Pseudoknot RNA Structures with Arc-Length $\ge 3$

TL;DR

This paper analyzes specific classes of RNA pseudoknot structures with arc-length constraints, deriving their enumeration formulas and growth rates for different crossing levels, providing insights into their combinatorial complexity.

Contribution

It introduces a novel functional equation for counting $k$-noncrossing RNA structures with arc-length ≥ 3 and derives explicit asymptotic formulas for their enumeration.

Findings

Derived exponential growth factors for all $k \\ge 3$

Obtained subexponential factors for the case $k=3$

Provided explicit asymptotic formula for ${\\sf S}_{3,3}(n)$

Abstract

In this paper we study $k$-noncrossing RNA structures with arc-length $\ge 3$, i.e. RNA molecules in which for any $i$, the nucleotides labeled $i$ and $i+j$ ($j=1,2$) cannot form a bond and in which there are at most $k-1$ mutually crossing arcs. Let ${\sf S}_{k,3}(n)$ denote their number. Based on a novel functional equation for the generating function $\sum_{n\ge 0}{\sf S}_{k,3}(n)z^n$, we derive for arbitrary $k\ge 3$ exponential growth factors and for $k=3$ the subexponential factor. Our main result is the derivation of the formula ${\sf S}_{3,3}(n) \sim \frac{6.11170\cdot 4!}{n(n-1)...(n-4)} 4.54920^n$.