Relaxation in NMR

Source of relaxation in NMR is fluctuation of nuclear

interaction with respect to the external magnetic field.

Dipolar interaction:!

N

HB0

CSA interaction:!

!11

!22

!33

B0

Other interactions: Quadrupolar, Paramagnetic, etc.

Relaxation in NMR

Where does the fluctuation come from?

•Random events:

! Rotational diffusion

! Translational diffusion

! Vibrational/Librational motions

! Conformational sampling/variability

•Non-random events:

! Magic angle spinning

! B0 quenching

!

Correlation function

The behavior of the random fluctuation can be described by a

correlation function.

What is a correlation function:

1. Correlation function of 2 variables is the expected value of

their product. This could be evaluated as a function of space or

time.

In liquid NMR random processes are typically defined as a

function of time.

2. It is a measure of how quickly the two variables change as a

function of time or space.

Correlation function

How does it relate to NMR relaxation?

Remember the master equation:

Taking the ensemble average, first term vanishes (H1(t) is random):

!

d ˜ " (t)

dt= #i ˜ H

i(t), ˜ " (0)[ ] # ˜ H

1(t), ˜ H

1(t'), ˜ " (t ')[ ][ ]

0

t

$ dt'

!

d ˜ " (t)

dt= # ˜ H

1(t), ˜ H

1(t'), ˜ " (t')[ ][ ]

0

t

$ dt '

Expand H1(t) in spin and time dependent operators:

!

H1(t) = V"Fa

"

# (t) = V"+F"*

"

# (t)

spin random function of time

Correlation function

How does it relate to NMR relaxation?

Substituting back:

!

d ˜ " (t)

dt= # ˜ V $ (t), ˜ V %

+(t'), ˜ " (t ')[ ][ ]F$ (t)F%

*(t ')dt '

0

t

&$,%

'

!

F" (t)F#*(t') =G"# ( t $ t' ) Correlation function

It describes the random function of time that contributes to changes in the spin density

Correlation function

How quickly the NH dipole changes as a function of time with

respect to the external magnetic field.

!

C(t) = Dq02( )*

"LF 0( )( )•Dq02( )"LF t( )( )

LF=Laboratory frame, B0 is static

In liquid it reduces to:

!

C(t) =1

5P2

r µ LF0( )•

r µ LF

t( )( )N

H

!

r µ

!

C(t) = d" d# p($,#,")P2

0

%

&0

2%

&0

2%

& (cos$0t )d$

p(",#,$) is the normalized distribution function of µ

Relaxation in NMR

In NMR observed quantities: T1, T2, and NOE are determined

by the Fourier transform of an appropriate time correlation function (the spectral density) evaluated at certain frequencies, thus certain magnetic fields.

C(t)

t

J(%)

%

F.T.

Relaxation in NMR

Motional model

Define correlationfunction

Calculate spectraldensity

Fit experimentaldata

Goodness of fit

Not accepta

ble

acceptable

Done

Approximate acorrelation function

Calculate spectraldensity

Fit experimentaldata

Pick models that can describe the fitted parameters

Lipari-Szabo Model

Free Approach

General approach

Lipari-Szabo Correlation functionConditions:

• Overall and internal motions are independent

• C(t) = Co(t) CI(t)

• CI(t) = &P2(µ(0) • µ(t))'! in the molecular frame.

• CI(0) = &P2(µ(0) • µ(0))' = 1

• CI(() = S2

• Area under the approximated correlation function is the same as the real one:

!

CI

A(t)" S2( )

0

#

$ dt = CI(t)" S2( )

0

#

$ dt!

CI

A(t) = S

2+ (1" S

2)e"t /#

e

!

"e(1# S2 ) = C

I(t)# S2( )

0

$

% dt

Lipari G & Szabo A, JACS (1982) 104:4546

N

H

!

r µ

N

H

N

H

N

H

Diso

Isotropic Diffusion

Lipari-Szabo Correlation functions

! CI(t) = & P2(µ(0) . µ(t)) ' = S2 + (1-S2) exp(-t/)e)

Isotropic Diffusion

! Co(t) = exp(-6Dmt) = exp(-t/)c)

! C(t) = Co(t) CI(t) = [S2 + (1-S2) exp(-t/)e)] exp(-t/)c)

Anisotropic Diffusion (Axial Symmetry)

! C(t) = (1/5) *m exp(-t [6D+ + m2 (D//-D+)]) [dmo2(,)]2

! ! ! [S2 + (1-S2) exp (-t/)e)]

Woessner, D. E., J. Chem. Phys. (1962) 3:647-654

N

H

N

H

N

H

D//

D+

Non-isotropic Diffusion

Correlation functions

Anisotropic Diffusion (Fully asymmetric)

! ! C(t) = (1/5) *m Am exp(-t/)m)

A1 = 6 m2n2! ! ! A4 = d - e

A2 = 6 l2n2! ! ! A5 = d + e

A3 = 6 l2m2

d = [3(l4 + m4 + n4)-1]/2

e = [-x(3l4 + 6m2n2 -1) + -y(3m4 + 6l2n2 - 1) + -z(3n4+6l2m2-1)]/6

-i = (Di - D)/ (D2-L2)1/2

D = 1/3 (Dx + Dy + Dz)! ! L2 = 1/3(DxDy + DxDz + DyDz)

)1 = (4Dx+Dy+Dz)-1! ! ! )4 = [6(D + (D2-L2)1/2)]-1

)2 = (4Dy+Dx+Dz)-1! ! ! )5 = [6(D - (D2-L2)1/2)]-1

)3 = (4Dz + Dx + Dy)-1

l, m, n = directional cosines with

respect to diffusion axes x, y,

and z

Woessner, D. E., J. Chem. Phys. (1962) 3:647-654

Correlation functions

with Slow Internal Motion

Isotropic Diffusion

C(t) = [Sf2 Ss

2 + Sf2(1-Ss

2) exp(-t/)s) + (1-Sf2) exp(-t/)e)] exp(-t/)c)

! Clore et. al. JACS, 1990, 112, 4989-4991

Anisotropic Diffusion (Axial Symmetry)

C(t) = (1/5) *m exp(-t [6D+ + m2 (D//-D+)]) [dmo2(,)]2

! ! ! [Sf2 + Sf

2(1-Ss2) exp(-t/)s) + (1-Sf

2) exp (-t/)e)]

Spectral density:

! ! J(%) = -" #" C(t) e-i%t dt

Isotropic Diffusion:

! ! J(%) = [ Sf2 )c / (1+(%)c)

2) + (1-Sf2) )’ / (1+(%)c)

2 ]

! ! 1/)’ = 1/)c + 1/)e

Relaxation Rates:

1/T1 = d2 [ J(%H - %N) + 3 J(%N) + 6 J(%H + %N) ] + c2 J(%N)

1/T2 = 0.5 d2 [4 J(0) + J(%H - %N) + 3 J(%N) + 6 J(%H) + 6 J(%H + %N)] +

! (1/6) c2 [ 3 J(%N) + 4 J(0)]

NOE = 1 + (.H / .N ) d2 (6 J(%H + %N) - J(%H - %N)] T1

d2 = 0.1 [(.H .N h) / (2 / r3NH)]2

c2 = (2/15) [%N 2 (!// - !+)2]

From correlation function to relaxation rates

Protein Backbone Relaxation Analysis

Protocol

! Estimate the overall correlation time

" Eliminate residues experiencing large amplitude fast internal motions (NOE < 0.6)

" Eliminate residues undergoing conformational exchange (low T2

values)

! Choose the proper diffusion model

" Isotropic (Diso)

" Axially symmetric (D// & D+)

" Fully asymmetric (Dx, Dy, & Dz)

! For non-isotropic model diffusion tensor must be defined

" Protein structure

! Local order parameter and fast internal correlation times

" Residue specific fitting

N

H

N

H

N

H

Diso

Isotropic Diffusion

N

H

N

H

N

H

D//

D+

Non-isotropic Diffusion

Human GAIP

Diffusion Tensor Fitting

"(°) 0(°) $(°) )c (ns) D⁄⁄/D+ Dy/Dx 12

96.8 -31.0 11.80 1.69 2.07

96.5 -31.6 61.1 11.75 1.70 1.05 2.05

94.4 -37.3 11.72 1.47 4.54

94.5 -37.3 107.6 11.70 1.48 1.01 4.54

Inertia ratio (Iz/I+): 2.49 ! ! Estimated D⁄⁄ /D+: 1.89

Human GAIP

Aggregation State

Concentration (mM) )c (ns)! ! % dimer

0.8! ! 8.7! ! 46

0.2! ! 8.7! ! 34

Robs = x R()cmono) + (1-x) R()c

dimer)

MA CA

NC

PR

p6

MA CA

HIV-1 Lenti

HTLV-I Onco

Pol

PR Pol

Myr p1p2

Myr

Transport

M domain

Assembly

I domain

Budding

L domain

NC

NC

MHR

MHR ZF ZF

ZF ZF

Immature

Immature

Mature

Mature

Things to consider in a single field

experiments

• Sample aggregation

– Measure T2 at different sample concentrations

– Substantially high average order parameters

– Poor overall fit of the diffusion parameters

• Error estimates

– Possible use of reproducibility

– Goodness of the fit

• Sample stability

– Interleave and scramble the data whenever possible

Things to consider in multiple fields

experiments

• Temperature effect

– Use a single TSP sample to calibrate the temperature of the probe under

the exact experimental conditions

– Interleave and scramble the data whenever possible

– Heat compensation

• Error estimates

– Consistent procedure through out the different field strengths

– Possible use of reproducibility and signal to noise normalization

– Goodness of the fit

• Sample stability

– No accidental correlation in sample decay and field strengths

Human Ubiquitin

C

N

2

1

1

1.6

T1(sec), 600 MHz

T1(sec), 800 MHz

20 60Resid No.

0.2

0.6

0.1

0.6

T2(sec), 600 MHz

T2(sec), 800 MHz

20 60Resid. No.

Temperature dependent of Ubiquitin 13C! relaxation

15°C - 47°C

Typical decay curves for 13C! relaxation data

Typical decay curves for 13C! relaxation data

E16, T12, 600 MHz, various temperatures

1. Interaction tensors

cross-correlated relaxation:

C!(CSA)/C!- C3(DD)

cross relaxation: C!-C3C! auto-correlated relaxation

!

R = RCSA

+ RDD, C '"N

+ RDD, C '"C#

+ RDD, C '"H$

Assumptions:

Uniform CSA

Isotropic overall diffusion

one effective correlation time and order parameter

!

" xx = #74.7, " yy = #11.8, " zz = #86.5 ppm

$ = 38o

!

R1

CSA =1

T1

CSA= " I

2B0

2g2

CS(#)[ ]

!

R2

CSA=

1

T2

CSA=1

2" I2B0

2g2

CS(#) +

4

3g2

CS(0)

$

% & '

( )

!

R1

DD=1

T1

DD= " I

2" S2h2 1

3g2

D(# I $#S ) + g

2

D(# I ) + 2g

2

D(# I +#S )

%

& ' (

) *

!

R2

DD=1

T2

DD=1

2" I2" S2h2 4

3g2

D(0) +

1

3g2

D(# I $#S ) + g

2

D(# I ) + g

2

D(#S ) + 2g

2

D(# I +#S )

%

& ' (

) *

General NMR relaxation rate equations:

!

2R2" R

1=4

3# I2B0

2g0

CS(0) + {

4

3# I2# S2S(S +1)[

4

3g2

D(0) + 2g

2

D($s)]}

dipolar

%

g0

cs(0) =

3

10(2

3&)2(1+

'(2

3) ) S2

& =* zz "* xx +* yy

2=3

2* zz

'( =* yy "* xx

* zz

, 0 +'( <1

Isolation of CSA contribution from field

dependent 2R2-R1 values

!

g2D("s) =

3

10rIS#6 $ cS

2

1+" 2$ c2

Human Ubiquitin 13C’ (2R2-R1) rates

Typical 2R2-R1 values as a function of B2

K27

T9

G53

Comparison of effective order parameters

I44

F45

H68

Protons within 3Å distance

Correlation times .vs. temperatures from 15N and 13C! data sets

Effective Model-Free order parameters

Changes in average order parameter as a function of temperature