Twist and writhing in short circular DNAs according to first-order elasticity.

Abstract

AbstractThe distribution of twist and writhing in a closed DNA shorter than its persistence length is examined. In this case, the only energy contribution is elastic. We have in tegrated the equations of elasticity for a homogeneous axially symmetric rod of undeformable infinitely small circular cross section with frictionless reactions, when there is no or only one self‐contact. In the absence of self‐contacts, the central line of the rod is drawn on a toroid. It makes ν turns around the axis of revolution of the toroid and m turns around its core. The integer, ν, is equal to one if the rod is unknotted. We prove that no infinitely thin rod with a positive Poisson ratio is stable in a toroidal conformation if there is no self‐contact. However, m‐leafed roses or rosettes, with but one multiple self‐contact, are shown to actually be stable when their writhing is not too great. When the integer, m, is equal to two, we have figure‐8 conformations. Buckling of the circle into a figure‐8 conformation occurs for the constraint such that the figure‐8 and the circular conformations have the same energy. This constraint is 1.845 turns for a bending‐to‐twisting elastic constants ratio of A/C = 1.5. For the same value of A/C, the figure‐8 conformation is unstable for a constraint greater than 2.4 turns. Corrections caused by a finite value of the radius ratio, a/L, of the cross section to the length of the rod, are estimated. For instance, both the circular conformation and the infinitesimally warped circle are simultaneous solutions for particular values of the β twist. β = A/C (m2 − ν2)½ [1 + (νπa/L)2/2]. The binding of ethidium to DNAs short enough to follow first‐order elasticity has been studied. Buckling occurs at an apparent average constraint of about 0.6. How the DNA molecules are distributed in figure‐8 conformations and circles has been determined as the ethidium concentration is varied.