Residual dipolar couplings

Transcription

1 B 0 1 H 15 N Residual dipolar couplings Markus Zweckstetter Department for NMR-based Structural Biology, Göttingen, Germany Max Planck Institute For Biophysical Chemistry 1

2 Background material Saupe A, Englert G (1963) Phys. Rev. Lett. 11: BothnerBy AA. (1996) In Grant DM, Harris RK (eds.), Encyclopedia of Nuclear Magnetic Resonance. Wiley, Chichester: pp Tjandra N, Bax A (1997) Direct measurement of distances and angles in biomolecules by NMR in a dilute liquid crystalline medium. Science 78: Bax A, Kontaxis G, Tjandra N. (001) Dipolar couplings in macromolecular structure determination. In Nuclear Magnetic Resonance of Biological Macromolecules, Pt B, Vol. 339, pp Prestegard JH, Al-Hashimi HM, Tolman JR (000) NMR structures of biomolecules using field oriented media and residual dipolar couplings. Q. Rev. Biophys. 33: Kramer F, Deshmukh MV, Kessler H, Glaser SJ (004) Residual dipolar coupling constants: An elementary derivation of key equations. Concepts Magn. Reson. Part A 1A: 10-1

3 NMR structure determination Bacteria 13C,15N-Source Sample Multi-D-NMR Assignment J Γ c NH,CH 1) Residual Dipolar Couplings ) N-H Relaxation Rates ω ω NOE/ROE~1/r (NOESY, ROESY) J Coupling (E.COSY, DQ/ZQ, FIDS, quant.j) Cross Correlated Relaxation H N φ N C' ψ k+1 N H N φ H α k C α k R ψ C' O ω N H k+1 Nk+1 Distance Restraints N Ηk+1 R Η α k C α k Nk+1 C' N φ ψ ω H N O Dihedral Restraints N H N φ H α k C α k R ψ C' O 1) Dipole-Dipole ) Dipole-CSA 3) CSA-CSA θ ω N H k+1 Nk+1 Projection Restraints R C α k Η α k 1) Dipole-Susceptibility Tensor ) Dipole-Mass Tensor N H N Η α k φ C α k R ψ C' ω N Η k+1 Nk+1 Chemical Shift Restraints O Structure Calculation (Simulated Annealing, Distance Geometry, Molecular Dynamics) 3D-Structure 3

4 Why do we want to use dipolar couplings in solution NMR? all these parameters give essentially local information: distances <5Å; dihedral angles; intraresidual/sequential ϕ/ψ values, long-range information is virtually non-existent this leads to NMR-specific problems in structure determination: what is the exact angle between two non-parallel α-helices in a protein? is a long helical structure (a-helical protein, DNA) straight or bent?? what is the relative orientation of two protein domains? 4

5 RDCs today Residual dipolar couplings (RDCs) can be observed in solution when a molecule is aligned with the magnetic field When alignment can be kept sufficiently weak NMR spectra remain simple as in isotropic solution quantitative measurement of a wide variety of RDCs Several dilute liquid crystalline media are now available RDC measurements and analysis are highly efficient Residual dipolar couplings are a generally applicable tool for NMR structure determination 5

6 The dipolar coupling 6

7 Theclassicaldipolarinteraction Nuclear spin magnetic dipole μ magnetic field B ( r ) = μ0 μ 3 5 ( ) ( ) 4π r μr r rr μ nd nuclear spin magnetic moment in B μ interaction energy E μμ 1 μ 1 3 = Bμ r μ = 1 μ μ μ r μ r 0 ( ) ( )( ) π r1 r1 Within strong external magnetic field B 0 alignment μ r / r = μ cosθ E μμ 1 μ 1 0 = μ 3 1μ 4π r 1 1 3cos θ 7

8 Theclassicaldipolarinteraction E μμ 1 μ 1 μ μ 1 3cos 0 = 3 1 4π r 1 θ molecular reorientation 1 H B 0 magic angle ~ N isotropic π ( θ) 0 1 3cos sinθdθ = 0 8

9 Quantum mechanical picture spin I μ = γ I hi H μ 1 x x γ γ hμ I x x S i j I S 0 i j D = 3 μ1i 3 δij μ j = 3 3 i δij j 4π r i, j= 1 r r 4πr i, j= 1 r r z IS z z :3 1 r xz IS x z + IS z x :3 r yz IS y z + IS z y :3 r xy IS x y + IS y x :3 r IS x IS y IS x y x :3 1 r y :3 1 r + IS :3 = 1 r r r x x y y 3x + 3y x y z z + x + y 3z 1 9

10 Quantum mechanical picture 1 z IS z z ( IS x x + IS y y) r γγ I Shμ0 xz yz HD = 3 ( IxSz IzSx) 3 ( IySz IzSy) 3 4π r r r + xy 3 ( IS x y + IS y x) 3 + ( IS x x + IS y y) r r ( x y ) spherical coordinates x y z = sinθ cos φ; = sinθsin φ; = cosθ r r r D m m m ( 1 ) ( θφ, ) 1 H = F T m coordinates spin operators where F γγ hμ 4π = 4π r 5 m I S 0 m Y 3 10

11 Spherical harmonics m component ( ) Y θφ, T m m = ( 3cos θ 1 ) 4π 1 m =± 1 15 sinθ cosθ exp( ± iφ) 8π 1 m =± 15 1 sin θ exp ( ± i φ) π IS z z ( IS x x + IS y y) 6 1 ± + [ IS IS] z ± ± z 1 IS ± ± = ( + + x y ) I = ( I x iiy ) I I ii ( i ) exp φ = cosφ + isinφ 11

12 Secular approximation γγ I Shμ0 1 H D = IS 3 z z IS + IS 4π r 4 ( 3cos θ 1) ( + + ) heteronuclear case 1 ( ) IS + IS = IS + IS + + x x y y H γ γ hμ 4π r ( θ ) I S 0 D = 3cos 1 I 3 zsz γ Iγ Shμ0 3 4π r r = D(N,H) = 1.7 khz 1

13 molecular tumbling/internal motion PQ D = γ γ hμ P( β, γ) F ( θ, φ) d( cosβ) dγ P Q πr 4π sampled orientations PQ D γ γ hμ 1 3cos 1 4π r ( θ ) P Q 0 = 3 13

14 Molecular alignment and RDCs 14

15 Molecular alignment and RDCs 3 ( ) ( () () () ) 1 P cosθ = cos βx t cosα x + cos βy t cosαy + cos βz t cosαz P C ( cosθ ) ( ) cos β ( ) = and c = cosα i t i t i i 3 Cx() t cx + Cy() t cy + Cz() t cz + 1 = C () () () () () () x t Cy t cxcy Cx t Cz t cxcz Cy t Cz t cyc + + z B 0 θ β x (t) β z (t) β y (t) α x x P z α z Q α y y P 3 cosθ = 1 T ( ) ( br) 0 0 Cx() t Cx() t Cy() t Cx() t Cz() t 0 rx ( x, y, z ) x() y() y() y() z() = r r r C t C t C t C t C t ry 0 r z Cx() t Cz() t Cy() t Cz() t Cz() t 15 orientational probability distribution

16 Molecular alignment and RDCs ( ) ( ) ( ) ( ) S = 3/ cosβ t cosβ t 1/δ = 3/ C t C t 1/δ ij i j ij i j ij C + C + C = 1 CC i j CjCi x y z P ( ) cosθ = Sij cosαi cosα j { x y z} i, j=,, = S is real, traceless, symmetric 5 independent elements P φ PQ z; S zz d θ PQ Q y; S yy d x; S xx d principal alignment frame, i.e. diagonalization of S S d 3 ( ) { } α, α, α = PQ x y z max x x y y z z PQ D D C c C c C c ( θpq = αz; cz = cos θpq; cx = sinθpq cos φpq; cy = sinθpq sinφpq ) C i PQ 3 PQ = 1/3+ A D ij ( α, α, α ) = D max cos θ A + sin θ cos φ A + sin θ sin φ A x y z PQ zz PQ PQ xx PQ PQ yy PQ 3 1 D D P A A A PQ ( θpq, φpq ) = max ( cosθpq ) zz + sin θpq cos φpq ( xx 16 yy )

17 A a =3/A zz =S zzd, A r =(A xx -A yy )=/3 (S xxd -S yyd ) PQ PQ 3 D ( θpq, φpq) = Dmax P ( cosθpq) Aa + Arsin θpqcosφpq 4 PQ PQ 3 D ( θpq, φpq ) = Da ( 3cos θpq 1) + Rsin θpq cosφpq γ γ hμ PQ 3 D θ φ S θ R θ φ P Q ( ) ( ) 0 PQ, PQ = LS 3cos 1 sin cos 3 PQ PQ PQ 4π r + PQ D a PQ = ½ D PQ max A a : magnitude of alignment tensor (A a =10-3 D a NH ~ 10 Hz) R = A a /A r : rhombicity of alignment tensor; R [0; /3] use only a very slight orientational preference ("partial alignment"), i.e., only ca. 1 out of 1000 solute molecules Generalized degree of order (GDO): Euclidean norm ϑ = ( / 3 4 / 5 π i,j S ij ) 1/ 17

18 How to Get Alignment 18

19 Partial Alignment dipolar couplings are LARGE in solids (~ khz for 1 D HN!), but fortunately average out for isotropically fast tumbling (solution NMR) full dipolar couplings are not desirable in high-resolution NMR Can we get orientational information WITHOUT messing up our nice NMR spectra? YES! use only a very slight orientational preference ("partial alignment"), i.e., only ca. 1 out of 1000 solute molecules result: dipolar coupling from the (0.1%) non-isotropic fraction is scaled down to residual dipolar couplings (RDCs) of ± 0 Hz max.! (0.1% of ±0 khz) 19

20 Alignment Anisotropic Tumbling To extract dipolar coupling data, the molecule must behave anisotropically! 1) large magnetic susceptibility anisotropy diamagnetic systems such as DNA (small anisotropy in each base) metalloproteins with paramagnetic centers lanthanide-binding tags ) anisotropic environment oriented liquid-crystalline phase anisotropically compresed gel field-dependent alignment of molecules field-independent alignment 0

21 Alignment media Requirements liquid crystalline at < 10% w/v order of biomolecules: ~ 0.00 aqueous uniform anisotropy over the whole sample volume, stable at different ionic strength, ph, temperature not too strongly charged < 0.5 e/nm, solute should not bind (significantly) to medium 1

22 Bicelles Diskshaped particles made from DMPC and DHPC (q = 3:1) concentration usually 5% (w/v) degree of protein alignment can be tuned by adjusting the bicelle concentration alignment is temperature dependent (liquid crystal > 37 C) aligning with their normal perpendicular to the direction of the magnetic field degree of alignment can be determined by measuring the H quadrupolar splitting in the HDO resonance.

23 Bicelles () Isotropic bicelles (q ~ 0.5) for solubilization of integral membrane proteins solution-state NMR Anisotropic bicelles for solid-state NMR and X-ray crystallography Disadvantages: unstable in the presence of certain proteins offers only a limited temperature and ph range 3

24 Alignment media - Toolbox bicelles alkyl poly(ethylene glycol) based media filamentous phage (Pf1,fd; e/nm ) polyacrylamide gel (charged, uncharged) cellulose crystallites purple membrane fragments cetylpyrimidinium-based media,... 4

25 Alignment of Membrane Proteins Acrylamide gels DNA nanotubes G-tetrad DNA Douglas et al. Lorieau et al. PNAS, 007 JACS, 008 5

26 Modulation of alignment tensor 6

27 The problem: Orientational degeneracy D PQ = D a PQ [(3 cos θ 1) + 3/ R sin θ cos(φ)] Ramirez & Bax JACS,

28 Strategies for Modulation of Alignment Alignment media with different properties (steric/electrostatic) Charged bicelles: doping with small charged amphiphiles to alter their charge Positive: CTAB / Negative: SDS Chargedgels variation of ph, ionic strength mutation (introduction of charged residues) paramagnetic alignment (different tags/lanthanides) 8

29 Attenuation of alignment strength by increasing the ionic strength 0 mg/ml Pf1 150 mm NaCl ppm 450 mm NaCl ubiquitin at 450 mm NaCl in 0 mg/ml Pf1 9

30 Modulation of alignment tensor orientation by ionic strength changes GB1 30

31 Liquid crystal theory 31

32 Liquid crystal theory Onsager (1949): concentration at which a solution undergoes a spontaneous first order phase transition from an isotropic to a chiral nematic phase B c p > c a B = π D eff L /4 first-order transition coexistence region c p [c i, c a ] 5 Hz for semi-flexible rods c i = [(1- x)b ] -1 c a = [(x- x)b ] -1 3

33 Onsager theory B c p > c a B = π D eff L /4 effective increase in rod diameter D eff = D + κ -1 (ln ω ) contact potential ω = π Z [βγx 0 K 1 (x 0 )] - Q κ -1 exp(- x 0 ) contribution of the polyions to the ionic strength κ = 8πQ(c s + Γz p c p ) 33

34 Liquid crystal theory density of most nematogens is higher than for the solvent average density of the nematic region is higher than for the isotropic region gravity causes it to occupy the bottom region of the sample Pf1 (1 mg/ml) 0.5M B c p > 4.19 and B c p >

35 Paranematic phase of Pf1 phage For a fully nematic phase, the degree of alignment is independent of field strength above a typically very low threshold Q cc ( H) = c Pf1 paranematic 600 MHz 800 MHz c Pf1 [mg/ml] 35

36 Measurement of RDCs 36

37 NOESY HSQC 37

38 Accuracy of measured splitting: ΔJ = LW/SN required accuracy < 5% * Da 1 J HN [1]: IPAP-HSQC, DSSE-HSQC, 3D HNCO 1 J C Cα [5]: 3D HNCO (CSA(C ) ~ 500 MHz optimum) 1 J C N & J C HN [8.3]: D HSQC, 3D TROSY-HNCO 1 J CαHα [0.5]: D J CH -modulated HSQC, (HA)CANH, HN(CO)CA 1 J CH (side-chain): D J CH -mod. HSQC, CCH-COSY, SPITZE-HSQC 1 H- 1 H: COSY, CT-COSY, HNHA, 3D SS-HMQC (long-range) Bax, Kontaxis & Tjandra Method Enzymol. 339, , 001; Chou & Bax JBNMR, 001; Delaglio et al. JMR 001; Wu & Bax, JACS, 00; 38

39 RDC measurement: J splitting ( 1 J HN ) IPAP-HSQC Ottiger et al. JMR,

40 RDC measurement: Quantitative J correlation ( 1 J C N ) Chou & Bax JBNMR,

41 Determination of a Molecular Alignment Tensor 41

42 Four Methods 1) RDC distribution analysis ) Back-calculation of alignment tensor 3) Prediction of alignment from structure 4) Prediction of alignment from structure and charge distribution 4

43 1) Estimate for alignment tensor 30 A D xx B 0. Count 0 10 D zz D yy R D anh [Hz] d(NH) [Hz] D zz PQ = D a PQ D yy PQ = D a PQ ( R) D xx PQ = D a PQ (1 1.5 R) log( L( d 1 n PQ D a PQ, R)) = i=1,..,n log (P(d i PQ )) with D ii PQ = D PQ max S ii d no structure necessary! 43

44 ) Back-calculation of alignment tensor if well-defined structure available singular value decomposition (SVD) very stable & with a minimum of five RDCs possible GLN HN GLN N ILE HN 3 ILE N PHE HN 4 PHE N VAL HN 5 VAL N LYS HN 6 LYS N THR HN 7 THR N LEU HN 8 LEU N GLY HN 10 GLY N LYS HN 11 LYS N iterative least squares procedure (Levenberg-Marquardt minimization) χ = i=1,..,n [d i PQ (exp) d i PQ (calc)] /(σ i PQ ) fixing of alignment parameters (e.g. rhombic component zero due to three-fold or higher symmetry) 44

45 Evaluation of uncertainty in backcalculated alignment tensors (I) Can you trust a back-calculated alignment tensor? Monte-Carlo type approach (RDC noise) repeat SVD calculation many times (~1000 times) each time add different Gaussian noise to experimental RDCs accept only those solutions for which all back-calculated RDCs are within a given margin of the original experimental dipolar couplings error in the data is dominated by the random measurement error in the dipolar couplings indirectly take into account uncertainties in the structure set the amplitude of the added noise two to three times higher than the measurement uncertainty 45

46 Evaluation of uncertainty in backcalculated alignment tensors (II) Structural noise Monte-Carlo type approach repeat SVD calculation many times (~1000 times) each time add different Gaussian noise to the original structure (match the RMSD between the experimental and back-calculated RDCs) spread in alignment parameters obtained for these noise-corrupted structures, when using the coupling constants calculated for the original structure (i.e., yielding a perfect fit if no structural noise were added) 46

47 3) Prediction of alignment from structure no RDCs necessary! 47

48 Computer experiment: PALES S ij =1/ <3cosΘ i cos Θ j - δ ij > (i,j=x,y,z) S mol linear average over all non-excluded S matrices Periodic boundary conditions r < d/( V f ) (wall model), or r < d/(4v f ) 1/ (cylinder) 48

49 Shape prediction of magnitude and orientation of alignment 1 DNH predicted [Hz] 1 D NH measured [Hz] Protein alignment in bicelles is sterically induced 49

50 Weak alignment in Pf1 bacteriophage 50

51 4) Prediction of alignment from structure and charge distribution p B = exp[ -ΔG el (r,ω)/k B T] ΔG el (r,ω) = i q i φ[r i (r,ω)] B 0 S ij mol = S ij p B (r,ω) dr dω / p B (r,ω) dr dω steric p = 0 p = 1 p = 1 51

52 How to calculate the electrostatic energy? Continuum electrostatic theory (Debye and Hueckel 193): protein embedded in a dielectric medium containing excess ions non-linear Poisson-Boltzmann equation (Chapman 1913; Gouy 1910) Further simplification: Protein = a particle in the external field of the liquid crystal many approximations! protein = charges of their ionizable residues static dielectric constant of water ε = 78.9 average surface charge of phages: e/nm 5

53 relative G 0.5 Electrostatic potential φ [kt/e] B distance from Pf1 [nm] 53

54 Experiment versus PALES prediction steric electrostatic + steric ubiquitin Zweckstetter et al., Biophys. J. 004 DinI GB3 GB1 DNA 54

55 Ionic strength dependence magnitude orientation ubiquitin DinI GB3 GB1 DNA 55

56 Weak alignment in surfactant liquid crystalline phases 3 nm Lα 10-0 nm PALES B

57 Residual dipolar couplings in cetylpyridinium bromide/hexanol/sodium bromide Zweckstetter, submitted steric and electrostatic interactions dominate weak alignment of biomolecules in polar liquid crystalline media 57

58 Partial alignment at ph 3 Barrientos et al., JMR 001 ph 3 ph 3 ph 7 ph 7 Zweckstetter, Eur. Biophys. J

59 PH dependence of alignment 59

60 RDCs are a sensitive probe of protein electrostatics DinI (PDB code: 1GHH) Zweckstetter et al., Biophys. J

61 Charge/Shape prediction: applications differentiation of monomeric and homodimeric states (Zweckstetter and Bax 000) conformational analysis of dynamic systems such as oligosaccharides (Azurmendi and Bush 00) refinement of nucleic acid structures (Warren and Moore, 001) determination of the relative orientation of protein domains (Bewley and Clore 000) validation of structures of protein complexes (Bewley 001) classify protein fold families on the basis of unassigned NMR data (Valafar and Prestegard 003) Probing surface electrostatics of proteins and nucleic acids, refinement of side chain orientations in proteins, 61

62 RDCs in Structure Calculation 6

63 RDC-based refinement of structures PROBLEM: Potential energy surface is very rough many local/false minima convergence problem RDC-based refinement of starting structures k dip force constant adjust such that dipolar RMS is equal to measurement error) 63

64 Crititical Points How to use the structural information obtained from molecular alignment 1. In order to use the information one needs to know the direction and the size of the tensor (susceptibility, alignment, etc).. Minimization of the deviation between the measured quantity and its calculated value from a given structure and set of tensor parameters. 3. One has to be able to evaluate the outcome of the minimization. Minimization procedure 1. Get a good estimate on the tensor parameters (magnitude and rhombicity).. Define structural constraints with respect to the arbitrary tensor coordinate system. 3. Turn on the force constant for a particular alignment potential such that the final RMS between the measured and calculated values reflect the experimental error. 4. During the calculation the tensor orientation will be automatically determined through global minimization with respect to the current structure. 5. Refine the initial estimate of the tensor parameters. This can be achieved through either grid search method or built into the minimization itself. 64

65 Define axis system representing the alignment tensor 65

66 From PALES to XPLOR rdc(nh) measured aligned - isotropic (PALES convention) (min(dc)=-16. Hz; max(dc)=1.7 Hz ==> estimated Da(NH)=-8.1*(-1) as correct measurement isotropic-aligned) ==> PALES gives: DATA Da e-04 ==> get correct Da by multiplying e-04* = 9.0 Hz (for XPLOR however use -9.0 Hz, i.e. sign of Szz/) rdc(nh) measured isotropic - aligned (SSIA convention) (min(dc)=-1.7 Hz; max(dc)=16. Hz ===> estimated Da(NH)=16./=8.1 ==> PALES gives: DATA Da e-04 ==> get correct Da by multiplying e-04* = 9.0 Hz (for XPLOR however use -9.0 Hz, i.e. (-1)*sign of Szz/) 66

67 How to evaluate structures produced by minimizing against alignment data 1. The final dipolar RMS between measured.vs. calculated values. Consistency between dipolar coupling data and NOE data (decrease in non-rdc energy terms) 3. If one has more than one class of dipolar couplings: cross validation with quality factor Q = rms (D obs -D calc ) rms (D obs ) 4. Use programs such as Procheck to look at the overall quality of the structure (distribution in Ramchandran plot should improve) 5. In most cases where one obtains a high degree of consistency one would also gain in the overall RMSD of the family of calculated structures. 67

68 Quality measure of calculated/simulated RDCs Q = rms (D obs -D calc ) rms (D obs ) 1 DNH predicted [Hz] 1 D NH measured [Hz] rms(d obs )= [ (D a norm ) (4 + 3R )/5] 1/ Q ~ 17% 1.8 Å X-ray O ~ 11% 1.1 Å X-ray use only for RDCs not included in structure determination! no translational validation 68

69 Impact of RDC refinement a) without rdc b) with rdc Zhou et al. Biopolymers ( ) 5,