ABSTRACT:We examine the motion of a knot along a tensioned chain whose backbone is corrugated due to excluded volume effects. At low applied tensions, the knot traverses the chain diffusively, while at higher tensions the knot makes slow, discrete hops that can be described as a Poisson process. In this "jammed" regime, the knot's long-time diffusivity decreases exponentially with increasing tension. We quantify how these measurements are altered by chain rigidity and the corrugation of the polymer backbone. We also characterize the energy barrier of the reptation moves that gives rise to the knot's motion. For the simple knot types examined thus far (3 1 , 4 1 , 5 1 , 5 2 ), the dominant contribution to the energy landscape appears in the first step of reptationi.e., polymer entering the knotted core. We hope this study gives insight into what physics contributes to the internal friction of highly jammed knots. E ver since the discovery of knots in DNA, 1−3 there has been immense interest in understanding how these structures affect the mechanical and dynamical properties of these molecules. 4 A knot is defined as a self-entanglement that cannot be undone when the polymer chain is closed. 5 The topology of a knot is well-defined as long as it is far from the ends of a polymer chain, and the topology can be determined by closing the chain and then computing Alexander polynomials.6 Experimentally, knots have been tied onto DNA 7 and actin 8 filaments via optical traps, and chemical synthesis techniques have been developed to create knotted loops up to five crossings. 9 Recently, researchers have also determined facile methods to create knotted DNA via an electric field, 10,11 and this work has led to microfluidic experiments examining how the coil−stretch transition is affected by the presence of knots.
12In some sense, knots are unavoidable for very long polymeric chains, as it has been proven that the knotting probability approaches unity as the chain size gets very large.13,14 Indeed, knots have been found in capsid DNA, 1 proteins, 15−17 and other biological systems.4 Bao et al. discovered that knots can self-reptate along a polymer contour due to thermal fluctuations.7 This discovery has led several computational studies to quantify the knot's motion along a chain, whether it be convection due to a directed force 18−20 or diffusion under uniform tension. 21−23 In most of these studies, the tension on the polymer is comparable to or smaller than the Brownian forcesi.e., f ∼ O(kT/l p ) or smaller, where kT is the thermal energy and l p is the persistence length of the chain. However, it is known that at large tensions knots can dynamically arrest this has been observed for knots jamming during translocation through a nanopore. 24−27 Huang and Makarov briefly study jamming in their simulations, 23 positing that the knot's arrest is caused by the "bumpiness of the energy landscape of the knot created by intrachain interactions". We explore this topic in this Letter. We note that this area of research is of ...