Geometry, Classification and Nomenclature of Capsids of Icosahedral Viruses

Abstract

Icosahedral (regular, spherical) viruses build highly ordered capsids of -3-5m and 235 symmetry point groups (s.p.g.’s). Geometrical principles in their construction from protein globules were found by Caspar and Klug in 1962. In general, the facets of mega-icosahedra look like triangles differently oriented for different capsids on the 2D dense hexagonal packing of protein globules. It was shown later that icosahedral capsids and fullerenes form homological series with combinatorial geometry of the latter being well investigated. This allowed us to describe the geometry of icosahedral capsids in more details. The icosahedral capsids can exist with triangulation numbers T = P f2 only, where P = h2 + hk + k2, where h and k—any pair of integers with no common factors, and f = 1, 2, 3, etc. The proof of the above statement was first published by Schmalz et al. in 1988. As a result, the whole variety of icosahedral capsids was divided as follows: P = 1 (i.e. h = 1, k = 0, any f, s.p.g. -3-5m), P = 3 (i.e. h = k = 1, any f, s.p.g. -3-5m), and skew classes (i.e. h > k > 0, s.p.g. 235). The series of capsids-isomers were found by the author, ex. for T = 49 (h = 5, k = 3, s.p.g. 235 and h = 7, k = 0, s.p.g. -3-5m), T = 91 (h = 6, k = 5 and h = 9, k = 1, both s.p.g.’s 235) and many others. That is, the nomenclature of icosahedral capsids should be based not on the triangulation numbers but on the (h, k) symbols which uniquely determine their geometry. The classification of icosahedral capsids by the s.p.g.’s -3-5m (with symmetry planes) and 235 (without them) is correct but very approximate. In more details, the -3-5m variety consists of the (t, 0) and (t, t) homological series (t = 1, 2, 3, etc.) connected by the dual transformations (h, k) → (h + 2k, h − k). The 235 variety of “skew classes” also consists of the (th, tk) homological series (t = 1, 2, 3, etc.), where (h, k) is capsid-generator with h and k—any pair of integers with no common factors and h − k being not divisible by 3. For any 235 homological series of capsids, another one is connected with it by the dual transformation (h, k) → (h + 2k, h − k). The simplest generators (1, 0), (2, 1) and (3, 1) are related to bacteriophage φX174, papovavirus and rotavirus, respectively. They generate the majority of the known icosahedral capsids as their homological series and duals. A matrix equation is found to describe the transition (h1, k1) → (h2, k2) between any two icosahedral capsids. This is a rare case of biological organization for which such a general result is obtained.