International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 12.2, pp. 257-258
Section 12.2.2. The Patterson function^{a}Institut für Pharmazeutische Chemie der Philipps-Universität Marburg, Marbacher Weg 6, D-35032 Marburg, Germany, and ^{b}Max-Planck-Institut für Biochemie, 82152 Martinsried, Germany |
Although the set of measured intensities contains no information regarding the phases, the Fourier transform of the intensities, the so-called Patterson function, contains valuable information. Patterson (1934) showed that the inverse Fourier transform of the intensity, is related to the electron density by
The Patterson function is an autocorrelation function of the density. For every vector u that corresponds to an interatomic vector, will contain a peak (Fig. 12.2.1.1). These are some properties of the Patterson function:
For simple crystals, the Patterson map can be used to solve the structure directly. For macromolecular structures, the Patterson map provides a vehicle for solving the phase problem.
If the crystal contains rotational symmetry elements, then the cross vectors between and its symmetry mate lie on a plane perpendicular to the symmetry axis – the Harker section (Harker, 1956). By way of example, the space group has two symmetry-related positions (Fig. 12.2.2.1),
Cross vectors between symmetry-related points will therefore have the form i.e. all cross vectors lie on the plane . For space group , the general coordinates give rise to cross vectors i.e. there are three Harker sections: , and . Peaks occurring on the Harker sections must reduce to a self-consistent set of coordinates (x, y, z), allowing reconstruction of the atomic positions.
If we have two isomorphous (see below) data sets and , then the difference in the two Patterson functions, will deliver information about the heavy-atom structure. Such a difference function gives rise to non-negligible peaks arising from interference between the and terms, however (Perutz, 1956). Rossmann (1960) showed that these interference terms could be reduced through calculation of the modified Patterson function
In the case of a single-site derivative, peaks should occur only at the Harker vectors corresponding to the heavy-atom position. Even so, there is a choice of positions for the heavy atom: e.g., in the case, coordinates , where ξ, ν and ζ can each take the value 0 or , will all give rise to the same Harker vectors. This in itself is not a problem, relating to equivalent choices of origin and of handedness, but has important ramifications for multisite derivatives or multiple isomorphous replacement (see below).
If there is more than one site, then there will be two sets of peaks: one set corresponding to the Harker sections (self-vector set) and one set corresponding to the difference vectors between different heavy-atom sites (the cross-vector set). In this case, the choice of one heavy-atom position determines the origin and the handedness to which all other peaks must correspond. Thus, in the example, only one cross vector will occur for
An alternative to the Harker-vector approach is Patterson-vector superposition (Sheldrick et al., 1993; Richardson & Jacobson, 1987). The Patterson map contains several images of the structure that have been shifted by interatomic vectors (Fig. 12.2.2.2). If this structure is relatively simple (as is to be hoped for in a `normal' heavy-atom derivative), then it should be possible to deconvolute the superimposed structures by vector shifts (Buerger, 1959).
References
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