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2ème école de RMN Cargèse
18-‐23 Mars 2013
Analysis of molecular motions by Nuclear Magnetic Resonance
(Mostly high resolution liquid state)
Carine van HeijenoortCNRS-‐ICSN
laboratoire de chimie et biologie structurales
[email protected]‐gif.fr0169823794
1
mailto:[email protected]:[email protected]:[email protected]:[email protected]
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013 Multiplicity of proteins states
Sequence
3D structure
disorder → ordermolecular recognitionvirus/phages assembly
stepping motors
order → disorder
α ↔ β
« lock & key »induced Eit
conformational switch
virus/pathogen penetrationmembrane insertionNucléosome activation
Elexible ensemble
Elexible linkersdisplay of sites
entropic bristles, springs and clocks
Folding
“Non folding”
Dunker et al., Journal of Molecular Graphics and Modelling 19, 26–59, 2001Dobson, C., Nature 426, 18-‐25, 2003
2
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013 Different techniques for the analysis of proteins dynamics
X-‐ray cristallography B factors time scalesstatic disorder, crystal contacts, ...
X-‐Ray, neutron scatteringDoniach, Chem. Rev. 2001, 101 ; Zacai, science 2000, 288.
size/shape modiSicationstimescales (ps-‐ns) for 1H positions
FluorescenceWeiss, Nat. Struct. Biol. 2000, 7 ; Yang, Science 2003, 302 ; Haustein, Curr. Opin. Struct. Biol. 2004, 14.
ensemble / single moleculecellular context
probes
Mass Spectroscopy (HX MS)radical footprintingWales, Mass. Spectrom. Rev. 2006, 25 ; Busenlehner, Arch. Biochem. Biophys. 2005, 433. Guan, Trends. Biochem. Sci. 2005, 30.
large molecular assemblies
Mössbauer, Raman, 2D infrared spectroscopy
Molecular dynamics ForceSields ...Short timescales ...
NMR
Boehr, Chem. Rev. 2006, 106, 3055. Palmer, Chem. Rev. 2004, 104, 3623.
➫ 10-‐12↔ 105 s➫ Site-‐speciSic information ➫ multiple atomic probes 1H, 2H, 15N, 13C, 31P, ...➫ Simultaneous monitoring of probes➫ kinetic & thermodynamic proSile of dynamic processes
➫ isotope labeling➫ quantities➫ size limitation➫ complexity of the method ?
3
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
How motions are « visible » in NMR ?
ü Molecular motions in/luence NMR parameters‣ Motions timescales versus NMR timescales ?
ü Three distinct timescales in NMRo Equilibrium constant time T1 / Signal lifetime T2
§ NMR experiment perturbation of spins system§ Typical timescale for (macro)molecules in solution : 100 ms -‐ s (T1) / 10ms-‐s (T2)§ Determine the lowest frequency of motions that can be characterized during one NMR experiment.
o Spectral range : τ=1/Δν§ Spectrum features : chemical shift range, couplings, …§ Averaging if the interactions by motions that have higher frequencies§ Perturbation of spectral appearance by motions/processes occuring around this timescale
o “Larmor” timescale: τ=1/ω0§ Precession frequency of the spins in the magnetic Sield B0=ω0/γ§ EfSiciency of spins state transitions during the relaxation processes is determined by the spectral density of molecular motions around these frequencies.
4
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
s ms nsμs
Three distinct typical timescales in NMRReturn to equilibrium : T1
Signal lifetime : T2Spectral range : τ=1/Δν
Larmor precession : τ=1/ω0
How motions are « visible » in NMR ?
5
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Internal motions
Types of motions
Folding ; order ↔ disorderEffect of m
otions on
spins interaction
s ms nsµs ps
Macroscopic diffusion
Magnetization exchange
Diffusionexperiments
Conformational exchange
Interactions
Catalytic processesRegulation, signalization
Molecular rotations
Molecular vibrations
Lineshape modiEications
Averaging of spectral
components
Spin relaxation averaging of non-‐secular interactions
T1, T2 1/Δω 1/ω0
Signal relaxation frequencies differencesnutation
frequencies
NMR parameters
Functions
Dipolar Couplings averaging
How motions are « visible » in NMR ?
6
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Wüthrich, « NMR of proteins and nucleic acids », WileyInterscience, 1986
Dynamical processes slower than T1«real time» kineticsfolding/unfoldingInteractions, exchangeH/ D exchange...
NMR timescales1. Relaxation times
ü Longitudinal relaxation time constant T1 characterizes the time it takes to the spin system to return to equilibrium.
ü It determines the delay beteen two scans
ü Motions slower than T1 can not be characterized by one NMR scan.
ü NB. T1 depends on the magnetic Sield strength B0, the type of spin, the size of the molecule, the local dynamics of the system, the temperature, etc.
7
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
1. Relaxation times
ü Transversal relaxation time constant T2 characterizes the lifetime of the signal.
ü It determines the linewidth of the resonances
ü Motions slower than T1 can not be characterized by one NMR scan.
ü NB. T1 depends on the magnetic Sield strength B0, the type of spin, the size of the molecule, the local dynamics of the system, the temperature, etc.
8
/2 /20
1/( )2
Mx(t) = �Meqsin(�0t)exp(�t/T2)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
1. Relaxation times
9
ü Relaxation contants depends on the magnetic Sield strength B0, the type of spin, the size of the molecule, the local dynamics of the system, the temperature, etc.
0,1
1
10
100
τc(s)
T1400,600,800MHz
rela
xati
on t
imes
(s)
10-12 10-11 10-10 10-9 10-8
Longitudinal relaxation time constant for a system of 2 proton spins with rHH=0.2nm
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Internal motions
Types of motions
Folding ; order ↔ disorderEffect of m
otions on
spins interaction
s ms nsµs ps
Macroscopic diffusion
Magnetization exchange
Diffusionexperiments
Conformational exchange
Interactions
Catalytic processesRegulation, signalization
Molecular rotations
Molecular vibrations
Lineshape modiEications
Averaging of spectral
components
Spin relaxation
T1, T2 1/Δω 1/ω0
Signal relaxation frequencies differencesnutation
frequencies
NMR parameters
Functions
Dipolar Couplings averaging
How motions are « visible » in NMR ?
averaging of non-‐secular interactions
10
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
11
Δω=13ppm Δν=7800HzB0=14.1T
Δν=12350Hz
B0=22.3T
τspect~100μs
Δω=0,2ppm Δν=120HzB0=14.1T
Δν=190Hz
B0=22.3T
τspect~5-‐10ms
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
12
Δω(15Ν)=25ppmΔν~1500Hzτspect~0,7ms
B0=14,1T
Δω(1Η)=3,5ppmΔν~2000Hzτspect~0,5ms
B0=14,1T
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
13
Δω(15Ν)=1,5ppmΔν~100Hzτspect~10ms
B0=14,1T
Δω(1Η)=0,5ppmΔν~300Hzτspect~3ms
B0=14,1T
Same nucleus ; two different conformations
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
14
Δω(15Ν)=1,5ppmΔν~100Hzτspect~10ms
B0=14,1T
Δω(1Η)=0,5ppmΔν~300Hzτspect~3ms
B0=14,1T
Same nucleus ; two different conformations
€
Δν Hz( ) =Δω ppm( )
2π∗ 10−6 ∗ γB0
€
τspect =1
Δν Hz( )
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
15
• Dipolar interaction between spins magnetic moment
E/ℏ ≤ 104-‐105 Hz
• Chemical shift anisotropy
E/ℏ ≤ 104-‐105 Hz
• Quadrupolar interaction spin/electric Sield
I > 1/2
E/ℏ ~ 2.105 Hz (2H) ; E/ℏ ~ 3.106 Hz (14N)
• Unpaired electron
€
ˆ H ISDD = −
µ0γ IγS
4πrIS3
3cos2 θIS −12
⎛
⎝ ⎜
⎞
⎠ ⎟ 3ˆ I z ˆ S z − ˆ I .ˆ S ( )
€
ˆ H ICSA = −γS
c|| − c⊥3
B0 3cos2 θ −1( ) ˆ I z
€
ˆ H IQ ≅ω I
Q 3ˆ I z2 − ˆ I .ˆ I ( ) ; ω IQ = 3eQI4 I 2I −1( ) Vzz
I θ( )
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
16
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
• Averaging of the secular interactions by the motions
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
17
• Averaging of the secular interactions by the motions
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
18
Averaging of the chemical shift anisotropy
(31P)
Burnell et al., Biochim. Biophys. Acta 603, 63 (1980)
• Averaging of the secular interactions by the motions
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
2. «Spectral» or «chemical shift» timescale
ü Largest observable difference of resonance frequency (spectral width)
ü Smallest observable difference of resonance frequency (resolution)
ü Motions slower than Δν have no effect on the appearance of the spectra.
ü Δν depends on the magnetic Sield strength B0, the type of the spins.
ü Different nuclei or same nuclei in different environments
ü Δν depends on the nature of the interactions between the spins.
19
• Using motions to average interactions
Davis, Auger & Hodges Biophysical Journal 69:1917-1932 (1995)Gross et al., J. Magn. Res. 106, 187-190 (1995)Carlotti, Aussenac & Dufourc Biochim. Biophys. Acta. 1564:156-164 (2002)
Glycine powder
Static
1H (kHz)
012
012
048
048
1H (kHz)
Lipids + water
Magic angle Rotation
ωr=5000Hz
�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡
⌧spect =1
⇡�⌫(Hz)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Internal motions
Types of motions
Folding ; order ↔ disorderEffect of m
otions on
spins interaction
s ms nsµs ps
Macroscopic diffusion
Magnetization exchange
Diffusionexperiments
Conformational exchange
Interactions
Catalytic processesRegulation, signalization
Molecular rotations
Molecular vibrations
Lineshape modiEications
Averaging of spectral
components
Spin relaxation
T1, T2 1/Δω 1/ω0
Signal relaxation frequencies differencesnutation
frequencies
NMR parameters
Functions
Dipolar Couplings averaging
How motions are « visible » in NMR ?
averaging of non-‐secular interactions
20
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
3. Larmor timescale
ü Resonance frequencies of the spin (i.e. difference of energy between the different observable states of the spin system)
ü Motions in this timescale have no effect on the appearance of the spectra.
ü Motions in this timescale are responsible for the efficiency of the relaxation processes.
ü The relationship between «motions» and relaxation rate constants is indirect.
21
ββ
αα
βα
αβ
ωI
ωI
ωS
ωS
ωI -ωS
ωI+ωS
B0=14,1T|ωI|=2π.600MHz ; τL(I)=265ps
|ωS|=2π.60MHz ; τL(S)=2,65ns
|!0| = |�B0|
⌧Larmor
=1
!0
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
3. Larmor timescale
ü Resonance frequencies of the spin (i.e. difference of energy between the different observable states of the spin system)
ü Motions in this timescale have no effect on the appearance of the spectra.
ü Motions in this timescale are responsible for the efficiency of the relaxation processes.
ü The relationship between «motions» and relaxation rate constants is indirect.
22
ex
ey
ezB0+Blocal(t)
Blocal(t)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013NMR timescales
3. Larmor timescale
ü Resonance frequencies of the spin (i.e. difference of energy between the different observable states of the spin system)
ü Motions in this timescale have no effect on the appearance of the spectra.
ü Motions in this timescale are responsible for the efficiency of the relaxation processes.
ü The relationship between «motions» and relaxation rate constants is indirect.
23
τc(s)
1 H r
elax
ation
times
(s)
0,01
0,1
1
10
100
T1
T2400,600,800MHz
400,600,800MHz
10-12 10-11 10-10 10-9 10-8
15N r
elax
ation
times
(s)
B0=14.1Teslas
0,01
0,1
1
10
100
10-12 10-11 10-10 10-9 10-8
T1
T2
τc(s)
|!0| = |�B0|
⌧Larmor
=1
!0
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Internal motions
Types of motions
Folding ; order ↔ disorderEffect of m
otions on
spins interaction
s ms nsµs ps
Macroscopic diffusion
Magnetization exchange
Diffusionexperiments
Conformational exchange
Interactions
Catalytic processesRegulation, signalization
Molecular rotations
Molecular vibrations
Lineshape modiEications
Averaging of spectral
components
Spin relaxation
T1, T2 1/Δω 1/ω0
Signal relaxation frequencies differencesnutation
frequencies
NMR parameters
Functions
Dipolar Couplings averaging
How motions are « visible » in NMR ?
averaging of non-‐secular interactions
24
k1A � B�A k�1 �B
A
BEa
ωAωB
ωA ωB
ωA
ωB
Kd = k1/k�1 = pB/pAkex = 1/�ex = k1 + k�1 = k1/pB = k�1/pA
ωA ωB
Δω
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
ü Processes that are on the spectral timescale (μs-‐ms range) affect the appearance of the spectra.
ü These processes generally correspond to conformational/chemical exchange.
ü The effects on spectra depend on the relative values of kex (τex) Δω/2 (2/Δω). One talk of slow or fast exchange at the chemical shift timescale.
1/πΔν
Coalescence
« fast » exchange
« slow » exchange
µsms
25
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
ü Conformational exchange corresponds to a sudden precession frequency change, with a probability kex to switch from one state to another.
ü These frequency changes induce a dephasing of the transverse magnetization, that adds to the «natural» loss of coherence of the spins.
ü R2(app) = R2 + Rex
kex=1/τex=k1+k-1=k1/pB=k-1/pA
A Bk1
k-1ωA ωB
ωA ωB
20 moléculesin state A
Total transverse magnetization for a large number of spins initially in the same environment.
Δν=1000Hz kex=500Hz pA=pB
Characterization of dynamic processes in the spectral timescaleconformational exchange
26
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Motional narrowing(Rétrécissement des raies par l’échange)
When kex increases from 0 to Δω, the loss of coherence becomes faster, resonances broaden and get closer, until one very enlarged resonance remains observable: this is the « coalescence » point.
Characterization of dynamic processes in the spectral timescaleconformational exchange
27
0,2
1,0
6,3
k (kHz)
0,5
k (kHz)
1,0
2,5
3,5
6,3
ν
-3kHz 3kHz
t
Δν=1000Hz
Case of a symetrical two-‐site exchange
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Motional narrowing(Rétrécissement des raies par l’échange)
When kex becomes greater than Δω, increasing kex induces a narrowing of the mean resonance, down to the «natural» linewidth.
One then observe only one apparent resonance, which does not correspond to the true resonance frequencies of the spins in the various sites.
Characterization of dynamic processes in the spectral timescaleconformational exchange
Δν=1000Hz
Case of a symetrical two-‐site exchange
6,3
20
50
10
k (kHz)
20
30
50
6,3
ν
-3kHz 3kHz
t
k (kHz)
28
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
k1A � B�A k�1 �B
AB
Ea
ωA ωB
ωA ωB
ωA
ωB
Kd = k1/k�1 = pB/pAkex = 1/�ex = k1 + k�1 = k1/pB = k�1/pA
ωA ωB
Δω
➫ Additional transverse magnetization dephazing ➫ Apparent increase of transverse relaxation
➫ R2app = R2 + Rex➫ Increased linewidths
➫ Apparent resonances frequencies
➫ Perturbation of spectra
that depends on the relative values of kex and Δω
29
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
Exchange regime
☞ “Fast” exchange : Δω ≪ kex➫ One peak at ω = pAωA + pBωB ➫ Rex ∝ B02
☞ “intermediate” exchange : Δω ≃ kex➫ One or several broadened peaks➫ Rex ∝ B0
☞ “slow” exchange : Δω≫ kex ➫ Several peaks➫ Rex independent of B0➫ Rex(B)→kBA=pAkex ; Rex(A)→kAB=pBkex
ν(Hz)
0
0.51.5
5.0
50
0.05
kex/∆ν
500
pA=pBR2A=R2B=5s-1
∆ν=200Hz
30
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
ν(Hz)
0
0.51.5
5.0
50
0.05
kex/∆ν
500
pA=pBR2A=R2B=5s-1
∆ν=200Hz
Exchange regime
☞ “Fast” exchange : Δω ≪ kex➫ One peak at ω = pAωA + pBωB ➫ Rex ∝ B02
☞ Number of states in exchange ?
☞ “intermediate” exchange : Δω ≃ kex➫ One or several broadened peaks➫ Rex ∝ B0
☞ “slow” exchange : Δω≫ kex ➫ Several peaks➫ Rex independent of B0➫ Rex(B)→kBA=pAkex ; Rex(A)→kAB=pBkex
31
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
Exchange regimeCase of unequal populations
☞ “Fast” exchange : Δω ≪ kex➫ One peak at ω = pAωA + pBωB ➫ Rex ∝ B02
☞ “intermediate” exchange : Δω ≃ kex➫ One or several broadened peaks➫ Rex ∝ B0
☞ “slow” exchange : Δω≫ kex ➫ Several peaks➫ Rex independent of B0➫ Rex(B)→kBA=pAkex ; Rex(A)→kAB=pBkex
pA ≫ pB ⇒ Rex(B) ≫ Rex(A)
☞ The minor peak can be undetectable even in a slow exchange regime.
-150 -100 -50 0 50 100 150
pA=0.8 ; pB=0.2R2A=R2B=5s-1
∆ν=200Hz
0
0.5
1.5
5.0
50
0.05
kex/∆ν
500
ν(Hz)
pAνA+pBνB
32
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
Unequal populationsAn «extreme» case: exchange of labile protons with the solvent
pA=105 ; R2A=1s-1, R2B=10s-1
Only the amide proton magnetization is depicted
kex = 0, 10, 100, 400, 1000, 5000 s-1
τex = inf, 100, 10, 2.5, 1, 0.2 mskex /Δν = 0 , 0.005, 0.05, 0.2, 0.5, 2.5
0
0.1
0.2
0.3
0.4
0.5
-1000 0 1000 2000
ν(HN) ν(H20)
1 10 102 103 104
kex (s-1)
I(H
N)/
I0(H
N)
ν(H
N)-ν 0
(HN
)
0
0.2
0.4
0.6
0.8
10
100
200
300
400
500
33
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
k1A � B�A k�1 �B
AB
Ea
ωA ωB
ωA ωB
ωAωB
ms μs1/πΔν
coalescence
“fast” intermediate
exchange
lineshape analysis
longitudinalmagnetization
exchange
R2app
R1ρapp
“slow” intermediate
exchange
R2app = R2 + Rex
thermodynamic parameters
Keq
= k1/k�1 = pB/pA
�G = �RT lnK
kinetic parameters
kex
= 1/⌧ex
= k1 + k�1 = k1/pB = k�1/pA
kex
(T ) = k0 exp (�Ea/kT )
34
d
dt�Mz(t) = (�R+K)�Mz(t)
d
dt
��!M+(t) = (i⌦�R+K)
��!M+(t)
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
€
M t( ) =
M 1 t( ) M 2 t( )
… M n t( )
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
General equation of exchange k1A � B�A k�1 �B
€
R −K =ρA −k1 −k−1
−k1 ρB −k−1
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
€
ddt
ΔMA+ t( )
ΔMB+ t( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=−iΩA −R2A
0 − pBkex pAkex
pBkex −iΩB −R2B0 − pAkex
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
MA+ t( )
MB+ t( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
MA+ t( )
MB+ t( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=aAA(t) aAB(t)
aBA(t) aBB(t)
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
MA+ 0( )
MB+ 0( )
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
☞ In the general case, the expression of aij coefSicients is “complicated”
two-‐site exchange
35
aAA(t) =1
2
(1 +
i!
�)e�(⇢+k��)t + (1� i!
�)e�(⇢+k+�)t
�
aBB(t) =1
2
(1� i!
�)e�(⇢+k��)t + (1 +
i!
�)e�(⇢+k+�)t
�
aAB(t) = aBA(t) =k
2�
he�(⇢+k��)t � e�(⇢+k+�)t
i
� =�k2 � !2
�1/2
M+1 (t) = M+A (t) +M
+B (t) = M
+(0)e�⇢t
k1A � B�A k�1 �B
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
The case of a two-‐site symmetrical exchange
“Slow” exchange limit
”Fast” exchange limit
-150 -100 -50 0 50 100 150
k/Δν
0
0.05
0.51.52.552550
500
k1=k-‐1ωA=-‐ωB=ωρΑ=ρΒ=ρ aAA(t) = aBB(t) =
1
2
h1 + 2e(�2kt)
ie�⇢t
aAB(t) = aBA(t) =1
2
h1� 2e(�2kt)
ie�⇢t
aAA(t) = e�(⇢+k�i!)t
aBB(t) = e�(⇢+k+i!)t
aAB(t) = aBA(t) ! 0
Two resonances at -‐ω and +ωlinewidth (ρ+ω)/π
One resonance at (ωA+ωB)/2Linewidth ρ/π
36
k1A � B�A k�1 �B
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
vs
AB
Ea
ωA ωB
thermodynamic parameters
Keq
= k1/k�1 = pB/pA
�G = �RT lnK
kinetic parameters
kex
= 1/⌧ex
= k1 + k�1 = k1/pB = k�1/pA
kex
(T ) = k0 exp (�Ea/kT )
ωAωB
kex = kon[P ] + koff
kex = kon[L] + koff
Kd =[P ][L][PL]
=koffkon
konP + L � PL
�Pf koff �Pb�Lf �Lb
ωf
ωb
Intramolecular Intermolecular processes
37
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
ωA ωBkex = kon[P ] + koff
kex = kon[L] + koff
Kd =[P ][L][PL]
=koffkon
konP + L � PL
�Pf koff �Pb�Lf �Lb
ωf
ωb
Intermolecular processesü The e x change r e g ime depends on t h e
concentration of the free speciesü Usually, it is considered that kon is controlled by the
diffusion and that the exchange regime is controlled by koff
ü This in general allows one to correlate ‘slow’ exchange regime with high afSinities (Kd < 10μM), and ‘fast’ exchange regime with low afSinities (Kd > 100μM)
This is not always true, especially when large conformational changes occur upon binding. Association rate constants can thus vary a lot and low afSinities can be associated with slow exchange regime. This is often the case for interactions involving peptides or intrinsically disordered proteins.
Beware of multi-‐sites binding: strong afSinities can then be associated with fast exchange regime.
!
38
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleconformational exchange
ωA ωBkex = kon[P ] + koff
kex = kon[L] + koff
Kd =[P ][L][PL]
=koffkon
konP + L � PL
�Pf koff �Pb�Lf �Lb
ωf
ωb
Intermolecular processes
Py tr
act “
stre
ngth
”
33 µM
18 µM
16 µM
7.1 µM
1.3 µM
ITCKd
Exchange regime & RNA binding affinity
U2AF65 conformation depends on Py tract strength
(Michael Sattlerpersonal communication)
39
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
1/πΔν
Coalescence “Fast”
intermediate exchange
“Slow” intermediate exchange
R2apparent = R2 + Rex
k1A � B�A k�1 �B
A
BEa
ωAωB
ωA ωB
ωA
ωB
ωA ωB
Δω
µsms
longitudinal magnetization exchange
Lineshapes analysis
δapp = pAδA +pBδB
weighted average values
thermodynamic parametersKd = k1/k�1 = pB/pA
�G = �RT lnK
kinetic parameterskex = 1/�ex = k1 + k�1 = k1/pB = k�1/pA
kex(T ) = k0 exp (�Ea/kT )
40
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
1/πΔν
Coalescence “Fast”
intermediate exchange
“Slow” intermediate exchange
µsms
€
Δω( )2−4k2
R2+k
€
S ω( ) = 12 1 +kD
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ L ω , R2 + k −D( )
+12
1 − kD
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ L ω , R2 + k + D( )
D = k2 − Δω2( )
2
€
k T( ) = k0 exp − EaRBT⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ Ross & True, J. Am. Chem. Soc. 106, 2451 (1984)
Lineshape perturbation
Changing the exchange regime by varying the temperature
{13C} N,N’ diméthylformamide gaz, B0=4,7T
Ea=90,1kJ.mol-‐1 ; k0=1,16.104Hz
41
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
Fig. 2. Observed (a) and calculated (b) 188.3MHz 19F-n.m.r. spectra of 3',5'-difluoromethotrexate bound to L. casei dihydrofolate reductase.The sample contained slightly less than one molar equivalent of difluoromethotrexate, so that all the ligand is bound to the enzyme. The small sharp resonance marked X represents a small amount ( Ea = 50,6kJ/mol
Lineshape perturbation
42
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
�obs = ff�f + fb�b
ms μs1/πΔν
“fast”“slow”Kd =
[P ][L][PL]
=koffkon
konP + L � PL
�Pf koff �Pb�Lf �Lb
kex = kA + kB
kB = koffkA = kon[P ]{kon[L]}
kex > ��
fbff
= VboundVbound+Vfree
�|(��)2 � 4k2|
R2 + k
Robs2 (P ) = R2(P ) + kon[L] = R2(P ) + kofffPLfP
Robs2 (L) = R2(L) + kon[P ] = R2(L) + kofffPLfL
Robs2 (PL) = R2(PL) + koff
Robs = ffRf + fbRb
R2,obs = pELR2,EL + pLR2,L
+pELp2L(�EL � �L)2
koff
43
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
fbff
= VboundVbound+Vfree
Robs2 (P ) = R2(P ) + kon[L] = R2(P ) + kofffPLfP
Robs2 (L) = R2(L) + kon[P ] = R2(L) + kofffPLfL
Robs2 (PL) = R2(PL) + koff
Temperature variation‣R2 decreases when T increases‣kex increases when T increases
Changing protein or ligand concentration.
Robs2 = R2,T exp(Ew/RbT ) + kT exp(�Ek/RbT )
Fig. 5. The temperature dependence of the linewidth of the N1-proton resonance of trimethoprim in its comp lex w i th d ihyd ro fo l a t e reductase, presented as an Arrhenius plot [ln (pi x linewidth) vs 1/T]. The points are experimental data, the error bars indicating the estimated maximum uncertainty in linewidth (+ 2 Hz). The line was calculated using the following parameters: W0 (273K)=32Hz; Ea(W0)=13 kJ/mole; kex(313K)=160s-1; Ea(exchange)=75 kJ/mole, where W0 is the natural linewidth (in the absence of exchange) and Ea denotes activation energy. (from Bevan et al. (1985), Eur. Biophys. J. 11, 211.
1/πΔν
Coalescence
µsms
“Slow” intermediate exchange
“Fast” intermediate exchange
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
44
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
1/πΔν
Coalescence
µsms
“Slow” intermediate exchange
“Fast” intermediate exchange
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
Titration : disappearance/appearance of peaks
Cerdan et al., FEBS 408, 235 (1997)
DNA
Protein
DNA + Protein ⇆ DNA-‐protein
45
�MAz(t)�MBz(t)
�=
aAA(t) aAB(t)aBA(t) aBB(t)
� �MAz(0)�MBz(0)
�
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
1/πΔν
Coalescence
µsms
“Slow” intermediate exchange
“Fast” intermediate exchange
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
t1 τmt2
[DNA]/[Prot]=2
➫ kex=14±2Hzτex=74±7ms
DNA + Protein ⇆ DNA-‐protein
V
b
(⌧m
) = xb
e
��⌧mcosh(D⌧
m
)� �D
sinh(D⌧m
)
�
V
f
(⌧m
) = xf
e
��⌧mcosh(D⌧
m
) +�
D
sinh(D⌧m
)
�
V
crosspeak
(⌧m
) = xb
x
f
k
D
e
��⌧msinh(D⌧
m
)
� =1
2(R1b +R1f ); � =
1
2(R1b �R1f );D =
q�
2 + xb
x
f
k
2
Cerdan et al., FEBS 408, 235 (1997)
46
�MAz(t)�MBz(t)
�=
aAA(t) aAB(t)aBA(t) aBB(t)
� �MAz(0)�MBz(0)
�
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
1/πΔν
Coalescence
µsms
“Slow” intermediate exchange
“Fast” intermediate exchange
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
V
b
(⌧m
) = xb
e
��⌧mcosh(D⌧
m
)� �D
sinh(D⌧m
)
�
V
f
(⌧m
) = xf
e
��⌧mcosh(D⌧
m
) +�
D
sinh(D⌧m
)
�
V
crosspeak
(⌧m
) = xb
x
f
k
D
e
��⌧msinh(D⌧
m
)
� =1
2(R1b +R1f ); � =
1
2(R1b �R1f );D =
q�
2 + xb
x
f
k
2
011
mt1 t2
1 3 52 4 6
( /2)1
( /2)2
( /2)3
rec, dig
2
1
Cross Peaks
DiagonalPeaks
Cross Peaks
2D longitudinal magnetization exchange experiment (EXSY)
2
1
k 2m
mincrease
mincrease
diagonal peak amplitudes
cross peak amplitudes
/s m /s m0 0.1 0.2 0.3 0.4 0.5
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5
0.2
0.4
0.6
0.8
1
47
Titration : Peaks shiftskon
P + N ⌦ PN�Nf
koff
�Nb
�obs
= ff
�f
+ fb
�b
Kd
=koff
kon
=[P ][N ]
[PN ]
Kd =(P0 � fbN0)(N0 � fbN0)
fbN0
fb =��
�b � �f
fb[P ]
= � 1Kd
fb +1
Kd
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
1/πΔν
Coalescence
µsms
“Slow” intermediate exchange
“Fast” intermediate exchange
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
Scatchard plot
Kd~170µM
Baleja et al. Biochemistry 33, 3071 (1994) 48
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleThe case of one observed peak
Case of unequal populations
☞ “Fast” exchange : Δω ≪ kex➫ One peak at ω = pAωA + pBωB ➫ Rex ∝ B02
☞ Number of states in exchange ?☞ “intermediate” exchange : Δω ≃ kex
➫ One or several broadened peaks➫ Rex ∝ B0
☞ “slow” exchange : Δω≫ kex ➫ Several peaks➫ Rex independent of B0➫ Rex(B)→kBA=pAkex ; Rex(A)→kAB=pBkex
pA ≫ pB ⇒ Rex(B) ≫ Rex(A)
☞ The minor peak can be undetectable even in a slow exchange regime.
-150 -100 -50 0 50 100 150
pA=0.8 ; pB=0.2R2A=R2B=5s-1
∆ν=200Hz
0
0.5
1.5
5.0
50
0.05
kex/∆ν500
ν(Hz)
pAνA+pBνB
Experiments to characterize thermodynamics, kinetics and structural parameters of the exchange when only one peaks is observed ?
49
thermodynamic parameters
Kd
= k1/k�1 = pB/pA
�G = �RT lnK
kinetic parameters
kex
= 1/⌧ex
= k1 + k�1 = k1/pB = k�1/pA
kex
(T ) = k0 exp (�Ea/kT )
structural parameters
�! ; !A
; !B
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
50
k1A � B�A k�1 �B
ABEa
ωA ωB
ωA ωB
ωAωB
ωA ωB
Δω
ΔG
ms μs1/πΔν
fast regimeslow regime
longitudinal magne7za7on
exchange Lineshape
relaxa7on dispersionR2app=R2+Rex
averaged values
One observed resonance, several states in exchange
15N
1H
Carine van Heijenoort -‐ RMN et mouvements moléculaires
Cargèse 2013
Characterization of dynamic processes in the spectral timescaleSome methods to characterize conformational exchange
51
ms μs1/πΔν
régime rapiderégime lent
échange d’aimanta7on longitudinale forme de raie
Dispersion de relaxa7onR2eff=R2+Rex
valeurs moyennes
Une seule résonance apparente, plusieurs états en échange
τcp/2τcp/2 τcp/2τcp/2
y x
2IzSy
2n2n
Tr/2
Sx
Tr/2
€
14JNH
€
14JNH
Sx2IzSy
RelaxationτCPMG
15Ndecoupling
Suppression ofDD-CSA cross-cor.
t1
1H
2HzNy Nx exp[-R2eff(15N)Tr]
acquisition
Tr
Reff2 = R02 + Rex(pB , kex, �!, ⌧CP , 2Necho)
200 400 600 800 1000 1200
5
10
15
20
25
30
35
νCPMG
(Hz)
R2
eff
(Hz)
residue 39 ; T=298K, pH=3.5, 700MHz
es
tim
ate
d R
ex
=1/2τCP
relaxation dispersion experiment
Afternoon Practical course
✓An example of relaxation dispersion experiment‣sequence analysis
✓Data processing‣nmrPipe / macro
✓Extraction of exchange parameters from the NMR data‣Fitting procedure writen in the software “octave”
✓Some typical examples