### Abstract

The elastic properties of a molecule of duplex DNA are strongly dependent on nucleotide sequence. In the theory developed here the contribution of the *n*th base-pair step to the elastic energy is assumed to be given by a function of six kinematical variables, called tilt, roll, twist, shift, slide, and rise, that describe the relative orientation and displacement of the *n * and base pairs. The sequence dependence of elastic properties is determined when one specifies the way depends on the nucleotides of the two base pairs of the *n*th step. Among the items discussed are the symmetry relations imposed on by the complementarity of bases, i.e., of A to T and C to G, the antiparallel nature of the DNA sugar–phosphate chains, and the requirement that be independent of the choice of the direction of increasing *n*. Variational equations of mechanical equilibrium are here derived without special assumptions about the form of the functions and numerical solutions of those equations are shown for illustrative cases in which is, for each *n*, a quadratic form and the DNA forms a closed, 150 base-pair, minicircle that can be called a DNA o-ring because it has a nearly circular stress-free configuration. Examples are given of noncircular equilibrium configurations of naked DNA o-rings and of cases in which the interaction with ligands induces changes in configuration that are markedly different from those undergone by a minicircle of intrinsically straight DNA. When a minicircle of intrinsically straight DNA interacts with an intercalating agent that upon binding to DNA causes a local reduction of intrinsic twist, the configuration that minimizes elastic energy depends on the number of intercalated molecules, but is independent of the spatial distribution of those molecules along the minicircle. In contrast, it is shown here that the configuration and elastic energy of a DNA o-ring can depend strongly on the spatial distribution of the intercalated molecules. As others have observed in calculations for Kirchhoff rods with intrinsic curvature, an o-ring that has its intrinsic twist reduced at a single base-pair step can undergo large deformations with localized untwisting and bending at remote steps, even when the amount α of twist reduction is less than the amount required to induce supercoiling in rings of intrinsically straight DNA. We here find that the presence in the functions of cross-terms coupling twist to roll can amplify the configurational changes induced by local untwisting to the point where there can be a value of α at which a first-order transition occurs between two distinct stable noncircular configurations with equal elastic energy.

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