Section 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry

If noncrystallographic symmetry is present, an atom at a general position within the relevant volume will imply the presence of others within the same crystallographic asymmetric unit. If the noncrystallographic symmetry is known, then the positions of equivalent atoms may be generated from a single atomic position. The additional vector interactions which arise from crystallographically and noncrystallographically equivalent atoms in a crystal may be predicted and exploited in an interpretation of the Patterson function.

An object in real space which has a closed point group may incorporate some of its symmetry in the crystallographic symmetry. If there are l such objects in the cell, then there will be equivalent positions within each object. The `self-vectors' formed between these positions within the object will be independent of the position of the objects. This distinction is important in that the self-vectors arising from atoms interacting with other atoms within a single particle may be correctly predicted without the knowledge of the particle centre position. In fact, this distinction may be exploited in a two-stage procedure in which an atom may be first located relative to the particle centre by use of the self-vectors and subsequently the particle may be positioned relative to crystallographic symmetry elements by use of the `cross-vectors' (Table 2.3.5.1).

Table 2.3.5.1 | top | pdf | Possible types of vector searches Self-vectors Cross-vectors Dimension of search, n (1) Locate single site relative to particle centre   (2) Use information from (1) to locate particle centre (3) Simultaneous search for both (1) and (2). In general this is a six-dimensional search but may be simplified when particle is on a crystallographic symmetry axis (4) Given (1) for more than one site, find all vectors within particle   (5) Given information from (3), locate additional site using complete vector set

The interpretation of a heavy-atom difference Patterson for the holo-enzyme of lobster glyceraldehyde-3-phosphate dehydrogenase (GAPDH) provides an illustration of how the known noncrystallographic symmetry can aid the solution (Rossmann et al., 1972; Buehner et al., 1974). The GAPDH enzyme crystallized in a cell (a = 149.0, b = 139.1, c = 80.7 Å) containing one tetramer per asymmetric unit. A rotation-function analysis had indicated the presence of three mutually perpendicular molecular twofold axes which suggested that the tetramer had 222 symmetry, and a locked rotation function determined the precise orientation of the tetramer relative to the crystal axes (see Table 2.3.5.2). Packing considerations led to assignment of a tentative particle centre near .

Table 2.3.5.2 | top | pdf | Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell Rotation axes Polar coordinates (°) Cartesian coordinates (direction cosines) u v w 1 45.0 −7.0 0.7018 0.7071 −0.0862 2 180.0–55.0 38.6 0.6402 −0.5736 0.5111 3 180.0–66.0 −70.6 0.3035 −0.4067 −0.8616

An isomorphous difference Patterson was calculated for the K₂HgI₄ derivative of GAPDH using data to a resolution of 6.8 Å. From an analysis of the three Harker sections, a tentative first heavy-atom position was assigned (atom at x, y, z). At this juncture, the known noncrystallographic symmetry was used to obtain a full interpretation. From Table 2.3.5.2 we see that molecular axis 2 will generate a second heavy atom with co-ordinates roughly (if the molecular centre was assumed to be at ). Starting from the tentative coordinates of site , the site related by molecular axis 1 was detected at about the predicted position and the second site generated acceptable cross-vectors with the earlier determined site . Further examination enabled the completion of the set of four noncrystallographically related heavy-atom sites, such that all predicted Patterson vectors were acceptable and all four sites placed the molecular centre in the same position. Following refinement of these four sites, the corresponding SIR phases were used to find an additional set of four sites in this compound as well as in a number of other derivatives. The multiple isomorphous replacement phases, in conjunction with real-space electron-density averaging of the noncrystallographically related units, were then sufficient to solve the GAPDH structure.

When investigators studied larger macromolecular aggregates such as the icosahedral viruses, which have 532 point symmetry, systematic methods were developed for utilizing the noncrystallographic symmetry to aid in locating heavy-atom sites in isomorphous heavy-atom derivatives. Argos & Rossmann (1974, 1976) introduced an exhaustive Patterson search procedure for a single heavy-atom site within the noncrystallographic asymmetric unit which has been successfully applied to the interpretation of both virus [satellite tobacco necrosis virus (STNV) (Lentz et al., 1976), southern bean mosaic virus (Rayment et al., 1978), alfalfa mosaic virus (Fukuyama et al., 1983), cowpea mosaic virus (Stauffacher et al., 1987)] and enzyme [catalase (Murthy et al., 1981)] heavy-atom difference Pattersons. A heavy atom is placed in turn at all plausible positions within the volume of the noncrystallographic asymmetric unit and the corresponding vector set is constructed from the resulting constellation of heavy atoms. Argos & Rossmann (1976) found a spherical polar coordinate search grid to be convenient for spherical viruses. After all vectors for the current search position are predicted, the vectors are allocated to the nearest grid point and the list is sorted to eliminate recurring ones. The criterion used by Argos & Rossmann for selecting a solution is that the sum of the lookup Patterson density values achieves a high value for a correct heavy-atom position. The sum is corrected for the carpet of cross-vectors by the second term in the sum.

An additional criterion, which has been found useful for discriminating correct solutions, is a unit vector density criterion where is the number of vectors expected to contribute to the Patterson density value (Arnold et al., 1987). This criterion can be especially valuable for detecting correct solutions at special search positions, such as an icosahedral fivefold axis, where the number of vector lookup positions may be drastically reduced owing to the higher symmetry. An alternative, but equivalent, method for locating heavy-atom positions from isomorphous difference data is discussed in Section 2.3.3.5.

Even for a single heavy-atom site at a general position in the simplest icosahedral or virus, there are 60 equivalent heavy atoms in one virus particle. The number of unique vectors corresponding to this self-particle vector set will depend on the crystal symmetry but may be as many as for a virus particle at a general crystallographic position. Such was the case for the STNV crystals which were in space group C2 containing four virus particles at general positions. The method of Argos & Rossmann was applied successfully to a solution of the K₂HgI₄ derivative of STNV using a 10 Å resolution difference Patterson. Application of the noncrystallographic symmetry vector search procedure to a K₂Au(CN)₂ derivative of human rhinovirus 14 (HRV14) crystals (space group ) has succeeded in establishing both the relative positions of heavy atoms within one particle and the positions of the virus particles relative to the crystal symmetry elements (Arnold et al., 1987). The particle position was established by incorporating interparticle vectors in the search and varying the particle position along the crystallographic threefold axis until the best fit for the predicted vector set was achieved.