A new series of polyhedra as blueprints for viral capsids in the family of Papovaviridae

In a seminal paper Caspar and Klug established a theory that provides a family of polyhedra as blueprints for the structural organisation of viral capsids. In particular, they encode the locations of the proteins in the shells that encapsulate, and hence provide protection for, the viral genome. Despite of its huge success and numerous applications in virology experimental results have provided evidence for the fact that the theory is too restrictive to describe all known viruses. Especially, the family of Papovaviridae, which contains cancer-causing viruses, falls out of the scope of this theory. In a recent paper we have shown that certain members of the family of Papovaviridae can be described via tilings. In this paper, we develop a comprehensive mathematical framework for the derivation of all surface structures of viral particles in this family. We show that this formalism fixes the structure and relative sizes of all particles collectively so that there exists only one scaling factor that relates the sizes of all particles with their biological counterparts. The series of polyhedra derived here complements the Caspar-Klug family of polyhedra. It is the first mathematical result that provides a common organisational principle for different types of viral particles in the family of Papovaviridae and paves the way for an understanding of Papovaviridae polymorphism. Moreover, it provides crucial input for the construction of assembly models.