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ABSTRACT
The configuration space available to randomly cyclized polymers is divided into subspaces accessible to individual knot types. A phantom chain utilized in numerical simulations of polymers can explore all subspaces, whereas a real closed chain forming a figure-of-eight knot, for example, is confined to a subspace corresponding to this knot type only. One can conceptually compare the assembly of configuration spaces of various knot types to a complex foam where individual cells delimit the configuration space available to a given knot type. Neighboring cells in the foam harbor knots that can be converted into each other by just one intersegmental passage. Such a segment-segment passage occurring at the level of knotted configurations corresponds to a passage through the interface between neighboring cells in the foamy knot space. Using a DNA topoisomerase-inspired simulation approach we characterize here the effective interface area between neighboring knot spaces as well as the surface-to-volume ratio of individual knot spaces. These results provide a reference system required for better understanding mechanisms of action of various DNA topoisomerases.
INTRODUCTION
A statistical ensemble of long circular polymers in solution like DNA, for example, can reach its highest entropy state only when the circular polymers are permitted to attain the topological equilibrium resulting in the production of a characteristic spectrum of various knots (Dean et al., 1985; Rybenkov et al., 1993; Shaw and Wang, 1993; Deguchi and Tsurusaki, 1994; Katritch et al.. 2000). In the case of circular DNA molecules specific enzymes called topoisomerases allow intersegmental passages and therefore may let the molecules reach the topological equilibrium, even if all molecules had the same topology before addition of the enzyme (Krasnow et al., 1983; Dean et al., 1985). Actually, only a subclass of topoisomerases, belonging to the class IA, can bring the system to the topological equilibrium since intersegmental passages mediated by these enzymes are driven by the free energy gradient (Stasiak, 2003), and topoisomerases of type IA can perform knotting and unknotting of single-stranded DNA as well as of double-stranded DNA if the latter contains at least one short interruption in one of the strands (Dean el al., 1985; Champoux. 2001).
In recent years several independent studies investigated how ATP-hydrolysis-driven type II topoisomerases can selectively lower the frequency of DNA knotting...