0 Thermal Fluctuations of Elastic Ring

We study the effects of thermal fluctuations on a small elastic ring. We derive analytical expressions for the correlation functions of the Euler angles, for the real space two–point correlation functions and for the probability distribution of writhe, as a function of the persistence lengths that determine the rigidity with respect to bending and twist. Fluctuation amplitudes diverge in the limit of vanishing twist rigidity. There is crossover from a small scale regime in which twist and writhe modes are uncoupled, to a large scale regime in which twist affects the spatial configurations of the ring. Small circular loops of extrachromosomal DNA (plasmids) play an important role in biological processes such as gene transfer between bacteria and in biothechnological applications where they are used as vectors for DNA cloning 3. The simplest model that captures both the topology and the physical properties of such an object is that of an elastic ring, and several studies of the writhing instability of twisted rings based on this model, were reported in recent years 4,5. However, these studies focused on the elastic response of the ring to mechanical forces and did not consider the effects of thermal fluctuations. These fluctuations dominate the physics of macromolecules and determine all their statistical properties, such as radii of gyration, dynamics in solution 6 , kinetics of loop formation, and dissociation of short DNA segments 7 and molecular beacons 8. Recently, we developed a theory of fluctuating elastic filaments, with arbitrary spontaneous curvature, torsion and twist in their 1