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

TY - JOUR

T1 - Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

AU - Twarock, Reidun

AU - Valiunas, Motiejus

AU - Zappa, Emilio

N1 - © International Union of Crystallography. 2015. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

PY - 2015

Y1 - 2015

N2 - 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 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 . 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.

AB - 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 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 . 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.

KW - computational group theory

KW - icosahedral viruses

KW - non-crystallographic symmetry

KW - orbits

UR - http://www.scopus.com/inward/record.url?scp=84946719640&partnerID=8YFLogxK

U2 - 10.1107/S2053273315015326

DO - 10.1107/S2053273315015326

M3 - Article

AN - SCOPUS:84946719640

SN - 2053-2733

VL - 71

SP - 569

EP - 582

JO - Acta crystallographica. Section A, Foundations and advances

JF - Acta crystallographica. Section A, Foundations and advances

ER -