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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