An algebro-geometric model for the shape of supercoiled DNA

TL;DR

This paper introduces a generalized algebro-geometric model incorporating thermal effects to better represent the complex shapes of supercoiled DNA, extending beyond traditional elastic rod models.

Contribution

It develops a novel algebro-geometric framework using hyperelliptic curves to model DNA shapes, capturing more realistic configurations than classical elastica models.

Findings

The model reproduces key features of supercoiled DNA shapes.

It extends elastica theory to include hyperelliptic curve data.

The approach accounts for thermal effects in DNA modeling.

Abstract

This article proposes a model including thermal effects for closed supercoiled DNA. Existing models include an elastic rod. Euler's elastica, ideal elastic rods on a plane, have only two kinds of closed shapes, the circle and a figure-eight, realized as minima of the Euler-Bernoulli energy. Even considering three dimensional effects, this elastica model provides much simpler shapes than observed via Atomic-Force Microscope (AFM), since the minimal points of the energy are expressed by elliptic functions. In this paper, by a generalization of elastica, we obtain shapes determined by data of hyperelliptic curves, which partially reproduce the shapes and properties of the DNA.