Asymptotics of Canonical and Saturated RNA Secondary Structures

TL;DR

This paper analyzes the asymptotic combinatorial properties of canonical and saturated RNA secondary structures, explaining computational speed-ups and providing new asymptotic counts and distributions for these subclasses.

Contribution

It introduces the asymptotic enumeration of saturated structures, explains the computational speed-up for canonical structures, and presents a stochastic sampling method for saturated structures.

Findings

Asymptotic number of saturated structures is computed.

Expected base pairs in saturated structures is approximately 0.337361 n.

Number of saturated stem-loop structures grows as 0.323954 1.69562^n.

Abstract

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 n^{-3/2} 2.618034^n$. In this paper, we study combinatorial asymptotics for two special subclasses of RNA secondary structures - canonical and saturated structures. Canonical secondary structures were introduced by Bompf\"unewerer et al., who noted that the run time of Vienna RNA Package is substantially reduced when restricting computations to canonical structures. Here we provide an explanation for the speed-up. Saturated secondary structures have the property that no base pairs can be added without violating the definition of secondary structure (i.e. introducing a pseudoknot or base triple). Here we compute the asymptotic number of saturated structures, we show that the asymptotic expected number of base pairs is $0.337361 n$, and the asymptotic number of saturated stem-loop structures is $0.323954 1.69562^n$, in contrast to the number $2^{n-2}$ of (arbitrary) stem-loop structures as classically computed by Stein and Waterman. Finally, we show that the density of states for [all resp. canonical resp. saturated] secondary structures is asymptotically Gaussian. We introduce a stochastic greedy method to sample random saturated structures, called quasi-random saturated structures, and show that the expected number of base pairs of is $0.340633 n$.