Continuum Theory of Retroviral Capsids

We present a self-assembly phase diagram for the shape of retroviral capsids, based on continuum elasticity theory. The spontaneous curvature of the capsid proteins drives a weakly first-order transition from spherical to spherocylindrical shapes. The conical capsid shape which characterizes the HIV-1 retrovirus is never stable under unconstrained energy minimization. Only under conditions of fixed volume and/or fixed spanning length can the conical shape be a minimum energy structure. Our results indicate that, unlike the capsids of small viruses, retrovirus capsids are not uniquely determined by the molecular structure of the constituent proteins but depend in an essential way on physical constraints present during assembly.

Figure 1

CK isometric construction of a spherical shell. (a) Folding template for an icosahedron. The template is indexed by the lattice vector $\overrightarrow{A}=h{\overrightarrow{a}}_1+k{\overrightarrow{a}}_2$ of a hexagonal lattice with basis vectors ${\overrightarrow{a}}_1$ and ${\overrightarrow{a}}_2$. (b) The case $h=3$ and $k=1$. (c) The corresponding icosahedron.

Figure 2

Nonicosahedral isometric shells. (a) Folding template of an isometric spherocylinder. The template is indexed by two perpendicular vectors $\overrightarrow{A}=n(h{\overrightarrow{a}}_1+k{\overrightarrow{a}}_2)$ and $\overrightarrow{B}=m(h{\overrightarrow{b}}_1+k{\overrightarrow{b}}_2)$, with $m>n$ two positive integers. (b) The four vectors ${\overrightarrow{a}}_1$, ${\overrightarrow{a}}_2$, ${\overrightarrow{b}}_1$, and ${\overrightarrow{b}}_2$. (c) Isometric spherocylinder with $(m,n,h,k)\equiv (4,2,1,0)$. (d) Folding template of an isometric 5–7 cone. The template is indexed by two parallel vectors $\overrightarrow{A}=n(h{\overrightarrow{a}}_1+k{\overrightarrow{a}}_2)$ and $\overrightarrow{B}=m(h{\overrightarrow{a}}_1+k{\overrightarrow{a}}_2)$. (e) Isometric 5–7 cone with $(m,n,h,k)\equiv (4,3,1,0)$.

Figure 3

Shape phase diagram at fixed area $S$. The vertical axis is the FvK number $\gamma =YS/\kappa$, and the horizontal axis the dimensionless spontaneous curvature $\alpha =2{\theta }_0{S}^{1/2}/\sqrt{3}a$. The buckling threshold ${\gamma }_B\simeq 3000$ separates spherical from polyhedral shells. The black squares show the phase boundary obtained from numerical simulation. On the right side of this boundary, the spherocylinder shape has the lowest energy. Three such shapes close to the transition line are shown for small, intermediate, and large $\gamma$ (all shapes have the same area but they are scaled to fit the allotted space). The FvK number of HIV-1 capsids is of the order of $2\times {10}^4$ (dotted line).

Figure 4

Transition from spherocylinder to cone at fixed area $S$ and fixed volume $V$. The horizontal axis is the ratio of the capsid volume to that of a reference icosahedral shell $V_o$. The vertical axis is the dimensionless spontaneous curvature. The FvK number is fixed at 25 000. At the indicated transition point, the $m/n$ ratio equals $98/31$ for the spherocylinder and $40/13$ for the cone; for larger (smaller) values of $V/V_o$, the $m/n$ ratio at the transition is smaller (larger). The dotted line shows the transition line predicted by the LMN analytical theory.