Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

Abstract

The architecture of infinite structures with non-crystallographic symmetries can be modelled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic group G is embedded into the point group P of a higher-dimensional lattice and the chains of all G-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensional G-invariant subspace consists of nested point sets with G-symmetry at each radial level. The number of different radial levels is bounded by the index of G in the subgroup of P. In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart.

Structural puzzles in virology solved with an overarching icosahedral design principle

“…It is difficult to predict the additional lattices that can occur at different radial levels, unless their structures are coupled to the lattices describing the organisation of the capsid core discussed here. Such coupling could be modelled via affine extended symmetry groups 45–47 or 3D tilings 48 , but this is beyond the scope of this paper. Interestingly, for the example of P22 the triangular positions correspond precisely to the trimer interactions between capsomers (cf.…”

Realizations of the abstract regular H 3 polyhedra

“…These diagrams represent a system of linked points decomposed into two child subsystems which, although they may act independently, behave the same symmetry-wise as their blend. Examples of systems that exhibit a similar structure or behavior are the cages of carbon atoms in a carbon onion or in a multi-wall carbon nanotube (Twarock, 2002) as well as the protein subunits in a viral capsid (Twarock et al, 2015). Finally, in terms of carrying out computations within a system of points arising from Wythoff construction, one has the advantage of manipulating vectors containing only integer entries, and applying symmetry operations on these vectors by simply permuting them (see Illustration 3.1).…”

A group theoretical approach to structural transitions of icosahedral quasicrystals and point arrays

“…( 4)), i.e. ρ 3 C ⊆ C. The projection operators π t given in (11) can be used to define a family of arrays C t , for t ∈ [0, 1], given by:…”

“…( 7)) for all t ∈ (0, 1), and moreover possess icosahedral symmetry for t = 0 and t = 1. We refer to [11] for a detailed method for the construction of finite nested icosahedral point sets via projection, in connection with the structure of viral capsids.…”

“…The lattice L in R 6 is taken as the simple cubic lattice with the standard basis in R 6 . The matrix R T (t) = M T ( βt)R encodes the projections π t and π ⊥ t as in (11), that define the family of model sets Σ t as in (12). In Figure 2 we show a patch of the resulting quasilattices for t = 0, 0.5 and 1.…”