Onset of writhing in circular elastic polymers

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Abstract
A perturbation analysis is presented of an exchange of stability between two mechanical equilibrium conformations of a torsionally deformed, linearly elastic, cylindrical polymer whose ends are held together to form a topologically circular structure. The imposed constraint is the twist angle ζ through which one end of the polymer is rotated relative to the other. The value ζ=0 corresponds to the relaxed state, in which no torsional deformation is imposed. Close to the relaxed state, the stable equilibrium conformation is a torsionally deformed circle in which the imposed constraint is expressed entirely as twist. A threshold value of ζ is shown to exist, whose location depends on the ratio of the polymer bending stiffness A to its torsional stiffness C. Beyond this threshold the stable equilibrium conformation involves nonplanar bending, also called writhing. As this threshold is surpassed, a smooth onset of writhing occurs as a form of torsional buckling. These results are applied to the analysis of torsional fluctuations in nicked circular DNA. A critical molecular length is found, at which the root-mean-square twist induced by thermal fluctuations just suffices to initiate writhing. Comparison with the analogous critical length for linear polymers shows that the constraint imposed by circularity greatly increases the stability of the unwrithed conformation. Although twisting and bending are not coupled either energetically or topologically in nicked circular molecules, the exchange of stability between unwrithed and writhed conformations provides a mechanical coupling between them. Because sufficiently large torsional deformations drive writhing, fluctuations in the parameters scrT and scrW are not statistically independent in nicked molecules, but instead are positively correlated.