Deuxième école RMN du GERM Cargèse 2008 Analysis of micro-millisecond timescale motions of macromolecules in liquid state by Nuclear Magnetic Resonance
Carine van Heijenoort ICSN laboratoire de chimie et biologie structurale
[email protected] 0169823794
A view of proteins multiple states « lock & key » induced fit
3D structure α↔β
conformational switch
order → disorder
virus/pathogen penetration membrane insertion Nucléosome activation
Folding
Sequence
“Non folding” flexible linkers display of sites entropic bristles, springs and clocks
flexible ensemble disorder → order
Deuxième école RMN du GERM, Cargèse 2008
molecular recognition virus/phages assembly stepping motors Dunker et al., Journal of Molecular Graphics and Modelling 19, 26–59, 2001 Dobson, C., Nature 426, 18-25, 2003
Techniques to monitor protein dynamics time scales static disorder, crystal contacts, ...
X-ray cristallography
B factors
X-Ray, neutron scattering
size/shape modifications
Doniach, Chem. Rev. 2001, 101 ; Zacai, timescales (ps-ns) for 1H positions science 2000, 288.
Fluorescence
ensemble / single molecule
Weiss, Nat. Struct. Biol. 2000, 7 ; Yang, Science 2003, 302 ; Haustein, Curr. Opin. cellular context Struct. Biol. 2004, 14.
Mass Spectroscopy (HX MS) radical footprinting Wales, Mass. Spectrom. Rev. 2006, 25 ; Busenlehner, Arch. Biochem. Biophys. 2005, 433. Guan, Trends. Biochem. Sci. 2005, 30.
probes
large moldecular assemblies
Mössbauer, Raman, 2D infrared spectroscopy Forcefields ... Short timescales ...
Molecular dynamics ➫ 10-12↔ 105 s ➫ Site-specific information ➫ multiple atomic probes 1H, 2H, 15
NMR Boehr, Chem. Rev. 2006, 106,3055. Palmer, Chem. Rev. 2004, 104, 3623.
N,
13
C,
31
P, ...
➫ Simultaneous monitoring of
probes ➫ kinetic & termodynamic profile of dynamic processes
➫ ➫ ➫ ➫
isotope labeling quantities size limitation complexity of the method ?
Deuxième école RMN du GERM, Cargèse 2008
Bibliography Theoretical NMR basis : A. Carrington et A. Mc Lachlan Introduction to Magnetic Resonance with appplications to chemistry and chemical physics, Harper International , 1967 C. Slichter Principle of Magnetic Resonance , 3 ème ed, 1990, Springer Verlag A. Abragam Principles of Magnetic Resonance, Oxford University Press, 1961 Ernst, Bodenhausen, Wokaun Principle of Nuclear Magnetic Resonance in one and two dimensions, Oxford Science 1987 M. Goldman* Quantum description of high resolution NMR in liquids. Oxford 1990 D. Canet* La RMN : concepts, méthodes et applications (2ème ed.), UniverSciences, Dunod 2002 M. H. Levitt*** Spin dynamics. Basic of Nuclear Magnetic Resonance (686 pages) J.Wiley 2001 John Cavanagh, Wayne J. Fairbrother, Arthur G., III Palmer*** Protein NMR Spectroscopy: Principles And Practice (912 pages) Academic Press; Édition : 2nd (novembre 2006) Experimental aspects, applications J. K. M. Sanders, B. K. Hunter* Modern NMR Spectroscopy: A Guide for Chemists 2nd Edition, Oxford 2003 D. Shaw Fourier transform NMR spectroscopy Elsevier 1984 K. Wüthrich* NMR of proteins and nucleic acids. Wiley interscience 1986 Deuxième école RMN du GERM, Cargèse 2008
How motions are visible in NMR ?
Molecular motions influence NMR parameters Time scale of motions compared to NMR characteristic timescales ➫ Three characteristic NMR timescales Return of the spin system to equilibrium : T1 / NMR signal lifetime : T2 NMR experiment spins system perturbation Liquid state: T1 ~ 100ms s ; T2 ~ 10ms s ➫ Minimal frequencies of motions that can be characterized during a single NMR experiment.
Spectral timescale : τ=1/Δν spectral features : chemical shifts range, couplings, … Hz
kHz
➫ Averaging of interactions by motions faster than the spectral dispersion due to these interactions. ➫ Perturbation of spectral features by motions in the same range than the spectral timescale.
Larmor timescale : ω0 precession frequency of spins in a magnetic field (B0=ω0/γ) ➫ transitions efficiency between spin states is determined by molecular fluctuations (spectral densities fo motions) at these frequencies.
Deuxième école RMN du GERM, Cargèse 2008
How motions are visible in NMR ? Molecular motions influence NMR parameters Time scale of motions compared to NMR characteristic timescales ➫ Three characteristic NMR timescales Return of the spin system to equilibrium : T1 / NMR signal lifetime : T2 Spectral timescale : τ=1/Δν Larmor timescale : ω0
s
ms
T1, T2 Signal lifetime
µs
1/2pΔω
frequencies differences
ns
1/ω0
nutation frequencies
Deuxième école RMN du GERM, Cargèse 2008
ps
fs
Motions & function & NMR disorder Interactions catalytic processes Regulation, signalization
Functions
Folding ; order
Effect of motions on spins interaction
Types of motions
s
NMR parameters
ms
µs
Internal motions
ns
Macroscopic diffusion
ps
fs
Molecular vibrations
Conformational exchange
Molecular rotations
Diffusion experiments Magnetization exchange Lineshape modifications
Spin relaxation
Averaging of spectral components
T1, T2 Signal lifetime
1/Δω frequencies differences
1/ω0 nutation frequencies
Deuxième école RMN du GERM, Cargèse 2008
Secular interactions averaging
NMR timescales -1- Longitudinal relaxation time T1
Dynamical processes slower than T1
✓T1 characterizes the time for a spin
Real time folding Transient / out of equilibrium experiments Protein - ligand interactions H-N D-N exchange rate constants
system to reach equilibrium.
✓Defines interscan delay ✓Motions slower than T1 cannot be
characterize by a single spectra.
✓NB. T1 depends on magnetic field
B0, nature of spins, molecule size, local flexibility, temperature, etc.
Wüthrich, « NMR of proteins and nucleic acids », Wiley Interscience, 1986
Deuxième école RMN du GERM, Cargèse 2008
NMR timescales -1- Longitudinal relaxation time T1
Dynamical processes slower than T1
✓T1 characterizes the time for a spin
Real time folding kinetic / out of equilibrium experiments Protein - ligand interactions H-N D-N exchange rate constants
system to reach equilibrium.
✓Defines interscan delay
holoP 1 + apoP 2
✓Motions slower than T1 cannot be
characterize by a single spectra.
✓NB. T1 depends on magnetic field
B0, nature of spins, molecule size, local flexibility, temperature, etc.
Deuxième école RMN du GERM, Cargèse 2008
! apoP 1 + holoP 2
NMR timescales -1- Longitudinal relaxation time T1
magnetic field, molecular size and correlation time dependance of T1 and T2
✓T1 characterizes the time for a spin system to reach equilibrium.
✓Defines interscan delay
Relaxation des azotes 15 amides à 500MHz (−) et 800MHz (−−)
✓Motions slower than T1 cannot be
2
10
characterize by a single spectra. B0, nature of spins, molecule size, local flexibility, temperature, etc.
T1,2 (s)
✓NB. T1 depends on magnetic field
1
T1
10
0
10
!1
T2
10
!2
10
!3
10
!2
10
Deuxième école RMN du GERM, Cargèse 2008
!1
10
!c (ns)
0
10
1
10
NMR timescales -2- Spectral or “chemical shift” timescale
✓Defined
by the observed spectral width
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
B0=14.1T
Δν=7800Hz τspect~100µs
Δω=13ppm ∗ γB 0
B0=18.8T
Δν=10400Hz
B0=14.1T
Δν=120Hz
1 Δν(Hz )
Motions slower than Δν have no effect on spectral€feature
✓Δν depends on magnetic field and on spins nature
τspect~5-10ms
Δω=0.2ppm B0=18.8T
Deuxième école RMN du GERM, Cargèse 2008
Δν=160Hz
NMR timescales -2- Spectral or “chemical shift” timescale
✓Defined
by the observed spectral width
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
∗ γB 0
1 Δν(Hz )
15
Δω( N)=25ppm
B0=14.1T
τspect~0.7ms
Motions slower than Δν have no effect on spectral€feature
✓Δν depends on magnetic field and on spins nature
Δω(1H)=3.5ppm
Deuxième école RMN du GERM, Cargèse 2008
Δν=1500Hz
B0=14.1T
Δν=2000Hz τspect~0.5ms
NMR timescales -2- Spectral or “chemical shift” timescale Same nucleus ; two different conformations
✓Defined
by the observed spectral width
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
∗ γB 0
1 Δν(Hz )
Motions slower than Δν have no effect on spectral€feature
Δω(15N)=1.5ppm
B0=14.1T
τspect~10ms
✓Δν depends on magnetic field and on spins nature Δω(1H)=0.5ppm
Deuxième école RMN du GERM, Cargèse 2008
Δν=100Hz
B0=14.1T
Δν=300Hz τspect~3ms
NMR timescales -2- Spectral or “chemical shift” timescale Averaging of secular interactions by motions
✓Defined
✓ Dipolar interaction between spins magnetic moments.
by the observed spectral width
µ γ γ h 3cos 2 θ IS −1 ˆ ˆ ˆ ˆ DD Hˆ IS = − 0 I 3S 3 I z S z − I.S 2 4 πrIS
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm
€
✓
−6
✓ Chemical shift anisotropy
c −c Hˆ ICSA = −γ S || ⊥ B0 3cos 2 θ −1 Iˆz 3
∗ γB 0
(
1 Δν(Hz )
)
E h < 104 −10 5 Hz €
Motions slower than Δν have no effect on spectral€feature
depends on the nature of the interactions between spins
✓ Interaction spin quadrupole/electric field
(
)
Hˆ IQ ≅ ω IQ 3 Iˆz2 − Iˆ.Iˆ ; ω IQ =
✓Δν depends on magnetic field and on spins nature ✓Δν
)
E h < 104 −10 5 Hz €
2π
τ spect =
(
I>
1 2
3eQI VzzI (θ ) 4 I (2I −1)
E h ≈ 2.10 5 Hz 2H and 3.10 6 14 N
€ ✓ Unpaired electron
Deuxième école RMN du GERM, Cargèse 2008
( )
( )
NMR timescales -2- Spectral or “chemical shift” timescale Averaging of secular interactions by motions
✓Defined
by the observed spectral width
soluble protein
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
∗ γB 0
1 Δν(Hz )
membrane protein
Motions slower than Δν have no effect on spectral€feature
✓Δν depends on magnetic field and on spins nature ✓Δν
depends on the nature of the interactions between spins
Deuxième école RMN du GERM, Cargèse 2008
NMR timescales -2- Spectral or “chemical shift” timescale Averaging of secular interactions by motions
✓Defined
by the observed spectral width
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
∗ γB 0
1 Δν(Hz )
Motions slower than Δν have no effect on spectral€feature
✓Δν depends on magnetic field and on spins nature ✓Δν
depends on the nature of the interactions between spins
Deuxième école RMN du GERM, Cargèse 2008
NMR timescales -2- Spectral or “chemical shift” timescale Averaging of secular interactions by motions
✓Defined
by the observed spectral width
✓More
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
Averaging of chemical shift anisotropy (31P)
∗ γB 0
1 Δν(Hz )
Motions slower than Δν have no effect on spectral€feature
✓Δν depends on magnetic field and on spins nature ✓Δν
depends on the nature of the interactions between spins
Burnell et al., Biochim. Biophys. Acta 603, 63 (1980)
Deuxième école RMN du GERM, Cargèse 2008
NMR timescales -2- Spectral or “chemical shift” timescale motions generated to average interactions by the observed spectral width
precisely by the difference of resonance frequency between two spins from two different nuclei or from the same nuclei in two different states.
( )
Δν Hz =
( ) ∗ 10
Δω ppm 2π
τ spect = €
✓
−6
∗ γB 0
1 Δν(Hz )
Motions slower than Δν have no effect on spectral€feature
Poudre de Glycine
Lipides + eau
Statique
✓More
Rotation à l’angle magique ωr=5000Hz
✓Defined
8
4
0
8
4
0
1H
(kHz)
2
1
0
2
1
0
1H
(kHz)
✓Δν depends on magnetic field and on spins nature ✓Δν
depends on the nature of the interactions between spins 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)
Deuxième école RMN du GERM, Cargèse 2008
NMR timescales -3- The “Larmor” timescale
✓Defined
by the resonance frequencies of spins
✓i.e.
the energy difference between spins states levels
ω0 τLarmor ✓
= −γB0 1 1 = = |ω0 | 2πν0
ββ ωI
in these timescales are responsible for the efficiency of spins relaxation processes
βα
ωI -ωS
αβ
ωI
ωS αα
Motions in these timescale have no direct effect on spectra.
✓Motions
ωI+ωS
ωS
B0=14,1T
✓ The
relationship between motions and relaxation rate/time constants is not simple
Deuxième école RMN du GERM, Cargèse 2008
|ωI|=2π.600MHz ; τL(I)=265ps |ωS|=2π.60MHz ; τL(S)=2,65ns
the energy difference between spins states levels
ω0 τLarmor ✓
= −γB0 1 1 = = |ω0 | 2πν0
Motions in these timescale have no direct effect on spectra.
✓Motions
in these timescales are responsible for the efficiency of spins relaxation processes
✓ The
relationship between motions and relaxation rate/time constants is not simple
1H
✓i.e.
relaxation times (s)
by the resonance frequencies of spins
15N
✓Defined
relaxation times (s)
NMR timescales -3- The “Larmor” timescale
100 10
T1
1 0,1 0,01 10-12
400,600,800MHz
T2
400,600,800MHz
10-11
10-10
τc(s)
10-9
10-8
B0=14.1Teslas
100 10
T1
1 0,1 0,01 10-12
T2 10-11
Deuxième école RMN du GERM, Cargèse 2008
10-10
τc(s)
10-9
10-8
Motions & function & NMR disorder Interactions catalytic processes Regulation, signalization
Functions
Folding ; order
Effect of motions on spins interaction
Types of motions
s
NMR parameters
ms
µs
Internal motions
ns
Macroscopic diffusion
ps
fs
Molecular vibrations
Conformational exchange
Molecular rotations
Diffusion experiments Magnetization exchange Lineshape modifications
Spin relaxation
Averaging of spectral components
T1, T2 Signal lifetime
1/Δω frequencies differences
1/ω0 nutation frequencies
Deuxième école RMN du GERM, Cargèse 2008
Secular interactions averaging
Analysis of µs-ms motions by NMR Conformational exchange ✓ ✓
Motions in the µs-ms range induce spectral modifications. These motions usually correspond conformational exchange conformationnel, chemical exchange.
to or
✓ Their effect on spectra depends of relative values
of kex (τex) and Δω (1/Δω). They are called either slow, intermediate or fast exchange processes at the chemical shift timescale.
A ωA
ωA
k1 ! k−1
B ωB
= k1 /k−1 = pB /pA
Kd
= 1/τex = k1 + k−1 = k1 /pB = k−1 /pA
kex
ωB
ωA ms “slow” exchange
1/Δω c o a l e s c e n c e
μs “fast” exchange
ωB
Ea
ωA
B ωB
A
Δω ωA Deuxième école RMN du GERM, Cargèse 2008
ωB
Analysis of µs-ms motions by NMR Conformational exchange ✓
Conformational exchange is characterized by a probability kex to switch from one state to the other, associated with an instantaneous change of precession frequency.
✓ These sudden frequency changes induce a dephasing of the transverse magnetization, which is added to the natural loss of coherence of spins.
A ωA Kd kex
k1 ! k−1
20 molecules in state A
B ωB
Δν=1000Hz kex=500Hz pA=pB
= k1 /k−1 = pB /pA
= 1/τex = k1 + k−1 = k1 /pB = k−1 /pA
Δν=1000Hz kex=500Hz pA=pB
Δω ωA
ωB
Resulting transverse magnetization for a great number of spins initially in the same state (effect of exchange only)
Deuxième école RMN du GERM, Cargèse 2008
Analysis of µs-ms motions by NMR Conformational exchange ➫ Modification of spectral features
A ωA
ωA
B ωB
= 1/τex = k1 + k−1 = k1 /pB = k−1 /pA
kex
ωB
ωA
20 molecules in state A
Ea
ωA
Δω/2π = 1000Hz kex=500Hz pA=pB
ωB
= k1 /k−1 = pB /pA
Kd ➫ depends of the relative values of kex and Δω
k1 ! k−1
B ωB
A
➫ Additional dephasing of the magnetization ➫ Apparent enhanced transverse relaxation ➫ R2app = R2 + Rex
Δω
➫ Larger linewidths
ωA Deuxième école RMN du GERM, Cargèse 2008
ωB
Analysis of µs-ms motions by NMR Conformational exchange ➫ Modification of spectral features A ωA
k1 ! k−1
Kd
B ωB
kex
= k1 /k−1 = pB /pA
= 1/τex = k1 + k−1 = k1 /pB = k−1 /pA
Δω ➫ depends of the relative values of kex and Δω kex/Δω = 1/10π
ωA
ωB kex(kHz)
Δω/2π = 1000Hz pA=pB
-3kHz
0
kex(kHz)
kex/Δω
0.5
1/4π =0.08
1.0
1/2π =0.16
2.5
5/4π =0.40
3.5
7/4π =0.56
6.3
6.3/2π =1.00
3kHz
Motional broadening
Motional narrowing Deuxième école RMN du GERM, Cargèse 2008
kex/Δω
6.3
6.3/2π =1.0
10
5/π =1.6
20
10/π =3.2
30
15/π =4.8
50
25/π =8,0
Analysis of µs-ms motions by NMR Conformational exchange The case of a two state exchange General equation for a system undergoing conformational exchange r M (t) r 1 v M (t) M(t) = 2 r… M t n ( )
r d r Δ M z (t) = (−R + K )Δ M z (t) dt r d r M + (t) = (iΩ −R +K )M + (t) dt
A ωA
k1 ! k−1
B ωB
ρ − k −k −1 A 1 R −K = −k ρB − k −1 1
+ 0 d ΔM A t −iΩA − R 2A − pBk ex €= dt ΔM+ t pBk ex B
() ()
M+ t A − p Ak ex M+B t
p Ak ex −iΩB − R 02B
+ + M A t a AA (t) a AB (t)M A 0 + = + MB t a BA (t) a BB (t)MB 0
() ()
€
€
() ()
In a general case, the expression of the matrix coefficients is complicated
Deuxième école RMN du GERM, Cargèse 2008
() ()
Analysis of µs-ms motions by NMR Conformational exchange + + M A t a AA (t) a AB (t)M A 0 + = + MB t a BA (t) a BB (t)MB 0
General equation
() ()
for a two state exchange
a AA (t) =
A ωA
k1 ! k−1
B ωB
Δω ωA
1 1− 2
() ()
−iωA + iωB + ρA − ρB + k1 − k−1 −iωA€+ iωB + ρA − ρB + k1 − k−1 exp −λ − t + 1 + exp −λ + t λ + − λ− λ + − λ−
(
( )
)
(
( )
)
€
−iωA + iωB + ρA − ρB + k1 − k−1 −iωA + iωB + ρA − ρB + k1 − k−1 a BB (t) = 1 1 + exp −λ t + 1 − exp −λ t − + 2 λ + − λ− λ + − λ−
€
a AB (t) =
€
a BA (t) =
(
ωB €
k−1
( )
)
(
(
)
exp −λ − t − exp −λ + t λ + − λ−
( )
k1
(
)
exp −λ − t − exp −λ + t λ + − λ−
λ± =
( )
1 2
( )
−iωA − iωB + ρA + ρB + k1 + k−1 ± −iωA + iωB + ρA − ρB + k1 − k−1
(
€
Deuxième école RMN du GERM, Cargèse 2008
) (
)
2
+ 4k1k−1
1
2
)
Analysis of µs-ms motions by NMR Conformational exchange
An easier case The two state symmetrical exchange €
A ωA
€
k ! B k ωB
iω iω a AA (t) = 1 1 + exp − ρ + k − Δ t + 1 − exp − ρ + k + Δ t 2 Δ Δ iω iω a BB (t) = 1 1 − exp − ρ + k − Δ t + 1 + exp − ρ + k + Δ t 2 Δ Δ k a AB (t) = a BA (t) = exp − ρ + k − Δ t − exp − ρ + k + Δ t 2Δ
{(
)}
{(
)}
{(
)}
{(
)}
{(
Δ = k2 − ω2
1
)}
{(
)}
2
€ €
Fast exchange kex >> ∆ω
a AB (t) = a BA
1
k/Δν
[1 + exp(−2kt)] exp(−ρt) (t) = [1 − exp(−2kt)] exp(−ρt) 2
a AA (t) = a AA (t) =
2 1
500 50
( )
+ M + (t) = MA (t) + MB+ (t) = M + (0) exp −ρt
-ω
0
€
ω
€
kex ~ ∆ω
() ()
Slow exchange kex 3.5 Localized conformational exchange ☞ Local Unfolding ?
Rex(Hz)
30
25
pH=5.8 pH=4.5 pH=3.5
20
15
10
5
0 0
10
20
30
40
sequence
Deuxième école RMN du GERM, Cargèse 2008
50
60
70
80
The method : relaxation dispersion NMR experiments First analysis : qualitative analysis of relaxation dispersion curves An example : variation of exchange contribution as a function of pH pH < 3 Global conformational exchange
(A)
(B)
40
pH > 3.5 Localized conformational exchange
30
I34
N42
35 25
30
(F)
pH=2.5
pH=2.7
Reff (Hz)
pH=3.0
pH=2.7
20
20
35
pH=3.5
pH=3.0
30
2
Reff (Hz) 2
pH=2.5 25
15
15
I34 N42 L68 H71
25 10
5 0
(C)
200
400
600
800
!CPMG (Hz)
1000
1200
5 0
1400
200
400
600
!
CPMG
(D)
35
800
(Hz)
1000
1400
20
15
40
L68
10
H71 35
30
pH=2.5
Reff (Hz) 2
pH=2.7 pH=3.0
20
5
30
pH=2.5
25
Reff (Hz) 2
1200
Reff (Hz) 2
10
pH=2.7
25
0 0
pH=3.0 20
200
400
600
800
!CPMG (Hz)
15
15 10
5 0
10
200
400
600
800
!CPMG (Hz)
1000
1200
1400
5 0
200
400
600
800
!CPMG (Hz)
1000
1200
1400
☞ Physical parameters of these exchange processes ? Deuxième école RMN du GERM, Cargèse 2008
1000
1200
1400
The method : relaxation dispersion NMR experiments Quantitative evaluation of physical parameters of exchange from the relaxation dispersion data
The analysis depends of the timescale of the process
☞
Fast exchange : ∆ω > kex ➫ ➫ ➫ ➫ ➫
#
50
# 2 φ + 2∆ω 2 φ + 2∆ω5.0 D+ = 0.5 1 + " ; D− = 0.5 −1 + " 2 φ2 + ζ 2 φ2 + ζ1.5 $ $ " " 0.5 τCP τ CP η+ = " φ + φ2 + ζ 2 ; η− = " −φ + φ2 + ζ 2 (2) (2)
ζ = 2∆ω(pB − pA )kex
(A)
Rex →kBA=pAkex ; Rex →kAB=pBkex
Rex(B) ≫ Rex(A)
☞
!
!
2 φ = [(pB − pA )kex ]2 − ∆ω 2 + 4pA pB kex
Two peaks Rex independant of B0 Tollinger equation ka, ∆ω, Rinf (B)
R2app = R2inf + 0.5kex − νCP acosh (D+ cosh η+ − D− cos 500 η− )
Detection of minor populations from the
0.05
!
R2app = R2inf + ka 1 − -150
-100
-50
broadenning of the major state resonance : use of Carver-Richards equation Deuxième école RMN du GERM, Cargèse 2008
0
sin ∆ω.τCP ∆ω.τCP 50
ν(Hz) pAνA+pBνB
100
" 150
0
The method : relaxation dispersion NMR experiments Quantitative evaluation of physical parameters of exchange from the relaxation dispersion data The analysis depends of the timescale of the process ☞ Fast exchange : ∆ω > kex ➫ Two peaks ! " sin ∆ω.τCP ➫ Rex independant of B0 R2app = R2inf + ka 1 − ∆ω.τCP ➫ Tollinger equation ➫ ka, ∆ω, Rinf ➫ Rex(B)→kBA=pAkex ; Rex(A)→kAB=pBkex Rex(B) ≫ Rex(A)
0.05
0 -150
-100
-50
0
50
100
ν(Hz) ☞ Detection of minor populations from the broadenning of the major state resonance : use of Carver-Richards equation Deuxième école RMN du GERM, Cargèse 2008
pAνA+pBνB
150
The method : relaxation dispersion NMR experiments Quantitative evaluation of physical parameters of exchange from the relaxation dispersion data pH < 3 Global conformational exchange (A)
(B)
40
pH > 3.5 Localized conformational exchange
30
I34
N42
35 25
30 pH=2.5
pH=2.7
Reff (Hz)
pH=3.0
(F)
pH=2.7
20
pH=3.0
35
pH=3.5
2
Reff (Hz) 2
pH=2.5 25
20
15
30
10
25
I34 N42 L68 H71
15
5 0
(C)
200
400
600
800
!CPMG (Hz)
1000
1200
5 0
1400
200
400
600
!
CPMG
(D)
35
800
(Hz)
1000
1400
40
L68
H71
pH=2.5
Reff (Hz) 2
pH=2.7 pH=3.0
20
15
30
pH=2.5
25
20
10
35
30
Reff (Hz) 2
1200
Reff (Hz) 2
10
5
pH=2.7
25
pH=3.0
0 0
20
15
200
400
600
800
!CPMG (Hz)
1000
1200
1400
15 10
5 0
10
200
☞
400
600
800
!CPMG (Hz)
1000
1200
1400
5 0
200
400
600
800
!CPMG (Hz)
1000
1200
Intermediate/Slow exchange regime pB ≪ pA
1400
☞
Deuxième école RMN du GERM, Cargèse 2008
Fast exchange regime pB ≃ pA
The method : relaxation dispersion NMR experiments relaxation dispersion experiments for the analysis of “invisible” states k1
A B k ωA -1 ωB
ωA
Ea
ωB
B
A Δω
ωA
ωB
pH < 3
☞
Individual fits
➫ Carver statistically favored ➫ τex : few ms ➫ pB
