A calculus for rational tangles: applications to DNA recombination

Abstract

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.

Tangle analysis of Gin site-specific recombination

“…[6]). As in [10], we are able to prove rationality at the key points, by using double branched cyclic covering spaces of tangles and by using known results [15] on Dehn surgeries on knots. We find that solving tangle equations based on reasonable mathematical and biological assumptions systematically gives the same Gin recombinational mechanism as that suggested by direct biological reasoning, and, in addition, shows the uniqueness of the mechanism.…”

“…Determining when the intervening tangles are rational, or sums of rational tangles is not easy. Ernst and Sumners have extensively studied this problem [10][11][12][13]. The next few concepts and results will be used in Section 4 to prove that the tangles O and R arising in the system corresponding to Gin recombination are rational.…”

“…In Ernst and Sumners [10] the authors successfully introduced and used tangle analysis (reviewed in [8,30]) to elucidate the topological mechanism of the Tn3 resolvase.…”

“…The work of Ernst and Sumners allows to find solutions for O and R that are rational or sums of rational tangles, when substrate and products of recombination are known 4-plats [10][11][12][13]. If only two equations are provided, it is sometimes impossible to find a unique solution pair (O, R) for the equations.…”

“…The tangle model provides mathematical tools to analyze such changes [26,27,28]. The detailed tangle formalism and an application of the model to the Tn3 resolvase system were first presented in [10].Here we analyze the enzymatic action of Gin, a site-specific recombinase encoded by bacteriophage Mu (a virus that infects bacteria). Gin inverts a DNA segment within the phage genome to extend the phage's range of infection.…”