Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. III. Nature of the double and relations between the alpha helix and the various forms of DNA structures

TL;DR

This paper explores the topological and algebraic structure of helicoidally-like biopolymers, such as alpha helices and DNA conformations, using algebraic topology and minimal surface theory to understand their stability and symmetry.

Contribution

It introduces a novel algebraic topological framework to analyze the structure and stability of biopolymers and DNA, linking minimal surfaces and bifurcation points to their configurations.

Findings

Topological stability relates to bifurcation points and minimal surface conditions.

Symmetry parameters of DNA structures are determined a priori.

Structures are modeled using algebraic polytopes and lattice E8.

Abstract

In the frameworks of algebraic topology {\alpha}-helix and different DNA-conformations are determined as the local latticed packing, confined by peculiar minimal surfaces which are similar to helicoids. These structures are defined by Weierstrass representation and satisfy to a zero condition for the instability index of the surface and availability of bifurcation points for these surfaces. The topological stability of such structures corresponds to removing of the configuration degeneracy and to existence of bifurcation points for these surfaces. The considered ordered non-crystalline structures are determined by homogeneous manifolds - algebraic polytopes, corresponding to the definite substructures the 8-dimensional lattice E8.The joining of two semi-turns of two spirals into the turn of a single two-spiral (helical) system is effected by the topological operation of a connected sum. The applied apparatus permits to determine a priori the symmetry parameters of the double spirals in A, B and Z forms DNA structures.