kaizuka, Y.; Groves, J. T. Biophys. J. 2004, 86, 905.(10) Lin, L. C.-L.; Groves, J. T.; Brown, F. L. H. Biophys. J. 2006, 91, 3600.(11) Parthasarathy, R.; Groves, J. T. Proc. Natl. Acad. Sci. U.S.A. 2004, 101,

12798.

(12) Statistical Mechanics of Membranes and Surfaces, 2nd ed.; Nelson, D.,Piran, T., Weinberg, S., Eds.; World Scientific Publishing: Singapore, 2004.

(13) Helfrich, W. Z. Naturforsch. 1978, 33a, 305.(14) Lipowsky, R. In Handbook of Biological Physics; Lipowsky, R., Sackmann,

E., Eds.; Elsevier: Amsterdam, 1995; Vol. 1, p 521.(15) Netz, R. R.; Lipowsky, R. Phys. Rev. Lett. 1993, 27, 3596.(16) Lipowsky, R.; Zielinska, B. Phys. Rev. Lett. 1989, 71, 1572.(17) Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986, 56, 2541.(18) Lipowsky, R. Europhys. Lett. 1988, 7, 255.(19) Grotehans, S.; Lipowsky, R. In Dynamical Phenomena at Interfaces,

Surfaces and Membranes; Beysens, D., Boccara, N., Forgacs, G., Eds.; Nova Science:New-York, 1993; p 267.

(20) Burkhardt, T. W.; Schlottmann, P. J. Phys. A 1993, 26, L501.

4058 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068

Article Manghi and Destainville

a solid substrate (formally a membrane with infinite rigidity).Double supported membranes belong to the latter category. Thephysics of asymmetric stacks studied by Monte Carlo simula-tions,15,21 or by analytical tools for one-dimensional strings,22

turned out to be very rich, showing for instance, over a certainrange of the parameters, a peeling process where successiveunbinding temperatures appear, the upper membrane unbindingfirst. This has to be compared to the case of symmetric stackswhere Tu is independent of the number of stacked membranes.However, notmuchwork has been done for real two-dimensionalmembranes embedded in a tridimensional space.

Moreover, the major part of these studies consider the effect ofbending rigidity in height-height fluctuations, without consider-ing the microscopic surface tension σ, which might play animportant role in supported bilayers. In such systems, σ is relatedto the chemical potential of the amphiphilic molecules (an energe-tic parameter associated with the full membrane area and not theprojected one).23 The value of this parameter is measured byfitting X-ray reflectivity spectra1, where it must be taken intoaccount for wave-vectors q < (σ/κ)1/2, and in flickering experi-ments.24 Moreover, according to the preparation method, thepresence of pinning defects can induce lateral tension.10

The surface tension always dominates bending rigidity at largedistances, and thus drastically reduces the membrane roughness.It leads to a fluctuation-induced interaction which decaysexponentially at large distances, instead of a power law in z-2

in the bending rigidity only case.25 By comparing it to the directinteraction, such as van der Waals attraction and screenedelectrostatic attraction, we thus enter in the weak- or intermedi-ate-fluctuation regimes. It has been shown, using renormalizationtechniques that, in theses regimes, the order of the transition canchange and discontinuous transitions might occur.18,19,26

The techniques presented above which include Monte Carlosimulations, renormalization group calculations, and numericaltransfer matrix methods prove successful for calculating quanti-tative values for critical exponent, but they are not always easy toimplement. Moreover, they often do not enhance our intuitiveand qualitative understanding of the problem. In this paper, wedevelop a variational approach of asymmetric unbinding wherethe intermembrane and substrate-membrane potentials are mod-eled by Morse potentials. The variational approach provides anapproximation that is analytically simple to implement and canprovide a direct link between quantitative calculations andqualitative pictures of the physics of the problem. This analyticalwork has thus similarities with the ones on single membrane

unbinding, which use self-consistent13,27-29 and variational30

calculations. Our goal is to understand the physical mechanismsand the role of the key physical parameters (adsorption potential,bending rigidity, surface tension) in the thermodynamics ofsupported stacked lipid membranes. What are the parameterranges for which double supported bilayers are stable? In parti-cular, nonlinearities of potentials are taken into account by thevariational parameters, which are effective disjoining pressuresand intermembrane distances, and their dependence with tem-perature is explicitly studied.

Fluctuating membranes are often studied by light-scatteringexperiments, which yield information about the bilayer dynamicsand the associated damping rates.31 Moreover, biomembranedynamics is crucial for the study of the diffusion of integralmembrane proteins,32 which is influenced both by fluctuationsdynamics (projection of the motion onto a reference plane33,34)and hydrodynamics.35 The dynamical fluctuations of a singlelipidic membrane in a viscous liquid had been studied in the earlyseventies by Kramer,36 who showed that the damping rate isdriven by the ratio between viscous damping and bending energy.The presence of an external “obstacle”, such as a second mem-brane24,37 or a solid substrate,38 modifies substantially the damp-ing rate as soon as they are close enough to induce hydrodynamicinteractions. Nonmonotonic behavior has been observed in thelast case. In this work, we generalize these results to doublesupported lipid bilayers and show how temperature and adsorp-tion potentials modify the wave-vector dependence of dampingrates.

The first section presents the variational approach in thesimplest case of one supported bilayer and a comparison is madewith previous works. This variational approach is then applied totwo stacked supported bilayers, where phase diagrams andcorrelation function for height-height membrane fluctuationsare computed. Finally, these variational parameters calculated inequilibrium serve to describe the normal modes and dampingrates of the system. The flow velocity field is established for thefirst time in this geometry and hydrodynamic friction and hydro-dynamic interactions between the two membranes turn out to becentral to understanding this complex dynamics. A discussion ofour results is given in the conclusion.

One Membrane Supported on a Solid Substrate

We first consider a single membrane lying on a horizontal solidplanar substrate at position z= 0. The membrane local positionvector is, in the Monge representation, R = xx þ yy þ h(x,y)z,where h(x,y) is the height of the membrane and η= Æh(x,y)æ is theaverage distance between the substrate and the membrane.

The Hamiltonian of a supported membrane is given by theusual Helfrich Hamiltonian of a single fluctuating membrane39,31

plus a term taking into account the interaction between themembrane and the substrate:

H ¼ZId2r

1

2½σðrhðrÞÞ2 þKðr2hðrÞÞ2 þV ½hðrÞ

ð1Þ

Figure 1. Sketch of a double supported lipid bilayer.

(21) Netz, R. R.; Lipowsky, R. Phys. Rev. E 1993, 47, 3039.(22) Hiergeist, C.; L€assig, M.; Lipowsky, R. Europhys. Lett. 1994, 28, 103.(23) David, F.; Leibler, S. J. Phys. II France 1991, 1, 959–976.(24) Brochard, F.; Lennon, J.-F. J. Phys. France 1975, 36, 1035.(25) Netz, R. R.; Lipowsky, R. Europhys. Lett. 1995, 29, 345.(26) Grotehans, S.; Lipowsky, R. Phys. Rev. A 1990, 41, 4574.(27) Evans, E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132.(28) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 2455.(29) Mecke, K. R.; Charitat, T.; Graner, F. Langmuir 2003, 19, 2080.(30) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557.

(31) Seifert, U. Adv. Phys. 1997, 46, 13.(32) Naji, A.; Brown, F. L. H. J. Chem. Phys. 2007, 126, 235103.(33) Reister, E.; Seifert, U. Europhys. Lett. 2005, 71, 859.(34) Gov, N. S. Phys. Rev. E 2006, 73, 041918.(35) Naji, A.; Atzberger, P. J.; Brown, F. L. H. Phys. Rev. Lett. 2009, 102,

138102.(36) Kramer, L. J. Chem. Phys. 1971, 55, 2097.(37) Frey, E.; Nelson, D. R. J. Phys. I France 1991, 1, 1715.(38) Seifert, U. Phys. Rev. E 1994, 49, 3124.(39) Helfrich, W. Z. Naturforsch. 1973, 28c, 693.

DOI: 10.1021/la903504n 4059Langmuir 2010, 26(6), 4057–4068

Manghi and Destainville Article

where r=xxþ yy is the projectionofR in the plane parallel to thesubstrate,r= x∂x þ y∂y and I is the projected membrane area.The membrane bending rigidity is κ (=20 to 50kBT for a lipidbilayer) and σ is the microscopic membrane surface tension.

Several contributions arise in the potential V(z): at shortdistances it is dominated by excluded volume interactions andhydrophobic or hydration repulsion (entropic interaction asso-ciated to the presence of water molecules inserted betweenhydrophilic lipid heads).27 At intermediate distances, it is domi-nated by screened electrostatic and van der Waals interactionswhich are essentially attractive. More precisely, it is usuallyadmitted that, if theDebye screening length is small, the potentialbetween the substrate and the membrane, assumed flat, isessentially the sum of two terms:14 a van der Waals attractivepart between a flat membrane of thickness ε and an semi-infinitemedium40

VvdWðzÞ ¼ -AH

12π

1

z2-

1

ðzþεÞ2" #

ð2Þ

where AH is the Hamaker constant between interfaces substrate/water and lipid/water (AH = 1-10 10-21 J), and a decreasingexponential repulsive part which takes both steric and hydrationcontributions into account:

VhydðzÞ ¼ Phyde-z=λhyd ð3Þ

where the values of the hydration pressure Phyd (1-10 10-2

J 3m-2) and λhyd (0.1-0.3 nm) are not precisely known.

To focus on the physical mechanisms of unbinding, we chooseto mimic the adsorption potential by a simpler one, the Morsepotential:

VMorseðzÞ ¼ D½e-2Rðz- dÞ - 2e-Rðz- dÞ ð4Þwhich contains the two essential features described above(repulsion at short length scales and attraction at intermediateones, see Figure 2). It is characterized by only three parameters, itsdepth D, its range R, and the position of its minimum d. Inprinciple, the following variational treatment could be done forthe potential defined in eqs 2 and3.However theFourier transformof eq 2, needed in the following calculations, leads to singular partsthat should be regularized using heuristic functions.30

A priori, replacing the potential eq 2 which varies as -z-3 forz . ε = 3 nm by an exponential decreasing potential eq 4 couldmodify the nature of the transition. Indeed, we move from theweak-fluctuation regime, |VvdW| . |Vf l|, to the intermediate-fluctuation regime, |VMorse|∼|Vf l| (where Vf l is the fluctuation-induced repulsion which decays exponentially at large distances).However, it has been shown that in these two regimes, first-ordertransitions can occur,18,26 and in the intermediate-fluctuationregime their occurrence is related to the comparison of the twodecay lengths (see below).

At finite temperature, thermal membrane shape fluctuationsinduce an entropic confinement of the bending modes (whichdepends on κ and σ) and modify in turn the repulsive componentof the interaction.13,27,30 Since the full calculation of the partitionfunction Z=

RD h exp(-βH [h]), where β=(kBT )-1, is untract-

able, we use in the following, a variational approachwhere the fullHamiltonian H is approximated by a trial Gaussian one:

H 0 ¼ 1

2

ZId2rfσðrhðrÞÞ2 þKðr2hðrÞÞ2 þA½hðrÞ- η2g ð5Þ

where the potential is harmonic with two variational parameters,the spring constant A, and the shifted equilibrium distance η.

The variational free energy reads in the Gibbs-Bogoliubov form

Fvar ¼ F0 þ ÆH -H 0æ0 ð6Þwhere βF0 = -lnZ0 is the free energy associated with the varia-tional Hamiltonian (5) and the subscript 0 refers to quantitiescalculated using eq 5. The Gibbs inequality ensures thatFvar g Fexact when Fvar is minimized with respect to the variationalparameters. Their values will thus be determined byminimizing eq 6with respect to A and η. It must be emphasized, however, that werestrict our choice of variational Hamiltonians to the subclass ofquadratic (and thus symmetrical) potentials, and by doing so, weobtain an approximate value of the trueminimumof the free energyor equivalently the true values of the unbinding transition tempera-ture and of the critical exponents.

The calculations of eq 6 with the Morse potential, eq 4, havebeen done in ref 41 in the context of DNA denaturation, that is,for a one-dimensional string in a space of two dimensions. In ourcase, the free energy F0 is

F0 ¼ -kBTI

2

Zd2q

ð2πÞ2 ln2πkBT

Aþσq2 þKq4

ð7Þ

and by defining u(r) = h(r)-η we get

ÆVMorseðhÞæ0 ¼ D ½expð2Rðd - ηÞÞ expð2R2Æu2æ0Þ- 2 expðRðd - ηÞÞ expðR2<u2>0=2Þ ð8Þ

Minimization of Fvar with respect to η yields

η- d ¼ 3

2RÆu2æ0 > 0 ð9Þ

where the average mean square value of u(r) is

Æu2æ0 ¼ 2

IDF0

DA¼ kBT

Zd2q

ð2πÞ21

Aþσq2 þKq4

¼ kBT

2πσg

4AKσ2

ð10Þ

Figure 2. Substrate-membrane potential as a function of thedistance to the substrate: sum of van derWaals eq 2 and hydrationeq3 contributions (solid blue line) andMorse potential (dashed redline). Parameters for the Morse potential are adjusted to havea goodmatch at theminimumwith the true potential (Phyd=1.510-2 J/m2, λhyd=0.1 nm,AH=1.5 10-21 J, ε=3nm,D=10-5

J/m2, R= 2 nm-1, and d= 1 nm).

(40) Parsegian, V. A. Van der Waals Forces, A Handbook for Biologists,Chemists, Engineers, and Physicists; Cambridge University Press: New York, 2006. (41) Dauxois, T.; Peyrard, M.; Bishop, A. R. Phys. Rev. E 1993, 47, 684.

4060 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068


and the function g(x) =R ¥0 dt=ðxþ 2tþ t2Þ is defined by14

gðxÞ ¼ arctanhð ffiffiffiffiffiffiffiffiffiffiffi1- x

p Þ= ffiffiffiffiffiffiffiffiffiffiffi1- x

pfor xe 1

arctanð ffiffiffiffiffiffiffiffiffiffiffix- 1

p Þ= ffiffiffiffiffiffiffiffiffiffiffix- 1

pfor xg 1

(ð11Þ

In the case of vanishing surface tension σ f 0, since

gðx f ¥Þ~- π

2ffiffiffix

p

eq 10 reduces to the classical result of Helfrich with no surfacetension13,30

A ¼ ðkBTÞ2ð8Æu2æ0Þ2K

ð12Þ

Helfrich closed the calculation self-consistently and found Æu2æ0=μsd

2/4 with μs ≈ 1/6 for a simple steric component at z = 0.13 Inour variational approach, the solution depends on the surfacetension σ and on the respective values of the Morse potentialparameters, D, R, and d.

Since

Fvar ¼ F0 þI ÆVæ0 -ADF0

DA

the minimization with respect to A yields

AD2F0

DA2¼ I

DÆVæ0DA

as already noticed by Podgornik and Parsegian30 and we finallyget an implicit equation for A

A ¼ 2R2D exp

-2

3Rðη- dÞ

¼ 2R2D exp -kBTR2

2πσg

4AKσ2

" #ð13Þ

Solving eq 13, which leads to the variational parameters A* andη* amounts to finding the lowest variational energy F var* =Fvar.(A*, η*) solution of the problem.

Let us denote by Fvar the variational free energy where η isreplaced by its expression in eq 9

βFvarðAÞI

¼ σ

16πKf

4AKσ2

-4AKσ2

g4AKσ2

" #

- βD exp -kBTR2

2πσg

4AKσ2

" #ð14Þ

where the function f ðxÞ ¼ R x0 gðtÞ dt is

f ðxÞ ¼- 2

ffiffiffiffiffiffiffiffiffiffiffi1- x

parctanhð ffiffiffiffiffiffiffiffiffiffiffi

1- xp Þ- ln

x

4

, xe1

2ffiffiffiffiffiffiffiffiffiffiffix- 1

parctanð ffiffiffiffiffiffiffiffiffiffiffi

x- 1p Þ- ln

x

4

, xg1

8>><>>: ð15Þ

The free energy of an unbound membrane which fluctuates freelyin the bulk is given by eq 14 with A = 0. Thus, the membraneremains weakly adsorbed on the substrate as long as Fvar(A) <Fvar(0) = 0. We introduce the renormalized parameters

~A ¼ 4AKσ2

, Ξ ¼ 8R2KDσ2

, θ ¼ 2πσ

kBR2ð16Þ

where the coupling parameter Ξ is simply the rescaledsecond derivative of the Morse potential at the minimum, and θis a temperature scale. Since thermal fluctuations decreasethe interaction between the membrane and the substrate,we expect that for a finite temperature, we have A(T ) e 2R2Dor ~A(T ) e Ξ.

From eq 9, one sees that the average height of the bilayer, η, isproportional to the average mean square of height fluctuationsÆu2æ0. Since g(x) is a monotonic decreasing function which tendsto 0 for large x and diverges for x=0, decreasing ~A, that is,decreasing A or κ or alternatively increasing σ, destabilizes thesupportedmembrane.This increase ofηhasbeen recently observedby heating double supported bilayers which makes κ decreasing(∼200kBT in the gel phase to (1-3)kBT at its minimum).2

It is interesting to note that naively one would expect themembrane todesorbwhen the fluctuation contribution in eq 9diver-ges, that is, for ~A f 0. Technically speaking, this type of transitionwould be continuous. However, as said in the Introduction, dis-continuous transitions can occur. Indeed, close to the transition wehave ~A, 1, and eqs 9 and 10 yield η- d=-3RkBT ln( ~A/4)/(8πσ).The variational free energy eq 14 can be written as

βFvarðηÞI

=σ

16πK2 exp -

2

3

2θ

TRðη- dÞ

(

-Ξθ

Texp -

2

3Rðη- dÞ

ð17Þ

and thus comparing both exponentials and prefactors, we find anunbinding transition in two cases: (i) a continuous transition forT/θe 2 andΞe 2T/θeΞc=4, and (ii) a discontinuous transition forT/θ> 2 and large Ξ> 2T/θ g Ξc = 4.

More quantitatively, the unbinding temperature Tu is definedby Fvar(A) = 0 together with eq 13, which yields

θ

Tu¼ f ð ~AÞ

~A- gð ~AÞ ð18Þ

~A ¼ Ξ exp -Tu

θgð ~AÞ

ð19Þ

The solution of this system is found numerically and the phasediagram is shown in Figure 3a keeping σ fixed. At low tempera-ture, the membrane is bound to the substrate and the bindingregion grows with the adsorption strength Ξ, which is in this caseproportional to D.

In the vanishing surface tension case, σf 0, one finds, from eqs18, 19, and 12, that the unbinding temperature is given byTuðσ f 0Þ=θ ¼ 2=ðπ ffiffiffi

ep Þ ffiffiffiffi

Ξp

that is,

Tu Tuðσ ¼ 0Þ ¼ 8

ffiffiffi2

e

r ffiffiffiffiffiffiffiKD

p

RkBð20Þ

DOI: 10.1021/la903504n 4061Langmuir 2010, 26(6), 4057–4068


Weobtain the correct scaling forTu.14 Equation 20, in unrescaled

units, corresponds to the dashed line in Figure 3a. Note thatwith the realistic parameter values given in Figure 2, we findT u* = 0.94Troom.

The same phase diagram is replotted in Figure 3b using theparametersT/Tu* andΞ. The adsorption strengthD is kept fixed, andthus the parameterΞ now controls the surface tension.One observesthat the unbinding temperature increases when Ξ decreases, that is,when the surface tension increases. Indeed, the system is entropicallystabilized when σ 6¼ 0, since more degrees of freedom are accessibleto the bound membrane at a given temperature. At low Ξ, theunbinding temperature follows the limiting law

Tu

Tu

~- πffiffiffie

pffiffiffiffiΞ

p

Indeed eq 18 leads to Tu f 2θ for ~A f 0. This limiting formcorresponds to the solid line in Figure 3b. At large Ξ, that is, forσf 0, T/T u* f1.

The disjoining pressure, or mechanical stress,27 is

pðTÞ ¼ -1

IDFvar

Dd¼ AðTÞ

3Rð21Þ

since ∂Fvar/∂d = ∂ÆVMorse(h)æ0/∂d. Hence the variational para-meter A(T), plotted in Figure 4 for various values of Ξ, readilygives the disjoining pressure as a function of temperature, whichcan be measured experimentally. To compare this case tothe vanishing surface tension case, we rescale the parameterA by the spring constant in the Morse potential 2R2D as A =A/(2R2D) = ~A/Ξ.

Equation 19 becomes

A¼ A

2R2D¼ exp -

2

πffiffiffie

p T

Tu

ffiffiffiffiΞ

pgðΞAÞ

!ð22Þ

Notehowever that our variational approach,whichbelongs to the“superposition of fluctuation-induced and direct forces” type ofapproaches, is not appropriate in this strong-fluctuating regime(with σ = 0 and thus |Vfl| . |VMorse|), since, in this regime, onlyrenormalization group approaches yield the correct (second)order of the transition.14

In Figure 4a,b are plotted ~A [eq 19] andA [eq 22], as a functionof T at constant σ and constant D, respectively. First of all, oneobserves that A (and η) depends on T which is a signature of thenonlinear potential as for thermal expansion in solids.We recoverthe straight result that atTf 0,A(T=0)=VMorse

00 (d ) = 2R2D,that is, the spring constant with no entropic induced repulsion[Figure 4b]. Furthermore, at fixed T, ~A increases with Ξ (or D)faster than Ξ, and A decreases slightly when Ξ increases (or σdecreases). This is an important result in the experimental context,since the surface tension of supportedmembranemay vary due todefects where the membrane is pinned or depending on theexperimental protocol (e.g., washing processes or superficialpressure exerted at the edge). The variational mean height ofthe bilayer η, which is the true order parameter of the transition, isdeduced following eq 13. One observes that, for Ξ= 3, A= 0 atthe unbinding temperature (or η diverges). More generally, assuggested in eq 17, our approach yields a continuous transition(or second order) for Ξ e 4 (solid line in the constant D phasediagram shown in Figure 3b). For Ξ > 4, A(Tu) is finite which

Figure 3. (a) Phase diagram of an adsorbed membrane in a Morse potential for a fixed value of the surface tension σ. The solid (red) linecorresponds to the rescaled unbinding temperatureTu/θ vs the coupling parameterΞwhichmeasures the bare adsorption strength. It dividesthe phase diagram in two parts, where the membrane is bound (bottom) and unbound (top). The dashed line corresponds to the systemwithvanishing surface tension given by eq 20. (b) Same phase diagram in the (Ξ, T/Tu*) space, that is, for constant adsorption strength, D, andvarying σ. The surface tension increases the unbinding temperature (dots) which behaves like 1/Ξ1/2 for 0eΞe 4 (solid line) and reaches theasymptote 1 for large Ξ.

Figure 4. (a) Variation of the variational parameter ~A rescaled by the adsorption strength Ξ as a function of rescaled temperature T/θ(σ constant) for different values ofΞ=3,30, 100. The parameter Ξ is in this case proportional to the adsorption strengthD. The fluctuationinduced repulsion increases when T increases thus leading to a decrease of ~A, until we reach the critical point at T= Tu. (b) Same as panel awith the temperature rescaled by the unbinding temperature at σ = 0. Ξ is thus proportional to σ-2, and the case Ξ f ¥ corresponds tovanishing surface tension, eq 22. The associated average distance between the substrate and the membrane, η, given by eq 13 is plotted in theinset.

4062 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068


makes the transition discontinuous (first order). The point (Ξc =4,Tc/θ=2) is thus a tricritical Lifshitz point, the surface tension σcontrolling the order of the transition.

Two Solid-Supported Membranes

In this section we consider two membranes superimposed on asubstrate, with an adsorption potential as eq 4 for the firstmembrane and a similar one for the intermembrane potential.Since the range of the adsorption potential is small (R-1≈ d, a fewnm), the second stacked bilayer does not directly feel the substrateand is adsorbed only through the intermembrane potential. Wemodel these two potentials as Morse potentials Vi defined in eq 4with parameters Di, di, and Ri where i = 1 refers to the lowermembrane and i=2 refers to the upper one. The Hamiltonian ofthe system is

H ¼ZId2r

X2i¼1

1

2½σðrhiðrÞÞ2 þKðr2hiðrÞÞ2 þ

Vi½hiðrÞ- hi- 1ðrÞ

ð23Þ

where h0=0 is the positionof the substrate plane. The variationalHamiltonian reads

H 0 ¼ 1

2

ZId2r

X2i¼1

fσðrhiðrÞÞ2 þKðr2hiðrÞÞ2 þAi½hiðrÞ

- hi- 1ðrÞ- ηi2g ¼ 1

2

ZId2ruTðrÞUuðrÞ ð24Þ

with uT (r) = (h1(r)-η1, h2(r)-η2-η1) and

U ¼ ð- σr2 þKr4ÞI2 þ A1 þA2 -A2

-A2 A2

!ð25Þ

where I2 is the identity matrix. In the eigenbasis the modes aredecoupled with eigenvalues (of the last matrix in eq 25)

λ1, 2 ¼ A1

2þA2 (

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1

2

4þA2

2

sð26Þ

and the Hamiltonian eq 24 becomes diagonal. The normal modesvT(r) = (ν1(r),ν2(r)) are defined through u(r) = Pv(r) whereP is amatrix of rotation of angle φ ∈ [0, π/2] defined by

tan φ ¼ A2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1

2

4þA2

2

r-A1

2

ð27Þ

One observes that λ1> λ2g 0where λ2=0 forA1= 0 orA2=0.The case A1 = 0 (φ = π/4) corresponds to a free stack of twomembranes remaining bound together, whereas A2 = 0 (φ = 0)corresponds to the case of a secondmembrane being freewhile thefirst one remains bound.

By proceeding as above, the free energy F0 is simply given bythe sum over the modes

F0 ¼ -kBTI

2

X2i¼1

Zd2q

ð2πÞ2 ln2πkBT

λi þσq2 þKq4

¼ kBTI σ

16πKf

4λ1Kσ2

þ f

4λ2Kσ2

" #ð28Þ

where f(x) is defined in eq 15.Minimization of the variational freeenergy with respect to the four variational parameters Ai and ηi,yields the generalization of eqs 9 and 13

ηi - di ¼ 3

2RiÆðui - ui- 1Þ2æ0 ð29Þ

Ai ¼ 2R2i Di exp½-R2

i Æðui - ui- 1Þ2æ0 ð30Þwhere

Æðui - ui- 1Þ2æ0 ¼ kBT

2πσ

X2j¼1

g4λjKσ2

DλjDAi

ð31Þ

By replacing theηi as a functionofAi andwritingdimensionlessquantities with a tilde, ~Ai = 4Aiκ/σ

2 and ~λi ¼ 4λiK=σ2, thevariational free energy Fvar(A1,A2), corresponding to eq 14 fora single supported membrane, is

βFvar

I¼ σ

16πKf ð~λ1Þþ f ð~λ2Þ-

X2i, j¼1

~λigð~λjÞD~λjD ~Ai

35

24

-X2i¼1

βDi exp -kBTRi

2

2πσ

X2j¼1

gð~λjÞD~λjD ~Ai

24

35 ð32Þ

where ~λ1, 2ð ~A1~A2Þ are given in eq 26.

The unbinding of the adsorbed stack can a priori follow threedifferent scenarios: (i) the upper membrane desorbs and diffusesfreely in the bulk while the lower one remains adsorbed (A2 = 0);(ii) the stack as a whole evolves freely in the bulk (A1= 0); or (iii)the two membranes are completely destacked and desorbed.Hence, to study quantitatively this unbinding transition, thevariational free energy, eq 32, should be compared to the varia-tional free energies of the system in the three configurationsdescribed above

ΔFvarðA1,A2Þ ¼ FvarðA1,A2Þ- Fvar ð33Þwhere Fvar is the variational free energy of the (possibly partially)unbound system. Since the variational free energy of a freelyfluctuatingmembrane in the bulk has been chosen as the referenceof energy, these free energies, Fvar, are respectively (i) eq 14 atits minimum, Fvar(~A1*); (ii) eq 32 with ~λ1 ¼ 2 ~A2 and ~λ1 ¼ 0 at itsminimum ~A2*

βFvarð ~A2Þ

I¼ σ

16πK½ f ð2 ~A

2Þ- 2 ~A2gð2 ~A

2Þ

- βD2 exp -kBTR2

2

πσgð2 ~A

2Þ" #

ð34Þ

and (iii) Fvar(0) = 0.Hence, the unbinding temperature, Tu, is solution of eq 30 and

ΔFvar(A1,A2) = 0, that is, the values of A1, A2, and Tu aresolutions of the three equations:

X2i¼1

f ð~λiÞ-X2j¼1

gð~λjÞ ~AiD~λjD ~Ai

-θ

Tu

~Ai

24

35

¼f ð ~A

1Þ- gð ~A1Þ ~A

1 -

θ

Tu

~A1

f ð~2A2Þ- 2gð~2 ~A

2Þ ~A2 -

θ

Tu

~A2

0

8>>>><>>>>:

ð35Þ

DOI: 10.1021/la903504n 4063Langmuir 2010, 26(6), 4057–4068


~Ai ¼ Ξi exp -Tu

θ

X2j¼1

gð~λjÞD~λjD ~Ai

24

35 i ∈ f1, 2g ð36Þ

The phase diagram is shown in Figure 5a for Ξ1 = 30 and inFigure 5b for Ξ1 = 100 (R1 = R2), and Ξ2/Ξ1 varying between 0and 10. The surface tension σ and bending modulus κ of bothmembranes are kept fixed, which sets the parameters Ξi to beproportional to the two adsorption strengths Di. Four distinctregions appear, related to the four free energies defined above.

For low Ξ2 < 2Ξ1 and by increasing temperature, the stack isprogressively peeled up, the upper bilayer unbinding at atemperature lower than for the bilayer close to the substrate(region 2). However, for intermediate values of Ξ2 [2Ξ1 < Ξ2 <235.5 in Figure 5a and 555 in Figure 5b], the stack unbindscompletely at the transition, defining one unique unbindingtemperature Tu(Ξ2). One then enters in region 3 where bothmembranes are unbound. This unbinding transition temperatureof the adsorbed stack is larger than for a single adsorbedmembrane. Indeed, the upper membrane reinforces the adsorp-tionof the lower one by “squeezing” it to the substrate.Moreover,the region in the phase diagram where the stack exists is largerthan when there is no adsorbing surface. This is due to the slightdecrease of the first membrane height fluctuations induced by thepresence of the wall. Finally, for large enough Ξ2 values, a fourthregion appears (region 4) where the stack unbinds as a whole, thetwomembranes remaining bound together because of their strongmutual attraction.

In Figure 6 are plotted the variational parameters as afunction of T/θ, for three different values of Ξ2/Ξ1 = 1, 5,and 10 (Ξ1 = 30), corresponding to the three regimes justdescribed and shown in Figure 5a. Clearly, one observes thatfor the upper membrane, A2(T ) increases when Ξ2/Ξ1 increases,as expected. For Ξ2/Ξ1= 1,A1(T ) >A2(T ) which is signature ofthe increase of height fluctuations: the uppermembrane fluctuatesmore than the lower thanks to the fluctuations of the lowermembranes which add up with its own ones. For Ξ2/Ξ1 = 5 wehave A1(T) = A2(T), which are both larger than above, and theassociated interlayer distances are almost the same. This is thereason why the stack unbinds completely at the transition.Finally, for Ξ2/Ξ1 = 10, A1(T ) < A2(T ) and the transition ofthe stack is reached when A1(Tu) = 0. We see that theadsorption strength of the substrateΞ1 compared to the interlayerpotential tuned by Ξ2 is central in computing membrane fluctua-tions.

More interestingly is the nature of the transition as a functionof Ξ2: for low Ξ2 < 2Ξ1 and very large Ξ2 the transition is

continuousoccurring respectively forA2(Tu)=0 andA1(Tu)=0,whereas for intermediate values of Ξ2 corresponding to the fullunbinding of the stack (from region 1 to region 3) the transitionoccurs for finite values of Ai, that is, it is discontinuous. This canbe related to the unbinding of one bilayer with a direct interactionincluding a potential barrier14,19 which exhibits discontinuous(or first order) unbinding transitions. In our case, the uppermembrane induces such a potential barrier felt by the “squeezed”membrane.

Once the variational parameters, Ai, are determined, one hasaccess to the fluctuations of the two membranes and theheight-height correlation functions. The variational Hamilto-nian being Gaussian, the structure factor is

ÆviðqÞvjðq0Þæ ¼ ð2πÞ2δijδðqþ q0Þ kBT

λi þ σq2 þKq4ð37Þ

and the three height-height correlation functions Cij(r) =Æui(r)uj(0)æ are given in the Appendix. In particular, we find

Æu12æ ¼ kBT

2πσcos2 φg

4λ1Kσ2

þ sin2 φg

4λ2Kσ2

" #ð38Þ

Æu1u2æ ¼ kBT

2πσcos φ sin φ g

4λ2Kσ2

- g

4λ1Kσ2

" #ð39Þ

Æu22æ ¼ kBT

2πσsin2 φg

4λ1Kσ2

þ cos2 φg

4λ2Kσ2

" #ð40Þ

which are plotted in Figure 7. Note that Æu12æ = 2(η1 - d)/(3R1)and Æ(u2- u1)

2æ=2(η2- d)/(3R2) [see eq 31] are already plotted inFigure 6. For Ξ2 = Ξ1 the fluctuations of the upper membraneincrease from one to two supported bilayers. In this case,fluctuations of the lower membrane add up to the upper-mem-brane ones leading to a substantial increase for T>2θ. Mem-brane 1 is thus very slightly perturbed by the presence ofmembrane 2 (compare lines to dots in Figure 6 and Figure 7).Finally, the correlations between both membranes are very low.When Ξ2 increases, correlations between both membranes in-crease and their correlation functions are almost identical (forΞ2/Ξ1 = 5 and 10).

In principle, the same calculations can be done for nmembranes(see Appendix) with eigenvalues λi where i ∈ 0, ..., n - 1. The

Figure 5. Phase diagram of two adsorbed membranes in a Morse potential, (a) Ξ1 = 30; (b) Ξ1 = 100. The colored dots correspond to therescaled unbinding temperature Tu/θ of the adsorbed stack vs the coupling parameter Ξ2 which measures the strength of the intermembranepotential. Solid lines (dashed for unstable case) correspond to the unbinding of the single membrane when the upper is unbound (red) and ofthe two membranes in the free stack (blue). The diagram is thus divided in four regions, where the stack is bound (region 1), the uppermembrane desorbs (2), both membranes are unbound (3), and the stack is unbound (4).

4064 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068


variational equations are the same as eqs 35 and 36, where the rhs.of eq 35 now contains all the free energies for free bundles madeof j membranes (An-jþ1 = 0) and adsorbed bundles made ofn- jmembranes. Eigenvalues and eigenmodes in the simplest casewhere all the variational parameters are equal, are given in theAppendix.

In this section, we have implicitly assumed R2 = R1. However,one might expect to have actually different potential ranges R2 >R1 since the intermembrane potential is at large distances in z-4

instead of z-3.14 It will introduce two temperatures θ1 > θ2 andthe results will remain qualitatively the same.

Dynamics of Two Supported Lipid Bilayers

Dynamics of supported membranes are governed by Langevinequations for height displacements of membranes 1 and 2, h1(r)and h2(r), written in q-space

Dhiðq, tÞDt

¼ -ΛijðqÞ δH 0

δhjð- q, tÞþ ζiðq, tÞ ð41Þ

where ζi(q,t) is the Fourier transform of the random noise obey-ing the fluctuation-dissipation theorem: Æζi(q,t)ζj(q0,t)æ =2(2π2)kBTΛij(q)δ(t- t0)δ(qþ q0). The dampingmatrixΛ(q) takeshydrodynamic interactions into account, both along the mem-brane and between adjacent surfaces (substrate-membrane 1 ormembrane 1-membrane 2).

It is known that for a single membrane in an infinite liquid, themobility is simply Λ(q) = (4μq)-1 and eq 41 leads directly to thedamping rate31

γ0ðqÞ ¼ ðKq3 þσqÞ=4μ ð42ÞIt is the ratio of the energy driving fluctuations and the viscousdamping. However, in the case considered in Figure 1, we expectthat both the presence of the secondmembrane and the solid wall,which imposes a no-slip condition, will substantially modifyeq 42. In the following, we compute Λ(q) in a simpler geometry,also shown in Figure 1 (dashed lines), where the membranes aresupposed planar, located at z = η1 and z = η2.Hydrodynamics of Two Supported Membranes. The

velocity flow field v(R) and pressure field p(R) in the bulk arefound using the Stokes equation for an incompressible fluid ofviscosity μ

r 3 v ¼ 0 ð43Þ

μr2v-rp ¼ 0 ð44Þwith the following conditions at the boundaries

pðr, z f ¥Þ ¼ p0 "r ð45Þ

vðr, z f ¥Þ ¼ 0 "r ð46Þ

Figure 7. Height-height correlation functions given by eqs 38, 39, and 40 (R= R1 = R2) as a function of temperature for increasingΞ2 (same values as in Figure 6). Solid lines correspond to one supported membrane (Ξ = 30) or a free stack of two bilayers(Ξ2 = 300).

Figure 6. Top:Variational parameter ~A/Ξ=A/(2R2D) vs temperatureT/θ forΞ2=30; 150; 300 (Ξ1=30).Reddots correspond to the lowerbilayer,blueones to theupper one.For comparison sake, solid lines correspond toone supportedmembrane forΞ=30anda free stackof twobilayers for Ξ2 = 300. Bottom: Corresponding average distance between membranes, R(η -d), given by eq 13, vs T/θ.

DOI: 10.1021/la903504n 4065Langmuir 2010, 26(6), 4057–4068


vðr, 0Þ ¼ 0 "r ð47Þ

vðr,η-1, 2Þ ¼ vðr, ηþ

1, 2Þ ð48Þ

r ) 3 v )ðr,η1, 2Þ ¼ 0 ð49Þ

Equation 47 imposes the no-slip condition at the substrate, andeq 49 ensures the membrane incompressibility. The two fluctuat-ing membranes impose normal forces f

1,2at z = η

1,2which

are balanced by fluid stress jumps

f1, 2 ¼ - σzzðr, ηþ1, 2Þþσzzðr, η-

1, 2Þ ð50Þ

where the stress tensor is σij= -pδ

ijþ μ(∂

jviþ ∂

ivj). From eqs

43-49, one finds easily that

f1, 2 ¼ δpðr,η1, 2Þ ð51Þ

Because of the linearity of eqs 43 and 44, one expects a linearrelation

f1f2

!¼ 4μqL v1

v2

!ð52Þ

between normal forces applied on membranes f1,2 and thez-component of flow velocity at the membranes, v1,2, whichdefines the resistance matrix 4μqL. Following Brochard andLennon24 and Seifert,38 one seeks for solutions of the type v|| =vx(x,z) = f(z)eiqx-γt, and vz(x,z) = g(z)eiqx-γt. From eqs 43-47we find, by writing ~z = qz and η~ = qη1

g1ð~z Þ ¼ A1 sinh ~z- ~z cosh ~zþ ~η1 sinh ~η1~z sinh ~z

sinh ~η1 þ ~η1cosh ~η1

ð53Þ

p1ð~z Þ ¼ 2μqA1

~η1sinh ~η1 sinh ~z

sinh ~η1 þ ~η1 cosh ~η1

- cosh ~z

ð54Þ

g2ð~zÞ ¼ ðA2 þB2~zÞe~z þðC2 þD2~zÞe-~z ð55Þ

p2ð~zÞ ¼ 2μqðB2e~z þD2e

-~zÞ ð56Þ

g3ð~zÞ ¼ ðC3 þD3~zÞe-~z ð57Þ

p3ð~zÞ ¼ 2μqD3e-~z ð58Þ

Finally by solving the system of four equations, eqs 48 and 49, wedetermine the four coefficients A2, B2, C2, and D2 and insertingthe result in eq 51, we find

Lð~η1, ~δÞ ¼ Að~η1ÞþBð~δÞ ð59Þwhere δ~ = qδ= q(η2 - η1) (δ= η2 - η1 is the average distancebetween membranes),

Að~η1 Þ ¼ 1

2

~η1 þ cosh ~η1 sinh ~η1

sinh2 ~η1 - ~η21

- 1 0

0 0

1A

0@ ð60Þ

and

Bð~δÞ ¼ 1

2

1þ~δþ cosh ~δ sinh ~δ

sinh2 ~δ- ~δ2

-sinh ~δþ ~δ cosh ~δ

sinh2 ~δ- ~δ2

-sinh ~δþ ~δ cosh ~δ

sinh2 ~δ- ~δ2

1þ~δþ cosh ~δ sinh ~δ

sinh2 ~δ- ~δ2

0BBBB@

1CCCCAð61Þ

The damping matrix in eq 41 is nothing but the inverse of theresistance matrix

Λðq, η1, η2Þ ¼ 1

4μqL-1ðqη1, qδÞ ð62Þ

The resistance matrix 4μqA(η~1) corresponds to the fluid frictioninduced by the substrate where the no-slip boundary conditionapplies, whereas the symmetric resistance matrix 4μqB(δ~) corre-sponds to the mutual hydrodynamic interactions between bothmembranes.

In the limit η~1f(¥, the fluid friction disappears (Af 0), andwe are left with the symmetric resistance matrix 4μqB(δ~), thelarge and low q limits of which were studied by Brochard andLennon.24,37 On the other hand, in the limit δ~ f ¥, membranesdecouple since Bf I2 and we find the result of Seifert for a singlemembrane close to a substrate.38 This is also the case in the limitsη~ f 0 or δ~ f 0 where the matrix L becomes a scalar

LðxÞ ¼ 1

21þ xþ cosh x sinh x

sinh2 x- x2

ð63Þ

where x = δ~ or x = η~1, respectively. The linearized lubricationapproximation x, 1 in eq 63, that is,L(x)=3/x3, corresponds toa flow parallel to the substrate.42

Damping Rates. Solving the coupled Langevin equations eq41 amounts to finding the eigenvalues and the eigenmodes of thedamping matrix γ(q) = Λ(q) U(q) where

UðqÞ σq2 þKq4 þA1 þA2 -A2

-A2 σq2 þKq4 þA2

!ð64Þ

By writing the two eigenvalues γ1(q) and γ2(q), eq 41 becomes, inthe eigenbasis

Dwiðq, tÞDt

¼ - γiðqÞ wiðq, tÞþ ðR-1Þijζjðq, tÞ ð65Þ

where R is the transformation matrix, u = Rw. The solution forthe temporal correlation function of wi(q,t) is

Æwiðq, tÞ wið-q, 0Þæ ¼ Æwiðq, 0Þ2æe-γiðqÞt ð66Þand the correlation function Æui(q,t) ui(-q,0)æ can then be com-puted coming back to the initial coordinates and using

Æju2ðqÞj2æ ¼ sin2 φ

λ1 þ σq2 þKq4þ cos2 φ

λ2 þσq2 þKq4

Æu2ðqÞ u1ð-qÞæ ¼ sin φ cos φ1

λ2 þσq2 þKq4-

1

λ1 þσq2 þKq4

ð67Þ

(42) Prost, J.; Manneville, J. -B.; Bruinsma, R. Eur. Phys. J. B 1998, 1, 465.

4066 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068


The eigenvalues γi(q) are increasing functions (see Figure 8)where limiting forms for q f 0 show a quadratic behavior

γ1ðqÞ ¼ A2δ3

12μq2 and γ2ðqÞ ¼ A1η31

12μq2 ð68Þ

For q f ¥ membranes decouple and we simply recover the freemembrane damping rate eq 42

γ1ðqÞ ¼ γ2ðqÞ ¼ K4μ

q3 ð69Þ

Note that without taking into account hydrodynamics, thedamping rates γi(q) are

γwoHIi ðqÞ ¼ Kq4 þ σq2 þ λi

4μqð70Þ

where λi are the eigenvalues given in eq 26. Hence hydrodynamicsinduce a large decrease of the damping rates for small andintermediate values of q, which is expected due to the no-slipcondition for the flow velocity at the substrate that slows downhydrodynamics.

Introducing the elastic decay length

ξ ffiffiffiffiffiffi2Kσ

rð71Þ

allows us to define the relevant dimensionless parameters in thedimensionless damping rates ~γ i defined as

~γi qξ, ~A1,~A2,

η1ξ,η2ξ

¼ 4μξ3

KγiðqÞ ð72Þ

where the ~Ai and ηi are fixed for a given couple of (Ξ1,Ξ2).Damping rates eq 72 are plotted in Figure 8 for (Ξ1,Ξ2) =

(30,3) at relative temperature T/θ = 1.5. Following the previoussection, it corresponds to variational parameters equal to ~A1 =18.6 and ~A2 = 0.4. From the phase diagram in Figure 5, we arethus in region 1 close to the unbinding of membrane 2 [R(η1 -d1)=0.72 andR(η2- d2)=4.76].However by choosing d1=2nm

and d2 = 3 nm, this yields η1/ξ = 0.07 and δ/ξ = 0.3, and thesubstrate-membrane 1 and intermembrane distances are smallenough to induce a sensible slowing down at small and inter-mediate wave-vectors q. Indeed, when hydrodynamic interactionsare neglected (thin lines in Figure 8) damping rates are largerby 4-6 orders of magnitude: for q f 0, γi

woHI(q) reduces toλi/4μq which should be compared to actual values in eq 68.Moreover, the damping rates of the two normal modes are lessdifferent for low q than expected without hydrodynamics. Fromeq 68 the low q values of the damping rates γi(q) are controlled byA1η1

3 formode 1 andA2δ3 for mode 2.Usually the values of these

quantities are quite close, which can be explained by the fact thatan increase ofAi implies a decrease of ηi as shown in the previoussection [see for instance eqs 30 and 31], and the decoupling of thetwo modes occurs only at intermediate q (see Figure 8).

At room temperature, the values used in Figure 8 yield ξ= 56nm (forσ=7 10-5 J 3m

-2 and κ=30kBT) withR1-1=R2

-1=2.5 nm, D1 = 10-6 J 3m

-2, and D2 = 10-7 J 3m-2 which are

reasonable values.1,10 Note that close to the unbinding transitionfor T j Tu, the average square value of height functions [eq 40]takes large values, the approximation of almost planar mem-brane fails, and dynamical renormalization techniques should beapplied.

Discussion

The variational approach that we have developed to describethe statistical physics of double supported membranes is acomplementary approach to numerical Monte Carlo simulationsand numerical transfer matrix methods. It allows an analyticaldetermination of the unbinding temperatures and the variationalparameters, the effective spring constants Ai(T) and the substra-te-membrane and intermembrane distances ηi(T), at any tem-perature and any values of the dimensionless adsorption strengths(Ξ1, Ξ2).

This is central for analyzing X-ray specular reflectivity spectrasince the Ai(T ) are necessary to fit the experimental structurefactors. In the experiments by Daillant et al.,1 the measuredeffective spring constant of the upper bilayerA2, which has a largeeffect on the spectrum shape, was found surprisingly weak, A2 ≈5 1010 J 3m

-4, much lower than the bare spring constant of thepotential estimated at 2R2D2≈ 2 1014 J 3m

-4. With these valueswe find ~A2/Ξ2 = 2 10-4, a value extremely low suggesting thatthe system is closed to the unbinding transition for the uppermembrane. Inserting their fitted parameter values for the surfacetension and the bending rigidity (σ = 5 10-4 J 3m

-2 and κ =1.5 10-19 J), we findΞ2= 200.Assuming for the firstmembraneΞ1= 100, the phase diagramFigure 5b shows that this would leadto T = 5θ. With R = 2 nm-1 we thus find T = 300 K which isroughly the temperature of the experiment. Although this is acrude estimation, we find the correct values, even if the para-meters Ai depend in a complex manner on both intermembraneand substrate-membrane potentials. Note that more generally,even if the transition is not perfectly described within ourvariational approach, the error made on the value of the unbind-ing temperature Tu is small since Ai(T ) varies rapidly close to Tu.

In recent experiments byStidder et al.,43 unbinding of the uppermembrane in DPPE double stacked bilayers with 10 mol % ofcholesterol has been observed between 52.2 and 56.4 C byneutron reflectivity. Their results suggest (i) a discontinuousunbinding transition since they observe both an hysteresis and alarger but finite roughness, Æu2æ1/2=0.6( 1.2 nm, at the transition,and (ii) that the presence of 10 mol % of cholesterol modifiesthe direct interaction potential and decreases the unbinding

Figure 8. Log-log plot of the (dimensionless) damping rates ~γi

versus adimensional wave-vector qξ for Ξ1 = 30 and Ξ2 = 3 attemperatureT=1.5θ (d1=2nmand d2=3nm). Thick solid linescorrespond to the full damping rates ~γi (i=1 in red and 2 in blue),whereas thin solid lines are damping rates without hydrodynamicseq 70, and the dashed line is the asymptotic result (qξ)3.

(43) Stidder, B.; Fragneto, G.; Roser, S. J. Soft Matter 2007, 3, 214.(44) Podgornik, R.; Hansen, P. L. Europhys. Lett. 2003, 62, 124.(45) Manghi, M.; Destainville, N.; Salome, L., in preparation.

DOI: 10.1021/la903504n 4067Langmuir 2010, 26(6), 4057–4068


temperature. We qualitatively obtain such unbinding transitionsfor Ξ2 j 2Ξ1, but values of the surface tension and the directadsorption potential are needed to do a quantitative comparison.Nevertheless, these experiments allow one to hope that a detailedcharacterization of double lipid bilayers will be achieved experi-mentally in the very near future.

When compared to numerical results of refs 15,21 where onlythe case Ξ1 = Ξ2 was explored, we also find sequential thermalunbinding transitions: for Ξ1 = Ξ2 = 30 the upper bilayerunbinds at Tu(2)/θ=2.6 and the remaining bilayer then unbindsat Tu(1)/θ = 3.3 (see Figure 6 and Figure 4). Moreover,membrane fluctuations increase when moving away from thesubstrate, η2 - d2 > η1 - d1. We show that this behavior is validwhenever Ξ2 < 2Ξ1. However, for higher values of Ξ2, oneobserve the reverse case where the stack of two membranesunbinds as a whole. This has been shown for bundles of stringsin two dimensions.22 Our phase diagram (Figure 5) resembles theone of ref 22 for strings. A way to compare them is to change thecoordinates of the phase diagram into Ξ1,2(T/θ)

2 T2/(κD1,2) inorder to eliminate the surface tension (which does not exist inthe string problem). Note that, for strings, the parameter κ

becomes an elastic (stretching) parameter. The qualitativediagram is then similar with a region corresponding to thepeeling process and another one where the bundle desorbs as awhole. However, contrary to these studies, we find a range ofparameters Ξ2 for which the double supported membranesdestack completely at the transition, this region being largerand larger when Ξ2 increases. The richness of the phasediagram is related to the fact that, contrary to refs 15,21,22we consider bilayers with a microscopic surface tension, whichis the more general case.

In this work, we assumed the substrate to be flat and did notconsider the substrate roughness which leads to pinned or hover-ing binding states for the first membrane.44 How the influence ofthe quenched substrate disorder propagates to the second mem-brane is an important issue. One might suggest that this effect forthe upper membrane is somehow annealed by thermal fluctua-tions of the lower one, and the pinning effect is then lesspronounced. Anyway, the influence of substrate roughness iscentral from an experimental point of view and deserves aquantitative study.

From an experimental perspective, we now discuss the biophy-sical interest of designing supported double (or even multiple)lipid bilayers. Ideally, for the upper bilayer to deserve to bequalified as “nearly floating” and weakly perturbed by thesubstrate, both δ and 1/A2 must be large so that this bilayerfluctuates nearly freely, while still being conveniently observableby spectroscopy techniques. One can in principle obtain a singleadsorbed bilayer with the same properties, by approaching itsunbinding transition. However, when the experimental goal is tomodel a biological situation (e.g., plasma membrane or cell-celljunction), both membrane composition and temperature areimposed by the biological context. This reduces the possibilityto design a single “nearly floating” bilayer. By contrast, our studysuggests that such a regime can be reached by stacking two (ormore) bilayers, because thermal fluctuations are enhanced in theupper one. Our goal is to help to anticipate in which regime ofparameters this situation is likely to occur.

Another experimental interest of studying thermal fluctua-tions, both at the equilibrium and dynamical levels, lies in thepossibility to study molecular diffusion in supported bilayers,

for example by single molecule tracking. On the one hand,fluctuations affect apparent diffusion coefficients because theprojected area is different from the real membrane one; therelationship between apparent and real diffusion coefficientsdepends on the relative time scales of membrane fluctuationsand diffusion.32 On the other hand, it has been recently shown bynumerical simulations35 that membrane elasticity decreaseprotein mobility, essentially due to the viscous drag of thedeformed membrane around the protein, but that membranefluctuations slightly increase protein mobility.33 However sucheffect is weak compared to viscous losses.35 It will be useful toquantify these effects, related to the amplitude of fluctuations, inthe case of stacked bilayers. This work is currently in progress.45

Acknowledgment. We thank Laurence Salome for her pre-cious knowledge of the experimental context and John Palmerifor illuminating discussions.

Appendix

Correlation Functions for Double Supported Bilayer. Byperforming an inverse Fourier transform of eq 37 and comingback to initial coordinates, one finds the following height-heightcorrelation functions

C11ðrÞ ¼ Æu1ðrÞ u1ð0Þæ ¼ kBT

2πcos2 φ

Z ¥

0

dqqJ0ðqjrjÞ

λ1 þσq2 þKq4

þ sin2 φ

Z ¥

0

dqqJ0ðqjrjÞ

λ2 þσq2 þKq4

ð73Þ

C12ðrÞ ¼ Æu1ðrÞ u2ð0Þæ

¼ kBT cos φ sin φ

2π-Z ¥

0

dqqJ0ðqjrjÞ

λ1 þσq2 þKq4

þZ ¥

0

dqqJ0ðqjrjÞ

λ2 þ σq2 þKq4

ð74Þ

C22ðrÞ ¼ Æu2ðrÞ u2ð0Þæ

¼ kBT

2πsin2 φ

Z ¥

0

dqqJ0ðqjrjÞ

λ1 þ σq2 þKq4

þ cos2 φ

Z ¥

0

dqqJ0ðqjrjÞ

λ2 þσq2 þKq4

ð75Þ

where J0 is the Bessel function of the first kind, and both φ and λidepend on the variational parametersA1(T) andA2(T), followingeqs 26 and 27.Stack of n Supported Membranes. In the case of n stacked

membranes on a substrate at z = 0, the full Hamiltonian can bewritten as the tensor product of the usualHelfrichHamiltonian ofa single fluctuating bilayermembrane and theHamiltonianwhichdescribes interactions between membranes:

H ½fhjðrÞg ¼ H Helfrich½hðrÞX1phonon þ 1HelfrichXH phonon½hj ð76Þ

where H Helfrich is the classical Helfrich Hamiltonian of a freemembrane, eq 1withV=0.The phononHamiltonian is describedby a succession of harmonic springs in the z-direction

H phonon½hj ¼ 1

2

Xn- 1

j¼0

Aj ½hjþ 1ðrÞ- hjðrÞ2 ð77Þ(46) Romanov, V. P.; Ul’yanov, S. V. Phys. Rev. E 2002, 66, 061701.(47) Constantin, D.; Mennicke, U.; Li, C.; Salditt, T. Eur. Phys. J. E 2003, 12, 283.

4068 DOI: 10.1021/la903504n Langmuir 2010, 26(6), 4057–4068


Taking thermal expansion into account would require a varia-tional approach where the Ai are different and computed var-iationally from, for instance, Morse potentials as defined in eq 4.However, even if possible in principle, this approach is hardlytractable in practice. In the following of this appendixwe considerall the Ai equal, Ai = A, "i.

Since the boundary conditions for the phonon modes areh0(r) = 0 and the last membrane free, we find from eq 77 thefollowing eigenmodes and eigenvalues

hlj ¼ -2ffiffiffiffiffiffiffiffiffiffiffiffiffi

2nþ 1p sin

2l þ 1

2nþ 1πj

ð78Þ

λl ¼ 4 sin22lþ 1

2nþ 1

π

2

ð79Þ

with 0e le n- 1. For n=2we find λ0 = 4 sin2(π/10) and λ1 =4 sin2(3π/10) which corresponds to λ2/A2 and λ1/A1, respectively,in eq 26 (with A1 = A2 = A).

The Hamiltonian (76) is diagonal in the Fourier (l, q) space,where l is the index for the deformation modes in the z-directionand q is the planar wave-vector in the (x,y) plane:

H ½hðl , qÞ ¼ 1

2

Zd2q

ð2πÞ2Xn- 1

l ¼0

ðAλl þ σq2 þKq4Þjhðl , qÞj2 ð80Þ

wherewedefine theFourier transformofheight fluctuationshj(r), by

hðl ,qÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffi2nþ 1

pXnj¼0

sin2l þ 1

2nþ 1πj

Zd2re-ir 3 qhjðrÞ ð81Þ

The Hamiltonian being Gaussian, the height-height correlationfunction written in Fourier space is

Æhðl ,qÞhðl 0, q0Þæ ¼ ð2πÞ2δl , l 0δðqþ q0Þ kBT

Aλl þσq2 þKq4ð82Þ

Theheight-height correlation functionCjk(r)=Æhj(r) hk(0)æbetweenmembrane j at r and membrane k at r = 0 is defined as

CjkðrÞ

¼ 2

π

kBT

2nþ 1

Xn- 1

l ¼0

sin2l þ 1

2nþ 1πj

sin

2l þ 1

2nþ 1πk

Zdq

qJ0ðqjrjÞAλl þσq2 þKq4

ð83Þ

This last equation is thus the generalization of eq 75 for nmembranes.

Similar but different results have been obtained in the contextsof smectic-A films46 with a different boundary condition for thelayer n, and of solid-supported multilayers47 using a continuousdescription of eq 79 valid for large n (in the continuum limit df 0,n f ¥ with L = nd held fixed).

2.3. Perspectives

2.3 Perspectives

L’utilisation de l’empilement des bicouches permet de s’approcher de la températurede transition pour la membrane supérieure, tout en gardant la température constante, cequi est indispensable pour étudier des processus biologiques (par exemple, les protéines sedénaturent sous l’augmentation de la température). Ainsi dès lors que la distance entre lesdeux membranes est suffisamment grande, la dynamique de la membrane supérieure estproche de celle d’une membrane libre, ce qui perment de se rapprocher de la situation invivo.

Nous entendons maintenant utiliser ce modèle pour décrire dans un futur proche lespropriétés diffusives des lipides ou des protéines de ces bicouches empilées. En particu-lier, une collaboration avec Laurence Salomé à l’IPBS concerne l’étude de la régulation dufonctionnement des récepteurs couplés aux protéines G. Un projet ANR du programme PI-RIBIO 2009 et intitulé «Régulation du fonctionnement des RCPG : hétéro-oligomerisationet organisation dynamique des récepteurs dans la membrane» est en cours. Il rassembledes biologistes (équipe de J.-M. Zajac, IPBS), des chimistes (équipe de J.-J. Bourguignon,Laboratoire d’Innovation Thérapeutique à Strasbourg), des bio-physiciens (équipe de L.Salomé, IPBS) et N. Destainville et moi-même du LPT.

Le but est de comprendre comment, lorsqu’il est activé par son ligand agoniste, lerécepteur Neuropeptide FF module les récepteurs µ aux opiacés dans la membrane cel-lulaire. Une hypothèse est un couplage entre les deux récepteurs médié par la membranelipidique, par exemple associé au changement de conformations qui induirait la modifica-tion des interactions hydrophobes entre les récepteurs et les lipides qui les entourent. Deplus, l’organisation dynamique de ces récepteurs dans la membrane est un point central àélucider pour comprendre le couplage entre ces récepteurs. Des expériences de retour defluorescence après photo-blanchiment (FRAP) à deux couleurs sont prévues, afin de suivrela dynamique des deux types de protéines, et une caractérisation de ces expériences sur lessystèmes modèles de bicouches supportées sera envisageable.

107


108

Chapitre 3

Ions aux interfaces : effetsdiélectriques et entropiques

3.1 Introduction

Les forces électrostatiques induites par des objets diélectriques immergés dans l’eaurégulent des phénomènes importants comme la stabilisation colloïdale [92, 27] et la sé-léctivité des ions par des membranes synthétiques [176] et biologiques [26]. Les forces decharge image induites par les sauts de la permittivité diélectrique entre les objets diélec-triques et l’eau les entourant joue un rôle majeur dans ces phénomènes. Ainsi l’équilibreentre des objets chargés de même nature dans un solvant est régi par la compétition entreles interactions électrostatiques répulsives et les interactions attractives de van der Waalsinduites par leur faible permittivité diélectrique. Ce mécanisme est l’essence de la théoriede Deryaguin-Landau-Verwey-Overbeek [190].

La perméabilité des canaux biologiques vis-à-vis des ions échangés entre l’extérieur etl’intérieur des cellules est gouvernée par un mécanisme similaire. Le discontinuité diélec-trique importante entre la membrane biologique (constante diélectrique 1 εm ' 2) et cellede l’eau dans le canal (εw ' 78) est à l’origine de la barrière de potentiel s’opposant à lapénétration d’un ion dans le pore. Ainsi Parsegian montra que l’énergie nécessaire à l’ionpour passer du réservoir à l’intérieur d’un nanopore cylindrique infini de rayon a = 0.2 nmest de l’ordre de 16 kBT [149]. Ce résultat suggère donc, de manière paradoxale, que lespores biologiques sont tous imperméables aux ions. En réalité, cette barrière est plus faiblepour des pores de taille finie [116] et la présence éventuelle de charges surfaciques peutégalement faciliter l’attraction des contre-ions dans le pore [115].

1. Il s’agit en fait de la permittivité relative en unités de la permittivité du vide ε0 = 8.85 ×10−12 C.V−1.m−1.

109

Chapitre 3. Ions aux interfaces : effets diélectriques et entropiques

Dans ce qui suit, nous développons une nouvelle approche permettant l’étude de la péné-tration des ions dans les nanopores en se concentrant sur les effets diélectriques, d’écrantageélectrostatique et de charge surfacique.

3.2 Théorie des champs pour les électrolytes

Le formalisme de théorie des champs s’écrit plus simplement dans l’ensemble grand-canonique, c’est-à-dire que l’on considère une assemblée de p espèces d’ions dans un solvant,l’eau dans la plupart des cas, à potentiel chimique fixé, le nombre d’ions pouvant varier.Les ions sont considérés comme ponctuels et l’eau comme un milieu diélectrique continudécrit par sa constante diélectrique εw ≈ 78. Il s’agit d’un cas limite du «modèle primitifrestrictif» dans lequel les ions sont modélisés par des sphères dures chargées avec la chargeplacée au centre. Ce modèle devient nécessaire pour des concentrations d’ions élevées, caril tient compte des interactions de volume exclu [58, 122, 72].

On suppose la solution en contact avec des objets (parois solides, membranes. . .) deconstante diélectrique différente notée εm. En pratique on s’intéresse à des objets carbonés,non-conducteurs pour lesquels εm ≈ 1− 4 et on s’attend donc à des effets de charge imageimportants associés au saut diélectrique aux interfaces. On note de façon générale ε(r)la distribution des permittivités diélectriques dans tout l’espace. La grande fonction departition s’écrit alors

Q =p∏

i=1

∞∑

Ni=0

λNiiNi!

∫ Ni∏

j=1

drj e−(H−Es) (3.1)

Par souci de simplicité, les énergies sont exprimées en unité de l’énergie thermique kBT . 2

La fugacité des ions est λi = eµi/Λ3i , où Λi = h/

√2πmikBT est la longueur d’onde de de

Broglie d’un ion i et µi est le potentiel chimique. Celui-ci est fixé dans l’électrolyte étudiépar la condition d’équilibre chimique λi = λi,b, où b désigne le réservoir des ions («bulk»).

L’interaction électrostatique dans l’éq. (3.1) s’écrit

H =12

∫dr′dr ρc(r)vc(r, r′)ρc(r′) (3.2)

où la distribution totale de charge s’écrit (en unités de la charge élémentaire e)

ρc(r) =p∑

i=1

Ni∑

j=1

qiδ(r− rj) + ρs(r) (3.3)

2. La température T est celle du solvant dans le réservoir.

110

3.2. Théorie des champs pour les électrolytes

La valence des ions est notée qi et la distribution surfacique de charge fixe, ρs(r). Lepotentiel d’interaction vc(r, r′) est le potentiel de Coulomb, solution de

v−1c (r, r′) = − 1

βe2∇[ε(r)∇δ(r− r′)

](3.4)

avec les conditions aux limites

εm∂vc∂n

∣∣∣∣m

= εw∂vc∂n

∣∣∣∣w

(3.5)

où n est la normale à la surface. Enfin l’énergie propre des ions que l’on soustrait dansl’éq. (3.1) est

Es =vbc(0)

2

p∑

i=1

Niq2i (3.6)

où vbc(r) = `B/r est le potentiel de Coulomb en solution et la longueur de Bjerrum, `B, estdéfinie comme la distance entre deux charges élémentaires qui interagissent électrostati-quement avec une énergie kBT , soit `B = e2/(4πεwkBT ) ' 0.7 nm dans l’eau à T = 300 K.

En appliquant une transformation de Hubbard-Stratonovitch à l’éq. (3.1), en introdui-sant un champ auxiliaire φ(r), puis en sommant sur les Ni, la grande fonction de partitions’écrit plus simplement comme

Q =∫Dφ e−H[φ] (3.7)

Le hamiltonien de type sine-Gordon apparaissant dans l’éq. (3.7) est

H[φ] = Hfree[φ] +Hsource[φ] +Hint[φ]

=∫

dr

[ε(r)2βe2

[∇φ(r)]2 − iρs(r)φ(r)−∑

i

λieiqiφ(r)

](3.8)

où sont introduites des fugacités renormalisées par l’énergie propre λi = λieq2i2vbc(0). Cet

hamiltonien est séparé en trois contributions : une énergie associée aux fluctuations duchamp, Hfree, une énergie due aux sources de charge fixes, Hsource et une énergie d’inter-action non-linéaire, Hint. Le potentiel électrostatique ψ(r) est relié au champ imaginairefluctuant φ(r) par ψ(r) = −i〈φ(r)〉.

La valeur moyenne de la densité locale ρi(r) se calcule simplement à partir de Ni ≡∫drρi(r) = λi

∂ lnQ∂λi

et

ρi(r) = λi〈eiqiφ(r)〉 (3.9)

Formellement, il est possible de faire apparaître un paramètre de couplage en renorma-lisant les longueurs par la longueur de Gouy-Chapman

`G = (2πq`Bσs)−1 (3.10)

111


qui est la longueur pour laquelle l’interaction entre un seul ion de valence q avec unesurface plane chargée, de densité surfacique de charge moyenne σs, est égale à kBT . Ainsien écrivant r = r/`G, ρs(r) = `Gρs(r)/σs et introduisant φ(r) = qφ(r) (en simplifiantqi = q), la fonction de partition (3.7) se réécrit [136] Q =

∫Dφ exp(−H[φ]/Ξ) et le

hamiltonien

H[φ] =∫

dr2π

[ε(r)4εw

[∇φ(r)]2 − iρs(r)φ(r)− 14

∑

i

Λieiφ(r)

](3.11)

La fugacité renormalisée s’écrit Λi = 8πλi`3GΞ et le paramètre de couplage est défini par

Ξ = 2πq3`2Bσs = q2 `B`G

(3.12)

On voit alors clairement que le paramètre de couplage joue le rôle d’un paramètre deboucles dans un développement autour du point col. Il permet de définir deux limites quipeuvent être étudiées analytiquement :

i) la limite de faible couplage, Ξ → 0, correspondant au hautes températures, faiblevalences ioniques et faible charges de surface. À l’ordre 0 en Ξ−1, on a alors lasolution champ moyen de Poisson-Boltzmann où les interactions électrostatiques sontdominées par les fluctuations thermiques. Des calculs perturbatifs en diagrammes deboucles autour du champ moyen ont été menés [5, 6, 153, 154, 55, 56]. En particulier,à l’ordre 1, les correlations sont calculées au niveau de Debye-Hückel [58, 122, 139].

ii) la limite de fort couplage, Ξ→∞, correspondant au faibles températures, grande va-lences et fortes charges de surface. Dans cette limite, les interactions surface chargée-ions mobiles contrôlent la distribution des ions dans la direction perpendiculaire àla surface. Leurs positions fluctuent peu dans les directions parallèles à la surface,on parle alors de cristallisation de Wigner-Seitz [166, 134]. Dans ce cas, à l’ordre 0,dans la direction perpendiculaire à la surface, les ions suivent une loi de distribu-tion «barométrique» d’un gaz parfait dans un champ électrique constant créé parla surface [175]. Les corrections à cette loi correspondent à un développement duViriel en puissance de Ξ−1, de manière semblable à celui des gaz réels [133, 136].Cependant, des travaux récents [87, 170] ont montré que la première correction dansle développement était en réalité en Ξ−1/2.

3.2.1 Approche champ moyen et limite de faible couplage

Le champ moyen ψPB = −iφPB(r) est défini par l’équation du point col δH[φ]/δφ(r) = 0qui conduit à l’équation de Poisson-Boltzmann

1βe2∇[ε(r)∇]ψPB(r) = −ρs(r)−

∑

i

qiλie−qiψPB(r) (3.13)

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Par exemple, dans le cas simple de contre-ions (q+ = q, q− = 0, λ+ = λ) au contactd’une surface chargée, l’équation de Poisson-Boltzmann (3.13) conduit à la solution deGouy-Chapman

ψPB = 2 ln(1 + z/`G) (3.14)

ρPB(z) =∑

i

qiλie−qiψPB(r) =

1`G(1 + z/`G)2

(3.15)

où z est la coordonnée dans la direction perpendiculaire à la surface et la valeur de λi aété fixée par l’électroneutralité q

∫ρPB(z)dz = |σs|.

3.2.2 Calcul à un boucle : théorie de Debye-Hückel

À l’ordre suivant en Ξ1/2 en φ dans le développement en boucles, la fonction de corré-lation à deux points est définie par

[− 1βe2∇ε(r)∇+

∑

i

q2i λi e

−qiψPB(r)

]G(r, r′) = δ(r− r′) (3.16)

La résolution des éqs. (3.13) et (3.16) pour des géométries particulières comme une surfaceplane chargée ou deux surfaces planes chargées de même signe permet d’obtenir les correc-tions au champ moyen pour le potentiel électrostatique et donc le densité d’ions à conditionde bien tenir compte des termes provenant du développement de la fugacité renormaliséeen fonction de Ξ. Les corrections au 1er ordre des solutions précédentes, qui nécessitentla résolution de l’éq. (3.16), ont été calculées dans le cas d’une seule surface chargée enprésence de contre-ions seuls par Podgornik et al. [153, 154] et Netz et Orland [138], et parAttard et al. dans le cas de deux interfaces [5, 6].

Pour un électrolyte en solution, le champ moyen est φPB(r) = 0 et le développementen φ2 du hamiltonien éq. (3.8) conduit à un hamiltonien quadratique avec pour noyau lepotentiel de Debye-Hückel [137, 139]

v−1DH(r, r′) = v−1

c (r, r′) +∑

i

q2i λiδ(r− r′) (3.17)

qui dans le cas où vc(r, r′) = `B/|r− r′| (ε(r) = εw) conduit à

vDH(r, r′) = `Be−κDH|r−r′|

|r− r′| avec κ2DH = 4π`B

∑

i

q2i λi (3.18)

qui est l’expression en grand canonique du potentiel classique de Debye-Hückel [58, 122].Lorsque des interfaces diélectriques sont présentes, l’expression de vDH est plus compli-

113


quée [5, 153]. Le grand-potentiel s’écrit alors, en incluant le calcul à une boucle 3

Ω =H[φPB]

Ξ− 1

2tr lnG+O(Ξ) (3.19)

3.2.3 Développement du viriel et limite de fort couplage

Dans la limite de fort couplage Ξ→∞, un développement en puissances de la fugacitérenormalisée λi est possible. Cela correspond d’après l’éq. (3.9) à un développement encumulants en concentration et est donc la version grand-canonique du développement duViriel. À l’ordre 0, on complète le carré dans l’éq. (3.8) et le hamiltonien devient

H[φ] = Hfree[φ] +12

∫drdr′σ(r)v(r, r′)σ(r′) (3.20)

où

Hfree[φ] =12

∫drdr′[φ(r)− iψfree(r)]v−1(r, r′)[φ(r′)− iψfree(r′)] (3.21)

ψfree(r) =∫

dr′σ(r′)v(r, r′) (3.22)

La dernière équation définit le potentiel électrostatique créé par la source σ(r), en l’ab-sence d’ions mobiles. En effectuant ensuite un développement en puissances de λ, le grandpotentiel Ω = − lnQ devient pour un électrolyte symétrique (q+ = −q− = q, λi = λ)

Ω = −12

tr ln v+12

∫drdr′σ(r)v(r, r′)σ(r′)−

∞∑

k=1

(2λ)k

k!

⟨[∫dr cos(qφ(r))

]k⟩

c,free

(3.23)

où l’indice c réfère à la partie connexe des espérances calculées en utilisant le hamiltonienlibre Hfree. Le premier terme correspond à l’énergie des fluctuations du vide. Le second estl’énergie associée aux sources électrostatiques. Enfin le dernier terme est la contributiondes interactions entre ions mobiles. Au 1er ordre dans le développement en cumulants,ce qui définit la limite de faible concentration ou de fort couplage (SC), le calcul de laconcentration donnée par l’éq. (3.9) conduit à

ρi,SC(r) = λi〈eiqiφ(r)〉c,free (3.24)

et la première correction au grand potentiel est

Ω(1) − Ω(0) = −∑

i

∫drρi,SC(r) (3.25)

3. Notons qu’une méthode de régularisation est nécessaire, car des divergences apparaissent dansl’éq. (3.19). En général on soustrait l’énergie propre, qui s’écrit dans le bulk [139] V vbc(0)λ =

V vbc(0)κ2DH/(8π`Bq

2).

114


ce qui correspond à la loi des gaz parfaits (pression osmotique) dans l’ensemble canonique.Dans le cas de contre-ions dans la région z > 0 et d’un plan chargé ρs(z) = σsδ(z),

la loi de Gauss à l’ordre 0, c’est-à-dire en négligeant les contre-ions, conduit à un champconstant E = e2σs

2εwz soit ψfree = 2π`Bσsz. La distribution des ions est alors simplement la

loi barométrique [175]ρSC(z) = 2π`Bσ2

s e−z/`G (3.26)

ce qui a été vérifié par des simulations Monte-Carlo [133].Un diagramme de phase a été calculé établissant les domaines de validité des approches

faible et fort couplage dans le cas d’une et de deux surfaces chargées [27]. En particulier, ilmontre que pour les valeurs intermédiaires Ξ ∼ 1, les deux approches présentées ci-dessusne sont pas satisfaisantes. De plus les calculs s’avèrent extrêmement compliqués aux ordressuivants dans les développements [134].

3.2.4 Images électriques et charges de polarisation

Hormis les distributions de charges que peuvent présenter les surfaces des objets méso-ou macroscopiques (colloïdes, membranes organiques,. . .), ces objets ont, en règle générale(pour ne pas dire toujours), une constante diélectrique différente de celle du solvant. Lapermittivité diélectrique indique comment le milieu se polarise sous l’influence d’un champélectrique. Ainsi l’eau, qui est un liquide polaire «écrante» d’un facteur 78 par rapport auvide, le champ créé par une source donnée.

Dans le cas le plus simple d’une interface plane entre la membrane et l’eau, la fonctionde Green vc(r, r′), solution de l’éq. (3.4), s’écrit en l’absence d’autres ions,

vc(r, r′) =`B|r− r′| −

∆`B√(r− r′)2 + 4zz′

(3.27)

où z, z′ > 0 et est introduit le paramètre associé au saut diélectrique

∆ =εw − εmεw + εm

(3.28)

Cette solution du potentiel créé par une charge élémentaire e en présence d’une interfacepolarisée peut être formellement vue comme le potentiel créé par deux charges : la chargeplacée en r = (x, y, z) et une charge image fictive, e∆ placée en (x, y,−z) [97]. Cette chargeimage est en réalité le résultat de la charge de polarisation crée à l’interface en z = 0 quivient du fait que la réponse du milieu de part et d’autre de l’interface est différente. Dans lecas d’interfaces courbes, la notion de charge image devient caduque (il y en a une infinité)et on utilise le concept de charge de polarisation. Cette interaction peut être attractive siεm > εw et répulsive dans le cas contraire indépendamment du signe de la charge.

115


Comme nous allons le voir plus loin, la présence d’électrolyte dans une des deux régionsva induire une interaction supplémentaire. En effet, dans le volume de la solution, un ionpar exemple chargé négativement, est entouré d’une atmosphère ionique composée d’ionspositifs. Cet effet de corrélation ion-ion abaisse l’énergie libre du système par rapport àun système de référence constitué de particules identiques mais non chargées. Or prèsde l’interface avec un corps solide, l’absence d’électrolyte de l’autre côté conduit à uneinteraction répulsive puisque les ions préfèrent être dans une région où l’écrantage estmaximal. Il s’agit d’un effet dit de solvatation [85, 86].

3.3 Approche variationnelle

Le principe de la méthode variationnelle consiste à approcher le hamiltonien exact (3.8)par un hamiltonien quadratique ce qui permet de faire des calculs analytiques. Nous décri-vons ici l’approche utilisée par Netz et Orland [140]. L’idée est d’introduire dans le grandpotentiel, Ω, un hamiltonien variationnel H0 dépendant de paramètres (ou fonctions) in-connus à déterminer

Ω = − ln∫Dφ e−H0 − ln〈eζ(H0−H)〉0 (3.29)

où ζ est un paramètre qui contrôle l’ordre du développement (pris égal à 1 à la fin ducalcul) et l’indice 0 signifie que les espérances sont calculées avec le hamiltonien d’essaiH0. En utilisant le développement en cumulants, on introduit

Ωn = Ω0 −n∑

m=1

ζm

m!〈(H0 −H)m〉c,0 (3.30)

Si la série converge, alors Ω = limn→∞Ωn. Au 1er ordre, la propriété de convexité del’exponentielle conduit à

Ω1 = Ω0 + 〈H −H0〉0 ≥ Ω (3.31)

ce qui définit une borne supérieure. L’idée est alors de minimiser Ω1 par rapport auxparamètres variationnels.

On choisit un hamiltonien gaussien très général

H0[φ] =12

∫drdr′ [φ(r)− iφ0(r)] v−1

0 (r, r′)[φ(r′)− iφ0(r′)

](3.32)

et Ω0 = −12tr ln v0. Les paramètres variationnels sont donc le noyau électrostatique v0(r, r′),

qui est une fonction de corrélation à 2 points prenant en compte les corrélations ion-ionet les discontinuités diélectriques, lorsqu’elles sont présentes, et le potentiel électrostatique

116

3.3. Approche variationnelle

moyen φ0(r). Les équations variationnelles δΩ1

δv−10 (r,r′)

= 0 et δΩ1δφ0(r) = 0 conduisent à deux

équations couplées (avec ε(r) = εw dans l’électrolyte) [140]

∆φ0(r) + 4π`B∑

i

qiλi e− q

2i2W (r)e−qiφ0(r) = −4π`Bρs(r) (3.33)

−∆v0(r, r′) + 4π`B∑

i

q2i λi e

− q2i2W (r)e−qiφ0(r)v0(r, r′) = 4π`Bδ(r− r′) (3.34)

où apparaît l’énergie propre nécessaire (ou «potentiel chimique local d’excès») pour amenerl’ion i du réservoir à la position r, q2

iW (r)/2, avec

W (r) ≡ limr→r′

[v0(r, r′)− vbc(r− r′)

](3.35)

On voit apparaître dans le premier membre de l’éq. (3.33) la concentration de chargelocale. Ainsi l’éq. (3.33) est une équation non-linéaire de type Poisson-Boltzmann, définieen (3.13), modifiée. En effet, la source de charge associées aux ions de l’électrolyte estlocale et évaluée selon la distribution de Boltzmann. Elle fait intervenir donc la quantitéq2iW (r)/2 + qiφ0(r) qui est le potentiel électrochimique local de l’ion i et dépend de v0

déterminé via l’éq. (3.34). En présence d’interfaces diélectriques et chargées,W (r) et φ0(r)varient avec la position de l’ion par rapport aux interfaces [119, 191].

La seconde éq. (3.34) a une structure hybride. Alors que les énergies qiW (r) et qiφ0(r)interviennent dans l’exponentielle provenant d’un poids de Boltzmann, qiv0(r, r′) est «li-néarisé». Cette approximation de type Debye-Hückel provient directement du choix d’unhamiltonien gaussien, éq. (3.32), similaire au Hamiltonien de Debye-Hückel. Notons quel’éq. (3.34) fait intervenir une constante de Debye locale, κ(r), définie selon

κ(r)2 = 4π`B∑

i

q2i λie

− q2i2W (r) e−qiφ0(r) (3.36)

qui tient compte à la fois de l’influence des sauts diélectriques et des distributions de chargesaux interfaces via respectivement W (r) et φ0(r). En particulier les effets de dépletionsinduits par la répulsion diélectrique et par le déficit de solvatation vont diminuer κ(r) prèsde l’interface. Ainsi l’éq. (3.34) variationnelle diffère donc de l’équation de Debye-Hückelcar elle traite l’écrantage de façon locale et non homogène.

Enfin, il est important de noter que, tout comme l’approche Debye-Hückel, cette ap-proche variationnelle considère à la fois les effets non-linéaires dans l’éq. (3.33), et va au-delàde l’approche champ moyen (Poisson-Boltzmann) puisqu’elle considère les fluctuations etles corrélations au niveau Gaussien. Elle va également au-delà de l’approche faible couplageen développement à une boucle, puisque dans les équations (3.13) et (3.16), le potentielchimique d’excès local, W (r), est absent.

117


Ces équations variationnelles ont été obtenues par Netz et Orland [140]. Elles ontégalement été établies par Avdeev et Martynov [7] en utilisant une procédure de fermeturedite de Debye des équations hiérarchiques BBGKY pour les fonctions de distributions.Yaroshchuk a ensuite trouvé une solution approchée des ces équations de fermeture pourdes électrolytes confinés afin d’étudier l’exclusion ionique des membranes [204].

Comment se situe cette approche variationnelle par rapport aux cas limites de faiblecouplage (Ξ→ 0) et fort couplage (Ξ→∞) ? Deux cas se présentent :

i) l’interface est chargée. Il est alors possible de renormaliser les longueurs par `G et,en introduisant v0 = v0`G/`B et W = W`G/`B, les éqs. (3.33)-(3.34) se réécrivent pour unélectrolyte symétrique

∆φ0(r)− Λe−Ξ2W (r) sinh φ0(r) = −2ρs(r) (3.37)

−∆v0(r, r′) + Λe−Ξ2W (r) cosh φ0(r)v0(r, r′) = 4πδ(r− r′) (3.38)

Dans la limite Ξ→ 0, le couplage entre φ0 et v0 disparaît dans l’éq. (3.37) et on retombesur les éqs. (3.13)-(3.16) ainsi que le grand potentiel (3.19) 4. Dans la limite Ξ → ∞, àcondition que W (r) reste positif ce qui est généralement le cas, les éqs. (3.37) et (3.38)conduisent directement à v0 = vc et au potentiel φ0 linéaire en z.

ii) Dans le cas où l’interface est neutre, les éqs. (3.33)-(3.34) ne peuvent pas être re-normalisées par `G mais seulement par la longueur de Bjerrum, `B. Le problème est alorsplus complexe, car comme nous le verrons par la suite un champ φ0 non nul peut appa-raître pour contrecarrer les inhomogénéités de charges induites par la répulsion diélectrique.

Notons qu’il existe d’autres types d’approches, non-variationnelles, permettant de consi-dérer les effets de charge image et des corrections associées aux correlations ioniques. Lesthéories des équations intégrales utilisent des relations de fermeture approchées de l’équa-tion de Ornstein-Zernicke, comme l’approximation «hyppernetted chain» utilisée par Kjel-lander et Marcelja [103] et l’approximation «mean spherical» [5, 6]. D’autres approchesmodifient directement l’équation de Poisson-Boltzmann en incluant une énergie propreapprochée des ions [143] ou en partant d’une fonctionnelle de la densité locale [189].

3.4 Application au cas d’une solution d’électrolyte

Considérons un électrolyte symétrique dans son volume. Dans ce cas, il n’y a pas d’autresources de charges que les ions de l’électrolyte, soit ρs = 0 et l’éq. (3.33) impose φ0 = 0.Il reste à résoudre l’éq. (3.34) pour trouver v0(r, r′). L’invariance par translation dans

4. Avec un terme supplémentaire 〈H0〉0 qui régularise le tracelog, ce qui permet de définir une approchede couplage faible donnant une borne supérieure (non-optimisée) sur Ω.

118

3.5. Electrolyte proche d’interfaces diélectriques planes

la solution impose v0(r, r′) = v0(|r − r′|) ce qui conduit à W (r) = cte. La solution del’éq. (3.34) est donc de la forme Debye-Hückel v0(r−r′) = `Be

−κb|r−r′|/|r−r′| avec κb qu’ilreste à déterminer. Puisque d’après l’éq. (3.35), W (r) = −κb`B, d’après l’éq. (3.34) κb estdéterminé de façon auto-cohérente par l’équation implicite

κ2b = 4π`B

∑

i

q2i λi e

q2i κb`B/2 (3.39)

Une autre façon équivalente de trouver l’éq. (3.39) est d’écrire le grand potentiel varia-tionnel en choisissant pour v0 la forme Debye-Hückel avec κb comme paramètre variationnel

Ω1(κb)V

= −p(κb) =κ3b

24π−∑

i

λi exp(q2i κb`B

2

)(3.40)

où −12tr ln v0+〈Hfree−H0〉0 = −1

2tr ln v0+ 12

∫r,r′ v0(v−1

c −v−10 ) = V κ3

b24π qui peut également se

réécrire en utilisant le processus de charge [58, 122], c’est-à-dire en construisant l’électrolytele long d’un chemin thermodynamique tel qu’à chaque étape, les ions portent un fractionζ de leur charge finale − κ2

bV8π`B

∫ 10 dζ[v0(κb

√ζ) − v0(κb)]. Nous voyons que la minimisation

de l’éq. (3.40) conduit directement à l’éq. (3.39). Cette équation conduit à des instabilitéspour les grandes valeurs de λi (c’est-à-dire les grandes concentrations) dans le cas d’ionsponctuels. L’inclusion des interactions de cœur dur régularise la théorie [85, 61]. De plus,l’éq. (3.39) n’a pas de solution pour q2

i κb(λ)`B > 4. Pour les faibles fugacités, elle conduità la loi limite de Debye-Hückel (DHLL). En effet, la concentration dans la solution estdonnée par

ρi,b(λ) = λi∂p

∂λi= λi e

q2i2κb`B (3.41)

ce qui conduit à réécrire l’éq. (3.40) comme

κ2b = 4π`B

∑

i

q2i ρi,b(λ) (3.42)

En résolvant l’éq. (3.41) pour µi, nous tombons sur le résultat DHLL dans l’ensemblecanonique 5 :

µi = ln(ρi,bΛ3i )−

q2i

2κb`B (3.43)

3.5 Electrolyte proche d’interfaces diélectriques planes

Nous présentons ici succinctement les résultats publiés dans l’article [37] présenté ci-après. Même pour des géométries simples, comme une interface plane ou un pore en

5. Notons que dans l’ensemble grand canonique, la constante de Debye-Hückel, valable au 1er ordre enλ, s’écrit selon l’éq. (3.18), donc le résultat variationnel de l’éq. (3.42) est non-perturbatif en λ.

119


forme de fente, la résolution des éqs. (3.33)-(3.34) s’avère très compliquée, d’une partparce qu’elles sont couplées via W (r) qui apparaît dans le poids de Boltzmann et d’autrepart parce qu’elles sont non-linéaires en φ0. Dans la suite nous choisissons de simplifierl’éq. (3.34) en modélisant le paramètre d’écrantage local κ(r) par un paramètre constantpar morceaux

κv(r) =

κv en presence d′electrolyte0 sinon

(3.44)

Cette approximation est une généralisation de l’approximation de Onsager-Samaras [142](où la fonction de Green est approchée par une forme de Debye-Hückel avec la constanteκb) au cas d’une constante d’écrantage quelconque, κv. Les paramètres variationnels sontdonc φ0(z) où z est la coordonnées dans la direction perpendiculaire aux interfaces (voirfigure 3.1) et κv. Pour décrire le réservoir, nous utiliserons par la suite plutôt ρb que λb quilui est fixé dans l’ensemble grand canonique via l’éq. (3.41).

La concentration des ions près de l’interface est alors simplement donnée par la loi deBoltzmann

ρi(r) = ρi,b exp [−Φi(r)] (3.45)

où le potentiel de force moyenne (PMF)

Φi(r) =q2i

2w(r) + qiφ0(r) (3.46)

a deux contributions, la différence entre le potentiel chimique d’excès en r et celui dans lasolution

w(r) = W (r)−Wb = (κv − κb) + δv0(r, r;κv) (3.47)

et l’énergie électrostatique de l’ion i dans la potentiel φ0(r). Le potentiel v0 peut être séparéselon

v0(r, r′) =e−κv |r−r′|

|r− r′| + δv0(r, r′) (3.48)

où le terme correctif, δv0(r, r′) est induit par la présence des surfaces. Le paramètre varia-tionnel κv est déterminé en minimisant Ω1(κv) [éq. (3.31)].

3.5.1 Paroi plane

Le but de cette étude est triple : i) tout d’abord valider cette approche dans les cas d’unesurface plane neutre et chargée ; ii) mettre en exergue le rôle joué par les effets diélectriqueset de solvatation près de l’interface, qui ne sont pas pris en compte au niveau Debye-Hückel,et iii) montrer que cette approche variationnelle interpole de façon satisfaisante les deuxcas limites de champ moyen (partie 3.2.1) et fort couplage (partie 3.2.3). Notons que dans

120


!

0 z

!w !!0 zd

!w

(a) (b)

Figure 3.1 – Géométrie du système formé d’un électrolyte au contact d’une interfacediélectrique plane (a) et à l’intérieur d’un pore en forme de fente (b).

la plupart des situations expérimentales, le paramètre de couplage est de l’ordre de l’unité 6

et le champ moyen n’est pas valable.La présence d’un volume d’électrolyte infini au contact de la paroi diélectrique fixe la

valeur de la constante d’écrantage κv = κb. On trouve en résolvant l’équation de Debye-Hückel dans la géométrie de la figure 3.1(a)

w(z) = `B∆e−2κbz

2z+ `Bκb

2ηη + 1

∫ ∞

1dx

x−√x2 − 1

x+ η√x2 − 1

e−2κbxz (3.49)

où η = εm/εw et ∆ est défini dans l’éq. (3.28). Le premier terme est un terme de répulsiondiélectrique semblable à celui que l’on obtient avec l’éq. (3.27) multiplié par un facteurd’écrantage e−2κbz et le second terme est dû à la fois à la répulsion diélectrique et audéficit de solvatation (puisqu’il subsiste pour η = 1). Le potentiel φ0(z) est solution del’éq. (3.33).

Le cas d’une surface neutre et d’un électrolyte symétrique a été examiné par Hatloet Lue en résolvant numériquement l’éq. (3.34). L’accord avec les simulations est remar-quable [85, 86]. Dans ce cas, l’interaction avec la surface n’étant pas sensible au signe de lacharge, les profils de concentration des anions et des cations sont identiques et φ0(z) = 0.

Le cas d’un électrolyte asymétrique est plus intéressant et a été étudié par Bravina [30]dans l’approximation de concentration faible dans le réservoir. Nous avons montré, enrésolvant numériquement l’éq. (3.33) pour φ0(z) qu’une zone de séparation de charge secrée près de l’interface (z ' `B) et |φ0(z)| décroît linéairement avant de relaxer vers la valeur0 pour des distances supérieures sur une longueur ' 1.4κ−1

b . Un choix de potentiel φ0(z)

6. Ce qui correspond, par exemple, pour un électrolyte monovalent à température ambiante à σs '0.3 e/nm2.

121


continu par morceaux avec 3 paramètres scalaires variationnels reproduit parfaitement bienla solution numérique. Dans la limite de faible ρb, nous retrouvons les résultats de Bravina.

Le cas d’un électrolyte symétrique au contact d’une surface chargée (négativement) n’apas été étudié auparavant avec ce type d’approche. Là encore deux zones apparaissent :près de l’interface, φ0 ' 2z/`G sur une région d’épaisseur h ' `G ln Ξ. Il s’agit d’une zonede fort couplage, où les ions sont exclus totalement à cause de la répulsion diélectriqueet du déficit de solvatation près de la surface. Du fait de l’attraction électrostatique descontre-ions par la surface, un maximum de ρ+(z) apparaît en bordure de cette zone. Pourdes distances plus grandes, z > h, nous retrouvons un régime champ moyen où φ0(z) estbien décrit par l’équation de Poisson-Boltzmann.

L’approche présentée permet donc d’interpoler simplement entre les deux régimes,champ moyen et fort couplage. Notons que Ξ mesure ici l’intensité du potentiel chimiqued’excès w(z) défini dans l’éq. (3.49). Les paramètres ∆, qui mesure la répulsion diélectrique,et Ξ sont donc multipliés. Du fait de la répulsion diélectrique et du déficit de solvatation,nous retrouvons donc bien le profil de Poisson-Boltzmann dans la limite Ξ → 0 mais pasle profil exponentiel dans la limite Ξ → ∞, puisque la répulsion près de la surface estexacerbée. En effet, alors que dans le cas d’une interface chargée sans saut diélectrique laconcentration est définie selon l’éq. (3.26), dans le cas de discontinuités diélectriques, elleest donnée par

ρSC(z) = ρ0 exp[−(1 + ∆)

z

`G− Ξ∆

`G4z

](3.50)

la constante ρ0 étant fixée par la condition d’électroneutralité 7.

3.5.2 Pore en forme de fente

La géométrie d’un pore en forme de fente de largeur d est présentée sur la figure 3.1b.Les cas champ moyen et fort couplage dans le cas de contre-ions seul ont été étudiés parKanduc et Podgornik [101]. L’expression du potentiel w(z) est alors

w(z) = (κb − κv)`B + 2κv`B∫ ∞

1dx∆(x)

e−xdκv∆(x) + cosh[x(d− 2z)κv]exdκv − e−xdκv∆2(x)

(3.51)

où ∆(x) = x−η√x2−1

x+η√x2−1

. On retrouve bien l’éq. (3.49) dans la limite d→∞.

7. Notons que l’on trouve alors pour le champ à la surface |Es| = 1+∆2

eσsεw

, ce qui ne correspond pas à lacondition attendue, |Es| = eσs

εw. En effet, cette dernière contient la condition d’électroneutralité (le champ

est nul pour z < 0) ce qui n’est pas le cas en fort couplage puisqu’il n’y a pas d’ions pour compenser lacharge de surface.

122


Pore neutre

L’équation variationnelle pour κv est résolue numériquement. À cause de la répulsiondiélectrique et du déficit de solvatation dans le pore, on a κv ≤ κb avec égalité dans lalimite d→∞. Notons que, pour des pores de largeur nanométrique (d = 2`B) et avec uneconcentration de 0.2 mol/L dans la solution, κv/κb croît en fonction de εm entre 0.7 (εm = 1)et 0.96 (εm = εw). La concentration ionique dans le pore chute brutalement dès que les effetsde répulsion diélectrique et de solvatation rentrent en jeu, soit grossièrement dès que lenuage ionique autour d’un ion central au centre du pore est déformé par les parois du pore,soit pour κ−1

b > d/2. Cette image est valable dans l’approximation dite de position centraleauto-cohérente, fréquemment utilisée dans les théories de nanofiltration [204, 185, 203].

Une des quantités mesurées dans les expériences de nanofiltration est le taux de réten-tion des sels, Σs, qui mesure la quantité de sel retenu par les membranes à haute pression.Il s’exprime dans l’approximation de la réponse linéaire comme

Σs = 1− 1Jρb

∫ d

0v||(z)ρ(z)dz = 1− 1

J

∫ d

0v||(z) e

−Φ(z)dz (3.52)

où v||(z) est le profil du champ de vitesse et J le flux de solvant. Deux cas limites sontenvisageables [106] : i) les parois sont hydrophiles et le profil de vitesse est le profil dePoiseuille, v||(z) = 6J

d3 z(d−z) ; ii) les parois sont très hydrophobes (la longueur de glissementest infinie) et l’écoulement est bouchon, soit v||(z) = J

d . D’après l’éq. (3.52), il est aisé devoir que Σs varie comme −e−q2w(xd)/2. Donc à ρb fixé, Σs diminue avec la largeur d puisquela répulsion se fait plus faible. Sur la figure 3.2 sont représentées les variations de Σs(ρb)pour les deux écoulements (ε = 0 et d/`B = 2 et 5). Nous observons que la rétention dansle cas d’un écoulement bouchon est plus importante car le champ de vitesse ne s’annulepas sur les parois, là où l’exclusion diélectrique est la plus grande.

La comparaison avec l’approche développée par Yaroshchuck montre que celle-ci minorel’effet car l’exclusion diélectrique au centre du pore (là où elle est évaluée) est la plus faible.Notons que cette approche consiste à écrire une équation auto-cohérente sur la constantede couplage κ en remplaçant le terme dans l’éq. (3.34) par sa valeur moyenne dans le pore

κ2sc = κ2

b〈e−Φ(r;κsc)〉pore (3.53)

L’approximation de la valeur centrale consiste à remplacer la moyenne dans le pore par lavaleur au centre du pore

κ2scmp = κ2

b e−Φ( d

2;κscmp) (3.54)

Les équations (3.53)-(3.54) sont à comparer à notre équation variationnelle sur κv quis’écrit en toute généralité

κ2v =

4π`B〈∂w(r;κv)

∂κv〉pore

∑

i

q2i ρb,i

⟨e−Φi(r;κv)∂w(r;κv)

∂κv

⟩

pore

(3.55)

123


-5.0 -4.0 -3.0 -2.0 -1.0

Log10(κb lB)

0.1

0.3

0.5

0.7

0.9

Σs

d=5, Variationald=5, Yaroschukd=2, Variationald=2, Yaroschukd=2, Variational, plug flow

Figure 3.2 – Coefficient de rétention des sels en fonction de log(κb) pour ε = 0 et deuxtailles de pores, d/`B = 2 et 5 dans le cas d’un écoulement de Poiseuille dans le pore. L’ap-proximation de la valeur centrale, éq. (3.54), correspond aux lignes rouges et les pointilléscorrespondent à un écoulement bouchon.

On voit donc d’où sort mathématiquement la différence : il y a une anti-corrélation forteprès des interfaces diélectriques entre e−Φi(r;κv) [éq. (3.45)] et le potentiel d’exclusion w(r),ce qui conduit à κv < κsc < κscmp.

Pore chargé

Lorsque les surfaces du pores sont chargées, les co-ions et contre-ions se distribuent demanière différente. Pour comprendre les variations de coefficients de partitions des co-ions,k− et des contre-ions, k+ définis selon

ki =⟨ρi(z)ρb,i

⟩

pore

(3.56)

nous nous sommes d’abord placés dans l’approximation de potentiel de Donnan effectif,c’est-à-dire que l’on suppose constant le potentiel électrostatique, noté φ0, créé par lessurfaces chargées. Cette approximation, qui permet des calculs analytiques simples, a étédéveloppée par Yaroshchuck dans le cadre de la nanofiltration en supposant que l’épaisseurdes pores étant très petites, φ0 pouvait être considéré comme constant. Nous verrons parla suite que cette approximation n’est pas toujours vérifiée mais elle conduit tout de même

124


a des résultats raisonnables puisque φ0 prend en charge l’électroneutralité dans le pore. Eneffet en minimisant Ω1 par rapport à φ0 on obtient

|σs| = −qρbd sinh(qφ0)〈e− q2

2w(z)〉pore (3.57)

Les coefficients de partition k± se comporte très différemment en fonction de d et σs.Lorsque σs augmente, k− diminue car la répulsion électrostatique des co-ions s’ajoute àla répulsion diélectrique. A contrario l’attraction des contre-ions vers la surface augmenteet k+ augmente. Il existe un régime de densité surfacique de charge intermédiaire, pourlequel les deux mécanismes, attraction électrostatique et répulsion diélectrique, s’opposent.Il en résulte une variation non-monotone de k+ en fonction de d : pour des tailles de poresinférieures à une valeur caractéristique, d < dc, l’attraction électrostatique domine et k+(d)décroît puis pour d > dc cette fonction devient croissante, tout comme dans un pore neutre,puisque la charge de surface est écrantée et la répulsion diélectrique décroît fortement avecz. Notons que ce régime correspond à σs < 0.07 e/nm2, des valeurs accessibles expérimen-talement.

Ainsi pour des faibles σs, un nouveau régime de bonne exclusion de co-ions est acces-sible. La répulsion diélectrique, par définition induite par la surface, domine et ce regimede faible σs est un régime de fort couplage avec la surface. Usuellement, dans les approchesutilisées en nanofiltration [204, 113, 202], le régime de bonne exclusion de co-ions est ob-tenu pour de grandes charges de surfaces, les co-ions étant alors exclus par la répulsionélectrostatique. Dans ce cas l’approche champ moyen est adéquate.

Enfin, nous sommes allés au-delà de l’approximation de potentiel de Donnan effectifen considérant pour φ0(z) une solution continue par morceaux semblable à celle proposéepour une seule interface diélectrique chargée. En comparant avec la solution numériquede l’éq. (3.33), nous montrons que ce choix est très bon et rend compte des variations deφ0 dans le pore. En particulier, le coefficient partition k+ est nul sur les surfaces mais aun maximum très proche de celles-ci, ce qui n’apparaît pas sur le profil champ moyen. Enconclusion, les solutions pour φ0(z), de type Donnan ou continue par morceaux, conduisentà des variations très proches de κv avec Ξ (même si κv est très légèrement sous-estimé dansl’approche Donnan) et rendent compte proprement de la répulsion diélectrique et de celleassociée au déficit de solvatation.

3.5.3 Article

Suit l’article :S. Buyukdagli, M. Manghi et J. Palmeri, Variational approach for electrolyte solutions :From dielectric interfaces to charged nanopores, Physical Review E 81 041601 (2010) (20pages)

125

Variational approach for electrolyte solutions: From dielectric interfaces to charged nanopores

Sahin Buyukdagli,* Manoel Manghi,† and John Palmeri‡

UPS, Laboratoire de Physique Théorique (IRSAMC), Université de Toulouse, F-31062 Toulouse, France and CNRS, LPT (IRSAMC),F-31062 Toulouse, France

Received 20 November 2009; revised manuscript received 11 February 2010; published 1 April 2010

A variational theory is developed to study electrolyte solutions, composed of interacting pointlike ions in asolvent, in the presence of dielectric discontinuities and charges at the boundaries. Three important andnonlinear electrostatic effects induced by these interfaces are taken into account: surface charge inducedelectrostatic field, solvation energies due to the ionic cloud, and image-charge repulsion. Our variationalequations thus go beyond the mean-field theory, or weak coupling limit, where thermal fluctuations overcomeelectrostatic correlations, and allows one to reach the opposite strong coupling limit, where electrostaticinteractions induced by interfaces dominate. The influence of salt concentration, ion valency, dielectric jumps,and surface charge is studied in two geometries. i A single neutral dielectric interface e.g., air-water orelectrolyte-membrane with an asymmetric electrolyte. A charge separation and thus an electrostatic field getestablished due to the different image-charge repulsions for coions and counterions. Both charge distributionsand surface tension are computed and compared to previous approximate calculations. For symmetric electro-lyte solutions close to a charged surface, two zones are characterized. In the first one, in contact with thesurface and with size proportional to the logarithm of the coupling parameter, strong image forces and strongcoupling impose a total ion exclusion, while in the second zone the mean-field approach applies. ii Asymmetric electrolyte confined between two dielectric interfaces as a simple model of ion rejection fromnanopores in membranes. The competition between image-charge repulsion and attraction of counterions bythe membrane charge is studied. For small surface charge, the counterion partition coefficient decreases withincreasing pore size up to a critical pore size, contrary to neutral membranes. For larger pore sizes, the wholesystem behaves like a neutral pore. For strong coupling and small pore size, coion exclusion is total and thecounterion partition coefficient is solely determined by global electroneutrality. A quantitative comparison ismade with a previous approach, where image and surface charge effects were smeared out in the pore. It isshown that the variational method allows one to go beyond the constant Donnan potential approximation, withdeviations stronger at high ion concentrations or small pore sizes. The prediction of the variational method isalso compared with MC simulations and good agreement is observed.

DOI: 10.1103/PhysRevE.81.041601 PACS numbers: 68.15.e, 03.50.De, 87.16.D

I. INTRODUCTION

The first experimental evidence for the enhancement ofthe surface tension of inorganic salt solutions compared tothat of pure water was obtained more than eight decades ago1,2. Wagner proposed the correct physical picture 3 byrelating this effect to image forces that originate from thedielectric discontinuity and act on ions close to the water-airinterface. He also correctly pointed out the fundamental im-portance of the ionic screening of image forces and formu-lated a theoretical description of the problem by establishinga differential equation for the electrostatic potential and solv-ing it numerically to compute the surface tension. Using se-ries expansions, Onsager and Samaras found the celebratedlimiting law 4 that relates the surface tension of symmetricelectrolytes to the bulk electrolyte density at low salt concen-tration. However, it is known that the consideration of chargeasymmetry leads to a technical complication. Indeed, image-charge repulsion, whose amplitude is proportional to thesquare of ion valency, leads to a split of concentration pro-

files for ions of different charge, which in turn causes a localviolation of the electroneutrality and induces an electrostaticfield close to a neutral dielectric interface. Bravina derivedfive decades ago a Poisson-Boltzmann type of equation forthis field 5 and used several approximations in order toderive integral expressions for the charge distribution and thesurface tension.

These image-charge forces play also a key role in slitlikenanopores which are model systems for studying ion rejec-tion and nanofiltration by porous membranes see the review6 and references therein, and 7 for a review of nanoflu-idics. Several results have been found in this geometry andalso for cylindrical nanopores beyond the mean-field ap-proach using the Debye closure and the Bogoliubov-Born-Green-Kirkwood-Yvon BBGKY hierarchical equationsand averaging all dielectric and charge effects over the porecross section. Within these two approximations, the salt re-flection coefficient has been studied as a function of the poresize, the bulk salt concentration and the pore surface charge.

More precisely, the strength of electrostatic correlations ofions in the presence of charged interfaces without dielectricdiscontinuity is quantified by one unique coupling parameter

= 2q3B2s, 1

where q is the ion valency, and s the fixed surface charge8–10. The Bjerrum length in water for monovalent ions,

*[email protected]†[email protected]‡[email protected]



B=e2 / 4wkBT0.7 nm w is the dielectric permittivityof water is defined as the distance at which the electrostaticinteraction between two elementary charges is equal to thethermal energy kBT. The second characteristic length is theGouy-Chapman length G=1 / 2qBs defined as the dis-tance at which the electrostatic interaction between a singleion and a charged interface is equal to kBT. The couplingparameter can be re-expressed in terms of these two lengthsas =q2B /G. On the one hand, the limit →0, called theweak coupling WC limit, is where the physics of the Cou-lomb system is governed by the mean-field or Poisson-Boltzmann PB theory, and thermal fluctuations overcomeelectrostatic interactions. It describes systems characterizedby a high temperature, low ion valency or weak surfacecharge. On the other hand, → is the strong coupling SClimit, corresponding to low temperature, high valency of mo-bile ions or strong surface charge. In this limit, ion-chargedsurface interactions control the ion distribution perpendicu-larly to the interface. For single interface and slab geom-etries, several perturbative approaches going beyond the WClimit 11,12 or below the SC limit 8,13,14 have been de-veloped. Although these calculations were able to captureimportant phenomena such as charge renormalization 15,ion specific effects at the water-air interface 16,17, Man-ning condensation 18, effect of multipoles 19, or attrac-tion between similarly charged objects, they also showedslow convergence properties, which indicates the inability ofhigh-order expansions to explore the intermediate regime,1. This is quite frustrating since the common experimen-tal situation usually corresponds to the range 0.110where neither WC nor SC theory is totally adequate.

Consequently, a nonperturbative approach valid for thewhole range of is needed. A first important attempt in thisdirection has been made by Netz and Orland 20 who de-rived variational equations within the primitive model forpointlike ions and solved them at the mean-field level inorder to illustrate the charge renormalization effect. Interest-ingly, these differential equations are equivalent to the clo-sure equations established in the context of electrolytes innanopores 6. They are too complicated to be solved ana-lytically or even numerically for general . A few yearslater, Curtis and Lue 21 and Hatlo et al. 22 investigatedthe partition of symmetric electrolytes at neutral dielectricsurfaces using a similar variational approach see also thereview 23. They have also recently proposed a new varia-tional scheme based on a hybrid low fugacity and mean-fieldexpansion 24, and showed that their approach agrees wellwith Monte Carlo simulation results for the counterions-onlycase. However, this method is quite difficult to handle, andone has to solve two coupled variational equations, i.e., asixth-order differential equation for the external potential to-gether with a second algebraic equation. Within this ap-proach, these authors generalized the study of ion-ion corre-lations for counterions close to a charged dielectric interface,first done by Netz in the WC and SC limits 25, to interme-diate values of . They also studied an electrolyte betweentwo charged surfaces without dielectric discontinuities at thepore boundary, in two cases: counterions only and added salt,handled at the mean-field level 26. Although this simplifi-cation allows one to focus exclusively on ion-ion correla-

tions induced by the surface charge, the dielectric disconti-nuity cannot be discarded in synthetic or biologicalmembranes. Indeed, it is known that image forces play acrucial role in ion filtration mechanisms 6. The main goalof this work is to propose a variational analysis which issimple enough to intuitively illustrate ionic exclusion in slitpores, by focusing on the competition between image-chargerepulsion and surface charge interaction. Moreover, our ap-proach allows us to connect nanofiltration studies 27–29with field-theoretic approaches of confined electrolyte solu-tions within a generalized Onsager-Samaras approximation4 characterized by a uniform variational screening length.This variational parameter takes into account the interactionwith both image charge and surface charge. We also comparethe prediction of the variational theory with Monte Carlosimulations 30 and show that the agreement is good.

The paper is organized as follows. The variational formal-ism for Coulombic systems in the presence of dielectric dis-continuities is introduced in Sec. II. Section III deals with asingle interface. We show that the introduction of simplevariational potentials allows one to fully account for thephysics of asymmetric electrolytes at dielectric interfacese.g., water-air, liquid-liquid, and liquid-solid interfaces, seeRef. 31, first studied by Bravina 5 using several approxi-mations, as well as the case of charged surfaces. In Sec. IV,the variational approach is applied to a symmetric electrolyteconfined between two dielectric surfaces in order to investi-gate the problem of ion rejection from membrane nanopores.Using restricted variational potentials, we show that due tothe interplay between image-charge repulsion and directelectrostatic interaction with the charged surface, the ionicpartition coefficient has a nonmonotonic behavior as a func-tion of pore size.

II. VARIATIONAL CALCULATION

In this section, the field theoretic variational approach formany body systems composed of pointlike ions in the pres-ence of dielectric interfaces is presented. Since the field theo-retic formalism as well as the first-order variational schemehave already been introduced in previous works 20,21, weonly illustrate the general lines.

The grand-canonical partition function of p ion species ina liquid of spatially varying dielectric constant r is

Z = i=1

p

Ni=0

eNii

Ni!t3Ni

j=1

Ni

drije−H−Es 2

where t is the thermal wavelength of an ion, i denotes thechemical potential and Ni the total number of ions of type i.For sake of simplicity, all energies are expressed in units ofkBT. The electrostatic interaction is

H =1

2 drdrcrvcr,rcr , 3

where c is the charge distribution in units of e

BUYUKDAGLI, MANGHI, AND PALMERI PHYSICAL REVIEW E 81, 041601 2010

041601-2

cr = i=1

p

j=1

Ni

qir − rij + sr , 4

and qi denotes the valency of each species, sr stands forthe fixed charge distribution and vcr ,r is the Coulombpotential whose inverse is defined as

vc−1r,r = −

kBT

e2 r r − r , 5

where r is a spatially varying permittivity. The self-energyof mobile ions, which is subtracted from the total electro-static energy, is

Es =vc

b02

i=1

p

Niqi2, 6

where vcbr=B /r is the bare Coulomb potential for r

=w. After performing a Hubbard-Stratonovitch transforma-tion and the summation over Ni in Eq. 2, the grand-canonical partition function takes the form of a functionalintegral over an imaginary electrostatic auxiliary field r,Z=D e−H . The Hamiltonian is

H = dr r2e2 r2 − isr r

− i

ieqi

2vcb0/2+iqi r 7

where a rescaled fugacity

i = ei/t3 8

has been introduced. The variational method consists in op-timizing the first order cumulant

Fv = F0 + H − H00, 9

where averages ¯ 0 are to be evaluated with respect to themost general Gaussian Hamiltonian 20,

H0 =1

2

r,r r − i 0rv0

−1r,r r − i 0r

10

and F0=− 12 tr ln v0. The variational principle consists in look-

ing for the optimal choices of the electrostatic kernelv0r ,r and the average electrostatic potential 0r whichextremize the variational grand potential Eq. 9. The varia-tional equations Fv /v0

−1r ,r=0 and Fv / 0r=0, for asymmetric electrolyte and r=w, yield

0r − 8Bqe−q2Wr/2 sinhq 0r = − 4Bsr ,

11

− v0r,r + 8Bq2e−q2Wr/2 coshq 0rv0r,r

= 4Br − r , 12

where we have defined

Wr limr→r

v0r,r − vcbr − r , 13

whose physical signification will be given below. The secondterms on the left-hand side lhs of Eq. 11 and of Eq. 12have simple physical interpretations: the former is 4Btimes the local ionic charge density and the latter is 4Bq2

times the local ionic concentration. The relations Eqs. 11and 12 are, respectively, similar in form to the nonlinearPoisson-Boltzmann NLPB and Debye-Hückel DH equa-tions, except that the charge and salt sources due to mobileions are replaced by their local values according to the Bolt-zmann distribution. On the one hand, Eq. 11 is a Poisson-Boltzmann-like equation where appears the local charge den-sity proportional to sinh 0. This equation handles theasymmetry induced by the surface through the electrostaticpotential 0, which ensures electroneutrality. This asymme-try may be due to the effect of the surface charge on anionand cation distributions see Sec. III B or due to dielectricboundaries and image charges at neutral interfaces, whichgive rise to interactions proportional to q2, and induce a localnonzero 0 for asymmetric electrolytes see Sec. III A. Onthe other hand, the generalized DH equation Eq. 12, whereappears the local ionic concentration proportional to cosh 0,fixes the Green’s function v0r ,r evaluated at r with thecharge source located at r and takes into account dielectricjumps at boundaries.

These variational equations were first obtained within thevariational method by Netz and Orland 20. They were alsoderived in Ref. 32 within the Debye closure approach andthe BBGKY hierarchic chain. Yaroshchuk obtained an ap-proximate solution of the closure equations for confinedelectrolyte systems in order to study ion exclusion frommembranes 6.

Equations 11 and 12 enclose the limiting cases of WC→0 and SC →. To see that, it is interesting torewrite theses equations by renormalizing all lengths and thefixed charge density, sr, by the Gouy-Chapman length ac-cording to r=r /G, sr=Gsr /s s is the average sur-face charge density. By introducing a new electrostatic po-tential 0r=q 0r, one can express the same set ofequations in an adimensional form

0r − e−Wr/2 sinh 0r = − 2sr , 14

− v0r, r + e−Wr/2 cosh 0rv0r, r = 4r − r ,

15

where v0=v0G /B, W=WG /B and we have also intro-duced the rescaled fugacity =8G

3 33. Now, one cancheck that, in both limits →0 and →, the couplingbetween 0 and v0 in Eq. 11 disappears and the theorybecomes integrable. Finally, it is important to note that thisadimensional form of variational equations allows one to fo-cus on the role of v0r ,r whose strength is controlled by in Eqs. 14 and 15. However, even at the numerical level,their explicit coupling does not allow for exact solutions forgeneral .

VARIATIONAL APPROACH FOR ELECTROLYTE … PHYSICAL REVIEW E 81, 041601 2010

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In the present work, we make a restricted choice forv0r ,r and replace the local salt concentration in the formof a local Debye-Hückel parameter or inverse screeninglength r in Eq. 12,

r2 = 8Bq2e−q2Wr/2 coshq 0r , 16

by a constant piecewise one vr=v in the presence of ionsand vr=0 in the salt-free parts of the system. Note that ithas been recently shown that many thermodynamic proper-ties of electrolytes are successfully described with a Debye-Hückel kernel 34.

The inverse kernel or the Green’s function v0r ,r isthen taken to be the solution to a generalized Debye-Hückelequation

− r + rv2rv0r,r =

e2

kBTr − r 17

with the boundary conditions associated with the dielectricdiscontinuities of the system

limr→−

v0r,r = limr→+

v0r,r , 18

limr→−

r v0r,r = limr→+

r v0r,r , 19

where denotes the dielectric interfaces. We now restrictourselves to planar geometries. We split the grand potential9 into three parts, Fv=F1+F2+F3, where F1 is the meanelectrostatic potential contribution,

F1 = S dz− 0z2

8B+ sz 0z

− i

ie−qi

2Wz/2−qi 0z , 20

F2 the kernel part and F3 the unscreened van der Waals con-tribution. The explicit forms of F2 and F3 are reported inAppendix A. The first variational equation is given byFv /v=F1+F2 /v=0. This equation is the restrictedcase of Eq. 12. As we will see below, its explicit formdepends on the confinement geometry of the electrolyte sys-tem as well as on the form of r. The variational equationfor the electrostatic potential 35 Fv / 0z=0 yields re-gardless of the confinement geometry

2 0

z2 + 4Bsz + i

4Bqiie−qi

2Wz/2−qi 0z = 0.

21

The second-order differential Eq. 21, which is simply thegeneralization of Eq. 11 for a general electrolyte in a planargeometry, does not have closed-form solutions for spatiallyvariable Wz. In what follows, we optimize the variationalgrand potential Fv using restricted forms for the electrostaticpotential 0z and compare the result to the numerical so-lution of Eq. 21 for single interfaces and slitlike pores.

The single ion concentration is given by

iz = ie−qi

2Wz/2−qi 0z 22

and its spatial integral by

dziz = − iFv

i. 23

We define the potential of mean force PMF of ions of typei, iz, as

iz − lnizb

. 24

By defining

wz Wz − Wb, 25

where Wb is the value of Wz in the bulk and comparingEqs. 22 and 24, we find

iz =qi

2

2wz + qi 0z , 26

qi2

2Wb = ln i

b i − lnbt3 . 27

Hence, qi2Wb /2 is nothing else but the excess chemical po-

tential of ion i in the bulk and qi2Wz /2=ln iz is its gen-

eralization for ion i at distance z from the interface. They arerelated to the activity coefficients i

b and iz. Note that thezero of the chemical potential is fixed by the condition that 0 vanishes in the bulk. The PMF, Eq. 26, is thus the meanfree energy per ion or chemical potential needed to bring anion from the bulk at infinity to the point at distance z fromthe interface, taking into account correlations with the sur-rounding ionic cloud.

Before applying the variational procedure to single anddouble interfaces, let us consider the variational approach inthe bulk. In this case, the variational potential 0 is equal to0, and the variational grand potential Fv only depends on v.Two minima appear: one metastable minimum v

0 at low val-ues of v, and a global minimum at infinity Fv→− forv→ which is unphysical since at these large concentra-tion values, finite size effects should be taken into account. Ithas been shown by introducing a cutoff at small distances21, that, for physical temperatures, this instability disap-pears and the global minimum of Fv is v

0 b given by theDebye-Hückel limiting law,

i = lnbt3 −

qi2

2bB,

b2 = 4B

i

qi2i,b. 28

From Eq. 27, we thus find Wb−bB and the potentialwz reduces to

wz v0z,z − vcb0 + bB, 29

which will be adopted in the rest of the paper. This choice isconsistent for bBq2<4. Furthermore, problems due to theformation of ion pairs do not enter at the level of the varia-


041601-4

tional approach we have adopted. Let us also report the fol-lowing conversion relations

Ib 0.19bB2 mol L−1

b 3.29I nm−1, 30

where Ib=iqi2i,b /2 is the ionic strength expressed in

mol L−1. Finally, the single-ion densities are given by

iz = i,be−qi2wz/2−qi 0z. 31

III. SINGLE INTERFACE

The single interfacial system considered in this sectionconsists of a planar interface separating a salt-free left half-space from a right half-space filled up with an electrolytesolution of different species Fig. 1. In the general case, thedielectric permittivity of the two half spaces may be differentwe note the permittivity in the salt-free part. The Green’sfunction, which is chosen to be the solution of the DH equa-tion with z=−z+wz and z=vz where zstands for the Heaviside distribution, is given for z0 by 5

wz = Bb − v + B0

kdk

k2 + v2k/ve−2k2+v

2z,

32

where

x =w

x2 + 1 − x

wx2 + 1 + x

. 33

and F2 Eq. A4 can be analytically computed 21

F2 = Vv

3

24+ S

v2

32, 34

where

x → =w −

w + . 35

The first term on the right-hand side rhs of Eq. 34 is avolumic contribution associated with a hypothetic bulk phasewith inverse Debye screening length v and the second termon the rhs involves interfacial effects, including the dielectricjump , and v.

For the single interface system, as seen in Sec. II, F3 isindependent of v and 0z, which means that it does notcontribute to the variational equations. By minimizing Eqs.20 and 34 with respect to v for fixed 0r and takingV→, one exactly finds the same variational equation for vas for the bulk case. Hence, as discussed above, we havev=b given by Eq. 28 and the first term of the rhs of Eq.32 vanishes. This result was obtained in 21 for the specialcase r=0. It is of course not surprising to end up with thesame result for finite r since we know that the electro-static potential should vanish in the bulk.

The potential wz given by Eq. 32 with v=b com-bines in an intricate way both the image charge and solvationcontributions due to the presence of the interface. The imageforce corresponds to the interaction of a given ion with thepolarized charges at the interface and is equivalent to theinteraction of the charged ion with its image located at theother side of the dielectric surface. As it is well known, theimage-charge interaction is repulsive for w e.g., water-air interface and attractive for w the case for anelectrolyte-metal interface 36. The interfacial reduction insolvation arises because an ion always prefers to be screenedby other ions in order to reduce its free energy. Hence, it isattracted toward areas where the ion density is maximum atleast at not too high concentrations for which steric repulsionmay predominate. This term is nonzero even for =w sincefor an ion close to the interface, there is a “hole” of screeningions in the salt-free region where v=0. Although ourchoice of homogeneous variational inverse screening lengthallows us to handle the deformation of ionic atmospheresnear interfaces that are impermeable to ions, it does not al-low us to treat in detail the local variations in ion solvationfree energy arising from ion-ion correlations except in anaverage way in confined geometries where v can differ fromthe bulk value of the inverse screening length, see Sec. IVbelow.

Equation 32 simplifies in three cases:1 For =0 =1, where the solvation effect vanishes

because the lines of forces are totally excluded from the airregion 36, Eq. 32 reduces to

w0z = Be−2bz

2z. 36

This is the case where the image-charge repulsion is thestrongest see Fig. 2.

2 A slightly better approximation for 0 can be ob-tained by artificially allowing salt to be present in the airregion. This gives rise to the “undistorted ionic atmosphere”approximation 6, for which wz in Eq. 36 is multipliedby ,

wz = Be−2bz

2z. 37

Solvation effects are now absent and salt exclusion arisessolely from dielectric repulsion. Equation 37 is exact forarbitrary b and =1, or arbitrary and b=0.

3 In the absence of a dielectric discontinuity =w =0, the potential can be expressed as

FIG. 1. Color online Geometry for a single dielectric interfacee.g., water–air a and double interfaces or slitlike pores b.


041601-5

wz = bBfbz ,

fx =1 + x2e−2x

2x3 −K22x

x, 38

where K2x is the Bessel function of the second kind. Onenotices that unlike the case 0, the potential has a finitevalue at the interface, i.e., w0=bB /3.

We note that in this case of one interface, we havelimz→ 0z=0 and the fugacity i of each species is fixedby its bulk concentration according to

i,b = limz→

iz = ieqi

2bB/2, 39

where we used Eq. 22.

A. Neutral dielectric interface

We investigate in this section the physics of an asymmet-ric electrolyte close to a neutral dielectric interface e.g.,water-air, liquid-liquid, or liquid-solid interface located atz=0 s=0. For the sake of simplicity, we assume =0,which is a very good approximation for the air-water inter-face characterized by =1 see the discussion in Ref. 4.Hence we keep the approximation wz given by Eq. 36unless otherwise stated. The electrolyte is composed of twospecies of bulk density + and − and charge q+e, −q−ewith q+q−. In order to satisfy the electroneutrality in thebulk, we impose +q+=−q−. According to Eq. 28, the bulkinverse screening length noted b is given by

b2 = 4Bq−−q− + q+ 40

and the variational Eq. 21 for the electrostatic potential is amodified Poisson-Boltzmann equation

2 0

z2 + 4Bchz = 0 41

with a local charge concentration

chz = −q−e−q+2wz/2−q+ 0z − e−q−

2wz/2+q− 0z . 42

Equation 41 cannot be solved analytically. Its numericalsolution, obtained using a fourth-order Runge-Kutta method,is plotted in Fig. 3a for asymmetric electrolytes with diva-lent and quadrivalent cations and the local charge density isplotted in Fig. 3b.

Figure 3 clearly shows that, very close to the dielectricinterface for za, where the depletion distance a is definedbelow, image-charge repulsion expulses all ions sincechzexp−1 /z has an essential singularity and 0 is flat.For za, but still close to the interface, there is a layerwhere the electrostatic field is almost constant 0 increaseslinearly, which is created by the charge separation of ions ofdifferent valency due to repulsive image interactions. Theintensity of image forces increases with the square of ionvalency and close to the interface, chz0 since we as-sumed q+q− the case for MgI2. To ensure electroneutral-ity, the local charge then becomes positive when we move

0.0 0.2 0.4 0.60.0

1.0

2.0

3.0

w(z)

z/lB

FIG. 2. Color online Potential wz in units of kBT for =0black solid curve, Eq. 36, and =w red dashed curve andbB=4.

(a)0.00 1.00 2.00 3.00 4.00

z/lB

-0.16

-0.12

-0.08

-0.04

0.00

electrolyte 1:2electrolyte 1:4

φ0

0.00 1.00 2.00 3.00 4.00z/lB

-0.03

-0.01

0.01

0.03

0.05

electrolyte 1:2electrolyte 1:4

l B3

ρ ch

(b)

FIG. 3. Color online a Electrostatic potential 0 in kBTunits for asymmetric electrolytes: numerical solution of Eq. 41symbols and variational choice, Eq. 44 solid lines, for divalentand quadrivalent ions and b=0.242 mol L−1. Variational param-eters are 1.4b, a /B=0.12;0.21 and =−0.10;−0.156. bAssociated local charge density profile thick lines and aniondashed lines and cation thin solid lines concentrations.


041601-6

away from the surface Fig. 3b, and the electrostatic po-tential goes exponentially to zero with a typical relaxationconstant . Moreover, in Fig. 3a one observes that whenthe charge asymmetry increases, the electrostatic potentialalso increases. Knowing that for symmetric electrolytes, 0=0, our results confirm that the charge asymmetry is thesource of the electrostatic potential 0. Figure 3b is quali-tatively similar to Fig. 1 of Bravina who had derived anintegral solution of Eq. 41 by using an approximation validfor bB1 5. In order to go further in the description ofthe interfacial distribution of ions, we look for a restrictedvariational function 0z which not only contains a smallnumber of variational parameters such as a and but alsois as close as possible to the numerical solution. As sug-gested by the description of Fig. 3, a continuous piecewise 0z is necessary to account for the essential singularity ofchz. To show this, let us expand Eq. 41 to order 0,

2 0

z2 − 4B−q−e−q+2wz/2 − e−q−

2wz/2

+ 4B−q−q+e−q+2wz/2 + q−e−q−

2wz/2 0. 43

This linearization is legitimate, as seen in Fig. 3: q+ 0z1 is satisfied for physical valencies. The first term on therhs of Eq. 43 corresponds to an effective local chargesource while the second term is responsible for the screeningof the potential. If we observe the charge distribution forq2wzq 0z and za, i.e., the first term of the rhs of Eq.43, we notice that it behaves like a distorted peak. Thesimplest function having a similar behavior is fz=cze− z,where c and are constants. Hence, we choose a restrictedvariational piecewise solution 0z

0z = for z a ,

1 + z − ae− z−a for z a . 44

whose derivation is explained in Appendix B. The variationalparameters are the constant potential , the depletion dis-tance a and the inverse screening length . The grand po-tential B5 derived for this solution was optimized with re-spect to the variational parameters using the MATHEMATICA

software. The restricted variational potential 44 is com-pared to the numerical solution of Eq. 41 in Fig. 3 forelectrolytes 1:2 and 1:4 and −B

3 =0.05. The agreement isexcellent. One notices that the screening of the effective sur-face charge created by dielectric exclusion enters into playwhen z

−1. Finally, let us note that since blB=1.37 andblB=1.77, respectively, for the monovalent and quadrivalentelectrolytes in Fig. 3, the method adopted by Bravina is notvalid.

To summarize, the charge separation is taken into accountby the potential which increases with q+ /q− and the re-laxation constant 1.4b is almost independent of q+ /q−.Interestingly, the variational parameter a /B0.1–0.2 is lessthan 1 nm. Indeed, for finite size ions, wz differs from Eq.36 very close to the interface and reaches a finite value atz=0. The size of this region exactly corresponds to a which

is of the order of an ion radius. This is thus an artifact of ourpointlike ion model and occurs only for asymmetric electro-lytes at neutral surfaces.

The surface tension is equal to the excess grand poten-tial defined as the difference between the grand potential ofthe interfacial system and that of the bulk system,

=b

2

32−

2

32B− −

0

dze−q−2wz/2+q− 0z − 1

+q−

q+e−q+

2wz/2−q+ 0z − 1 . 45

The surface tension for electrolytes characterized by q−=1and q+=1 to 4 is plotted in Fig. 4 as a function of −, becausethe anion density is an experimentally accessible parameter.Unlike symmetric electrolytes 21, a plot with respect to b

2

may lead to a different behavior. One notices that the in-crease in valency asymmetry leads to an important increasein the surface tension. This is of course mainly due to thereduction of the cation density in the bulk by a factor ofq− /q+ necessary to satisfy the bulk electroneutrality see thesecond term in the integral of Eq. 45.

B. Charged surfaces

We now consider a symmetric electrolyte in the proximityof an interface of constant surface charge s0 located atz=0. The variational Eq. 21 simplifies to

2 0

z2 = 2z + b2e−wz/2 sinh 0. 46

The mean-field limit →0 of this equation corresponds tothe NLPB equation, whose solution reads

0z = 4 arctanhbe−bz 47

where b= b−1+ b2. In this section, we show that a piece-

wise solution for the electrostatic potential similar to the one

0.01 0.06 0.11 0.16 0.21 0.26

lB3ρ-

0.02

0.07

0.12

0.17

0.22

electrolyte 1:1electrolyte 1:2electrolyte 1:3electrolyte 1:4

βlB2σ

FIG. 4. Color online Surface tension B2 /kBT for asymmetric

electrolytes vs the anion bulk concentration, for increasing asym-metry q+ /q−=1 to 4 from bottom to top.


041601-7

introduced in Sec. III A agrees very well with the numericalsolution of Eq. 46. Inspired by the existence of a salt-freelayer close to the interface and a mean-field regime far fromthe interface WC, we propose two types of piecewise varia-tional functions see Appendix C. The first variationalchoice obeys the Poisson equation in the first zone of size hand the nonlinear Poisson-Boltzmann solution in the secondzone,

0NLz =4 arctanh + 2z − h for z h ,

4 arctanhe− z−h for z h , 48

where = −1+ 2 . Variational parameters are h and an

effective inverse screening length . The second type oftrial potential obeys the Poisson equation with a chargerenormalization in the first zone and the linearized Poisson-Boltzmann solution in the second zone,

0Lz = −

2

+ 2z − h for z h ,

−2

e− z−h

for z h . 49

Variational parameters are h, , and the charge renormaliza-tion , which takes into account the nonlinear effects at themean-field level 20. The explicit form of the associatedvariational free energies is reported in Appendix C. The insetof Fig. 5 displays the size of the SC layer h against . Our

approach predicts a logarithmic dependence h ln , thefactor in front of the logarithm being b

−1 for b1. Therestricted choices for 0 are compared with the full numeri-cal solution of Eq. 46 in the same figure for =0. We seethat, as in the previous section, the numerical solution and

the restricted ones match perfectly. Hence salt-exclusion ef-fects are essentially carried by the parameter h. Furthermore,one notices that 0z relaxes to zero between z= h and z= h+2

−1. At bG=4 we are in the linear regime of the PBequation and therefore one has 1. The charge renormal-ization idea was introduced by Alexander et al. 15, whoshowed that the nonlinearity of the PB equation can be ef-fectively taken into account at long distances by renormaliz-ing the fixed charge source and extending the linearized zonewhere 01 to the whole domain. A linear solution of theform Eq. 49 can be very helpful for complicated geom-etries or in the presence of a nonuniform charge distributionwhere the NLPB equation does not present an analytical so-lution even at the mean-field level. These issues will be dis-cussed in a future work.

Figure 6 displays the ion concentrations iz /i,b=e−i,which are related to the ion PMF Eq. 24, computed withthe restricted solution Eq. 48 for several values of . Asalready said in the Introduction, in rescaled distance, the cou-pling parameter measures the strength of the excess

φ0

0.0 0.5 1.0 1.5 2.0

z/lG

-1.5

-1.0

-0.5

0.0

NumericalNon-linear PBLinear PB

0 2 4 6 8

Log Ξ

0.0

0.5

1.0

1.5

2.0

2.5

h/lG

FIG. 5. Color online Electrostatic potential, 0 in units ofkBT: numerical solution of Eq. 46 symbols and restricted varia-tional choices Eqs. 48 and 49 for =0, bG=4, and =1, 10,100, and 1000 from top to bottom. The variational parameters are,respectively, =3.83,3.74,3.69,3.66 and 1. Markers on the

x-axis denote, for each curve, the size, h, of the SC zone, plotted vsln in the inset, where NLPB and LPB solutions aresuperimposed.

0.0 0.5 1.0 1.5 2.0

z/lG

0.00

0.25

0.50

0.75

1.00

1.25

1.50

ρ/ρb

(a)

ρ/ρb

0.0 0.5 1.0 1.5 2.0

z/lG

0.00

0.25

0.50

0.75

1.00

1.25

1.50

(b)

FIG. 6. Color online Ion densities for bG=4, and a =0and b =w, for increasing coupling parameter: from left to right,=1, 10, 100, and 1000. Solid lines correspond to counterions,dashed lines to coions and dashed-dotted lines to the Poisson-Boltzmann result for counterions 47.


041601-8

chemical potential, wz. We first see that for coions as wellas for counterions, the depletion layer in rescaled units in theproximity of the dielectric interface increases with due tothe image-charge repulsion and/or solvation effect, i.e., theterm e−wz/2 in Eq. 46. Furthermore, one notices that thecounterion density exhibits a maximum. This concentrationpeak is due to the competition between the electrostatic at-tractive force toward the charged wall and the repulsive im-age and solvation interactions. It is important to note that inthe particular case =w, there is no depletion layer for 10.

IV. DOUBLE INTERFACE

In this section, the variational method is applied to adouble interface system which consists of a slitlike pore ofthickness d, in contact with an external ion reservoir at itsextremities Fig. 1b. The dielectric constant is w insidethe pore and in the outer space. The electrolyte occupiesthe pore and the external space is salt free. The solution ofthe DH Eq. A2 in this geometry is 6

wz = b − vB + B0

kdk

k2 + v2

k/v

e2dk2+v2

− 2k/v

2k/v + e2d−zk2+v2

+ e2zk2+v2 50

where x is given in Eq. 33. The variational parameter ofthe Green’s function is the variational inverse screeninglength v which is taken uniform generalized Onsager-Samaras approximation, see 5,21. A more complicated ap-proach has been previously developed in Ref. 22 where theauthors introduced a piecewise form for the variationalscreening length, i.e., z=v over a layer of size h andv=b in the middle of the pore. Although this choice ismore general than ours, the minimization procedure with re-spect to v is significantly longer than in our case and thevariational equation is much more complicated. Conse-quently, this piecewise approach is not very practical whenone wishes to study a charged membrane where the externalfield created by the surface charge considerably complicatesthe technical task see Sec. IV B. We show that the simplevariational choice adopted here captures the essential physicswith less computational effort.

As in Eq. 32, the integral on the rhs of Eq. 50 takesinto account both image-charge and solvation effects due tothe two interfaces, whereas the first term is the Debye resultfor the difference between the bulk and a hypothetic bulk ofinverse screening length v. We should emphasize that, in thepresent case, the spatial integrations in Eqs. A3 and A4run over the confined space, that is from z=0 to z=d. Bysubstituting the solution Eq. 50 into Eqs. 20 and A5 andperforming the integration over z, one finds 23

F2 + F3

S=

dv3

24+v

2

16+v

2

4

1

dxx ln1 − 2xe−2vdx

+v

2

8

1

dxx − 3x/x − 2vd2x

e2dvx − 2x, 51

where we have defined x=x2−1.

The limiting case =0 allows for closed-form expres-sions. This limit is a good approximation for describing bio-logical and artificial pores characterized by an external di-electric constant much lower than the internal one. In thefollowing part of the work, we will deal most of the timewith the special case =0, unless stated otherwise. In thislimit, Eq. 51 simplifies to

F2 + F3

S=v

3d

24+v

2

161 + 2 ln1 − e−2dv −

v

8dLi2e−2dv

−Li3e−2dv

16d2 , 52

where Linx stands for the polylogarithm function and xthe Riemann zeta function see Appendix D. Within thesame limit =0, x=1 and we obtain an analytical ex-pression for the Green’s function Eq. 50

w0z = b − vB −B

dln1 − e−2dv +

B

2d

e−2dv;1 −z

d,0

+d

ze2

−2vz2F11,

z

d,1 +

z

d,e−2dv , 53

where x ;y ,z is the incomplete Beta function and

2F1a ,b ;c ;d the hypergeometric series. The definitions ofthese special functions are given in Appendix D. At this step,the PMF thus depends on three adimensional parameters,namely dv, db, and d /B.

For the system with a single interface, the ion fugacity iwas fixed by the bulk density. In the present case where theconfined system is in contact with an external reservoir, i isfixed by chemical equilibrium,

i = i,b = i,be−qi2bB/2, 54

where b and i,b are, respectively, the inverse Debye screen-ing length and the fugacity in the bulk reservoir see Eq.28. Once this constraint is taken into account, the last termof the electrostatic part of the variational grand potential Eq.

20 can be written as −ii,b0ddze−qi

2wz/2−qi 0z.Equation 21 then becomes for a symmetric q :q electro-

lyte,

2 0

z2 − b2e−q2wz/2 sinh 0 = − 4qBsz + z − d .

55

The optimization of Fv=F1+F2+F3 given by Eq. 20 and52 with respect to the inverse trial screening length vleads to the following variational equation for v,


041601-9

dv2 + dv tanhdv

= db20

1

dxe−q2w0xd/2cosh 0xd

1 +cosh2x − 1dv

coshdv . 56

Within the particular choice that fixed the functional form ofthe v dependent Green’s function Eq. 53, the two coupledEqs. 55 and 56 are the most general variational equations.In the following, we first consider the case of neutral poresand then the more general case of charged pores.

A. Neutral pore, symmetric electrolyte

In the case of a symmetric q :q electrolyte and a neutralmembrane, s=0, the solution of Eq. 55 is naturally 0=0. The variational parameter v is solution of Eq. 56 with 0=0 and wz=w0z given by Eq. 53 when =0, whichcan be written as dv, a function only of db and B /d. Letus note that Eq. 56 can be solved with the Mathematicasoftware in a fraction of a second.

Within the Debye-Hückel closure approach, Yaroshchuksee Eq. 59 of Ref. 6 obtains a self-consistent approxi-mation for constant v by replacing the exponential term ofEq. 12 with its average value in the pore,

v2 = b

20

1

dxe−q2wxd/2, 57

which should be compared with Eq. 56 with 0=0. In or-der to simplify the numerical task, Yaroshchuk introduces afurther approximation in which he replaces the potential wzinside the depletion term of Eq. 57 by its value in themiddle of the pore, wd /2. Then Eq. 57 takes the simplerform

v2 = b

2e−q2wd/2/2. 58

The self-consistent midpoint approximation is frequentlyused in nanofiltration theories 6,29,37. For =0, the mid-point potential has the simple form wd /2= b−vB−2B ln1−e−vd /d. This approach is compared with the fullvariational treatment in Fig. 7 where the adimensional in-verse screening length in the pore v /b is plotted as a func-tion of the pore size d. We first note that as d decreasesbelow a critical value d, the pore is empty of salt and v=0. The inset of Fig. 7 shows d versus the inverse bulkscreening length. Searching for d such that v=0 in Eq. 56leads to the same equation as Eq. 57, thus the value of d isidentical within both approaches. However, Fig. 7 shows thatthe midpoint approximation, Eq. 58, overestimates the in-ternal salt concentration as well as the abruptness of thecrossover to an ion-free regime for decreasing pore size. In-deed, this approximation is equivalent to neglecting thestrong ion exclusion close to the pore surfaces which islarger than in the middle of the pore. A similar behavior wasalso observed in Fig. 6 of Ref. 22 for the screening lengthin the neighborhood of the dielectric interface.

The effect of the dielectric discontinuity is illustrated inFig. 8a where the inverse internal screening length is com-

pared for between 0 and w=78 where the image-chargerepulsion is absent and the solvation effect is solely respon-sible for ion repulsion. First of all, one observes that the totalexclusion of ions in small pores is specific to the case =0.Moreover, in the solvation only case, the inverse screeninglength inside the pore only slightly deviates from the bulkvalue, 0.8v /b1. This clearly indicates that, within thepointlike ion model considered in this work, the image-charge interaction brings the main contribution to salt rejec-tion from neutral membranes. Roughly speaking, the image-charge and solvation effects come into play when the surfaceof the ionic cloud of radius b

−1 around a single ion located atthe pore center touches the pore wall, i.e., for b

−1d /2. Thissimple picture fixes a characteristic length dch2b

−1 belowwhich the internal ion density significantly deviates from thebulk value and ion rejection takes place. This can be verifiedfor intermediate salt densities in the bottom plot of Fig. 7 andthe top plot of Fig. 8.

Since image-charge effects are proportional to q2, we il-lustrate in Fig. 8b the effect of ion valency q. At pore sized2.5B1.8 nm, where the inverse internal screeninglength for monovalent ions is close to 80% of its saturationvalue b, the exclusion of divalent ions from the membraneis total. This effect driven by image interactions is evenmuch more pronounced for trivalent ions. Since the typicalpore size of nanofiltration membranes ranges between 0.5and 2 nm, we thus explain why ion valency plays a centralrole in ion selectivity, even inside neutral pores.

The salt reflection coefficient, frequently used in mem-brane transport theories to characterize the maximum salt

0.0

0.2

0.4

0.6

0.8

1.0

VariationalYaroschuk

κv/κb

0 1 2 3

d/lB

0.0

0.2

0.4

0.6

0.8

1.0

(b)

(a)

κblB0.1 0.60

1

2

3

d*/lB

κv/κb

FIG. 7. Color online Inverse screening length inside the neu-tral membrane monovalent ions normalized by b vs the pore sized /B for =0 and a bB=0.1 b=1.926 mmol L−1, b bB

=1 b=0.1926 mol L−1. Dashed lines correspond to the midpointapproximation, Eq. 58. The inset shows the characteristic poresize corresponding to total ionic exclusion as a function of the in-verse bulk screening length. The bottom curve corresponds tomonovalent ions and the top curve to divalent ions.


041601-10

rejection obtained at high pressure is related to the ratio ofthe net flux of ions across the membrane to that of the sol-vent volume flux J per unit transverse surface,

s 1 −1

Jb

0

d

vzzdz

= 1 – 121/2

1

x1 − xe−q2wxd/2dx . 59

where we have used, in the second equality, the Poiseuillevelocity profile, vz= 6J

d3 zd−z in the pore and the PMFgiven by Eq. 24. It depends only on the parameters bBand d /B. In certain nanopores with hydrophobic surfaces,the solvent flux may considerably deviate from the Poiseuilleprofile see 38. In this case, the velocity profile is flat,vz= J

d plug flow. We emphasize that since the velocityprofile is normalized in both cases, the midpoint approxima-tion is unable to distinguish between a Poiseuille and a plugflow velocity profile. Figure 9 displays s as a function of

the inverse bulk screening length for two pore sizes d=2Band d=5B. As seen by Yaroshchuk, decreasing the pore sizeshifts the curves to higher bulk concentration and thus in-creases the range of bulk concentration where nearly totalsalt rejection occurs. However, quantitatively, the differencebetween the variational and midpoint approaches becomessignificant at high bulk concentrations and this difference isaccentuated in the case of plug-flow for which s is higherwhen compared to the Poiseuille case because the flow ve-locity no longer vanishes at the pore wall where the saltexclusion is strongest. This deviation is again due to themidpoint approximation of Eq. 58 in which the image in-teractions are underestimated. However since the velocityprofile vanishes at the solid surface for the Poiseuille flow,the deficiencies of the midpoint approximation are less vis-ible in s than in v in this case.

Finally, we compute the disjoining pressure within ourvariational approach. We compare in Appendix E the resultwith that of the more involved variational scheme presentedin Ref. 22 and show that one gets a very similar behavior,revealing that the simpler variational method is able to cap-ture the essential physics of the slit pore.

As stressed above, the main benefit obtained from thesimpler approach proposed in this work is that the minimi-zation procedure is much less time consuming. This pointbecomes crucial when considering the fixed charge of themembrane, which is thoroughly studied in the next section.

B. Charged pore, symmetric electrolyte

In this section, we apply the variational approach to aslitlike pore of surface charge s0. In the following, wewill solve Eqs. 55 and 56 numerically in order to test, asin the case of a single charged surface, the validity of re-stricted trial forms for 0z. We define the partition coeffi-cients in the pore for counterions and coions, k+ and k−, as

0.5 1.0 1.5 2.0 2.5 3.0 3.5

d/lB

0.0

0.2

0.4

0.6

0.8

1.0

κv/κb

(a)

0.0 0.4 0.8 1.2

log10 (d/lB)

0.0

0.2

0.4

0.6

0.8

1.0

κv/κb

monovalent ionsdivalent ionstrivalent ions

(b)

FIG. 8. Color online Inverse screening length inside the mem-brane vs the pore size d /B w=78,bB=1. a From bottom totop: =0 =1, =3.2 =0.92, =39 =1 /3, and =78 =0. b Log-linear plot for monovalent, divalent and trivalent ions,from left to right =0.

-5.0 -4.0 -3.0 -2.0 -1.0

Log10(κb lB)

0.1

0.3

0.5

0.7

0.9

Σs

d=5, Variationald=5, Yaroschukd=2, Variationald=2, Yaroschukd=2, Variational, plug flow

FIG. 9. Color online Salt reflection coefficient dimensionlessagainst the logarithm of the inverse bulk screening length for =0and two pore sizes, d /B=2 and 5 light red lines correspond tothe midpoint approximation, Eq. 58.


041601-11

k b

= 0

d dz

de− z. 60

where z is given by Eq. 26.

1. Effective Donnan Potential

When one considers a charged nanopore, because of itssmall size, gradients of the potential 0 can be neglected as afirst approximation. We thus assume a constant potential 0.The so-called effective Donnan potential 0 introduced byYaroshchuk 6 will be fixed by the variational principle. Bydifferentiating the grand potential Eq. 20 with respect to 0or equivalently integrating Eq. 55 from z=0 to z=d with 0=0, we find

2s = − 2qb sinhq 00

d

dze−q2wz/2, 61

which is simply the electroneutrality relation in the pore,taken in charge by the electrostatic potential 0. By defining

! = 0

1

dx exp− q2Bwxd/2d , 62

=0

1

dx exp−wxd/2d , 63

where wxwxd /B, we have k =! exp"q 0 and Eq.61 can be rewritten as

k+ − k− = 2s

qbd=

Xm

qb=

8

b2dG

=8

b2d

, 64

where we introduce in the third equality the Gouy-Chapmanlength G and the quantity Xm=2s /d, frequently used innanofiltration theories, corresponds to the volume chargedensity of the membrane. Hence, the partition coefficient ofthe charge, k+−k−, does not depend on , i.e., charge imageand solvation forces. By using Eq. 61 in order to eliminatethe potential 0 from Eq. 60, one can rewrite the partitioncoefficients in the form

k = !e"q 0 =!2 + 4

b2d2

4

b2d

. 65

By substituting into Eq. 56 the analytical expression for 0obtained from Eq. 61 or Eq. 65, one obtains a singlevariational equation for v to be solved numerically,

dv2 + dv tanhdv

= db2!2 + 4

b2d2

1 + 0

1

dxe−wxd/2cosh2x − 1dv

! coshdv .

66

The numerical solution of Eq. 66 is plotted in Fig. 10 as afunction of the coupling parameter . We see that as wemove from the WC limit to the SC one by increasing , thepore evolves from a high to a low salt regime. This quiterapid crossover, which results from the exclusion of ionsfrom the membrane, is mainly due to repulsive image-chargeand solvation forces controlled by ! whose effects increasewith increasing .

In Fig. 11 are plotted the partition coefficients of counte-rions and coions, Eq. 60, as a function of . Here again, k decreases with increasing . Moreover, we clearly see thatthe rejection of coions from the membrane becomes total for

FIG. 10. Color online Inverse internal screening length vagainst for bG=2, =0 and a d=3G and b d=10G. Com-parison of various approximations: Yaroshchuk, Eq. 71 dia-monds, variational Donnan potential dashed line, piecewise solu-tions solid line, and numerical results squares. Horizontal linescorresponds to the WC limit, Eq. 67 top, and SC limit, Eq. 70bottom.

1.0 2.0 3.0 4.0

Ξ0.0

0.2

0.4

0.6

0.8

1.0

<ρ/ρb>

0.0

0.2

0.4

0.6

0.8

1.0

1.2 CounterionsCoions

(a)

(b)

<ρ/ρb>

FIG. 11. Color online Ionic partition coefficients, k , vs forbG=2, =0, and a d=3G and b d=10G. The horizontal linecorresponds to the SC limit for counterions. As explained in thetext, we note that k+−k−=8 / b

2dG.


041601-12

4. In other words, even for intermediate coupling param-eter values, we are in a counterion-only state. This is obvi-ously related to the electrical repulsion of coions by thecharged surface.

In the asymptotic WC limit →0, !=1 and we find theclassical Donnan results in mean-field where k−=k+

−1=eq 0

with q 0=arcsinh4 / b2d. The variational Eqs. 66 and

65 reduce to

v2 = b

21 + 4

b2d2

, 67

k =1 + 4

b2d2

4

b2d

, 68

Quite interestingly, the relation Eq. 67 shows that, even inthe mean-field limit, due to the ion charge imbalance createdby the pore surface charge, the inverse screening length islarger than the Debye-Hückel value b. In the case of smallpores or strongly charged pores or at low values of the bulkionic strength, i.e., b

2Gd1 or db s /q, we find v2 /Gd and −=0 and +=2s / dq. We thus find theclassical Poisson-Boltzmann result for counterions only 25.The counterion-only case is also called good coion exclusionlimit GCE, a notion introduced in the context of nanofiltra-tion theories 6,39,40. Hence, in this limit the quantity ofcounterions in the membrane is independent of the bulk den-sity and depends only on the pore size d and the surfacecharge density s. In the case of a pore of size d1 nm andfixed surface charge s0.03 nm−2, this limit can bereached with an electrolyte of bulk concentration b50 mmol L−1. In the opposite limit b

2Gd1, one findsvb and =b.

In the SC limit →, !=0 and Eq. 66 simplifies to

dv2 + dv tanhdv = 4d1 + sechdv . 69

For dG d1, the solution of Eq. 69 yields with a highaccuracy

v 1 + 16d − 1

2d. 70

The partition coefficients simplify to k−=0 and k+=8 / db2

=2s / dqb and we find the counterion only case or GCElimit without image-charge forces discussed by Netz 25.Partition coefficients in the SC limit and variational inversescreening length in both limits, Eqs. 67 and 70, are illus-trated in Figs. 10 and 11 by dotted reference lines. Conse-quently, one reaches for =0 the GCE limit exclusively forlow salt density or small pore size, while the SC limit leadsto GCE for arbitrary bulk density. It is also important to notethat although the pore-averaged densities of ions are thesame in the GCE limit of WC and SC regimes, the densityprofiles are different since when one moves away from thepore center, the counterion densities close to the interfaceincrease in the WC limit due to the surface charge attraction

and decrease in the SC limit due to the image-charge repul-sion.

It is interesting to compare this variational approach to theapproximate midpoint approach of Yaroshchuk 6. Forcharged membranes, he considers a constant potential andreplaces the exponential term of Eqs. 11 and 12 by itsvalue in the middle of the pore. He obtains the followingself-consistent equations:

2 = b2e−q2wd/2/2 coshq 0 , 71

2s = − 2qdb sinhq 0e−q2wd/2/2. 72

The above set of equations are frequently used in nanofiltra-tion theories 6,29,37. By combining these equations in or-der to eliminate 0, one obtains an approximate nonlinearequation for v approximation CYar in Fig. 10. In the limitof a high surface charge, the nonlinear Eqs. 71 and 72depend only on the pore size d and the surface charge densitys,

2 8Bqs

d=

4

Gd s. 73

One can verify that in the regime of strong surface charge,Eq. 73 is also obtained from the asymptotic solution Eq.70 since the dependence of the PMF on z is killed when→ and only the midpore value contributes. The numeri-cal solution of Eq. 71 and 72 is illustrated as a function of in Fig. 10, and as a function of the surface charge in Fig.12, together with the asymptotic formula Eq. 73. For theparameter range considered in Fig. 10, the solution of Eq.71 strongly deviates from the result of the full variational

σ lB2

0.0

1.0

2.0

3.0

κvlB Donnan variationalCYarasymptotic limit

(a)

0.0 0.1 0.2 0.3 0.40.0

1.0

2.0

3.0 (b)

(a)

κvlB

FIG. 12. Color online Inverse internal screening length vagainst the reduced surface charge =B

2s for d=B, =0 and abB=1, b bB=2: constant variational Donnan approximationsolid line, asymptotic result Eq. 73 dotted line and Yaroshchukapproximation Eq. 71 dashed line.


041601-13

calculation. For 2, the midpoint approach follows an in-correct trend with increasing . It is clearly seen that atsome values of the coupling parameter, Eqs. 71 and 72 donot even present a numerical solution. Using the relationsd /B= d / and Bb=b for monovalent ions, one canverify that the regime where the important deviations takeplace corresponds to high ion concentrations. This is con-firmed in Fig. 12: the error incurred by the approximate mid-point solution of Yaroshchuk increases with the electrolyteconcentration.

In Sec. IV A on neutral nanopores, it has been underlinedthat, due to the image-charge repulsion, the ionic concentra-tion inside the pore decreases with the pore size d see Fig.8. In the present case of charged nanopores, this result ismodified: Eqs. 67, 70, and 73 show that for stronglycharged nanopores the concentration of ions inside the poredecreases with d. Moreover, the very high charge limit is acounterion-only state and Eq. 61 shows that, for a fixedsurface charge density, electroneutrality alone fixes the num-ber of counterions, N+, in a layer of length d joining bothinterfaces, and image-charge interactions play a little role.This is the reason why v

2+=N+ / Sd decreases for in-creasing d.

Hence, we expect an intermediate charge regime whichinterpolates between image force counterion repulsion caseof neutral pores, see Sec. IV A and counterion attraction bythe fixed surface charge. This is illustrated in Fig. 13 wherethe partition coefficients are plotted vs d for increasing s.As expected, coions are electrostatically pushed away by thesurface charge which adds to the repulsive image forces,leading to a stronger coion exclusion than for neutral pores.The issue is more subtle for counterions: obviously, increas-ing the surface charge, s, at constant pore size, d, increasesk+. However, for small fixed s, a regime where imagecharge and direct electrostatic forces compete, k+ is non-monotonic with d. Below a characteristic pore size, ddcr,the electrostatic attraction dominates over image-charge re-pulsion and due to the mechanism explained above, k+ de-creases for increasing d. For ddcr, the effect of the surfacecharge weakens and k+ starts increasing with d. In this re-gime, the pore behaves like a neutral system. The inset ofFig. 13 shows that dcr increases when s increases. Forhighly charged membranes lB

2s0.1, there is no minimumin k+d, and the average counterion density inside the mem-brane monotonically decreases toward the bulk value. Ex-perimental values for surface charges are 0s0.5 nm−2

or 0B2s0.25, which corresponds to physically attain-

able values of dcr. The interplay between image forces anddirect electrostatic attraction is thus relevant to the experi-mental situation.

The variational Donnan potential approximation is thus ofgreat interest since it yields physical insight into the exclu-sion mechanism and allows a reduction in the computationalcomplexity. However, membranes and nanopores are oftenhighly charged and spatial variations in the electrostatic po-tential inside the pore may play an important role. In thefollowing we seek a piecewise solution for 0z.

2. Piecewise solution

The variational modified PB Eq. 55 for 0 shows that asone goes closer to the dielectric interface, wz increases and

the screening experienced by the potential 0 gradually de-creases because of ionic exclusion. This nonperturbative ef-fect which originates from the strong charge-image repulsioninspires our choice for the variational potential 0z. We optfor a piecewise solution as in Sec. III: a salt-free solution inthe zone 0zh and the solution of the linearized PB equa-tion for hzd /2, with a charge renormalization parameter taking into account nonlinear effects. By inserting theboundary conditions 0 /z z=0=2 /G and 0 /z z=d/2=0and imposing the continuity of 0 and its first derivative atz=h Eq. B3, the piecewise potential, solution of Eq. 55with b

2 exp−q2wz /2 replaced by 2 , takes the form

0z = −2

Gz −

d

2 for 0 z h ,

−2

G

cosh d/2 − zsinh d/2 − h

for h z d/2,74

where

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

k-

d/lB

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.5

1.0

1.5

2.0

2.5

d/lB

0.01 0.03 0.05 0.07

σ (nm-2 )

0

5

10

15

20

25

dcr (nm)

k+

(b)

(a)

FIG. 13. Color online Partition coefficient in the pore ofcoions a and counterions b vs the pore size d /B for increasingsurface charge density, sB

2 =0,0.004,0.02,0.04,0.08,0.12, fromleft to right, and bB=1. Inset: critical pore size dcr vs the surfacecharge density s =0.


041601-14

= +2

Gd

2− h −

2

G coth d

2− h 75

is imposed by continuity, and , , h, and are the varia-tional parameters. By injecting the piecewise solution Eq.74 into Eq. 20, we finally obtain

F1

S= −

2sq − 2

h

G−

2d/2 − h2G sinh2 d/2 − h

+ − 42G

coth d/2 − h + −

b2

4Bq20

d

dze−q2wz/2 cosh 0z . 76

The solution to the variational problem is found by optimi-zation of the total grand potential F=F1+F2 with respect to , , h, , and v, where F2+F3 is given by Eq. 51 for ageneral value of and by Eq. 52 for =0. This was easilycarried out with Mathematica software.

A posteriori, we checked that two restricted forms for 0,homogeneous with h=0 and piecewise with =0, were goodvariational choices. Figure 14 compares the ion densities ob-tained from the variational approach with homogeneous 0with the predictions of the MC simulations 30 and the

NLPB equation for =W, d=2 and =1. Two variationalchoices are displayed in this figure, namely, the homoge-neous approach with four parameters v=1.68, =1.36,=0.16,=0.97 and a simpler choice with =1, =v andtwo variational parameters: v=1.69,=−0.18. In the lattercase, one can obtain an analytical solution for and injectingthis solution into the free energy, one is left with a singleparameter v to be varied in order to find the optimal solu-tion. We notice that with both choices, the agreement be-

tween the variational method and MC result is good. It isclearly seen that the proposed approach can reproduce with agood quantitative accuracy the reduced solvation inducedionic exclusion, an effect absent at the mean-field level.Moreover, we verified that with the single parameter choice,one can reproduce at the mean-field variational level the iondensity profiles obtained from the numerical solution of theNLPB equation dashed lines in Fig. 14 almost exactly. Wefinally note that the small discrepancy between the predic-tions of the variational approach and the MC results close tothe interface may be due to either numerical errors in thesimulation, or our use of the generalized Onsager-Samarasapproximation our homogeneous choice for the inverse ef-fective screening length appearing in the Green’s function v0does not account for local enhancement or diminution ofionic screening due to variations in local ionic density.

For =0, the piecewise and homogeneous solutions arecompared with the full numerical solution of Eqs. 55 and56 in Fig. 15 for =1 and 100. First of all, one observesthat for =1, both variational solutions match perfectly wellwith the numerical solutions. For =100, the piecewise so-lution matches also perfectly well with the numerical one,whereas the matching of the homogeneous one is poorer. Theoptimal values of the variational parameters v , , ,h forthe piecewise choice are 2.57,2.6,0.98,0.15 for =1 and0.83,0.13,0.97,1.37 for =100.

The form of the electrostatic potential 0z is intimatelyrelated to ionic concentrations. Ion densities inside the poreare plotted in Fig. 16 for =1 and =100. We first noticethat even at =1, the counterion density is quite differentfrom the mean-field prediction. Furthermore, due to image-charge and electrostatic repulsions from both sides, the coion

2πΞ

l G3

ρ(z

)

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

z/lG

Counterion, MC

Coion, MC

Counterion, variational with four parameters

Coion, variational with four parameters

Counterion, variational with a single parameter

Coion, variational with a single parameter

Counterion, NLPB

Coion, NLPB

FIG. 14. Color online Ion densities in the nanopore for =W, =1, and h /G=2. The continuous lines correspond to theprediction of the variational method with four parameters, thedashed-dotted line the variational solution with a single parametersee the text, the symbols are MC results Fig. 2 of 30 and thedashed lines denote the numerical solution of the nonlinear PBresult.

-1.0

-0.5

0.0

NumericalHomogeneousPiecewise

φ0

(a)

0.0 1.0 2.0 3.0 4.0z/lG

-111.5

-110.0

-108.5(b)

φ0

FIG. 15. Color online Variational electrostatic potential inunits of kBT in the nanopore. Comparison of the numerical solutionof Eq. 55 with the homogeneous h=0 and piecewise solution ofEq. C5 for Gb=3 and a =1, b =100. The horizontal lineis the Donnan potential obtained from Eq. 61 =0.


041601-15

density has its maximum in the middle of the pore. On theother hand, the counterion density exhibits a double peak,symmetric with respect to the middle of the pore, whichoriginates from the attractive force created by the fixedcharge and the repulsive image forces. When increases,we see that the counterion density close to the wall shrinksand becomes practically flat in the middle of the pore. Hencethe potential 0 linearly increases with z until the counterionpeak is reached and then it remains almost constant since thecounterion layer screens the electrostatic field created by thesurface charge since in Fig. 15, z is renormalized by theGouy-Chapman length which decreases with increasing s,one does not see the increase of the slope 0z=0. Inagreement with the variational Donnan approximation above,coions are totally excluded from the pore for large . Hence,the piecewise potential allows one to go beyond the varia-tional Donnan approximation within which the density pro-file does not exhibit any concentration peak.

The inverse screening length v obtained with the piece-wise solution is compared in Fig. 10 with the prediction ofthe Donnan approximation and that of the numerical solu-tion. The agreement between piecewise and numerical solu-tions is extremely good. Although the Donnan approximationslightly underestimates the salt density in the pore, its pre-dictions follow the correct trend.

V. CONCLUSION

In this study, we applied the variational method to inter-acting pointlike ions in the presence of dielectric discontinui-ties and charged boundaries. This approach interpolates be-tween the WC limit 1 and the SC one 1,

originally defined for charged boundaries without dielectricdiscontinuity, and takes into account image-charge repulsionand solvation effects. The variational Green’s function v0 hasa Debye-Hückel form with a variational parameter v and theaverage variational electrostatic potential 0z is eithercomputed numerically or a restricted form is chosen withvariational parameters. The physical content of our restrictedvariational choices can be ascertained by inspecting the gen-eral variational equations Eqs. 11 and 12 for symmetricsalts. The generalized Onsager-Samaras approximation thatwe have adopted for the Green’s function replaces a localspatially varying screening length by a constant variationalone; although near a single interface this screening length isequal to the bulk one, in confined geometries the constantvariational screening length can account in an average wayfor the modified ionic environment as compared with theexternal bulk with which the pore is in equilibrium and cantherefore strongly deviate from the bulk value. This modifiedionic environment arises both from dielectric and reducedsolvation effects present even near neutral surfaces encodedin the Green’s function and the surface charge effects en-coded in the average electrostatic potential. Our restrictedvariational choice for 0z is based on the usual nonlinearPoisson-Boltzmann type solutions with a renormalized in-verse screening length that may differ from the one used forv0 and a renormalized external charge source. The couplingbetween v0 and 0 arises because the inverse screeninglength for v0 depends on 0 and vice versa. The optimalchoices are the ones that extremize the variational free en-ergy.

In the first part of the work, we considered single inter-face systems. For asymmetric electrolytes at a single neutralinterface, the potential 0z created by charge separationwas numerically computed. It was satisfactorily compared toa restricted piecewise variational solution and both chargedensities and surface tension are calculated in a simpler waythan Bravina 5 and valid over a larger bulk concentrationrange. The variational approach was then applied to a singlecharged surface and it was shown that a piecewise solution,characterized by two zones, can accurately reproduce thecorrelations and nonlinear effects embodied in the more gen-eral variational equation. The first zone of size h is governedby a salt-free regime, while the second region corresponds toan effective mean-field limit. The variational calculation pre-dicts a relation between h and the surface charge of the formh c+lns / s where the parameter c depends on thetemperature and ion valency.

In the second part, we dealt with a symmetric electrolyteconfined between two dielectric interfaces and investigatedthe important problem of ion rejection from neutral andcharged membranes. We illustrated the effects of ion valencyand dielectric discontinuity on the ion rejection mechanismby focusing on ion partition and salt reflection coefficients.We computed within a variational Donnan potential approxi-mation, the inverse internal screening length and ion parti-tion coefficients, and showed that for 4 one reaches theSC limit, where the partition coefficients are independent ofthe bulk concentration and depend only on the size andcharge of the nanopore. This result has important experimen-tal applications, since it indicates that complete filtration can

0.0

0.5

1.0

1.5coioncounterionmean-field

ρ/ρ b

ulk

z/lG0.0 1.0 2.0 3.0 4.0 5.0

0.00

0.10

0.20

0.30

0.40

(b)

(a)

ρ/ρ b

ulk

FIG. 16. Color online Local ionic partition coefficient in thenanopore same parameters as in Fig. 15 and =0 computed withthe piecewise solution. a =1, b =100. The dotted line in thetop plot corresponds to the mean-field prediction for counteriondensity.


041601-16

be done at low bulk salt concentration and/or high surfacecharge. Furthermore, we showed that, due to image interac-tions, the quantity of salt allowed to penetrate inside a neu-tral nanopore increases with the pore size. In the case ofstrongly charged membranes, this behavior is reversed forthe whole physical range of pore size. We quantified theinterplay between the image-charge repulsion and the surfacecharge attraction for counterions and found that even in thepresence of a weak surface charge, the competition betweenthem leads to a characteristic pore size dcr below which thecounterion partition coefficient rapidly decreases with in-creasing pore size. On the other hand, for nanopores of sizelarger than dcr the system behaves like a neutral pore. Ourvariational calculation was compared to the Debye closureapproach and the midpoint approximation used by Yarosh-chuk 6. We have not yet been able to find analytical ornumerical solutions to the exact variational equations Eqs.11 and 12. Our approach, based on restricted variationalchoices, shows significant deviations from Yaroshchuk’smidpoint approach at high ion concentrations and small poresize. Finally, the introduction of a simple piecewise trialform for 0, which perfectly matches the numerical solutionsof the variational equations, enabled us to go beyond thevariational Donnan potential approximation and thus accountfor the concentration peaks in counterion densities. We com-puted ion densities in the pore and showed that for 4, theexclusion of coions from the pore is nearly total. We alsocompared the ionic density profiles obtained from the varia-tional method with MC simulation results and showed thatthe agreement is quite good, which illustrates the accuracy ofthe variational approach in handling the correlation effectsabsent at the mean-field level.

The main goal in this work was first to connect two dif-ferent fields in the chemical physics of ionic solutions focus-ing on complex interactions with surfaces: field-theoretic cal-culations and nanofiltration studies. Moreover, on the onehand, this variational method allows one to consider, in anonperturbative way, correlations and nonlinear effects; onthe other hand the choice of one constant variational Debye-Hückel parameter is simple enough to reproduce previousresults and to illuminate the mechanisms at play. This ap-proach is also able to handle, in a very near future, morecomplicated geometries, such as cylindrical nanopores, or anonuniform surface charge distribution.

The present variational scheme also neglects ion-size ef-fects and gives rise to an instability of the free energy atextremely high salt concentration. Second-order correctionsto the variational method may be necessary in order to prop-erly consider ionic correlations leading to pairing 41,42 andto describe the physics of charged liquids at high valency,high concentrations or low temperatures. Introducing ionsize will also allow us to introduce an effective dielectricpermittivity p for water confined in a nanopore intermediatebetween that of the membrane matrix and bulk water, leadingnaturally to a Born-self-energy term that varies inverselywith ion size and depends on the difference between 1 /wand 1 /p 43,44. Furthermore, the incorporation of the ionpolarizability 45 will yield a more complete physical de-scription of the behavior of large ions 46. Charge inversionphenomena for planar and curved interfaces are another im-

portant phenomenon that we would like to consider in thefuture 47. Note, however, that our study of asymmetricsalts near neutral surfaces reveals a closely related phenom-enon: the generation of an effective nonzero surface chargedue to the unequal ionic response to a neutral dielectric in-terface for asymmetric salts. A further point that possessesexperimental relevance is the role played by surface chargeinhomogeneity. Strong-coupling calculations show that aninhomogeneous surface charge distribution characterized bya vanishing average value gives rise to an attraction of ionstoward the pore walls, but this effect disappears at the mean-field level 48. For a better understanding of the limitationsof the proposed model, a more detailed comparison withMC/MD simulations is in order 49,50. Finally, dynamicalhindered transport effects 28,49 such as hydrodynamicforces deserve to be properly included in the theory for prac-tical applications.

ACKNOWLEDGMENTS

We would like to thank David S. Dean for numerous help-ful discussions. This work was supported in part by theFrench ANR Program NANO-2007 SIMONANOMEMproject, Grant No. ANR-07-NANO-055.

APPENDIX A: VARIATIONAL FREE ENERGY

For planar geometries charged planes, the translationalinvariance parallel to the plane, allows us to significantlysimplify the problem by introducing the partial Fourier-transformation of the trial Green’s function in the form

v0z,z,r − r = dk

22eik·r−rv0z,z,k . A1

By injecting the Fourier decomposition A1 into Eq. 17,the DH equation becomes

−

zz

z+ zk2 + v

2zv0z,z,k;vz

=e2

kBTz − z . A2

The translational symmetry of the system enables us toexpress any thermodynamic quantity in terms of the partiallyFourier-transformed Green’s function v0z ,z ,k. The averageelectrostatic potential contribution to Fv that follows fromthe average H0 reads

F1 = S dz− 0z2

8B+ sz 0z

− i

ie−qi

2Wz/2−qi 0z , A3

the kernel part is

F2 =S

1620

1

d0

dkk dzv

2zB

v0z,z,k;vz − v0z,z,k;vz A4

where the first term in the integral follows from F0 and the


041601-17

second term from H00. Finally, the unscreened van derWaals contribution, which comes from the unscreened partof F0, is given by

F3 =S

8

0

1

d dz 1

Bz−

1

B 2

− ln D e−dr 2/8B, A5

where Bz=e2 / 4zkBT. The technical details of thecomputation of F3 can be found in Ref. 51. The last term ofEq. A5 simply corresponds to the free energy of a bulkelectrolyte with a dielectric constant w. In the above rela-tions, S stands for the lateral area of the system. The dummy“charging” parameter is usually introduced to compute theDebye-Hückel free energy 52. It multiplies the Debyelengths of v0z ,z ,k ;vz in Eq. A4 and the dielectric per-mittivities contained in the thermal average of the gradient inEq. A5. This latter is defined as

2 = − 02

+ dk

22 k2 + zzvcz,z,k;zz=z

A6

where we have introduced −1zB

−1+B−1z−B

−1 andvcz ,z ,k ;z stands for the Fourier transformed Coulomboperator given by Eq. 5 with Bjerrum length z. Thequantity F3 defined in Eq. A5 does not depend on the in-verse screening length v. Moreover, in order to satisfy theelectroneutrality, 0z must be constant in the salt-free partsof the system where BzB. Hence, F3 does not dependon the potential 0z.

APPENDIX B: VARIATIONAL CHOICE FOR THENEUTRAL DIELECTRIC INTERFACE

We report in this appendix the restricted variational piece-wise 0z for a neutral dielectric interface which is a solu-tion of

2 0

z2 = 0 for z a , B1

2 0

z2 − 2 0 = cze− z for z a , B2

where 0z in both regions is joined by the continuity con-ditions

0a = 0

a, 0

z

z=a

= 0

z

z=a

. B3

We also tried to introduce different variational screeninglengths in the second term of the lhs and in the rhs of Eq.B2 without any significant improvement at the variationallevel. For this reason, we opted for a single inverse varia-tional screening length, . The solution of Eqs. B1 andB2 is

0z = for z a ,

1 + z − ae− z−a for z a . B4

where the coefficient c disappears when we impose theboundary and continuity conditions, Eq. B3. The remainingvariational parameters are the constant potential , the dis-tance a and the inverse screening length . By substitutingEq. B4 into Eq. A3, we obtain the variational grand po-tential

Fv = Vb

3

24+

S

32b

2 − B2 − S−

0

dz

e−q−2wz/2+q− 0z +

q−

q+e−q+

2wz/2−q+ 0z . B5

APPENDIX C: VARIATIONAL CHOICE FOR THECHARGED DIELECTRIC INTERFACE

The two types of piecewise variational functions used forsingle charged surfaces are reported below.

i The first trial potential obeys the salt-free equation inthe first zone and the NLPB solution in the second zone,

2 0NL

z2 = 2z for z h , C1

2 0NL

z2 − 2 sinh 0 = 0 for z h ,

whose solution is

0NLz =4 arctanh + 2z − h for z h ,

4 arctanhe− z−h for z h , C2

where = −1+ 2 . Variational parameters are h and ,

and the electrostatic contribution of the variational grand po-tential Eq. A3 is

F1

S=

h + − 4 arctanh

2−b

2

4 dze−wz/2 cosh 0

NL.

C3

ii The second type of trial potential obeys the salt-freeequation with a charge renormalization in the first zone andthe linearized Poisson-Boltzmann solution in the secondzone,

2 0L

z2 = 2z for z h ,

2 0L

z2 − 2 0

L = 0 for z h , C4

whose solution is given by


041601-18

0Lz = −

2

+ 2z − h for z h ,

−2

e− z−h

for z h . C5

Variational parameters introduced in this case are h, , andthe charge renormalization , which takes into account non-linearities at the mean-field level 20. The variational grandpotential reads

F1

S=

21 + h − 21/2 + h 2

−

2

4 dze−wz/2 cosh 0

Lz . C6

In both cases, the boundary condition satisfied by 0 isthe Gauss law

0

z

z=0

= 2 , C7

where =1 for the nonlinear case. It is important to stressthat in the case of a charged interface, Eq. C7 holds even if0. In fact, since the left half-space is ion-free, 0z mustbe constant for z0 in order to satisfy the global electroneu-trality in the system.

APPENDIX D: DEFINITION OF THE SPECIALFUNCTIONS

The definition of the four special functions used in thiswork are reported below.

Linx = k1

xk

kn , n = Lin1 , D1

x;y,z = 0

x

dtty−11 − tz−1, D2

2F1a,b;c;x = k0

akbkckxk

k!, D3

where ak=a ! / a−k!.

APPENDIX E: DISJOINING PRESSUREFOR THE NEUTRAL PORE

The net pressure between plates is defined as

P = −1

S

Fv

d− 2b −

b3

24 , E1

where the subtracted term on the rhs is the pressure of thebulk electrolyte. The total van der Waals free energy, whichis simply the zeroth order contribution F0 to the variationalgrand potential Eq. 9, is with the constraint v=b there isno renormalization of the inverse screening length at thisorder,

FvdW =db

3

24−b

8dLi2e−2db −

1

16d2Li3e−2db

E2

and

PvdW = −1

S

FvdW

d+b

3

24. E3

We illustrate in Fig. 17 the difference between the van derWaals pressure and the prediction of the variational calcula-tion for bB=0.5, 1, and 1.5. We notice that the predictionof our variational calculation yields a very similar behaviorto that illustrated in Fig. 8 of Ref. 22. The origin of theextra-attraction that follows from the variational calculationwas discussed in detail in the same article. This effect origi-nates from the important ionic exclusion between the platesat small interplate separation, an effect that can be capturedwithin the variational approach.

1 A. Heydweiller, Ann. Phys. 4, 33145 1910.2 P. K. Weissenborn and R. J. Pugh, Langmuir 11, 1422 1995.3 G. Wagner, Phys. Z. 25, 474 1924.4 L. Onsager and N. Samaras, J. Chem. Phys. 2, 528 1934.5 V. E. Bravina, Sov. Phys. Dokl. 120, 381 1958.

6 A. E. Yaroshchuk, Adv. Colloid Interface Sci. 85, 193 2000.7 R. B. Schoch, J. Han, and P. Renaud, Rev. Mod. Phys. 80, 839

2008.8 A. G. Moreira and R. R. Netz, Phys. Rev. Lett. 87, 078301

2001.

0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0

d/lB

0.1

0.2

0.3

0.4

0.5

-βl B

3(P

var-

Pvd

W)

FIG. 17. Color online Difference between the pressure and thescreened van der Waals contribution vs d /B for blB=0.5,1 and1.5, from left to right =0.


041601-19

9 A. G. Moreira and R. R. Netz, Europhys. Lett. 52, 705 2000.10 H. Boroudjerdi et al., Phys. Rep. 416, 129 2005.11 R. R. Netz and H. Orland, Eur. Phys. J. E 1, 203 2000.12 D. Dean and R. Horgan, Phys. Rev. E 65, 061603 2002.13 A. G. Moreira and R. R. Netz, Eur. Phys. J. E 8, 33 2002.14 A. Naji, S. Jungblut, A. Moreira, and R. R. Netz, Physica A

352, 131 2005.15 S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, P. Pincus,

and D. Hone, J. Chem. Phys. 80, 5776 1984.16 Y. Levin, Rep. Prog. Phys. 65, 1577 2002.17 Y. Levin, J. Chem. Phys. 113, 9722 2000.18 G. S. Manning, J. Chem. Phys. 51, 924 1969.19 M. Kanduč, A. Naji, Y. S. Jho, P. A. Pincus, and R. Podgornik,

J. Phys.: Condens. Matter 21, 424103 2009.20 R. R. Netz and H. Orland, Eur. Phys. J. E 11, 301 2003.21 R. A. Curtis and L. Lue, J. Chem. Phys. 123, 174702 2005.22 M. M. Hatlo, R. A. Curtis, and L. Lue, J. Chem. Phys. 128,

164717 2008.23 M. M. Hatlo and L. Lue, Soft Matter 4, 1582 2008.24 M. M. Hatlo and L. Lue, Soft Matter 5, 125 2009.25 R. R. Netz, Eur. Phys. J. E 5, 557 2001.26 The case with dielectric discontinuity has been studied without

added salt in M. Hatlo and L. Lue, Europhys. Lett 89, 250022010.

27 V. M. Starov and N. V. Churaev, Adv. Colloid Interface Sci.43, 145 1993.

28 X. Lefebvre, J. Palmeri, and P. David, J. Phys. Chem. B 108,16811 2004.

29 A. Yaroshchuk, Sep. Purif. Technol. 22-23, 143 2001.30 C. Li and H. R. Ma, J. Chem. Phys. 121, 1917 2004.31 I. Benjamin, Annu. Rev. Phys. Chem. 48, 407 1997.32 S. M. Avdeev and G. A. Martynov, Colloid J. USSR 48, 632

1968.33 Note that the factor −1 in front of v0 in Eq. 12 found in 19

should be cancelled.

34 J. Janeček and R. R. Netz, J. Chem. Phys. 130, 074502 2009.35 This is in fact a maximization with respect to 0 or to any

variational parameter associated to a restricted choice for 0.Indeed, the physical electrostatic potential 0 corresponds to apure imaginary auxiliary field in the functional integral Z.Hence the complex minimum of H−H00 with respect to corresponds to a real maximum with respect to 0.

36 J. D. Jackson, Classical Electrodynamics, 2nd ed. Wiley, NewYork, 1975.

37 A. Szymczyk and P. Fievet, J. Membr. Sci. 252, 77 2005.38 F. Fornasiero et al., Proc. Natl. Acad. Sci. U.S.A. 105, 17250

2008.39 X. Lefebvre and J. Palmeri, J. Phys. Chem. B 109, 5525

2005.40 A. Yaroshchuk, Adv. Colloid Interface Sci. 60, 1 1995.41 M. E. Fisher and Y. Levin, Phys. Rev. Lett. 71, 3826 1993.42 J. P. Simonin et al., J. Phys. Chem. B 103, 699 1999.43 S. Senapati and A. Chandra, J. Phys. Chem. B 105, 5106

2001.44 J. Marti et al., J. Phys. Chem. B 110, 23987 2006.45 S. Buyukdagli, D. S. Dean, M. Manghi and J. Palmeri unpub-

lished.46 P. Jungwirth and B. Winter, Annu. Rev. Phys. Chem. 59, 343

2008.47 C. D. Lorenz and A. Travesset, Phys. Rev. E 75, 061202

2007.48 A. Naji and R. Podgornik, Phys. Rev. E 72, 041402 2005.49 Y. Ge et al., Phys. Rev. E 80, 021928 2009.50 K. Leung and S. B. Rempe, J. Comput. Theor. Nanosci. 6,

1948 2009.51 R. R. Netz, Eur. Phys. J. E 5, 189 2001.52 D. A. McQuarrie, Statistical Mechanics University Science

Book, New York, 2000, Chap. 15.


041601-20


3.6 «Évaporation capillaire» des ions dans les nanopores cy-lindriques

Alors que près d’une seule surface diélectrique, un ion a une seule charge image, dans unpore en forme de fente, l’ion en a une infinité discrète dans la direction perpendiculaire auxsurfaces. Dans la géométrie cylindrique, on s’attend en quelque sorte à une infinité d’imagesdistribuées dans le plan perpendiculaire à l’axe et donc à une interaction diélectriquedémultipliée. Le potentiel induit par le cylindre diélectrique s’écrit

δv0(r, r;κv) =4`Bπ

∫ ∞

0dk∑′

m≥0

κKm(ka)K ′m(κa)− ηkKm(κa)K ′m(ka)ηkIm(κa)K ′m(ka)− κKm(ka)I ′m(κa)

I2m(κ|r|) (3.58)

où κ =√κ2v + k2, Im et Km sont les fonctions de Bessel modifiées et le prime signifie que

le 1er terme de la somme est multiplié par 1/2. Pour des rayons de pores a = 0.84 nmet sans sel dans le pore, ce potentiel est de l’ordre de 11 kBT au centre, mais il chutebrutalement autour de 3 à 7 kBT pour les valeurs de κv déterminées après minimisationde Ω1. Dans cette partie, nous avons fait l’approximation du potentiel effectif de Donnan,soit φ0(z) = cste lorsque la surface cylindrique est chargée.

3.6.1 Pore neutre

Dans le cas d’un pore neutre (φ0 = 0), nous observons, pour des valeurs raisonnablesde la constante diélectrique de la membrane (1 ≤ εm ≤ 4) et pour certains rayons de pores(a . 1 nm) et potentiels chimiques (ou concentrations dans le réservoir ρb ≈ 0.2 mol/L),deux minima dans Ω1(κv) à κL

v et κVv < κL

v . Ces deux minima résultent de la compétitionentre :

i) la déplétion des ions induite par la répulsion diélectrique et qui favorise les grandsκv (car dans ce cas δv0 diminue dans le pore ce qui facilite la pénétration des ionsdans le pore), et

ii) le coût de tension de surface associé à la déformation des nuages ioniques dans lepore, qui globalement favorise les faibles κv (car lorsque κv augmente, le nombre denuages déformés augmente également).

Ainsi lorsque ρb diminue de 0.77 mol/L à 0.12 mol/L, le minimum du grand potentiel variede `BκLv ' 1.1 (〈ρ〉 = 0.2524 mol/L) à `BκVv = 0.037 (〈ρ〉 = 3 × 10−4 mol/L) : le poreévolue donc d’un état de pénétration ionique (phase «liquide», L) à un état d’exclusionionique quasi totale (phase «vapeur», V).

Il s’agit d’un transition discontinue du 1er ordre induite par la répulsion diélectriqueprès des surfaces. Dans la phase V, le pore est rempli d’une solution très pauvre en ionsalors que la concentration en ions libres dans le réservoir est élevée et correspond donc à

146

3.6. «Évaporation capillaire» des ions dans les nanopores cylindriques

! = 1m

0.90.80.7 1 1.21.1 1.3 1.40.60

0.2

0.4

0.6

0.8

1

a (nm)

" (mol/L)

b

! = 2m! = 3m! = 4m

L

V

Figure 3.3 – Diagramme de phase pour différentes valeurs de εm montrant les lignes decoexistence qui délimitent les phases liquides (L) et vapeur (V).

une phase L concentrée. On parle donc d’«évaporation capillaire» des ions dans le pore,par analogie avec l’évaporation capillaire de fluides de Lennard-Jones dans des pores hy-drophobes, observée dans des simulations de dynamique moléculaire [19], des simulationsMonte-Carlo dans l’ensemble grand-canonique [76] et des calculs numériques de théorie dela fonctionnelle de la densité [165]. Ce mécanisme est observable expérimentalement avecdu mercure et est utilisé pour mesurer la porosité de solides poreux [80].

Pour des pores de rayon supérieur a à une valeur critique a∗, le coefficient de partitionk est non-nul et croît continûment en fonction de ρb, ce qui correspond à la phase liquideL. Pour de rayons plus faibles, a < a∗, une transition apparaît pour une concentration decoexistence ρc

b : pour ρb < ρcb, k ≈ 0 et le pore est dans la phase V alors que pour ρb > ρc

b,les ions pénètrent le pore. Le diagramme de phase dans le plan (a, ρb) est représenté surla figure 3.3. Nous observons que la phase vapeur est stable pour les petites tailles de poreet les faibles concentrations ρb. De plus, lorsque la constante diélectrique de la membraneaugmente, cette phase disparaît au profit de la phase liquide. Pour εm > 4, elle n’est plusaccessible pour des tailles de pores expérimentales (a > 0.7 nm).

Nous avons vérifié que cette transition discontinue est due à la courbure des surfacesdiélectriques. En étudiant un pore situé entre deux cylindres concentriques [57] et en faisantvarier le rayon du cylindre inférieur tout en maintenant l’épaisseur du pore fixe, nous avonsdécrit continûment tous les types de pore allant d’un pore en forme de fente jusqu’à unpore cylindrique.

147


-1

-2

-3

-4

-5

00.4

0.3

0.2

0.1

00 0.2 0.4 0.6 0.8

coionscounterionsGCE

potential

(mol/L)!b

"0

k+

k#

Figure 3.4 – Variations du potentiel de Donnan effectif φ0 et des coefficients de partitiondes co-ions k− et contre-ions k+ en fonction de la concentration en ions dans le réservoirρb dans le cas d’un pore chargé (εm = 2, a = 0.84 nm et σs = −5.4× 10−4 e/nm2).

3.6.2 Nanopores faiblement chargés

Il est notoire qu’il est difficile d’obtenir des pores strictement neutres d’un point devue expérimental. Par exemple, lors de la synthèse de membranes nano-poreuses à l’aidede macromolécules organiques (comme le polyéthylène terephtalate [114, 150]), un cer-tain nombre de groupe carboxyliques se trouvent à la surface des pores. En fonction del’acidité du milieu aqueux, ceux-ci peuvent se dissocier chimiquement et conduire à unpore faiblement chargé dont la densité surfacique de charge est inconnue mais contrôléeexpérimentalement par le pH de la solution.

Nous nous sommes donc intéressés au cas de pores faiblement chargés, dans un pre-mier temps en fixant la densité surfacique de charge σs, puis en décrivant, à l’aide d’unmécanisme de régulation de charge, la dépendance de σs avec le pH dans le réservoir.

Dans l’approximation du potentiel effectif de Donnan, l’électroneutralité dans le poreest assurée par φ0. La figure 3.4 représente les variations de φ0 et des coefficients departition k± en fonction de ρb. Nous observons que la transition discontinue subsiste pourune valeur de coexistence ρc

b légèrement plus faible que dans le cas neutre. Pour ρb > ρcb le

pore est dans la phase L, φ0 ' 0 et la charge nette des ions compense la faible valeur deσs, k+ − k− = 2|σs|/(qρba). Le potentiel de Donnan, |φ0|, augmente de façon discontinueà la transition pour ρb = ρc

b puis fortement lorsque ρb diminue, car les co-ions sont exclus.Une faible quantité de contre-ions subsiste dans le pore malgré la répulsion diélectrique

148


-3 -2 -1 20.0

0.1

0.2

0.3

0.4

0.5

0.1

0.3

0.7

0.9

k±

0 1-4

aKp-Hp

!s/!0

Figure 3.5 – Coefficients de partition ioniques des contre-ions (courbe noire) et des co-ions(rouge) et densité surfacique de charge (bleue) en fonction de pH-pKa dans l’approche derégulation de charge (εm = 2, εw = 78, ρb = 1 mol/L, |σ0| = 10−2 nm−2 et a = 0.617 nm).

afin de maintenir l’électroneutralité. Il ne s’agit donc plus d’une phase d’exclusion ioniquemais de pénétration faible d’ions, correspondant à une bonne exclusion de co-ions par effetdiélectrique. Pour |σs| > 10−2 e/nm2, la transition disparaît, car l’énergie électrostatiquegénérée par le pore chargé, attractive pour les contre-ions, devient trop forte pour êtrecontrecarrée par la répulsion diélectrique.

Dans la plupart des expériences, la densité surfacique de charge des nanopores nepeut être mesurée. Cependant, pour des surfaces comportant des groupes carboxyliques,σs augmente avec le pH [89]. En effet, le relarguage d’ions oxonium (H3O+) des groupesacides conduit alors à une charge de surface qui peut varier. Nous présentons donc unmécanisme de régulation de charge [89, 124] qui revient à considérer l’équilibre chimiqueacido-basique suivant à la surface du pore

[surface]− COOH + H2O [surface]− COO− + H3O+ (3.59)

caractérisé par se constante d’équilibre Ka. En considérant l’équilibre chimique entre lesions oxonium près de la surface du pore et dans le réservoir [38], la densité surfacique decharge effective s’écrit 8

σs =σ0

1 + 10pKa−pH e−φ0(3.60)

8. Le mécanisme de régulation de charge est traité au niveau champ moyen effectif, c’est-à-dire queles forces diélectriques et non purement électrostatiques sont prises en compte de façon implicite dans laconstante d’équilibre Ka.

149


où pH = − log ρb (H3O+) (pKA = − logKa) et σ0 < 0 est la charge limite dans l’étatcomplètement dissocié. Nous considérons, comme dans les expériences de Lev et al. [114],des traces d’ions oxoniums (ions spectateurs), dont la concentration dans le réservoir estbien plus faible que celle des ions (2 < pH < 12). Sur la figure 3.5 sont représentés σs etles coefficients de partition ioniques k± en fonction de pH − pKa. Pour les faibles pH, laforme stable est la forme (acide) carboxylique et σs ' 0. L’état du pore est semblable àcelui du pore neutre, soit pour les paramètres considérés, l’état de pénétration faible desions. Lorsque le pH augmente à partir de la valeur critique pHC , pour lequel la transitiondiscontinue apparaît, le système évolue dans l’état de pénétration ionique. Une prédictionintéressante de ce modèle est l’augmentation brutale de σs à la transition. Ce comportementest relié au saut du potentiel électrostatique φ0 (voir figure 3.4) : les ions écrantent lepotentiel dans la phase liquide, ce qui conduit à un relarguage soudain des ions oxoniumsdu pore vers le réservoir et donc à un saut de la charge surfacique.

3.6.3 Traces expérimentales de cette transition discontinue

Existe-t-il une trace expérimentale de cette évaporation capillaire des ions dans lesnanopores cylindriques ?

Un certains nombres de travaux ont été menés sur les canaux biologiques dans lesmembranes, tel le canal gramicidine A [91, 90], qui permettent l’échange d’ions et d’eauentre le milieu intracellulaire et extracellulaire. Ils ont mis en évidence un mécanisme de«barrage» (ou «gating») qui contrôle le passage des ions, conduisant à une conductivitéqui fluctue entre deux états au cours du temps. À la suite de travaux numériques, plusieursmécanismes possibles ont été proposés, parmi lesquels une modification de la géométrie dupore contrôlée par un mécanisme actif, ou encore la présence d’un goulet d’étranglementtrès hydrophobe dans le canal, qui empêcherait le passage des molécules d’eau (voir parexemple le livre de Hille [90] ou les références [19, 18]).

Plus surprenant encore a été la découverte par Lev et al. que de telles fluctuations dela conductivité ont été observées à travers des nanopores artificiels (de rayon a ' 1 nm etlongueur L ' 5 µm) synthétisés dans des membranes de polyéthylène terephtalate [114,150]. La conductivité fluctue entre deux états bien distincts de haute conductivité (HC) etde faible (mais non-nulle) conductivité (LC) avec un temps caractéristique de l’ordre de laseconde (voir la figure 3.6).

Nous proposons que cette bistabilité est associée à un état du système proche de latransition discontinue forte/faible pénétration des ions décrite ci-dessus. La phase liquidecorrespondrait à l’état HC et la phase vapeur à l’état LC et les oscillations entre ces deuxétats HC et LC correspondrait à la fenêtre de σs pour laquelle les branches métastables dela transition du 1er ordre existent. Il a d’ailleurs été montré que ce régime bistable existe

150


on January 7, 2010rspb.royalsocietypublishing.orgDownloaded from

Figure 3.6 – (a) Fluctuations de courant due à une solution 0.1 mol/L de KCl (pH = 7.4)observées à travers des membranes de polyéthylène terephtalate et (b) histogramme dela conductivité pour différentes concentrations d’ions divalents CaCl2 (de haut en bas 0 ;3× 10−4 ; 3× 10−3 mol/L et sans CaCl2 mais à pH = 8). Tiré de Lev et al. [114].

pour une petite fenêtre de pH [114]. Enfin le quotient des temps de résidence dans les étatsHC et LC ont été mesurés expérimentalement [114]. Dans l’approximation à deux états,nous sommes en mesure de calculer le quotient des temps de résidence dans les états faibleet forte pénétration ionique,

τLτV

= exp[Ω(κVv )− Ω(κLv )] (3.61)

en fonction de |σs|. Les variations sont qualitativement similaire à celle observées expéri-mentalement [114].

De manière similaire au mécanisme de régulation de charge par le pH, l’augmentationde la concentration de cations divalents à l’état de traces va diminuer la charge de surfacecar ils s’adsorbent à la surface du pore et le nanopore peut alors se trouver dans l’étatbistable (voir figure 3.6). Une comparaison plus quantitative nécessite une déterminationprécise des paramètres inconnus tels que σ0 ou Ka, ainsi qu’une étude plus précise desécoulements dans le pore en tenant compte d’éventuels effets électrophorétiques. Notonsenfin qu’une forte corrélation entre la conductivité à travers le pore et la charge surfa-cique a été mesurée expérimentalement [13]. Cette corrélation s’explique également par le

151


mécanisme de régulation de charge comme le montre la figure 3.5.

3.6.4 Article

Suit l’article :S. Buyukdagli, M. Manghi et J. Palmeri, Ionic Capillary Evaporation in Weakly ChargedNanopores, Physical Review Letters 105 158103 (2010) (4 pages)

152

Ionic Capillary Evaporation in Weakly Charged Nanopores

Sahin Buyukdagli, Manoel Manghi, and John Palmeri

Laboratoire de Physique Theorique - IRSAMC, CNRS and Universite de Toulouse, UPS, F-31062 Toulouse, France(Received 9 April 2010; revised manuscript received 1 July 2010; published 6 October 2010)

Using a variational field theory, we show that an electrolyte confined to a neutral cylindrical nanopore

traversing a low dielectric membrane exhibits a first-order ionic liquid-vapor pseudo-phase-transition

from an ionic-penetration ‘‘liquid’’ phase to an ionic-exclusion ‘‘vapor’’ phase, controlled by nanopore-

modified ionic correlations and dielectric repulsion. For weakly charged nanopores, this pseudotransition

survives and may shed light on the mechanism behind the rapid switching of nanopore conductivity

observed in experiments.

DOI: 10.1103/PhysRevLett.105.158103 PACS numbers: 87.16.dp, 82.45.Gj

Electrolytes near charged surfaces are omnipresent insoft matter (charged colloidal suspensions and polyelec-trolyte mixtures), biology (DNA, proteins, and cell mem-branes) [1], and nanofiltration (ion channels) [2,3]. Forbulk ionic fluids, the existence of an ionic first-orderliquid-vapor (L-V) phase transition is now well established[4] with the driving mechanism being the competitionbetween the short range steric repulsion and the attractivecorrelations arising from the long range Coulomb interac-tion. However, for aqueous bulk electrolytes composed ofconventional inorganic salts, the critical temperature iswell below freezing (Tbulk

c 50 K), so that it can bereached experimentally only with special liquids [4].When electrolytes are in contact with low dielectric, andpossibly charged, mesoscopic bodies, interactions betweenmobile ions and the body surface come into play andstrongly modify ion-ion interactions [5,6]. More generally,the influence of confinement on L-V transitions is of broadfundamental and technological interest [7].

In this Letter, we show that when an electrolyte is con-fined to a neutral or weakly charged cylindrical nanopore,and in thermal and chemical equilibrium with an externalsalt solution reservoir, a novel type of ionic liquid-vaporpseudotransition occurs for conventional electrolytes atroom temperature, in contrast to the bulk, and within theexperimental salt concentration, pore size, and pore wallsurface charge density range. This pseudo-phase-transition(occurring in a quasi-one-dimensional infinite system)presents parallels with capillary evaporation of water inhydrophobic nanopores [7]. The driving mechanism is acompetition between the enhanced screening with ionicconcentration of the dielectric repulsion and the increaseof the surface tension associatedwith the deformation of theionic cloud, as sketched in Fig. 1 (steric interactions do notseem to play an important role, and therefore we adopt thepoint ion approximation). By using a field-theoretic varia-tional approach, ionic correlations and polarization chargeeffects can be taken into account nonperturbatively [8,9]. Inthe bulk, if 1=ai is introduced as a cutoff in momentumspace, this approach correctly predicts the existence of a

first-order ionic L-V transition [10]. In neutral nanopores,we also find a (pseudo)transition between a high concen-tration conducting ionic liquid phase and a very low con-centration insulating ionic-exclusion vapor phase; forweakly charged nanopores, however, the pseudotransitionis to a low conductivity counterion-only vapor phase, wherecoions are nearly entirely excluded from the pore and due toelectroneutrality the counterion concentration is fixed al-most entirely by the surface charge density. We finallypropose that the underlying mechanism controlling con-ductivity fluctuations as a function of pH and divalent ionconcentration in certain artificial and biological nanopores[11–13] is a manifestation of the pseudotransition proposedhere in a weakly charged nanopore.We consider an electrolyte at T ¼ 300 K of dielectric

permittivity w ¼ 78, confined in a cylindrical nanopore ofradius a, length L, and surface charge densitys; the spaceoutside the pore is salt-free with a dielectric permittivitym < w (Fig. 1). The electrolyte is in contact with anexternal ion bulk reservoir at the end boundaries of thepore, which fixes the fugacity of ions inside the pore accord-ing to chemical equilibrium:i ei=3 ¼ i;b, wherei

is the chemical potential of ion i ¼ 1; . . . ; (energies are inunits of the thermal energy kBT ¼ 1=) and is thede Broglie wavelength of an ion. Although ions interactthrough the bare Coulomb potentialvb

Cðr; r0Þ ¼ ‘B=jr r0jin a bulk electrolyte, where ‘B ¼ e2=ð4wÞ 0:7 nm,

FIG. 1 (color online). Sketch of a cylindrical nanopore (radiusa) filled with counterions and coions. The surface charge densityis s; v and b are screening parameters in the pore and in thebulk, respectively (cylindrical coordinates r; z).

PRL 105, 158103 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

8 OCTOBER 2010

0031-9007=10=105(15)=158103(4) 158103-1 2010 The American Physical Society

dielectric jumps at the nanopore boundaries yield amodified Coulomb potential vCðr; r0Þ obeyingrðrÞrvCðr;r0Þ¼e2ðrr0Þ. After performing aHubbard-Stratonovich transformation and introducing afluctuating field ðrÞ, the grand-canonical partition func-

tion for ions in the nanopore can be written as Q ¼RDeH½=QC where the Hamiltonian is [8,9]

H½ ¼Z

dr

ðrÞ2e2

½rðrÞ2 iðrÞðrÞ

X

i¼1

~ieiqiðrÞ

; (1)

where qi is the ion valency, QC ¼ 12 tr lnvC, ðrÞ ¼

sðr aÞ, and ~i ¼ ieq2i v

bCð0Þ=2. The electrostatic poten-

tial is given by ihðrÞi.Evaluating the partition function by using Eq. (1) is

intractable due to nonlinear terms, and we use theGaussian variational method, which consists in computingthe variational grand potential v ¼ 0 þ hHH0i0,where the expectation value is evaluated with a variationalGaussian Hamiltonian

H0½ ¼ 1

2

Z

r;r0½ðrÞ i0ðrÞv1

0 ðr;r0Þ½ðr0Þ i0ðr0Þ;(2)

and 0 ¼ 12 tr lnðv0=vCÞ. We then extremize v with

respect to the variational functions, namely, the Green’sfunction v0ðr; r0Þ and the electrostatic potential0ðrÞ. Thisyields two intractable coupled nonlinear differential equa-tions, similar in form to a generalized Poisson-Boltzmannequation for 0ðrÞ and a generalized Debye-Huckel (DH)one for v0ðr; r0Þ [8,9]. To make progress we consider therestricted case of a constant 0 in the pore [2] andtake v0ðr; r0Þ as the solution of ½rðrÞr þðrÞ2ðrÞv0ðr; r0Þ ¼ e2ðr r0Þ with a screening pa-rameter ðrÞ ¼ vða rÞ [2,6]. We are then left withtwo variational parameters: the effective Donnan potential0 and the DH parameter in the pore v [9].

The variational grand potential becomes v ¼pa2Lþ s

v2aL, where p ¼ Piie

q2i ‘Bv=2 3v=

ð24Þ is the pressure of a bulk electrolyte with screeningparameter v [14] and s

v a surface contribution given by

sv ¼ s0 a

2

X

i¼1

ieq2i ‘Bv=2heq2i v0ðr;rÞ=2qi0 1i

þ a2v

16‘B

Z 1

0dhv0ðr; r;v

ffiffiffi

p Þ v0ðr; r;vÞi;(3)

where h. . .i is the average over the nanopore and

v0ðr; r0Þ ¼ ‘Bevjr0rj=jr0 rj þ v0ðr; r0Þ with

v0ðr; r;vÞ ¼ 4‘B

Z 1

0dk

X

m0

0Fmðk;vÞI2mðßrÞ; (4)

where ß2 ¼ k2 þ 2v,

P0means that the term m ¼ 0 is

divided by 2,

Fmðk;vÞ ¼ wßKmðkaÞK0mðßaÞ mkKmðßaÞK0

mðkaÞmkImðßaÞK0

mðkaÞ wßKmðkaÞI0mðßaÞ ;

(5)

and Im and Km are modified Bessel functions [15,16]. Thefirst term in Eq. (3) is the electrostatic energy of the surfacecharge, the second term is a depletion term, and the lastterm is the cost of ionic cloud deformation.In the following, we consider low to moderate ion con-

centrations in the bulk reservoir such that the physicalminimum of v in the bulk (for a ! 1) is given by the

DH result [2,9] i;b ¼ i;beq2i b‘B=2, where 2

b ¼4‘B

Pi¼1 q

2i i;b. The ion concentration averaged over

the pore section is h ii ¼ i;bieqi0 with i

heq2i wðrÞ=2i, where the potential wðrÞ incorporates the sol-vation energy due to the ionic cloud and polarizationcharge-ion interactions

wðrÞ ¼ ðb vÞ‘B þ v0ðr; r;vÞ (6)

andiðrÞ ¼ q2i wðrÞ=2þ qi0 is the potential of the meanforce of ion i in the nanopore [9].For a symmetric electrolyte (q ¼ q, ¼ ), by ex-

tremizing v with respect to 0, one obtains the electro-neutrality condition s ¼ q ba sinhðq0Þ, which, in themean-field Poisson-Boltzmann limit [wðrÞ ¼ 0] leads tothe usual Donnan potential [2]. By injecting the solutionfor 0 in v, we are left with a single variational parame-ter vð b; a; m=wÞ. The averaged partition coefficients ofcounterions and coions are

k h i b

¼ eq0 : (7)

In a neutral nanopore, 0 ¼ 0 and k ¼ . For mono-valent ions (q ¼ 1) and low enough membrane permittiv-ities (m < 5), pore radii (a < 1 nm), and bulk ionconcentrations, the variational grand potential vðvÞ ex-hibits a minimum at V

va ’ 5 102. By increasing thereservoir concentration, this minimum jumps discontinu-ously to a finite value L

v with Vv < L

v < b, and the poreundergoes a first-order pseudo-phase-transition from anionic-exclusion state to an ionic-penetration one. As illus-trated in Figs. 2(a) and 2(b) for m ¼ 2, where k is plotted,respectively, vs b and a, for a < a ¼ 0:989 nm, a criti-cal pore radius, a jump occurs for a specific coexistencevalue of b larger than

b ¼ 0:074 mol=l. At the transi-

tion, there is phase coexistence vðVv Þ ¼ vðL

vÞ, whereLv=

Vv ’ 10. When increasing b beyond this coexistence

value, the ionic-exclusion stable state becomes metastableand the pore becomes penetrable to ions [Fig. 2(a)]. Fora ¼ a, the pseudotransition is continuous.Because of the strong ion depletion near the pore sur-

face, an instructive analogy with capillary evaporation ofwater in hydrophobic nanopores favoring the vapor phasecan be drawn [7]. For ionic fluids, in contrast to water, we


8 OCTOBER 2010

158103-2

find that the critical temperature increases with decreasingpore radius. Whereas the critical temperature Tc of the L-Vtransition in bulk electrolytes is extremely low (reducedtemperature ~Tbulk 2ai=‘B ’ 0:05 with ai ’ 0:1 nm theion radius), the phase diagram in Fig. 2(c) shows that, atroom temperature, the exclusion pseudo-phase-transitiontakes place for experimentally accessible parameter values(w ¼ 78, m ¼ 1 to 4, and reduced temperature ~Tpore 2a=‘B ’ 1): The coexistence lines separate the L (abovethe line) from the V state (below the line). Beyond thecritical point [aðmÞ,

bðmÞ], we enter the ‘‘fluid’’ phasewhere the pseudotransition disappears and is replaced by asmooth crossover (as in slit pores [2,9,17]). Using a sim-plified self-consistent approach, Dresner also found a first-order pseudo-phase-transition for electrolytes confined inneutral spherical pores [17], but Yaroshchuk may havewrongly argued that this is an artifact arising fromthe use of the self-consistent DH equation [2]. Whenincreasing m and thus decreasing the repulsive surface

polarization charge, the parameter range where the phaseseparation is observable is considerably reduced. Hence,for neutral pores the ion penetration pseudotransition isdriven by the competition between the last two terms ofEq. (3), which favor, respectively, high and low v.What happens for slightly charged nanopores? In

Fig. 3(a) are plotted the partition coefficients of counter-ions kþ and coions k and the variational Donnan potential0 vs b for m ¼ 2, a ¼ 0:84 nm, and a weak surfacecharge s ¼ 8:6 105 C=m2. One observes that thediscontinuous pseudo-phase-transition survives and phasecoexistence occurs at b ¼ 0:17 mol=l, a value slightlylower than for the neutral case (0:18 mol=l). At this coex-istence value, 0 also exhibits a jump, and for larger b,0 ’ 0 and we recover the neutral case. A slight dif-ference between kþ and k remains due to electroneutral-ity, rewritten by using Eq. (7) as kþ k ¼ 2jsj=ðq baÞ.For smaller b, k kþ while kþ rapidly increases. Wethus reach a low surface charge density counterion-only

FIG. 2 (color online). Partition coefficient k ¼ h i= b inside a neutral nanopore (q ¼ 1, m ¼ 2, w ¼ 78) vs (a) the bulkconcentration b for three pore radii with metastable branches (thin solid lines) and a window (vertical lines) for a ¼ 0:84 nmand (b) the pore radius a for, from left to right, b ¼ 0:7, 0.3, 0.156, and 0:08 mol=l. The black solid line shows stable states, blackdotted (gray or red solid) lines show metastable (unstable) branches, light gray (blue) lines (guide for the eye) are the ‘‘boiling point’’curve (bottom) and the ‘‘dew point’’ curve (top), and the dot is the critical point,

b ¼ 0:074 mol=l, a ¼ 0:989 nm. (c) Phase diagram

for various m (w ¼ 78). Critical lines correspond to phase boundaries between the ionic-penetration phase (L, above) and the ionic-exclusion one (V, below).

FIG. 3 (color online). (a) Partition coefficients of coions k and counterions kþ (left) and effective Donnan potential0 (right) vs b

for charged pores (m ¼ 2, a ¼ 0:84 nm, and fixed surface charge density s ¼ 0:9 104 C=m2). The dotted line is thecounterion-only approximation for counterions. (b) Phase diagram [similar to Fig. 2(c)] for several values of jsj. Inset: k vsjsj ( b ¼ 1 mol=l, a ¼ 0:617 nm) showing the window (horizontal arrow) with stable (solid lines) and metastable branches (dashedlines); the predicted HC to LC conductivity ratio / P

kLi =P

kVi ’ 5 is close to that seen in experiments [11] (a value difficult to

explain via water capillary evaporation [7]). (c) Logarithm of the ratio of resident times in L and V states of the inset of (b), L=V ¼exp½ðV

v Þ ðLvÞ vs jsj (pore length L ¼ 5 m [11]).


8 OCTOBER 2010

158103-3

regime, with kþ ’ 2jsj=q ba [Fig. 3(a), dotted line] andk ¼ 2=kþ [9]. In this regime, h þi ’ 2jsj=ðqaÞ, isindependent of b, and determined solely by global elec-troneutrality, and coions are excluded mainly by dielectricrepulsion and not charge. Hence the vapor phase is nolonger an ionic-exclusion phase but a weak ionic-penetration one, governed by the weak surface charge den-sity. For larger surface charge densities jsj> 103 C=m2,the transition disappears. The phase diagram for severalvalues ofs is illustrated in Fig. 3(b). Increasing the surfacecharge favors the L phase by reducing the coexistence lineand shifting the critical point towards smaller a and larger b (because s increases s

v, but due to screening, thisenhancement of s

v decreases with v). Comparison ofFigs. 2(c) and 3(b) clearly shows that an increase in s

plays qualitatively the same role as an increase in m, i.e., adecrease in the repulsive polarization charge density.

Can the trace of this first-order pseudotransition be ob-served experimentally in finite-sized open-ended pores? Bystudying ionic conductivity in nanopores (a ’ 1 nm, L ’5 m) produced in polyethylene terephthalate membranes[11], Lev et al. observed three regimes: a high conductivity(HC) regime, a low conductivity (LC) one, and a two-stateHC-LC regime where the conductivity switches rapidlybetween both states. Interestingly, the dynamic character-ization of the HC-LC regime, performed by measuring theratio of resident times, leads to the identification of threeparts: HC more stable, HC and LC at coexistence, and LCmore stable. We argue that one possible mechanism forthese current fluctuations on the scale of seconds is thatthe system is close to the phase coexistence presented here,where the HC and LC regimes are identified, respectively,with the ionic-penetration ‘‘liquid’’ and ionic-weak pene-tration ‘‘vapor’’ phases, and the HC-LC regime with thewindow of s for which metastable branches exist [inset inFig. 3(b)]. For finite open-ended nanopores, we expect theL-V pseudotransition to be rounded, with phase separationand hysteresis replaced by two-state fluctuations betweenpseudostable or metastable phases [7]. For surfaces carry-ing carboxylic acid groups, jsj is an increasing function ofpH [18] (for more details, see [16]), and Lev et al. [11]showed that the HC-LC regime exists only within a narrowpHwindow. At high pH, jsj is high and only the HC stateexists; at low pH, jsj is low and only the LC state exists.Within this ‘‘fluctuation’’ pH window, by using the two-state approximation [7], our model yields trends for ratio ofresident times vs jsj [Fig. 3(c)] in qualitative agreementwith the experimental ones vs pH (Fig. 4 of Ref. [11]).Similarly, increasing the concentration of trace divalentcations decreases the bare negative surface charge [19],and the nanopore can likewise switch from the HC to theLC state [11]. Observations revealing a strong correlationbetween s and conductivity fluctuations [12] can also beexplained via the charge-regulation mechanism [18], be-cause jsj depends in a self-consistentway on local solutioncharacteristics [16]. Although other mechanisms have beenproposed to explain nanopore conductivity fluctuations

(gas bubbles [13] or salt occlusions [20]), we believe thatthe one proposed here provides a natural explanation for ahost of experimental trends [11,12] and deserves furtherdetailed investigation [16]. One open question concerns therole of ion pairing; such effects appears naturally in asecond-order variational approach and are currently understudy. Using the Fisher and Levin [4] bulk approach as aguideline, we expect higher order corrections to lead toquantitative, but not qualitative, changes to our results.To corroborate our predictions, it would also be interestingto perform simulations, although it may be extremely diffi-cult to properly include dielectric discontinuities and reachsufficiently long time scales ( 1 s).We thank B. Coasne for helpful discussions. This work

was supported in part by the French ANR (ProjectSIMONANOMEM No. ANR-07-NANO-055).

[1] C. Holm, P. Kekicheff, and R. Podgornik, ElectrostaticEffects in Soft Matter and Biophysics (Kluwer Academic,Dordrecht, 2001); H. Boroudjerdi et al., Phys. Rep. 416,129 (2005).

[2] A.E.Yaroshchuk,Adv.Colloid InterfaceSci.85, 193 (2000).[3] J. Zhang, A. Kamenev, and B. I. Shklovskii, Phys. Rev.

Lett. 95, 148101 (2005).[4] G. Stell et al., Phys. Rev. Lett. 37, 1369 (1976); R. R.

Singh and K. S. Pitzer, J. Chem. Phys. 92, 6775 (1990);M. E. Fisher and Y. Levin, Phys. Rev. Lett. 71, 3826(1993); A.-P. Hynninen and A. Z. Panagiotopoulos, Mol.Phys. 106, 2039 (2008).

[5] G. Wagner, Phys. Z. 25, 474 (1924); A. L. Loeb, J. ColloidSci. 6, 75 (1951).

[6] L. Onsager and N. Samaras, J. Chem. Phys. 2, 528 (1934).[7] O. Beckstein and M. S. P. Sansom, Proc. Natl. Acad. Sci.

U.S.A. 100, 7063 (2003); R. Roth and K.M. Kroll, J. Phys.Condens. Matter 18, 6517 (2006); L. D. Gelb et al., Rep.Prog. Phys. 62, 1573 (1999).

[8] R. R. Netz and H. Orland, Eur. Phys. J. E 11, 301 (2003).[9] S. Buyukdagli, M. Manghi, and J. Palmeri, Phys. Rev. E

81, 041601 (2010).[10] G. Ding and B. Xu, Chin. Phys. Lett. 13, 500 (1996); A.

Diehl, M. C. Barbosa, and Y. Levin, Phys. Rev. E 56, 619(1997).

[11] A. A. Lev et al., Proc. R. Soc. B 252, 187 (1993); C. A.Pasternak et al., Colloids Surf. A 77, 119 (1993).

[12] C. L. Bashford, G.M. Alder, and C.A. Pasternak, Biophys.J. 82, 2032 (2002).

[13] R.M.M. Smeets et al., Phys. Rev. Lett. 97, 088101(2006); R. Roth et al., Biophys. J. 94, 4282 (2008).

[14] D. A. McQuarrie, Statistical Mechanics (UniversityScience, New York, 2000), Chap. 5.

[15] B. Jancovici and X. Artru, Mol. Phys. 49, 487 (1983).[16] S. Buyukdagli, M. Manghi, and J. Palmeri (unpublished).[17] L. Dresner, Desalination 15, 39 (1974).[18] H. J.M. Hijnen and J. A.M. Smit, Biophys. Chem. 41, 101

(1991).[19] T.W. Healy and L. R. White, Adv. Colloid Interface Sci. 9,

303 (1978).[20] M. R. Powell et al., Nature Nanotech. 3, 51 (2008).


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158103-4

3.7. Perspectives

3.7 Perspectives

Dans cette dernière section, on comprend pourquoi l’approche variationnelle est néces-saire pour décrire la transition de phase exclusion (ou faible pénétration si σs 6= 0) ionique/forte pénétration ionique. C’est une méthode simple et non-perturbative qui permet d’in-terpoler entre les limites de faible et fort couplage et permet d’isoler de façon systématiqueles états stables et métastables. Alors que la phase de forte pénétration ionique est prochede la solution faible couplage (équivalent au choix κv = κb) et la phase d’exclusion ioniqueest proche de la solution fort couplage (κv = 0), la méthode variationnelle permet d’in-terpoler entre ces deux limites et de déterminer la coexistence L-V en cherchant l’égalitédes potentiels, Ω(κV

v ) = Ω(κLv ). De plus, notre approche montre clairement, comme l’ont

également souligné Hatlo et al. [86], qu’à cause de l’exclusion diélectrique près des inter-faces, des solutions de type fort couplage existent même pour des nanopores non-chargés(un résultat non-perturbatif clairement inaccessible avec les développements standards enfort couplage [101]).

L’approche variationnelle présentée ci-dessus, et plus généralement le formalisme dethéorie des champs pour les électrolytes près d’interfaces, ouvrent de nombreuses perspec-tives. Une partie de celles présentées ci-dessous sont, ou vont être étudiées en collaborationavec John Palmeri, Sahin Buyukdagli et David Dean du laboratoire de Physique Théorique.Quelques résultats préliminaires sont évoqués.

Approche variationnelle développée au second ordre : corrélations et for-mation de paires d’ions

L’évaporation capillaire des ions dans les nanopores cylindriques a son pendant aucœur de l’électrolyte en solution, tout comme l’évaporation (ou la condensation) capil-laire des fluides simples est une transition de type liquide-vapeur qui existe loin des in-terfaces mais dont la température critique est déplacée [165, 68]. Ainsi, l’existence d’unetransition liquide-vapeur ionique dans les électrolytes en solution a été montrée théorique-ment [180, 72, 95]. Cette transition, et en particulier la courbe de coexistence, a été mesuréeexpérimentalement pour la première fois en 1990 pour un électrolyte exotique [177] 9. Eneffet, la température critique pour des électrolytes aqueux simples, Tbulk, est très faible, del’ordre de 50 K et donc pas mesurable.

Cette transition est la conséquence de la compétition entre les corrélations attractivesde type Debye-Hückel et la répulsion stérique à courte distance associée à la taille de l’ionai. La phase fluide pour T > Tbulk est constituée d’un mélange homogène d’ions libres,

9. Il s’agit du sel de triéthyle-n–hexylammonium triéthyle-n-hexylboride (N2226B2226).

157


tandis que dans la phase vapeur, pour T < Tbulk, l’attraction électrostatique domine devantl’entropie des ions. Ceux-ci forment alors des paires d’ions globalement neutres et seule unfaible fraction des ions restent libres. La température critique réduite s’écrit Tbulk = 2ai/`Bqui est très faible, de l’ordre de 0.05 pour des électrolyte simples (ai = 0.1 nm). Dans lepore, en revanche, la longueur importante est le rayon du pore critique a∗ ≈ 1 nm, cequi conduit à une température réduite bien plus grande Tpore = 2a∗/`B ' 1 et rend latransition observable à température ambiante.

La question sous-jacente est de savoir si la formation de paires d’ions joue un rôlecentral dans le phénomène d’évaporation capillaire ionique. En effet, le potentiel v0(r, r′)dans le nanopore est important (par exemple au centre du pore, v0(z − z′) est de l’ordrede 2kBT dans la phase L et 9kBT dans la phase V pour |z − z′| = 3 Å) et peut conduire àune quantité non négligable de paires d’ions à la transition. L’étude de cette question peutêtre envisagée en suivant deux pistes :

i) En poussant l’approche variationnelle au second ordre [voir l’éq. (3.30) pour n = 2].Dans ce cas le grand potentiel se généralise, dans le cas d’un électrolyte symétriquedans un pore neutre, selon :

Ω2 = −12

tr ln v0 −∫

r

κ2v

8π`Bv0(r, r) + 2λ e−

q2

2∆v0(r,r)

+λq2κ2

v

4π`B

∫

r,r′v2

0(r, r′) e−q2

2∆v0(r,r)

− κ4v

(8π`B)2

∫

r,r′2v2

0(r, r′) + ∆v20(r, r′)−∆v0(r, r)∆v0(r′, r′)

−2λ2

∫

r,r′e−

q2

2[∆v0(r,r)+∆v0(r′,r′)]

[evHC(r,r′) cosh(q2v0(r, r′))− 1

](3.62)

où ∆v0(r, r′) = v0(r, r′) − vc(0) et vHC(r, r′) est le potentiel de cœur dur. Les troisderniers termes sont les termes de second ordre qui prennent en compte les corré-lations entre ions au-delà de l’approche de Debye-Hückel. En particulier, le dernierterme de l’éq. (3.62) fait intervenir les effets de volume exclu et est nécessaire pourcalculer la concentration des paires d’ions. Les calculs sont lourds mais des approxi-mations pour v0(r, r′) peuvent être envisagées afin de quantifier le nombre de pairesd’ions au sein du pore.

ii) En étudiant les paires d’ions selon l’approche développée par Bjerrum [21], Ebe-ling [69] et Fisher et Levin [72] d’un point de vue «équilibre chimique». Les pairesd’ions sont vues comme des particules idéales et neutres et la quantité de paire d’ionsest déterminée par la constante d’équilibre de la dissociation des paires en anion pluscation. Cette approche a été utilisée dans le cas d’électrolytes en solution [72] eta conduit à des résultats quantitativement corrects. Dans le cas d’un nanopore, on

158

3.7. Perspectives

peut montrer que la concentration moyenne en paires dans le pore est alors donnéepar :

ρpore2 = ρbulk

2

ζ0

ζDHou ζ0 =

2Vpore

∫

r,r′

[cosh(q2v0(r, r′))− 1− v2

0(r, r′)2

](3.63)

et ζDH est égal à ζ0 en remplaçant v0 par le potentiel de Debye-Hückel vDH [éq. (3.42)].Notons que l’intégration se fait pour |r − r′| > ai. Ce dernier terme correspondsimplement au dernier terme de l’éq. (3.62) dans l’approximation homogène où ρ(r)est constant dans le pore.

Cette étude permettra également de déterminer plus précisément les limites du calculvariationnel au premier ordre.

Au delà de l’approximation de Onsager-Samaras généralisée

Dans les travaux présentés précédemment, le potentiel v0 déterminé par les équationsvariationnelles (3.33)-(3.34) est choisi de type Debye-Hückel avec une constante d’écran-tage κv fixée variationnellement. Dans le cas simple d’interfaces planes non-chargées, Lueet al. ont fait des choix un peu moins restreint en proposant un κv nul près de l’interfacesur une épaisseur h et non nul pour des distances plus grandes que h (h et κv sont dé-terminés variationnellement) [85, 86]. Cependant ces calculs sont extrêmement lourds etn’apportent pas des résultats très différents. Nous envisageons de relaxer ces contraintesdans des situations particulières.

Ainsi pour des contre-ions seuls en regard d’une interface chargée, les éqs. (3.37)-(3.38)se réécrivent comme

∆φ0(r) + 2ρ(r) = −2ρs(r) (3.64)

∆v0(r, r′)− 2ρ(r)v0(r, r′) = −4πδ(r− r′) (3.65)

où ρ(r) = Λ4 e−Ξ

2fW (r)−φ0(r). Plusieurs méthodes approchées peuvent être envisagées pour

résoudre ces équations :i) Utiliser l’approximation de Wagner [191] qui consiste à écrire une fonction de Green

de type Debye-Hückel, donnée par exemple par l’éq. (3.49) dans le cas d’une seuleinterface, puis ensuite remplacer la constante κv par la valeur locale κeff(z) =

√2ρ(z).

Ainsi dans le cas où il n’y a pas de saut diélectrique, ∆ = 0, on trouve une formegénérale :

wWag(z) = κeff(z)f(κeff(z) z)− (κeff(z)− κb) (3.66)

où

f(x) =e−2x(1 + x)2

2x3− K2(2x)

x(3.67)

159


et K2 est une fonction de Bessel modifiée. Le couplage de cette équation avecl’éq. (3.37) devient plus simple car il n’y a plus qu’une seule équation aux dérivéespartielles.

ii) Résoudre l’éq. (3.34) dans l’approximation de Wentzel–Kramers–Brillouin (WKB).Les calculs au second ordre dans le développement WKB sont envisageables. Unepremière étape consisterait à comparer les résultats dans les cas où la solution exacteest possible, par exemple dans la limite de fort couplage.

iii) La troisième voie est la résolution du système d’équations in silico par la méthodenumérique des éléments finis.

Dipôles et ions polarisables aux interfaces

Une perspective à plus long terme est l’étude des interactions entre une interface di-électrique et des molécules polarisables en solution. La concentration locale des chargesportées par les ions s’écrit alors très généralement

ρc(r) =∑

i

Ni∑

j=1

[qiδ(r− rj)− pj · ∇δ(r− rj)] (3.68)

où les pj sont les dipôles fluctuants. Le hamiltonien de la théorie de champ, éq. (3.8), setransforme alors selon

H[φ] =∫

dr

[ε(r)2βe2

[∇φ(r)]2 − iρs(r)φ(r)−∑

i

λieiqiφ(r) e−

αi2

(∇φ(r))2

](3.69)

où αi est la polarisabilité des ions de type i et λi est la fugacité renormalisée.Une question centrale dans cette approche est celle du signe de la polarisabilité α des

ions dans l’eau. Cette question a été soulevée par Onsager dans les années 1930 [141]lorsqu’il développa une approche de type Debye-Hückel pour les dipôles constituant lesolvant. Localement, près d’un dipôle les autres dipôles sont exclus à cause des interactionsrépulsives à courte portée, ce qui conduit à la création d’une cavité sphérique diélectrique deconstante εi εw. Le dipôle fluctuant de polarisabilité αi est lui placé au centre de la cavité.La polarisabilité effective αeff de l’ion dans l’eau est alors la somme de la polarisabilité αiet de celle de la cavité (déterminée par la formule de Clausius-Mossotti [97]), et devientdonc négative.

Des résultats préliminaires montrent qu’il faut tenir compte de trois types d’interactionsinduites par le saut diélectrique qui dépendent du signe de αeff . Dans le cas d’un ionpolarisable dans l’eau près d’une surface solide de constante diélectrique faible (εm εw),on trouve : i) l’interaction usuelle charge/charge image, qui est répulsive ; ii) l’interactiondipôle/dipôle image qui est attractive ; et enfin iii) les interactions charge/dipôle image

160

3.7. Perspectives

et charge image/dipôle qui sont répulsives. Des modifications sensibles du PMF et doncde la concentration des ions apparaissent alors très près de la surface, pour des distancescomprises entre 2 Å (la distance minimale d’approche des ions) et 1 nm.

Le cas de l’interface eau-air est également intéressant car de nombreuses simulationsde dynamique moléculaire ont été effectuées pour ces systèmes. Citons en particulier lestravaux pionniers de Dang et Chang [51] et de Jungwirth et Tobias [99] pour les anionsmonovalents de la famille des halogènes (ions fluorure F−, chlorure Cl−, bromure Br− etiodure I−).

161


162

Chapitre 4

Un phénomèned’élasto-hydrodynamique :propulsion par un flagelle élastique

Dans cette partie, nous nous intéressons à un phénomène qui met en jeu l’hydrodyna-mique d’objets semi-flexibles loin de l’équilibre. Dans le domaine de la matière molle, lesinteractions hydrodynamiques ont été bien étudiées pour comprendre la dynamique descolloïdes [83, 120, 94, 168] et des polymères [62] proche de l’équilibre. Depuis quelquesannées, l’étude de la propulsion aux petits nombres de Reynolds fait l’objet de nombreuxtravaux (voir par exemple les revues récentes de Powers [159] et Lauga et Powers [110]).C’est dans ce contexte que nous avons étudié l’hydrodynamique d’un flagelle en rotation. Ilreprésente un des systèmes biomimétiques les plus simples permettant la propulsion et sertde modèle pour comprendre la propulsion de bactéries telles que l’E. coli qui est activéepar un moteur trans-membranaire rotatif.

4.1 Bifurcation élastique et propulsion à petit nombre deReynolds

A l’aide de simulations numériques de dynamique brownienne incluant les interactionshydrodynamiques, nous nous sommes intéressés au couplage entre les interactions hydrody-namiques (à cette échelle la dissipation visqueuse domine les effets inertiels), l’élasticité et lacourbure du flagelle. Le flagelle, de longueur L, est modélisé comme une tige semi-flexibledroite (de longueur de persistance `p L), constituée de sphères reliées élastiquemententre elles, et faisant un angle θ avec l’axe de rotation. Les interactions hydrodynamiquesentre les différentes sphères sont prises en compte via le tenseur de Rotne-Prager.

163

Chapitre 4. Un phénomène d’élasto-hydrodynamique :propulsion par un flagelle élastique

(a)

side

AA

A

BB

B top(b)

Vz!lp/L("10 )3

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4!NL/lp

Figure 4.1 – (a) Vues projetées du flagelle avant (état rigide A) et après la bifurcation(«hélice» B). (b) Vitesse de propulsion du flagelle dans la direction de son axe de rotation enfonction du couple appliqué pour 3 longueurs de persistance différentes : `p/L = 2500(×),5000 () et 20000 (), L étant la longueur du flagelle. On observe un saut dans la vitessede propulsion associé à la bifurcation.

En appliquant un couple constant à la base du flagelle initialement droit, celui-ci tourneet se déforme sous l’action du frottement hydrodynamique [voir figure 4.1(a)]. Lorsque l’onaugmente le couple, une bifurcation discontinue, d’un état où le flagelle est quasi-droitvers un état où il prend la forme d’une hélice, est observée. En effet, afin de minimiserla dissipation sous forme de friction hydrodynamique, l’enroulement hélicoïdal autour del’axe de rotation diminue la distance du flagelle à l’axe de rotation et donc permet dediminuer la vitesse translationnelle, et donc la friction, tout en conservant la même vitesseangulaire ω.

À cause des interactions hydrodynamiques, la géométrie hélicoïdale induit une propul-sion significative [129], un peu à la manière d’un tire-bouchon qui se translate lorsqu’il estvissé dans le liège. La figure 4.1(b) montre ainsi l’augmentation brutale de cette vitessede propulsion à la transition. De plus le sens de propulsion est indépendant du sens derotation, car la chiralité de l’hélice est elle-même induite par le sens de rotation.

L’étude de l’hydrodynamique des flagelles a été étendue vers l’étude du propulseur com-plet, tête plus flagelle propulsif. En effet, il a été observé sur les bactéries utilisant ce modede propulsion que la tête était elle aussi en rotation mais dans le sens inverse du flagelleafin de conserver le moment cinétique total. On peut donc s’attendre à ce que la rotation

164

4.2. Confirmation expérimentale

de la tête ait deux conséquences : i) elle contribue à un frottement visqueux important etii) elle perturbe l’écoulement et donc les interactions hydrodynamiques responsables de lapropulsion. Nous avons donc étudié numériquement l’influence de ces phénomènes sur lapropulsion de l’ensemble du propulseur. Nous avons montré que la vitesse de propulsionvarie linéairement avec le ratio taille du flagelle sur taille caractéristique de la tête. De plusil s’avère que seule une petite fraction de puissance est nécessaire pour maintenir la têteen rotation, l’essentiel étant utilisé pour la propulsion [128].

Ce mécanisme de bifurcation pourrait jouer un rôle dans les transformations poly-morphes des bactéries qui sont observées et qui apparaissent lorsque les moteurs changentde sens de rotation. Il pourrait également expliquer l’évolution de la géométrie du flagellevers une géométrie hélicoïdale comme le suggère Buchanan [33].

4.2 Confirmation expérimentale

Ces travaux théoriques ont suscité deux expériences à l’échelle centimétrique et enutilisant les propriétés de similitude des équations de l’hydrodynamique. Qian et al. [160]ont utilisé dans leur expérience un flagelle de rayon a = 1.25 mm et de longueur L = 20 à30 cm, mis en rotation dans l’huile de silicone (nombre de Reynolds Re ≈ 0.1) tandis quedans l’expérience de Coq et al. [48], les paramètres du flagelle sont a = 0.4 mm, L = 2 à10 cm et le liquide visqueux est la glycérine (Re ≈ 0.01). Comme le montre la figure 4.2,dans les deux cas, la bifurcation élastique ainsi que l’hystérèse a été observée et les photosexpérimentales du flagelle correspondent de façon surprenante aux résultats numériques.La relation entre le couple critique et le couple élastique `p/L

Nc '`pL

sin θ (4.1)

où θ est l’angle de déflexion du flagelle a été également validée expérimentalement (oùθ = 26 [160] et 15 [48]).

La mesure de la vitesse de propulsion n’est pas possible dans ces expériences. En re-vanche, la mesure de la poussée axiale a été faite par Coq et al. [48] en fonction de la vitesseangulaire appliquée ou du couple appliqué. L’hystérèse est observable dans le second cas,comme le montre la figure 4.2.

Comme le suggère Coq et al. [48], dans un perspective de fabrication de nano-nageursartificiels, choisir un point de fonctionnement du moteur près de la transition de formepermettrait des «accélérations» pour une faible variation du couple appliqué.

165

Chapitre 4. Un phénomène d’élasto-hydrodynamique :propulsion par un flagelle élastique

[Fig. 1(b)]. To illustrate the physics, we first present asimple analysis of this shape transition using the lumpedparameter model shown in Fig. 2. The rod is modeled bytwo rigid links of unit length connected by a torsionalspring. The link OP is fixed at angle ! between the rotationaxis z and the base of the rod in our experiment. Since theReynolds number is small, we take Re ! 0. Thus, we maywork in the rod’s rotating rest frame without introducingfictitious forces. The flow in this frame at point r is !z" r.The torsional spring represents the bending resistance andis only sensitive to changes in the angle between thevectors OP and PQ. Assuming ! # 1 and K is sufficientlylarge, the moment about P on PQ from the spring is Mb $K%OP" PQ& $ K%y; 2!' x; 0&, where K is the torsionalspring constant, and %x; y; 2& $ rQ is the position of thepoint Q to leading order in !. To find the steady-stateposition of Q, equate the moment on PQ due to the tor-sional spring to the moment on PQ due to the flow.Assuming all drag on PQ is concentrated at Q (Fig. 2),the viscous moment about P is Mv $ '"!%x; y; 0&, where" is a resistance coefficient. Solving moment balance for xand y yields x ! 2!=(1) %"!=K&2* and y ! x"!=K. As

! increases from zero, the link PQ deflects and y in-creases, which causes Q to experience a viscous force inthe negative x direction. These forces push Q toward therotation axis, and tend to cause the rod in our experiment towrap around the z axis. As ! increases further, Q movescloser to the rotation axis, and y begins to decrease. Thereis also some drag on the link OP, concentrated a distance dfrom O. The moment about O due to flow is

MO ! "!!2!d2 ) 4

1) "2!2=K2

": (1)

For d2 < 1=2, we find that the moment first increases with!, then decreases as the link folds in toward the rotationaxis where the flow is slow. The moment then increasesagain as the drag from the base link OP dominates. If MOis plotted vs !, then we find an S-shaped curve, just as inour experiment (Fig. 3), with discontinuous transitions inshape and speed as moment varies.

We now turn to a more complete quantitative analysis.We will continue to prescribe ! rather than motor torqueMm, and we limit the analysis to steady-state shapes.Unlike Manghi et al. [10,11], we disregard hydrodynamicinteractions between distant parts of the rod and use resis-tive force theory to model the force per unit length f actingon the rod [1,12]:

f ! "?%v' rsrs + v& ) "krsrs + v; (2)

where "? ! 4#$=(log%L=a& ) 1=2* and "k !2#$=(log%L=a& ' 1=2* $ "?=2, r%s; t& is the position ofthe point on the rod center line with arclength s at time t,v%s; t& is the velocity of the undisturbed flow relative to thevelocity of the rod at s, and rs ! @r=@s is the tangentvector to the rod center line. There is also a hydrodynamic

! P

Q

x

z

yOd

FIG. 2. Lumped parameter model consisting of two rigid linksconnected by a torsional spring (open circle). The top link isclamped. All drag is concentrated at the two filled circles.

100

100

101

102

103

" =

(#$

L4 )/A

(MmL)/A10

-0.810

-0.5

FIG. 3 (color online). Dimensionless motor torque MmL=Awas measured as a function of dimensionless speed % for L !250 mm (,) and L ! 290 mm (4) with angle ! ! 26-. ForL ! 250 mm, speed was measured as a function of increasingtorque (!) and decreasing torque ("). Note the hysteresis. Thelinear (dashed line) and nonlinear (solid line) predictions areshown. The insets show examples of the steady-state filamentshapes in the low (left) and high (right) speed regimes.

!

FIG. 1. Orthogonal images of steady-state shapes of rotatingrod with torque just below (a) and just above (b) the criticaltorque. The motor (not shown) is at the top, with rotation axisalong z. Gravity points down. In (a) and (b), the left panel is theside view, and the right panel is the front view. The rod is markedwith white dots for contrast. The axes in (b) are the same as in(a). The curved arrow in (b) denotes the sense of rotation of therod.

PRL 100, 078101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending22 FEBRUARY 2008

078101-2

correlation algorithm to detect the coordinates of the twocorresponding projected profiles, which allows for the deter-mination of the full three dimensional !3D" shape of thedistorted rods with a submillimeter accuracy !see Fig. 1".After a transient regime, the rotating filament reaches a sta-tionary shape and undergoes a rigid body rotation. The short-est rods are hardly deformed by the viscous flow even whenthe rotation speed is increased by three orders of magnitude.They rapidly adopt a slightly chiral shape close to the initialstraight and tilted conformation. The rods with intermediatelengths display a continuous but sharp transition from analmost straight to an helical shape when increasing the an-gular velocity !see Fig. 2". The longest rods are significantlybent by the viscous drag; after a long transient regime!#1 h", they are tightly wrapped around the rotation axiseven at the slowest rotation speed. In all that follows, werestrain our attention only to the final stationary shapes.

To go beyond the above qualitative observations, the di-mensionless distance to the rotation axis, d!L" /L and thepolar angle !!L" of the rod end are plotted in Fig. 2 as afunction of the nondimensional rotation speed Sp$"#!L4 /$. Sp is commonly referred to as the sperm num-ber; it compares the period of angular rotation to the elasto-viscous relaxation time %=#!L4 /$, of the bending mode ofwavelength L,9,15 where $=&Ea4 /4 is the bending modulusof the filament and #! is the drag coefficient for normalmotion. First of all, it is worth noticing that all of the experi-mental data collapse on the same master curve, which im-plies that the deformation of the rods results from the com-petition between viscous and elastic forces. This is consistent

with the highest Reynolds numbers that we have measured:Re#10!2. Below this value, inertial effects are negligible.

At low Sp, the polar angle !!L" increases linearly with", whereas d!L" /L remains constant over more than twodecades. Above Sp%10, the variation of the polar angle be-comes much weaker. Conversely, the distance to the rotationaxis drops down to a very small value. Surprisingly, a quan-titative description of this wrapping transition can be per-formed, ignoring both geometrical nonlinearities and long-range hydrodynamic coupling. To determine the filamentshape, we compute the elastic and the viscous forces actingon the flexible rod. Using a local drag description, the vis-cous force is

fv = !#& ! #!"!t · v"t + #!v ,

with t as the tangent vector and #!=4&# /'log!L /a"+ 12( and

#& =2&# /'log!L /a"! 12( as the drag coefficients in the slen-

der body approximation.16 The elastic force fe is derivedfrom the bending energy functional that is written within thesmall deformation approximation E= ! 1

2")$!!s

2r"2ds, with sas the curvilinear coordinate. r!s"= 'r1!s" ,r2!s"( is the dis-placement field normal to the undeformed filament !see Fig.3". Ignoring the incompressibility constraint, which wouldonly add extra nonlinear contributions to the linearized elas-tic force, fe=!$!s

4r, the filament shape can then be exactlycomputed by solving the force balance equation fe=!fv inthe frame rotating at " around the z axis. By introducing thepenetration length of the bending modes l!""$'$ / !#!" cos '"(1/4, this equation can be written in thecompact form

l4!""!s4r1 = ! r2 ! s tan ' , !1"

l4!""!s4r2 = r1, !2"

with the torque and force free conditions at s=L: !s2r!L"

=!s3r!L"=0 and the geometrical constraints on the rotation

axis: r!0"=!sr!0"=0. The excellent agreement between thetheoretical and the measured geometrical parameters plottedin Fig. 2 demonstrates that this simplified approach correctlycaptures the main features of the filament dynamics. Al-though this linear equation can be analytically solved, theform of the exact solution is so complex that it is not reallyinsightful. We rather detail here the two asymptotic regimes

!(L)(rad)

Sperm Number

d(L)/L

a b c d ge f

a b c d ge f

FIG. 2. !Color online" Polar angle in the !x ,y" plane !circles" and distance tothe axis !diamonds" of the free end of the filament as a function of thedimensionless angular velocity Sp. Solid lines are solutions of the linearizeddeformation equation. Dashed lines are analytical solution in the low andhigh Sp limits !see main text". On top are the corresponding shapes of therotating filament !superimposed pictures at different times".

yr

-z

r1

yx

x

-z -z

r1

r2r2

FIG. 3. !Color online" Sketch of the filament deformations in the !x ,z" and!y ,z" planes illustrating the definition of the displacement field r!s"= 'r1!s" ,r2!s"(. !Solid lines" Filament shape in 3D. !Dashed lines" Unde-formed filament. !Dotted lines" Projections of the filament on the !x ,z" and!y ,z" planes.

051703-2 Coq et al. Phys. Fluids 20, 051703 "2008#

Downloaded 15 Sep 2008 to 130.120.228.142. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

[Fig. 1(b)]. To illustrate the physics, we first present asimple analysis of this shape transition using the lumpedparameter model shown in Fig. 2. The rod is modeled bytwo rigid links of unit length connected by a torsionalspring. The link OP is fixed at angle ! between the rotationaxis z and the base of the rod in our experiment. Since theReynolds number is small, we take Re ! 0. Thus, we maywork in the rod’s rotating rest frame without introducingfictitious forces. The flow in this frame at point r is !z" r.The torsional spring represents the bending resistance andis only sensitive to changes in the angle between thevectors OP and PQ. Assuming ! # 1 and K is sufficientlylarge, the moment about P on PQ from the spring is Mb $K%OP" PQ& $ K%y; 2!' x; 0&, where K is the torsionalspring constant, and %x; y; 2& $ rQ is the position of thepoint Q to leading order in !. To find the steady-stateposition of Q, equate the moment on PQ due to the tor-sional spring to the moment on PQ due to the flow.Assuming all drag on PQ is concentrated at Q (Fig. 2),the viscous moment about P is Mv $ '"!%x; y; 0&, where" is a resistance coefficient. Solving moment balance for xand y yields x ! 2!=(1) %"!=K&2* and y ! x"!=K. As

! increases from zero, the link PQ deflects and y in-creases, which causes Q to experience a viscous force inthe negative x direction. These forces push Q toward therotation axis, and tend to cause the rod in our experiment towrap around the z axis. As ! increases further, Q movescloser to the rotation axis, and y begins to decrease. Thereis also some drag on the link OP, concentrated a distance dfrom O. The moment about O due to flow is

MO ! "!!2!d2 ) 4

1) "2!2=K2

": (1)

For d2 < 1=2, we find that the moment first increases with!, then decreases as the link folds in toward the rotationaxis where the flow is slow. The moment then increasesagain as the drag from the base link OP dominates. If MOis plotted vs !, then we find an S-shaped curve, just as inour experiment (Fig. 3), with discontinuous transitions inshape and speed as moment varies.

We now turn to a more complete quantitative analysis.We will continue to prescribe ! rather than motor torqueMm, and we limit the analysis to steady-state shapes.Unlike Manghi et al. [10,11], we disregard hydrodynamicinteractions between distant parts of the rod and use resis-tive force theory to model the force per unit length f actingon the rod [1,12]:

f ! "?%v' rsrs + v& ) "krsrs + v; (2)

where "? ! 4#$=(log%L=a& ) 1=2* and "k !2#$=(log%L=a& ' 1=2* $ "?=2, r%s; t& is the position ofthe point on the rod center line with arclength s at time t,v%s; t& is the velocity of the undisturbed flow relative to thevelocity of the rod at s, and rs ! @r=@s is the tangentvector to the rod center line. There is also a hydrodynamic

! P

Q

x

z

yOd

FIG. 2. Lumped parameter model consisting of two rigid linksconnected by a torsional spring (open circle). The top link isclamped. All drag is concentrated at the two filled circles.

100

100

101

102

103

" =

(#$

L4 )/A

(MmL)/A10

-0.810

-0.5

FIG. 3 (color online). Dimensionless motor torque MmL=Awas measured as a function of dimensionless speed % for L !250 mm (,) and L ! 290 mm (4) with angle ! ! 26-. ForL ! 250 mm, speed was measured as a function of increasingtorque (!) and decreasing torque ("). Note the hysteresis. Thelinear (dashed line) and nonlinear (solid line) predictions areshown. The insets show examples of the steady-state filamentshapes in the low (left) and high (right) speed regimes.

!

FIG. 1. Orthogonal images of steady-state shapes of rotatingrod with torque just below (a) and just above (b) the criticaltorque. The motor (not shown) is at the top, with rotation axisalong z. Gravity points down. In (a) and (b), the left panel is theside view, and the right panel is the front view. The rod is markedwith white dots for contrast. The axes in (b) are the same as in(a). The curved arrow in (b) denotes the sense of rotation of therod.


078101-2

tinuous shape transition would allow for strong accelerationstriggered by a slight variation of the torque command. Aninteresting issue that goes beyond the scope of this letterdeals with the efficiency of such a propulsive mechanism,both in the pumping and the swimming regimes.17

While we were completing this work, we became awareof a very similar study from Qian et al.18

H. Stone and R. Netz are gratefully acknowledged forstimulating discussions. We thank N. Champagne, E. Laïk,and L. Gani for help with the experiments. N. Coq is sup-ported by a DGA-CNRS grant.

1D. Bray, Cell Movements: From Molecules to Motility !Garland, NewYork, 2001".

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3N. Hirokawa, Y. Tanaka, Y. Okada, and S. Takeda, “Nodal flow and thegeneration of left-right asymmetry,” Cell 125, 33 !2006".

4V. N. Manoharan, M. T. Elsesser, and D. J. Pine, “Dense packing andsymmetry in small clusters of microspheres,” Science 301, 483 !2003".

5C. Goubault, P. Jop, M. Fermigier, J. Baudry, E. Bertrand, and J. Bibette,“Flexible magnetic filaments as micromechanical sensors,” Phys. Rev.Lett. 91, 260802 !2003".

6A. W. Feinberg, A. Feigel, S. S. Shevkoplyas, S. Sheehy, G. M. White-sides, and K. K. Parker, “Muscular thin films for building actuators andpowering devices,” Science 317, 1366 !2007".

7R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J.Bibette, “Microscopic artificial swimmers,” Nature !London" 437, 862!2005".

8G. I. Taylor, “Analysis of the swimming of microscopic organisms,” Proc.R. Soc. London, Ser. A 209, 447 !1951".

9E. Lauga, “Floppy swimming: Viscous locomotion of actuated elastica,”Phys. Rev. E 75, 041916 !2007".

10M. Kim, J. C. Bird, A. J. Van Parys, K. S. Breuer, and T. R. Powers, “Amacroscopic scale model of bacterial flagellar bundling,” Proc. Natl. Acad.Sci. U.S.A. 100, 15481 !2003".

11C. H. Wiggins, D. Riveline, A. Ott, and R. E. Goldstein, “Trapping andwiggling: Elastohydrodynamics of driven microfilaments,” Biophys. J. 74,1043 !1998".

12T. S. Yu, E. Lauga, and A. E. Hosoi, “Experimental investigations ofelastic tail propulsion at low Reynolds number,” Phys. Fluids 18, 091701!2006".

13M. Manghi, X. Schlagbergerand, and R. Netz, “Propulsion with a rotatingelastic nanorod,” Phys. Rev. Lett. 96, 068101 !2006".

14A. Vilfan and F. Julicher, “Hydrodynamic flow patterns and synchroniza-tion of beating cilia,” Phys. Rev. Lett. 96, 058102 !2006".

15C. H. Wiggins and R. E. Goldstein, “Flexive and propulsive dynamics ofelastica at low Reynolds number,” Phys. Rev. Lett. 80, 3879 !1998".

16G. K. Batchelor, “Slender-body theory for particles of arbitrary cross-section in Stokes flow,” J. Fluid Mech. 44, 419 !1970".

17O. Raz and J. E. Avron, “Swimming, pumping, and gliding at low Rey-nolds numbers,” New J. Phys. 9, 437 !2007".

18B. Qian, T. Powers, and K. Breuer, “Shape transition and propulsive forceof an elastic rod rotating in a viscous fluid,” Phys. Rev. Lett. 100, 078101!2008".

FIG. 5. !Color online" Normalized propulsive force vs normalized torquefor three different filament lengths !!: L=48 mm, ": L=52.5 mm, #: L=96 mm". !Solid line" Force-torque relation obtained according to our lin-earized model.

051703-4 Coq et al. Phys. Fluids 20, 051703 !2008"

Downloaded 15 Sep 2008 to 130.120.228.142. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

Figure 4.2 – À gauche : en haut, photographies de la tige en rotation pour des couplesjuste en dessous (a) et juste au dessus (b) du couple critique ; en bas, vitesse angulaireω en fonction du couple appliqué Mm (sans dimensions, η est la viscosité et A = kBT`p)pour différentes longueurs L. D’après Qian et al. [160]. À droite : en haut, photographiessuperposées de la tige en rotation pour des vitesses angulaires croissantes de (a) à (g). Labifurcation de la forme d’une tige à celle d’une hélice se fait entre les images b et e ; enbas, force de poussée F en fonction du couple appliqué T (Fe = kBT`p/L

2 et Te = FeL).D’après Coq et al. [48].

4.2.1 Article

Suit l’article :M. Manghi, X. Schlagberger et R.R. Netz, Propulsion with a Rotating Elastic Nanorod,Physical Review Letters 96 068101 (2006) (4 pages)

166

Propulsion with a Rotating Elastic Nanorod

Manoel Manghi,1 Xaver Schlagberger,2 and Roland R. Netz2

1Laboratoire de Physique Theorique, IRSAMC, Universite Paul Sabatier, 31062 Toulouse, France2Physik Department, Technical University Munich, 85748 Garching, Germany

(Received 7 January 2005; revised manuscript received 23 September 2005; published 15 February 2006)

The dynamics of a rotating elastic filament is investigated using Stokesian simulations. The filament,straight and tilted with respect to its rotation axis for small driving torques, undergoes at a critical torque astrongly discontinuous shape bifurcation to a helical state. It induces a substantial forward propulsionwhatever the sense of rotation: a nanomechanical force-rectification device is established.

DOI: 10.1103/PhysRevLett.96.068101 PACS numbers: 87.19.St, 05.45.a, 47.15.G

Propulsion of a micrometer sized object through a vis-cous fluid, where friction dominates and motion is invari-ably overdamped, calls for design strategies very differentfrom the ordinary macroscopic world, where inertia playsan important role [1]. Biology has come up with a largenumber of concepts, ranging from active polymerization ofgel networks [2], molecular motors moving on railwaytracks formed by protein filaments [3], to rotating or beat-ing propellerlike appendices [4]. Those propellers involvestiff polymers that are moved by molecular motors and relyon two different basic designs: (i) flagella of bacteria(prokaryotic cells) are helical stiff polymers set in motionat their base by a rotary motor. Hydrodynamic frictionconverts the rotational motion of the helix into thrust alongthe helix axis [1,4]. (ii) Cilia and flagella of sperms (eu-karyotic organisms) are rodlike polymers that are anchoredto a surface and beat back and forth driven by internalmotors. They are used by small cells to swim but also byinternal organs to pump liquids [4].

For a number of biomedical applications, e.g., for di-rected motion of artificial viruses through cells or nano-devices through the bloodstream, as well as for mixingstrategies in nanochips, it is desirable to develop similarsynthetic propulsion mechanisms or to incorporate biologi-cal single-molecule motors into synthetic environments.Recent discoveries opened the route to the synthetic manu-facture of rotary single-molecule motors driven by chemi-cal [5] or optical [6,7] energy. An ATPase can also be fixedto different substrates and used to rotate metallic [8] ororganic nanorods [9]. This raises the question about theminimal design necessary to convert the rotational powerof such nanoengines into directed thrust in a viscous envi-ronment. In this Letter, we show that an elastic rod thatrotates around a point at a constrained azimuthal anglegives rise to a substantial forward thrust regardless of thesense of its rotation. It thus acts as a rectification device andproduces net thrust even when it is stochastically rotatedback and forth. It makes usage of a helical polymer un-necessary, allowing a selection from the much wider classof straight stiff polymers. A few theoretical works haveelucidated the coupling between hydrodynamics in a vis-

cous medium and elasticity of soft polymers [10–12]. Itwas shown that finite stiffness of beating straight filamentsbreaks time-reversal symmetry and enables propulsion[10,12]. Conversely, reversing the rotation sense of bacte-rial flagella gives rise to pronounced shape polymorphictransformations [13] and shows that elasto-hydrodynamicsare important in biology. Likewise, hydrodynamic inter-actions between rotating flagella are involved in bundlingand are important for switching between tumbling andrunning modes of E. coli bacteria [14,15].

We study an elastic rod whose base rotates on a conedriven by an external torque applied to its anchoring point.We use Stokes-simulation techniques where hydrodynam-ics are treated on the Rotne-Prager Green’s function levelincluding the full coupling between thermal, elastic, andhydrodynamic forces. We obtain the nonlinear relationbetween driving torque and angular frequency in the pres-ence of an external force load and estimate the powerconversion efficiency. The filament exhibits a stronglydiscontinuous shape bifurcation at a finite torque value atwhich the rod jumps closer to the rotation axis. The elasticpolymer is modeled as a chain ofM 1 connected spheresand the time evolution of the sphere positions rit isgoverned by the position-Langevin equation [16]

_r it XMj0

ijrijrrjUrk i2Frot2 it; (1)

where the potential Urk PM1i0

4a rii1 2a2

"2a 1 cosi is the discrete version of the extensiblewormlike chain model, a the sphere radius, rij rj ri,and i is the angle between neighboring bonds of sphere i.The parameters and " are the stretching and bendingmoduli, which are related by "= a2=4 (isotropic rod).Hydrodynamic interactions between two spheres i and jare implemented via the Rotne-Prager mobility tensor

ijrij 1

8rij

1

rijrijrijrij

2a2

r2ij

13

rijrijrijrij

; (2)

which is an approximation valid for large sphere separa-tions [16]. 1 denotes the unit 3 3 matrix and the


0031-9007=06=96(6)=068101(4)$23.00 068101-1 © 2006 The American Physical Society

viscosity of the aqueous solvent. For the self-mobilitywe choose the Stokes mobility of a sphere ii 0 1=6a. To mimic a rotary motor, we apply an ex-ternal force Frot

2 on monomer 2, which is related to thetorque N by Frot

2 N r12=r212. The Langevin random

displacement t models the action of the heat bath andobeys the fluctuation-dissipation relation hitjt

0i

2kBTijt t0 numerically implemented by a Cholesky

decomposition [16]. The persistence length is given by‘p "=kBT. Twist and torsional degrees are omittedsince for most synthetic polymers, free rotation aroundthe polymer backbone is possible. Two different forceensembles are investigated: the stalled case, where the firsttwo monomers forming the base are held fixed in space byapplying virtual forces which exactly cancel all otherelastic and hydrodynamic forces. In the moving case, welet the polymer move along the z axis and apply virtualforces on the first two monomers only laterally such thatthe polymer base moves along a vertical line. Hydro-dynamic boundary effects are considered further below.A finite tilt angle at the base is imposed by a spontaneouscurvature term in the elastic energy "=2a1 cos1 1; in this Letter we show data for spontaneous curvature1 45. For the numerical iterations, we discretizeEq. (1) with time step . By rescaling time, space, and en-ergies, we arrive at rescaled parameters ~ kBT0=a

2,‘p=L "=2aMkBT, and ~N N=kBT. For sufficient nu-merical accuracy we choose time steps in the range ~

103–108. Output values are calculated every 103–104

steps; total simulation times are 108–109 steps.When a torque is applied to the filament base, it rotates

and after some time exhibits a stationary shape. Since thefriction is larger at the free end, the filament bends into acurved structure. The stationary angular velocity ~! !a2=kBT0 (in degrees) is plotted in Fig. 1(c) as afunction of the applied torque ~N showing a nonlinearincrease, which is caused by the finite bending rigidity.Indeed, for an infinitely stiff polymer, this relation isstrictly linear. For a critical torque, ~Nc, a shape bifurcationoccurs and the angular frequency jumps dramatically.Figure 1(a) shows the stationary shape of the polymerbefore (A) and after the bifurcation (B). The high-torquestate is characterized by a smaller distance from the rota-tion axis. Figure 1(b) shows the actual time-dependentbehavior of the rotational speed as the rod crosses thetransition and settles in the stationary state. Pronouncedhysteresis (which becomes slightly weaker with decreasingtorque sweep rate) is observed when ~N is varied across thecritical region, as illustrated in Fig. 1(d). Figure 1(e) showsthe dependence of the critical torque on the persistencelength of the rod. Since the torque change rate is finite,increasing () and decreasing torque () gives rise toslightly different bifurcation values. For very small persis-tence lengths, conformational fluctuations suppress thetransition. The shape bifurcation can be understood bysimply balancing elastic and driving torques: the bendingtorque due to the filament deformation, projected along the

p/L

e)

4

8

12

16

20

00 2 4 6 8 10 12 14

CN∼

(/10 )3

(/10 )3

2

4

6

8

10

00 4 8 12 16 20

0

0.4

0.8

1.2

0 1 2 3

c)

N∼

(/10 )3

ω ∼(/10 )3

t/(∆×10 )5

1

2

3

144 146 148 150

b)

ω ∼(/10 )3

0

1

2

3

4

4 6 8 10 12 14 16 18

d)

N∼

(/10 )2

ω ∼(/10 )2

a)

05 AB

Y

X 0 10 20 30

θ∗1

A

B

Z

0

20

40

60

0 5

A

B

Y

Z

Z

XY

A

B

FIG. 1. (a) Sketch in 3D and projected views of the rotating filament before (A, ~N 600) and after (B, ~N 800) the bifurcation(M 30 beads, ‘p=L 103). (b) Angular velocity ~! across the shape transition as a function of time ( ~N 1:33 103, ‘p=L 2 104). (c) Angular velocity as function of external torque, ~N, for various ‘p=L: 3333 (), 6667 (), 1:33 104 (), and 2 104

() [inset: 333 (), 1000 () and 2500 ()]. (d) Hysteresis cycle for ‘p=L 1667 for increasing () and decreasing torque () atconstant rate ~N=t= 1:25 106. (e) Persistence length vs critical torque for increasing () and decreasing torque () bothobeying a linear law according to Eq. (3) (All data for stalled case, Vz 0.)


068101-2

vertical axis, reads " sin1=R where R is the bendingradius of the filament, which has to be balanced by theexternal torque N. The transition occurs when the bendingradius reaches the length of the filament, R L. Thisyields the critical torque

~N c ’‘p

Lsin1; (3)

which is shown in Fig. 1(e) as a solid line and agrees verywell with the numerical data. Neither hydrodynamicalparameters nor temperature appear in Eq. (3); the bifurca-tion is thus purely elastic in origin and indeed can beobtained within free-draining or slender-body approxima-tion (as will be published separately). Using the rotationalmobility rot 1=L3, the critical angular frequencyreads !c rotNc "=L

4. This threshold turns outto be much lower than for the continuous twirling-whirlingtransition of a rotating rod with torsional elasticity, !tw

c "=a2L2, which was obtained using linear analysis [12].

We define the translational and rotational mobilities ofthe filament as [1]

!Vz

rr rt

tr tt

NFext

; (4)

where Vz is the translational velocity downwards, Fext thecorresponding external force applied at the propeller base.In general, the mobilities depend on torque N and forceFext in a nonlinear fashion. For perfectly stiff propellers,the mobility matrix is constant and symmetrical, i.e.,rt tr [1]. However, in our case, due to flexibility, the pro-peller shape changes with external efforts and the matrix isasymmetric. The propulsion velocity along the rotationaxis is plotted in Fig. 2(a) as a function of ~NL=‘p fordifferent persistence lengths for the z-moving case (Fext 0). At the transition, a jump in the propulsion velocity isobserved and ~Vz is almost linear for ~N > ~Nc, revealing thatthe shape remains almost fixed in this range of torquevalues. The inset of Fig. 2(a) shows the ratio of Vz afterand before the transition versus the same ratio for !. Thevariation is roughly linear, meaning that the jump of ~Vz isdirectly due to the jump in !. To test our propeller under

load, we applied an external force, ~Fext, which we define tobe positive when it pushes against its natural swimmingdirection. Figure 2(b) shows the variation of ~Vz with ~Fext attwo different torques just below and above the bifurcation.The laws are almost linear in both cases, meaning that tt

is almost independent of Fext. The efficiency of the powerconverter can be defined as the ratio of the propulsivepower output and the rotary power input,

FextVzN!

: (5)

By inserting Eq. (4) in Eq. (5), we obtain Fext trNFext ttF2

ext=rrN2 rtNFext. We havechecked that rt is negligibly small. Hence, the efficiencybecomes parabolic as a function of the external force asshown in Fig. 2(c). The highest efficiency is obtained forFext trN=2tt Fstall=2 and is only of the order of1% after the transition and 3 times smaller before.

Up to now we considered an isolated rotating filamentto which external torque and forces were applied. In re-ality, the filament is attached to a base, and rotation ofthe filament is caused by some relative torque generatedbetween base and filament. As for any self-propellingobject in viscous solvent, the total force and torqueadd up to zero [1]. If the filament rotates counterclock-wise at frequency !, then the base rotates clockwise withan angular velocity . Within our Stokes simulation, wemodel the base by three elastic arms of length La andpersistence length ‘ap which are coplanar and connectedto the filament at one point [see Fig. 3(a)]. Hydrodynamicscreening and lubrication are accurately accounted foron scales larger than the monomer radius a. The torqueacting on the filament is balanced by a counter torqueon the base. The resulting complex helical trajectoriesof this nanomachine make calculation of the mean propul-sion velocity difficult due to thermally caused reorienta-tion of the complete device. In the following, we thereforeturn off the thermal noise (which plays a minor role forstiff polymers). Figure 3 shows the propulsion velocity ~Vpand the ratio of rotational speeds of the base and the fila-ment, ~= ~! versus the ratio L=La at fixed relative torque

Fext

∼

Vz∼

-8

-6

-4

-2

0

2

0 10 20 30 40 50

b)

Fext

∼-1

-0.5

0

0.5

1

1.5

0 10 20 30 40 50

c)

η(%)

Vz∼

p/L(×10 )3

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

a)

5

6

7

8

4 5 6 7 8 9 10

/V V zc zc> <

ω /ωc c> <

∼NL/ p

FIG. 2. (a) Propulsion velocity parallel to the rotation axis vs ~NL=‘p in the z-moving case for ‘p=L 2500 ( ), 5000 (), and2 104 (). The inset shows the variations of the ratio of velocities after and before the transtion vs the ratio of angular velocities (,‘p=L 1667). (b) ~Vz vs the external force ~Fext applied on monomer 2 ‘p=L 1667 before the bifurcation for ~N 0:78‘p=L ()and after ~N 0:84‘p=L (). (c) Efficiency, , vs ~Fext [same parameters as (b)]. The solid line is a parabolic fit.


068101-3

~N 2000 (L 30a, ‘p=L 33, ‘ap=‘p 10). For basesizes comparable or larger than the filament length,L=La < 1, the propulsion velocity is highly reduced com-pared to the case of an isolated filament (dashed line),mostly due to hydrodynamic friction caused by the base.But for decreasing base size, the propulsion velocity ap-proaches quite closely the velocity of an isolated filament,calculated according to the preceding model. Indeed, whenit is very small, the counter-rotating base causes negligiblehydrodynamic effects on the filament. At the same time,the base rotational speed (rescaled by !, see Fig. 3)increases when the base size decreases while remainingmuch smaller than the filament one. This is because, forsuch a torque much larger than the critical torque, thefilament is close to the filament axis and thus gives littlerotational resistance. It also means that only a small frac-tion of power input is invested into the base rotation. Inconclusion, we show that a simple straight elastic filamentcan induce propulsion when rotated at one end. The inter-play between elastic deformations and hydrodynamic in-teractions results in a substantial directed thrust due to asubcritical dynamic bifurcation for control parameters~N= ~Nc > 1. The mobility tr changes its sign when thetorque is reversed, which implies a forward thrust whateverthe sense of rotation. This propulsion occurs even in thepresence of an explicit base, being larger for small bases.This work provides a clue for the synthetic manufacture ofbiomimetic micropropeller, by using simple semiflexiblepolymers instead of rigid helices. This bifurcation could beexperimentally investigated using a macroscopic scalemodel similar to the one developed by Powers group[17]. With their experimental setup, the bifurcation wouldoccur for torques on the order of 0.05 Nm and angularvelocities larger than 0.01 Hz, which are accessible.

With regard to bacteria propulsion, flagella are helicaland their physics is therefore much more involved.

However, flagellar motors generate sufficient torques forthe nonlinear elastic phenomena discussed here: they arepowered by a proton-motive force which yields torques103kBT [13,18]. Moreover, the flagellum length is L ’10 m, leading to ‘p=L ’ 103–104 and ~NL=‘p ’ 0:1–1.Hence, the shape bifurcation discussed in this model isprobably biologically relevant and might be directly ob-served with straight biopolymers attached to flagellar mo-tors. For a bacterial flagellum at physiological conditions,the diameter is roughly a ’ 20 nm and the stall forces areroughly 10 pN at the bifurcation. Critical angular andpropulsion velocities (with no external force) then followfrom our results as !c ’ 2 105–106 rad=s and Vzc ’6–40 mm=s. These values are large but if we actuallyconsider a thicker bundle formed by approximately 7flagella [13], we come up with smaller velocities whichare comparable to the ones observed for E. coli (! ’104 rad=s and Vz ’ 30 m=s).

Financial support of the German Science Foundation(DFG, SPP1164) is acknowledged.

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401, 150 (1999).[6] N. Koumura et al., Nature (London) 401, 152 (1999).[7] T. Harada and K. Yoshikawa, Appl. Phys. Lett. 81, 4850

(2002).[8] R. K. Soong et al., Science 290, 1555 (2000).[9] H. Noji, R. Yasuda, M. Yoshida, and K. Kinosita, Nature

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3879 (1998).[11] S. Camalet, F. Julicher, and J. Prost, Phys. Rev. Lett. 82,

1590 (1999).[12] C. W. Wolgemuth, T. R. Powers, and R. E. Goldstein,

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2793 (2000).[14] M. Ramia, D. L. Tullock, and N. Phan-Thien, Biophys. J.

65, 755 (1993).[15] M. Kim and T. R. Powers, Phys. Rev. E 69, 061910 (2004).[16] D. L. Ermack and J. D. McCammon, J. Chem. Phys. 69,

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L

LaΩ

ω

0

0.2

0.4

0.6

0 1 2 3 4 4

4.4

4.8

5.2V∼

aL/L

(x10 )2−Ω/ωp

a) b)

FIG. 3. (a) Sketch of the self-propelling nanomachine whosebase is made of 3 arms of length La. (b) Propulsion velocity ~Vp() and ratio of the base, , and filament, !, angular velocites() against L=La for L 30a, ‘p=L 33, ‘ap=‘p 10, andN 2000. The dashed line is the asymptotic velocity reached byan isolated filament.


068101-4

Conclusion générale

Dans la première partie de ce mémoire, j’ai présenté mes travaux de recherche portantsur la modélisation de l’ADN. Ce modèle de physique statistique qui couple l’état internedes paires de bases aux conformations de la chaîne ADN dans l’espace à trois dimensionsa conduit à déterminer certaines caractéristiques du profil de dénaturation de l’ADN ensolution ainsi que la dépendance des caractéristiques mécaniques de la chaîne en fonctionde la température. Il a également permis de montrer l’importance de la prise en comptede ce couplage dans l’analyse des images AFM des ADN adsorbés sur des substrats. Cetteméthode étant de plus en plus utilisée pour observer, par exemple l’interaction entre ADNet protéines, nous montrons qu’il faut rester prudent quant aux conclusions que l’on peuttirer des ces expériences sur les mêmes situations en solution ou in vivo.

Les directions vers lesquelles j’entends poursuivre mes recherches sur la physique del’ADN sont développées à la fin du chapitre 1. Il s’agit i) de la prise en compte de la disso-ciation des brins, qui nécessitera probablement des simulations numériques, ii) d’étendrenotre approche aux ADN sous contrainte extérieure (force, couple ou protéines induisantleur cyclisation), iii) d’étudier l’influence de la diffusion des parties de l’ADN double-brinssur la dynamique de fermeture d’une bulle de dénaturation. Enfin, depuis deux ans, noustravaillons en collaboration avec le biologiste P. Rousseau et les expérimentatrices L. Sa-lomé et C. Tardin à la mise en place d’une expérience de suivi de particule unique pourobserver la dénaturation sur un ADN unique. La mise au point de l’expérience prend dutemps mais des résultats préliminaires sont encourageants.

Le second chapitre est dédié à mes travaux sur les membranes fluctuantes. Nous mon-trons comment l’hétérogénéité de la composition lipidique de la membrane peut conduireà des interactions répulsives entre domaines lipidiques, médiées par les fluctuations de lamembrane. Le mécanisme proposé peut conduire à la formation de mésophases lipidiques.Je me suis également intéressé au cas de deux membranes empilées, supportées par unsubstrat plan. Nous avons établi le diagramme de phase de ces systèmes en fonction desdifférents paramètres les caractérisant et nous avons étudié l’influence de la friction hydro-dynamique, associée à cette configuration de membranes supportées, sur la dynamique desmembranes.

171

Conclusion

La modélisation de ces systèmes a été rendue nécessaire par le développement de mem-branes modèles afin d’étudier de façon contrôlée la dynamique des ses constituants. Encollaboration avec L. Salomé et des biologistes de l’IPBS, nous entendons modéliser ainsila dynamique des interactions entre deux protéines membranaires impliquées dans la régu-lation de récépteurs aux opiacés.

La troisième partie est consacrée à l’étude de la répulsion des ions d’un éléctrolyte auvoisinage d’une surface faiblement diélectrique et dans des pores nanométriques. Ces effetssont importants en nanofiltration et ce travail s’insère dans une large collaboration avec deschimistes de Montpellier et de Marseille qui synthètisent et caractérisent les membranesde filtration et des numériciens de Montpellier qui simulent ces systèmes.

Les perspectives dans ce domaine de recherche, développées à la fin du chapitre 3,sont très nombreuses. Elles concernent aussi bien des avancées possibles sur les modèlesthéoriques utilisés pour étudier les électrolytes aux interfaces, que la compréhension deseffets de polarisabilité des ions qui semblent importants à l’échelle du nanomètre et qui ontété observés dans des expériences [144] et dans des simulations de dynamique moléculaire.

Enfin, les effets hydrodynamiques sont récurrents dans la physique statistique des objetsbiologiques, et je développe dans le dernier chapitre une partie de mon travail postdoctoralsur la propulsion à petits nombres de Reynolds. Ce travail me permet encore aujour-d’hui de mieux comprendre les effets des interactions hydodynamiques, omnisprésents enbiophysique, et j’envisage de considérer l’hydrodynamique aussi bien dans l’étude de ladynamique de l’ADN et des membranes que dans la modélisation des flux d’ions à traversles nanopores, dont la mesure caractérise expérimentalement la qualité des membranes denanofiltration.

172

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