Open AccessMathematical aspects of molecular replacement. I. Algebraic properties of motion spaces
Author(s) -
Chirikjian Gregory S.
Publication year - 2011
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
eISSN - 1600-5724
pISSN - 0108-7673
DOI - 10.1107/s0108767311021003
Subject(s) - rigid body , group (periodic table) , quasigroup , algebraic structure , coset , space (punctuation) , motion (physics) , unit (ring theory) , space group , crystallography , macromolecule , algebraic number , diffraction , molecular replacement , position (finance) , euclidean space , crystal structure , mathematics , set (abstract data type) , pure mathematics , chemistry , physics , x ray crystallography , combinatorics , classical mechanics , computer science , mathematical analysis , quantum mechanics , operating system , biochemistry , programming language , economics , mathematics education , finance
Molecular replacement (MR) is a well established method for phasing of X‐ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid‐body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid‐body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right‐coset spaces of the full group of rigid‐body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.