Biocomplexity Institute of Virginia Tech, Blacksburg, VA, 24061, USA.
Bull Math Biol. 2018 Jun;80(6):1514-1538. doi: 10.1007/s11538-018-0411-9. Epub 2018 Mar 14.
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 [Formula: see text] 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.
在本文中，我们分析了 RNA 二级结构中的彩虹长度谱。二级结构中的彩虹是指相对于嵌套引起的偏序的最大弧。我们表明，在这个长度谱中存在显著的间隔。我们将证明，存在一个渐近几乎必然存在的唯一最长彩虹，其长度至少为[Formula: see text]，并且具有高概率的任何其他彩虹都具有有限的长度。我们表明，最长彩虹长度的分布收敛于一个离散的极限定律，并且对于有限的 k，长度为 k 的彩虹的分布对于大的 n 成为负二项分布。然后，我们将本文的结果置于上下文中，将分析结果与在 RNA 最小自由能结构、生物 RNA 结构中观察到的结果进行比较，并将我们的发现与折叠算法的稀疏化联系起来。
