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Advanced NMR

Advanced NMR. Relaxation and Dynamics. Overview. Relaxation Solution state (mainly 15 N) Relation to Protein Dynamics Orientation Order TROSY (large proteins) Residual Dipolar Couplings Protein Folding. Where do relaxation rates matter?. Recycle delay between scans: consider T 1

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Advanced NMR

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  1. Advanced NMR Relaxation and Dynamics

  2. Overview • Relaxation • Solution state (mainly 15N) • Relation to Protein Dynamics • Orientation Order • TROSY (large proteins) • Residual Dipolar Couplings • Protein Folding

  3. Where do relaxation rates matter? • Recycle delay between scans: consider T1 • Acquisition times: consider T2 • Magentisation transfers in multidimensional experiments • Mixing times in NOE and ROE experiments • Relaxation probes motion

  4. Relaxation: Return of a System to Equilibrium Various levels of description: • Phenomenological (Bloch equations) • Thermodynamic description (eg Solomon Equations for NOE) • Semiclassical or quantum description of the processes that cause relaxation (solution NMR: Redfield Theory)

  5. Simplified Bloch equations The return to equilibrium is generally (mono) exponential T1 is the spin-lattice relaxation time constant

  6. Inversion Recovery (T1)

  7. Transverse Relaxation • The return to equilibrium is governed by the Bloch equation • T2 is called the spin-spin relaxation time constant

  8. Spin Echo (T2)

  9. T1 and T2 are different • T1 requires exchange of energy with the “lattice” to return spin populations to the Boltzmann equilibrium • T2 is the time constant describing the loss of (phase) coherence between the spins.

  10. QM description of Relaxation • Hamiltonian is time dependent • Remove the time independent part of the interaction:

  11. Liouville equation is time dependent • Solution requires expansion: Shown Here to second order

  12. provided H(t) fluctuates much faster then density matrix evolves we can make the following approximation (Redfield limit):

  13. H(t) can be separated in time independent spin operators A and time dependent space fluctuations F • After a bit of maths we can write:

  14. Where the functions J(w) are actually the fourier transfroms for the auto-correlation function of the spacial fluctuations F

  15. Frequencies of transitions for a two ½-spin system

  16. Relaxation Mechanisms: 1 • Dipolar Relaxation: • A magnetic field created by a dipole A • Leads to a fluctuating field at site B. Hence causes relaxation

  17. Relaxation Mechanisms 2 • Chemical Shift Anisotropie: if shielding varies with orientation w.r.t the the applied B field (Boz) the effective field at the nucleus will fluctuate. Hence this will cause relaxation. Interaction energy:

  18. Auto-relaxation of a 15N nucleus • Longitudinal relaxation rate constant R1=1/T1 • Transverse relaxation rate constant R2=1/T2

  19. Cross-relaxation: 1H-15N-NOE • Dipolar interactions cause cross-relaxation • We measure the ratio between the steady-state and equilibrium magnetisation

  20. Cross-correlation: Relaxation Interference • Provided two relaxation mechanisms are governed by the same dynamics they can interfere constructively and destructively. • e.g. carbons in ring systems • 1H-15N

  21. Transverse Relaxation Optimised Spectroscopy: TROSY

  22. TROSY application: ligand complex 15N-OSM 21.5kDa HSQC TROSY 1:1 15N-OSM: CHRgp130 46.7kDa

  23. Dynamics and Spectral Densities • J(w) is the energy density of the system at a specific frequency w. • Establish the relationship between J(w) and dynamics • Suggest models that predict J(w) at any frequency

  24. Auto-correlation function c(t) • For any (random) function a(t) we may define • For a random process c(t) is an exponential decay: • With a characteristic correlation time tc.

  25. Relationship between c(t) and J(w) • The spectral density J(w) is the Fourier Transform of the autocorrelation function c(t). The Fourier Transform of an exponential decay is:

  26. 15N-T1, 15N- T2, 1H-15N-NOE

  27. Interpretation of Relaxation Rates • Established so far: - Relaxation mechanisms (dipolar, CSA) - Dependence of relaxation rate (time) constants with correlation time(s) tc • Still to do: Interpretation of rates (T1,T2, NOE) via diffusion models.

  28. Relaxation of cbEGF32-33:with and without Ca2+ cbEGF32 cbEGF33 cbEGF32 cbEGF33

  29. N Ca2+ bound Ca2+ free N C C Dynamics analysis: Rex > 8 Hz Rex 4-8 Hz Rex 2-4 Hz

  30. relaxation time D|| te fast slow correlation time H t N S2  fast motion: te D T1 slow motion (ms-ms) T2 Relaxation times Dynamics

  31. H N tm Rotational Diffusion: Sphere All NH have the same mono- exponetial correlation function tm= 1/6D

  32. Rotational Diffusion:Ellipsoid

  33. Modelfree Approach (Lipari&Szabo) • Divide dynamics in overall co(t) and internal motion ci(t): c(t) = c0(t)ci(t) • overall dynamics: c0(t)=1/5exp(-t/tm). • internal dynamics (faster than tm):

  34. Modelfree: Internal Dynamics internal dynamics: Observe two extreme cases: Define te such that the area of capproxbecomes exact:

  35. Together…. with

  36. Order Parameter S2 • The order parameter describes the spatial restriction of the internal motion: S2= 0: internal motion is not restricted S2= 1: internal motion is fully restricted In a diffusion in a cone motion: S = <1/2(3cos2q -1)> where q is the opening angle of the cone.

  37. J(w) with Internal Dynamics

  38. Conformational Exchange • Contribution to the transverse relaxation rate R2 from slow processes (ms-ms time scale): • Two site exchange in fast exchange regime: p1 and p2 are the populations in the two states tex =p1/k-1 = p2/k1 and DW=|W1-W2|

  39. Rex from field dependence

  40. D|| te H N S2  fast motion: te D slow motion (ms-ms) Relaxation rates and correlation times fast slow in addition: slow motion ms–ms (Rex)

  41. T1/T2 determine the diffusion tensor

  42. Dzz 1.0 0.8 0.6 0.4 E O N 0.2 - N 0.0 5 - 1 ] H -0.2 [ 1 -0.4 -0.6 -0.8 -1.0 0 10 20 30 40 50 60 70 80 Dxx Dyy Determination of the diffusion tensor fast motion 1H-15N NOE NH: T1 and T2 Rex from field dependence use well ordered NHs to fit diffusion tensor Werner et al. (2001) Meth.Mol.Biol 173, 285

  43. Models of increasing complexity Models of increasing complexity isotropic Symmetric top asymmetric isotropic axial prolate oblate asymmetric Dxx=Dyy=Dzz Dxx=DyyDzz Dxx  DyyDzz

  44. xyz not available Do the individual domains give the same tensor ? Do the domains have different tensors? Application to multiple domains Overall properties can be inferred from partial co-ordinate sets

  45. NMR of linked modules D D//

  46. Parametric Plot of T1 and T2 T1/T2 ratio depends primarily on overall motion: ~tm

  47. Combine anisotropy and modelfree • Obtain: • shape information from first terms • ps-ns dynamics from the last term

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