Beyond Quasi-Equivalence: New Insights Into Viral Architecture via Affine Extended Symmetry Groups

Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence provide protection for the viral genome, it has been recognised that icosahedral symmetry is crucial for the prediction of viral architecture. In particular, their theory of quasi-equivalence invokes icosahedral symmetry to pinpoint the surface structures of viral capsids in terms of triangulations that schematically encode the locations of the protein subunits in the capsids. Whilst this approach is capable of predicting the relative locations of the proteins in the capsids, information on their tertiary structures and the organisation of the viral genome within the capsid are inaccessible. We present here a mathematical framework based on affine extensions of the icosahedral group that has been developed to predict a wide spectrum of features of the three-dimensional structure of simple viruses. This approach implies that the predictions of Caspar and Klug's quasi-equivalence theory are the consequences of a deeper level of structural organisation in viruses that orchestrates the full three-dimensional structure of simple viruses.