The extension and refinement of the 1.9 Å spacing isomorphous phases to 1.5 Å spacing in 2Zn insulin by determinantal methods

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, F_obs, 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.