非纽结自回避多边形中局部链通过后的纽结概率
Knotting probabilities after a local strand passage in unknotted self-avoiding polygons
摘要：
我们研究了简单立方晶格上的无结自回避多边形中的局部链通过后的打结概率。我们假设，两个多边形段已经被带到一起执行一个股通道的目的，并使用Theta-SAP，多边形，包含在一个固定位置的模式Theta模型。证明了n边Theta-SAP的数量以与n边未打结的自避免多边形的总数相同的速率呈指数增长（随n），并且对于产生特定的后链通道结类型的n边Theta-SAP的子集也是如此。因此，一个给定的后链通道结类型的概率不增长（或衰减）与n呈指数关系，我们推测，相反，它接近结类型依赖的振幅比严格介于0和1之间。这是支持的临界指数估计值从一个新的极大似然方法生成的复合（又名多）马尔可夫链蒙特卡罗BFACF算法的Theta-SAP。我们还给出了强有力的数值证据，后链通道打结的概率取决于周围的链通道网站的局部结构。考虑到当地的结构和交叉的迹象，在股通道网站，我们观察到，更“紧凑”的当地结构，不太可能后股通道多边形是打结。这一趋势与其他链通道模型的结果一致，然而，我们是第一个注意到交叉标志信息的影响。使用了两种“紧凑性”的度量：包含结构的最小多边形的大小和结构的“开口”角度。开角的定义与单分子DNA实验中可测量的开角一致。
英文摘要：
We investigate the knotting probability after a local strand passage is performed in an unknotted self-avoiding polygon on the simple cubic lattice. We assume that two polygon segments have already been brought close together for the purpose of performing a strand passage, and model this using Theta-SAPs, polygons that contain the pattern Theta at a fixed location. It is proved that the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate as the total number of n-edge unknotted self-avoiding polygons, and that the same holds for subsets of n-edge Theta-SAPs that yield a specific after-strand-passage knot-type. Thus the probability of a given after-strand-passage knot-type does not grow (or decay) exponentially with n, and we conjecture that instead it approaches a knot-type dependent amplitude ratio lying strictly between 0 and 1. This is supported by critical exponent estimates obtained from a new maximum likelihood method for Theta-SAPs that are generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF algorithm. We also give strong numerical evidence that the after-strand-passage knotting probability depends on the local structure around the strand passage site. Considering both the local structure and the crossing-sign at the strand passage site, we observe that the more "compact" the local structure, the less likely the after-strand-passage polygon is to be knotted. This trend is consistent with results from other strand-passage models, however, we are the first to note the influence of the crossing-sign information. Two measures of "compactness" are used: the size of a smallest polygon that contains the structure and the structure's "opening" angle. The opening angle definition is consistent with one that is measurable from single molecule DNA experiments.
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