Knots and Random Walks in Vibrated Granular Chains

Ben‐Naim, E.; Daya, Zahir A.; Vorobieff, Peter; Ecke, Robert E.

doi:10.1103/physrevlett.86.1414

Cited by 92 publications

(92 citation statements)

References 12 publications

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“…Such projections are well known from knot theory where they are at the basis of the determination of topological invariants [11]. Experimentally flat knots can be realized by adsorbing polymers like DNA [12], or even macroscopic chains [13] on a plane. In the former case thermal fluctuations can allow the system to reach equilibrium in two dimensions.…”

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confidence: 99%

Polymer θ-point as a knot delocalization transition

2003

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We study numerically the tightness of prime flat knots in a model of self-attracting polymers with excluded volume. We find that these knots are localised in the high temperature swollen regime, but become delocalised in the low temperature globular phase. Precisely at the collapse transition, the knots are weakly localised. Some of our results can be interpreted in terms of the theory of polymer networks, which allows to conjecture exact exponents for the knot length probability distributions.PACS numbers:36.20. Ey, 64.60.Ak, 87.15.Aa, 02.10.Kn The presence of knots in single ring macromolecules, or of chain entanglement in polymer melts, has fundamental consequences at the physical, chemical and biological level. For example, the dynamics of a ring polymer in a gel depends crucially on its topology [1]. The replication and the transcription of circular DNA are controlled by enzymes affecting the topology, the topoisomerases [2]. Knots have even been identified in some proteins in their native state [3]. It is easy to imagine the importance that the degree of localisation of knots can have. In the folding of a protein it must make quite a difference whether the knot is tight and localised within a restricted region of the backbone, or not. A loose knot poses less restrictions to the exploration of configuration space in the search for the native state. Similarly, one could expect that the degree of knot localisation can strongly influence the function of topoisomerases.These examples all concern heteropolymers in nonequilibrium situations. Here we investigate the interplay between topology and temperature in a simpler context, by studying the equilibrium properties of (prime) knots in homopolymers when these are cooled below their theta temperature. Under such conditions we find that knots definitely loose their usual property of being localized within restricted portions of the chains.It is very difficult to include topological constraints within a statistical mechanical description of a polymer, since they imply a global control of its conformations [4]. So far, most work has concentrated on the probability of occurence of knots [5] and much less has been done on the physically relevant problem of precisely quantifying the knot size. Besides attempts to measure knot size directly [6], most studies have given only indirect, and often incomplete information on knot localisation. For a ring polymer in the infinite temperature, good solvent regime, it was found numerically that the presence of a prime knot leads to a simple multiplication of the partition sum with a factor proportional to the length L of the macromolecule [7]. Moreover, for the relation between the radius of gyration and L (Eq.(1)), evidence was given that neither the amplitude A [8] nor the critical exponent ν depends on the topology [8,9]. These results suggest that knots are somehow localised within the chain. On the other hand, in a recent study of the force-extension relation for knotted polymers in good solvent, corrections to scaling ar...

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confidence: 99%

Polymer θ-point as a knot delocalization transition

2003

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“…This polymer system is well suited for studying topological constraints as it allows control of the chain size and the constraint type, as well as direct observations of the chain conformation in contrast with traditional polymer systems. Recent studies have successfully used vibrated chains to study diffusive relaxation [17] and spontaneous formation [18] of knots.We considered the simplest possible topology, a "figure-8": a once-twisted ring consisting of two loops, separated by a single crossing point, which functions as the topological constraint (see Fig. 1).…”

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confidence: 99%

Entropic tightening of vibrated chains

et al. 2002

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We investigate experimentally the distribution of configurations of a ring with an elementary topological constraint, a "figure-8" twist. Using vibrated granular chains, which permit controlled preparation and direct observation of such a constraint, we show that configurations where one of the loops is tight and the second is large are strongly preferred. This agrees with recent predictions for equilibrium properties of topologically-constrained polymers. However, the dynamics of the tightening process weakly violate detailed balance, a signature of the nonequilibrium nature of this system. 81.05.Rm, 83.10Nn Topological constraints such as knots and entanglements constantly form and relax in polymer systems and in biomolecules such as DNA [1][2][3][4][5][6]. These constraints increase relaxation time scales and additionally restrict the phase space accessible by the macromolecule. Whereas the role entanglements play in chain dynamics is well appreciated [7,8], even more basic questions concerning effects of topological constraints on static properties such as the chain structure remain largely unanswered. Recent theoretical studies predict that in equilibrium, a knotted polymer will generally favor configurations where the knot is "tight", i.e., localized to a small region of the chain [9,10]. Numerical simulations and scaling analysis support this prediction [11], but direct experimental tests are difficult [12]. In this study, we examine this interesting prediction experimentally using vertically vibrated granular chains.Granular chains consist of spherical beads connected by rods, and their backbone enforces the same geometrical constraints as in a polymer system. A vibrating plate supplies the system with energy, balancing the energy dissipation due to inelastic collisions experienced by beads [13][14][15][16]. This polymer system is well suited for studying topological constraints as it allows control of the chain size and the constraint type, as well as direct observations of the chain conformation in contrast with traditional polymer systems. Recent studies have successfully used vibrated chains to study diffusive relaxation [17] and spontaneous formation [18] of knots.We considered the simplest possible topology, a "figure-8": a once-twisted ring consisting of two loops, separated by a single crossing point, which functions as the topological constraint (see Fig. 1). Under appropriate vibration amplitude, the system is effectively twodimensional and the crossing point hops along the chain without flipping open the figure-8. Surprisingly, despite the highly nonequilibrium drive applied to the system, we observed strong entropic tightening. The microscopic degrees of freedom, the beads, experience periodic drive and dissipative collisions with the plate, rods, and other beads, as well as frictional forces. Remarkably the macroscopic observable, the loop size, obeys a statistical mechanics. Detailed balance is only weakly violated and the empirical loop-size distribution is close to that conjectured on...

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“…There have been a number of recent studies on the dynamics of spontaneous knotting in the macroscopic world, in either a shaken hanging chain (14) or a chain bouncing on a vibrating plate (18,20). However, the study by Raymer and Smith (8) is remarkable for the sheer scope of its statistics: Ͼ3,000 knots were tied.…”

Section: Spontaneous Knotting-not So Randommentioning

confidence: 99%

“…For instance, there is only one open knot realization of the 6 1 but several for the 6 3 . These open knots are closer to what can tie and untie (in a string or chain) (14,18) and what was studied by Raymer and Smith (8).…”

Section: Knot Theory Vs Knotted Thingsmentioning

confidence: 99%

“…This is treated as an ''equilibrium'' process, in that a rate of untying is also included by the opposite process: unbraiding by the free end. In other physical systems, different untying mechanisms have been observed, such as slipping (14,15) or a diffusion-like process (18). All of these processes should be included in developing a Physical Theory of Knots, meaning essentially the proper mechanics and dynamics of knotted things.…”

Section: Spontaneous Knotting-not So Randommentioning

confidence: 99%

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