Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact

Configurations of protein-free DNA miniplasmids are calculated with the effects of impenetrability and self-contact forces taken into account by using exact solutions of Kirchhoff’s equations of equilibrium for elastic rods of circular cross section. Bifurcation diagrams are presented as graphs of excess link, $\Delta \mathcal{L},$ versus writhe, W, and the stability criteria derived in paper I of this series are employed in a search for regions of such diagrams that correspond to configurations that are stable, in the sense that they give local minima to elastic energy. Primary bifurcation branches that originate at circular configurations are composed of configurations with ${\mathrm{D}}_m$ symmetry $(m=2,3,\dots ).\mathrm{}$ Among the results obtained are the following. (i) There are configurations with ${\mathrm{C}}_2$ symmetry forming secondary bifurcation branches which emerge from the primary branch with $m=3,$ and bifurcation of such secondary branches gives rise to tertiary branches of configurations without symmetry. (ii) Whether or not self-contact occurs, a noncircular configuration in the primary branch with $m=2,$ called branch α, is stable when for it the derivative $d\Delta \mathcal{L}/d\mathcal{W},$ computed along that branch, is strictly positive. (iii) For configurations not in α, the condition $d\Delta \mathcal{L}/d\mathcal{W}>0\mathrm{}$ is not sufficient for stability; in fact, each nonplanar contact-free configuration that is in a branch other than α is unstable. A rule relating the number of points of self-contact and the occurrence of intervals of such contact to the magnitude of $\Delta \mathcal{L},$ which in paper I was found to hold for segments of DNA subject to strong anchoring end conditions, is here observed to hold for computed configurations of protein-free miniplasmids.