Algorithms and programs for the shell decomposition of oscillating functions in space

Abstract

Real-space refinement of atomic models in macromolecular crystallography and cryo-electron microscopy fits a model to a map obtained with experimental data. To do so, the atomic model is converted into a map of limited resolution and then this map is compared quantitatively with the experimental one. For an appropriate comparison, the atomic contributions comprising the model map should reflect the resolution of the experimental map and the atomic displacement parameter (ADP) values. Such contributions are spherically symmetric oscillating functions, different for chemically different kinds of atoms, different ADPs and different resolution values, and their derivatives with respect to atomic parameters rule the model refinement. For given parameter values, every contribution may be calculated numerically using two Fourier transforms, which is highly time consuming and makes calculation of the respective derivatives problematic. Alternatively, for an atom of each required type its contribution can be expressed in an analytical form as a sum of specially designed terms. Each term is different from zero essentially inside a spherical shell, and changing the ADP value does not change its form but rather changes the value of one of its arguments. In general, these terms become a convenient tool for the decomposition of oscillating spherically symmetric functions. This work describes the algorithms and respective software, named dec3D, to carry out such a shell decomposition for density contributions of different kinds of atoms and ions.