GENERAL EXPRESSION FOR PROBABILISTIC ESTIMATION OF MULTIPHASE STRUCTURE-INVARIANTS IN THE CASE OF A NATIVE PROTEIN AND MULTIPLE DERIVATIVES- APPLICATION TO ESTIMATES OF THE 3-PHASE STRUCTURE-INVARIANTS

Concise probabilistic formulae with definite crystallographic implicat ions are obtained from the distribution for eight three-phase structur e invariants (3PSIs) in the case of a native protein and a heavy-atom derivative [Hauptman (1982). Acta Cryst. A38, 289-294] and from the di stribution for 27 3PSIs in the case of a native and two derivatives [F ortier, Weeks & Hauptman (1984). Acta Cryst. A40, 646-651]. The main r esults of the probabilistic formulae for the four-phase structure inva riants are presented and compared with those for the 3PSIs. The analys is directly leads to a general formula of probabilistic estimation for the n-phase structure invariants in the case of a native and m deriva tives. The factors affecting the estimated accuracy of the 3PSIs are e xamined using the diffraction data from a moderate-sized protein. A me thod to estimate a set of the large-modulus invariants, each correspon ding to one of the eight 3PSIs, that has the largest \Delta\ values an d relatively large structure-factor moduli between the native and deri vative is suggested, which remarkably improves the accuracy, and thus a phasing procedure making full use of all eight 3PSIs is proposed.