Abstract
This paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses. Such structures are often modular, consisting of one or two types of units arranged in a symmetric pattern. In particular, the polyoma virus has a structure apparently related to the snub dodecahedron. Here, we consider the problem of how such patterns might grow in time, starting from a given number N of randomly placed circles on the surface of a sphere. The circles are first jostled by random perturbations, then their radii are enlarged, then they are jostled again, and so on. This ‘yin–yang’ method of growth can result in some very close packings. When N =12, the closest packing corresponds to the snub tetrahedron, and when N =24 the closest packing corresponds to the snub cube. However, when N =60 the closest packing does not correspond to the snub dodecahedron but to a less-symmetric arrangement. Special attention is given to the structure of the human polyoma virus, for which N =72. It is shown that the yin–yang procedure successfully assembles the observed structure provided that the 72 circles are pre-assembled in clusters of six. Each cluster consists of five circles arranged symmetrically around a sixth at the centre, as in a flower with five petals. This has implications for the assembly of the capsomeres in a polyoma virus.
Highlights
The protein sheath of a virus particle commonly consists of a number of nearly identical subunits, arranged in a symmetrical pattern, for reasons first suggested by Crick & Watson (1957)
It was proposed by Klug & Finch (1965) that a papilloma virus has the symmetry of a snub icosahedron, 60 of the units being surrounded each by six neighbours, and the remaining 12 units each surrounded by five
Snub polyhedra belong to the class of Archimedean and Platonic solids illustrated in Coxeter et al (1954, p. 439)
Summary
The protein sheath of a virus particle commonly consists of a number of nearly identical subunits, arranged in a symmetrical pattern, for reasons first suggested by Crick & Watson (1957) It was proposed by Klug & Finch (1965) that a papilloma virus (figure 1) has the symmetry of a snub icosahedron, 60 of the units being surrounded each by six neighbours, and the remaining 12 units each surrounded by five. The most complete coverage corresponds to the small rhombic dodecahedron, illustrated in figure 4 In this case, we find aZ25.888 (see appendix A) and pZ0.76176; only 76 per cent of the spherical surface is covered. We address a somewhat different but related problem: given N equal, non-overlapping circles placed at random on the surface of a sphere, suppose they are subjected to small random displacements and are allowed at the same time to increase in size at equal rates. We shall describe some applications of this method to the problem of virus growth
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