Direct-space methods in phase extension and phase determination. III. Phase refinement using Sayre"s equation

An algorithm is described for refining a set of phases to agree with the Sayre equation. All operations are carried out using Fourier transforms with modest computer-store requirements even for very large systems. The procedure is tested with two moderate-sized proteins, one containing heavy atoms, and is found to give good refinement with data at more than atomic resolution (1.17 Å) and useful, if less good, refinement when the data resolution is lower (1.5 Å). It is concluded that at atomic resolution, or slightly below, the Sayre equation still has something to offer both for phase refinement and phase extension, especially if used cautiously with weighted multiple isomorphous replacement phases acting as a constraint on the phase changes. Even when the Sayre equation on its own refines phases badly, or not at all, it may still make an important contribution in conjunction with other real-space refinement procedures.