DNA coby polyelektrolyt: Manningova teorie kondenzace protiiontů

Manning, G. S. (1978). "The molecular theory of polyelectrolyte solutions with aplications to the electrostatic properties of polynulceotides." Quart. Rev. Biophys. 11: 179-246.

Jiří Kozelka Université Paris DescartesLaboratoire de Chimie et Biochimie Pharmacologiques et ToxicologiquesUMR CNRS 860145 rue des Saints-Pères75270 Paris FRANCE******Masaryk University of BrnoInstitute of Condensed Matter PhysicsFaculty of ScienceKotlářská 261137 Brno CZECH REPUBLICTel. +420 549 49 4048Port. +420 720 379 379

DNA double helix

Double-stranded DNA can be considered as a linear long rod composed of segments, each carrying one unit charge. The segments repel each other. The repulsion is diminished if positively charged counter-ions associate with the DNA surface.

__

_

__

_

__

__

__

1 segment: charge -1e

b = 1.7 Å

distance between segments b defines the charge density of the polymerb-1

+

+

++

+

+

+

+

+

+

+

+

At each segment, a fraction of positive counterions „condenses“ within a volume VP

G/ = 0

VP

1 equation2 unknowns

Univalent salt in water (e.g., NaCl)

DNA double helix__

_

__

_

__

__

__

G(, VP ) / = 0

The charge of 1 segment is reduced to (-1+)e

Gain in enthalpy H, since repulsion between segments diminishedLoss in entropy S, since order increases

Nature will „choose“ such a fraction, that G = H – TS minimal

qnet qnet qnet qnet qnet qnet qnet qnet

Calculation of the electrostatic repulsion term Gel

Phosphate N°: 1 2 3……………..s………………………np

Each pair contributes:

b

|i-j| = 1: number of pairs = np-1|i-j| = 2: number of pairs = np-2

2b

.

.|i-j| = s: number of pairs = np-s

|i-j| > s: interaction negligible

if np>> s

≈ np

≈ np

≈ np

Gij≈ 0

Charge:

rD = „Debye screening length“proportional to c-1/2

c: concentration of univalent salt

qnet = qe(1- )D

i j b2 2re e

ij

k q (1- )ΔG e

ε i- b

θ

j

D

i-j b2 2 sAV e e r

eli-j

p

=1

L k q (1- ) 1ΔG = e

ε

n θ

b i-j

All pairs contribute:

Coulomb Debye-Hückelscreening factor

efekt vody

efekt soli

D D D D

i-j b b 2b 3bsr r r r

i-j =1

1 1 1e = e + e + e +...

i-j 2 3

Simplifying the sum over pairwise electrostatic repulsion terms

D

i-j b2 2 sp AV e e r

eli-j =1

n L k q (1-θ) 1ΔG = e

εb i-j

Looks like a Taylor series….

f(x) =

Cvičení 1: odvoďte Taylorův rozvoj pro funkci f(x)=ln(1+x)(vyžadováno je řešení, nikoliv jen výsledek)

Taylorův rozvoj :

List of Taylor series of some common functions

http://en.wikipedia.org/wiki/Taylor_series

Simplifying the sum over pairwise electrostatic repulsion terms

D

i-j b2 2 sp AV e e r

eli-j =1

n L k q (1-θ) 1ΔG = e

εb i-j

D D D D

- i-jb -b -2b -3bsr r r r

i-j=1

1 1 1e = e + e + e +...

i-j 2 3

xe = 1+x for x<<1

2 3 4x x x x

e = 1+ x + + + +...2 6 24

Cvičení 2: vypočtěte sumu

=

Cvičení 3: zjednoduštesumu (rD>>b)

D D D D

i-j b b 2b 3bsr r r r

i-j =1 D

1 1 1 be = e + e + e +...= -ln( )

i-j 2 3 r

Simplifying the sum over pairwise electrostatic repulsion terms

Sum has become a simplelogarithmic term!D

i-j b2 2 sp AV e e r

eli-j =1

n L k q (1-θ) 1ΔG = e

εb i-j

2

Del p

Del

bΔG = -ξRTn (1-θ) ln( )

bln( ) < 0; ΔG

rr0

is dimensionless charge density2

e ek q 1with ξ=

εb kT

2 2p AV e e

elD

n L k q (1-θ) bΔG =- ln( )

εb r

Cvičení 4: Vypočtěte pro dvojšroubovicovou DNA (b = 1.7 Å) při 25 °C

2e ek q 1

with ξ=εb kT

k = 1.38x10-23 J.K-1 T = 298 K ke = 9.0x109 Nm2C-2 = 78 qe = 1.60x10-19 C

+

+

++

+

+

+

+

+

+

+

+

A fraction of positive counterions „condenses“ within a volume VP

VP

Loss in entropy S, since order increases

DNA double helix__

_

__

_

__

__

__

cbulk

clocal= nP(nPVP)-1 = VP-1

cbulk = c = concentration [mol/L] of univalent salt KNOWN

Let‘s minimize the total free energy with respect to

Mixing entropy term dependent on

Vp UNKNOWN → HOPELESS CALCULATION???

VP [L/mol] = volume to which the counterions/phosphate are confined

(ΔG)/ θ = 0

i.e., loss in entropy when the fraction of counterions associates with each segment

)c

θVRTln(θn)

cc

RTln(θnG1

PP

bulk

localPmix

number of moles of couterions that is confined from cbulk to clocal

)]c

θVln(θ)

r

bln(θ)ξ(1[RTnΔGΔGΔG

1P

D

2Pmixel

Cvičení 5: Vypočtěte první derivaci následují funkce podle x :(řešení a výsledek)

)2x1

1(ln)(

xf

Minimizing total free energy

-3

0D 2

AV e

ε εRT 10r m =

2(L q ) c

2e ek q 1

b =εξ kT

From Debye-Hückel Theory of electrolytes

)]c

θVln(θ)

r

bln(θ)ξ(1[RTnΔGΔGΔG

1P

D

2Pmixel

)]c

θVln(1)

rb

ln(θ)(1[2RTn

)]c

θVln(1

θ1

θ)rb

ln((-1)θ)(12[RTn

1P

DP

1P

DP

δθ

δ(ΔG)Cvičení 6: vypočtěte derivaci. Naznačte postup výpočtu.

Dependence of Gel on c Dependence of Gmix on c

-3

0D 2

AV e

ε εRT 10r m =

2(L q ) c

2e ek q 1

b =εξ kT

1 12 2 2 2 6e e AV e e e AV2 2

-3 2 3 -3D 0 AV 0

k q 2L q c 2k q L cb 1( ) ( )

r εξ kT ε εL kT 10 ξ ε (εkT) 10

0)θV

cln(1]

10kT)(εξcLq2k

ln[21

θ)(12 1P

330

2AV

6e

2e

0)θV

cln(1]

10kT)(εξcLq2k

ln[θ)(1 1P

330

2AV

6e

2e

0)]c

θVln(1)

rb

ln(θ)(1[2RTnδθ

δ(ΔG) 1P

DP

k Boltzmann‘s constant dimensionless charge density of polymerke electrostatic constant c concentration of univalent electrolyteqe electron charge dielectric coefficient of electrolyte vacuum permittivity T absolute temperatureLAV Avogadro‘s number fraction of counterions condensed

VP volume to which counerions condense

knowns

unknowns

We want to know at the low concentrationlimit, c→0

(1- ) - 1 ξ θ = 0

-1 = 1 ξ-θ

→Singular ln c terms

must vanish!

ln c↓

-∞

ln c↓

-∞

Now we have got a second

equation!

0)θV

cln(1]

10kT)(εξcLq2k

ln[θ)(1 1P

330

2AV

6e

2e

We want to know at the low concentrationlimit, c→0

(1- ) - 1 ξ θ = 0

-1 = 1 ξ-θDNA in water:

= θ4.2 = 0.76

→Singular ln c terms

must vanish!

ln c↓

-∞

ln c↓

-∞

Now we have got a second

equation!

0)θV

cln(1]

10kT)(εξcLq2k

ln[θ)(1 1P

330

2AV

6e

2e

1

-1θ

no association of counterions

association of counterions

76% of the negative charge of the phosphates

is neutralized by condensed counterions!

k = 1.38x10-23 J.K-1 = 8.85x10-12 C2N-1m-2 DNA in water at 25°Cke = 9.0x109 Nm2C-2 LAV = 6.02x1023 mole-1 = 4.2qe = 1.60x10-19 C e = 2.718 =78

T = 298 K= 0.76

(1- ) - 1 ξ θ = 0

Now we can determine VP, the volume to which the θ counterions/nucleotide condense in double-stranded DNA

0)θV

cln(1]

10kT)(εξcLq2k

ln[θ)(1 1P

330

2AV

6e

2e

1

mole/m6.46x10L/mole646.010kT)(εξ

eθLq2kV 34-

330

2AV

6e

2e

P

= 6.46x1026LAV-1 Å3/nucleotide = 1073 Å3/nucleotide

base of natural logarithms

-1 = 1 ξ-θ

Counterion Condensation Theory: Manning, G. S. (1978) Quart. Rev. Biophys. 11: 179-246

20 Å

7 Å

7 Å

The volume to which the condensed univalent counter-cationsare confined, VP

Per mole: 646 cm3

Per nucleotide: 1073 Å3

This corresponds to a layer of ~7 Å around the DNA double-helix(which corresponds ~to a cylinder with a diameter of 20 Å)

Experimental and theoretical results supporting the Counterion Condensation Theory

1. Titration of Carboxymethylcellulose (CMC) with different hydroxides monitored by ultrasonic absorption

CMC

Titration with XOH; X = Na, K, Li, N(CH3)4…….

Upon titration, the COOH groups are successively deprotonated, become COO-, and the negative charge density increases. This modifies the structure of the water layer at the surface of the polymer, which can be monitored by measuring the absorption of ultrasound. If there is no association of cations with the polyanion, the structure of the water layer should be independent of the cation.

O

OH

OH

O

OHO

OO

OH

OH

O

OHO

O

OH

OH

O

OHO

O

O

O

OH

OH

O

OHO

+ OH-

O-

O+ H2O

< 1: no difference between the various hydroxides no association between the counterions and the polymer

> 1: great differences between the various hydroxides association between the counterions and the polymer, as predicted by the Manning theory

R. Zana et al., J. Chim. Phys, 68, 1258-1266 (1971)

Radial distribution of Na+ counterions around a DNA duplex 12 base-pairs long,from a molecular dynamics simulation (Yang et al., J. Am. Chem. Soc. 1997, 119, 59-69). The calculation predicts 82% of the phosphate charge to be neutralized,with cations moving within a radius of 15 Å from the helix axis. Manning theorypredicts 76% of cations within 17 Å from the helix axis.

2. Molecular dynamics simulations of a solvated DNA duplex

[Å]

% phosphatecharge

20 Å

7 Å

7 Å

-1 θ = 1- ξ

= fraction of monovalent cations predicted by Counterion Condensation Theory to condense on each unit charge of the poly-anion

If then a fraction of 1-ξ-1 counterions condense at each unit charge

If , there is no counterion condensation.

duplex DNA in H2O at 298 K: = 4.2 single-stranded DNA: = 2.1

In a diluted solution of counterions, is concentration-independent!

Remember:

http://matematika-online-a.kvalitne.cz/derivace-funkce.htm