The two-dimensional histogram as a constraint for protein phase improvement.

The joint distribution of electron density and its gradient in a protein electron-density map was examined. This joint distribution was represented by a two-dimensional histogram (2D histogram) of electron-density values and the modulus of the gradient. 16 structures representing distinct protein-fold families were selected to study the dependence of the 2D histogram on resolution, overall temperature factor, structural conformation and phase error. The similarity between the histograms for a pair of structures was measured by correlation coefficient, and the residual provided a measure of the difference. The 2D histogram was found to vary with resolution and overall temperature factor, but was found to be insensitive to structure conformation. The average correlation coefficient between pairs of 2D histograms at three different resolutions examined was 0.90 with a standard deviation of 0.04. The average residual for the same condition was 0.13 with a standard deviation of 0.03. The 2D histogram was also found to be sensitive to phase error. The average correlation coefficient and residual between 2D histograms with 10 degrees phase difference are 0.71 and 0.18, respectively. The variation of the 2D histogram resulting from structure-conformation changes was estimated to be equivalent to that of a 4 degrees phase error. This establishes the minimal phase error that a 2D histogram-matching method could achieve. The conservation of the 2D histogram with respect to structure conformation enables the prediction of the ideal 2D histogram for unknown structures. The sensitivity of the 2D histogram to phase error suggests that it could be used as a target for the density-modification method and also could be used as a figure of merit for phase selection in ab initio phasing.