Direct Phasing in Crystallography Fundamentals and Applications

About this book

Direct methods are, at present, applied to a large variety of cases: X-ray, neutron or electron data; single crystal and powder data; small molecules and macromolecules. While direct methods solved in practice the phase problem for small molecules, their application to macromolecules is recent and still undergoing strong development. The fundamentals of the methods are described: in particular it is shown how the methods can be optimized for powder, neutron or electron data, and how they can be integrated with isomorphous replacement, molecular replacement and anomalous dispersion techniques. Maximum Entropy methods are also described and discussed. Sets of test structures are used to verify, throughout the various chapters, the mathematical techniques there described and to provide practical examples of applications. This book will appeal to a wide variety of readers - offering both a comprehensive description of direct methods in crystallography and an invaluable reference tool. The first three chapters can be considered as an introduction to the field, with sufficient material to constitute a university course and for allowing the expert use of most direct methods programs. Subsequent chapters are aimed at graduate students and working crystallographers. Basic results are described and discussed in the main body of the text, while the appendices compliment these with in depth mathematical details. The quoted literature is extremely wide and the interested reader can find suggestions for future work and further reading throughout the book.

Contents

1. Wilson statistics; 1.A About pseudosymmetries; 1.B Fourier series of representation; 1.C A modified Wilson plot procedure; 2. Structure invariants and semi-invariants and the origin problem; 3. Triplet invariant estimation and classical direct phasing procedures; 3.A Other multi-solution methods; 4. Direct methods and real space properties; 4.A Direct methods and Patterson map properties: additional approaches; 4.B Estimating structures semi-invariants from Harker sections; 4.C Patterson deconvolution methods; 5. The method of joint probability distributions; 5.A P(E) and the saddle point method; 6. Representations of structure invariants and semi-invariants; 6.A Two algebraic propositions; 7. The probabilistic estimation of triplet invariants; 7.A The probabilistic estimation of quartet invariants; 8.A Mathematical derivation of the phase relationship; 8.B Mathematical derivation of the sign relationship; 8.C Special quartets; 9. The probabilistic estimation of quintet invariants; 10. The probabilistic estimation of 1- and 2- phase structure semi-invariants; 10.A Some mathematical details about the (1 estimates; 11. Probabilistic determinantal approaches; 12. Phasing neutron and electron data; ABOUT THE ELASTIC SCATTERING OF ELECTRONS: THE KINEMATICAL APPROXIMATION; 12.B About HREM image formation; 13. Direct phasing from powder data; 14. Macromolecular crystallography techniques and traditional direct methods; 14.A Crick and Magdoff relation; 14.B Protein phase estimate incorporating the treatment of errors; 14.C Solvent flattening; 14.D About Fourier synthesis of large use in macromolecular crystallography; 14.E Solvent content according to Matthews; 14.F Ab initio calculation of low-resolution envelopes; 15. The integration of direct methods with isomorphous replacement techniques; 15.A Karle's Algebraic rule Riso; 15.B About local scaling; 16. The integration of direct methods with anomalous dispersion techniques; 16.A Probabilistic treatment of the errors in the SIRAS case; 16.B Coefficients in the distribution; 16.C The maximal SAS principle; 17. Molecular replacement techniques and direct methods

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