Section 25.2.1.10. Noncrystallographic symmetry calculations

NC symmetry operators are specified in terms of the parameters φ, ψ, χ, , , and t, which refer to a Cartesian coordinate system in Å, obtained by orthogonalization of the unit cell as in the Protein Data Bank (Bernstein et al., 1977). The angles φ and ψ determine the direction of the NC rotation axis, while χ determines the amount of rotation about it. , and are coordinates of a point through which the rotation axis passes, and t is a post-rotation translation parallel to the rotation axis. The relationships between the angles, orthogonal reference axes X, Y, Z and the unit cell are given in Fig. 25.2.1.2. Coordinates for a pair of points related by NC symmetry are then expressed in the orthogonal system by where and are three-element column vectors containing coordinates for the related points, is a 3 × 3 rotation matrix derived from the angles, O is a three-element column vector containing coordinates for a point through which the rotation axis passes, t is the post-rotation translation scalar in Å and is a three-element column vector containing direction cosines of the rotation axis. This type of parameterization simplifies transfer of information from self-rotation functions, which are usually calculated in spherical polar angles anyway, and also makes obvious pseudo-space-group symmetry type operations such as pseudo-screw axes. For convenience, a program O_TO_SP is provided to convert from a 3 × 3 rotation matrix and 1 × 3 column vector representation of the NC symmetry operation, as used in some programs, to the parameters described here.

Figure 25.2.1.2| top | pdf | Relationships between noncrystallographic symmetry rotation axis direction, orthogonal reference system axes X, Y, Z and crystallographic axes. The X axis is aligned with the crystal a. The Y axis is parallel to . The Z axis is parallel to , i.e. . ψ is the angle between the NC rotation and +Y axes. φ is the angle between the projection of the NC rotation axis in the XZ plane and the +X axis, with +φ counterclockwise when viewed from +Y toward the origin. χ is the amount of rotation about the directed axis, with +χ clockwise when viewed from the axis toward the origin.

Refinement of the NC symmetry operator parameters is achieved by least-squares minimization of the squared difference in electron density for all NC-symmetry-related points. Thus, one minimizes with respect to the operator parameters, where the sum is taken over all points within the appropriate averaging envelope(s). One starts refinement with low-resolution data (∼6 Å) on a coarse (∼2 Å) grid and monitors progress by following the correlation coefficient between the related electron-density values. Once convergence is obtained, the calculation is resumed with higher-resolution data on a finer grid. Typically, a correlation coefficient of around 0.4 or higher (for a 3 Å MIR map, 1 Å grid) indicates that the operator has been correctly located. The operator refinement is confined to submaps and is facilitated by use of an orthogonal grid. A submap containing the molecules to be averaged is obtained from the programs MAPVIEW or EXTRMAP and can be converted to an orthogonal grid, if needed, by the program MAPORTH, as described earlier.

Masks encompassing the region(s) to be averaged are usually created by hand in the interactive program MAPVIEW. Here, one reads in a submap comprising the desired region of whatever type of map is available, usually an MIR map. An appropriate contour level and initial section are selected and the contoured electron density for that section appears on the screen. One then selects the `add next section' menu item two or three times to create a projection over several sections of the map, since in the projection the molecular boundary is usually more obvious. Selecting the `trace mask' menu item then allows the user to hand-contour the molecular envelope by directing the cursor tied to a mouse or other pointing device. One then moves to an adjacent section and repeats the process until the complete 3D mask is obtained. To simplify matters and speed up the process, there are `copy next mask' and `copy previous mask' menu items allowing one to take advantage of the fact that the mask is a slowly changing function, particularly when near the centre of the molecule. One can use this feature to copy a mask from the previous or following section and apply it to the current section. Up to twelve distinct 3D masks can be selected. Each mask is colour-coded and can be simultaneously displayed superimposed on the contoured electron-density section. Once the mask is completed, the `make asu' menu item is selected to apply crystallographic symmetry operations to all points within the generated envelope masks. If these operations generate a point also within the envelope masks, the point is flagged in red to indicate that it is redundant, indicating that when tracing the mask, one inadvertently strayed into a symmetry-related molecule. After this check for redundancy, all points within the submap distinct from but related to points within the molecular envelopes by crystal symmetry are flagged in green. This enables one to detect packing contacts and also to ensure that all significant electron density has been assigned to some envelope. Upon completion, the mask is written to a file suitable for use either in averaging, solvent flattening (after expansion by BLDCEL), or operator refinement. In cases of `proper' NC symmetry, it is often desirable to trace the averaging envelope mask in a `skewed' map, such that one is looking directly down the NC rotation axis. In this case, it is usually very obvious where the NC symmetry breaks down, simplifying identification of the averaging envelope. If the averaging mask is created in a skewed submap, then the batch program TRNMSK can be used to transform it so as to correspond to the original, unskewed submap for use in averaging calculations (which do not require skewing).

All averaging calculations are carried out by the program MAPAVG (batch), which requires the submap to be averaged along with the envelope masks and NC symmetry operators. A copy of the input submap is made and each grid point in the mask is examined in turn. If the grid point lies within any averaging envelope, then all points related to it by NC symmetry are generated from the operators and examined. If the generated points also lie within the appropriate envelope mask, the electron density there is interpolated, as described earlier, and the density values for all related points are summed. The average value of the electron density is then inserted at the original point in the submap copy. Upon completion, the averaged version of the submap is written to a file and correlation coefficients for regions related by the various NC symmetry operations are output. The averaged submap is then passed to the program BLDCEL along with the averaging mask and the original unaveraged FSFOUR map from which the submap was created. For all points within the averaging envelope(s), their electron-density values and those at points related by crystallographic symmetry are inserted into the full-cell map, and it is written to a file. This file then contains the NC symmetry averaged electron density expanded to a full-cell map that obeys space-group symmetry. As an option, the averaging mask can also be expanded in BLDCEL to a full-cell mask, which could then be used for solvent flattening.

During NC symmetry averaging, phase combination and extension is carried out precisely as described during solvent flattening and negative-density truncation. The only difference is that after generation of each electron-density map, the NC symmetry averaging is carried out on the appropriate submap region, which is then expanded back to a full-cell map prior to each solvent-flattening calculation.