Fig. 1.

The geometry of icosahedral lattices. (A) Different equilateral triangular facets can be constructed on a hexagonal lattice by moving integer numbers of steps along each of the ĥ and k̂ lattice vectors. (B) Construction of a T=3 lattice. Twenty copies of the triangular facet (left) obtained by moving one step along each of the ĥ and k̂ lattice vectors are arranged as shown in the middle panel, and then folded to obtain the icosahedral structure shown on the right. To connect this construction to a capsid, note that each pentagon will comprise five proteins in identical environments and each hexagon will comprise six subunits in two different types of local environments, resulting in a total of 180 proteins in three distinct local environments. (C) Example icosahedral capsid structures. From left to right are shown the T=1 satellite tobacco mosaic virus capsid (STMV) PDBID 1A34 [152], the T=3 cowpea chlorotic mottle virus capsid (CCMV) PDBID 1CWP [238], and the T=4 human hepatitis B viral capsid (HBV) PDBID 1QGT [276]. Structures are shown scaled to actual size, and the protein conformations are indicated by color. In each image the 60 pentameric subunits are colored blue. The images of capsids in (C) were obtained from the Viper database [212]. The images in (A) and (B) were reprinted from J. Mol. Biol, Johnson and Speir, 269, 665-675 (1997) Quasi-equivalent viruses: A paradigm for protein assemblies, with permission from Elsevier.