**The architecture of viral capsids based on tiling theory**

A vital constituent of a virus is its protein shell, called the viral capsid, that encapsulates and hence
provides protection for the viral genome. Mathematical models for the surface structures of viral capsids are important for the analysis of experimental
data, and are a crucial step for the understanding of the viral replication
cycle. In a landmark paper, Caspar and Klug have developed a theory for the classification of icosahedral viral capsids based on the principle of
quasi-equivalence. From a mathematical point of view, quasi-equivalence implies that the architecture of the capsids corresponds to triangulations that are compatible with their icosahedral
symmetry. In particular, the protein subunits in the capsids are assumed to be located in the corners of the triangular facets of these
triangulations, and viral capsids corresponding to a triangulation in terms of *T*
facets per face of the icosahedron are hence predicted to be formed from precisely
60*T* protein subunits.

Caspar-Klug Theory models the locations of the protein subunits accurately for a large number of
viruses, and it has hence become one of the fundamental concepts in virology.
However, experimental evidence has shown that the assumptions of Caspar-Klug Theory are too restrictive to accommodate all icosahedral viruses. In
particular, Papovaviridae - a family of viruses linked to cancer - fall out of the remit of this theory.
Moreover, Caspar-Klug Theory does not make any predictions about the locations of the inter-subunit
bonds, which is important in particular for the construction of assembly models.

Our group has developed a generalisation of Caspar-Klug Theory based on tiling theory. This approach solves a structural puzzle concerning the surface structure of the (pseudo-)
*T*=7 particles in the family of Papovaviridae, which cannot be explained in the framework of Caspar-Klug
Theory. Moreover, the tiling approach has been applied to tubular malformations in this family. The tiling approach also
predicts, besides the locations of the protein subunits, the locations of the inter-subunit bonds that stabilize the
capsids. Therefore, it also explains cross-linking structures as we have
explored in joint work with Roger Hendrix. Moreover, it provides a basis for the study of assembly
models.