Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. II. {\alpha}-helix and DNA structures

TL;DR

This paper uses algebraic topology to model helicoidally-like biopolymer structures, including alpha-helix and DNA, revealing their topological stability and potential transformations between different DNA conformations.

Contribution

It introduces a novel topological model based on algebraic polytopes to describe and analyze the structure and stability of helicoidally-like biopolymers.

Findings

Topologically stable rod substructures corresponding to alpha-helix and Z-DNA.

Model predicts possible transformations between DNA structures.

Identification of bifurcation points indicating structural stability changes.

Abstract

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra (algebraic polytopes) was defined. This sequence is started by the polytope {240}, discovered by Coxeter, and is determined by the second coordination sphere of 8-dimensional lattice E8. The second polytope of sequence allows to determine a topologically stable rod substructure that appears during multiplication by a non-crystallographic axis 40/11 of the starting union of 4 tetrahedra with common vertex. When positioning the appropriate atoms tin positions of special symmetry of the staring 4 tetrahedra, such helicoid determines an {\alpha}-helix. The third polytope of sequence allows to determine the helicoidally-like union of rods with 12-fold axis, which can be compare with Z-DNA structures. This model is defined as a local lattice rod packing, contained within a surface of helicoidally similar type, which ensures its topological stability, as well as possibility for it to be transformed into other forms of DNA structures. Formation of such structures corresponds to lifting a configuration degeneracy, and the stability of a state - to existence of a point of bifurcation. Furthermore, in the case of DNA structures, a second "security check" possibly takes place in the form of local lattice (periodic) property using the lattices other than the main ones.