Geometry and mechanics of DNA superhelicity

This paper analyzes the elastic equilibrium conformations of duplex DNA constrained by the constancy of its molecular linking number, Lk . The DNA is regarded as having the mechanical properties of a homogeneous, linearly elastic substance with symmetric cross section. Integral representations of the writhing number Wr and of Lk are developed, in terms of which the equilibria are given as solutions to an isoperimetric problem. It is shown that the Euler angles defining equilibrium conformations must obey equations identical to those governing unconstrained equilibria. A scaling law is developed stating that molecules supercoiled the same amount Δ Lk will have geometrically similar elastic equilibria regardless of their length. Thus, comparisons among molecules of properties related to their large‐scale tertiary structure should be referred to differences in Δ Lk rather than to their superhelix densities. Specific conditions on the elastic equilibrium conformations are developed that are necessary for ring closure. The equilibrium superhelical conformations accessible to closed‐ring molecules are shown to approximate toroidal helices. Questions relating to the stability and nonuniqueness of equilibria are treated briefly. A comparison is made between these toroidal conformations and interwound configurations, which are shown to be stable, although they are not equilibria in the present sense. It is suggested that entropic factors are responsible for favouring the toroidal conformation in solution.