Molecular structure Patterson maps

The crux of the method is that the relative positions of the heavy atoms in the two different crystals must be known. When nothing detailed is known of the molecular structure, it is not easy to obtain this information. Perutz (1956) devised methods based on Fourier syntheses of the Patterson type referred to in a later section, which give interatomic vector maps the combined data for the two heavy-atom derivatives, in special correlation functions, give the relative positions... 

I will show, in a two-dimensional example, how to construct the Patterson map from a simple crystal structure and then how to use a calculated Patterson map to deduce a structure (Fig. 6.10). The simple molecular structure in Fig. 6.10a contains three atoms (dark circles) in each unit cell. To construct the Patterson map, first draw all possible vectors between atoms in one unit cell, including vectors between the same pair of atoms but in opposite directions. (For example, treat 1 —> 2 and 2 — 1 as distinct vectors.) Two of the six vectors (1 — 3 and 3 —> 2) are shown in the figure. Then draw empty unit cells around an origin (Fig. 6.10b), and redraw all vectors with their tails at the origin. The head of each vector is the location of a peak in the Patterson map, sometimes called a Patterson "atom" (light circles). The coordinates (u,v,w)... 

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. 

Heavy-atom method Relative phases calculated for a heavy atom in a location determined from a Patterson map are used to calculate an approximate electron-density map. Further portions of the molecular structure may be identified in this map and used to calculate better relative phases, and therefore a more realistic electron-density map results. Several cycles of this process may be necessary in order to determine the entire crystal structure. 

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. 

This alternative way of looking at a Patterson map is illustrated by a four atom and a five atom structure in Figures 9.5 and 9.6. This is sometimes a useful way of considering the Patterson map because it provides the basis for various kinds of Patterson search methods where the objective is to find the image of a known part of a molecule in Patterson space. It is also the basis for the rotation and translation functions used in molecular replacement procedures (see Chapter 8). 

The molecular replacement method used for protein structure determination (50,51) involves determining the orientation and the position in the unit cell of a known structure such as that of a homologous protein that has previously been determined or the same protein in a different unit cell (a polymorph). For the rotation function the Patterson map is systematically laid down upon itself in all possible orientations (Fig. 23). Six parameters that define the position and orientation of the protein in the unit cell are found from maxima in a function that describes the extent of overlap between the two placements of the Patterson function. This function will reveal the relative orientations of protein molecules in the unit cell. The rotation function is thus a computational tool used to assess the agreement or degree of coincidence of two Patterson functions, one from a model and the other from the diffraction pattern. 

Alternative ways of exploiting the information in a Patterson map are known as molecular replacement methods, being able to locate structural fragments and using this initial phase information to proceed as in the case of heavy atoms. 

For macromolecules such as proteins, the numbers of atoms that compose molecules are huge, therefore the crystal cells contain large numbers of atoms. It is not possible to apply the methods for small molecules, such as the direct method or Patterson map searching, in the structure determinations of proteins. The methods for retrieving the phases of protein crystal diffractions are molecular replacement, isomorphous replacement and anomalous scattering. In recent years, the direct method, which has been widely and successfully used in the determination of small-molecule structures, has also been applied in protein crystallography. 

The sketch map of finding the rotation in molecular replacement. The target structure is unknown and a similar model structure is known. By rotating the Patterson map of model structure and calculating the overlap of the two Patterson maps, the rotation function is defined. The max-imums of the rotation function yield the orientation of the target molecules in the crystals. 

Solution and Refinement of the Structure of Siliconate 6. All calculations were performed using the TEXSAN (14) crystallographic software package of Molecular Structure Corporation. The structure was solved by a combination of the Patterson method and direct methods (15). The non-hydrogen atoms were refined anisotropically. The final cycle of full-matrix least-squares refinement (16) was based on 2956 observed reflections [I > 3.00a(I)] and 478 variable parameters and converged to R = 0.051 and Rw = 0.059 The maximum and minimum peaks on the final difference Fourier map corresponded to 0.46 and -0.61 e"/A, respectively. 

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