Asymptotic Enumeration of RNA Structures with Pseudoknots

TL;DR

This paper develops a mathematical framework to asymptotically enumerate RNA structures with pseudoknots, deriving explicit formulas for their growth rates and subexponential factors, especially for 2- and 3-noncrossing structures.

Contribution

It introduces a general method for asymptotic enumeration of $k$-noncrossing RNA pseudoknot structures using generating functions and singularity analysis.

Findings

Derived explicit asymptotic formulas for 2- and 3-noncrossing RNA structures.

Established the existence of singular expansions for arbitrary $k$-noncrossing structures.

Provided the asymptotic growth rate for 3-noncrossing RNA structures as $n$ grows large.

Abstract

In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for $k$-noncrossing RNA structures. Our results are based on the generating function for the number of $k$-noncrossing RNA pseudoknot structures, ${\sf S}_k(n)$, derived in \cite{Reidys:07pseu}, where $k-1$ denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function $\sum_{n\ge 0}{\sf S}_k(n)z^n$ and obtain for $k=2$ and $k=3$ the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary $k$ singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula ${\sf S}_3(n) \sim \frac{10.4724\cdot 4!}{n(n-1)...(n-4)} (\frac{5+\sqrt{21}}{2})^n$.