viral protein (the subunit) corresponds to each triangular face of the icosahedron: as shown in Fig. 4.9B, such trimers interact with one another at the five-, three-, and twofold axes of rotational symmetry that define an icosahedron. As an icosahedron has 20 faces, 60 identical subunits (3 per face × 20 faces) is the minimal number needed to build a capsid with icosahedral symmetry.
Large capsids and quasiequivalent bonding. In the simplest icosahedral packing arrangement, each of the 60 subunits (structural or asymmetric units) consists of a single molecule in a structurally identical environment (Fig. 4.9B). Consequently, all subunits interact with their neighbors in an identical (or equivalent) manner, just like the subunits of helical particles such as that of tobacco mosaic virus. As the viral proteins that form such closed shells are generally <~100 kDa in molecular mass, the size of the viral genome that can be accommodated in this simplest type of particle is restricted severely. To make larger capsids, additional subunits must be included. Indeed, the capsids of the majority of animal viruses are built from many more than 60 subunits and can house very large genomes. In 1962, Donald Caspar and Aaron Klug developed a theoretical framework accounting for the properties of larger particles with icosahedral symmetry. This theory has had enormous influence on the way virus architecture is described and interpreted.
The triangulation number, T. A crucial idea introduced by Caspar and Klug was that of triangulation, the description of the triangular face of a large icosahedron in terms of its subdivision into smaller triangles, termed facets (Fig. 4.10). This process is described by the triangulation number, T, which gives the number of small “triangles” (called structural units) per face (Box 4.3). Because the minimum number of structural units required is 60, the total number of subunits in the structure is 60T.
Figure 4.10 The principle of triangulation: formation of large capsids with icosahedral symmetry. The formation of faces of icosahedral particles by triangulation is illustrated by comparison of structural units, organization of structural units at fivefold axes of icosahedral symmetry, and in capsids with the T number indicated below. In each case, the protein subunits are represented by trapezoids, with those that interact at the vertices colored purple and all others tan. It is important to appreciate that protein subunits are not, in fact, flat, as shown here for simplicity, but highly structured (see, for examples, Fig. 4.11 and 4.13). The interaction of subunits around the fivefold axes of symmetry and the capsid, with an individual face outlined in red, are shown for each value of T, to illustrate the increase in face and particle size with increasing T.
Quasiequivalence. A second cornerstone of the theory developed by Caspar and Klug was the proposition that when a capsid contains >60 subunits, each occupies a quasiequivalent position; that is, the noncovalent bonding properties of subunits in different structural environments are similar, but not identical. This property is illustrated in Fig. 4.9C for a particle with 180 identical subunits. In the small, 60-subunit structure, 5 subunits make fivefold symmetric contact at each of the 12 vertices (Fig. 4.9B). In the larger assembly with 180 subunits, this arrangement is retained at the 12 vertices, but the additional subunits are interposed to form clusters with sixfold symmetry (hexamers). In such a capsid, each subunit can be present in one of three different structural environments (designated A, B, or C in Fig. 4.9C). Nevertheless, all subunits bond to their neighbors in similar (quasiequivalent) ways, for example, via head-to-head and tail-to-tail interactions.
Capsid architectures corresponding to various values of T, some very large, have been reported. The triangulation number and quasiequivalent bonding among subunits describe the structural properties of many small and large viruses with icosahedral symmetry. However, it is now clear that the molecular arrangements adopted by specific segments of capsid proteins can govern the packing interactions of identical subunits. The resulting large conformational differences between small regions of chemically identical subunits were not anticipated in early considerations of virus structure, because these principles were formulated when little was known about the conformational flexibility of proteins. As we discuss in the next sections, the architectures of both small and more-complex viruses can depart radically from the constraints imposed by quasiequivalent bonding. For example, the capsid of the small polyomavirus simian virus 40 is built from 360 subunits, corresponding to the T = 6 triangulation number excluded by the rules formulated by Caspar and Klug (Box 4.3). Furthermore, a capsid stabilized by covalent joining of subunits to form viral “chain mail” has been described (Box 4.4). Our current view of icosahedrally symmetric virus structures is therefore one that includes greater diversity in the mechanisms by which stable capsids can be formed than was anticipated by the pioneers in this field.
BACKGROUND
The triangulation number, T, and how it is determined
In developing their theories about virus structure, Caspar and Klug used graphic illustrations of capsid subunits, such as the net of flat hexagons shown at the top left of panel A in the figure. Each hexagon represents a hexamer, with identical subunits shown as equilateral triangles. When all subunits assemble into such hexamers, the result is a flat sheet, or lattice, which can never form a closed structure. To introduce curvature, and hence form three-dimensional structures, one triangle is removed from a hexamer to form a pentamer in which the vertex and faces project above the plane of the original lattice (A, far right). As an icosahedron has 12 axes of fivefold symmetry, 12 pentamers must be introduced to form a closed structure with icosahedral symmetry. If 12 adjacent hexamers are converted to pentamers, an icosahedron of the minimal size possible for the net is formed. This structure is built from 60 equilateral-triangle asymmetric units and corresponds to a T = 1 icosahedron (Fig. 4.9B). Larger structures with icosahedral symmetry are built by including a larger number of equilateral triangles (subunits) per face (Fig. 4.10). In the hexagonal lattice, this is equivalent to converting 12 nonadjacent hexamers to pentamers at precisely spaced and regular intervals.
To illustrate this operation, we use nets in which an origin (O) is fixed and the positions of all other hexamers are defined by the coordinates along the axes labeled h and k, where h and k are any positive integer (A, left). The hexamer (h, k) is therefore defined as that reached from the origin (O) by h steps in the direction of the h axis and k steps in the direction of the k axis. In the T = 1 structure, h = 1 and k = 0 (or h = 0 and k = 1), and adjacent hexamers are converted to pentamers. When h = 1 and k = 1, pentamers are separated by one step in the h and one step in the k direction. Similarly, when h = 2 and k
