Programs and algorithms for the shell decomposition of oscillating functions in space

TL;DR

This paper introduces software and algorithms for shell decomposition of oscillating functions in space, aiding in the refinement of atomic models in macromolecular imaging by accounting for local resolution variations.

Contribution

It presents a novel method and software for decomposing oscillating functions into spherical shells, improving model map calculations in structural biology.

Findings

Effective decomposition of oscillating functions into spherical shells.

Enhanced accuracy in local resolution modeling.

Software implementation for practical use.

Abstract

Real-space refinement of atomic models in macromolecular crystallography or in cryo electron microscopy fits a model to a map obtained experimentally. This requires generating model maps of a limited resolution which moreover may vary from one molecular region to another. Calculating such map as a sum of atomic contributions requires that these contributions reflect the local resolution of the experimental map. A possibility to refine the parameters of these contribution means to express it as a function of atomic coordinates, displacement factor and eventually of resolution. Recently, Urzhumtsev & Lunin (BioRxiv, 10.1101/2022.03.28.486044) suggested to decompose finite-resolution atomic images, and more generally spherically symmetric oscillating functions in space, into a sum of specially designed terms analytically dependent on all atomic parameters. Each term is a spherically symmetric function concentrated in a spherical shell. Here we describe the software and respective algorithms to carry out such shell decomposition of oscillating functions.