The rainbow-spectrum of RNA secondary structures

TL;DR

This paper investigates the length distribution of rainbows in RNA secondary structures, revealing a dominant longest rainbow and a probabilistic distribution of shorter rainbows, with implications for RNA folding analysis.

Contribution

It provides a rigorous asymptotic analysis of rainbow lengths in RNA structures, including distribution laws and their biological relevance.

Findings

Existence of a unique longest rainbow of length at least n - O(n^{1/2})

Distribution of the longest rainbow converges to a discrete limit law

Rainbow lengths of fixed size follow a negative binomial distribution for large n

Abstract

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.