Generating random walks and polygons with stiffness in confinement

Diao, Yuanan; Ernst, Claus; Saarinen, Sam; Ziegler, Uta

doi:10.1088/1751-8113/48/9/095202

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…An example of the connection between ascents, descents, crossing signs, and linking number is shown in Figure 4 and Table 1. Observe in Table 1 that σ(5) < σ (6). Thus j = 5 would be an ascent.…”

Section: Linking Numbers Of Disjoint Monotonic Cyclesmentioning

confidence: 96%

“…By comparing the topological invariants of DNA before and after enzymes act on it, we can learn more about mechanisms of these enzymes and their effects on the structure of DNA [15]. Because many polymers are too small to image in detail, several authors have used mathematical models to study configurations of long polymer chains by introducing versions of uniform random distributions of polygonal chains in a cube [1,2,6,7,18,20,22]. Even-Zohar, et al introduced a random model based on petal diagrams of knots and links where the distribution of links can be studied in terms of random permutations, achieving an explicit description of the asymptotic distribution for the linking number [11].…”

Section: Introductionmentioning

confidence: 99%

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Linking number of monotonic cycles in random book embeddings of complete graphs

Yasmin¹,

Burkholder²,

Cheng³

et al. 2023

Preprint

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A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is i 6 , where i is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of n-cycles in K 2n grows linearly in n.

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Section: Linking Numbers Of Disjoint Monotonic Cyclesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Linking number of monotonic cycles in random book embeddings of complete graphs

Yasmin¹,

Burkholder²,

Cheng³

et al. 2023

Preprint

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“…Below we will define two variants of our simulation protocol. In contrast to more sophisticated methods [31] which aim at generating random walks in a confined volume, in protocol 1 we simply place 25, 719 particles in a unit cube using a random, uniform distribution. To ensure maximum randomness particles are connected randomly to form 20 linear chains corresponding to 20 chromosomes with lengths ranging from 582 and 1925 particles.…”

Section: Model and Simulation Protocolsmentioning

confidence: 99%

A minimal Gō-model for rebuilding whole genome structures from haploid single-cell Hi-C data

Wettermann

Brems

Siebert

et al. 2020

Computational Materials Science

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We present a minimal computational model, which allows very fast, onthe-fly construction of three dimensional haploid interphase genomes from single-cell Hi-C contact maps using the HOOMD-blue molecular dynamics package on graphics processing units. Chromosomes are represented by a string of connected beads, each of which corresponds to 100,000 base pairs, and contacts are mediated via a structure-based harmonic potential. We suggest and test two minimization protocols which consistently fold into conformationally similar low energy states. The latter are similar to previously published structures but are calculated in a fraction of the time. We find evidence that mere fulfillment of contact maps is insufficient to create experimentally relevant structures. Particularly, an excluded volume term is required in our model to induce the formation of chromosome territories. We also observe empirically that contact maps do not capture the chirality of the underlying structures. Depending on starting configurations and protocol details, one of two mirror images emerges. Finally, we analyze the occurrence of knots in a particular chromosome. The same knot appears in (almost) all structures irrespective of minimization protocols or even details of underlying potentials providing further evidence for the existence of knots in interphase chromatin. 1 S.W. and M.B. have contributed equally arXiv:1904.04209v2 [cond-mat.soft]

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“…Many authors have considered uniform random distributions of open and closed polygonal chains in a cube as a model for long molecular chains in a confined region (see for example [1], [2], [5], [4], [13], [14], [15]). Of particular note, Arsuaga et al [1] obtained a formula for the mean squared linking number of two uniform random n-gons in a cube, and showed that the probability of linking between a given simple closed curve in the cube and a uniform random n-gon grows at a rate of at least 1−O 1 √ n .…”

Section: Introductionmentioning

confidence: 99%

Linking number and writhe in random linear embeddings of graphs

Flapan

Kozai

2016

J Math Chem

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Abstract. In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of Kn in a cube, the mean sum of squared linking numbers and the mean sum of squared writhes are of the order of θ(n(n!)). We obtain a similar result for the mean sum of squared linking numbers in linear embeddings of graphs on n vertices, such that for any pair of vertices, the probability that they are connected by an edge is p. We also obtain experimental results about the distribution of linking numbers for random linear embeddings of these graphs. Finally, we estimate the probability of specific linking configurations occurring in random linear embeddings of the graphs K6 and K3,3,1.

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Generating random walks and polygons with stiffness in confinement

Cited by 6 publications

References 15 publications

Linking number of monotonic cycles in random book embeddings of complete graphs

Linking number of monotonic cycles in random book embeddings of complete graphs

A minimal Gō-model for rebuilding whole genome structures from haploid single-cell Hi-C data

Linking number and writhe in random linear embeddings of graphs

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