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Research Papers

# Kinematics Meets Crystallography: The Concept of a Motion Space1

[+] Author and Article Information
Gregory S. Chirikjian

Robot and Protein Kinematics Laboratory,
Department of Mechanical Engineering,
Johns Hopkins University,
Baltimore, MD 21218
e-mail: gregc@jhu.edu

In reality such crystals exist only in three-dimensional Euclidean space, but for the purpose of generality in this introduction, the dimension n will be allowed to be general, with realistic examples having n = 2 or 3.

In the symmorphic case $Pu=Fns=Pns$.

1This paper was originally presented at the ASME 2014 International Design Engineering Technical Conferences as Paper No. DETC2014-34243.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 23, 2014; final manuscript received August 28, 2014; published online February 2, 2015. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 15(1), 011012 (Mar 01, 2015) (7 pages) Paper No: JCISE-14-1259; doi: 10.1115/1.4028922 History: Received August 23, 2014; Revised August 28, 2014; Online February 02, 2015

## Abstract

In this paper, it is shown how rigid-body kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using x-ray 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 three-dimensional structure of molecules from x-ray diffraction patterns. A computational approach called molecular replacement (MR) is a well-established method for phasing of x-ray 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 symmetry-related molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigid-body 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 rigid-body 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 spaces” (which are compact manifolds) are investigated here.

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