The geometric role of the time parameter in DNA base-pair genetic information exchange and assembly: applications of Frenet–Serret formulas to circular and elliptical helix models

Abstract

This study employs tools from differential geometry to quantitatively reexamine the relationship between the chemical structure of biomacromolecules, such as DNA, and their geometric organization relevant to genetic information storage. We introduce the concept of an action-time parameter, t, and its correlation with the arclength, s, of three-dimensional curves, as exemplified by the canonical double helix, to provide a geometric framework for assessing the robustness of structural descriptors commonly used in chemical and biochemical studies of DNA. Using the Frenet–Serret formulas, we demonstrate a clear linear relationship between t and s. This highlights a consistent geometric parametrization of molecular structure. The invariance of curvature and torsion under geodesic curve conditions is a key geometric feature of structural regularity, highlighting the interplay between topological constraints and chemical bonding networks. Numerical analysis of an elliptical helix model indicates that minor perturbations in chemical geometry change do not disrupt this linearity, underscoring the system's structural tolerance. We expect that this study will serve as a valuable perspective for future research in physical-chemistry, where the interplay between molecular geometry, bonding interactions, and energetic stability can be quantitatively explored.