With few exceptions, the shells (capsids) of sphere-like viruses have the symmetry of an icosahedron and are composed of coat proteins (subunits) assembled in special motifs, the T-number structures. Although the synthesis of artificial protein cages is a rapidly developing area of materials science, the design criteria for self-assembled shells that can reproduce the remarkable properties of viral capsids are only beginning to be understood. We present here a minimal model for equilibrium capsid structure, introducing an explicit interaction between protein multimers (capsomers). Using Monte Carlo simulation we show that the model reproduces the main structures of viruses in vivo (T-number icosahedra) and important nonicosahedral structures (with octahedral and cubic symmetry) observed in vitro. Our model can also predict capsid strength and shed light on genome release mechanisms. Icosahedral symmetry is ubiquitous among spherical viruses (1). A classic example is the cowpea chlorotic mottle virus (CCMV), a well studied RNA virus with a shell composed of exactly 180 identical proteins (subunits) (2, 3). Fig. 1a is a cryo-transmission electron microscopy reconstruction showing 5-and 6-fold morphological units (capsomers), and Fig. 1b shows the arrangement of the individual subunits within the capsomers. The capsid has 6 5-fold rotation axes, 10 3-fold axes, and 15 2-fold axes, the symmetry elements of an icosahedron. The subunits can be divided into 12 capsomers that contain five subunits (pentamers) and 20 capsomers that contain six subunits (hexamers).The current understanding of sphere-like virus structures, like that adopted by CCMV, is based on the Caspar and Klug (CK) ''quasi-equivalence'' principle (4), which provides the foundation of modern structural virology. CK showed that closed icosahedral shells can be constructed from pentamers and hexamers by minimizing the number T of nonequivalent locations that subunits occupy, with the T-number adopting the particular integer values 1, 3, 4, 7, 12, 13, . . . (T ϭ h 2 ϩ k 2 ϩ hk, with h, k equal to nonnegative integers). These CK shells always contain 12 pentamers plus 10 (T-1) hexamers, and these structures have indeed been found to characterize a predominantly large fraction of sphere-like viruses (the CCMV capsid, for example, being a T ϭ 3 structure). Nevertheless, exceptions among WT capsids have been documented (5, 6), including the retroviruses like HIV (7); assembly of subunits with point mutations can also produce breakdown of CK-type capsid icosahedral symmetry (8).It is significant that the icosahedral point group generates the maximum enclosed volume for shells comprised of a given size subunit (4). But the fact that many viruses, including CCMV, self-assemble spontaneously from their molecular components under in vitro conditions (9) indicates that both icosahedral symmetry and the CK construction should be generic features of the free energy minima of aggregates of viral capsid proteins. Because of current computational limitations, direct eval...
A theoretical description of the shape of a random object is presented that is analytically simple in application but quantitatively accurate. The asymmetry of the object is characterized in terms of the invariants of a tensor, analogous to the moment-of-inertia tensor, whose eigenvalues are the squares of the principal radii of gyration. The complications accompanying ensemble averaging because of random processes are greatly reduced when the object is embedded in a space of high dimensionality, d. Exact analytical expressions are presented in the case of infinite spatial dimensions, and a procedure for developing an expansion in powers of l/d is discussed for linear chain and ring-type random walks. The first two terms in such an expansion lead to results for various shape parameters that agree remarkably well with those calculated by computer simulation. The method can be extended to yield an approximate, but extremely accurate, expression for the probability distribution function directly. The theoretical approach discussed here can, in principle, be used to describe the shape of other random fractal objects as well.
The protein shells, or capsids, of all sphere-like viruses adopt icosahedral symmetry. In the present paper we propose a statistical thermodynamic model for viral self-assembly. We find that icosahedral symmetry is not expected for viral capsids constructed from structurally identical protein subunits and that this symmetry requires (at least) two internal "switching" configurations of the protein. Our results indicate that icosahedral symmetry is not a generic consequence of free energy minimization but requires optimization of internal structural parameters of the capsid proteins.
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