Transcript Residual Dipolar Couplings II

Residual Dipolar Couplings II
Joel R. Tolman
Department of Chemistry
Johns Hopkins University
EMBO Course 2009
Rosario, Argentina
Overview
• The dipolar interaction
• Molecular alignment
• Interpretation of residual dipolar couplings
• Measurement of residual dipolar couplings
• Example applications
• Use of multiple alignment media
The dipolar coupling interaction depends on both angle and distance
H DD
    h
   0  I 2S 3  I z S z  14 I  S   14 I  S 
 4  2 rIS
 23
 43

 3cos  1 Can influence line positions
I S  I S sin cos exp i  I S  I S sin cos exp i 
I S sin  exp 2i  I S sin  exp 2i 


z


3
2
z

2
1
2
2
3
4



z
z
2
Nonsecular – contributes only to relaxation
B0
θ
15
N
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
    i j h
Dij    0  2 3
 4  2 rij
Isotropic solution
1
2
3cos  1
Dij = 0
2
Anisotropic solution
The Dij are referred to as residual dipolar couplings
Dij ≠ 0
Residual dipolar couplings will contribute to line splittings much like J couplings
H0
  I I z   S Sz  J  D  I z Sz  J  12 D  I x Sx  I y Sy 
h
 k l h 3cos2  1
D
 rkl3
2
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
Scalar and dipolar coupling between equivalent spins
D coupling is observed between
equivalent spins
J coupling not observed between
equivalent spins
Spontaneous alignment in the magnetic field due to anisotropy of
the magnetic susceptibility
Diamagnetic
Alignment of a DNA strand with
respect to the static magnetic field, B0
Paramagnetic
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 < 0
Dc > 0
Alignment governed by induced magnetic dipole-magnetic field interaction:
E = -B·c·B
Alignment induced by employing a highly ordered solvent environment
Some examples of aqueous media compatible with biomolecules
Bicelles
Purple Membrane
- - - - - - - - - - - - -
B0
- - - - - - Bacteriophage Pf1
Non-aqueous alignment media
Poly--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’-methylenebisacrylamide
+ 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
The Saupe order tensor formalism
D
res
ij
    i j h
 0  2 3
 4  2 rij,eff
Skl 
3
2


   
Skl cos  kij cos  lij
kl  xyz
 
cos(k t )cos l t  12 kl
The Saupe order tensor, S, is used to
describe the alignment of the molecule
relative to the magnetic field.
Angles n: used to describe Saupe tensor
Angles n: used to describe orientation of
dipolar interaction vector, r
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 (, , )
Magnitudes:
Azz and h = (Axx – Ayy)/Azz
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)
D
res
ij
 0   i j h
   2 3 Azz
 4  2 rij
 3 cos
1
2
2
  1  12 h sin 2  cos 2 
For axially symmetric alignment, permissible
orientations will lie along the surface of a
cone with semi-angle θ
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
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
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:
+/-
In-Phase doublet
(HSQC only)
Anti-Phase doublet
(HSQC+open pulses)
Addition/Subtraction
allows up-field and downfield peaks to be
separated into two
different spectra -increasing resolution
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,
T=n/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
Assignment of diasteriomeric configuration for dihydropyridone derivatives
cis
trans
iso
Aroulanda et al, Chem. Eur. J. 2003, 9, 4536-4539
Determination of Sagittamide stereochemistry using RDCs
Four possibilities consistent
with J couplings:
1) A, C
2) A, D
3) B, C
4) B, D
A, C
A, D
B, C
B, D
Schuetz, et al JACS 2007, 129, 15114
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
Prediction of alignment in biomolecules
Dot products among the normalized tensors
The collision tensor:
Tmj 
j
T
 m (W)rc dW
 r dW

c
j
T
 m (W)rc
W
r
c
W
Each orientation W,
weighted proportional to rc
PALES program: Zweckstetter and Bax JACS 2000, 122, 3791
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(,
y) from RDCs, NOEs and J couplings
with adjustable parameters lxy and exy
Stevensson, et al JACS 2002, 124, 5946
Determination of the relative orientation of domains
1) Measure RDCs for each domain – assignments required
2) Determine Saupe tensor for each domain – a structure is required for each
domain
3) Rotate Principal Axes into coincidence. Solution is fourfold ambiguous
Multi-alignment residual dipolar couplings
D
res
ij
 0   i j h
   2 3 Azz
 4  2 rij
 3 cos
1
2
2
  1  12 h sin 2  cos 2 
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
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
Theoretical formulation
The alignment
tensors and the
individual dipolar
interaction tensors
are written in
irreducible form and
combined into a
single matrix
equation
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
(, ): mean orientation
(, Szz, h): generalized order parameter
+ direction and magnitude of motional asymmetry
NMR tools for studying molecular dynamics
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
Predicted RDCs (Hz)
Bicelles
Charged bicelles
Purple membrane
Measured RDCs (Hz)
Predicted RDCs (Hz)
RDC measurements were carried out for ubiquitin under 11
different aligning conditions, using 6 distinct media
C12E5/n-hexanol
Pf1 phage
Measured RDCs (Hz)
CPBr/n-hexanol
Construction of 5 independent datasets for ubiquitin
1
2
Singular values
Noise vectors (6-11):
11
6
3 4
5
6
11
Signal vectors (1-5):
1
2
3
4
5
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
8.0
°
Angular RMSDs
between different
ubiquitin models
7.2
°
X-ray crystal
structure (1UBQ)
NMR structure
(1D3Z)
5.6
°
15N-1H
bond
orientations from
DIDC
7.3
°
2.2
°
15N-1H
RDC-refined
bond orientations
starting from X-ray
5.8
°
2.6
°
2.1
°
RDC-refined 15N-1H
bond orientations
starting from X-ray
RDCs measured in …
5 independent
alignment media
3 independent
alignment media
Ubiquitin
Singular Value
Protein G, B1 domain
100
10
1
0
2
4
6
8
10
12
Index
Mean internuclear vector
orientations + dynamics
Rigid internuclear vector
orientations; no dynamics
14
Internuclear vector orientations are overdetermined with three independent
RDC datasets
Two RDC
measurements
Prior knowledge of
alignment tensors is
required.
Three RDC
measurements
Internuclear vector orientations
are overdetermined. Not all
possible choices for alignment
tensors are consistent
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.
Our approach to the problem consists of three phases
Input:
RDC data
(3 tensors)
Output:
Generate initial
estimates for A
Bond orientations +
alignment tensors
Minimization
Minimize all bond
orientations
Iterate to
convergence
Minimize all
alignment tensors
Choose best
solution based on
RMSD and
magnitude of A
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
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
For some vectors, there is more than one orientation which
agrees with the RDC data
The global minimum RMSD between experimental and calculated
RDCs does not always correspond to the best solution!
Dynamic case
Merr
3
2
Rigid case
Upper
bound
1
Estimate
from data
0
M err 
   
i
est
1
i
 i  i
3 i upper
est


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.
Experimental application to Ubiquitin and Protein GB1
Ub
GB1
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