Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces
Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces
复制标题
分子替换的数学方面。
DOI：
10.1107/s2053273319014797
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Chirikjian, Gregory S.
中科院分区：
文献类型：
--
作者：
Shiffman, Bernard;Lyu, Shengnan;Chirikjian, Gregory S.
本文对全分子置换(MR)搜索中巨大的6D旋转平移空间中的微小可行域进行了数学刻画。先验分离这些区域的能力对于提高大分子结晶学(MX)中计算相变的稳健性和效率具有潜在的重要意义。这一系列的前四篇论文集中讨论了在晶体对称性约束下相对于彼此运动的刚体的完整构型空间的性质。特别地，该问题的构形空间是右陪集空间Γ\G，其中Γ是手性大分子晶体的空间群，G是刚体运动群，基本域FΓ\G可以用许多具有有趣的代数和几何性质的方法来实现。MR方法中的成本函数可以被视为这些基本域上的函数。这是本系列的第五篇也是最后一篇论文，阐明了具有晶体对称性的物体必须遵守的限制条件。结果表明，这些约束在运动空间中定义了一个薄的可行集，它们分为两类：(I)物体不能相互穿透，因此在MR搜索中排除了所谓的‘碰撞区域’；(Ii)物体必须与足够数量的邻居接触，以便形成一个刚性网络，从而通向物理上可实现的晶体。在本文中，这些约束被应用于使用蛋白质的椭球代理来限制可行域。结果表明，这些可行区域的体积相对于运动空间的总体积很小，这证明了在MR搜索中使用椭球作为复杂蛋白质的替代品是合理的，这一点在P1(最简单的空间群)和P212121(MX中最常见的空间群)中得到了证明。
This paper mathematically characterizes the tiny feasible regions within the vast 6D rotation–translation space in a full molecular replacement (MR) search. The capability to a priori isolate such regions is potentially important for enhancing robustness and efficiency in computational phasing in macromolecular crystallography (MX). The previous four papers in this series have concentrated on the properties of the full configuration space of rigid bodies that move relative to each other with crystallographic symmetry constraints. In particular, it was shown that the configuration space of interest in this problem is the right-coset space Γ\G, where Γ is the space group of the chiral macromolecular crystal and G is the group of rigid-body motions, and that fundamental domains FΓ\G can be realized in many ways that have interesting algebraic and geometric properties. The cost function in MR methods can be viewed as a function on these fundamental domains. This, the fifth and final paper in this series, articulates the constraints that bodies packed with crystallographic symmetry must obey. It is shown that these constraints define a thin feasible set inside a motion space and that they fall into two categories: (i) the bodies must not interpenetrate, thereby excluding so-called `collision zones' from consideration in MR searches; (ii) the bodies must be in contact with a sufficient number of neighbors so as to form a rigid network leading to a physically realizable crystal. In this paper, these constraints are applied using ellipsoidal proxies for proteins to bound the feasible regions. It is shown that the volume of these feasible regions is small relative to the total volume of the motion space, which justifies the use of ellipsoids as proxies for complex proteins in MR searches, and this is demonstrated with P1 (the simplest space group) and with P212121 (the most common space group in MX).