Joel R. Tolman Department of Chemistry Johns Hopkins University

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

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

3. 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. θ 1H r 15N Isotropic solution Dij = 0 Dij ≠ 0 Anisotropic solution The Dij are referred to as residual dipolar couplings

4. 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

5. Scalar and dipolar coupling between equivalent spins D coupling is observed between equivalent spins J coupling not observed between equivalent spins

6. 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 < 0 Dc > 0 Alignment governed by induced magnetic dipole-magnetic field interaction: E = -B·c·B

7. Alignment induced by employing a highly ordered solvent environment Some examples of aqueous media compatible with biomolecules - - - - - - - Bicelles Purple Membrane - - - - - - - - - - - - - - B0 - - - - - - - Bacteriophage Pf1

8. 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 JACS1994, 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

9. 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

10. 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 andh = (Axx – Ayy)/Azz

11. 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 θ

12. 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

13. 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 Lett2003, 5, 705

14. 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 (HSQC+open pulses) In-Phase doublet (HSQC only)

15. 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/JNHnominal 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

16. Assignment of diasteriomeric configuration for dihydropyridone derivatives trans cis iso Aroulanda et al, Chem. Eur. J.2003, 9, 4536-4539

17. 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, C B, D Schuetz, et al JACS2007, 129, 15114

18. 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 JACS2002, 124, 9986

19. 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 JACS2000, 122, 3791

20. 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 JACS2002, 124, 5946

21. 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

22. 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

23. 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

24. Theoretical formulation The alignment tensors and the individual dipolar interaction tensors are written in irreducible form and combined into a single matrix equation

25. 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

26. NMR tools for studying molecular dynamics

27. 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

28. 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 C12E5/n-hexanol Pf1 phage CPBr/n-hexanol Predicted RDCs (Hz) Measured RDCs (Hz)

29. Construction of 5 independent datasets for ubiquitin 1 Noise vectors (6-11): Singular values 2 11 6 3 4 5 6 11 Signal vectors (1-5): 1 2 3 4 5

30. 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

31. 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° 2.2° 2.6° RDC-refined 15N-1H bond orientations starting from X-ray RDC-refined 15N-1H bond orientations starting from X-ray 2.1°

32. RDCs measured in … 5 independent alignment media 3 independent alignment media Ubiquitin Rigid internuclear vector orientations; no dynamics Mean internuclear vector orientations + dynamics

33. 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.

34. 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

35. 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

36. 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

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

38. 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.

39. Experimental application to Ubiquitin and Protein GB1 Ub GB1

40. 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