International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 13.4, pp. 284-285
Section 13.4.7. Finding the averaged density
aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA, and bBiomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854-5638, USA |
Electron density can be averaged (1) among the N NCS-related molecules in the p-cell (the real crystal unit cell), thus creating a new and improved map of the p-cell; (2) among the N NCS-related molecules in the p-cell and placing the results into a standard orientation in the h-cell; or (3) among the N NCS-related molecules in different unit cells and placing the results back into the original different unit cells or into a standard h-cell. Before averaging commences, the matrices and translation vectors must be evaluated [see (13.4.5.9) and (13.4.5.11a )]. Here, N is the noncrystallographic redundancy and M is the number of molecules that impinge on the crystallographic asymmetric unit of the p-cell. Associated with each grid point in the p-cell asymmetric unit will be (1) the value of m designating which molecular centre is to be associated with that grid point (a special value of m is for solvent) and (2) the p-cell electron density at that point.
The grid points within the asymmetric unit are then examined one at a time. If the grid point is within the mask, it is averaged among the N noncrystallographically related equivalent positions belonging to molecule m. If the grid point is solvent, the density can be set to the average solvent density.
The N noncrystallographically equivalent non-integral grid points can be computed from (13.4.5.11a ). Some of these will lie outside the crystallographic asymmetric unit. These will, therefore, have to be operated on by unit-cell translations and crystallographic symmetry operations to bring them back into the asymmetric unit before the corresponding interpolated density can be calculated.
Averaging into the h-cell can be done by a procedure similar to averaging in the p-cell, except that the rotation and translation matrices are given by (13.4.5.13). Furthermore, no mask is required as all the averaging into the h-cell (from p-cell electron density) can be done with respect to the reference molecule centred at in the p-cell. Each grid point is taken in turn in the h-cell. The electron density at any grid point that is further away from than from is set to zero. Other grid-point positions are expanded into the N equivalent positions in the p-cell surrounding . The interpolated density is then found, averaged over the N equivalent positions, and stored at the original h-cell grid point in successive sections, in the same way as in the p-cell averaging. As in averaging within the p-cell, a record is kept of as a function of (Table 13.4.7.1). In general, the local NCS is valid only within the molecule. Hence, the h-cell density will show the molecular envelope and can be used to recompute an improved p-cell density mask. The rate of build up of signal within the molecule should be roughly proportional to N, while the rate outside the molecule should be proportional to about .
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