Symmetry of hydrocarbon chains of phospholipids and hyperbolic honeycomb

^{1}Talis A.L., ^{2}Rabinovich A.L.

^{1}Institute of organoelement compounds RAS ^{2}IKarelian Research Centre of Russian Academy of Sciences

Hydrocarbon chains play the key role in various processes of biological membranes; in particular, they are fatty acid constituents of phospholipid molecules. Most parts of each chain are tetracoordinate structures, i.e. can be approximated by a combination of regular tetrahedral chains. To reveal the “hidden” symmetry of such chain molecules in 3-dimensional Euclidean space, the symmetry of their analogs in 3-dimensional Riemannian spaces can be used. In this work, the space of constant negative curvature, tessellated by ideal hyperbolic tetrahedra is considered (hyperbolic honeycomb)

Molecular structures that can be approximated by chains of regular tetrahedra: the basic structural unit

^{1}Talis A.L., ^{2}Rabinovich A.L.

^{1}Institute of organoelement compounds RAS ^{2}IKarelian Research Centre of Russian Academy of Sciences

If the substructure of an ordered structure in 3-dimensional Euclidean space is linear, then the arrangement of it’s atoms (along the definite length) can be determined by the symmetries of non-Euclidean mathematical constructions. In this work, the space of constant positive curvature, tessellated by ideal tetrahedra (4-dimensional polytopes) is considered, and the basic structural unit of chains of regular tetrahedra is determined

Symmetry of tetrahedral and tetracoordinate structures and combinatorial constructions

^{1}Rabinovich A.L., ^{2}Talis A.L.

^{1}Karelian Research Centre of Russian Academy of Sciences ^{2}Institute of organoelement compounds RAS

To elucidate the cause of existence of some structure features which are typical of hydrocarbon constituents of lipid molecules that form the basis of biomembranes, the combinatorial construction is considered that can be related to linear unions of regular face-sharing tetrahedra; a diamond-like union of two tetrahedral chains results in a tetracoordinate structure. The combinatorial construction allows studying non-crystallographic symmetry properties of tetrahedral chains

Mapping of symmetry of hydrocarbon chains: a tetrabloc is the universal structural unit

^{1}Rabinovich A.L., ^{2}Talis A.L.

^{1}Karelian Research Centre of Russian Academy of Sciences ^{2}Institute of organoelement compounds RAS

A 7-vertex linear union of 4 face-sharing regular tetrahedra (a tetrabloc) is shown to be the universal maximally possible "unit" that allows one to map a “non-crystallographic” (“hidden”) symmetry of the linear chains of regular tetrahedra in 3-dimensional Euclidean space (the space with zero curvature) which can be embedded into finite unions of face-sharing tetrahedra in 3-dimensional Riemannian spaces of constant positive and negative curvature