Minimum-variance Fourier coefficients from the isomorphous replacement method by least-squares analysis

A theory for the simultaneous least-squares refinement of protein phases and heavy-atom parameters is presented. Weights are utilized which include experimental error in the heavy-atom structure amplitude as well as in the protein structure amplitude. Direct refinement of the cosine and sine functions of the phase, α, automatically furnishes the best phase thereby avoiding the calculation of the best phase via the Blow-Crick probability frequency function postulated for phase error. The refinement of protein phases is constrained such that cos² α + sin² α = 1 for all protein phases. A minimum-variance Fourier synthesis analogous to the best Fourier synthesis is formulated which has as weights a figure of merit not only based upon phase error as calculated by the new method but also including experimental error in the protein structure amplitude. Approximations show that correlations between protein phases and heavy-atom parameters can be neglected provided the ratio of heavy-atom parameters to protein phases is sufficiently small. Comparison of the new refinement with a conventional refinement utilizing two heavy-atom derivatives measured to 3 Å resolution shows that the new refinement gives better closure and a stronger heavy-atom signal. Comparison of electron density maps shows that the minimum-variance Fourier synthesis yields an electron density map of improved resolution with respect to the conventional best Fourier synthesis.