Phasing macromolecular structures via structure-invariant algebra

Abstract

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 2rr 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 2rr ambiguity in the estimate toHr of the triplet ~H + tpK + ~p-n-~:, solves by least squares the resulting redundant system of linear equations ~n + ~pK + ~p-a-K = tonK. 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.

High-resolution Structure of the Catalytic Domain of Avian Sarcoma Virus Integrase

“…After normalization (unpublished results), the two data sets were converted into a DE corresponding to the difference structure. In all, 189 DE values (e1.76) in the resolution range 20 to 3.7 Å were used to locate the four Se atoms by means of the minimal function (Hauptman & Hahn, 1993;Weeks et al, 1993Weeks et al, , 1994De Titta et al, 1994) and a program (Langs et al, 1995) especially adapted for this purpose. The positions and occupancies of the Se atoms determined by direct methods agree remarkably well with those obtained by MAD techniques and in phase-model refinements (see below).…”

Computational challenges for macromolecular structure determination by X-ray crystallography and solution NMRspectroscopy

“…Recently, the structure of crambin was solved by this method at atomic resolution with 16 converging solutions out of 1200 trials (Hauptman, personal communication). The minimal principle also shows promise for phase refinement of macromolecules when it is combined with anomalous dispersion diffraction data (Hauptman & Han, 1993). Roth (1991) has examined the application of direct methods to low-resolution data of crystals consisting of multi-protein complexes.…”