INTERPRETING PATTERSON MAPS

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

The mathematical function that must be interpreted in order to deduce the heavy atom coordinates, a puzzle really, is called a Patterson function or Patterson synthesis (Patterson, 1935). It has a form similar to the equation for electron density except that all phases are effectively zero. It yields, also in a similar manner, a three-dimensional density distribution. The peaks in this map, however, do not correspond to electron density centers but mark the interatomic vectors relating those centers. 

The Patterson function has been employed since its formulation in 1935 for determining the locations of heavy atoms in crystals of conventional compounds. This alone made possible application of the heavy atom technique (see Chapter 8) for structure determination. For conventional molecules the information for the heavy atom positions is contained entirely within the native diffraction data, unlike macromolecules, where the information is embedded in differences between two independent data sets, or differences between Friedel mates. Aside from the coefficients employed, use of the function is virtually identical in all cases. Perhaps the major difference arises from the fact that diffraction data from macromolecular crystals, and therefore corresponding difference Patterson maps, contain more noise than 

Introduction to Macromolecular Crystallography, Second Edition By Alexander McPherson Copyright 2009 John Wiley Sons, Inc. 

It should be noted that there are alternatives in some cases to Patterson syntheses for locating heavy atom positions, and these are direct methods (see Chapter 8). Indeed direct methods have achieved sufficient power, in a mathematical sense, that they are now frequently competitive with Patterson methods. In cases where vast constellations of selenium atoms must be found for MAD analyses, often equivalent itself to solving a challenging conventional molecule structure, direct methods have become the methods of choice occasionally they are the only choice. 

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The interpretation of Patterson maps requires knowledge of crystallographic symmetry and space groups. Chapter 4 of Blundell and Johnson (1976) offers a concise review of these topics. The ease of interpretation of these maps depends on the quality of the data, the degree of isomorphism, the number of heavy-atom sites per macromolecule and the... 

Difference Fourier techniques are most useful in locating sites in a multisite derivative, when a Patterson map is too complicated to be interpretable. The phases for such a Fourier must be calculated from the heavy-atom model of other derivatives in which a difference Patterson map was successfully interpreted, and should not be obtained from the derivative being tested, in order not to bias the phases. Also, difference Fourier techniques can be used to test the correctness of an already identified heavy-atom site by removing that site from the phasing model and seeing whether it will appear in... 

Figure 6.10 Construction and interpretation of a Patterson map. (a) Structure of unit cell containing three atoms. Two of the six interatomic vectors are shown, (h) Patterson map is constructed by moving all interatomic vectors to the origin. Patterson "atoms" (peaks in the contour map) occur at the head of each vector. (c) Complete Patterson map, containing all peaks from (b) in all unit cells. Peak at origin results from self-vectors. Image of original structure is present (origin and two darkened peaks) amid other peaks, (d) Trial solution of map (c).lf origin and Patterson atoms a and b were the image of the real unit cell, the interatomic vector a - b would produce a peak in the small box. Absence of the peak disproves this trial solution.

I have shown that, in simple systems, Patterson functions can give us valuable clues about distances, even when we know nothing about phases (see Chapter 6, Section III.C). Diffraction from the randomly oriented molecules in a solution or powder would give a spherically averaged diffraction pattern, from which we can compute a spherically averaged Patterson map. Is this map interpretable ... 

Interpretation of interatomic vectors. Use of known atomic positions for an initial trial structure (a preliminary postulated model of the atomic structure) can be made, by application of Equations 6.21,4 and 6.21.5 (Chapter 6), to give calculated phase angles. Methods for obtaining such a trial structure include Patterson and heavy-atom methods. Such methods are particularly useful for determining the crystal structures of compounds that contain heavy atoms (e.g., metal complexes) or that have considerable symmetry (e.g., large aromatic molecules in which the molecular formula includes a series of fused hexagons). The Patterson map also contains information on the orientation of molecules, and this may also aid in the derivation of a trial structure. 

With a knowledge of the space-group symmetry, strong vectors can be analyzed to give the fractional atomic coordinates of those atoms in the structure that have the highest atomic numbers. If there is only one such heavy atom in the asymmetric unit, the interpretation of the Patterson map is simplified because the map is dominated by heavy-atom-hea ... 

Sharpened Patterson function A Patterson map computed with values of I F p enhanced as sin0/A increases. As a result the map contains sharper peaks and it easier to interpret. The Patterson map with — 1 as coefficients is commonly... 

Tong, L., and Rossmann, M. Patterson-map interpretation with noncrystallo-graphic symmetry. J. Appl. Cryst. 26, 15-21 (1993). 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

As with the isomorphous replacement technique it is necessary to identify the positions, the x, y, z coordinates of the anomalous scatterers. This can be done by anomalous difference Patterson maps, which are Patterson syntheses that use the anomalous differences Fhki — F—h—k—i as coefficients (Blow and Rossmann, 1961). These maps are interpreted identically to isomorphous difference Patterson maps (see Chapter 9). Rapidly surpassing Patterson approaches, particularly for selenomethionine problems and others where the number of anomalous scatterers tends to be large, are direct methods (see below). These are strictly mathematical methods that have proved to be surprisingly effective in revealing the constellation of anomalous scatterers in a unit cell. 

Unless direct methods are used to locate heavy atom positions, an understanding of the Patterson function is usually essential to a full three-dimensional structure analysis. Interpretation of a Patterson map has been one of two points in a structure determination where the investigator must intervene with skill and experience, judge, and interpret the results. The other has been the interpretation of the electron density map in terms of the molecule. Interpretation of a Patterson function, which is a kind of three-dimensional puzzle, has in most instances been the crucial make or break step in a structure determination. Although it need not be performed for every isomorphous or anomalous derivative used (a difference Fourier synthesis using approximate phases will later substitute see Chapter 10), a successful application is demanded for at least the first one or two heavy atom derivatives. 

Occasionally more than one pair of atoms in a structure may be related by identical vectors, for example, atoms that lie on opposite sides of a benzene ring. Therefore overlap can occur in Patterson space. For some points u, v, w the Patterson value P(u, v, w) will be the sum of more than one nonzero product. This is what makes interpretation of Patterson maps particularly challenging, and obviously this becomes more so as the number of atoms increases. In general, however, every pair of atoms in the unit cell will give rise to two centrosymmetrically related, nonzero vectors in Patterson space. Hence, if there are N atoms in the unit cell, there will be N(N — 1) peaks in the Patterson map, as well as the vector from every atom to itself, but this vector is always u = (0, 0, 0). 

Another problem that frequently arises with multiple isomorphous derivatives is that of handedness. In space group P2i2i2i, Patterson maps for two independent derivatives may be interpreted to yield a set of symmetry related sites for one derivative and, independently, a second set for the other. Because handedness is completely absent in a Patterson map (because it contains a center of symmetry), there is an equal chance that the heavy atom constellation for the first will be right handed, and the constellation for the other will be left handed, and vice versa. This won t do. The two heavy atom sets will not cooperate when used to obtain phase information. There are ways of unraveling this problem too, and once again, it involves difference Patterson maps between the two derivative data sets and cross vectors. This case can also be resolved by calculating phases based on only one derivative and then computing a difference Fourier map (see Chapter 10) for the other. 

The unit cell of a protein is assigned with respect to a right-handed system of axes. Once a heavy atom has been located, its phase angle may be +a or —a for FH, since it is not known whether the interpreted peak in the Patterson map is from atom 1 to atom 2 or atom 2 to atom 1. Several methods have been developed to remove this ambiguity of which the most decisive are those that involve the preparation of a derivative containing both heavy atoms and/or anomalous dispersion measurements (5). 

O to O by a vector OO = r. Therefore, the Patterson function will be large if the strong regions of electron density are separated by the vector r and if there are several strong regions of electron density separated by a vector r, the P(r) will show a total effect of it and will also be large. The Patterson function will be a superposition of peaks derived from all pairs of atoms in the unit cell. If there is no overlap of Patterson peaks, then the function P(r) will show the position of all interatomic vectors but usually the resolution of Patterson function is very poor and in fact the Patterson peaks are more diffuse than the electron density peaks. Despite this overcrowded nature of the Patterson maps, useful information can be derived and complex structures can be solved from an interpretation of Patterson function. 

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