Principles of Virology. Jane Flint

the symmetry with which the structural units interact is that of an icosahedron.

Figure 4.9 Icosahedral packing in simple structures. (A) An icosahedron, which comprises 20 equilateral triangular faces characterized by positions of five-, three-, and twofold rotational symmetry. The three views at the bottom illustrate these positions. (B and C) A comma represents a single protein molecule, and axes of rotational symmetry are indicated as in panel A. In the simplest case, T = 1 (B), the protein molecule forms the structural unit, and each of the 60 molecules is related to its neighbors by the two-, three-, and fivefold rotational axes that define a structure with icosahedral symmetry. In such a simple icosahedral structure, the interactions of all molecules with their neighbors are identical. In the T = 3 structure (C) with 180 identical protein subunits, there are three modes of packing of a subunit (shown in orange, yellow, and purple): a trimer (outlined in blue) is now the asymmetric unit, which, when replicated according to 60-fold icosahedral symmetry, generates the complete structure. The orange subunits are present in pentamers, formed by tail-to-tail interactions, and interact in rings of three (head to head) with purple and yellow subunits, and in pairs (head to head) with a purple or a yellow subunit. The purple and yellow subunits are arranged in rings of six molecules (by tail-to-tail interactions) that alternate in the particle. Despite these packing differences, the bonding interactions in which each subunit engages are similar, that is, quasiequivalent: for example, all engage in tail-to-tail and head-to-head interactions. Adapted from Harrison SC. 1984. Trends Biochem Sci 9:345–351, with permission.

      In solid geometry, each of the 20 faces of an icosahedron is an equilateral triangle, and five such triangles interact at each of the 12 vertices (Fig. 4.9A). In the simplest protein shells, a trimer of a single 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.

BOX 4.3

      BACKGROUND

      The triangulation number, T, and