Statistical mechanics of elastica for the shape of supercoiled DNA: Hyperelliptic elastica of genus three

Abstract

This article studies the statistical mechanics of elastica as a model of the shapes of the supercoiled DNA, and shows that its excited states can be characterized by the focusing modified KdV (MKdV) equation due to thermal fluctuation. Following the previous paper (Matsutani and Previato, 2022), the hyperelliptic solutions of the focusing MKdV equation of genus three are considered. Numerical evaluation shows that the excited states given by the solution has a pattern as a repetition of the modulation of figure-eight and the inverse ‘S’ as a thermal fluctuation of elastica, called the S-eight mode. Our model states that the excited states of elastica due to the thermal effect have the S-eight mode. Since in the shapes of the AFM image of the supercoiled DNAs observed by Japaridze et al. (2017), the S-eight mode appears, our model is a novel model that mathematically represents the shape of the supercoiled DNA.