Direct methods and isomorphous replacement. The triplet invariant estimate when heavy atoms are located

Abstract

The probabilistic theory of the three-phase structure invariants for isomorphous pairs has been generalized to the case in which a heavy-atom structure model is available. The rigorous method of joint probability distributions has been applied: it is able to handle errors in measurements and in the heavy-atom structure model, as well as the lack of isomorphism. The conclusive formulas have been successfully applied to experimental data. NotationF p jF p j expi0 p : structure factor of the protein F d jF d j expi0 d : structure factor of the isomorphous derivative F H F d À F p : structure factor of the heavy-atom structure (i.e. the atoms added to the native protein)pseudo-normalized structure factor of the derivative (normalized with respect to the native protein structure) E H : pseudo-normalized structure factor of the heavy-atom structure (normalized with respect to the native protein structure) IntroductionThe probability theory of the three-phase structure invariants for isomorphous pairs was initiated by Hauptman (1982;Hauptman et al., 1982) who derived the joint probability distributionThe Hauptman approach was revisited by Giacovazzo et al. (1988), who derived an ef®cient and simple formula for estimating the triplet phase È p . Their conclusive expression may be written asd j are the structure-factor moduli of the protein and of the derivative, respectively, normalized with respect to the heavy-atom structure. The formulas (2) and (3a) were implemented into a direct-methods procedure aimed at phasing protein structure factors without any information on the heavy-atom positions (Giacovazzo,