Variance of electron-density maps in space group P1.

The expected mean-square error of electron-density maps (observed and difference) is traditionally estimated as a function of the variance of the observed amplitudes. The usual purpose is to evaluate the reliability of the structural parameters suggested by the final electron-density maps. Accordingly, such calculations are performed after the refinement stage, when the phases are considered perfectly determined. In this paper a mathematical expression for the variance (observed, difference and hybrid) is obtained for each point of an electron-density map for the space group P1 under a different hypothesis: the current phases are distributed on the trigonometric circle about the correct values, according to von Mises distributions. The variance calculation may then be performed at any stage of the phasing process, starting from a random up to a highly correlated model. It has been shown that the variance does not change dramatically from point to point of the map; therefore emphasis has been given to the concept of map variance, which allows an easier study of its properties. When the model is highly correlated with the target structure the conclusive formulas reduce to those previously described in the literature. The properties of the variance are discussed: it is shown that they are the basis for the most successful phasing procedures.