Combinatorics of locally optimal RNA secondary structures

TL;DR

This paper analyzes the asymptotic count of RNA secondary structures that are locally optimal under the base stacking energy model, extending previous work on saturated structures and employing combinatorial enumeration techniques.

Contribution

It introduces a combinatorial approach to determine the asymptotic number of locally optimal RNA structures in the base stacking energy model, building on prior models and algorithms.

Findings

Asymptotic number of locally optimal structures is derived.

Structures with unextendable stems are characterized.

Enumeration methods are applied to structures with annotated dangles.

Abstract

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$. Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes -1 towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07427\cdot n^{-3/2} \cdot 2.35467^n$ many saturated structures for a sequence of length $n$. In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes -1 toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles).