Joel R. TolmanDepartment of ChemistryJohns Hopkins University

Residual Dipolar Couplings II

EMBO Course 2009Rosario, 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

15N

1Hθ

B0

rIsotropic solution

Dij

0

4

i

jh

2 2rij3

12

3cos2 1

HDD

0

4

I

Sh

2 2rIS3

IzS

z 1

4I S 1

4I S 1

23cos2 1

32

I Sz I

zS sin cos exp i 3

2I S

z I

zS sin cos exp i

34

I S sin2 exp 2i 34

I S sin2 exp 2i Nonsecular – contributes only to relaxation

Dij = 0

Can influence line positions

The dipolar interaction is averaged by molecular reorientation and in the solution state will generally not contribute line positions in the NMR spectrum.

Anisotropic solution Dij ≠ 0

The Dij are referred to as residual dipolar couplings

J = 7 Hz

D = -204 Hz

H0

h I I z SSz J D I zSz J 1

2D I xSx I ySy

D

klhrkl

3

3cos2 1

2

Residual dipolar couplings will contribute to line splittings much like J couplings

1H spectrum of uracil in Cesium perfluorooctanoate. Shown is the spectral region encompassing the H5 and H6 protons

Quantum mechanical energy level diagram for a weakly coupling two spin system

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

Alignment of a DNA strand with respect to the static magnetic field, B0 Alignment of cyanometmyoglobin (low spin Fe (S = ½))

Diamagnetic Paramagnetic

Alignment governed by induced magnetic dipole-magnetic field interaction:

E = -B··B

< 0 > 0

Orients with principal axis of susceptibility tensor perpendicular to the field

Orients with principal axis of susceptibility tensor parallel to the field

Alignment induced by employing a highly ordered solvent environment

B0

Bicelles Purple Membrane - - - - - - -

- - - - - - -

- - - - - - -

- - - - - - -

Bacteriophage Pf1

Some examples of aqueous media compatible with biomolecules

Poly--benzyl-L-glutamate

Non-aqueous alignment media

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

Dijres

0

4

i

jh

2 2rij ,eff3

Skl

cos kij cos

lij

klxyz

S

kl 3

2cos(

kt)cos

lt 1

2

kl

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 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

Structural coordinates + RDC data

Least squares fit

Alignment tensor (5 parameters)

Orientation: 3 Euler angles ()

Magnitudes:Azz and = (Axx – Ayy)/Azz

Determination of the alignment tensor

Description of alignment

AZZ(1)

For axially symmetric alignment, permissible orientations will lie along the surface of a cone with semi-angle θ

Any single measured RDC (Dij) corresponds to a continuum of possible

bond orientations

2cossin1cos324

D 2212

21

320res

ij

zz

ij

ji Ar

h

The alignment tensor provides the basis for interpretation of RDCs

Residual dipolar couplings provide long-range orientational constraints

For each internuclear vector, there is a corresponding cone of possible orientations, all related to a common reference coordinate system

The reference coordinate axes are determined according to the nature of molecular alignment (the alignment tensor)

from Thiele and Berger Org Lett 2003, 5, 705

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!

Frequency domain measurement of 15N-1H RDCs using 2D IPAP-HSQC

Couplings are measured as splittings in the frequency domain

In-Phase doublet (HSQC only)

Addition/Subtraction allows up-field and down-field peaks to be separated into two different spectra -- increasing resolution

Ottiger, M.; Delaglio, F.; Bax A. J. Magn. Reson., 1998, 131, 373-378

Two spectra are collected:

Anti-Phase doublet (HSQC+open pulses)

+/-

constant time period, T=n/J

NH

nominal

Quantitative J-type experiment (coupling is encoded in signal phase or intensity)

HSQC-PEC2 (HSQC with Phase-Encoded Couplings and Partial Error Correction)

Cutting, B.; Tolman, J.R.; Nanchen, S.; Bodenhausen, G. J. Biomol. NMR, 2001, 23, 195-200

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

trans

iso

cis

Assignment of diasteriomeric configuration for dihydropyridone derivatives

Aroulanda et al, Chem. Eur. J. 2003, 9, 4536-4539

Schuetz, et al JACS 2007, 129, 15114

Determination of Sagittamide stereochemistry using RDCs

Four possibilities consistent with J couplings:1) A, C2) A, D3) B, C4) B, D

A, C A, D B, DB, C

Shape based prediction of the alignment tensor

Burnell and de Lange Chem. Rev. 1998, 98, 2359

Circumference model

Calculates a mean field potential, U(), according to:

Equivalent ellipsoid models

An equivalent ellipsoid is derived from the gyration tensor R with eigenvalues k. Under this model, the order tensor shares the same principal axes and has the following eigenvalues:

Almond and Axelson JACS 2002, 124, 9986

Tm

j T

mj ()r

cd

rcd

T

mj ()r

c

r

c

Dot products among the normalized tensors

Each orientation weighted proportional to rc

The collision tensor:

PALES program: Zweckstetter and Bax JACS 2000, 122, 3791

Prediction of alignment in biomolecules

Additive Potential/ Maximum Entropy (APME) approach

Stevensson, et al JACS 2002, 124, 5946

with RDCs without RDCs

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

Maximum entropy determination of P(, ) from RDCs, NOEs and J couplings with adjustable parameters xy and xy

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

2cossin1cos324

D 2212

21

320res

ij

zz

ij

ji Ar

h

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, ): 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

Bicelles Charged bicelles

Pf1 phage

Purple membrane

CPBr/n-hexanolC12E5/n-hexanol

Pre

dict

ed R

DC

s (H

z)

Measured RDCs (Hz)

Pre

dict

ed R

DC

s (H

z)

RDC measurements were carried out for ubiquitin under 11 different aligning conditions, using 6 distinct media

Measured RDCs (Hz)

3 41 2 5

6 11

Construction of 5 independent datasets for ubiquitin

Noise vectors (6-11):

Signal vectors (1-5):

Singular values12

5

6

11

3 4

Residual dipolar tensors

Remaining 25 unknown parameters

The DIDC approach selects the solution with minimum overall motional amplitude

5 orthogonal RDC datasets

Direct Interpretation of Dipolar Couplings (DIDC)

X-ray crystal structure (1UBQ)

NMR structure (1D3Z)

RDC-refined 15N-1H bond orientations starting from X-ray

15N-1H bond orientations from DIDC

2.1°

2.6°2.2°

7.2°

7.3°5.8°

5.6°

8.0°

Angular RMSDs between different ubiquitin models

RDC-refined 15N-1H bond orientations starting from X-ray

5 independent alignment media

Mean internuclear vector orientations + dynamics

Rigid internuclear vector orientations; no dynamics

0 2 4 6 8 10 12 14

1

10

100

Protein G, B1 domain

Sin

gula

r V

alue

Index

RDCs measured in …

3 independent alignment media

Ubiquitin

Internuclear vector orientations are overdetermined with three independent RDC datasets

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.

Two RDC measurements

Three RDC measurements

Internuclear vector orientations are overdetermined. Not all possible choices for alignment tensors are consistent

Our approach to the problem consists of three phases

Minimize all bond orientations

Minimize all alignment tensors

Iterate to convergence

Input:

RDC data (3 tensors)

Generate initial estimates for A

Minimization

Choose best solution based on RMSD and magnitude of A

Output:

Bond orientations + alignment tensors

Phase I: Initial estimation of alignment tensors

Alignment tensor magnitudes are estimated from the extrema of the RDC distribution

Focus on vectors corresponding to the max and min RDCs observed in each set

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 second iteration

At the global minimum for A

For some vectors, there is more than one orientation which agrees with the RDC data

Upper bound

0

1

2

3

Merr

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.

Rigid case

Dynamic case

The global minimum RMSD between experimental and calculated RDCs does not always correspond to the best solution!

1

3

ii est

err i ii upper est

M

Estimate from data

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