Stacks in canonical RNA pseudoknot structures

TL;DR

This paper analyzes the distribution of stacks in RNA pseudoknot structures using generating functions, establishing a central limit theorem for stack distribution, which advances understanding of RNA folding patterns.

Contribution

It derives a bivariate generating function for $<k, au>$-structures and proves a central limit theorem for the number of stacks in these structures.

Findings

Unique dominant singularity in the generating function

Central limit theorem for stack distribution

Insights into RNA pseudoknot structure patterns

Abstract

In this paper we study the distribution of stacks in $k$-noncrossing, $\tau$-canonical RNA pseudoknot structures ($<k,\tau> $-structures). An RNA structure is called $k$-noncrossing if it has no more than $k-1$ mutually crossing arcs and $\tau$-canonical if each arc is contained in a stack of length at least $\tau$. Based on the ordinary generating function of $<k,\tau>$-structures \cite{Reidys:08ma} we derive the bivariate generating function ${\bf T}_{k,\tau}(x,u)=\sum_{n \geq 0} \sum_{0\leq t \leq \frac{n}{2}} {\sf T}_{k, \tau}^{} (n,t) u^t x^n$, where ${\sf T}_{k,\tau}(n,t)$ is the number of $<k,\tau>$-structures having exactly $t$ stacks and study its singularities. We show that for a certain parametrization of the variable $u$, ${\bf T}_{k,\tau}(x,u)$ has a unique, dominant singularity. The particular shift of this singularity parametrized by $u$ implies a central limit theorem for the distribution of stack-numbers. Our results are of importance for understanding the ``language'' of minimum-free energy RNA pseudoknot structures, generated by computer folding algorithms.