Kinematics Meets Crystallography: The Concept of a Motion Space1

Abstract

In this paper, it is shown how rigidbody kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using xray crystallography, which is a powerful method for structure determination. The importance of determining molecular structures for understanding biological processes and for the design of new drugs is well known. Phasing is a necessary step in determining the threedimensional structure of molecules from xray diffraction patterns. A computational approach called molecular replacement (MR) is a wellestablished method for phasing of xray diffraction patterns for crystals composed of biological macromolecules. In MR, a search is performed over positions and orientations of a known biomolecular structure within a model of the crystallographic asymmetric unit, or, equivalently, multiple symmetryrelated molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigidbody motions known to kinematicians, the set of motions over which MR searches are performed does not form a group. Rather, it is a coset space of the group of continuous rigidbody motions, SE(3), with respect to the crystallographic space group of the crystal, which is a discrete subgroup of SE(3). Properties of these “motion spacesâ€‌ (which are compact manifolds) are investigated here.