General formalism for phase combination and phase refinement: a statistical thermodynamics approach in reciprocal space

The mean-field optimization methodology has been used to recast in a single formalism the problem of phase optimization using an arbitrary energy func tion in the presence of an experimentally determined phase probability dist ribution function. It results naturally in the generalization of the notion s of figure of merit and centroid phase where the weight of the energy refi nement is controlled by an effective temperature in a self-consistent manne r. In the limit of high temperature, the formalism reduces of course to the Blow & Crick [Acta Cryst. (1959), 12, 794-802] classical treatment. If a m odel is available, Sim's [Acta Cryst. (1960), 13, 511-512] weighting scheme for a combined map appears as the first step of a refinement to be conduct ed until self-consistency is achieved. Assuming that MIR phases exist and t hat they agree reasonably well with the phases of the model, a first-order expansion gives an estimate of the changes of weights and phases to be perf ormed for the Fourier synthesis. This provides for a new way of doing phase combination that might prove useful in challenging cases of model refineme nt, e.g. in large macromolecular complexes. Thermodynamic considerations ha ve been used to discuss the best determination of weights in phase refineme nt; they also suggest that a variational expression of maximum likelihood i s best suited as a target for refinement because it is the free energy of t he system. The formalism readily allows use of solvent flattening, density averaging and the atomicity criterion to refine phases, and automatically a ssigns a figure of merit to each reflection. Numerical tests of the method are presented in an attempt to resolve the phase-ambiguity problem of prote in crystallography in the centrosymmetric P (1) over bar space group using an energy derived from the Sayre equation.