Direct calculation of cryo-EM and crystallographic model maps for real-space refinement

This work addresses the problem of the calculation of limited-resolution maps from an atomic model in cryo-electron microscopy and in X-ray and neutron crystallography, including cases where the resolution varies from one molecular region to another. Such maps are necessary in real-space refinement for comparison with the experimental maps. For an appropriate numeric comparison, the calculated maps should reproduce not only the structural features contained in the experimental maps but also the principal map distortions. These model maps can be obtained with no use of Fourier transforms but, similar to density distributions, as a sum of individual atomic contributions. Such contributions, referred to as atomic density images, are atomic densities morphed to reflect distortions of the experimental map, in particular the loss of resolution. They are described by functions composed of a central peak surrounded by Fourier ripples. For practical calculations, atomic images should be cut at some distance. It is shown that to reach a reasonable accuracy such a distance should be significantly larger than the distance customarily applied when calculating density distributions. This is a consequence of the slow rate with which the amplitude of the Fourier ripples decreases. Such a large distance means that at least a few ripples should be included in calculations in order to obtain a map that is sufficiently accurate. Oscillating functions describing these atomic contributions depend, for a given atomic type, on the resolution and on the atomic displacement parameter values. To express both the central peak and the Fourier ripples of the atomic images, these functions are represented by the sums of especially designed terms, each concentrated in a spherical shell and depending analytically on the atomic parameters. In this work, the strength of the dependence of the accuracy of resulting map on the accuracy of the atomic displacement parameters and on the truncation distance, i.e. the number of ripples included in atomic density images, is analyzed. This analysis is completed by practical aspects of the calculation of maps of inhomogeneous resolution. Tests show that the calculation of limited-resolution maps from an atomic model as a sum of atomic contributions requires a large truncation radius extending beyond the central peak of an atomic image and the first Fourier ripples. The article discusses the practical details of such calculations expressing atomic contributions as analytic functions of the atomic coordinates, the atomic displacement parameters and the local resolution.