Viral Self-Assembly as a Thermodynamic Process

The protein shells, or capsids, of nearly all spherelike viruses adopt icosahedral symmetry. In the present Letter, 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.

Figure 1

First row: The $T=1$, $T=3$, $T=4$, and $T=7$ capsids produced by the Caspar-Klug construction with 12 pentamers and 10 ($T-1$) hexamers per capsid. $T=1$ is a dodecahedron and $T=3$ a truncated icosahedron. Second row: Optimal packing arrangements of $N$ disks covering a sphere, known as the Tammes problem. Only $N=12$ has icosahedral symmetry; $N=24$ is an Archimedean solid known as the snub cube; $N=32$ could adopt $T=3$ icosahedral symmetry but in fact has $D_5$ symmetry. Third row: The improved coverage for $N=32$, $N=42$, and $N=72$ when 12 of the disks adopt a diameter smaller than that of the other disks.

Figure 2

(a) The sphere coverage for the Tammes problem. The maximum coverage, $\pi /(2\sqrt{3})={\rho }_{\max }$, is indicated by the dashed line. The structures encountered in the self-assembly diagram are indicated by arrows. $N=24$ is the snub cube. (b) Self-assembly phase diagram for the single-radius model with $B$ equal to $100k_{\mathrm{B}}T$ per disk and $\kappa$ equal to $10k_{\mathrm{B}}T$. The horizontal axis is the spontaneous curvature ${\theta }^{*}$; the vertical axis is $[(k_{\mathrm{B}}T)/\kappa ]\ln (\Phi /{\Phi }_0)$, with $\Phi$ the capsomer mole fraction and ${\Phi }_0\propto {exp}^{1/2[zV(0)/k_{\mathrm{B}}T]}$.

Figure 3

Comparison between the self-assembly phase diagram of the two-radius model (a) with that measured for the cowpea chlorotic mottle virus (a small, $T=3$, plant virus) (b). The cross in (b) indicates physiological conditions. The horizontal axis in (a) is the pentamer/hexamer switching energy per protein (in units of $\kappa$) while the vertical axis is the spontaneous curvature. The dashed line in (a) and (b) shows the proposed correspondence between $\Delta E$, $p\mathrm{H}$, and ${\theta }^{*}$, salinity. (b) is adapted from the phase diagram in Ref. 25.