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

TL;DR

This paper introduces a group theoretical method to construct finite nested point sets with non-crystallographic symmetry, providing insights into viral capsid organization and offering a finite group framework for virus architecture modeling.

Contribution

It presents a novel systematic construction of finite non-crystallographic symmetric structures using subgroup chains, applicable to virology and structural biology.

Findings

Constructed nested point sets with non-crystallographic symmetry.

Applied method to model viral capsid organization.

Provided explicit subgroup chains for icosahedral symmetry.

Abstract

The architecture of infinite structures with non-crystallographic symmetries can be modeled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. We present here a group theoretical method for the construction of finite nested point set with non-crystallographic symmetry. Akin to the construction of quasicrystals, we embed a non-crystallographic group $G$ into the point group $\mathcal{P}$ of a higher dimensional lattice and construct the chains of all $G$-containing subgroups. We determine the orbits of lattice points under such subgroups, and show 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 $\mathcal{P}$. In the case of icosahedral symmetry, we determine all subgroup chains explicitly and illustrate that these point sets in projection provide blueprints that approximate the organisation of simple viral capsids, encoding information on the structural organisation 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 for the modelling of its dynamic properties than its infinite dimensional counterpart.