Topological interactions in DNA catenanes

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 ≫ RG (RG being the radius of gyration of a random walk ring) is found, while for R ≪ RG the interaction is strongly repulsive. Treating the attractive interaction as the dominant one, distribution functions for the distance between segments located on different rings for several linking numbers are derived which are in qualitative agreement with distributions functions obtained experimentally from electron micrographs of DNA catenanes (S. D. Levene et al., Biophys. Knots and links made from DNA beautifully elucidate the general role of topology in nature. More specifically, DNA rings from bacteria such as Escherichia coli form so-called catenanes (the chemical term for entangled rings called links by knot theorists) as intermediate products of DNA replication and recombination [1]. Together with single knotted DNA rings they form the class of so-called topoisomers, i.e. topologically distinct isomers. They can be isolated experimentally by manipulating certain enzymes (topoisomerases) switching back between various topologies [1,2]. Several catenanes and knots have been identified e.g. by electron microscopy and electrophoresis [1,3]. Most recently, the conformation of open-circular, multiply linked dimeric DNA catenanes has been studied by electron microscopy [4]. From the tracing of the molecular contours on the micrographs, in particular, distribution functions for the distance (in the projection plane) between segments on separate rings have been established. As the linking number-measuring the degree of concatenation between pairs of rings-is increased, the catenane conformation becomes more and more compact due to an effective attractive interaction-of topological origin-between segments. In this letter, we present a theory to explain the experimental results of ref. [4] mentioned above. From a theoretical point of view, the statistical mechanics of entangled rings is a generally unsolved problem [5,6]. The essential difficulty is how to specify the topological state of the system, which is assumed to remain unchanged with respect to conformational fluctuations. The mathematical answer to this question is a so-called link invariant, various types of which are discussed in knot theory while none of them is one-to-one [7,8]. The most simple, yet most crude invariant is the Gauss integral for two given …