Joel R. Tolman Department of Chemistry Johns Hopkins University Residual Dipolar Couplings II 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 15 N 1H1H θ B0B0 r Isotropic solution Nonsecular – contributes only to relaxation D ij = 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 D ij ≠ 0 The D ij are referred to as residual dipolar couplings

J = 7 Hz D = -204 Hz Residual dipolar couplings will contribute to line splittings much like J couplings 1 H 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, B 0 Alignment of cyanometmyoglobin (low spin Fe (S = ½)) DiamagneticParamagnetic 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 B0B0 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 CHCl 3, CH 2 Cl 2, 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 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: A zz and  = (A xx – A yy )/A zz Determination of the alignment tensor Description of alignment

A ZZ (1) For axially symmetric alignment, permissible orientations will lie along the surface of a cone with semi-angle θ Any single measured RDC (D ij ) corresponds to a continuum of possible bond orientations 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 15 N- 1 H 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, 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-PEC 2 (HSQC with Phase-Encoded Couplings and Partial Error Correction) Cutting, B.; Tolman, J.R.; Nanchen, S.; Bodenhausen, G. J. Biomol. NMR, 2001, 23, 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,

Schuetz, et al JACS 2007, 129, 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, CA, DB, 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

Dot products among the normalized tensors Each orientation  weighted proportional to r c 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 RDCswithout 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 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 ( , S zz,  ): 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

BicellesCharged bicelles Pf1 phage Purple membrane CPBr/n-hexanolC 12 E 5 /n-hexanol Predicted RDCs (Hz) Measured RDCs (Hz) Predicted RDCs (Hz) RDC measurements were carried out for ubiquitin under 11 different aligning conditions, using 6 distinct media Measured RDCs (Hz)

Construction of 5 independent datasets for ubiquitin Noise vectors (6-11): Signal vectors (1-5): Singular values

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

5 independent alignment media Mean internuclear vector orientations + dynamics Rigid internuclear vector orientations; no dynamics 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 M err M err 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 M err 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! Estimate from data

Experimental application to Ubiquitin and Protein GB1 UbGB1

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