Patterson function molecular replacement

The molecular replacement method assumes similarity of the unknown structure to a known one. This is the most rapid method but requires the availability of a homologous protein s structure. The method relies on the observation that proteins which are similar in their amino acid sequence (homologous) will have very similar folding of their polypeptide chains. This method also relies on the use of Patterson functions. As the number of protein structure determinations increases rapidly, the molecular replacement method becomes extremely useful for determining protein phase angles. 

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. 

DeLano, W. L. and Brunger, A. T. (1995) The direct rotation function Patterson correlation search applied to molecular replacement. Acta Crystallogr. D 51, 740-748. 

A GA method has been developed [92, 93] for ab initio phasing of low-resolution X-ray diffraction data from highly symmetric structures. The direct-space parameterization used incorporates information on structural symmetry, and has been applied to study the structures of viruses, with resolution as high as 3 A [93]. A GA has also been introduced [94] to speed up molecular replacement searches by allowing simultaneous searching of the rotational and translational parameters of a test model, while maximizing the correlation coefficient between the observed and calculated diffraction data. An alternative GA for sixdimensional molecular replacement searches has been described [95,96] and GA methods have also been used [97] to search for heavy atom sites in difference Patterson functions. 

It is possible, as shown by Rossmann and Blow (1962), to search for redundancies in Patterson space that correspond to the multiple copies of molecular transforms. Rossmann and Blow show, however, that the Patterson map does not need to be computed and used in any graphical sense, but that an equivalent search process can be carried out directly in diffraction or reciprocal space. Using such a search procedure, called a rotation function, they showed that noncrystallographic relationships, both proper and improper rotations, could be deduced in many cases directly from the X-ray intensity data alone, and in the complete absence of phase information. Translational relationships (only after rotations have been established) can also be deduced by a similar approach. Rotation functions and translation functions constitute what we call molecular replacement procedures. Ultimately the spatial relationships among multiple molecules in an asymmetric unit can be defined by their application. 

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. 

The Patterson function of a twinned stmcture is the sum of the Patterson functions of both domains. Therefore procedures using Patterson methods are in principle possible. There are several examples in the literature of stmctures solved by molecular replacement using twinned data (e.g. see Breyer et al, 1999). 

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. 

---