NMR (CHEM8028) Solid-state NMR: Anisotropic interactions and how we use them. Dr Philip Williamson January 2015

Transcription

1 NMR (CEM808) Solid-state NMR: Anisotropic interactions and how we use them Dr Philip Williamson January 015

2 NMR: From Molecular to Cellular Level Cell Solid State NMR Mitochondrion Membrane Liquid NMR Solution! Tissues Organelles Lipid/protein assemblies Key proteins Increasing complexity

3 Solid-state NMR spectra 3

4 Solid-state NMR Anisotropic Interactions What are they, what do they do (to our spectra) ow can we manipulate them Oriented samples Magic angle spinning ow can we exploit them Cross polarization Dipolar recoupling ow can we use them to probe structure/dynamics ( nd series of lectures) 4

5 Outline (1) What is anisotropy ow does it effect NMR spectra What interactions give rise to anisotropic properties? Describing interactions: tensors Chemical Shielding Anisotropy Orientational dependence of resonance frequency Powder spectra Dipolar interactions Quadrupolar interactions 5

6 What is anisotropy Something whose properties depend on its orientation e.g. stress 6

7 ow does it effect the NMR spectrum Each molecular orientation gives rise to a difference resonance frequency In powder we have the sum of all distributions In the liquid state these anisotropic properties are averaged on the NMR timescale 7

8 Which interactions in NMR Isotropic Anisotropic CS CS J CSA CS J CS J CSA Dipolar CS J CSA Dipolar Q 8

9 Describing interactions: tensors (1) We are concerned with 3 flavours Zero rank tensors Physical property independent of coordinate system in which it is described (scalar, distance) First rank tensors Coordinate, depends on frame of reference (vector, has size and direction) Second rank tensors Multiple first rank tensors e.g. stress (matrix) igher rank exist but we will not be considering 9

10 Describing interactions: tensors () Rank zero tensor Rank one tensor B 0 r Isotropic chemical shift, J-coupling (0,0,B z ) 10

11 Describing interactions: tensors (3) Second rank tensors k i j i j k xx yx zx xy yy zy xz yz zz z zz x xx yy y PAS 0 0 xx 0 0 yy 0 0 zz 11

12 Parameterizing nd rank tensors In cartesian notation tensors defined by principle components, A xx, A yy anda zz Frequently parameterized with a Tr A A xx A yy A zz A zz a 3 A yy A xx This assumes A zz a 3 A xx a 3 A yy a 3 Thus the asymmetry 0.0<η<1.0 and anisotropy can be both positive and negative 1

13 Chemical Shielding Anisotropy (1) Perturbation of the magnetic field due to interaction with surrounding electrons Inherently asymmetric (e.g. electron distribution surrounding carbonyl group) 13

14 Chemical Shielding Anisotropy () We can describe the perturbation of the main field (B 0 ), by the second rank tensor, B S xx yx zx xy yy zy xz 0 yz 0 B zz 0 The amiltonian which describes the interaction with the modified field is: CSA k k Which can be written in a simplified form as: Iˆ kx, Iˆ ky, Iˆ kz xx yx zx xy yy zy xz 0 yz 0 B zz 0 CSA Iˆ k k k B 0 k 14

15 Chemical Shielding Anisotropy (3) Thus the chemical shielding amiltonian simplifies to: ˆ ( k ) I B CSA k k kz zz 0 and the resonance frequency of the line is: ( k ) 1 zz 0 Thus the resonance frequency is proportional to zz in the laboratory frame. owever, is usually defined in the principle axis system (PAS) not in the lab frame (LF). Therefore, we need to transform from the PAS to LF. 15

16 Transformations Principle Axis System z zz xx yy y x Lab Frame z y Rotation characterized by the three Euler angles (α,β,γ) x Multiple by rotation matrix R 16

17 Transformation matrix Can derive a rotation matrix which bring about the rotation described above: cos cos cos sin sin R,, sin cos cos cos sin cos sin sin cos cos cos sin sin cos sin cos cos sin sin sin cos sin sin cos To determine in the laboratory frame, need to apply to the chemical shielding tensor in the principle axis system: LAB R(,, ) (,, ) This can be simplified to give general amiltonian for CSA in lab frame of: CS k B PAS R 1 3cos 1 sin cos Iˆ kz 0 iso 17

18 Effect on resonance position z zz =3000z / xx =-1500z yy =-1500z y x σ iso = 1/3(σ xx +σ yy +σ zz ) = 0z δ = σ zz -σ iso = 3000 z = (σ yy -σ xx )/ CS k B 3cos 1 sin cos Iˆ kz 0 iso 18

19 Powder Patterns In powders we have a random distribution of molecular orientations. Thus the lineshape is the weighted superposition of all the different orientations: s( t) s(,,, t) sin.. 19

20 Empirical relation between PAS and MF 1) Methyl carbons axially symmetric, axis along threefold symmetry axis ) Ring carbons three distinct tensor elements, most shielded perpendicular to plane, least shielded bisecting C-C-C angle of ring 3) Most shielded direction: 1) Perpendicular to ring in aromatic carbons ) Along C3 axis for methyl carbons 3) Perpendicular to the sp plane for carbonyl/carboxylic acids 4) Least shielded direction: 1) In the ring plane, bisecting C-C-C angle ) Perpendicular to C3 axis for methyl groups 3) In the sp place for carbonyl/carboxylic acids 5) Intermediate shielding 1) Tangential to ring for aromatic systems ) In the sp plane and perpendicular to the C-C bond for COO 0

21 Dipolar Interaction Classical interpretation Classical interaction energy between two magnetic (dipole) moments when both are aligned with the magnetic field: Quantum mechanical where: Symmetric second rank axially symmetric tensor. Again we need to rotate from the PAS to LF to obtain resonance frequency. 1 B 0 1 ) 3cos ( r E ˆ ˆ. ˆ. ˆ 3 ˆ ˆ 4 DI I r I r I r I I r D r D

22 Orientation dependence of dipolar interaction omo-nuclear Dipolar amiltonian: etero-nuclear Dipolar amiltonian: D, II 0 1 (1 3cos ) 3 3Iˆ ˆ 1 4 r Iˆ ˆ 1I 0 1 (1 3cos ) Iˆ Iˆ ziz 1 D, IS 3 1z z 4 r1 dip dip dip =0 kz

23 Quadrupolar Interaction (1) If spin>1/, nucleus contains an electronic quadrupole moment (Q). Electronic quadrupole moment interacts with surrounding electron cloud (electric field gradient(efg), V). Iˆ QIˆ Q k k where: V xx 0 0 eq Q 0 Vyy 0 I(I 1) 0 0 V zz Again we can define the anisotropy and asymmetry: Q Q Q V ZZ yy V V e qq I(I 1) zz xx 3

24 Quadrupolar Interaction () To calculate the resonance frequency, we must transform from the PAS of the EFG to the laboratory frame. Retaining only the secular terms gives the following amiltonian in the LF: Q Q 3 ˆ 3cos 1 sin cos I I( I 1) Q Z Powder spectrum of Ala-d 3 Q Orientation dependence of a single crystal of Ala-d 3 4

25 Exploitation of anisotropic interaction Oriented samples Single Crystal studies Oriented Biological Membranes Dynamics Averaging of anisotropic interaction Local electronic environment Perturbation in chemical shielding anisotropy 5

26 Dynamics: averaging of anisotropy Axis of rotational averaging Gel Phase Liquid Crystalline Phase Rotational diffusion: Scaling of interaction by 3cos 1 where is the angle between axis of motional averaging and the PAS of the interaction 1 6

27 Oriented samples Necessary to introduce macroscopic alignment: 1) Crystallization ) Oriented membranes 3) Fibres (Silk/DNA) Field (B 0 ) Orientation 0 C3 Cys19/193 C3 90 7

28 Oriented samples ligand orientations B 0 B 0 Orientation±5 Mosaic Spread±5 Orientation±5 Mosaic Spread±5

29 Protein Backbone Orientation B o 15 N chemical shielding anisotropy 15 N- 1 hetero-nuclear dipolar coupling Opella et al. 1998

30 Local electronic environment Cl As we shall see next week, typically these parameters are obtained under conditions of magic-angle spinning to enhance signal to noise. 30

31 An aside: spherical tensors Make the calculations a lot easier to handle Frequently used in papers 31

32 Sensitivity and resolution enhancement in solid-state NMR 3

33 Resume Isotropic Anisotropic CS CS J CSA CS J CS J CSA Dipolar CS J CSA Dipolar Q 33

34 Oriented samples Increase resolution by orienting interactions, therefore all spins resonate at the same frequency As all spins resonate with the same frequency the sensitivity of the measurements is higher 34

35 Magic-angle spinning 35

36 Magic Angle Spinning Seeks to reintroduce averaging process through mechanical rotation Magic Angle Spinning Probehead (Doty) Sample rotors (Varian) 36

37 Averaging of anisotropic interactions 37

38 Averaging of anisotropic interactions The amiltonian becomes time dependent: ˆ CS I Z We can deconvolute this into the iso- and anisotropic contributions: where ( t) (,,, t) ˆ (,,, t) (,, ) iso 0 iso and (,, ) C CSA 1 C CSA cos( t ) S cos(t ) S sin( t ) sin(t ) Where C 1, C, S 1 and S relate the anisotropic interaction to magnetic field (Appendix 1). 1 38

39 Analysis of MAS spectra All anisotropic interactions become time dependent To analyze spectra need to treat these time dependencies Several mathematical descriptions that allow us to do this Average amiltonian Treatment Floquet Theory Piece wise integration 39

40 Slow speed spinning Rotational echoes apparent in fid which characterise the anisotropy of the interaction At lower spinning speed the intensity of the sidebands characterises the anisotropic interaction ( and ) 40

41 erzfeld-berger Analysis Expression exist to calculate the intensity of sidebands for a given anisotropic interaction: where and 41 1) erzfeld and Berger, J.Chem.Phys 73 (1980) ),, ( sin 4 1 ) ( r r N F N I ) exp( ),, ( CSA r i N F ) sin( ) sin( ) cos( ) sin( ),,, ( ) ( ),,, ( 1 1 t S t C t S t C t t t t r r r r r r r r CSA iso iso CSA iso CSA

42 CSA analysis in reality Several programs now available that now facilitate this task: 1) Tables Paper by erzfeld and Berger ) matnmr (routines for analysis of both CSA and quadrupolar interactions in bothe static and MAS spectra) (requires matlab) 3) MAS sideband analysis (Levitt group homepage) (requires mathematica) 4

43 Effect of off-angle MAS Anisotropic interaction scaled by ½(3cos -1) Useful for characterizing anisotropy whilst gaining some sensitivity Indicates why magic angle should be carefully set! 43

44 When does MAS not work? omogeneous interactions e.g. omonuclear dipolar interactions eterogeneous line-broadening e.g. Samples with conformational heterogeneity (lyophilized solids) Nuclei with large quadrupolar interactions When samples are not solid 44

45 Applications of MAS Resolution/Sensitivity Enhancement Low speed spinning characterisation of anisotropy Isotropic chemical shifts in the protein backbone are sensitive to secondary structure Analysis of the principle components of the chemical shielding tensor reveals that larger changes are seen in making it a sensitive probe of protein secondary structure. Wei et al. 001 JACS 13:

46 Applications of MAS Low speed spinning anisotropymobility Amyloid precursor protein in differing lipid environments has different propensity to oligomerise. Sideband analysis reveals changes in peptide mobility Marenchino et al. Biophysical Journal

47 Magic-angle spinning and metabolomics 47

48 Tale of a hungry worm 4kz MAS spectrum of C.elegans (400Mz)

49 Carbohydrate metabolism

50 Fatty acid metabolism

51 MAS-NMR and metabolism Spectroscopically: Simple markers for metabolites Observe changes in metabolite levels Labelling possible to aid in assignment Biologically: Genetics of C.elegans well characterized Large library of mutations Development well understood Genes linked to phenotype Behavioral differences

52 A molecular view of biological systems Structure of high affinity ligands Membrane protein complexes Regulation of intra-cellular trafficking Whole organisms