A simple topological method for describing stereoisomers of DNA catenanes and knots

Although linking number is an effective topological invariant for describing supercoiled DNA, it is inadequate for the additional interwinding in catenated or knotted DNA. We explain how the two-bridge theory of Schubert provides a powerful yet simple method for analyzing these forms by associating them with two integral invariants, alpha and beta, that measure their geometric complexity. These integers can either be determined graphically or computed with the aid of standard tables, and they allow tabulation of all possible stereoisomers of a given knot or catenane . A complete classification can then be made via a simple theorem. Stereoisomers of representative knots and catenanes are tabulated for easy reference. There are four stereoisomers of regularly interlocked catenanes that we designate right-handed parallel, right-handed antiparallel, left-handed parallel, and left-handed antiparallel according to the helical intertwining of the rings. The biological processes that form catenanes --replication, recombination, and topoisomerase action--predict distinctly different isomers.