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Understanding Crystallography: Phases, Amplitudes, and Solutions

Explore the intricate world of crystallography and the phase problem. Learn about direct and reciprocal spaces, Patterson maps, phasing techniques, and more. Discover how to solve the phase problem and unveil the structure of complex molecules.

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Understanding Crystallography: Phases, Amplitudes, and Solutions

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  1. Pattersons The “third space” of crystallography

  2. The “phase problem”

  3. The “phase problem” Phases Amplitudes

  4. The “phase problem” Phases Amplitudes

  5. The “spaces” of crystallography • Direct/real space • Distances are in Å, Angles are in degrees • Reciprocal space • Distances in 1/Å, Angles are different • Patterson space • Distances are in Å , Angles are in degrees • Relative distances only, origin lost • “direction” is preserved

  6. scattering detector detector x-ray beam sample

  7. How to make a Patterson map: • Set all phases to zero • Square all structure factors • Calculate Fourier transform

  8. One atom in unit cell Electron density Fraction across unit cell

  9. Patterson: one atom Electron density Fraction across unit cell

  10. Patterson: one atom Electron density Fraction across unit cell

  11. Patterson: one atom Electron density Fraction across unit cell

  12. Patterson: two atoms Electron density Fraction across unit cell

  13. Patterson: three atoms Electron density Fraction across unit cell

  14. Better resolution! Patterson: three atoms Electron density Fraction across unit cell

  15. Patterson: five atoms Electron density Fraction across unit cell

  16. Patterson: five atoms Electron density Fraction across unit cell

  17. scattering from a structure sample detector

  18. forward Fourier Transform inverse Fourier Transform no phase Patterson map!

  19. Snapshot from single virus particle 200 nm Reconstructed image 2 keV LCLS 200 fs Mimi virus single-shot. Resolution 20nm TEM Seibert, et al. (2011). Nature470, 78-81.

  20. lysozyme: real and reciprocal

  21. forward Fourier Transform 9 atoms

  22. forward Fourier Transform 10 atoms

  23. Patterson map 10 atoms

  24. Patterson map 9 atoms

  25. Difference Patterson Still no phases!

  26. forward Fourier Transform 9 atoms

  27. Harker Section of a Patterson

  28. X-ray data are 3D!

  29. Patterson: five atoms + 3-fold symmetry Electron density Fraction across unit cell

  30. Patterson: five atoms + 3-fold symmetry Electron density Fraction across unit cell

  31. Major Phasing techniques • Molecular Replacement • Multiple Isomorphous Replacement • Anomalous Diffraction • Direct methods

  32. 2Fo-Fc maps Electron density Fraction across unit cell

  33. 2Fo-Fc maps Electron density Fraction across unit cell

  34. 2Fo-Fc maps Electron density Fraction across unit cell

  35. 2Fo-Fc maps Electron density Fraction across unit cell

  36. 2Fo-Fc maps Electron density Fraction across unit cell

  37. 2Fo-Fc maps Electron density Fraction across unit cell

  38. 2Fo-Fc maps Electron density Fraction across unit cell

  39. 2Fo-Fc maps Electron density Fraction across unit cell

  40. 2Fo-Fc maps Electron density Fraction across unit cell

  41. 2Fo-Fc maps Electron density Fraction across unit cell

  42. 2Fo-Fc maps Electron density Fraction across unit cell

  43. 2Fo-Fc maps Electron density Fraction across unit cell

  44. 2Fo-Fc maps Electron density Fraction across unit cell

  45. 2Fo-Fc maps Electron density Fraction across unit cell

  46. 2Fo-Fc maps Electron density Fraction across unit cell

  47. The “phase problem” Phases Amplitudes Fobs

  48. The “phase problem” Phases & Amplitudes Amplitudes Fobs-Fcalc

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