Structural Description of Viral Particles Based on Affine Extensions of Non-Crystallographic Coxeter Groups

An essential part of a virus is its viral capsid, a protein shell that encapsulates and hence provides protection for the viral genome. Due to the fact that the majority of the known viruses has capsids with overall icosahedral symmetry, group theory lends itself for their structural description. In partic- ular, icosahedral symmetry determines the arrangement of the protein subunits in the capsid up to the fundamental domain of the icosahedral group. Thus, for larger viruses consisting of more than 60 protein subunits, icosahedral sym- metry does not specify their arrangement completely. This has already been observed by Caspar and Klug who suggested in their seminal paper from the 1960s to model the locations of the additional subunits based on the concept of quasi-equivalence. However, their theory is applicable only to a subset of the known viruses. In particular, it does not apply to the cancer-causing Papovaviri- dae, which are of crucial importance to the public health sector. We present here a theory - based on affine extensions of non-crystallographic Coxeter groups - that closes this gap and provides a structural description also for the missing cases. Recent results concerning applications of this theory to the description of tubular malformations, the modelling of crosslinking structures and the assem- bly process are also discussed.