The joint probability distribution of structure factors incorporating anomalous-scattering and isomorphous-replacement data

Abstract

A method to derive joint probability distributions of structure factors is presented which incorporates anomalous-scattering and isomorphous-replacement data in a unified procedure. The structure factors F, and F_n, whose magnitudes are different due to anomalous scattering, are shown to be isomorphously related. This leads to a definition of isomorphism by means of which isomorphous-replacement and anomalous-scattering data can be handled simultaneously. The definition and calculation of the general term of the joint probability distribution for isomorphous structure factors turns out to be crucial. Its analytical form leads to an algorithm by means of which any particular joint probability distribution of structure factors can be constructed. The calculation of the general term is discussed for the case of four isomorphous structure factors in P1, assuming the atoms to be independently and uniformly distributed. A main result is the construction of the probability distribution of the 64 triplet phase sums present in space group P1 amongst four isomorphous structure factors F,, four isomorphous FK and four isomorphous F-H-K. The procedure is readily generalized in the case where an arbitrary number of isomorphous structure factors are available for FH, FK and F_n_K.

On direct-methods phase information from differences between isomorphous structure factors

“…The evaluations of (13) and (14) can be performed in a standard way (see, for example, Peschar, 1991) and result in…”

“…An important goal of the paper is the derivation of a new expression to estimate the triplet phase sums present among isomorphous data. It will be shown that the new procedure, supplemented by optimal doublet phase-sum estimates that use difference Patterson information [Kyriakidis, Peschar & Schenk (1993b), from now on referred to as KPS2], leads to far better results than obtainable by other j.p.d.-based expressions (Hauptman, 1982a, b;Giacovazzo, 1983;Giacovazzo, Cascarano & Zheng, 1988;Fortier & Nigam, 1989;Peschar & Schenk, 1991;hereafter P&S), in particular if the DR is small (Kyriakidis, Peschar & Schenk, 1993a; from now on KPS 1). In contrast with other DM techniques, the final triplet distributions in the SAS and 2DW cases seem to be of sufficient quality to be used in a normal DM procedure.…”

On the doublet phase sums of isomorphous data sets

“…Let us define a set of p isomorphous structure factors, (2) in accordance with Peschar & Schenk (1991) …”

“…Fortier & Nigam (1989) rationalized the similar expressions for the joint probability distributions (j.p.d.s) to be a result of isomorphous data sets. Recently, the full probabilistic integration of DM with any number and type of isomorphous data sets has been accomplished (Peschar & Schenk, 1991). Although test results (Furey, Chandrasekhar, Dyda & Sax 1990) exist which suggest that DM may be applicable in solving protein structures ab initio, the full potential of DM in this respect seems not to have been realized as yet.…”

“…For example, for a correct estimation of the quartet phase sum (order N-l), the triplets (of order N -1/2) that add up to the quartet are required. A recent investigation of two probabilistic formulae for the ab initio determination of protein structures (Peschar & Schenk, 1991) has revealed that correct doublet phase sums [which are of order O(N°)] are important for a correct triplet-phase-sum evaluation. For this purpose, a new diffraction ratio (DR) was developed which shows a linear relationship with the ideal doublet phase sum as calculated from the atomic coordinates (Kyriakidis, Peschar & Schenk, 1993).…”