Topological Interactions in Multiply Linked DNA Rings

The elasticity of DNA catenanes, i.e., multiply linked DNA rings, is investigated using the Gauss invariant as a minimal model for topology conservation. An effective elastic free energy as a function of the distance $R$ between segments located on different rings is obtained. An anharmonic part at large distances, growing as $R^4$, if $R\gg R_G$ ( $R_G$ being the radius of gyration), is found, for $R\ll R_G$ the interaction is strongly repulsive. Calculated distribution functions for the distance between segments located on different rings for several linking numbers are in qualitative agreement with distribution functions obtained experimentally from electron micrographs of DNA catenanes [S. D. Levene et al., Biophys. J. 69, 277 (1995)].