On integrating the techniques of direct methods and isomorphous replacement. III. The three-phase invariant for the native and two-derivative case

The probabilistic theory of the three-phase structure invariant for a triplet of isomorphous structures is worked out. In particular, when diffraction data are available for a native protein and two derivatives, the conditional distributions of the three-phase structure invariants, given the nine magnitudes in their first neighbourhoods, are derived for the special case that the heavy atoms of the two derivatives are located in different positions in the unit cell. The distributions have the form P(Ω) = (1/K) exp (A cos Ω), where the parameters K and A are functions of the nine magnitudes in the first neighborhood. In the favorable case that the variance of a distribution happens to be small, a reliable estimate, 0 or π, of the invariant is obtained. An example shows that these distributions, which employ simultaneously the diffraction data from a triple of isomorphous structures, yield more accurate estimates for the three-phase structure invariants than are obtainable from earlier distributions, which employ diffraction data from only a pair of isomorphous structures [Hauptman (1982). Acta. Cryst. A38, 289-294]. Unique origin and enantiomorph specification in direct-methods applications to all three structures is an advantage of the present approach.