# Geometry, Classification and Nomenclature of Capsids of Icosahedral Viruses

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## 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 f^{2} only, where P = h^{2} + hk + k^{2}, 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 (h_{1}, k_{1}) → (h_{2}, k_{2}) between any two icosahedral capsids. This is a rare case of biological organization for which such a general result is obtained.

## Keywords

Icosahedral viruses Classification Nomenclature Transformation of capsids## References

- Caspar DLD, Klug A (1962) Physical principles in the construction of regular viruses. In: Cold spring harbor symposia on quantitative biology, vol 27, pp 1–24CrossRefGoogle Scholar
- Luria SE, Darnell JE, Baltimore D, Campell A (1978) General virology, 3rd edn. Wiley, New YorkGoogle Scholar
- Rees AR, Sternberg MJE (1984) From cells to atoms: an illustrated introduction to molecular biology. Blackwell Sci. Publ., OxfordGoogle Scholar
- Schmalz TG, Seitz WA, Klein DJ, Hite GE (1988) Elemental carbon cages. J Am Chem Soc 110(4):1113–1127CrossRefGoogle Scholar
- Voytekhovsky YL (2015) Biomineral analogues in ontogeny and phylogeny. Paleontol J 49(14):1691–1697CrossRefGoogle Scholar
- Voytekhovsky YL (2016) Homological series of icosahedral viruses and fullerenes. Paleontol J 50(13):1505–1509CrossRefGoogle Scholar
- Voytekhovsky YL, Stepenshchikov DG (2016) Fullerene transformations as analogues of radiolarian skeleton microevolution. Paleontol J 50(13):1544–1548CrossRefGoogle Scholar