A crystallographic approach to structural transitions in icosahedral viruses

Viruses with icosahedral capsids, which form the largest class of all viruses and contain a number of important human pathogens, can be modelled via suitable icosahedrally-invariant finite subsets of icosahedral 3D quasicrystals. We combine concepts from the theory of 3D quasicrystals, and from the theory of structural phase transformations in crystalline solids, to give a framework for the study of the structural transitions occurring in icosahedral viral capsids during maturation or infection. As 3D quasicrystals are in a one-to-one correspondence with 6D icosahedral Bravais lattices, we study systematically the 6D-analogs of the classical Bain deformations in 3D, characterized by minimal symmetry loss at intermediate configurations, and use this information to infer putative viral-capsid transition paths in 3D via the cut-and-project method used for the construction of quasicrystals. We apply our approach to the Cowpea Chlorotic Mottle virus (CCMV) and show that the putative transition path between the experimentally-observed initial and final CCMV structures is most likely to exhibit $D_6$ symmetry, i.e. we conclude from our analysis that one three-fold axis is preserved throughout the transition. Our procedure suggests a general method for the investigation and prediction of symmetry constraints on the capsids of icosahedral viruses during structural transitions, and thus provides insights into the mechanisms underlying structural transitions of these pathogens.