$k$-noncrossing RNA structures with arc-length $\ge 3$

TL;DR

This paper provides enumeration formulas and asymptotic estimates for $k$-noncrossing RNA pseudoknot structures with minimum arc-length 3 and stack-length 2, aiding RNA structure prediction algorithms.

Contribution

It introduces new enumeration formulas and asymptotic growth rates for constrained $k$-noncrossing RNA structures with specific arc and stack lengths, extending prior combinatorial models.

Findings

Asymptotic formulas for 3, 4, and 5-noncrossing RNA structures.

Growth rate estimates: approximately 2.57, 3.03, and 3.41.

Enumeration results support RNA pseudoknot prediction algorithms.

Abstract

In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length $\ge 3$, stack-length $\ge \sigma$ and in which there are at most $k-1$ mutually crossing bonds, denoted by ${\sf T}_{k,\sigma}^{[3]}(n)$. In particular we prove that the numbers of 3, 4 and 5-noncrossing RNA structures with arc-length $\ge 3$ and stack-length $\ge 2$ satisfy ${\sf T}_{3,2}^{[3]}(n)^{}\sim K_3 n^{-5} 2.5723^n$, ${\sf T}^{[3]}_{4,2}(n)\sim K_4 n^{-{21/2}} 3.0306^n$, and ${\sf T}^{[3]}_{5,2}(n)\sim K_5 n^{-18} 3.4092^n$, respectively, where $K_3,K_4,K_5$ are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.