Heteronuclear Relaxation and
Macromolecular Structure and Dynamics
Outline:
Note: refer to lecture on “Relaxation & nOe”
•
Information Available from Relaxation Measurements
•
Relaxation Mechanisms
•
Relaxation Rates
•
Experimental Methods
•
Data Analysis
•
Case Studies
1.
Fushman, D., R. Xu, et al. (1999). "Direct determination of
changes of interdomain orientation on ligation: Use of the
orientational dependence of N

15 NMR relaxation in Abl
SH(32)."
Biochemistry
38
(32): 10225

10230.
1.
Eisenmesser, E. Z., D. A. Bosco, et al. (2002). "Enzyme
dynamics during catalysis."
Science
295
(5559): 1520

1523.
1.
Lee, A. L., S. A. Kinnear, et al. (2000). "Redistribution and
loss of side chain entropy upon formation of a calmodulin

peptide complex."
Nature Structural Biology
7
(1): 72

77.
1.
Ishima, R., D. I. Freedberg, et al. (1999). "Flap opening and
dimer

interface flexibility in the free and inhibitor

bound
HIV protease, and their implications for function."
Structure
with Folding & Design
7
(9): 1047

55.
•
References
Biomolecules are not static:
•
rotational diffusion (
t
c
)
•
translational diffusion (D)
•
internal dynamics of backbone and sidechains (
t
i
)
•
degree of order for backbone and sidechains (S
2
)
•
conformational exchange (R
ex
)
•
interactions with other molecules (k
on
,k
off
)
Biomolecules are often not globular spheres:
•
anisotropy (D
xx
,D
yy
,D
zz
)
Structure/Dynamics
Function
t
c
S
2
t
i
R
ex
D
k
on
k
off
NMR Relaxation and Dynamics
NMR relaxation measurements provide information on structure and
dynamics at a wide range of time scales that is site specific:
•
Dynamics on Different Time Scales
time scale
example
experiment type
ns
–
ps
bond librations
lab frame relaxation
reorientation of protein
T
1
, T
2
motions of protein main chain
side chain rotations
(case study #2)
us
–
ms
rapid conformational exchange
lineshape analysis
(case study #4)
rotating frame relax.
T
1
r
ms
–
s
interconversion of discrete
magnetization exch.
conformations
> s
protein folding
exchange rates
opening of 2
o
structures
(H/D exchange)
•
Structural Information from Relaxation
•
anisotropy of overall shape (case study #1)
•
distance information from cross

correlation relaxation
•
Thermodynamics from Relaxation
•
relationship to entropy (case study #3)
Relaxation
Bloch equations
–
introduce relaxation to account for return of
magnetization to equilibrium state:
excite
relax
treat relaxation as a first order process:
d
M
/dt =
g
M
x
B
–
R
(
M

M
o
)
where
T
1
(longitudinal or spin

lattice relaxation time) is the time constant used to
describe rate at which M
z
component of magnetization returns to
equilibrium (the Boltzman distribution) after perturbation.
T
2
(transverse or spin

spin relaxation time) is the time constant used to
describe rate at which M
xy
component of magnetization returns to
equilibrium (completely dephased, no coherence) after perturbation.
R =
1/T
2
0 0
0 1/T
2
0
0 0 1/T
1
so far, all we have is a time constant; is it possible to get a “picture” of
what is causing relaxation?
•
consider spontaneous emission of photon:
transition probability
a
1/
l
3
= 10

20
s

1
for NMR
•
consider stimulated emission:
the excited state couples to the EMF inducing transitions
–
this
phenomenon is observed in optical spectroscopy (eg. lasers) but its
effect is negligible in RF fields.
•
in a historic paper, Bloembergen, Purcell and Pound (Phys. Rev.
73
, 679

712 (1948)) found that relaxation is related to
molecular motion
(NMR
relaxation time varied as a function of viscosity or temperature). They
postulated that relaxation is caused by fluctuating fields caused by
molecular motion.
RF
photon
•
relaxation is dependent on motion of molecule
•
Zeeman interaction is independent of molecular motion therefore “local
fields” exist that are orientation dependent and couple the magnetic
moment with the external environment (the “lattice”)
•
time dependence of interaction determines how efficiently the moment
couples to the lattice
•
it is the fluctuating “local fields” that induce transitions between energy
levels of spins:
RF
source of local fields?
timescale of fluctuation?
Relaxation Mechanisms
The relaxation of a nuclear spin is governed by the fluctuations of local
fields that result when molecules reorient in a strong external magnetic
field. Although a variety of interactions exist that can give rise to a
fluctuating local field, the dominant sources of local fields experienced by
15
N and
13
C nuclei in biomolecules are dipole

dipole interactions and
chemical shift anisotropy:
•
Magnetic Dipole

Dipole Interaction

the dipolar interaction is a
through

space coupling between two nuclear spins:
q
I
S
r
IS
The local field experienced by spin I is:
H
loc
=
g
S
h/r
3
IS
((3cos
2
q
–
1)/2)
•
Chemical Shift Anisotropy

the CSA interaction is due to the distribution of
electrons surrounding the nucleus, and the local magnetic field generated by
these electrons as they precess under the influence of the applied magnetic
field. The effective field at the nucleus is:
H
loc
= H
o
(1

s
)
where H
o
is the strength of the applied static magnetic field and
s
is the
orientationally dependent component of the CSA tensor.
Expressions for Relaxation Rates
The relaxation rate constants for dipolar, CSA and quadrupolar
interactions are linear combinations of spectral density functions, J(
w
).
For example, one can derive the following equations for dipolar relaxation
of a heteronucleus (i.e.
15
N or
13
C) by a proton
R
1,N
= 1/T
1,N
= (d
2
/4)[J(
w
H

w
N
) + 3J(
w
N
) + 6J(
w
H
+
w
N
)]
R
2,N
= 1/T
2,N
= (d
2
/8)[4J(0) + J(
w
H

w
N
) + 3J(
w
N
) + 6J(
w
H
) + 6J(
w
H
+
w
N
)]
NOE
15N{1H}
= 1 + (d
2
/4)(
g
H
/
g
N
) [6J(
w
H
+
w
N
)

J(
w
H

w
N
)] x T
1,N
where d = (
g
H
g
N
(h/8
p
)/r
HN
3
)
The J(
w
) terms are “spectral density” terms that tell us what frequency of
motions are going to contribute to relaxation. They have the form
J(
w
) =
t
c
/(1+
w
2
t
c
2
)
and allow the motional characteristics of the system (the correlation time
t
c
) to be expressed in terms of the “power” available for relaxation at
frequency
w
:
J(
w
⤠
=
w
=
=
10
6
=
10
7
=
10
8
=
10
9
=
10
10
=
t
c
=10

7
t
c
=10

8
t
c
=10

9
Measurement of Relaxation Rates
•
spin lattice relaxation is measured using an inversion recovery
sequence:
180
t
I
=
t
I
t
= I
o
(1

2exp(

t
/T
1
))
•
spin

spin relaxation is measured using a “spin echo” sequence
(removes effect of field inhomogeneity):
90
t
t
ㄸ1
=
I
t
= I
o
exp(

t
/T
2
)
I
t
Measurement of Relaxation Rates
The inversion

recovery sequence and spin

echo sequence can be
incorporated into a 2D
1
H

15
N HSQC pulse sequence in order to measure
15
N T
1
and T
2
for each crosspeak in the HSQC:
Experimental techniques for
15
N (a) R
1
, (b) R
2
, and (c) {
1
H}
15
N NOE spin
relaxation measurements using two

dimensional, proton

detected pulse
sequences. R
1
and R
2
intensity decay curves are recorded by varying the
relaxation period T in a series of two dimensional experiments. The NOE
is measured by recording one spectrum with saturation of
1
H
magnetization and one spectrum without saturation.
Data Analysis
Analysis of the relaxation data provides dynamical parameters (amplitude
and timescale of motion) for each bond vector under study and parameters
related to the overall shape of the molecule (rotational diffusion tensor):
Dynamical parameters in proteins.
(a) Overall rotational diffusion of the molecule is represented using an axially
symmetric diffusion tensor for an ellipsoid of revolution. The diffusion
constants are D

for diffusion around the symmetry axis of the tensor and
D
perp.
for diffusion around the two orthogonal axes. For isotropic rotational
diffusion, D

= D
perp.
. The equilibrium position of the
i
th N

H bond vector is
located at an angle
q
i
with respect to the symmetry axis of the diffusion tensor.
Picosecond

nanosecond dynamics of the bond vector are depicted as
stochastic motions within a cone with amplitude characterized by S
2
and time
scale characterized by
t
e
.
(b) The value of S
2
is graphed as a function of (

)
q
o
calculated using Equation
22 for diffusion within a cone or (



)
s
f
calculated using Equation 23 with
q
= 70.5
°
for the GAF (Gaussian Axial Fluctuation) model.
from: Palmer, A. G. (2001). “NMR probes of molecular dynamics: Overview
and comparison with other techniques.”
Annual Review of Biophysics and
Biomolecular Structure
30
: 129.
“Model Free” analysis of relaxation based on Lipari, G. and A. Szabo
“Model

Free Approach to the Interpretation of Nuclear Magnetic
Resonance Relaxation in Macromolecules. 1. Theory and Range of
Validity.”
Journal of the American Chemical Society
104
: 4546 (1982).
Internal dynamics characterized by:
•
internal correlation time,
t
e
•
spatial restriction of motion of bond vector, S
2
S
2
= 1 highly restricted
S
2
= 0 no restriction
•
R
ex
, exchange contribution to T
2
The spectral density terms in the relaxation equations are modified with
terms representing internal dynamics and spatial restriction of bond
vector:
J(
w
) ~ { S
2
t
c
/(1+
w
2
t
c
2
) + (1

S
2
)
t
/(1+
w
2
t
2
) }
where
t
=
t
e
t
c
/(
t
e
+
t
c
).
Analysis of relaxation data using software package (eg. Model

Free or
DASHA) allows the dynamical parameters to be calculated:
Data Analysis
t
c
R
ex
,S
2
,t
e
15
N
1
H
measure:
15
N T
1
15
N T
2
15
N{
1
H} NOE
calculate
relaxation
data for a
given
t
c
recalculate
by varying
values of S
2
,
t
e
and R
ex
Compare
measured
vs. calc.
value
Defining Regions of Structure using NMR Relaxation Measurements
Case study #1
Red indicates chemical shift
changes observed upon ligand
binding
Case study #2
Case study #3
goal:
measure effects of inhibitor
binding on conformational fluctuations
of HIV protease on
m
s

ms timescale.
sample:
0.3mM protease dimer +
DMP323 inhibitor
experiments:
1H and 15N T2 and T1
r
at 500MHz
Case study #4
result:
inhibitor binding enhances
dyanamics on the ms timescale of
the
b

sheet interface, a region that
stabilizes the dimeric structure of
the protease (residues 95

98).
Relaxation behavior of the flap
(residues 48

55) indicates a
transition from a slow dynamic
equilibrium between semi

open
conformations on the 100
m
s
timescale to a closed conformation
upon inhibitor binding.
References
Palmer, A. G. (2001). “NMR probes of molecular dynamics: Overview and
comparison with other techniques.”
Annual Review of Biophysics and
Biomolecular Structure
30
: 129.
Palmer, A. G., C. D. Kroenke and J. P. Loria (2001). “Nuclear magnetic resonance
methods for quantifying microsecond

to

millisecond motions in biological
macromolecules.”
Nuclear Magnetic Resonance of Biological Macromolecules, Pt
B
339
: 204.
Brutscher, B. (2000). “Principles and applications of cross

correlated relaxation in
biomolecules.”
Concepts in Magnetic Resonance
12
(4): 207.
Engelke, J. and H. Ruterjans (1999). Recent Developments in Studying the
Dynamics of Protein Structures from 15N and 13C Relaxation Time
Measurements.
Biological Magnetic Resonance
. N. R. Krishna and L. J. Berliner.
New York, Kluwer Academic/ Plenum Publishers.
17:
357

418.
Fischer, M. W. F., A. Majumdar and E. R. P. Zuiderweg (1998). “Protein NMR
relaxation: theory, applications and outlook.”
Progress in Nuclear Magnetic
Resonance Spectroscopy
33
(4): 207

272.
Daragan, V. A. and K. H. Mayo (1997). “Motional Model Analyses of Protein and
Peptide Dynamics Using 13C and 15N NMR Relaxation.”
Progress in Nuclear
Magnetic Resonance Spectroscopy
31
: 63

105.
Cavanagh, J., W. J. Fairbrother, A. G. Palmer and N. J. Skelton (1996).
Protein
NMR Spectroscopy: Principles and Practice
, Academic Press.
Chapter 5 “Relaxation and Dynamic Processes”
Nicholson, L. K., L. E. Kay and D. A. Torchia (1996). Protein Dynamics as
Studied by Solution NMR Techniques.
NMR Spectroscopy and Its Application to
Biomedical Research
. S. K. Sarkar.
Peng, J. W. and G. Wagner (1994). “Investigation of protein motions via relaxation
measurements.”
Methods in Enzymology
239
: 563

96.
Wagner, G., S. Hyberts and J. W. Peng (1993). Study of Protein Dynamics by
NMR.
NMR of Proteins
. G. M. Clore and A. M. Gronenborn, CRC Press
:
220

257.
Mini Reviews:
Ishima, R. and D. A. Torchia (2000). “Protein dynamics from NMR.”
Nature
Structural Biology
7
(9): 740

743.
Kay, L. E. (1998). “Protein dynamics from NMR.”
Nature Structural Biology
5
:
513

7.
Palmer, A. G., 3rd (1997). “Probing molecular motion by NMR.”
Current Opinion
in Structural Biology
7
(5): 732

7.
Case Studies:
Fushman, D., R. Xu, et al. (1999). "Direct determination of changes of interdomain
orientation on ligation: Use of the orientational dependence of N

15 NMR
relaxation in Abl SH(32)."
Biochemistry
38
(32): 10225

10230.
Eisenmesser, E. Z., D. A. Bosco, et al. (2002). "Enzyme dynamics during
catalysis."
Science
295
(5559): 1520

1523.
Lee, A. L., S. A. Kinnear, et al. (2000). "Redistribution and loss of side chain
entropy upon formation of a calmodulin

peptide complex."
Nature Structural
Biology
7
(1): 72

77.
Ishima, R., D. I. Freedberg, et al. (1999). "Flap opening and dimer

interface
flexibility in the free and inhibitor

bound HIV protease, and their implications for
function."
Structure with Folding & Design
7
(9): 1047

55.
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