A calculus for rational tangles: applications to DNA recombination

There exist naturally occurring enzymes (topoisomerases and recombinases), which, in order to mediate the vital life processes of replication, transcription, and recombination, manipulate cellular DNA in topologically interesting and non-trivial ways [24, 30]. These enzyme actions include promoting the coiling up (supercoiling) of DNA molecules, passing one strand of DNA through another via a transient enzyme-bridged break in one of the strands (a move performed by topoisomerase), and breaking a pair of strands and recombining them to different ends (a move performed by recombinase). An interesting development for topology has been the emergence of a new experimental protocol, the topological approach to enzymology [30], which directly exploits knot theory in an effort to understand enzyme action. In this protocol, one reacts artificial circular DNA substrate with purified enzyme in vitro (in the laboratory); the enzyme acts on the circular DNA, causing changes in both the euclidean geometry (supercoiling) of the molecules and in the topology (knotting and linking) of the molecules. These enzyme-caused changes are experimental observables, using gel electrophoresis to fractionate the reaction products, and rec A enhanced electron microscopy [15] to visualize directly and to determine unambiguously the DNA knots and links which result as products of an enzyme reaction. This experimental technique calls for the building of knot-theoretic models for enzyme action, in which one wishes mathematically to extract information about enzyme mechanism from the observed changes in the DNA molecules.