Elasticity theory and shape transitions of viral shells

Recently, continuum elasticity theory has been applied to explain the shape transition of icosahedral viral capsids—single-protein-thick crystalline shells—from spherical to “buckled” or faceted as their radius increases through a critical value determined by the competition between stretching and bending energies of a closed two-dimensional (2D) elastic network. In the present work we generalize this approach to capsids with nonicosahedral symmetries, e.g., spherocylindrical and conical shells. One key additional physical ingredient is the role played by nonzero spontaneous curvature. Another is associated with the special way in which the energy of the 12 topologically required fivefold sites depends on the “background” local curvature of the shell in which they are embedded. Systematic evaluation of these contributions leads to a shape “phase” diagram in which transitions are observed from icosahedral to spherocylindrical capsids as a function of the ratio of stretching to bending energies and of the spontaneous curvature of the 2D protein network. We find that the transition from icosahedral to spherocylindrical symmetry is continuous or weakly first order near the onset of buckling, leading to extensive shape degeneracy. These results are discussed in the context of experimentally observed variations in the shapes of a variety of viral capsids.

Figure 1

(Color online) CK construction of icosahedral shells. (a) Folding template for an icosahedron consisting of 20 equilateral triangles. The triangles are indexed by a lattice vector $\vec{A}=h{\vec{a}}_1+k{\vec{a}}_2$ of a hexagonal lattice with basis vectors ${\vec{a}}_1$ and ${\vec{a}}_2$. (b) shows the case $h=3$ and $k=1$. (c) shows an icosahedron obtained from folding the template for this lattice vector, which corresponds to $T=h^2+k^2+hk=13$. Note that there are six hexagons for each face of the icosahedron, and that there are $10(T-1)=120$ hexagons in total.

Figure 2

Elastic energy $E$ of an icosahedral shell expressed in units of the bending constant $\kappa$, for the case of zero spontaneous curvature. The horizontal axis is the Föppl–von Kármán number $\gamma =YS∕\kappa$ with $S$ the surface area of the shell, $Y$ the Young’s modulus, and $\kappa$ the bending constant. For FvK numbers near a critical value ${\gamma }_B$ around 3000 (arrow) a buckling transition takes place, with the shell shape transforming from spherical to icosahedral. The dotted line shows the result of a fit to the LMN theory (see Sec. 3C). On a linear scale, $E\left(S\right)$ has a negative curvature for FvK numbers significantly above the buckling threshold.

Figure 3

(Color online) Construction of an isometric spherocylinder. (a) Folding template for an isometric spherocylinder. (b) The basis vectors of the template are $\vec{A}=n(h{\vec{a}}_1+k{\vec{a}}_2)$ and $\vec{B}=m(h{\vec{b}}_1+k{\vec{b}}_2)$. They are perpendicular to another with $(h,k)$ and $(n,m)$ any two pairs of non-negative integers with $m>n$. For $m=n$, the spherocylinder reduces to an icosahedron. (c) Isometric spherocylinder with $n=2$, $m=0$, and $h=3$, $k=0$.

Figure 4

Construction of an isometric cone. (a) Folding template of an isometric cone. The two lattice vectors $\vec{A}=n(h{\vec{a}}_1+k{\vec{a}}_2)$ and $\vec{B}=m(h{\vec{a}}_1+k{\vec{a}}_2)$ are parallel, with $(h,k)$ and $(m,n)$ any two pairs of non-negative integers with $m>n$. (b) Isometric cone with $h=1$, $k=0$, $m=3$, and $n=2$.

Figure 5

Elastic energy of isometric shells as computed from the Lobkovsky scaling relation Eq. (3.6). The energy is normalized with respect to that of an icosahedral $\left(S\right)$ shell of the same area. (a) Elastic energy of a spherocylinder (SC) as a function of the ratio $m∕n$ of the isometric construction. This ratio is about 2.5 times the aspect ratio $h∕\rho$ for a long spherocylinder. (b) Elastic energy of an isometric 5-7 cone $\left(C\right)$ as a function of the $m∕n$ ratio of the isometric construction. The $m∕n$ ratio is approximately the ratio $(R_l∕R_s)$ of the radii of the larger and smaller caps of the cone.

Figure 6

Elastic energy of an icosahedral shell as a function of the Föppl–von Kármán number $\gamma$ for different values of the spontaneous curvature $C_0$ as predicted by Eqs. (3.8, 3.9, 3.10, 3.11, 3.12). Solid line, $C_0=0$. Above the buckling threshold, the energy has a negative second derivative with respect to area. In this case, the capsid size distribution would be highly polydisperse in a self-assembly experiment. Dashed line, $C_0{R}_B=0.2$. The region of negative second derivative is reduced to a finite interval. Dotted line, $C_0{R}_B=0.8$. The energy has a positive second derivative with respect to area. In a self-assembly experiment, capsids with Fvk numbers near this minimum would dominate.

Figure 7

(a) Construction of a smooth conical shell by joining a larger sphere of radius $R_l$ to a smaller sphere of radius $R_s$ by a cone that is cotangent to the two spheres. The cone aperture angle is $2\theta$. The parts of the spheres inside the cone are then removed leaving two spherical cap portions. Note that the surface has a discontinuity in the curvature along the two matching circles. (b) Spherocylinder of height $h$ and cylinder radius $\rho$.

Figure 8

Analytical shape phase diagram. The vertical axis is the Föppl–von Kármán number $\gamma =YS∕\kappa$, with $S$ the area of the shell. The horizontal axis is $\alpha =C_0{S}^{1∕2}$, the spontaneous curvature $C_0$ in dimensionless units. Solid line, transition from a spherical shell to a spherocylindrical shell. The transition is discontinuous, with the aspect ratio of the spherocylinder along the transition line ranging from 4.6 to 5.8. Conical shells with $M\ne 6$ do not arise. For small FvK numbers, the transition takes place close to the critical spontaneous curvature of the Helfrich theory (see Sec. 3A). For small values of $\alpha$, the transition takes place, as a function of $\gamma$, just below the numerically computed buckling threshold of the icosahedron of about 3269 (see Sec. 4).

Figure 9

Elastic energy $E$ of a spherocylindrical shell with $m=80$, $n=38$, $h=1$, and $k=0$ (thick line) and an icosahedral shell with $h=55$ and $k=0$ (thin line) for the case of zero spontaneous curvature (in units of the bending constant $\kappa$) as a function of the FvK number $\gamma =YS∕\kappa$, with $S$ the area of the shell $Y$ the Young’s modulus and $\kappa$ the bending constant. The central portion of the spherocylinder develops a negative Gauss curvature (“waistlike region”) for $\gamma$ values around 1000, followed by a buckling transition for $\gamma$ near ${\gamma }_B\approx 2967$ that is smaller than the buckling threshold of a spherical shell $(\approx 3269)$. The dotted line shows the result of a fit to Eq. (3.14) with adjusted values for $B$ and ${\gamma }_B$ (see text).

Figure 10

Dependence of the buckling threshold parameter ${\gamma }_B$ of a spherocylinder (SC)—relative to that for a sphere $\left(S\right)$—on the $m∕n$ ratio, which is proportional to the size aspect ratio. The values of ${\gamma }_B$ were obtained from a fit of Eq. (3.14) to the results of numerical energy minimization.

Figure 11

Elastic energy $E∕\kappa$ of a conical shell with $m=42$, $n=22$, $h=1$, $k=0$ (thick line) and the icosahedral shell with $h=55$ and $k=0$ (thin line) for the case of zero spontaneous curvature obtained by numerical minimization as a function of the FvK number $\gamma =YS∕\kappa$. Note the pronounced negative Gaussian curvature of the shell. The dotted line shows the result of a fit to Eq. (3.14) with $B\approx 1.36$ and ${\gamma }_B\approx 4486$.

Figure 12

Dependence of the buckling threshold value ${\gamma }_B$ of a conical shell $\left(C\right)$—relative to that for a sphere $\left(S\right)$—on the $m∕n$ ratio, which is proportional to the cap size ratio. The values of ${\gamma }_B$ were obtained from a fit of Eq. (3.14) to the results of numerical energy minimization.

Figure 13

(Color online) Shape phase diagram. The vertical axis is the FvK number $\gamma =YS∕\kappa$; the horizontal axis is the dimensionless spontaneous curvature $\alpha =(2{\theta }_0∕\sqrt{3}a)S^{1∕2}$. For low $\alpha$, icosahedral shells are stable for all FvK numbers. The buckling threshold ${\gamma }_B$ separates spherical from polyhedral shells. For increasing $\alpha$ and FvK numbers below the buckling threshold, a first-order transition line separates spherical and spherocylindrical shells. The $m∕n$ ratio of the spherocylindrical shell along the transition line equals $121∕25$. For increasing $\alpha$ and FvK numbers above the buckling threshold, the aspect ratio of the spherocylinder at the transition line is reduced and the transition is either weakly first order or continuous. The solid line shows the phase boundary between sphere and spherocylinder according to the theory described in Sec. 3 (see Fig. 8).

Figure 14

Dependence of the elastic energy $E$, in units of the bending constant $\kappa$, of the spherocylinder (black line) and cone (gray line) on the $m∕n$ ratio at the transition point. (a) The FvK number $\gamma =873$ is below the buckling threshold; the energy barrier separating sphere and spherocylinder is of the order of $\kappa$. (b) The Fvk number $\gamma =27289$ is above the buckling threshold; the energy barrier is of the order of $0.1\kappa$. The energy of the spherocylinder (black line) is nearly independent of the $m∕n$ ratio.

Figure 15

Dependence of the elastic energy of the cone (gray line) and the spherocylinder (black line) on the $m∕n$ ratio at about twice the critical spontaneous curvature and with an FvK number $\gamma$ about twice the buckling threshold. The cone energy has a minimum for $m∕n=21∕11$ and the spherocylinder energy has a minimum at $m∕n=121∕25$.

Figure 16

In vitro self-assembly of the VP1 capsid proteins of SV40 (from Ref. 38). (a) Electron micrograph of VP1 assembly in $2M$ ammonium sulfate and $2\mathrm{m}M$ $\mathrm{Ca}{\mathrm{Cl}}_2$ ($p\mathrm{H}$ 7.2) at $4°\mathrm{C}$. $V$, virus-sized shell; $I$, intermediate particle; Ti, tiny particle. (b) Electron micrograph of VP1 assembly in $1M$ NaCl and $2\mathrm{m}M$ $\mathrm{Ca}{\mathrm{Cl}}_2$ ($p\mathrm{H}$ 7.2) at room temperature. Tu, tubular structure. Arrow, conical structure. (c) Higher magnification of virus-sized shells, intermediate particles, tiny particles, and a tubular structure observed in (b).

Figure 17

(Color online) CryoTEM tomography images of several tens of noninfectious HIV-1 virions, each shown along three orthogonal directions. A variety of conical, rodlike, and other shapes are found from a single cell culture.