Direct methods with single isomorphous replacement data. I. Reduction of systematic errors

The direct-methods procedure for single isomorphous replacement (SIR) data [Hauptman (1982). Acta Cryst. A38, 289-294], as modified by Fortier, Moore & Fraser [Acta Cryst. (1985), A41, 571-577] has been implemented and tested with a large number of known structures. It was found that the modified procedure greatly reduces the bias toward 'unresolved' SIR invariant values associated with estimates of 0 or pi, but does not remove it entirely. If the heavy atoms are not in a centrosymmetric array the centroid of the distribution of invariant estimates is not centered on true protein values, but is biased toward conventional SIR values by up to 15 degrees, thus errors in the estimates are not random but systematic. When the heavy atoms are in a centrosymmetric array (or single heavy-atom site in space group P21), the distribution of estimates is often sharply bimodal, with peaks centered at both true invariant values and pure 'unresolved' SIR values. Simple procedures are given which can be applied in both situations to reduce significantly the bias with no overall loss of accuracy. An additional correction factor is then described which can be used to remove nearly all of the bias, and improve the accuracy as well. The result is that errors in the corrected invariant estimates are small in magnitude, but are now also random instead of systematic. Since the number of estimates greatly exceeds the number of phases, the remaining random errors should have little impact in phasing processes.