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

2015 ◽  
Vol 71 (6) ◽  
pp. 569-582 ◽  
Author(s):  
Reidun Twarock ◽  
Motiejus Valiunas ◽  
Emilio Zappa
Keyword(s):  
Dynamic Properties ◽  
Group Structure ◽  
Point Group ◽  
Theoretical Method ◽  
Lattice Points ◽  
Crystallographic Group ◽  
Icosahedral Symmetry ◽  
Point Sets ◽  
Finite Structures ◽  
Crystallographic Symmetry

The architecture of infinite structures with non-crystallographic symmetries can be modelledviaaperiodic 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 groupGis embedded into the point group {\cal P} of a higher-dimensional lattice and the chains of allG-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensionalG-invariant subspace consists of nested point sets withG-symmetry at each radial level. The number of different radial levels is bounded by the index ofGin the subgroup of {\cal 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.

Homological functor of a torsion free crystallographic group of dimension five with a nonabelian point group

2014 ◽  
Author(s):  
Tan Yee Ting ◽  
Nor'ashiqin Mohd. Idrus ◽  
Rohaidah Masri ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Crystallographic Group ◽  
Homological Functor ◽  
Torsion Free

scholarly journals The nonabelian tensor square of a crystallographic group with quaternion point group of order eight

2017 ◽  
Vol 893 ◽  
pp. 012006
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Crystallographic Group ◽  
Nonabelian Tensor ◽  
Tensor Square

A group theoretical method for the construction of the nonequivalent configurations of finite clusters with cellular disorder

1984 ◽  
Vol 62 (9) ◽  
pp. 904-910 ◽  
Author(s):  
H. Bonadeo
Keyword(s):  
Point Group ◽  
Theoretical Method ◽  
Group Theoretical Method

The nonequivalent configurations of finite clusters with cellular disorder may be constructed by combining subconfigurations of atoms located at lattice sites that are equivalent by symmetry. The combination laws depend only on the symmetry of the cluster and are obtained from point group theoretical arguments. The method may be applied to clusters of arbitrary symmetry and composition, and is illustrated with a simple example.

Second-order piezomagnetism in polychromatic crystals

1990 ◽  
Vol 46 (2) ◽  
pp. 130-133
Author(s):  
K. Rama Mohana Rao
Keyword(s):  
Point Group ◽  
Physical Property ◽  
Theoretical Method ◽  
Second Order ◽  
Acta Cryst ◽  
Theoretical Methods ◽  
Group Theoretical Method ◽  
Group 4 ◽  
Independent Number

The group-theoretical method established for obtaining the non-vanishing independent number of constants required to describe a magnetic/physical property in respect of the 18 polychromatic crystal classes [Rama Mohana Rao (1987). J. Phys. A, 20, 47-57] has been explored to enumerate the second- order piezomagnetic coefficients (n i ′) for the same classes. The advantage of Jahn's method [Jahn (1949). Acta Cryst. 2, 30-33] is appreciated in obtaining these n i ′ through the reduction of a representation. The different group-theoretical methods are illustrated with the help of the point group 4. The results obtained for all 18 classes are tabulated and briefly discussed.

scholarly journals Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesBn

2015 ◽  
Vol 71 (2) ◽  
pp. 175-185 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koc
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Icosahedral Symmetry ◽  
The Novel ◽  
Voronoi Cells ◽  
Aperiodic Tilings ◽  
Coxeter Number ◽  
Hypercubic Lattice ◽  
Higher Dimensional ◽  
Coxeter Plane

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter–Weyl groupWa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroupDhofW(Bn) withh= 2nrepresenting the Coxeter number describes theh-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed forn= 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroupsW(A3),W(H2) ×W(A1) andW(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the latticeB4onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. TheB5lattice is used to describe both fivefold and tenfold symmetries. The latticeB6can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Approximation of virus structure by icosahedral tilings

2015 ◽  
Vol 71 (4) ◽  
pp. 410-422 ◽  
Author(s):  
D. G. Salthouse ◽  
G. Indelicato ◽  
P. Cermelli ◽  
T. Keef ◽  
R. Twarock
Keyword(s):  
Goodness Of Fit ◽  
Virus Assembly ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Three Dimensions ◽  
Coarse Grained ◽  
Icosahedral Symmetry ◽  
Point Sets ◽  
Viral Capsids ◽  
Project Scheme

Viruses are remarkable examples of order at the nanoscale, exhibiting protein containers that in the vast majority of cases are organized with icosahedral symmetry. Janner used lattice theory to provide blueprints for the organization of material in viruses. An alternative approach is provided here in terms of icosahedral tilings, motivated by the fact that icosahedral symmetry is non-crystallographic in three dimensions. In particular, a numerical procedure is developed to approximate the capsid of icosahedral viruses by icosahedral tilesviaprojection of high-dimensional tiles based on the cut-and-project scheme for the construction of three-dimensional quasicrystals. The goodness of fit of our approximation is assessed using techniques related to the theory of polygonal approximation of curves. The approach is applied to a number of viral capsids and it is shown that detailed features of the capsid surface can indeed be satisfactorily described by icosahedral tilings. This work complements previous studies in which the geometry of the capsid is described by point sets generated as orbits of extensions of the icosahedral group, as such point sets are by construction related to the vertex sets of icosahedral tilings. The approximations of virus geometry derived here can serve as coarse-grained models of viral capsids as a basis for the study of virus assembly and structural transitions of viral capsids, and also provide a new perspective on the design of protein containers for nanotechnology applications.

A molecular theory of electromagnetic wave propogation in two-component ansiotropic polymer systems

1962 ◽  
Vol 254 (1044) ◽  
pp. 397-440 ◽  
Keyword(s):  
Electric Field ◽  
Group Structure ◽  
Incident Light ◽  
Pair Correlation ◽  
Internal Field ◽  
Field Vector ◽  
Lattice Points ◽  
Electric Vector ◽  
Internal Electric Field ◽  
Principal Axes

It has been shown by Bragg (1924) that the birefringence of anisotropic crystalline materials can be ascribed in part to the anisotropic dependence of the magnitude of the induced internal electric field on the electric vector of an incident light wave set at differing orientations to the crystalline axes. The internal field depends on positional correlation between pairs of particles, and if this is anisotropic the induced field depends on the relative orientation of the electric vector to the symmetry axes of the pair correlation function. The square, of the refractive index m of the material depends on the ratio of the induced electric field to the applied field, and, when this ratio depends on the orientation of the applied electric field vector, m2 will have tensor-like properties—at least in so far as it will have three (in general) principal axes and values. In condensed phases the spherical symmetry of individual isolated atoms is lost and a second source o f birefringence resides in the ordered orientation o f individually anisotropically polarizable particles. In so far as it is also mathematically convenient, when treating condensed systems, to deal with the polarizability of any group of atoms which retains its group structure over long periods of time as that of a single entity, birefringence must a fortiori be ascribed also to an intrinsic anisotropy of polarizability of individual particles. Nitta (1940) therefore described the observed birefringence in certain tetragonal crystals in terms of an anisotropically polarizable unit corresponding to the content of one unit cell localized on tetragonal lattice points

The symmetry origin of the austenite-cementite orientation relationships in steels

2019 ◽  
Vol 234 (4) ◽  
pp. 237-245 ◽  
Author(s):  
Valentin Kraposhin ◽  
Alexander Talis ◽  
Nenad Simich-Lafitskiy
Keyword(s):  
Crystal Structure ◽  
Orientation Relationship ◽  
Formation Mechanism ◽  
Mutual Orientation ◽  
Crystallographic Group ◽  
Mutual Transformation ◽  
Orientation Relationships ◽  
Nucleus Formation ◽  
Crystallographic Symmetry ◽  
Initial Crystal

Abstract The connection between austenite/cementite orientation relationships and crystal structure of both phases has been established. The nucleus formation mechanism at the mutual transformation of austenite and cementite structures has been proposed. Mechanism is based on the interpretation of the considered structures as crystallographic tiling onto triangulated polyhedra, and the said tiling can be transformed by diagonal flipping in a rhombus consisting of two adjacent triangular faces. The sequence of diagonal flipping in the fragment of the initial crystal determines the orientation of the fragment of the final crystal relative to the initial crystal. In case of the mutual austenite/cementite transformation the mutual orientation of the initial and final fragments is coinciding to the experimentally observed in steels Thomson-Howell orientation relationships: ${\left\{ {\bar 103} \right\}_{\rm{C}}}||{\left\{ {111} \right\}_{\rm{A}}};{\rm{}} < {\kern 1pt} 010{\kern 1pt} { > _{\rm{C}}}{\rm{||}} < {\kern 1pt} 10\bar 1{\kern 1pt} { > _{\rm{A}}};\; < {\kern 1pt} 30\bar 1{\kern 1pt} { > _{\rm{C}}}\;||\,\, < {\kern 1pt} \bar 12\bar 1{\kern 1pt} { > _{\rm{A}}}{\rm{}}$ The observed orientation relationship between FCC austenite and cementite is determined by crystallographic group-subgroup relationship between transformation participants, and non-crystallographic symmetry which is determining the transformation of triangulated clusters of transformation participants.

Symmetry-adapted digital modeling I. Axial symmetric proteins

2016 ◽  
Vol 72 (3) ◽  
pp. 298-311 ◽  
Author(s):  
A. Janner
Keyword(s):  
Rna Binding ◽  
Rotation Axis ◽  
Point Group ◽  
Lattice Points ◽  
Group Symmetry ◽  
Coarse Grained ◽  
Mitochondrial Creatine Kinase ◽  
Fine Grained ◽  
Coarse Grained Model ◽  
Digital Modeling

Considered are axial symmetric proteins exemplified by the octameric mitochondrial creatine kinase, the Pyr RNA-binding attenuation protein, the D-aminopeptidase and the cyclophilin A–cyclosporin complex, with tetragonal (422), trigonal (32), pentagonal (52) and pentagonal (52) point-group symmetry, respectively. One starts from the protein enclosing form, which is characterized by vertices at points of a lattice (the form lattice) whose dimension depends on the point group. This allows the indexing of Cα's at extreme radial positions. The indexing is extended to additional residues on the basis of a finer lattice, the digital modeling lattice Λ, which includes the form lattice as a sublattice. This leads to a coarse-grained description of the protein. In the crystallographic point-group case, the planar indices are obtained from a projection of atomic positions along the rotation axis, taken as thezaxis. The planar indices of a Cαare then those of the nearest projected lattice point. In the non-crystallographic case, low indices are an additional requirement. The coarse-grained bead follows from the condition imposed on the residues selected to have azcoordinate within a band of value δ above and below the height of lattice points. The choice of δ permits a variation of the coarse-grained bead model. For example, the value δ = 0.5 leads to a fine-grained indexing of the full set of residues, whereas with δ = 0.25 one gets a coarse-grained model which includes only about half of these residues. Within this procedure, the indexing of the Cαonly depends on the choice of the digital modeling lattice and not on the value of δ. The characteristics which distinguish the present approach from other coarse-grained models of proteins on lattices are summarized at the end.

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