0')&(1%(( 2.-$,#/(3#4,0'-(5,&)#-, - Freethcgerm.free.fr/IMG/pdf/CvanHeijenoort_dynamique1.pdf · Carinevan’Heijenoort’-’RMN’et’mouvements’moléculaires Cargèse2013 Wüthrich,

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

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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 ?

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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


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


5


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


6


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


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)



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


Cargèse 2013

Internal motions

Types of motions


otions on

spins interaction

s ms nsµs ps





Interactions


Molecular rotations




components

Spin relaxation

T1, T2 1/Δω 1/ω0


frequencies

NMR parameters

Functions



averaging of non-‐secular interactions

10



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)










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)










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)




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( )










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)










16

�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡

⌧spect =1

⇡�⌫(Hz)

• Averaging of the secular interactions by the motions










17


�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡

⌧spect =1

⇡�⌫(Hz)










18

Averaging of the chemical shift anisotropy

(31P)

Burnell et al., Biochim. Biophys. Acta 603, 63 (1980)


�⌫(Hz) = ��(ppm) ⇤ 10�6 ⇤ �B02⇡

⌧spect =1

⇡�⌫(Hz)










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)


Cargèse 2013

Internal motions

Types of motions


otions on

spins interaction

s ms nsµs ps





Interactions


Molecular rotations




components

Spin relaxation

T1, T2 1/Δω 1/ω0


frequencies

NMR parameters

Functions




20



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



3. Larmor timescale





22

ex

ey

ezB0+Blocal(t)

Blocal(t)



3. Larmor timescale





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


Cargèse 2013

Internal motions

Types of motions


otions on

spins interaction

s ms nsµs ps





Interactions


Molecular rotations




components

Spin relaxation

T1, T2 1/Δω 1/ω0


frequencies

NMR parameters

Functions




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


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


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


26


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.


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


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.


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


Cargèse 2013


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


Cargèse 2013


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


Cargèse 2013


ν(Hz)

0

0.51.5

5.0

50

0.05

kex/∆ν

500

pA=pBR2A=R2B=5s-1

∆ν=200Hz

Exchange regime


☞ Number of states in exchange ?



31


Cargèse 2013


Exchange regimeCase of unequal populations




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


Cargèse 2013


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


Cargèse 2013


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)


Cargèse 2013


€

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


Cargèse 2013


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


Cargèse 2013


vs

AB

Ea

ωA ωB


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


Cargèse 2013


ωA ωBkex = kon[P ] + koff

kex = kon[L] + koff

Kd =[P ][L][PL]

=koffkon

konP + L � PL


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


Cargèse 2013


ωA ωBkex = kon[P ] + koff

kex = kon[L] + koff

Kd =[P ][L][PL]

=koffkon

konP + L � PL


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


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


Cargèse 2013


1/πΔν

Coalescence “Fast”

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


Cargèse 2013


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


Cargèse 2013


�obs = ff�f + fb�b

ms μs1/πΔν

“fast”“slow”Kd =

[P ][L][PL]

=koffkon

konP + L � PL


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


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


“Fast” intermediate exchange


44


Cargèse 2013

1/πΔν

Coalescence

µsms




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)

�


Cargèse 2013

1/πΔν

Coalescence

µsms




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)

�


Cargèse 2013

1/πΔν

Coalescence

µsms




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


Cargèse 2013

1/πΔν

Coalescence

µsms




Scatchard plot

Kd~170µM

Baleja et al. Biochemistry 33, 3071 (1994) 48


Cargèse 2013

Characterization of dynamic processes in the spectral timescaleThe case of one observed peak

Case of unequal populations


☞ Number of states in exchange ?☞ “intermediate” exchange : Δω ≃ kex

➫ One or several broadened peaks➫ Rex ∝ B0


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


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


Cargèse 2013


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


Cargèse 2013


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

Documents

0')&(1%(( 2.-$,#/(3#4,0'-(5,&)#-, - Freethcgerm.free.fr/IMG/pdf/CvanHeijenoort_dynamique1.pdf · Carinevan’Heijenoort’-’RMN’et’mouvements’moléculaires Cargèse2013 Wüthrich,