The writhe of a self-avoiding walk

Orlandini, Enzo; Tesi, Maria Carla; Whittington, S G; Sumners, D. W.; Rensburg, E J Janse van

doi:10.1088/0305-4470/27/10/006

Cited by 50 publications

(56 citation statements)

References 19 publications

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“…Numerical simulations gave α = 1.122 ± 0.005, but it was argued that this might actually be a lower estimate, the true value being higher [1]. In later simulations, Arteca [6] found a value 1.20 ± 0.04 for SAWs and 1.34 to 1.4 for protein backbones [7].…”

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confidence: 88%

“…Indeed an increase of the value of C had also been seen in [1] for SAWs with self-attraction, and it was conjectured in [7,8] that C is a useful observable for detecting the coil-globule transition. Due to its supposed importance, C was called the "entanglement complexity" in [9], and was shown there (by non-rigorous arguments) to be < 1.4 for random configurations.…”

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confidence: 91%

“…Topological properties of linear polymer chains are, strictly spoken, only defined for closed rings where the theory of knots can be applied with several non-trivial results [1][2][3]. But even for open chains quantities like linking, twist and writhe can be applied.…”

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confidence: 99%

“…The number of crossings C was introduced in [5,1] as a simplified version of the writhe. In it, all crossings contribute with the same sign.…”

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Opacity and entanglement of polymer chains

Grassberger

2001

J. Phys. A: Math. Gen.

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We argue that the mean crossing number of a random polymer configuration is simply a measure of opacity, without being closely related to entanglement as claimed by several authors. We present an easy way of estimating its asymptotic behaviour numerically. These estimates agree for random walks (theta polymers), self-avoiding walks, and for compact globules with analytic estimates giving log N, a − b/N 2ν−1 , and N 1/3 , respectively, for the average number of crossings per monomer in the limit N → ∞. While the result for compact globules agrees with a rigorous previous estimate, the result for SAWs disagrees with previous numerical estimates. One important quantity for studying entanglement is the writhe which is defined as the number if signed crossings in a projection of a 3-d non-selfintersecting oriented curve. If two parts of the curve seem to cross when seen from a particular angle, this crossing contributes +1 or −1 to the writhe, depending on whether the direction of the front part is obtained by a right or left turn from the direction of the part behind it. Its interest stems from the fact that for closed loops it is related to linking which is a topological invariant [3].The number of crossings C was introduced in [5,1] as a simplified version of the writhe. In it, all crossings contribute with the same sign. Of particular interest is its average value C , averaged over all angles of projection, called the mean crossing number. In these papers it was shown that for a self avoiding random walk (SAW) of N straight bondswith 1 ≤ α ≤ 2. Numerical simulations gave α = 1.122 ± 0.005, but it was argued that this might actually be a lower estimate, the true value being higher [1].In later simulations, Arteca [6] found a value 1.20 ± 0.04 for SAWs and 1.34 to 1.4 for protein backbones [7]. Indeed an increase of the value of C had also been seen in [1] for SAWs with self-attraction, and it was conjectured in [7,8] that C is a useful observable for detecting the coil-globule transition. Due to its supposed importance, C was called the "entanglement complexity" in [9], and was shown there (by non-rigorous arguments) to be < 1.4 for random configurations.It is the purpose of this note to show that C can be easily estimated by using well known formulae for generic intersections of random fractals [10]. Take If this is positive, the average number of intersections between the line of view and X increases as m ∼ N DX∩Y /DX for N → ∞. Actually, for this to be true we either have to assume that X is a true fractal without lower length cutoff (which is not true for random walks with finite step size a), or we have to fatten Y . Thus we consider instead of a single line of view a cylinder whose thickness is of the order of the step size a (the precise value is irrelevant), and the above number of intersections has to be interpreted as the number of crossings between projected bonds within a distance O(a). This scales in the same way as the number of crossings per bond. Thus we obtain immediatelyThis is the case for compa...

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confidence: 88%

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confidence: 91%

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confidence: 99%

“…The number of crossings C was introduced in [5,1] as a simplified version of the writhe. In it, all crossings contribute with the same sign.…”

mentioning

confidence: 99%

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Opacity and entanglement of polymer chains

Grassberger

2001

J. Phys. A: Math. Gen.

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“…Indeed, such attempts (explicit and implicit) have been made. 9,12,14,15,16,17,18,19,20,21 It happens that the definitions of the writhe suggested by different authors are not consistent. One of the first works where it was proposed how to compute the writhing number for an open (and non-smooth) curve, is Ref.…”

Section: Introductionmentioning

confidence: 99%

On the Writhing Number of a Non-Closed Curve

Starostin

2005

Series on Knots and Everything

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Shape transitions in polymer mushrooms compressed by a finite-size obstacle

Arteca

1997

Int. J. Quant. Chem.

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ABSTRACT:We study the interrelation between molecular-shape changes and Ž . configurational transitions in isolated grafted chains ''polymer mushrooms'' under geometrical confinement. The confinement effects are due to a finite-size obstacle, representing a simple model for the tip of an atomic-force microscope. Using an off-lattice polymer model, we monitor how the molecular shape of a short chain is affected by the geometry of the tip. We show the occurrence of shape transitions as the chain ''tries to dodge'' the approaching obstacle. These transitions can be correlated with changes in the nature of the dominant chain conformations. Using a descriptor of self-entanglements in the polymer chain, we propose a ''phase diagram'' indicating the regions characterized by free chains, confined chains, chains that can ''escape'' the tip, and chains whose escape is ''frustrated.'' The results help in better understanding polymer behavior under compression, and they can be useful for designing interphases with targeted properties.

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The writhe of a self-avoiding walk

Cited by 50 publications

References 19 publications

Opacity and entanglement of polymer chains

Opacity and entanglement of polymer chains

On the Writhing Number of a Non-Closed Curve

Shape transitions in polymer mushrooms compressed by a finite-size obstacle

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