Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria

Local heterogeneity analysis of crystallographic and cryo-EM maps using shell-approximation

In X-ray crystallography and cryo-EM, experimental maps can be heterogeneous, showing different level of details in different regions. In this work we interpret heterogeneity in terms of two parameters, assigned individually for each atom, combining the conventional atomic displacement parameter with the resolution of the atomic image in the map. We propose a local real-space procedure to estimate the values of these heterogeneity parameters, assuming that a fragment of the density map and atomic positions are given. The procedure is based on an analytic representation of the atomic image, as a function of the inhomogeneity parameters and atomic coordinates. In this article, we report the results of the tests both with maps simulated and those derived from experimental data. For simulated maps containing regions with different resolutions, the method determines the local map resolution around the atomic centers and the values of the displacement parameter with reasonable accuracy. For experimental maps, obtained as a Fourier synthesis of a given global resolution, estimated values of the local resolution are close to the global one, and the values of the estimated displacement parameters are close to the respective values of the closest atoms in the refined model. Shown successful applications of the proposed method to experimental crystallographic and cryo-EM maps can be seen as a practical proof of method.

On the suitability of using the diagonal Gaussian approximation in the calculation of the magnitude-based likelihood function

Statistical likelihood maximization is currently one of the main tools in computational procedures in biological crystallography. In these procedures, the likelihood function is calculated, as a rule, within the framework of a diagonal Gaussian approximation (DGA) of the joint probability distribution of the real and imaginary parts of a set of structure factors. This approximation assumes pairwise uncorrelated values of various structure-factor components. In this paper, exact formulas are derived for pairwise correlations of structure factors, and conditions under which these correlations can be considered to be negligible are discussed. It is shown that in the case where the probability distribution of the atomic coordinates is related to the region of the molecule or its domains, the correlation of the structure factors of reflections s and w is determined mostly by the magnitudes of the Fourier transform of the probability distribution calculated at the points 2s, 2w, s - w and s + w. However, in the case where the probability distribution describes small corrections to the coordinates of the existing preliminary atomic model, the correlation is determined by the values of the structure factors of the preliminary model that correspond to the 2s, 2w, s - w and s + w reflections rather than by the Fourier transform of the probability distribution. Test cases demonstrate that the practice of using the DGA for calculation of the likelihood when based on sets containing neighbouring reflections may be unjustified in some crystallographic applications, especially in single-particle studies.