The heavy-atom problem: a statistical analysis. I.A prioridetermination of best scaling, level of substition, lack of isomorphism and phasing power

Abstract

The classical problem of determining heavy-atom parameters in single or multiple isomorphous replacement methods is reconsidered in two related papers. This first paper systematically examines how to derive a priori statistical information concerning heavy atoms and lack of isomorphism (LOI). By a priori is meant without any knowledge other than that of the measured intensities (and their estimated σ's) of a `native' and `derivative' crystal pair, that is to say before any potential site of substitution has been determined. First, both the terms ΣH ΣNi =1 fi2, where fi is the scattering factor of the ith heavy atom and N is the number of sites and, simultaneously, the best scale factor between the `native' and `derivative' data are estimated a priori as a function of sin θ/λ. It is then shown how to derive a quantitative estimate of the respective contributions to ΣH of `signal' due to heavy atoms and `noise' from LOI. The actual heavy-atom contribution is obtained from estimates of both a global isotropic temperature factor and a global absolute occupancy factor. The noise contribution is obtained as a `LOI factor' analogous to a Debye–Waller term as shown by Read [Acta Cryst. (1990), A46, 900–912; Crystallographic Computing 5 (1991), edited by D. Moras, A. D. Podjamy & J. C. Thierry, pp. 158–167. IUCr/Oxford Univ. Press]. As an important consequence, the variation with resolution of both the `lack of closure' and derivative phasing power can be estimated.