Viral self-assembly pathway and mechanical stress relaxation

The final shape of a virus is dictated by the self-assembly pathway of its constituents. Using standard thin-shell elasticity, we highlight the prominent role of the viral shell's spontaneous curvature in determining the assembly pathway. In particular, we demonstrate that the mechanical stress inherent to the growth of a curved surface can be relaxed in two different ways in the early steps of assembly, depending on the value of the spontaneous curvature of the surface. This important result explains why most viral shells have either a compact shape with icosahedral symmetry or an elongated shape lacking this symmetry.

Figure 1

Elastic energy for 12 disclinations as a function of the intermediate size. (a) Energetic cascade for disclination inclusion. Elastic energy for 12 disclinations with icosahedral symmetry (thick red line), compared to the energy of $n=1–12$ disclinations (thin black lines). Dashed vertical lines indicate the radial positions of disclinations. The elastic energy without disclinations is shown by the blue line. (b) Elastic energy for 12 disclinations with icosahedral symmetry (thick red line), compared to 10 realizations of random locations for 12 disclinations (thin dashed colored lines). Inset: Example of locations of six disclinations regularly localized on the shell growing at a constant curvature.

Figure 2

Comparison of growth modes as a function of the intermediate size. (a) Elastic energy for isotropic growth with 12 disclinations with icosahedral symmetry (thick red line) and for cylindrical growth without defects at different curvatures, $R_0/d_0=5$, 25, and 50 (dashed orange, blue, and green lines, respectively). Cylinders represent the typical geometry for anisotropic growth, rather than the actual configuration. (b) Zoom-in on the elastic energies previously mentioned, with an identical legend, during the early steps of growth. Additional thin lines are the energies of the hemisphere-cone configuration with one defect for the same curvatures, $R_0/d_0=5$, 25, and 50 (thin orange, blue, and green lines, respectively), or a hemisphere with multiple defects (thin black lines).