Analytic representation of inhomogeneous-resolution maps of three-dimensional scalar fields

Refinement of macromolecular atomic models versus experimental maps in cryo-electron microscopy and crystallography is a critical step in structure solution. For an appropriate comparison, model maps should mimic imperfections of the experimental ones, mainly atomic disorder and its limited resolution, often inhomogeneous over the molecule. We construct these model maps as a sum of atomic contributions expressed through a specially designed function describing a solitary spherical wave. Thanks to this function, atomic contributions analytically depend on both an atomic disorder and the local resolution, a value associated now with each atom. Such fully analytic dependence of inhomogeneous-resolution map values on model parameters permits an efficient refinement of all these parameters together and, beyond structural biology, opens a way to solve similar problems in other research domains. One-Sentence SummaryAn analytic decomposition of 3D-oscillating functions results in efficient tools to calculate maps and refine atomic models.