Abstract
The topological state of covalently closed, double-stranded DNA is defined by the knot type, K, and the linking number difference from relaxed DNA, ΔLk. DNA topoisomerases are essential enzymes that regulate topological states of DNA in vivo: type-I topoisomerases (topo-Is) change ΔLk, thereby regulating the torsional tension, whereas type-II topoisomerases (topo-IIs) change both (ΔLk, K) by passing one DNA helix through another. A critical biological function of type-II enzymes is the elimination of knots in DNA because their presence impedes transcription and replication. It has been a long-standing puzzle how small type-II enzymes select passages that unknot large DNA molecules, since topology is a global property which cannot be determined by local DNA-enzyme interactions. Previous studies addressing this question have focused on the equilibrium distribution P(ΔLk, K). Motivated by the fact that topo-IIs reduce the knotting level below equilibrium at the expense of ATP hydrolysis, we set out to study topoisomerase activity in the framework of non-equilibrium thermodynamics. We consider the dynamics of transitions in a network of topological states (ΔLk, K) induced by type-II and type-I action by solving the master equation for the time-dependent probability distribution P(ΔLk, K; t). We fully characterize non-equilibrium steady states generated by injecting DNA molecules in a given topological state in terms of stationary probability distributions and currents in the network. This allows us, for the first time, to predict detailed kinetic pathways of topoisomerase action as a function of geometry of the enzyme. In particular, we find that unknotting activity of topo-II is significantly enhanced in DNA molecules which maintain a supercoiled state with constant torsional tension; this is relevant for bacterial cells in which the torsional tension is maintained by a homeostatic mechanism using topo I and DNA gyrase.
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