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1.  Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces 
Search spaces in the method of molecular replacement are shown to be coset spaces of the Lie group of rigid-body motions by the chiral space group of a crystal. The resulting ‘motion space’ can be endowed with a quasigroup operation that has interesting properties which are explored here.
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.
PMCID: PMC3171898  PMID: 21844648
rigid-body motion; coset space; quasigroup; fundamental domain; molecular replacement

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