Blueprints for viral capsids in the family of polyomaviridae.

In a seminal paper, Caspar and Klug [1962. Physical principles in the construction of regular viruses. Cold Spring Harbor Symp. Quant. Biol. 27, 1-24] derived a family of surface lattices as blueprints for the structural organisation of the protein shells, called viral capsids, which encapsulate and hence protect the viral genome. These lattices schematically encode, and hence predict, the locations of the proteins in the viral capsids. Despite the huge success and numerous applications of this theory in virology, experimental results have provided evidence for the fact that it is too restrictive to describe all known viruses [Casjens, S., 1985. Virus Structure and Assembly. Jones and Bartlett, Boston, MA]. Especially, the family of Polyomaviridae, which contains cancer-causing viruses, falls out of the scope of this theory. In [Twarock, R., 2004. A tiling approach to virus capsid assembly explaining a structural puzzle in virology. J. Theor. Biol. 226, 477], we have shown that a member of the family of Polyomaviridae can be described via an icosahedrally symmetric tiling. We show here that all viruses in this family can be described by tilings with vertices corresponding to subsets of a quasi-lattice that is constructed based on an affine extended Coxeter group, and we use this methodology to derive their coordinates explicitly. Since the particles appear as different subsets of the same quasi-lattice, their relative sizes are predicted by this approach, and there hence exists only one scaling factor that relates the sizes of all particles collectively to their biological counterparts. It is the first mathematical result that provides a common organisational principle for different types of viral particles in the family of Polyomaviridae, and paves the way for modelling Polyomaviridae polymorphism.

Assembly models for Papovaviridae 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. Assembly models are developed for viral capsids built from protein building blocks that can assume different local bonding structures in the capsid. This situation occurs, for example, for viruses in the family of Papovaviridae, which are linked to cancer and are hence of particular interest for the health sector. More specifically, the viral capsids of the (pseudo-) T = 7 particles in this family consist of pentamers that exhibit two different types of bonding structures. While this scenario cannot be described mathematically in terms of Caspar-Klug theory (Caspar D L D and Klug A 1962 Cold Spring Harbor Symp. Quant. Biol. 27 1), it can be modelled via tiling theory (Twarock R 2004 J. Theor. Biol. 226 477). The latter is used to encode the local bonding environment of the building blocks in a combinatorial structure, called the assembly tree, which is a basic ingredient in the derivation of assembly models for Papovaviridae along the lines of the equilibrium approach of Zlotnick (Zlotnick A 1994 J. Mol. Biol. 241 59). A phase space formalism is introduced to characterize the changes in the assembly pathways and intermediates triggered by the variations in the association energies characterizing the bonds between the building blocks in the capsid. Furthermore, the assembly pathways and concentrations of the statistically dominant assembly intermediates are determined. The example of Simian virus 40 is discussed in detail.

Mathematical models for tubular structures in the family of Papovaviridae.

An important part of a virus is its protein shell, called the viral capsid, that protects the viral genome. While the viral capsids of viruses in the family of Papovaviridae are usually spherical, their protein building blocks are known to assemble also as tubular structures [Kiselev, N.A., Klug, A., 1969. J. Mol. Biol. 40, 155]. In Twarock [2004. J. Theor. Biol. 226, 477] Viral Tiling Theory has been introduced for the structural description of the protein stoichiometry of the spherical capsids in this family. This approach is extended here to the tubular case and is used to classify the surface lattices of tubular structures in the family of Papovaviridae. The predictions of the theory are compared with the experimental results in Kiselev and Klug [1969. J. Mol. Biol. 40, 155].