Mathematical virology: a novel approach to the structure and assembly of viruses.

Understanding the structure and life cycle of viruses is a fascinating challenge with a crucial impact on the public health sector. In the early 1960s, Caspar & Klug (Caspar & Klug 1962 Cold Spring Harbor Symp. Quant. Biol. 27, 1-24) established a theory for the prediction of the surface structures of the protein shells, called viral capsids, which encapsulate and hence provide protection for the viral genome. It is of fundamental importance in virology, with a broad spectrum of applications ranging from the image analysis and classification of experimental data to the construction of assembly models. However, experimental results have provided evidence for the fact that it is incomplete and, in particular, cannot account for the structures of Papovaviridae, which are of particular interest because they contain cancer-causing viruses. This gap has recently been closed by the viral tiling theory, which describes the locations of the protein subunits and inter-subunit bonds in viral capsids based on mathematical tools from the area of quasicrystals. The predictions and various recent applications of the new theory are presented, and it is discussed how further research along these lines may lead to new insights in virology and the design of anti-viral therapeutics.