Bounds for the minimum step number of knots confined to slabs in the simple cubic lattice
Bounds for the minimum step number of knots confined to slabs in the simple cubic lattice
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DOI：
10.1088/1751-8113/45/6/065003
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发表时间：
2012-02-17
影响因子：
2.1
通讯作者：
Shimokawa, K.
中科院分区：
文献类型：
--
作者：
Ishihara, K.;Scharein, R.;Shimokawa, K.
体积限制是聚合物的拓扑结构和几何形状的关键决定因素。然而，两者之间的直接关系并没有得到充分的理解。例如，最近的实验研究已经构建了P4互补链，即其基因组序列和长度已经被人工工程化的P4噬菌体，并且已经显示在提取时它们的DNA结分布不同于野生型噬菌体P4的DNA结分布。特别地，观察到结的复杂性随着包装的基因组的长度而急剧降低。这一问题正是本文的写作动机。在这里，聚合物被建模为简单立方晶格上的自回避多边形，并且限制条件是使得多边形被限制在两个平行平面之间（即，被限制在板内）。我们估计这样的多边形所需的最小长度，以实现一个结类型。我们的数值模拟表明，为了实现一个素结（与多达10个交叉）在1-slab（即板的高度为1），需要一个多边形的长度严格长于所需的最小长度，以实现相同的结时，没有限制条件。在三叶结的情况下，我们实际上可以通过证明在1-slab中系三叶结所需的最小长度为26来分析地建立这个结果，该长度大于24，即在没有限制条件的情况下系三叶结所需的已知最小长度。此外，我们发现，在1-板坯不是所有的几何实现的一个给定的结类型下BFACF移动是等价的。这表明，在某些受限体积中，知道聚合物的拓扑结构不足以描述其所有状态。
Volume confinement is a key determinant of the topology and geometry of a polymer. However, the direct relationship between the two is not fully understood. For instance, recent experimental studies have constructed P4 cosmids, i.e. P4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their DNA knot distribution differs from that of wild-type bacteriophage P4. In particular, it was observed that the complexity of the knots decreases sharply with the length of the packed genome. This problem is the motivation of this paper. Here, a polymer is modeled as a self-avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes (i.e. bounded within a slab). We estimate the minimum length required for such a polygon to realize a knot type. Our numerical simulations show that in order to realize a prime knot (with up to ten crossings) in a 1-slab (i.e. a slab of height 1), one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition. In the case of the trefoil knot, we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1-slab is 26, which is greater than 24, the known minimum length required to tie a trefoil without a confinement condition. Additionally, we find that in the 1-slab not all geometrical realizations of a given knot type are equivalent under BFACF moves. This suggests that in certain confined volumes, knowing the topology of a polymer is not enough to describe all its states.