## Abstract

This chapter is intended to give an overall view of the process of structure solution with some of the basic theory behind it. It is possible to skip the most mathematical section, at any rate, on a first reading. There is a bibliography at the end of this chapter that should provide further reading matter for readers at every level of crystallographic experience.

### Keywords

Nickel Crystallization Quartz Graphite Hydrate### References

- 1.Eisenberg, D and Crothers, D (1979)
*Physical Chemistry with Applications to the Life Sciences*, Benjamin-Cummings, Redwood City, CAGoogle Scholar - 2.Bernal, J. D and Crowfoot, D (1934) Use of the centrifuge in determining the density of small crystals.
*Nature***134**, 809,810CrossRefGoogle Scholar - 3.Green, D. W Ingram, V M., and Perutz, M F (1954) The structure of haemoglobin IV Sign determination by the isomorphous replacement method.
*Proc Roy Soc***A225**, 287–307Google Scholar - 4.Bricogne, G (1984) Maximum entropy and the foundations of Direct methods.
*Acta Cryst***A40**, 410–445Google Scholar - 5.Bricogne, G. (1988) A Bayesian statistical theory of the phase problem I A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors.
*Acta Cryst***A44**, 517–545Google Scholar - 6.Bricogne, G (1991) Maximum entropy as a common statistical basis for all phase determination methods, in
*Crystallographic Computing 5*(Moras, D, Podjarny, A D, and Thierry, J C, eds), Oxford Umversity Press, Oxford, UKGoogle Scholar - 7.Gilmore, C J and Bricogne, G. (1991) Maximun entropy, likelihood, and the phase problem in single crystal and powder diffraction, in
*Crystallographic Computing 5*(Moras, D., Podjarny, A D, and Thierry, J C., eds.), Oxford University Press, Oxford, UKGoogle Scholar - 8.Patterson, A L (1935) A direct method for the determination of the components of interatomic distances in crystals.
*Z Krist***90**, 517–542Google Scholar - 9.Terwilliger, T. C, Kim, S-H, and Eisenberg, D. (1987) Generalized method of determining heavy-atom positions using the difference Patterson function.
*Acta Cryst***A43**, 1–5Google Scholar - 10.Sheldrtck, G M (1990) Phase annealing in SHELX-90 Direct methods for larger structures.
*Acta Cryst***A46**, 467–473Google Scholar - 11.Germam, G., Main, P, and Woolfson, M M (1971) The application of phase relationships to complex structures III The optimum use of phase relationships.
*Acta Cryst***A21**, 410–445Google Scholar - 12.Sheldrick, G M (1991) Chapter 13, in
*Crystallographic Computing 5*(Moras, D, Podjarny, A D., and Thierry, J C, eds.), Oxford University Press, Oxford, UKGoogle Scholar - 13.Jones, T A, Zou, J-Y, Cowan, S W, and Kjeldegaard, M. (1991) Improved methods for building protein models in electron density maps and the location of errors in these models.
*Acta Cryst***A47**, 110–119Google Scholar - 14.Jones, T. A and Kjeldegaard, M. (1994) Chapter 1, in
*From First Map to Final Model*(Bailey, S, Hubbard, R, and Waller, D, eds), CCP4 WorkgroupGoogle Scholar

### Bibliography Mathematical Texts

- Arfken, G (1970)
*Mathematical Methods for Physicists*Academic, New YorkGoogle Scholar - Bamberg, P and Sternberg, S (1991)
*A Course in Mathematics for Students of Physics*, vols 1 and 2 Cambridge University Press, New York.Google Scholar - Boas, M L (1983)
*Mathematical Methods for the Physical Sciences*Wiley, New YorkGoogle Scholar - DuChateau, P C (1992)
*Advanced Math for Physicists and Engineers*Harper Collins outline series, Harper Collms, New YorkGoogle Scholar - Fitts, D. D. (1974)
*Vector Analysis in Chemistry*McGraw Hill, New York.Google Scholar - Hirst, D M (1994)
*Mathematics for Chemists*Macmillan, New York. (This covers vector manipulation and Fourier transformation and is particularly recommended as an introduction)Google Scholar - Janin, J. (1985) Chapter 5, in
*Methodes Biophysiques pour l’etude des Macromolecules*, Hermann, ParisGoogle Scholar - Margeneau, H. and Murphy, G M (1961)
*The Mathematics of Physics and Chemistry*van Nostrand, Princeton, NJGoogle Scholar - Prince, E (1994)
*Mathematical Techniques in Crystallography and Maternal Science*, 2nd ed, Springer Verlag, New York.Google Scholar - Stephenson, G. (1979)
*Mathematical Methods for Science Students*, 2nd ed., Longman, London.Google Scholar - Stephenson, G. (1985)
*Worked Examples in Mathematics for Scientists and Engineers*. Longman, LondonGoogle Scholar

### Basic Introductions

- Dressier, D. and Potter, H (1991)
*Discovering Enzymes*W H Freeman, New York.Google Scholar - Matthew, C. K and van Holde, K E (1990)
*Biochemistry*, Benjamin-Cummings, Redwood City, CAGoogle Scholar - Stryer, L (1995)
*Biochemistry*, 4th ed, W H Freeman, New YorkGoogle Scholar

### Short Introductions to X-Ray Structure Determination

- Branden, C-I and Tooze, J (1991) Chapter 17, in
*Introduction to Protein Structure*, Garland, New YorkGoogle Scholar - Cantor, C R and Schimmel, P. R. (1980) Part II of
*Biophysical Chemistry*, W. H Freeman, New YorkGoogle Scholar - Eisenberg, D. and Crothers, D (1979) Chapters 16 and 17, in
*Physical Chemistry with Applications to the Life Sciences*Benjamin-Cummings Publishing Company, Redwood City, CA. (Contains clear description of X-ray diffraction, which includes interesting short biographies of J D Bernal and J-B J Fourier.)Google Scholar - Holmes, K C and Blow, D M (1965)
*Methods of Biochemical Analysts*, vol 13, Wiley, New York, 113–239CrossRefGoogle Scholar - Janin, J. (1985) Chapters 1–4, in
*Methodes Biophysiques pour l’etude des Macromolecules*. Hermann, Paris (This is a very good introduction for the French reader)Google Scholar - Perutz, M. (1992) Chapter 1 and Appendix 1, in
*Protein Structure, New Approaches to Disease and Therapy*. W. H. Freeman, New York.Google Scholar - Sawyer, L and Turner, M A (1992) Chapter 12, in
*Crystallization of Nucleic Acids and Proteins*(Ducruix, A and Geige, R., eds), IRL, New YorkGoogle Scholar - Stuart, D. and Jones, Y (1993) Chapter 9, in
*Protein Engineering*(Sternberg, M, ed), IRL, New York.Google Scholar - van Holde, E (1985)
*Physical Biochemistry*, 2nd ed, Prentice-Hall, Englewood Cliffs, NJGoogle Scholar

### Texts on X-Ray Structure Determination

- Blundell, T B and Johnson, L H (1976)
*Protein Crystallography*Academic, New York (This is an excellent, indispensible guide to the subject, although the data collection sections are now dated)Google Scholar - Buerger, M J (1959)
*Vector Space*Wiley, New YorkGoogle Scholar - Buerger, M J (1976)
*Contemporary Crystallography*McGraw Hill, New YorkGoogle Scholar - Bunn, C W (1961)
*Chemical Crystallography*, 2nd ed., Oxford University Press, New YorkGoogle Scholar - Drenth, J (1994)
*The Principles of Protein X-ray Crystallography*Springer-Verlag, New York.Google Scholar - Dunitz, J D (1979)
*X-ray Analysis and the Structure of Organic Molecules*Cornell University Press, Ithaca, N Y (This is a thorough treatment of small molecule crystallography)Google Scholar - Giaccovazzo, C, Monaco, H L, and Viterbo, B (1992)
*Fundamentals of Crystallography*Oxford University Press, New YorkGoogle Scholar - Glazer, A M (1987)
*The Structure of Crystals*, Adam Hilger, Bristol, UKGoogle Scholar - Glusker, J P and Trueblood, K N (1985)
*Crystal Structure Analysis, A Primer*, 2nd ed, Oxford University PressGoogle Scholar - Ladd and Palmer, R (1989)
*X-ray structure determination A Practical Guide*, 2nd ed, Wiley, New York (This is a very good on symmetry and space group derivations, covers both small-molecule and macromolecular crystallography )Google Scholar - Lifson, H and Taylor, C A (1958)
*Fourier Transforms and X-ray Diffraction*G Bell, LondonGoogle Scholar - Lipson, H S. (1970)
*Crystals and X-rays*Wykeham Publications Ltd, London (This is a clear elementary introduction )Google Scholar - Lipson, H and Cochran, W (1957)
*The Determination of Crystal Structures*G Bell, LondonGoogle Scholar - McRee, D E (1993)
*Practical Protein Crystallography*Academic, New YorkGoogle Scholar - Rhodes, C (1993)
*Crystallography Made Crystal Clear A Guide for Users of Macromolecular Models*Academic, New York.Google Scholar - Sherwood, D. (1976)
*Crystals, X-rays and Proteins*Wiley, New York (This is a very understandable treatment, which derives all the mathematical aspects of the subject)Google Scholar - Stout, G H and Jensen, L H. (1989)
*X-ray Crystal Structure Determination*, 2nd ed Wiley, New York (This provides good coverage of the basics of X-ray crystallography, primarily from a small-molecule perspective )Google Scholar - Wilson, H R (1966)
*Diffraction of X-rays by Proteins, Nucleic Acids, and Viruses*Edward Arnold, London. (This gives a description of diffraction and especially of helical diffraction by a member of the King’s College DNA group.)Google Scholar - Woolfson, M M. (1978)
*An Introduction to X-ray Crystallography*Cambridge University Press, Cambridge, UK. (This develops diffraction theory from a scattering theory perspective, rather than starting with the Bragg equation. It is an excellent treatment of Direct methods by a leader in the field and recommended for readers stronger in physics)Google Scholar - Woolfson, M M (1961) Direct Methods in Crystallography, Oxford University Press, New YorkGoogle Scholar

### Advanced Texts

- Dodson, G, Glusker, J P, and Sayre, D (eds.) (1981)
*Structural Studies on Molecules of Biological Interest*Oxford University Press, New YorkGoogle Scholar - Moras, D., Podjarny, A. D, and Thierry, J. C. (eds) (1991)
*Crystallographic Computing 5*Oxford University Press, New York (This is an extensive series of articles on all aspects of macromolecular structure solution )Google Scholar - Rollett, J. S. (1965)
*Computing Methods in Crystallography*Pergamon Press, OxfordGoogle Scholar - Rossman, M G. (ed.) (1972)
*The Molecular Replacement Method*Gordon and Breach, New YorkGoogle Scholar - Wyckoff, H. W, Hirs, C H W, and Timasheff, S N (eds.) (1985)
*Diffraction Methods for Biological Macromolecules, Methods in Enzymology*, vols. 114 and 115 Academic, New YorkGoogle Scholar

### CCP4 Weekend Workshops

- Helliwell, J R, Machin, P A, and Papiz, M Z (1987) Computational Aspects of Protein Crystal Data AnalysisGoogle Scholar
- Bailey, S, Dodson, E, and Phillips, S (1988) Improving Protein PhasesGoogle Scholar
- Goodfellow, J, Hendrick, K., and Hubbard, R (1989) Molecular Simulation and Protein CrystallographyGoogle Scholar
- Hendrick, K, Moss, D S, and Tickle, I J (1990) Accuracy and Reliability of Macromolecular Crystal StructuresGoogle Scholar
- Wolf, W, Evans, P R, and Leslie, A G. W (1991) Isomorphous Replacement and Anomalous ScatteringGoogle Scholar
- Dodson, E J, Gover, S, and Wolf, W. (1992) Molecular ReplacementGoogle Scholar
- Sawyer, L., Isaacs, N, and Bailey, S (1993) Data Collection and ProcessingGoogle Scholar
- Bailey, S, Hubbard, R, and Walter, D (1994) From First Map to Final ModelGoogle Scholar

## Copyright information

© Humana Press Inc. 1996