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

Abstract

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, DeltaL, 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 D(m) symmetry (m=2,3,...). Among the results obtained are the following. (i) There are configurations with C2 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 alpha, is stable when for it the derivative dDeltaL/dW, computed along that branch, is strictly positive. (iii) For configurations not in alpha, the condition dDeltaL/dW>0 is not sufficient for stability; in fact, each nonplanar contact-free configuration that is in a branch other than alpha is unstable. A rule relating the number of points of self-contact and the occurrence of intervals of such contact to the magnitude of DeltaL, 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.

Helices

“…The modeling of contact problems in rods and helical structures has been considered by many different authors in the mechanics, engineering, and physics literature (32)(33)(34)(35)(36)(37)(38)(39)(40); however, these analyses are restricted to rods in contact with external objects, rods in self-contact at discrete points, or very particular helical solutions. Here, we concentrate on self-interactions of uniform helices.…”

Self-Contact for Rods on Cylinders

“…(4) Given that ω c = ∅, what is the structure of ω c ? Is the contact simply contained in a single interval, or is the structure more intricate, as in the examples of contact-skip-contact at the end of a ply [6] and in a (ropelength minimizing) clasp [3]? (5) What form do the contact forces take?…”

“…The study of self-contact in elastic rods has seen some remarkable progress over the last ten years, with highlights such as the numerical work of Tobias, Coleman, and Swigon [22,6,5], the introduction of global curvature by Gonzalez and co-workers [10], and the derivation of the Euler-Lagrange equations for energy minimization by Schuricht and Von der Mosel [19]. Parallel advances have been made on the highly related ideal knots and Gehring links, where ropelength is minimized instead of elastic energy [4,18,3].…”

Dynamics of Elastic Rods in Perfect Friction Contact

“…Another avenue of studies of stationary states of elastic rods with self-contact [6][7][8][9][10] explicitly computes the contact forces from the existence of constraints. In our case, the rolling contact comes from friction, which does not admit any potential description.…”