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Theory, Pulse sequences a nd SIMPSON

Advanced Practical Course in NMR Techniques for Membrane Proteins Rutherford Appleton Laboratory, May 2009. Theory, Pulse sequences a nd SIMPSON. Outline: Two lectures. Lecture 1 A dvanced experiments rely on Tayloring Hamiltonians The basic tools

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Theory, Pulse sequences a nd SIMPSON

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  1. Advanced Practical Course in NMR Techniques for Membrane ProteinsRutherford Appleton Laboratory, May 2009 Theory, Pulse sequences and SIMPSON

  2. Outline: Two lectures • Lecture 1 • AdvancedexperimentsrelyonTayloringHamiltonians • The basictools • MAS, Decoupling, Dipolarrecoupling, etc • Optimal control • Lecture 2 • -Introduction to SIMPSON • -Examplesonapplications of SIMPSON

  3. High-resolution solid-state NMR:Sample rotation or oriented samples 15Nanisotropic chemical shift 15N-1H dipolarcoupling 1 1 Powder 54.7° 1 Magic-angle Spinning (MAS) Oriented Samples (crystals or uniaxial) B0 MAS: High-resolution: yes Lost information: yes

  4. Structural contraints Assignment Needs for structure/dynamics analysis B0 H=Hs+HD+HQ+...

  5. Dipole-dipole coupling The Hamiltonian is quite complex -- tailoring is important! Chemical shielding tensor Bjerring, Vosegaard, Malmendal & Nielsen Concepts Magn. Reson. 18A, 111-129 (2003)

  6. MAS, decoupling, and recoupling

  7. “Dozens” of Re- and Decoupling experiments available forBiological Solid-State NMR How do these experiments work? How do we make the best experiment?

  8. iso iso iso II, SS II, SS IS IS aniso aniso H=Hs + Hs + HD + HD H=Hs + Hs + HD + HD H=Hs + H= k HD IS Tailoring of the Hamiltonian Lee, Kurur, Helmle, Johannesen, Nielsen & Levitt, Chem.Phys.Lett. 242, 304 (1995).

  9. OCC7 2D CACB, CACX & 3D NCACB Low-power C7 Designed by Optimal Control Nielsen, Bjerring, Nielsen, and Nielsen J. Chem. Phys.., in press, 2009

  10. rf = UriU+ U=exp{-iH(t)dt} Quantum control U rf Final Spin State Hamiltonian H(t): External parameters B0, B1, Rf pulses, gradients, sample spinning Internal Interactions HJ, HD Hs, HQ ri Initial Spin State Quantum control => Design of Ū

  11. Spin-EngineersToolbox To understand, design, and optimally exploit solid-state NMR spin engineering, we need tools Operations Simplifications - understanding

  12. Our handle to information and manipulation The equation of motion Liouville-von-Neuman equation Hamiltonian Density operator Propagator

  13. Externalpart: Zeeman, rf Hamiltonian Zeeman Rf Cartesian operators: Ix, Iy, Iz, ... Larmor fr. Rf ampl rf carrier fr. phase We will remove this time-dependence later by moving into rotating frame!

  14. The nuclear spin Hamiltonian: Internal Hamiltonian H = Hs + HJ + HD + HQ Hs = wsIz (Chemical Shielding) (J coupling) HJ = pJ 2IzSz HD = wD 2IzSz (Heteronuclear Dipole-Dipole coupling) HD = wD (3IzSz-I·S) (HomenuclearDipole-Dipolecoupling) HQ = wQ (3Iz2-I2) (Quadrupole coupling)

  15. Cyclic commutation {Ix, Iy, Iz} {Ix, 2IySz, 2IzSz} {2IzSx, Sy, 2IzSz} Rotations in spinspace - Cartesian Known as ”product operator formalism” (Sørensen et al.)

  16. The evolution angle {Ix, Iy, Iz} {Ix, 2IySz, 2IzSz} {2IzSx, Sy, 2IzSz} Product operator calculationsin solid-state NMR Single-pulse: q Ix IzcqIz – sqIy Evolution under anisotropic shielding: FIz IxcF Ix+sFIy Evolution under heteronuclar Dipolar coupling: F 2IzSz IxcF Ix+sF 2IySz

  17. Internal part: Shift, J, dip, quad Spatial Spin InternalHamiltonian Irreducibletensor operators: Rj,m andTj,m (j = rank) Constant

  18. Interaction h h Fundamental constants Spatial part Spin part InternalHamiltonian – all in oneTable • Note: • rank j=1 not relevant • shiftspintensor is rank 1

  19. Euler angles Reduced Wigner Matrix Wigner Matrix Rotations in spin and spatialspace:WignerRotations – irreducibletensors

  20. H(t) → H = Heff EffectiveHamiltonian H(t) Refined variants: scBCH, EEHT, ....

  21. Corriolis term:Takes the big part out! Interactionframesorrotatingframes Could be Zeeman interaction, a rf pulse sequence, ....

  22. Analysis/development of NMR experiments:Universal procedure

  23. Example 1:The Zeemaninteractionframe– high-fieldapproximation Rotate with frequency w0 around z External Hamiltonian Take the average over one period

  24. Internal, simpler, only 1/5 of the terms! Example 1:The Zeeman interaction frame – high-field approximation InternalHamiltonian Rotate with frequency w0 around z Take the average over one period

  25. Example 2:Heteronuclear Dipolar Truncation But we know that it should be: 2IzSz -- How comes? wS wI

  26. Isolatedinfluence from MAS Spatial transformations – the effect of sample rotation We have to bring establish a link between the known form of the interaction in PAS frame to the LAB frame ≡Coordinate Transformations

  27. h Static: P → L h gPR Rot: P → R → L bRL bPR Fourier components Dipole-dipole coupling:Static and MAS

  28. The evolution angle Rotor synchronized sampling: t = 0, tr, 2tr, .... gPR+wrt bRL bPR Fast MAS Averaging of dipole-dipole coupling:Rotor synchronized sampling or fast MAS

  29. MAS averaging of anisotropicinteractions ... Homonuclear Spin-Pair System Proceed with demonstration of heteronuclear and homonuclear RECOUPLING methods.

  30. Interaction frame of rf 2. Average Hamiltonian C.W. F.I.D HeteronuclearDipolarDecoupling:CW Irradiation Much more advanced schemes: TPPM, XiX, SPINAL64, eDUMBO, .....

  31. Dipolar Recoupling:How do we get the anisotropy back – selectively? Sample rotation averages this one!! Add in some rfirrad. wrfI = qwr wrfS = pwr - 0 RECOUPLING

  32. → Rf interaction frame wrfS = pwr wrfI = qwr OK, we have now modulations – but to appreciate recoupling we recast in terms of exponentials HeteronuclearDipolarRecoupling:

  33. →Average Hamiltonian over rotor period n = 1, 2 Four experiments that provide recoupling r > |2| Heteronuclear Dipolar Recoupling:Recoupling conditions

  34. p-q p+q I -n S n -n 4wr n 3wr n=1, p=4 Which n? -- the scaling factor equals the dipolar Fourier component => n=1 largest scaling The DCP experiment HeteronuclearDipolarRecoupling: The Four Experiments: Note: only the difference in the rf field strengths matters – but try to avoid other resonances!

  35. → Tiltedframe -p/2 around y -x -x -x -x ZQy ZQx The DCP Experiment:Let’s have a look at the Hamitonian 4wr I S 3wr = 2Ix23 2IxSx+2IySy = = 2Iy23 2IySx-2IxSy =

  36. The DCP Experiment:Let’s have a look at the initial and destination operators Iz = ½ (Iz-Sz) + ½(Iz+Sz) = ½ + ½ = 2Iz23+2Iz14 Sz = -½ (Iz-Sz) + ½(Iz+Sz) = -½ + ½ = -2Iz23+2Iz14 Transfer from Iz to Sz amounts to inverting Iz23

  37. → Tilted frame -p/2 around y -x -x -x -x Fictitious spin-1/2 ZQy ZQx HeteronuclearDipolarRecoupling:The DCP experiment Experiment D: Full transfer of Iz to Sz for knt=p

  38. HeteronuclearDipolarRecouplingDCP is just an inversion Ix Iz 4wr I Rf S 3wr Sz Sx Dipole

  39. CompositeDipolarRecoupling:COMB DCP DCP I II III IV V COMB3 DCP COMB6 DCP

  40. Practical Performance Hartman-Hahn matching Experiments Simulations Experiments 13C,15N-Glycine @ 700 MHz 10 kHz spinning

  41. HomonuclearDipolarRecoupling:RotationalResonance (R2) Known as Rotational Resonance – is established when n wr = Dwiso Note the difference But how does it work?

  42. Ignore anisotropic shifts Interaction frame: Recoupling if n = ±1, ±2 First-order average: HomonuclearDipolarRecoupling:RotationalResonance (R2) Hamiltonian:

  43. HomonuclearDipolarRecoupling:RotationalResonance (R2) inversion Hamiltonian: • You need to have the energy levels of the Initial Operator and the • Hamilonian matching • If they share both energy levels => full effect • If they share one, the commutator is scaled by ½ => half effect • If they share no energy levels => no effect Iz23

  44. a As usual: First appropriate interaction frame b a. Tilted frame b. Rotating frame Second Average Hamiltonian wrf = nwr RecouplinganalyzedusingWigner rotations:HomonuclearRotaryResonance (HORROR) p/2 p/2 x -y y Selection rules: {m,n,m} = {1,-1/2,2} or {-1,1/2,2}

  45. Transformation for 2Sz(14) : Transformation for 2Sz(23): invariant kt = p Net Effect: HomonuclearDipolarRecoupling:HORROR The evolutions is going on in the 2Q subspace

  46. HomonuclearDipolarRecouplingHORROR is just an inversion + p/2 p/2 Rf x -y y Dipole + +

  47. COMB “Excitation” pulses in RecouplingHomonuclear 2Q Recoupling

  48. 1a. Tilted frame (brings effective field along z): IS 1b. Rotating frame (z-rotation around effective field): q woff wrf 2. Average Hamiltonian (over one cycle of the effective field) HomonuclearDipolarDecoupling:Lee-GoldburgDecoupling

  49. recoupling 2IxSx±2IySy (planar) Invisible Solution! recoupling 2IzSz (Ising) 2IzSz-IxSx-IySy [2IzSz, 2RzSz] = 0 The Dipolar Truncation Problem 2IzSz-IxSx-IySy [2IxSx-2IySy, 2RxSx-2RySy] ≠ 0

  50. ? ? ? ? Dipolar trunction is not solved by introducing a third spin – and sequential transfers OBS: Planar vs Ising Dephasing Planar Hamiltonian 2IxSx-2IySy (2Q) Ising Hamiltonian 2IzSz w/o D13

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