X‐ray Diffraction: Principles

Jan Drenth, University of Groningen, The Netherlands

Published online: May 2003

DOI: 10.1038/npg.els.0002721

The properties of biological macromolecules cannot be fully understood without knowledge of their three-dimensional structure. For mutation of proteins and for rational drug design it is essential to know the three-dimensional structure. X-ray diffraction is one technique for structure determination. It requires the macromolecules to be present in a well-organized arrangement as crystals or as fibres; amorphous material does not provide sufficient information.

Keywords: X-rays; crystallography; crystals; diffraction

Figure 1. A unit cell in a crystal. The vectors a, b and c indicate the repeating distances in the crystal structure. Depending on relation between, and the values of, the six parameters a, b, c, , and , the crystal belongs to one of the seven crystal systems (see Table 1).
Figure 2. The beams reflected by a set of parallel planes amplify each other when the path difference between reflections from successive planes is equal to 1, 2, etc. Because the path difference is also 2d sin , the Bragg law is obtained: 2d sin = n × . Reproduced, in modified form, from Drenth (1999), with permission of Springer-Verlag.
Figure 3. A wave can be represented in the ligand diagram as a vector E in a plane with horizontal and vertical axes. The wave E can be regarded as the sum of two waves, one along the horizontal axis with amplitude E cos , and the other along the vertical axis with amplitude E sin . The mathematical expression for E is (E cos ) + i (E sin ) = E exp(i). Reproduced, in modified form, from Drenth (1999) with permission of Springer-Verlag.
Figure 4. A simple system consisting of two electrons, 1 and 2. The position of electron 2 with respect to electron 1 is given by vector r. Vectors s0 and s1 indicate the direction of the incident and scattered beams. The path difference between the wave involving electron 1 and the wave involving electron 2 is p1 + p2. Reproduced, in modified form, from Drenth (1999) with permission of Springer-Verlag.
Figure 5. Vectors s0 and s1 indicate the directions of the primary and diffracted beams respectively. Vector S = s1 – s0 is perpendicular to a plane that can be regarded as a plane reflecting the primary beam.
Figure 6. Each arrow in the ligand diagram represents the scattering by one unit cell. In the figure the scattering by nine unit cells (n = 0 to 8) is shown. In fact n is a very large number and the vectors point in all directions, the scattering by a crystal would be zero. However, in the special case that a • S is an integer, all vectors point to the right and the crystal does scatter. Reproduced, in modified form, from Drenth (1999) with permission of Springer-Verlag.
Figure 7. The construction of the diffracted beam by means of the Ewald sphere and the reciprocal lattice. M is the center of the sphere and O is the origin of the reciprocal lattice. The primary beam passes through M and is directed towards O. If the crystal is regarded as being situated at M, a diffracted beam leaves the crystal in the direction of MP where P is any reciprocal lattice point on the surface of the Ewald sphere.
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 References
    Bragg WL (1913) The diffraction of short electromagnetic waves by a crystal. Proceedings of the Cambridge Philosophical Society 17: 43–57.
    book Drenth J (1999) The theory of X-ray diffraction by a crystal. Principles of Protein X-ray Crystallography, 2nd edn, pp. 83–84. New York: Springer Verlag.
    Ewald PP (1913) Zur Theorie der Interferenzen der Röntgenstrahlen in Kristallen. Physikalische Zeitschrift 14: 465–472.
 Further Reading
    book Blundell TL and Johnson LN (1976) Protein Crystallography. London: Academic Press.
    book Drenth J (1999) Principles of Protein X-ray Crystallography, 2nd edn. New York: Springer Verlag.
    book Glusker JP, Lewis M and Rossi M (1994) Crystal Structure Analysis for Chemists and Biologists. New York: VCH.
    book International Union for Crystallography (1995) International Tables for Crystallography. Dordrecht: Kluwer Academic Publishers.
    book McRee DE (1993) Practical Protein Crystallography. San Diego: Academic Press.
    book Rhodes G (1993) Crystallography Made Crystal Clear. San Diego: Academic Press.

Related Sub-Topics:

X-Ray Crystallography | Biochemical and Biophysical Techniques
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Article Title: X‐ray Diffraction: Principles
 
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Drenth, Jan(May 2003) X‐ray Diffraction: Principles. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1038/npg.els.0002721]