![]() The infinite series of icosahedral blueprints introduced by Caspar and Klug is currently the major tool for the classification of virus structures 12. These two principles have dominated structural virology over the last 60 years. This allows larger viruses to form, requiring even smaller relative portions of their genomic sequences to code for their capsids, thus generating coding capacity for other viral components that are not present in smaller viruses and enabling more complex infection scenarios. Caspar and Klug extended this idea by introducing the principle of quasi-equivalence 11, which explains how proteins can adopt locally equivalent, or quasiequivalent, positions in a capsid, by repeating this local configuration across the capsid surface. This favours such symmetric architectures, because icosahedral symmetry has 60 different symmetry operations 10, reducing the cost of coding for the capsid by 1/60th, whilst creating a container with sufficient volume to store the viral genetic material. Viruses exhibit this high degree of symmetry as a consequence of a principle that Crick and Watson termed genetic economy, namely, the limited capacity in the viral genome to code for the CPs forming its surrounding capsid 9. The example shown corresponds to a \(T=4\) layout, formed from 240 CPs that are organised as 12 pentamers (red) and 30 hexamers CPs are positioned in the corners of the triangular faces (shown here as dots), and result in clusters (capsomers) of six (hexamers) and five (pentamers) CPs. d The CP positions are indicated with reference to the dual polyhedron, that is, the triangulated structure obtained by connecting midpoints of adjacent hexagons and pentagons in the surface lattice. The corresponding polyhedra (right) have 20 identical triangular faces corresponding to the triangles (left), one of which is shown in each case. ![]() c One of the 20 triangles of the icosahedral frame is shown superimposed on the hexagonal grid for the four smallest polyhedra that can be constructed in CK theory. Dark grey areas indicate parts of hexagons in the lattice that do not form part of the surface lattice of the final polyhedral shape. b Construction of an icosahedral polyhedron via replacement of hexagons in a hexagonal lattice by the equivalent of 12 equidistant pentagons (red). a Viruses exhibit the characteristic 5-, 3- and 2-fold rotational symmetry axes of icosahedral symmetry, indicated here with reference to the vertices, edges and faces of an icosahedral frame superimposed on the crystal structure of the \(T=1\) STNV shell (figure based on PDB-ID 2BUK). They also apply to other icosahedral structures in nature, and offer alternative designs for man-made materials and nanocontainers in bionanotechnology.Ĭapsid architecture according to Caspar and Klug theory. ![]() These surface structures encompass different blueprints for capsids with the same number of structural proteins, as well as for capsid architectures formed from a combination of minor and major capsid proteins, and are recurrent within viral lineages. We show that this theory is a special case of an overarching design principle for icosahedral, as well as octahedral, architectures that can be formulated in terms of the Archimedean lattices and their duals. The principle of quasi-equivalence is currently used to predict virus architecture, but improved imaging techniques have revealed increasing numbers of viral outliers. The geometric constraints defining these container designs, and their implications for viral evolution, are open problems in virology. Viruses have evolved protein containers with a wide spectrum of icosahedral architectures to protect their genetic material. ![]()
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