Central and Local Limit Theorems for RNA Structures

TL;DR

This paper establishes central and local limit theorems for the distribution of the number of bonds in k-noncrossing RNA structures, extending previous asymptotic results to probabilistic distributions.

Contribution

It provides the first probabilistic limit theorems for the distribution of bonds in k-noncrossing RNA structures, generalizing prior asymptotic enumeration results.

Findings

Proves a central limit theorem for the number of bonds in 3-noncrossing RNA structures.

Derives a local limit theorem for the distribution of bonds.

Predicts distributions for k-noncrossing RNA folding algorithms.

Abstract

A k-noncrossing RNA pseudoknot structure is a graph over $\{1,...,n\}$ without 1-arcs, i.e. arcs of the form (i,i+1) and in which there exists no k-set of mutually intersecting arcs. In particular, RNA secondary structures are 2-noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3-noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of \cite{Reidys:07rna1} and \cite{Reidys:07rna2}, where the generating function of k-noncrossing RNA pseudoknot structures and the asymptotics for its coefficients have been derived. The results of this paper explain the findings on the numbers of arcs of RNA secondary structures obtained by molecular folding algorithms and predict the distributions for k-noncrossing RNA folding algorithms which are currently being developed.