In this paper we derive polynomial time algorithms that generate random $k$-noncrossing matchings and $k$-noncrossing RNA structures with uniform probability. Our approach employs the bijection between $k$-noncrossing matchings and oscillating tableaux and the $P$-recursiveness of the cardinalities of $k$-noncrossing matchings. The main idea is to consider the tableaux sequences as paths of stochastic processes over shapes and to derive their transition probabilities.
A combinatorial framework for RNA tertiary interaction
在本文中，我们展示了如何表达RNA三级相互作用通过纠缠图的概念。缠结图允许制定RNA碱基三元组和假结相互作用，并控制相互交叉的弧的最大数量。特别是我们研究两个子集的纠缠图：3-noncrossing纠缠图与$\ell$顶点的程度2和2-定期，3-noncrossing分区（即没有弧的形式$(i,i+1)$）。我们的主要结果是2-正则，3-非交叉划分，记为$p_{3,2}(n)$，3-非交叉划分在$[n]$上的一个渐近公式。利用Birkhoff-Trjitzinsky的奇异差分方程的解析理论，导出了渐近公式。证明了公式$p_{3,2}(n+1)\sim K 8^{n}n^{-7}(1+c_{1}/n+c_{2}/n^2+c_3/n^3)$，其中$K,c_i$，$i=1,2,3$为常数.
Background: RNA exhibits a variety of structural configurations. Here we consider a structure to be tantamount to the noncrossing Watson-Crick and \pairGU -base pairings (secondary structure) and additional cross-serial base pairs. These interactions are called pseudoknots and are observed across the whole spectrum of RNA functionalities. In the context of studying natural RNA structures, searching for new ribozymes and designing artificial RNA, it is of interest to find RNA sequences folding into a specific structure and to analyze their induced neutral networks. Since the established inverse…
Background: RNA exhibits a variety of structural configurations. Here we consider a structure to be tantamount to the noncrossing Watson-Crick and \pairGU -base pairings (secondary structure) and additional cross-serial base pairs. These interactions are called pseudoknots and are observed across the whole spectrum of RNA functionalities. In the context of studying natural RNA structures, searching for new ribozymes and designing artificial RNA, it is of interest to find RNA sequences folding into a specific structure and to analyze their induced neutral networks. Since the established inverse…
在本文中，我们介绍了多项式时间算法，产生随机的3-不相交的划分和2-正规，3-不相交的划分具有均匀的概率。一个3-非交叉划分在其规范表示中不包含任何三个相互交叉的弧，并且如果后者不包含$(i,i+1)$形式的弧，则它是2-正则的。使用Chen的双射{ \it et al.} \cite{Chen,Reidys:08tan}，我们把3-不相交的划分和2-正则3-不相交的划分解释为受限制的广义摆动表.此外，我们解释的tableaux作为抽样路径的马尔可夫过程的形状，并得出其转移概率。
最近Yoffe { \it et al.}观察到RNA分子的$5'-3'$末端之间的平均距离非常小，并且在很大程度上与序列长度无关。这种观察是基于数值计算以及最大化某些熵泛函的理论论证。本文对任意有限$n$，计算了RNA二级结构的$5'-3'$距离的精确分布。此外，我们计算极限分布，并表明，已经为$n=30$的确切分布和极限分布是非常接近的。我们的结果表明，随机RNA二级结构的距离明显低于随机RNA序列的最小自由能结构的距离。
