Patterson maps described

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 Patterson map, commonly designated P(uvw), is a Fourier synthesis that uses the indices, h,k,l, and the square of the structure factor amplitude, F(hkl), of each diffracted beam. It is usual to describe the Patterson map in vector space defined by u, v, and w, rather than x,y,z as used in electron-density maps. 

In a minimum-function map, the origin of the Patterson map is put in turn on each of the known symmetry-related positions of a heavy atom that has already been located from a Patterson map. On each superposition of the origin of the Patterson map onto the various symmetry-related heavy-atom positions, the lowest value at each superimposed grid point in the pairs of maps is recorded. This superposition process is repeated until the structure is revealed. In this way the lighter atoms can be located. The method is an alternative to the heavy-atom method just described and has proved useful in many cases. 

The method of isomorphous replacement is the primary method used to determine the relative phases of protein crystal structures. The phenomenon of isomorphism was first described by Mitscherlich in 1819. and is described in Chapter 2. Isomorphous crystals have, by definition, almost identical structures, but with one or more atoms replaced by chemically similar ones (with different X-ray scattering power). The method by which relative phases are determined for a pair of isomorphous crystals depends on a knowledge of the intensity differences between the data sets for the two isomorphous crystals and the location of the varied atom, a quantity that is available from an analysis of the Patterson map or difference map. 

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. 

It may not be obvious how we would locate the x, y, z coordinates of the heavy atom in the unit cell. Indeed it is sometimes not a simple matter to find those coordinates, but as for the heavy atom method described above, it can be achieved using Patterson methods (described in Chapter 9). As we will see later, Patterson maps were used for many years to deduce the positions of heavy atoms in small molecule crystals, and with only some modest modification they can be used to locate heavy atoms substituted into macromolecular crystals as well. Another point. It is not necessary to have only a single heavy atom in the unit cell. In fact, because of symmetry, there will almost always be several. This, however, is not a major concern. Because of the structure factor equation, even if there are many heavy atoms, we can still calculate Juki, the amplitude and phase of the ensemble. This provides just as good a reference wave as a single atom. The only complication may lie in finding the positions of multiple heavy atoms, as this becomes increasingly difficult as their number increases. 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

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. 

The model protein is used to search the crystal space until an approximate location is found. This is, in a simplistic way, analogous to the child s game of blocks of differing shapes and matching holes. Classical molecular replacement does this in two steps. The first step is a rotation search. Simplistically, the orientation of a molecule can be described by the vectors between the points in the molecule this is known as a Patterson function or map. The vector lengths and directions will be unique to a given orientation, and will be independent of physical location. The rotation search tries to match the vectors of the search model to the vectors of the unknown protein. Once the proper orientation is determined, the second step, the translational search, can be carried out. The translation search moves the properly oriented model through all the 3-D space until it finds the proper hole to fit in. 

---