The growth of the mean average crossing number of equilateral polygons in confinement
The growth of the mean average crossing number of equilateral polygons in confinement
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DOI：
10.1088/1751-8113/42/46/465202
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发表时间：
2009-11
期刊：
影响因子：
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通讯作者：
J. Arsuaga;B. Borgo;Y. Diao;R. Scharein
中科院分区：
文献类型：
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作者：
J. Arsuaga;B. Borgo;Y. Diao;R. Scharein
已知折叠的长聚合物链以及高度浓缩的生物聚合物（例如所有生物体中的 DNA）的物理和生物特性至少部分地由它们的拓扑和几何特性决定。为了表征这种凝聚系统的拓扑性质，通常使用限制在有限体积内的等边随机多边形。然而，已知的分析结果很少。在本文中，我们研究了体积限制对等边随机多边形的平均交叉数（ACN）的影响。限制下的结和链接的平均 ACN 为统计意义上的结和链接的拓扑复杂性提供了一种简单的替代测量。对于无任何体积约束的 n 段等边随机多边形，已知其均值 ACN ⟨ACN⟩ 的量级为 。在这里，我们将限制体积建模为半径为 R 的简单球体。我们提供了一个分析论证，表明 n 段等边随机多边形的 ⟨ACN⟩ 在极端限制下（即 R ≪ n）增长为 O(n2)。我们建议在不太极端的约束条件下将 ⟨ACN⟩ 的增长建模为 a(R)n2 + b(R)nln(n)，其中 a(R) 和 b(R) 是 R 的函数，R 是约束球体的半径。执行的计算机模拟显示使用该模型相当合适。
The physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers (such as DNA in all organisms) are known to be determined, at least in part, by their topological and geometrical properties. With this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used. However, very few analytical results are known. In this paper, we investigate the effect of volume confinement on the mean average crossing number (ACN) of equilateral random polygons. The mean ACN of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense. For an equilateral random polygon of n segments without any volume confinement constrain, it is known that its mean ACN ⟨ACN⟩ is of the order . Here we model the confining volume as a simple sphere of radius R. We provide an analytical argument which shows that ⟨ACN⟩ of an equilateral random polygon of n segments under extreme confinement (meaning R ≪ n) grows as O(n2). We propose to model the growth of ⟨ACN⟩ as a(R)n2 + b(R)nln(n) under a less-extreme confinement condition, where a(R) and b(R) are functions of R with R being the radius of the confining sphere. Computer simulations performed show a fairly good fit using this model.