A tale of two symmetrons: Rules for construction of icosahedral capsids from trisymmetrons and pentasymmetrons

The capsids of large, icosahedral dsDNA viruses are built from well-ordered aggregates of capsomers, known as trisymmetrons and pentasymmetrons, which are centered on the icosahedral 3-fold and 5-fold axes, respectively. We derive the complete set of rules for constructing an icosahedral structure from these symmetrons when the T lattice symmetry is odd and show that there are three classes of solutions, each of which follows from a different relationship between the size of the pentasymmetron and the values of the h and k icosahedral lattice parameters. Together, these three classes account for all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons. Also, every icosahedral lattice with odd T number has solutions from exactly two of these three classes, with the set of allowed classes dependent on which of the two lattice parameters is odd. For symmetric lattices (if h = k or h = 0), the two solutions yield the same symmetron sizes, but when the lattice parameters are equal (h = k) the solutions can be distinguished by the relative orientations of the symmetrons. We discuss these results in the context of known virus structures and explore the implications for viruses whose structures have not yet been solved.