International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 
International Tables for Crystallography (2012). Vol. F, ch. 13.4, pp. 357358
Section 13.4.7. Finding the averaged density ^{a}Department of Biological Sciences, Purdue University, West Lafayette, IN 47907–1392, USA, and ^{b}Biomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854–5638, USA 
Electron density can be averaged (1) among the N NCSrelated molecules in the pcell (the real crystal unit cell), thus creating a new and improved map of the pcell; (2) among the N NCSrelated molecules in the pcell and placing the results into a standard orientation in the hcell; or (3) among the N NCSrelated molecules in different unit cells and placing the results back into the original different unit cells or into a standard hcell. 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 pcell. Associated with each grid point in the pcell 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 pcell 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 nonintegral 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 unitcell translations and crystallographic symmetry operations to bring them back into the asymmetric unit before the corresponding interpolated density can be calculated.
Averaging into the hcell can be done by a procedure similar to averaging in the pcell, 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 hcell (from pcell electron density) can be done with respect to the reference molecule centred at in the pcell. Each grid point is taken in turn in the hcell. The electron density at any grid point that is further away from than from is set to zero. Other gridpoint positions are expanded into the N equivalent positions in the pcell surrounding . The interpolated density is then found, averaged over the N equivalent positions, and stored at the original hcell grid point in successive sections, in the same way as in the pcell averaging. As in averaging within the pcell, 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 hcell density will show the molecular envelope and can be used to recompute an improved pcell 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 .
