  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/faceted as their radius increases through a
critical value determined by the competition between stretching and bending
energies of a closed 2D elastic network. In the present work we generalize this
approach to capsids with non-icosahedral symmetries, e.g., spherocylindrical
and conical shells. One key new physical ingredient is the role played by
nonzero spontaneous curvature. Another is associated with the special way in
which the energy of the twelve topologically-required five-fold 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.
