Supercoiling transitions of closed DNA

Conformations of closed DNA molecules are considered within a simple elastic model. The interplay of elastic energies of bending and twisting can lead to supercoiling induced by variations of the linking difference. These shape transformations are studied in the limit of a large length to thickness ratio of the molecule. Stationary shapes and energy diagrams are obtained by solving shape equations for closed rings. Four different families of stationary shapes can be distinguished: (i) planar circles, (ii) nonplanar rings, (iii) self-interacting rings, and (iv) interwound configurations. They all occur as shapes of minimal energy in the phase diagram of supercoiling. The transitions between the different regions in this phase diagram can be either continuous or discontinuous. The sequence of shape transitions turns out to be sensitive to the precise values of the elastic parameters. The buckling instability of the circle changes within the range of physically accessible values from continuous to discontinuous behavior.