Joel R. Tolman

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Residual Dipolar Couplings II
Joel R. Tolman Department of Chemistry Johns
Hopkins University
EMBO Course 2009 Rosario, Argentina
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Overview

• The dipolar interaction
• Molecular alignment
• Interpretation of residual dipolar couplings
• Measurement of residual dipolar couplings
• Example applications
• Use of multiple alignment media

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The dipolar coupling interaction depends on both
angle and distance
Can influence line positions
Nonsecular contributes only to relaxation
B0
The dipolar interaction is averaged by molecular
reorientation and in the solution state will
generally not contribute line positions in the
NMR spectrum.
?
1H
r
15N
Isotropic solution
Dij 0
Dij ? 0
Anisotropic solution
The Dij are referred to as residual dipolar
couplings
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Residual dipolar couplings will contribute to
line splittings much like J couplings
J 7 Hz D -204 Hz
Quantum mechanical energy level diagram for a
weakly coupling two spin system
1H spectrum of uracil in Cesium
perfluorooctanoate. Shown is the spectral region
encompassing the H5 and H6 protons
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Scalar and dipolar coupling between equivalent
spins
D coupling is observed between equivalent spins
J coupling not observed between equivalent spins
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Spontaneous alignment in the magnetic field due
to anisotropy of the magnetic susceptibility
Diamagnetic
Paramagnetic
Alignment of a DNA strand with respect to the
static magnetic field, B0
Alignment of cyanometmyoglobin (low spin Fe (S
½))
Orients with principal axis of susceptibility
tensor perpendicular to the field
Orients with principal axis of susceptibility
tensor parallel to the field
Dc lt 0
Dc gt 0
Alignment governed by induced magnetic
dipole-magnetic field interaction E -BcB
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Alignment induced by employing a highly ordered
solvent environment
Some examples of aqueous media compatible with
biomolecules
- - - - - - -
Bicelles
Purple Membrane
- - - - - - -
- - - - - - -
B0
- - - - - - -
Bacteriophage Pf1
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Non-aqueous alignment media
Poly-g-benzyl-L-glutamate
Forms a chiral phase a compatible with CHCl3,
CH2Cl2, DMF, THF, 1,4-dioxane
ref Meddour et al JACS 1994, 116, 9652
DMSO-compatible polyacrylamide gels
N,N-dimethylacrylamide N,N-methylenebisacrylami
de 2-(acrylamido)-2-methylpropanesulfonic acid
ref Haberz et al, Angew. Chem. 2005, 44, 427
Alignment in polyacrylamide gels is achieved by
stretching or compressing the gel within the NMR
tube. The resulting elongated cavities bias the
orientation of the solute molecule
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The Saupe order tensor formalism
The Saupe order tensor, S, is used to describe
the alignment of the molecule relative to the
magnetic field.
Angles bn used to describe Saupe tensor Angles
an used to describe orientation of dipolar
interaction vector, r
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Molecular alignment is described by means of the
alignment tensor
Determination of the alignment tensor
Structural coordinates RDC data
Least squares fit
Alignment tensor (5 parameters)
Description of alignment
Orientation 3 Euler angles (a, b, g)
Magnitudes
Azz and h (Axx Ayy)/Azz
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The alignment tensor provides the basis for
interpretation of RDCs
Any single measured RDC (Dij) corresponds to a
continuum of possible bond orientations
AZZ(1)
For axially symmetric alignment, permissible
orientations will lie along the surface of a cone
with semi-angle ?
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Residual dipolar couplings provide long-range
orientational constraints
The reference coordinate axes are determined
according to the nature of molecular alignment
(the alignment tensor)
For each internuclear vector, there is a
corresponding cone of possible orientations, all
related to a common reference coordinate system
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Measurement of residual dipolar couplings
The simplest way to measure RDCs is by difference
between line splittings measured in both
isotropic solution and in the aligned state
-- Determination of the absolute sign of D could
be a problem!
from Thiele and Berger Org Lett 2003, 5, 705
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Frequency domain measurement of 15N-1H RDCs using
2D IPAP-HSQC
Couplings are measured as splittings in the
frequency domain
Ottiger, M. Delaglio, F. Bax A. J. Magn.
Reson., 1998, 131, 373-378
Two spectra are collected
Addition/Subtraction allows up-field and
down-field peaks to be separated into two
different spectra -- increasing resolution
/-
Anti-Phase doublet (HSQCopen pulses)
In-Phase doublet (HSQC only)
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HSQC-PEC2 (HSQC with Phase-Encoded Couplings and
Partial Error Correction)
Quantitative J-type experiment (coupling is
encoded in signal phase or intensity)
constant time period, Tn/JNH nominal
The experiment produces two spectra with peak
intensities modulated as a function of the
coupling of interest and the length of the
constant time period, T
Cutting, B. Tolman, J.R. Nanchen, S.
Bodenhausen, G. J. Biomol. NMR, 2001, 23, 195-200
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Assignment of diasteriomeric configuration for
dihydropyridone derivatives
trans
cis
iso
Aroulanda et al, Chem. Eur. J. 2003, 9, 4536-4539
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Determination of Sagittamide stereochemistry
using RDCs

• Four possibilities consistent with J couplings
• A, C
• A, D
• B, C
• B, D

A, C
A, D
B, D
B, C
Schuetz, et al JACS 2007, 129, 15114
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Shape based prediction of the alignment tensor
Circumference model
Equivalent ellipsoid models
An equivalent ellipsoid is derived from the
gyration tensor R with eigenvalues rk. Under
this model, the order tensor shares the same
principal axes and has the following eigenvalues
Calculates a mean field potential, U(W),
according to
Burnell and de Lange Chem. Rev. 1998, 98, 2359
Almond and Axelson JACS 2002, 124, 9986
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Prediction of alignment in biomolecules
Dot products among the normalized tensors
The collision tensor
Each orientation W(q,f) weighted proportional to
rc
PALES program Zweckstetter and Bax JACS 2000,
122, 3791
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Additive Potential/ Maximum Entropy (APME)
approach
Additive potential model assumes each ring makes
a distinct and conformation independent
contribution to overall alignment. The total
tensor is a simple sum of the two ring specific
tensors
with RDCs
without RDCs
Maximum entropy determination of P(f, y) from
RDCs, NOEs and J couplings with adjustable
parameters lxy and exy
Stevensson, et al JACS 2002, 124, 5946
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Determination of the relative orientation of
domains

• 1) Measure RDCs for each domain assignments
  required
• Determine Saupe tensor for each domain a
  structure is required for each domain
• Rotate Principal Axes into coincidence.
  Solution is fourfold ambiguous

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Multi-alignment residual dipolar couplings
RDCs measured in a single alignment A continuum
of possible internuclear vector orientations
Ambiguity can be lifted by acquisition of RDCs
using two or more alignment media
Possible internuclear vector orientations
correspond to the intersection of cones
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Multi-alignment RDC methodology

• Determination of NH bond orientations and
  mobility from RDCs measured under 5 independent
  aligning conditions

• Determination of de novo bond orientations from
  RDCs measured in 3 independent alignment media

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Theoretical formulation
The alignment tensors and the individual dipolar
interaction tensors are written in irreducible
form and combined into a single matrix equation
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How do we relate this to structural and dynamic
properties?
5 parameters are obtained for each internuclear
vector. In analogy to the alignment tensor, they
can be related to physical properties
(a, b) mean orientation
(g, Szz, h) generalized order parameter
direction and magnitude of motional asymmetry
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NMR tools for studying molecular dynamics
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Singular value decomposition of the RDC data
SVD of the data matrix D allows one to judge
independence of the RDC data and to signal
average across datasets. It is also the basis by
which independent orthogonal linear combination
(OLC-) RDC datasets can be constructed
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Bicelles
Charged bicelles
Purple membrane
Predicted RDCs (Hz)
Measured RDCs (Hz)
RDC measurements were carried out for ubiquitin
under 11 different aligning conditions, using 6
distinct media
Pf1 phage
CPBr/n-hexanol
C12E5/n-hexanol
Predicted RDCs (Hz)
Measured RDCs (Hz)
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Construction of 5 independent datasets for
ubiquitin
Noise vectors (6-11)
1
Singular values
2
11
6
3
4
5
6
11
Signal vectors (1-5)
1
3
4
2
5
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Direct Interpretation of Dipolar Couplings (DIDC)
5 orthogonal RDC datasets
Residual dipolar tensors
Remaining 25 unknown parameters
The DIDC approach selects the solution with
minimum overall motional amplitude
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8.0
Angular RMSDs between different ubiquitin models
NMR structure (1D3Z)
5.6
7.2
X-ray crystal structure (1UBQ)
15N-1H bond orientations from DIDC
5.8
7.3
2.6
2.2
RDC-refined 15N-1H bond orientations starting
from X-ray
RDC-refined 15N-1H bond orientations starting
from X-ray
2.1
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RDCs measured in
5 independent alignment media
3 independent alignment media
Ubiquitin
Rigid internuclear vector orientations no
dynamics
Mean internuclear vector orientations dynamics
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Internuclear vector orientations are
overdetermined with three independent RDC datasets
Two RDC measurements
Three RDC measurements
Internuclear vector orientations are
overdetermined. Not all possible choices for
alignment tensors are consistent
Prior knowledge of alignment tensors is required.
The requirement that the corresponding 3 cones
must share a common intersection for a rigid
molecule provides a route by which the need for
prior knowledge of alignment can be overcome.
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Our approach to the problem consists of three
phases
Output Bond orientations alignment tensors
Input RDC data (3 tensors)
Generate initial estimates for A
Minimization
Choose best solution based on RMSD and magnitude
of A
Minimize all bond orientations
Iterate to convergence
Minimize all alignment tensors
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Phase I Initial estimation of alignment tensors
Focus on vectors corresponding to the max and min
RDCs observed in each set
Alignment tensor magnitudes are estimated from
the extrema of the RDC distribution

• Vectors corresponding to the max and min
  observed RDCs are assumed to be collinear with
  the Z and Y principal axes of alignment
• Minimization is carried out to find 9 unknown
  angles given 18 RDC measurements
• At least 500 initial guesses of the 9 angles
  are made All unique results are stored and used
  in the subsequent stage

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Phase II Least squares minimization of both
bond vectors and alignment tensors
At the initial estimate for A
At the global minimum for A
At the second iteration
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For some vectors, there is more than one
orientation which agrees with the RDC data
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The global minimum RMSD between experimental and
calculated RDCs does not always correspond to the
best solution!
Merr
3
Dynamic case
2
Upper bound
1
Estimate from data
0
Rigid case
Merr is a measure of how far the average
generalized magnitude of alignment exceeds the
upper bound predicted assuming a uniform vector
distribution and given an estimate for
experimental errors. A value of Merr between 0
and 1 is within expectation.
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Experimental application to Ubiquitin and Protein
GB1
Ub
GB1
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Amide N-H bond results for Ubiquitin and protein
GB1
Ubiquitin Mean deviation 6.5
Protein GB1 Mean deviation 8.9
Open circles denote second solutions which are
within experimental error