Abstract
Topological entanglements are abundant, and often detrimental, in polymeric systems in biology and materials science. Here we theoretically investigate the topological simplification of knots by diffusing slip-links (SLs), which may represent biological or synthetic molecules, such as structural maintenance of chromosome proteins or cyclodextrins in slide-ring gels. We find that SLs entropically compete with knots and can localize them, greatly facilitating their downstream simplification by transient strand-crossing. We further show that the efficiency of knot localization depends strongly on the topology of the SL network, and, informed by our findings, we discuss potential strategies to control the topology of biological and synthetic materials.
Highlights
Knots and topological entanglements are often found in physical and biological systems [1,2,3,4]
And in marked contrast with previous works on entropic competition, we show that the efficiency of entanglement localization depends on the particular topology of the SL network, and that including the action of topoisomerases leads to extremely fast and efficient simplification of complex knots
Our semianalytical results are in line with previous works, which showed that nested SLs (NC topology) promote growth, or extrusion, of the outer loop via a ratcheting process in which the diffusive dynamics of the outer loop is rectified by the entropic pressure of the inner ones [23]
Summary
Knots and topological entanglements are often found in physical and biological systems [1,2,3,4]. While it is typical to treat abundant topological constraints at the mean-field level—for instance in the tube model of polymer melts [10]—exact and scaling results can be obtained via theories that replace entanglements with slip-links (SLs), which enforce contacts between polymer segments while allowing them to slide past each other [11,12,13,14,15] Because of this physically appealing analogy, systems of polymers with slip-links have been theoretically and numerically explored in the field of statistical and polymer physics, for instance to estimate the size of knots [16,17] and the effective tube size in polymer melts [18]. SLs can be realized in synthetic and supramolecular chemistry using cyclodextrins [33] and are employed for instance to make molecular machines [34] and slide-ring gels [35] In these cases, the benefit of using SLs is that they effectively act as mobile cross-links, imposing strong, yet plastic, topological constraints on the polymeric substrates. Our results suggest an entropy-driven mechanism through which generic SL-like molecules can regulate the topology of DNA or synthetic polymers
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