Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids.

There are circumstances under which it is useful to model a molecule of duplex DNA as a homogeneous, inextensible, intrinsically straight, impenetrable elastic rod of circular cross-section obeying the theory of Kirchhoff. For such rods recent research has yielded exact analytical solutions of Kirchhoff's equations of mechanical equilibrium with the effects of impenetrability taken into account, and criteria have been derived for determining whether an equilibrium configuration is stable in the sense that it gives a strict local minimum to the elastic energy. This paper contains a summary of published results on equilibrium configurations for the case in which a rod has been pre-twisted and closed to form a knot-free ring. Emphasis is placed on the way the writhe Wr of the ring, the number of its discrete points of self-contact, and the presence or absence of lines of contact, depend on the excess link, Delta Lk, which is a measure of the amount the rod was twisted before its ends were joined. Bifurcation diagrams are presented and a summary is given of the properties of the primary, secondary and tertiary branches that arise by successive bifurcations from the 'trivial branch' comprised of configurations for which the axial curve is a circle. New results are presented in the theory of equilibrium configurations of closed rods with the topology of torus knots. It is remarked that examples of equilibrium configurations of closed rods of one knot type can be obtained from examples of other knot types using methods previously employed to calculate isolas of equilibrium configurations of knot-free rings. Bifurcation diagrams are shown for supercoiled (2,3) torus knots (trefoil knots). It is observed that for sufficiently large and sufficiently small Delta Lk the minimum elastic energy configuration of a trefoil knot contains plectonemic loops with straight contact lines, although the configuration that minimizes the elastic energy of a general (2,q) torus knot over the entire range of Delta Lk has self-contact along a closed curve. As the ratio of the diameter of the rod to its length approaches zero, that contact curve becomes a circle, and there is an open interval of values of Delta Lk for which stable equilibrium configurations with such circular contact curves exist. Examples of minimum energy configurations are presented for both torus knots and catenates formed by linking two unknots.