Section 18.5.5.1. Block calculations

The full-matrix inversions described in the previous section require massive calculations. The length of the calculations is more a matter of the order of the matrix, i.e., the number of parameters, than of the number of observations. When restraints are applied, it is the diffraction-cum-restraints full matrix which should be inverted.

With the increasing power of computers and more efficient algorithms (e.g. Tronrud, 1999; Murshudov et al., 1999), a final full matrix should be computed and inverted much more regularly – and not just for high-resolution analyses. Low-resolution analyses have a need, beyond the indications given by B values, to identify through estimates their regions of tolerable and less tolerable precision.

If full-matrix calculations are impractical, partial schemes can be suggested. As far back as 1973, Watenpaugh et al. (1973), in a study of rubredoxin at 1.5 Å resolution, effectively inverted the diffraction full matrix in 200 parameter blocks to obtain individual s.u.'s. A similar scheme for restrained refinements could also use overlapping large blocks. A minimal block scheme in refinements of any resolution is to calculate blocks for each residue and for the block interactions between successive residues. The inversion process could then use the matrices in running groups of three successive residues, taking only the inverted elements for the central residue as the estimates of its variances and covariances.

For low-resolution analyses with very large numbers of atoms, it might be sufficient to gain a general idea of the behaviour of as a function of B by computing a limited number of blocks for representative or critical groups of residues. The parameters used in the blocks should include the B's, since atomic images overlap at low resolution, thus correlating the position of one atom with the displacement parameters of its neighbours.