Phasing macromolecular structures via structure-invariant algebra

Owing to the breakdown of Friedel's law when anomalous scatterers are present, unique values of the three-phase structure invariants in the whole range from 0 to 2π are determined by measured values of diffraction intensities alone. Two methods are described for going from presumed known values of these invariants to the values of the individual phases. The first, dependent on a scheme for resolving the 2π ambiguity in the estimate ω_HK of the triplet φ_H + φ_K + φ_−H−K, solves by least squares the resulting redundant system of linear equations φ_H + φ_K + φ_−H−K = ω_HK. The second attempts to minimize the weighted sum of squares of differences between the true values of the cosine and sine invariants and their estimates. The latter method is closely related to one based on the `minimal principle' which determines the values of a large set of phases as the constrained global minimum of a function of all the phases in the set. Both methods work in the sense that they yield values of the individual phases substantially better than the values of the initial estimates of the triplets. However, the second method proves to be superior to the first but requires, in addition to estimates of the triplets, initial estimates of the values of the individual phases.