Geometric considerations in virus capsid size specificity, auxiliary requirements, and buckling.

Spherical capsids are shells of protein subunits that protect the genomes of many viral strains. Although nature displays a range of spherical capsid sizes (reflected by the number of subunits in the formation), specific strains display stringent requirements for forming capsids of specific sizes, a requirement that appears crucial to infectivity. Despite its importance in pathogenicity, little is known regarding the determinants of capsid size. Still less is known about exactly which capsids can undergo maturation events such as buckling transitions--postcapsid-assembly events that are crucial to some virus strains. We show that the exclusive determinant of capsid size is hexamer shape, as defined by subunit-subunit dihedral angles. This conclusion arises from considering the dihedral angle patterns within hexamers belonging to natural canonical capsids and geometric capsid models (deltahedra). From simple geometric models and an understanding of endo angle propagation discussed here, we then suggest that buckling transitions may be available only to capsids of certain size (specifically, T < 7 capsids are precluded from such transformations) and that T > 7 capsids require the help of auxiliary mechanisms for proper capsid formation. These predictions, arising from simple geometry and modeling, are backed by a body of empirical evidence, further reinforcing the extent to which the evolution of the atomistically complex virus capsid may be principled around simple geometric design/requirements.