Enantiomorphous phase doublets from isomorphous replacement or anomalous scattering

A direct-methods procedure has been proposed for separating the phase doublet resulting from the use of either isomorphous replacement or anomalous scattering techniques. The phase doublet is expressed as ~#u= ~h±lA~.l. Formulae combining the structure-factor relationships with the phase-doublet information are given. Problems concerning the practical applications are also discussed. A test calculation with the error-free data for the protein insulin showed a satisfactory result. Introduction The idea of combining direct methods with isomorphous replacement or anomalous scattering data was first introduced in the 1960s by several authors independently. Coulter (1965) suggested the use of the tangent formula with starting phases extracted from the single isomorphous replacement data. This method does not make full use of the information which could be obtained from a single isomorphous pair. Fan (1965a) and Karle (1966) suggested the use of 'component relationships', i.e. the relationships among the real and imaginary components of the structure factors. If the arrangement of the heavy atoms is centrosymmetric then this makes the problem of breaking the phase ambiguities just that of finding the signs for the real or imaginary components of the structure factors by direct-method procedures. However, if the arrangement of heavy atoms is noncentrosymmetric then the component relation is not convenient to use. A phase-difference relation is given here together with the associated probability formula. This enables one to treat the problem of phase ambiguities arising from single isomorphous replacement (SIR), as well as that from one-wavelength anomalous scattering (OAS), by a simple and unified manner, no matter what the arrangement of the heavy atoms is. Recently, Hauptman (1982a, b) integrated the probabilistic theory of the three-phase structure invariants with the techniques of isomorphous replacement and anomalous scattering leading to a series of complex formulae. Our method differs from * Part of this paper was presented at the IUCr Summer School on Crystallographic Computing, Kyoto, Japan, 1983. Hauptman's in that, instead of two sets of diffraction data, e.g. one for the native protein and one for the derivative in the SIR case, only one set of diffraction data but with the phase-doublet information are introduced into the probability formula. Enantiomorphous phase doublets from isomorphous replacement or anomalous scattering In the case of SIR (see Blundell & Johnson, 1976, for details), for a given reciprocal vector H, we have FH.p = FH.~-FH.Q, (1) where FH,p is the structure factor of the native protein, FH,po is that of the heavy-atom derivative and FH,Q the heavy-atom contribution to FH,pO. From experiment the magnitudes of FH,p and FH,PO can be obtained. Accordingly, the parameters of the heavy atoms can be found and FH,O be calculated. Hence, we have two ways for drawing the triangle of (1) leading to an enantiomorphous phase doublet for FH, v or for FH,pQ in the phase-vector diagram, as shown in Fig. 1. The phase doublets are of the form ~.= ~h±lA~.l, (2) where ~H is the phase of the structure factor FH for the native protein or for the heavy-atom derivative,