Maximum-likelihood density modification

Abstract

A likelihood-based approach to density modification is developed that can be applied to a wide variety of cases where some information about the electron density at various points in the unit cell is available. The key to the approach consists of developing likelihood functions that represent the probability that a particular value of electron density is consistent with prior expectations for the electron density at that point in the unit cell. These likelihood functions are then combined with likelihood functions based on experimental observations and with others containing any prior knowledge about structure factors to form a combined likelihood function for each structure factor. A simple and general approach to maximizing the combined likelihood function is developed. It is found that this likelihood-based approach yields greater phase improvement in model and real test cases than either conventional solvent flattening and histogram matching or a recent reciprocal-space solvent-flattening procedure [Terwilliger (1999), Acta Cryst. D55, 1863-1871].

Figure 1

Sections of electron density obtained before and after density modification of phases obtained for IF-5A (Peat et al., 1998 ▶) phased using one Se atom in the asymmetric unit. Density modification was carried out by real-space solvent flattening and histogram matching or by maximum-likelihood solvent flattening. Values for real-space density modification were carried out using the program dm (Cowtan & Main, 1996 ▶), version 1.8, using solvent flattening with histogram matching. Starting phases were calculated with SOLVE (Terwilliger & Berendzen, 1999 ▶). The correlation coefficient between the map calculated based on the refined model of IF-5A and the starting map was 0.37, for the real-space modifed map it was 0.65 and for the maximum-likelihood map it was 0.79.