Average size of random polygons with fixed knot topology

Matsuda, Hiroshi; Yao, Akihisa; Tsukahara, Hiroshi; Deguchi, Tetsuo; Furuta, Ko; Inami, Takeo

doi:10.1103/physreve.68.011102

Cited by 42 publications

(75 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…He argued that the radius of gyration would then scale as n with Ϸ 0.588, the value appropriate for self-avoiding polygons or for rings with excluded volume. While this topological expansion might seem surprising at first sight there is growing evidence that the prediction is correct Matsuda et al, 2003;Moore et al, 2004͒. The asymptotic behavior of the average size of ring polymers with fixed topology has also been studied for an off-lattice model in which self-avoiding polygons consist of N cylinders with radius r. By varying r Deguchi ͑2001, 2002͒ showed that the topological expansion strongly depends on the radius r. For small r ͑where the model is closer to the Gaussian random poly-gon͒ the effect is indeed significant, but it becomes less important as the radius r increases, i.e., as the selfavoidance becomes more and more relevant.…”

Section: B Metric Propertiesmentioning

confidence: 96%

“…This implies that ͑͒ = ͑͒ and A ͑͒ = A ͑͒ for all . There is general agreement that ͑͒ is independent of ͑Janse van Rensburg and Whittington, 1991a;Quake, 1994Quake, , 1995Matsuda et al, 2003͒. Although there has been some disagreement about the constancy of the amplitude, there is a growing amount of numerical evidence that it is constant.…”

Section: B Metric Propertiesmentioning

confidence: 99%

See 1 more Smart Citation

Statistical topology of closed curves: Some applications in polymer physics

2007

View full text Add to dashboard Cite

Topological entanglement in polymers and biopolymers is a topic that involves different fields of science such as chemistry, biology, physics, and mathematics. One of the main issues in this topic is to understand how the entanglement complexity can depend on factors such as the degree of polymerization, the quality of the solvent, and the temperature or the degree of confinement of the macromolecule. In this respect a statistical approach to the problem is natural and in the last few years there has been a lot of work on the study of the entanglement complexity of polymers within the statistical mechanics framework. A review on this topic is given here stressing the main results obtained and describing the tools most used with this approach

show abstract

Section: B Metric Propertiesmentioning

confidence: 96%

Section: B Metric Propertiesmentioning

confidence: 99%

Statistical topology of closed curves: Some applications in polymer physics

2007

View full text Add to dashboard Cite

show abstract

“…In all the works (7)(8)(9), to extract the value of scaling exponent from the data, which (particularly in ref. 7) is almost entirely restricted to the crossover range, authors fitted the data using the formula:…”

Section: [2]mentioning

confidence: 99%

Topologically driven swelling of a polymer loop

Moore

Lua

Grosberg

2004

Proc. Natl. Acad. Sci. U.S.A.

104

117

View full text Add to dashboard Cite

Numerical studies of the average size of trivially knotted polymer loops with no excluded volume were undertaken. Topology was identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius were generated for loops of up to N ‫؍‬ 3,000 segments. Gyration radii of trivially knotted loops were found to follow a power law similar to that of self-avoiding walks consistent with earlier theoretical predictions.A lthough knots in polymers have been studied for several decades, they remain perhaps the least understood subject in polymer physics. Most of the work in this area has been directed at classification of knots, finding efficient topological invariants, and the probabilistic questions, e.g., ''What is the probability to obtain a certain knot type under given conditions (e.g., on loop closure)?'' Much less is known about the more physical aspects, which are how knots influence the properties of polymers. The simplest question to ask about physical properties is what the average spatial size is of a polymer loop whose knot type is quenched. To this end, des Cloizeaux (1) conjectured as early as 1981 that the size of a trivially knotted loop (i.e., an unknot) scales with the number of segments, N, in the same way as in the case of a self-avoiding walk, which is N , where ϭ SAW Ϸ 0.589 Ϸ 3͞5. We should emphasize that the polymer in question is not phantom in the sense that segments cannot cross each other, but it is assumed to have a negligible excluded volume (or thickness). Thus, according to des Cloizeaux's conjecture, exclusion of all knots acts effectively as volume exclusion. More systematic arguments, albeit still only at a scaling level, to support this conjecture were presented more recently (2), yielding the following prediction for the (mean square) average gyration radius of a trivially knotted loop.Here, ᐉ is the segment length, and the parameter N 0 is sometimes called the characteristic length of random knotting; it appears in the probability of observing a trivially knotted conformation (an unknot) in a fluctuating phantom (i.e., freely crossing itself) loop:When N is smaller than N 0 , a phantom loop has few conformations with nontrivial knots. Therefore, the set of allowed conformations for an unknotted nonphantom loop is not significantly different from that of a phantom loop. This condition is why at N Ͻ N 0 the Gaussian scaling of gyration radius is expected. For this case, the ᐉ 2 ͞12 prefactor results from the facts that (i) the mean square gyration radius for the linear chain is 1͞6 of its mean square end-to-end distance, ᐉ 2 N, and (ii) ͗R g 2 ͘ for the loop is half that for the linear chain (3). Prefactor A for the N Ͼ N 0 regime in Eq. 1 must provide for smooth crossover between regimes at N ϳ N 0 , which meansGiven the fundamental character of the problem, and given that theoretical arguments remain far short of mathematically rigorous, it is vital to look at the simul...

show abstract

“…These works belong to the direction [26,27,28,29,30] addressing the spatial statistics of polymer loops restricted to remain in a certain topological knot state. It turns out that even for loops with no excluded volume and thus are not self-avoiding, N 0 marks the crossover scale between mostly Gaussian (N < N 0 ) and significantly non-Gaussian (N > N 0 ) statistics.…”

Section: Introduction: Formulation Of the Problemmentioning

confidence: 99%

The abundance of unknots in various models of polymer loops

Moore

Grosberg

2006

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract. A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of N segments follows a decaying exponential form, ∼ exp (−N/N 0 ), where N 0 marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of N 0 for a variety of polymer models. Among models examined, N 0 is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.

show abstract

Average size of random polygons with fixed knot topology

Cited by 42 publications

References 25 publications

Statistical topology of closed curves: Some applications in polymer physics

Statistical topology of closed curves: Some applications in polymer physics

Topologically driven swelling of a polymer loop

The abundance of unknots in various models of polymer loops

Contact Info

Product

Resources

About