Keef T, Twarock R, Elsawy K M
Department of Mathematics, University of York, York Y010 5DD, UK.
J Theor Biol. 2008 Aug 21;253(4):808-16. doi: 10.1016/j.jtbi.2008.04.029. Epub 2008 May 4.
In a seminal paper, Caspar and Klug [1962. Physical principles in the construction of regular viruses. Cold Spring Harbor Symp. Quant. Biol. 27, 1-24] derived a family of surface lattices as blueprints for the structural organisation of the protein shells, called viral capsids, which encapsulate and hence protect the viral genome. These lattices schematically encode, and hence predict, the locations of the proteins in the viral capsids. Despite the huge success and numerous applications of this theory in virology, experimental results have provided evidence for the fact that it is too restrictive to describe all known viruses [Casjens, S., 1985. Virus Structure and Assembly. Jones and Bartlett, Boston, MA]. Especially, the family of Polyomaviridae, which contains cancer-causing viruses, falls out of the scope of this theory. In [Twarock, R., 2004. A tiling approach to virus capsid assembly explaining a structural puzzle in virology. J. Theor. Biol. 226, 477], we have shown that a member of the family of Polyomaviridae can be described via an icosahedrally symmetric tiling. We show here that all viruses in this family can be described by tilings with vertices corresponding to subsets of a quasi-lattice that is constructed based on an affine extended Coxeter group, and we use this methodology to derive their coordinates explicitly. Since the particles appear as different subsets of the same quasi-lattice, their relative sizes are predicted by this approach, and there hence exists only one scaling factor that relates the sizes of all particles collectively to their biological counterparts. It is the first mathematical result that provides a common organisational principle for different types of viral particles in the family of Polyomaviridae, and paves the way for modelling Polyomaviridae polymorphism.
在一篇具有开创性的论文中，卡斯帕和克鲁格[1962年。规则病毒构建中的物理原理。《冷泉港定量生物学研讨会论文集》第27卷，第1 - 24页]推导了一族表面晶格，作为蛋白质外壳（称为病毒衣壳）结构组织的蓝图，病毒衣壳包裹并因此保护病毒基因组。这些晶格示意性地编码并预测了病毒衣壳中蛋白质的位置。尽管该理论在病毒学中取得了巨大成功并有着众多应用，但实验结果证明，用它来描述所有已知病毒过于局限[卡斯延斯，S.，1985年。病毒结构与组装。琼斯和巴特利特出版社，马萨诸塞州波士顿]。特别是，包含致癌病毒的多瘤病毒科超出了该理论的范围。在[特沃罗克，R.，2004年。一种用于病毒衣壳组装的平铺方法，解释病毒学中的一个结构难题。《理论生物学杂志》第226卷，第477页]中，我们已经表明多瘤病毒科的一个成员可以通过二十面体对称平铺来描述。我们在此表明，该科中的所有病毒都可以用与基于仿射扩展考克斯特群构建的准晶格子集相对应的顶点平铺来描述，并且我们使用这种方法明确推导它们的坐标。由于粒子表现为同一准晶格的不同子集，这种方法可以预测它们的相对大小，因此只存在一个缩放因子，将所有粒子的大小与它们的生物学对应物联系起来。这是第一个为多瘤病毒科中不同类型的病毒粒子提供共同组织原则的数学结果，并为多瘤病毒科多态性建模铺平了道路。
