Elastic rod model of a supercoiled DNA molecule

We study the elastic behaviour of a supercoiled DNA molecule. The simplest model is that of a rod-like chain, involving two elastic constants, the ben ding and the twist rigidities. Writing this model in terms of Euler angles, we show that the corresponding Hamiltonian is singular and needs a small d istance cut-off, which is a natural length scale giving the limit of validi ty of the model, of the order of the double-helix pitch. The rod-like chain in the presence of the cut-off is able to reproduce quantitatively the exp erimentally observed effects of supercoiling on the elongation-force charac teristics, in the small supercoiling regime. An exact solution of the model , using both transfer matrix techniques and its mapping to a quantum mechan ics problem, allows to extract, from the experimental data, the value of th e twist rigidity. We also analyse the variation of the torque and the writh e-to-twist ratio versus supercoiling, showing analytically the existence of a rather sharp crossover regime which can be related to the excitation of plectoneme-like structures. Finally we study the extension fluctuations of a stretched and supercoiled DNA molecule, both at fixed torque and at fixed supercoiling angle, and we compare the theoretical predictions to some pre liminary experimental data.