Equations of Motion Laboratory Frame

1. Equations of MotionLaboratory Frame Classical vector model Bloch equations Quantum mechanical model Liouville-von Neumann Solution i.e. series of rotations Solution i.e. series of rotations Graphical interpretation Graphical interpretation

2. Equations of MotionRotating Frame Bloch equations Liouville-von Neumann Solution i.e. series of rotations Solution i.e. series of rotations Graphical interpretation Graphical interpretation

3. Observable Expectation value Transverse magnetization Fourier transformation to a spectrum

4. Description of a System Density matrix (operator) Magnetization Visualization Realization (example)

5. Hamiltonian of a Coupled Two-Spin System Weak coupling Evolution

6. Evolution in the Product Operator Formalism For a set of operators that satisfy evolve Illustration

7. Evolution of a Coupled Two-Spin System Observable

8. Evolution of a Coupled Two-Spin System Observable Trace of sF+ Spectrum

9. Operators for the Two-Spin System Basis Populations and transitions Populations of all leves Polarization across the single quantum I-spin transitions

10. Evolution in the Product Operator Formalism For a set of operators that satisfy evolve Illustration

11. Evolution of the Two-Spin System Free precession (chemical shift) Pulses Free precession (scalar coupling)

12. Evolution of the Two-Spin System Piecewise solution Illustration Order is immaterial when operators commute.

13. Spin-echo Echo during chemical shift evolution Refocused chemical shift evolution!

14. Spin-echo Echo during scalar coupling evolution Chose t = 1/2J to select refocusing and t = 1/4J antiphase magnetization.

15. HSQC (Heteronuclear single quantum correlation spectrum)

16. Two-dimensional spectrum

17. Three-dimensional spectrum

18. Three-dimensional Experiment

19. Theoretical Description of NMR Experiment(without relaxation)

20. NMR of Biological MacromoleculesMultidimensional Multinuclear Spectroscopy Structural Biology

21. How to Interpret Spectra? ? • Structural implications • Atom type (and near neighbours) • Spatially near neighbours • Chemically bonded neighbours • Dynamic consequences • Fluctuating magnetic environment • Spectral parameters • Resonance frequency • Modulation of frequency • Correlation via dipolar field • Correlation indirectly via electrons (scalar coupling) • Relaxation

22. Magnetic EnvironmentDispersion of resonances • External magnetic field of the NMR-spectrometer • Local fields due to • -adjacent nuclei • -surrounding electron clouds • Chemical shift • = g(1 - s)B s is shielding (tensor)

23. Assignment of Resonances Proteins display large dispersion because they contain distinct magnetic microenvironments.

24. Assignment of ResonancesIdentification of Residues by Characteristic Chemical Shifts Aliphatic carbon shifts are particularly characteristic for the residues.

25. Assignment of Backbone ResonancesPrinciple – Sequential Walk HNCA H R H R H R | | | | | | -N–Ca– C –N– Ca– C –N– Ca– C- | || | || | || H O H O H O Ca S( Hi, Ni, Cai, Cai-1 ) N H HN(CO)CA H R H R H R | | | | | | -N–Ca– C –N– Ca– C –N– Ca– C- | || | || | || H O H O H O Ca S( Hi, Ni, Cai-1 ) N H

26. HNCA HN(CO)CA

27. Assignment of ResonancesSequential Walk via HNCO and HN(CA)CO

28. The redundancy in many alternatives for sequential assignment is important for automated assignment.

29. Spectra Contain Implict Structural DataNOEs  Short Distances NOEs Nuclear Overhauser Enhancement i.e. dipole-dipole relaxation.

30. Short Range Distances (NOEs) ri = rref(Sref/Si)1/6

31. Spectra Contain Implict Structural DataScalar Couplings  Dihedrals Karplus curve

32. How to Convert Spectral Parameters to Explicit Structural Data? • Short (<5-7Å) distances • via nuclear Overhauser spectroscopy (NOE) • Torsion angles • via scalar couplings (J-couplings) • Angles • via residual dipolar couplings (RDC) • Hydrogen bonds • via correlation spectroscopy • Secondary structures • via chemical shifts (resonance frequences)

33. T t Computation of Structure Conversion of structural data to restraints expressed as pseudo potentials Restrained molecular dynamics (MD) (Cartesian or torsion angle)

34. Result – Family of Structures All structures that satisfy restraints (within experimental error) are possible.

35. Evaluation of Structure • Accuracy • Restraint violations • Inconsitancies • Ramachandran violations • Precision • Spread of the family • Number of restraints • per residue

36. Direct Inspection of Spectra Observing binding Mapping binding epitopes Detecting conformational changes

37. Signal Decay – RelaxationLocal Fluctuating Fields

38. About Field Fluctuations ”Reasons” ”Spectral Manifestations” • Bond vibrations • from pico to nano seconds • Conformational changes • from micro to milli seconds • Chemical exchange • from micro seconds to days • Relaxation measurements • -> rate constants, order • parameters, correlation times • Relaxation measurements • -> dispersion of parameters • Line width analysis • -> rate constants Motional model

39. Conformational Exchange

40. Conformational Exchange

41. Conformational Exchange

42. Relaxation Dispersion Transverse relaxation rates vs. effective field and temperature Frans A.A. Mulder et al.Nature Structural Biology 8, 932 - 935 (2001)

43. Hydrogen Exchange Monitoring signal intensity after dissolving to D2O Denis Canet et al.Nature Structural Biology - Published online: 11 March 2002,

44. Hydrogen Exchange Denis Canet et al.Nature Structural Biology - Published online: 11 March 2002,

45. Reaction Dynamics Elan Zohar Eisenmesser,1 Daryl A. Bosco,1 Mikael Akke,2 Dorothee Kern1* Science - Feb 2002,

46. Reaction DynamicsHot Spots

47. Reaction DynamicsProlyl Cis-Trans Isomerase

48. Dilute Liquid CrystalAnisotropic Medium

49. Distribution of Molecular Orientations

50. Vfree Net Alignment Vrestricted D = Dmax(3cos2q-1)/2