Affine extensions of the icosahedral group with applications to the three-dimensional organisation of simple viruses

Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence protect the viral genome, it has been recognised that icosahedral symmetry is crucial for the structural organisation of viruses. In particular, icosahedral symmetry has been invoked in order to predict the surface structures of viral capsids in terms of tessellations or tilings 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 develop here a mathematical framework based on affine extensions of the icosahedral group that allows us to describe those aspects of the three-dimensional structure of simple viruses. This approach complements Caspar-Klug theory and provides details on virus structure that have not been accessible with previous methods, implying that icosahedral symmetry is more important for virus architecture than previously appreciated.