The use of Sayre's equation with solvent flattening and histogram matching for phase extension and refinement of protein structures

A new method for phase refinement and extension, which combines Sayre's equation with solvent flattening and histogram matching, has been developed. Equations which express electron-density constraints are solved using discrete Fourier transforms to give a new approximation to the electron density. The formulation of the equations is in real space, which allows any set of constraints to be entered directly into the calculation. An application to the known structure of 2Zn insulin refined the 3 Å MIR phases from a mean phase error of 46 to 39° and extended the phases to 2 Å resolution with a mean overall phase error of 57°. Analysis of the phase errors shows that, for the strong reflexions, the new method determines phases with half the mean error of MIR phases.