Mathematical modelling of interwound DNA supercoils

Small loops of DNA are affected by a variety of enzymes which remove turns of twist relative to the underlying double-helical structure. The molecule adopts a complex three-dimensional shape known as a supercoil in order to relieve the resulting internal stresses. This article describes an approach to modelling the overall shape of the supercoiled structure using elastic rod theory, which leads to simple expressions for predicting the shape of the structure. Predictions on the number of crossings in the balanced ply and the length of the end loops are compared to data in the literature and show reasonable agreement. The effect of the charged phosphate groups along the backbone of the DNA on the resulting supercoiled shape are also examined, and it is shown that this shape is very sensitive to the ionic concentration of the solution.