Direct-space methods in phase extension and phase refinement. V. The histogram moments method

Any distribution is completely defined by its moments. It is shown that a process of phase refinement can be carried out, based on Fourier transforms, which modifies the moments of electron density, separately in the protein and solvent regions, towards target values. Tests have been carried out on two moderate-sized proteins with 800-900 atoms in the asymmetric unit, one containing heavy atoms and the other not. It has been found that refinement using the third moment about zero in the protein region is most effective and that refinement with higher moments, or in the solvent region, adds nothing useful. Two kinds of weights are necessary in the method. One is for giving a weighted mixture of new phase indications with original phase estimates from, say, multiple isomorphous replacement. The other weights are applied to the Fourier coefficients of density maps to give the best possible signal:noise ratio. These weights have been explored empirically and the best ones found are described. It is concluded that since the moments method, which changes phases in reciprocal space, is independent of other histogram-matching procedures, which change density in real space, it has something to offer in a refinement package containing several procedures.