Advertisement

Recent Advances in the Direct Methods of X-Ray Crystallography

  • Herbert A. Hauptman
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 62)

Abstract

The electron density function ρ(r) in a crystal determines its diffraction pattern, that is, both the magnitudes and phases of its X-ray diffraction maxima, and conversely. If, however, as is always the case, only magnitudes are available from the diffraction experiment, then the density function ρ(r) cannot be recovered. If one invokes prior structural knowledge, usually that the crystal is composed of discrete atoms of known atomic numbers, then the observed magnitudes are, in general, sufficient to determine the positions of the atoms, that is the crystal structure.

The intensities of a sufficient number X-ray diffraction maxima determine the structure of a crystal. The available intensities usually exceed the number of parameters needed to describe the structure. From these intensities a set of numbers |E H| can be derived, one corresponding to each intensity. However, the elucidation of the crystal structure also requires a knowledge of the complex numbers E H = |E H|exp( H), the normalized structure factors, of which only the magnitudes |E H| can be determined from experiment. Thus, a “phase” φ H, unobtainable from the diffraction experiment, must be assigned to each |E H|, and the problem of determining the phases when only the magnitudes |E H| are known is called “the phase problem.” Owing to the known atomicity of crystal structures and the redundancy of observed magnitudes |E H|, the phase problem is solvable in principle.

Keywords

Global Maximum Joint Probability Distribution Minimal Principle Structure Invariant Conditional Probability Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Burla, M.C., Camalli, M., Cascarano, G., Giacovazzo, C., Palidori, G., Spagna, R., and Viterbo, D. (1989). SIR88–a direct-methods program for the automatic solution of crystal structures. Journal of Applied Crystallography, 22: 389–393.CrossRefGoogle Scholar
  2. Deacon, A.M., Weeks, C.M., Miller, R., and Ealick, S.E (1998). The Shake-and-Bake structure determination of triclinic lysozyme. Proceedings of the National Academy of Sciences of the USA, 95: 9284–9289.CrossRefGoogle Scholar
  3. Debaerdemaeker, T., Tate, C., and Woolfson, M.M. (1985). On the application of phase relationships to complex structures. XXIV. The Sayre tangent formula. Acta Crystallographica, A41: 286–290.Google Scholar
  4. Debaerdemaeker, T. and Woolfson, M.M. (1983). On the application of phase relationships to complex structures. XXII. Techniques for random phase refinement. Acta Crystallographica, A39: 193–196.Google Scholar
  5. Debaerdemaeker, T. and Woolfson, M.M. (1989). On the application of phase relationships to complex structures. XXVIII. XMY as a random approach to the phase problem. Acta Crystallographica, A45: 349–353.Google Scholar
  6. DeTitta, G., Weeks, C., Thuman, P., Miller, R., and Hauptman, H. (1994). Structure solution by minimal-function phase refinement and fourier filtering. I. Theoretical basis. Acta Crystallographica, A50: 203210.Google Scholar
  7. Gilmore, C.J. (1983). A computer program for the automatic solution of crystal structures from X-ray diffraction data. Technical report, University of Glasgow, Glasgow, U.K.Google Scholar
  8. Han, F., DeTitta, G., and Hauptman, H. (1991). TELS: least-squares solution of the structure-invariant equations. Acta Crystallographica, A47: 484–490.Google Scholar
  9. Hauptman, H. (1982). On integrating the techniques of direct methods and isomorphous replacement. I. The theoretical basis. Acta Crystallographica, A38: 289–294.MathSciNetGoogle Scholar
  10. Hauptman, H. (1982). On integrating the techniques of direct methods with anomalous dispersion. I. The theoretical basis. Acta Crystallographica, A38: 632–641.Google Scholar
  11. Hauptman, H. (1991). A minimal principle in the phase problem. In Moras, D., Podjarny, A.D., and Thierry, J.C., editors, Crystallographic Computing 5 from Chemistry to Biology,pages 324–332, Oxford, U.K. International School of Crystallographic Computing, Bischenberg, France, 1990, IUCr Oxford University Press.Google Scholar
  12. Hauptman, H. (1996). The SAS maximal principle: a new approach to the phase problem. Acta Crystallographica, A52: 490–496.Google Scholar
  13. Hauptman, H. and Han, F. (1993). Phasing macromolecular structures via structure-invariant algebra. Acta Crystallographica, D49: 3–8.Google Scholar
  14. Hauptman, H. and Karle, J. (1972). Phase determination from new joint probability distribution: space group P1. Acta Crystallographica, 11: 149–157.CrossRefGoogle Scholar
  15. Hauptman, H., Velmurugan, D., and Fusen, H. (1991). The minimal principle solves some crystal structures. In Schenk, H., editor, Direct Methods of Solving Crystal Structures, pages 324–332, New York, NY. International School of Crystallography, Erice, Trapani, Italy, April 1990, Plenum Publishers.Google Scholar
  16. Hauptman, H.A. (1972a). Crystal Structure Determination, The Role of the Cosine Seminvariants. Plenum Press, New York, NY.Google Scholar
  17. Hauptman, H.A. (1972b). A joint probability distribution of seven structure factors. Acta Crystallographica, A31: 671–679.Google Scholar
  18. Hauptman, H.A. (1972c). A new method in the probabilistic theory of the structure invariants. Acta Crystallographica, A31: 680–687.Google Scholar
  19. Heinerman, J.J.L., Krabbendam, H., Kroon, J., and Spek, A.L. (1978). Direct phase determination of triple products from Bijvoet inequalities. II. A probabilistic approach. Acta Crystallographica, A34: 447–450.CrossRefGoogle Scholar
  20. Hendrickson, W.A. and Teeter, M.M. (1981). Structures of the hydrophobic protein Crambin determined directly from the anomalous scattering of Sulphur. Nature, 290: 107–113.CrossRefGoogle Scholar
  21. Housset, D., Habersetzer-Rochat, C., Astier, J., and Fontecilla-Camps, J.C. (1994). Crystal structure of Toxin II from the scorpion Androctonus australis hector refined at 1.3A resolution. Journal of Molecular Biology, 238: 88.CrossRefGoogle Scholar
  22. Hughes, E.W. (1953). The signs of products of structure factors. Acta Crystallographica, 6: 871.CrossRefGoogle Scholar
  23. Karle, J. and Hauptman, H. (1972). Phase determination from new joint probability distribution; space group Pi. Acta Crystallographica, 11: 264–269.CrossRefGoogle Scholar
  24. Kroon, J., Spek, A.L., and Krabbendam, H. (1977). Direct phase determination of triple products from Bijvoet inequalities. Acta Crystallographica, A33: 382–385.CrossRefGoogle Scholar
  25. Miller, R., Gallo, S.M., Khalak, H.G., and Weeks, C.M. (1994). SnB: crystal structure determination via shake-and-bake. Journal of Applied Crystallography, 27: 613–621.CrossRefGoogle Scholar
  26. Okaya, Y. and Pepinsky, R. (1956). New formulation and solution of the phase problem in x-ray analysis of noncentric crystals containing anomalous scatterers. Physical Review, 103 (6): 1645–1647.MATHCrossRefGoogle Scholar
  27. Peerdeman, A.F. and Bijvoet, J.M. (1956). Proc. K. Ned. Akad. Wet., B59: 312–313.Google Scholar
  28. Ramachandran, G.N. and Raman, S. (1956). Current Science, 25: 348–351.Google Scholar
  29. Rossman, M.G. (1961). The position of anomalous scatterers in protein crystals. Acta Crystallographica, 14: 383–388.CrossRefGoogle Scholar
  30. Sayre, D. (1952). The squaring method: a new method for phase determination. Acta Crystallographica, 5: 60–65.CrossRefGoogle Scholar
  31. Sheldrick, G.M. (1990). Phase annealing in SHELX-90: direct methods for larger structures. Acta Crystallographica, A46: 467–473.Google Scholar
  32. Smith, G.D., Blessing, R.H., Ealick, S.E., Fontecilla-Camps, J.C., Hauptman, H.A., Housset, D., Langs, D.A., and Miller, R. (1997). Ab initio structure determination and refinement of a scorpion protein toxin. Acta Crystallographica, D53: 551–557.Google Scholar
  33. Weeks, C., DeTitta, G., Hauptman, H., Thuman, P., and Miller, R. (1994). Structure solution by minimal-function phase refinement and fourier filtering. II. Implementation and applications. Acta Crystallographica, A50: 210–220.Google Scholar
  34. Woolfson, M.M. and Yao, J.-X. (1990). On the application of phase relationships to complex structures. XXX. Ab initio solution of a small protein by saytan. Acta Crystallographica, A46: 409.Google Scholar
  35. Yao, J.-X. (1981). On the application of phase relationships to complex structures. XVIII. RANTAN-random MULTAN. Acta Crystallographica, A37: 642–644.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Herbert A. Hauptman
    • 1
  1. 1.Hauptman-Woodward Medical Research Institute, Inc.BuffaloUSA

Personalised recommendations