An algebro-geometric model for the shape of supercoiled DNA

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.