Structure-factor probabilities for related structures

Probability relationships between structure factors from related structures have allowed previously only for either differences in atomic scattering factors (isomorphous replacement case) or differences in atomic positions (coordinate error case). In the coordinate error case, only errors drawn from a single probability distribution have been considered, in spite of the fact that errors vary widely through models of macromolecular structures. It is shown that the probability relationships can be extended to cover more general cases. Either the atomic parameters or the reciprocal-space vectors may be chosen as the random variables to derive probability relationships. However, the relationships turn out to be very similar for either choice. The most intuitive is the expected electron-density formalism, which arises from considering the atomic parameters as random variables. In this case, the centroid of the structure-factor distribution is the Fourier transform of the expected electron-density function, which is obtained by smearing each atom over its possible positions. The centroid estimate has a phase different from, and more accurate than, that obtained from the unweighted atoms. The assumption that there is a sufficient number of independent errors allows the application of the central limit theorem. This gives a one- (centric case) or two-dimensional (non-centric) Gaussian distribution about the centroid estimate. The general probability expression reduces to those derived previously when the appropriate simplifying assumptions are made. The revised theory has implications for calculating more accurate phases and maps, optimizing molecular replacement models, refining structures, estimating coordinate errors and interpreting refined B factors.