Section 15.1.4.1. Sim and weighting

The phase probability distribution for the density-modified phase is conventionally generated under assumptions that were made for the combination of a partial atomic model with experimental data. It assumes that the calculated amplitudes and phases arise from a density map in which some atoms are present and correctly positioned, and the remainder are completely absent (Sim, 1959). Thus, the difference between the true structure factor and the calculated value must be the effective structure factor due to the missing density alone. If the phase of this quantity is random and the amplitude is drawn from a Wilson distribution (Wilson, 1949), the following expression is obtained: where and where is the variance parameter in the Wilson distribution for the missing part of the structure. The figure of merit, w, can be derived from where and are zero- and first-order modified Bessel functions. A similar argument follows for centric reflections.

The error estimate for the phase depends on the effective amount of missing structure that is estimated on the basis of the agreement of the modified amplitudes with their measured values, where may be estimated by a number of means, for example (Bricogne, 1976), where the average is normally taken over all reflections at a particular resolution. A more sophisticated approach is the method of Read (1986), which allows for errors in the atomic model and has also been used in density modification (Chapter 15.2 ).

Although these approaches have been applied with some success, the assumption in equation (15.1.4.1) that the density-modified amplitudes and phases are independent of the initial values is invalid. Since the density constraints are typically under-determined, it is possible to achieve an arbitrarily good agreement between the model amplitudes and their observed values without improving the phases. As a result, phase weights from density modification are typically overestimated.

This problem has traditionally been addressed by limiting the number of cycles of density modification in which weakly phased reflections are included. Typically, density modification is started with only some subset of the data, such as those reflections well phased from MIR data. Only these reflections are included in the phase recombination, with other reflections set to zero. As the calculation progresses, more reflections are introduced until all the data are included. The figures of merit of reflections that undergo fewer cycles of phase recombination will be correspondingly smaller (e.g. Leslie, 1987; Zhang & Main, 1990a). In averaging calculations where considerable phase information is available from high-order NCS, it is still typically necessary to perform phase extension over hundreds of cycles and to add a very thin resolution shell of new reflections at each cycle.

The phases and figure of merit generated from density modification are more suited to the calculation of weighted maps than maps. The map is designed to aid the structure completion from a partial model (Main, 1979). The map will restore features missing from the current model at full weight if the following conditions are fulfilled. First, the model phases must be close to their true values. Secondly, the difference between the model and observed amplitudes is a good indicator of the phase error and the difference between the calculated and observed amplitudes decreases as the phases approach their true values. Neither of these assumptions are necessarily true for density modification, since it may be applied to very poor maps with almost random phases, and under most density-modification schemes the structure-factor amplitudes may be over-fitted to the observed values.