Unique solutions for the SIR and SAS phase problems and the use of partial structural information in phase refinement

Algebraic formulae are presented which permit a unique phased solution for diffraction data measured from a single isomorphous pair of crystals. Trial calculations performed on an SIR (single isomorphous replacement) data set from an 84-atom structure demonstrate that complete phasing can be achieved from a single chirally positioned replacement atom representing less than one percent of the total scattering power of the derivative structure. Similar phase refinements employing error-free SIR data for 2Zn pig insulin are less remarkable, and converge to an average phase error of 50°. The phase convergence of the formulae can be markedly improved if estimates of the cosine invariants from the SIR data are available [Hauptman (1982). Acta Cryst. A38, 289-294; Fortier, Moore & Fraser (1985). Acta Cryst. A41, 571-577]. The precision of these cosine estimates was found not to be critical; modular estimates of +1 or -1 were sufficient to allow the SIR phase refinement of the insulin structure to converge to an average phase error of 6°, which compares favorably with the value of 3° produced if the cosine invariants were known precisely. The derived formulae are also shown to be applicable to single-crystal analyses which utilize one-wavelength anomalous dispersion or partial structural fragments to initiate phasing. Test examples indicate that tangent-formula recycling procedures based on the derived formulae compare favorably with the traditional tangent-formula methods to exploit partial structure information.