Optimal 3D angular sampling with applications to cryo-EM problems

The goal of cryo-EM experiments in the biological sciences is to determine the atomic structure of a molecule and deduce insights into its functions and mechanisms. Despite improvements in instrumentation for data collection and new software algorithms, in most cases, individual atoms are not resolved. Model building of proteins, nucleic acids, or molecules in general, is feasible from the experimentally determined density maps at resolutions up to the range of 3–4 Angstroms. For lower-resolution maps or parts of maps, fitting smaller structures obtained by modelling or experimental techniques with higher resolution is a way to resolve the issue. In practice, we have an atomic structure, generate its density map at a given resolution, and translate/rotate the map within a region of interest in the experimental map, computing a measure-of-fit score with the corresponding areas of the experimental map. This procedure is computationally intensive since we work in 6D space. An optimal ordered list of rotations will reduce the angular error and help to find the best-fitting positions faster for a coarse global search or a local refinement. It can be used for adaptive approaches to stop fitting algorithms earlier once the desired accuracy has been achieved. We demonstrate how the performance of some fitting algorithms can be improved by grouping sets of rotations. We present an approach to generate more efficient 3D angular sampling, and provide the computer code to generate lists of optimal orientations for single and grouped rotations and the lists themselves.