Chirikjian Gregory S
Department of Mechanical Engineering, Johns Hopkins University, 223 Latrobe Hall, 3400 N. Charles Street, Baltimore, Maryland, MD 21218, USA.
Acta Crystallogr A. 2011 Sep;67(Pt 5):435-46. doi: 10.1107/S0108767311021003. Epub 2011 Jul 6.
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.
分子置换（MR）是一种成熟的方法，用于对由化学结构已知但构象未知的生物大分子组成的晶体的X射线衍射图谱进行相位测定。在分子置换中，起始点是已知的结构域，假定其形状与待确定的大分子结构中的结构域相似。然后在晶体学不对称单元模型内对已知结构域的位置和取向进行搜索，以使计算出的衍射图谱与实验数据最佳匹配。与欧几里得空间中的连续刚体运动和离散的晶体学空间群不同，分子置换搜索所执行的运动集合在合成运算下不构成一个群，本文证明其缺乏结合律。然而，不对称单元中的刚体运动集合形成了另一种数学结构，称为拟群，它可以与大分子晶体的手性空间群相对于刚体运动全群的右陪集空间相识别。本文阐述了这个运动空间的代数性质。
