DNA superhelical structures with non-constant helical pitch

Abstract

We develop an elastic–isotropic rod model for superhelical DNA structures where the helical angle is varying as a function of the arc-length. Our motivation for a variable helical angle comes from some experiments and simulations on DNA braids where complex superhelical structures have been observed. The helical solutions are minimizers of a free energy consisting of elastic, entropic and electrostatic terms. These minimizers are obtained within a variational framework where the end-points of the helices are allowed to be variable so that the length of the superhelix is computed as part of the solution. Considering variable curvature solutions brings up the possibility of finding more complex DNA structures because for two (or more) interwound helices there is a geometrical lock-up helical angle which puts a limit on the length of a superhelix. We perform calculations with different ionic concentrations and study the effects of lock up for braided structures. We also extend the variable curvature model to study the formation of plectonemes in the presence of multivalent salts where the supercoiling radius can be regarded as a constant prescribed by the balance of attractive and repulsive forces in DNA–DNA interactions, and provide analytical solutions in terms of elliptic functions for the supercoil parameters.