A new mathematical description of phase relationships which connects different approaches, both in reciprocal and direct space, is formulated. It leads to the development of a novel algorithm for phase extension and refinement based on a probability function for atomic presence. This function, calculated from the elements of the Karle-Hauptman inverse matrix, is used in an iterative procedure. Various tests have been performed on an idealized set of calculated structure factors for an insulin model structure. The method has been applied to experimental data, Fobs, and the isomorphous phases for 2Zn insulin. An assessment of the quality of the phase refinement and calculation has been made by comparison with the crystallographically refined phases. IntroductionDirect methods, at the present stage, are not powerful enough to allow the ab initio determination of protein structures. But they can be used in order to improve and/or extend a set of approximate phases at medium resolution. This extension may lead to a set of phases at higher resolution corresponding to a Fourier series which can provide a better set of atomic coordinates for subsequent refinement. Different mathematical approaches have been developed and used in practical work: the tangent formula method (Hendrickson, Love & Karle, 1973); the Sayre (1972Sayre ( , 1974 equations; the determinantal methods CasteUano, Podjarny & Navaza, 1973 ;Podjarny, Yonath & Traub, 1976;Podjarny & Yonath, 1977;Podjarny, Schevitz & Sigler, 1981); and the filter type methods (Gassmann, 1976;Raghavan & Tulinsky, 1979;Schevitz, Podjarny, Zwick, Hughes & Sigler, 1981;Collins, 1982).A brief description of the application of the determinantal methods to actual protein structures has * Present address: SERC Daresbury Laboratory, Warrington WA4 4AD, England.0108-7673/85/010003-!5501.50 been the object of several communications (Xth Int. Congr. Crystallogr., Amsterdam, 1975; IVth Eur. Crystallogr. Meet., Oxford, 1977; Xlth Int. Congr.. Crystallogr., Warsaw, 1978; de Rango, Mauguen, ~'soucaris, Dodson, Dodson & Taylor, 1979).Here, we give a new mathematical description, including the connection between different-mathematical approaches: the probability of atomic presence ~'(r) function, the regression method and the forbidden domain theory of von Eller. Then, we describe the algorithms and the practical procedure. Next, we give the results of preliminary tests on an idealized set of data constituted of calculated structure factors for an insulin model structure. Finally, we apply the procedure to actual problems using the insulin crystal data and report a detailed analysis of the results of phase extension of the 1.9/~ isomorphous data to 1.5/~ resolution. I. Theoretical aspects of the regression equation ( a) The probability of atomic presence functionsLet us consider a periodic function, p(r), whose Fourier coefficients are known up to a certain resolution, R. The Fourier series p0(r) = ~ En exp (-27rill. r), IHI < 1 / R, H is an approximation to p(r). If the set o...