Structural constraints on the three-dimensional geometry of simple viruses: case studies of a new predictive tool

Abstract

Understanding the fundamental principles of virus architecture is one of the most important challenges in biology and medicine. Crick and Watson were the first to propose that viruses exhibit symmetry in the organization of their protein containers for reasons of genetic economy. Based on this, Caspar and Klug introduced quasi-equivalence theory to predict the relative locations of the coat proteins within these containers and classified virus structure in terms of T-numbers. Here it is shown that quasi-equivalence is part of a wider set of structural constraints on virus structure. These constraints can be formulated using an extension of the underlying symmetry group and this is demonstrated with a number of case studies. This new concept in virus biology provides for the first time predictive information on the structural constraints on coat protein and genome topography, and reveals a previously unrecognized structural interdependence of the shapes and sizes of different viral components. It opens up the possibility of distinguishing the structures of different viruses with the same T-number, suggesting a refined viral structure classification scheme. It can moreover be used as a basis for models of virus function, e.g. to characterize the start and end configurations of a structural transition important for infection.

Viruses and fullerenes – symmetry as a common thread?

“…The reason for this choice of polyhedral shapes, called start configurations in Keef & Twarock (2009), stems from the fact that they correspond to the projections of the standard bases of the three Bravaislattice types with icosahedral symmetry in six dimensions: the icosahedron obtained from a six-dimensional simple cubic lattice, the dodecahedron from a six-dimensional bodycentred cubic lattice and the icosidodecahedron from a sixdimensional face-centred cubic lattice. In particular, the use of these shapes in the construction of the affine groups ensures that the point arrays are subsets of the vertex sets of quasilattices with icosahedral symmetry [see Keef et al (2013) for a two-dimensional example and Salthouse (2013) for a threedimensional one]. In particular, this implies that the affineextended non-crystallographic groups are by construction related to aperiodic tilings, in analogy to the relation between affine-extended crystallographic groups and lattices.…”

“…A full classification of the affine extensions of the icosahedral group based on the three polyhedral start configurations given by the icosahedron, dodecahedron and icosidodecahedron has been provided in Keef & Twarock (2009), and applications of these point arrays to viruses have been discussed in Keef et al (2013). In particular, since the relative scalings between all the points in the array are fixed by the extended group, there is only one global scaling factor that maps all points collectively onto the biological system.…”

“…In particular, in this reference the reflection groups H 3 and H 4 , which are the only reflection groups containing icosahedral symmetry as a subgroup, have been extended by an affine reflection, and it has been shown that in combination with generators of the finite groups the affine reflections act as translations. Subsequently, affinizations of icosahedral symmetry via translation operators have been classified (Keef & Twarock, 2009;Keef et al, 2013;Dechant et al, 2012Dechant et al, , 2013Wardman, 2012). These contain the affinizations derived previously, but in addition provide a much wider spectrum of extended group structures.…”

“…In particular, since the relative scalings between all the points in the array are fixed by the extended group, there is only one global scaling factor that maps all points collectively onto the biological system. For example, in the case of Pariacoto virus discussed in Keef et al (2013) and shown in Fig. 2, the length scales are determined by the structure of the genomic RNA such that array points map into the minor grooves of the molecule.…”

“…It therefore lends itself to the modelling of carbon onions (Patera & Twarock, 2002;Twarock, 2002). Recently we introduced new affine extensions for the icosahedral group (Twarock, 2006;Keef & Twarock, 2009) and demonstrated that these can be used to model virus structure at different radial levels (Keef et al, 2013;Wardman, 2012). In particular, this work revealed a previously unappreciated molecular scaling principle in virology, relating the structure of the viral capsid of Pariacoto virus to that of its packaged genome.…”