Heteronuclear Relaxation and

1

• 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

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NMR Relaxation and Dynamics NMR relaxation
measurements provide information on structure and
dynamics at a wide range of time scales that is
site specific
S2 ti Rex
D
kon
koff
tc

• Biomolecules are not static
• rotational diffusion (tc)
• translational diffusion (D)
• internal dynamics of backbone and sidechains
  (ti)
• degree of order for backbone and sidechains (S2)
• conformational exchange (Rex)
• interactions with other molecules (kon,koff)
• Biomolecules are often not globular spheres
• anisotropy (Dxx,Dyy,Dzz)
• Structure/Dynamics Function

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• Dynamics on Different Time Scales
• time scale example experiment type
• ns ps bond librations lab frame relaxation
• reorientation of protein T1, T2
• motions of protein main chain
• side chain rotations
• (case study 2)
• us ms rapid conformational exchange lineshape
  analysis
• (case study 4) rotating frame relax.

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Relaxation Bloch equations introduce
relaxation to account for return of magnetization
to equilibrium state
treat relaxation as a first order process dM/dt
gM x B R(M-Mo) where T1
(longitudinal or spin-lattice relaxation time) is
the time constant used to describe rate at which
Mz component of magnetization returns to
equilibrium (the Boltzman distribution) after
perturbation. T2 (transverse or spin-spin
relaxation time) is the time constant used to
describe rate at which Mxy component of
magnetization returns to equilibrium (completely
dephased, no coherence) after perturbation.
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• 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/l3 10-20 s-1 for NMR
  
• consider stimulated emission

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• 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

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• 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 15N and 13C 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

• The local field experienced by spin I is
• Hloc gSh/r3IS ((3cos2q 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
• Hloc Ho(1-s)
• where Ho is the strength of the applied static
  magnetic field and s is the orientationally
  dependent component of the CSA tensor.

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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. 15N or 13C)
by a proton R1,N 1/T1,N (d2/4)J(wH-wN)
3J(wN) 6J(wHwN) R2,N 1/T2,N (d2/8)4J(0)
J(wH-wN) 3J(wN) 6J(wH) 6J(wHwN) NOE15N
1H 1 (d2/4)(gH/gN) 6J(wHwN) - J(wH-wN) x
T1,N where d (gHgN(h/8p)/rHN3) 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)
tc/(1w2tc2) and allow the motional
characteristics of the system (the correlation
time tc) to be expressed in terms of the power
available for relaxation at frequency w
tc 10-7
J(w)
tc 10-8
tc 10-9
w
106
107
108
109
1010
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• Measurement of Relaxation Rates
• spin lattice relaxation is measured using an
  inversion recovery sequence

180
t
I
It Io(1-2exp(-t/T1))
t

• spin-spin relaxation is measured using a spin
  echo sequence (removes effect of field
  inhomogeneity)

90
t
t
180
I
It Ioexp(-t/T2)
t
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Measurement of Relaxation Rates The
inversion-recovery sequence and spin-echo
sequence can be incorporated into a 2D 1H-15N
HSQC pulse sequence in order to measure 15N T1
and T2 for each crosspeak in the HSQC
Experimental techniques for 15N (a) R1, (b) R2,
and (c) 1H15N NOE spin relaxation measurements
using two-dimensional, proton-detected pulse
sequences. R1 and R2 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 1H magnetization and one spectrum
without saturation.
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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 Dperp. for
diffusion around the two orthogonal axes. For
isotropic rotational diffusion, D Dperp.. The
equilibrium position of the ith N-H bond vector
is located at an angle qi 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 S2 and time scale
characterized by te. (b) The value of S2 is
graphed as a function of (-) qo calculated using
Equation 22 for diffusion within a cone or (- -
-) sf 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.
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Data Analysis

• 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, te
• spatial restriction of motion of bond vector, S2
• S2 1 highly restricted
• S2 0 no restriction
• Rex, exchange contribution to T2
• The spectral density terms in the relaxation
  equations are modified with terms representing
  internal dynamics and spatial restriction of bond
  vector
• J(w) S2tc/(1w2tc2) (1-S2)t/(1w2t2)
• where t tetc/(te tc).
• Analysis of relaxation data using software
  package (eg. Model-Free or DASHA) allows the
  dynamical parameters to be calculated

tc
measure 15N T1 15N T2 15N1H NOE
calculate relaxation data for a given tc
recalculate by varying values of S2, te and Rex
Compare measured vs. calc. value
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Defining Regions of Structure using NMR
Relaxation Measurements
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Case study 1
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Red indicates chemical shift changes observed
upon ligand binding
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Case study 2
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Case study 3
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Case study 4
goal measure effects of inhibitor binding on
conformational fluctuations of HIV protease on
ms-ms timescale. sample 0.3mM protease dimer
DMP323 inhibitor experiments 1H and 15N T2 and
T1r at 500MHz
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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 100ms
timescale to a closed conformation upon inhibitor
binding.
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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.
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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.