By performing strand-passages on DNA, type II topoisomerases are known to resolve topological constraints that impede normal cellular functions. The full details of this enzyme–DNA interaction mechanism are, however, not completely understood. To better understand this mechanism, researchers have proposed and studied a variety of random polygon models of enzyme-induced strand-passage. In the present article, we review results from one such model having the feature that it is amenable to combinatorial and asymptotic analysis (as polygon length goes to infinity). The polygons studied, called Θ-SAPs, are on the simple-cubic lattice and contain a specific strand-passage structure, called Θ, at a fixed site. Another feature of this model is the availability of Monte Carlo methods that facilitate the estimation of crossing-sign-dependent knot-transition probabilities. From such estimates, it has been possible to investigate how knot-reduction depends on the crossing-sign and the local juxtaposition geometry at the strand-passage site. A strong relationship between knot-reduction and a crossing-sign-dependent crossing-angle has been observed for this model. In the present article, we review these results and present heuristic geometrical arguments to explain this crossing-sign and angle-dependence. Finally, we discuss potential implications for other models of type II topoisomerase action on DNA.

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