THE HEAVY-ATOM PROBLEM - A STATISTICAL-ANALYSIS .1. A-PRIORI DETERMINATION OF BEST SCALING, LEVEL OF SUBSTITUTION, LACK OF ISOMORPHISM AND PHASING POWER

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 sigma'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 SIGMA(H) = SIGMA(i = 1)N f(i)2, where f(i) 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 theta/lambda. It is then shown how to derive a quantitative estimate of the respective contributions to SIGMA(H) 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. Podjarny & 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.