Description of the topological entanglement of DNA catenanes and knots by a powerful method involving strand passage and recombination.

We utilize a recently discovered, powerful method to classify the topological state of knots and catenanes. In this method, each such form is associated with a unique polynomial. These polynomials allow a rigorous determination of whether knotted or catenated DNA molecules that appear distinct actually are, and indicate the structure of related molecules. A tabulation is given of the polynomials for all possible stereoisomers of many of the knotted and catenated forms that are found in DNA. The polynomials for a substrate DNA molecule and the products obtained from it by either recombination or strand passage by a topoisomerase are related by a simple theorem. This theorem affords natural applications of the polynomial method to these processes. Examples are presented involving site-specific recombination by the transposon Tn3-encoded resolvase and the phage lambda integrase, in which product structure is predicted as a function of crossover mechanism.