Patterson maps derivation

Fig. 11 (a) Experimental Patterson map derived from the in-plane scattering from an air annealed NiO(lll) single crystal (b) Calculated Patterson map for a relaxed octopolar reconstruction (c) Comparison between the experimental (right half circles) and calculated (left half circles) structure factors. 

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... 

Since protein BLl 1 is nearly globular its location may be determined in a Patterson map with coefficients of [F(wild)-F(mutant)] and may serve, by itself, as a giant heavy-atom derivative. At preliminary stages of structure determination this approach may provide phase information and reveal the location of the lacking protein. 

An entire data set must be collected for each of these derivatives. The evaluation of the phases from these data is a complex mathematical process which usually involves the calculation first of a "difference Patterson projection."406 This is derived by Fourier transformation of the differences between the scattering intensities from the native and heavy atom-containing crystals. The Patterson map is used to locate the coordinates of the heavy metal atoms which are then refined and used to compute the phases for the native protein. 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

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. 

Isomorphous replacement method A method for deriving relative phases by comparing the intensities of corresponding Bragg reflections from two or more isomorphous crystals. If the locations in the unit cell of atoms that vary between each isomorph have been located, for instance from a Patterson map, then the relatiw phase of each Bragg reflection can be assessed if a sufficient number of isomorphs is studied (at least 1 for a centrosymmetric crystal, at least 2 for a noncentrosymmetrir crystal). 

Superposition method Analysis of the Patterson map by setting the origin of the Patterson map in turn on the positions of certain atoms whose positions may already be known, and then recording those areas of the superposed maps in which peaks appear that are derived from both maps. As a result, it may be possible to derive the atomic arrangement. 

Soak complexes of heavy atoms of various kinds into a crystal. Screen diffraction data from heavy-atom derivatives of the protein for differences in intensities from those in the native data set. Calculate difference Patterson maps and locate the positions of the heavy atoms in the 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. 

FIGURE 9.2 A section from a difference Patterson map calculated between a heavy atom derivative and native diffraction data (known as a difference Patterson map). This map is for a mercury derivative of a crystal of bacterial xylanase. The plane of Patterson density shown here corresponds to all values of u and w for which v =. Because the space group of this crystal is P2, this section of the Patterson map is a Harker section containing peaks denoting vectors between 2t symmetry related heavy atoms. 

FIGURE 9.11 The w = j plane of the difference Patterson map for the K2HgI4 heavy atom derivative of the hexagonal crystal form of the protein canavalin. The space group is P6, so w = is a Harker section. The derivative crystal contained two major K2HgI4 substitution sites and one minor substitution site per asymmetric unit. The Patterson peaks corresponding to those sites are marked with crosses. Note that the Patterson peak corresponding to the minor site cannot be discriminated from noise peaks in the Patterson map as is often the case. 

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 above features can make it difficult to derive the atomic positions from analysis of the Patterson map. The most traditional approach involves the use of the so-called Marker sections they contain the interatomic vectors between an atom and its symmetry equivalents. For instance, let us consider the space group P2j, with equivalent positions ... 

Fig. 21. Vectors in a Patterson map. (a) A peak in a Patterson map indicates that the vector defined between the origin of the Patterson map and the peak in it must be found between atoms in the crystal structure, (b) Harker sections of the Patterson map for a heavy-atom derivative of D-xylose isomerase (Courtesy of H. L. Carrell).

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). 

The largest non-origin peaks on the Patterson map (figure 2.12) should be due to the heavy atom vectors provided the derivative protein crystal is isomorphous with the unmodified or native protein crystal. 

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. 

Except for the iodinated derivative, the major site of substitution of the heavy atoms was found from three-dimensional difference Patterson maps and verified by cross difference-Fourier maps phased on other derivatives. Alternate cycles of phase calculation and refinement of heavy atom parameters were carried out until convergence was reached. Minor sites of substitution of the heavy atoms were located on Fourier maps in which the lack of closure error for the derivative was used as coefficients (11). Table I summarizes the refinement statistics as well. The final figure-of-merit (12) for the 2.5 A data is 0.77. 

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