The molecular replacement method. I. The rotation function problem, application to bovine liver catalase and STNV

Abstract

A method to predict all peaks of the rotation function corresponding to a given molecular crystal is formulated in a rigorous mathematical manner. The applicability of this method is shown both in the analysis of rotation function data and in testing the validity of model molecular crystals proposed on the basis of limited rotation function data. Possible models of crystalline bovine liver catalase are determined assuming that all peaks of the rotation function are contained in the set of peaks determined by Eventoff and Gurskaya. All peaks of the rotation function corresponding to a model of crystalline satellite tobacco necrosis virus (STNV) proposed by Klug and Akervall et aL are determined and tabulated. Experimental determination of all these peaks would be a final test of the validity of this generally accepted model of crystalline STNV.

The molecular replacement method. II. The translation function problem; a new translation function

“…In the case where the Patterson function of the crystal of unknown structure is compared with itself, a method to determine the orientation and point group of the molecules from the peaks of the rotation function, equation (1), has been formulated by Litvin (1975). This formulism has been used in the analysis of the rotation function of crystalline satellite tobacco necrosis virus (STNV) and crystalline bovine liver catalase (Litvin, 1975). As shown in the case of crystalline bovine liver catalase, the solution of the rotation function problem may not be unique (Eventoff & Gurskaya, 1975;Litvin, 1975).…”

“…This formulism has been used in the analysis of the rotation function of crystalline satellite tobacco necrosis virus (STNV) and crystalline bovine liver catalase (Litvin, 1975). As shown in the case of crystalline bovine liver catalase, the solution of the rotation function problem may not be unique (Eventoff & Gurskaya, 1975;Litvin, 1975). Similar methods for solving the translation problem have been used when all or part of the molecule's structure is known (Tollin, 1966;Crowther & Blow, 1967;Huber, 1969), with the use of known molecules of similar structure (Tollin, 1969;Lattman & Love, 1970), and when none of the molecule's structure is known (Rossmann, Blow, Harding & Coller, 1964;Dodson, Harding, Hodgkin & Rossmann, 1966).…”

“…We denote by {R(jk)} the set of all rotations which rotate the molecule at rj into the orientation of the molecule at rk. It has been shown (Litvin, 1975) that:…”

“…When y II = 0 the centre of the volume U of the Patterson is centred at x--Ajk and the inter-* It is easily shown that rotations contained in equation (8) satisfy equation (5), and we havc assumed that there are no additional rotations with this property. Such an assumption is basic to this analysis of the translation function and no justification will be given other than to point out similar assumptions which have been implicitly used in the analysis of the translation function T(x,m) and of the rotation function (Litvin, 1975).…”

Erratum to: Allowable irreducible representations of the point groups with five-fold rotational axes†

“…Using a new approach to the group representation theory -which in turn paved way for eigenfunction method -Chen [12] obtained the characters, IRs and isoscalar factors for the icosahedral point groups which have been of interest in connection with the vibrational and electronic propagation [13,14]. Interest in the icosahedral point groups also stemmed from the icosahedral symmetry of biological macromolecules [15].…”