Sparsity of the normal matrix in the refinement of macromolecules at atomic and subatomic resolution

The normal matrix in the least-squares refinement of macromolecules is very sparse when the resolution reaches atomic and subatomic levels. The elemen ts of the normal matrix, related to coordinates, thermal motion and charge- density parameters, have a global tendency to decrease rapidly with the int eratomic distance between the atoms concerned. For instance, in the case of the protein crambin at 0.54 Angstrom resolution, the elements are reduced by two orders of magnitude for distances above 1.5 Angstrom. The neglect a priori of most of the normal-matrix elements according to a distance criter ion represents an approximation in the refinement of macromolecules, which is particularly valid at very high resolution. The analytical expressions o f the normal-matrix elements, which have been derived for the coordinates a nd the thermal parameters, show that the degree of matrix sparsity increase s with the diffraction resolution and the size of the asymmetric unit.