Capsid

Icosahedral capsid of an adenovirus

Virus capsid T-numbers

The icosahedral structure is extremely common among viruses. The icosahedron consists of 20 triangular faces delimited by 12 fivefold vertexes and consists of 60 asymmetric units. Thus, an icosahedral virus is made of 60N protein subunits. The number and arrangement of capsomeres in an icosahedral capsid can be classified using the "quasi-equivalence principle" proposed by Donald Caspar and Aaron Klug.[11] Like the Goldberg polyhedra, an icosahedral structure can be regarded as being constructed from pentamers and hexamers. The structures can be indexed by two integers h and k, with ${\displaystyle h\geq 1}$ and ${\displaystyle k\geq 0}$; the structure can be thought of as taking h steps from the edge of a pentamer, turning 60 degrees counterclockwise, then taking k steps to get to the next pentamer. The triangulation number T for the capsid is defined as:

${\displaystyle T=h^{2}+h\cdot k+k^{2}}$

In this scheme, icosahedral capsids contain 12 pentamers plus 10(T − 1) hexamers.[12][13] The T-number is representative of the size and complexity of the capsids.[14] Geometric examples for many values of h, k, and T can be found at List of geodesic polyhedra and Goldberg polyhedra.

Many exceptions to this rule exist: For example, the polyomaviruses and papillomaviruses have pentamers instead of hexamers in hexavalent positions on a quasi-T=7 lattice. Members of the double-stranded RNA virus lineage, including reovirus, rotavirus and bacteriophage φ6 have capsids built of 120 copies of capsid protein, corresponding to a "T=2" capsid, or arguably a T=1 capsid with a dimer in the asymmetric unit. Similarly, many small viruses have a pseudo-T=3 (or P=3) capsid, which is organized according to a T=3 lattice, but with distinct polypeptides occupying the three quasi-equivalent positions [15]

T-numbers can be represented in different ways, for example T = 1 can only be represented as an icosahedron or a dodecahedron and, depending on the type of quasi-symmetry, T = 3 can be presented as a truncated dodecahedron, an icosidodecahedron, or a truncated icosahedron and their respective duals a triakis icosahedron, a rhombic triacontahedron, or a pentakis dodecahedron.[16]

The prolate structure of a typical head on a bacteriophage

An elongated icosahedron is a common shape for the heads of bacteriophages. Such a structure is composed of a cylinder with a cap at either end. The cylinder is composed of 10 elongated triangular faces. The Q number (or T_mid), which can be any positive integer,[17] specifies the number of triangles, composed of asymmetric subunits, that make up the 10 triangles of the cylinder. The caps are classified by the T (or T_end) number.[18]

3D model of a helical capsid structure of a virus

Many rod-shaped and filamentous plant viruses have capsids with helical symmetry.[19] The helical structure can be described as a set of n 1-D molecular helices related by an n-fold axial symmetry.[20] The helical transformation are classified into two categories: one-dimensional and two-dimensional helical systems.[20] Creating an entire helical structure relies on a set of translational and rotational matrices which are coded in the protein data bank.[20] Helical symmetry is given by the formula P = μ x ρ, where μ is the number of structural units per turn of the helix, ρ is the axial rise per unit and P is the pitch of the helix. The structure is said to be open due to the characteristic that any volume can be enclosed by varying the length of the helix.[21] The most understood helical virus is the tobacco mosaic virus.[19] The virus is a single molecule of (+) strand RNA. Each coat protein on the interior of the helix bind three nucleotides of the RNA genome. Influenza A viruses differ by comprising multiple ribonucleoproteins, the viral NP protein organizes the RNA into a helical structure. The size is also different; the tobacco mosaic virus has a 16.33 protein subunits per helical turn,[19] while the influenza A virus has a 28 amino acid tail loop.