In this paper we analyze the length-spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this length-spectrum. We shall prove that there asymptotically almost surely exists a unique longest rainbow of length at least $n-O(n^{1/2})$ and that with high probability any other rainbow has finite length. We show that the distribution of the length of the longest rainbow converges to a discrete limit law and that, for finite $k$, the distribution of rainbows of length $k$, becomes for large $n$ a negative binomial distribution. We then put the results of this paper into context, comparing the analytical results with those observed in RNA minimum free energy structures, biological RNA structures and relate our findings to the sparsification of folding algorithms.
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。
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…
蛋白质折叠的接触图是表示折叠中接触模式的图。众所周知，接触图可以分解为堆栈和队列。RNA二级结构是一种特殊的堆积结构，其中每个顶点的度至多为1，每个弧的长度至少为2。Waterman和Smith推导出了长度为$n$且恰好为$k$弧的RNA二级结构的数量公式。Höner zu Siederdissen等人开发了一种扩展RNA二级结构的折叠算法，其中每个顶点的最大度为2。Müller和Nebel利用上下文无关文法方法得到了扩展RNA二级结构的生成函数方程，并由此得到了一个渐近公式。本文考虑了$m$-正则线性堆叠，其中每个弧的长度至少为$m$，每个顶点的度以2为界.延伸的RNA二级结构是精确的$2$-规则线性堆叠。对于任意$m\geq 2$，我们得到了$m$-正则线性叠加的生成函数的一个方程.对于给定的$m$，我们可以推导出一个递推关系和一个关于$n$顶点上$m$-正则线性栈个数的渐近公式.为了建立方程，我们使用Chen，Deng和Du的约化操作将$m$-正则线性叠加变换为$m$-约化的锯齿形（或交替）叠加.然后，我们找到一个方程$m$减少锯齿形堆栈导致一个方程$m$-定期线性堆栈。
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…
