Direct phasing by binary integer programming

In the absence of phase information, a variety of electron-density distributions is consistent with the observed magnitudes. This ambiguity may be reduced significantly if the distribution values are restricted to 0 or 1, i.e. when the object of search is an envelope rather than a continuous electron-density distribution. The binarizing in both real (the grid-point density values) and reciprocal (the phases) spaces allows the usual structure-factor equations to be replaced by a system of linear inequalities with binary unknowns. A special computer procedure is applied to obtain several sets of values, which satisfy or almost satisfy these inequalities. The averaging of the found phase sets allows the final map to be calculated. The approach was tested with calculated and experimental data for a known protein structure. The size of the grid for the envelope calculation is at the moment the major limitation of the approach. Nevertheless, even for a very small grid, some structure information can be extracted and used as a starting point for further phase improvement or as a way to solve the molecular replacement problem.