Mannige Ranjan V, Brooks Charles L
Department of Molecular Biology and Center for Theoretical Biological Physics, The Scripps Research Institute, 10550 North Torrey Pines Road, TPC 6, La Jolla, California 92037, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 May;77(5 Pt 1):051902. doi: 10.1103/PhysRevE.77.051902. Epub 2008 May 1.
Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present paper we ask to what extent these structures can be viewed as mathematically tilable objects using a single two-dimensional tile. We find that spherical viruses from a large number of families-eight out of the twelve studied-qualitatively possess properties that allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit. This we did by characterizing the extent to which individual spherical capsids display subunit-subunit (1) holes, (2) overlaps, and (3) gross structural variability. All capsids with T numbers greater than 1 from the Protein Data Bank, with homogeneous protein composition, were used in the study. These monohedral tilings, called canonical capsids due to their platonic (mathematical) form, offer a mathematical segue into the structural and dynamical understanding of not one, but a large number of virus capsids. From our data, it appears as though one may only break the long-standing rules of quasiequivalence by the introduction of subunit-subunit structural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in understanding structural aspects of such assemblies, we used graph theory and discrete geometry to enumerate the types of shapes that the tiles (and hence the subunits) must possess. We show that topology restricts the shape of the face to a limited number of five-sided prototiles, one of which is the "bisected trapezoid" that is a platonic representation of the most ubiquitous capsid subunit shape seen in nature (the trapezoidal jelly-roll motif). This motif is found in a majority of seemingly unrelated virus families that share little to no host, size, or amino acid sequence similarity. This suggests that topological constraints may exhibit dominant roles in the natural design of biological assemblies, while having little effect on amino acid sequence similarity.
病毒衣壳是高度特异性的聚集体，由大量通常化学性质相同的衣壳亚基组成。在本文中，我们探讨了这些结构在何种程度上可以被视为使用单个二维平铺块的数学可平铺物体。我们发现，来自大量病毒科（在所研究的十二个病毒科中有八个）的球形病毒在质量上具有一些特性，使其能够表示为有界表面的二维单形平铺，其中每个平铺块代表一个亚基。我们通过表征单个球形衣壳在以下方面的程度来做到这一点：(1) 亚基 - 亚基之间的孔洞，(2) 重叠，以及 (3) 总体结构变异性。研究使用了蛋白质数据库中所有T数大于1且蛋白质组成均匀的衣壳。这些单形平铺，由于其柏拉图式（数学）形式而被称为规范衣壳，为理解大量病毒衣壳的结构和动力学提供了一个数学切入点。从我们的数据来看，似乎只有通过在衣壳中引入亚基 - 亚基结构变异性、孔洞和大量重叠，才可能打破长期以来的准等效规则。为了探索规范衣壳在理解此类聚集体结构方面的实用性，我们使用图论和离散几何来枚举平铺块（进而亚基）必须具有的形状类型。我们表明，拓扑结构将面的形状限制为有限数量的五边形原平铺块，其中之一是“二等分梯形”，它是自然界中最普遍存在的衣壳亚基形状（梯形果冻卷基序）的柏拉图式表示。这种基序存在于大多数看似不相关的病毒科中，这些病毒科在宿主、大小或氨基酸序列相似性方面几乎没有或没有相似之处。这表明拓扑约束可能在生物聚集体的自然设计中发挥主导作用，而对氨基酸序列相似性影响很小。
