Mathematical aspects of molecular replacement. IV. Measure‐theoretic decompositions of motion spaces

• Published in: Acta crystallographica. Section A, Foundations and advances Vol. 73; no. 5; pp. 387 - 402
• Main Authors: Chirikjian, Gregory S., Sajjadi, Sajdeh, Shiffman, Bernard, Zucker, Steven M.
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.09.2017
• Subjects: coset space; discrete subgroup; fundamental domain; measure theory; molecular replacement; coset space; measure theory; fundamental domain; molecular replacement; discrete subgroup
• ISSN: 2053-2733; 2053-2733
• DOI: 10.1107/S2053273317007227

In molecular‐replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid‐body models of macromolecules in crystallographic asymmetric units. The properties of the space of non‐redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three‐dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension. The minimal‐volume search space in molecular replacement can be expressed in multiple ways wherein there is a trade‐off between the sizes of rotation and translation subspaces. In particular, if the space group is a semi‐direct product of a point group and a Bieberbach group, then the search space can be represented as a product of two manifolds: a spherical space form and a Euclidean space form.