Random words in free groups, non-crossing matchings and RNA secondary structures

TL;DR

This paper investigates the expected fraction of unpaired bases in random RNA sequences and relates it to properties of words in free groups, establishing convergence to a constant and bounds for it.

Contribution

It establishes the convergence of the unpaired base fraction in random RNA structures and connects this to free group word length ratios, extending results to all non-abelian free groups.

Findings

Expected unpaired base fraction converges to a constant between 0 and 1.

The ratio of shortest word length in conjugate generators to standard generators grows linearly.

Results hold for all non-abelian finitely generated free groups.

Abstract

Consider a random word $X^n=(X_1,\ldots ,X_n)$ in an alphabet consisting of $4$ letters, with the letters viewed either as $A$, $U$, $G$ and $C$ (i.e., nucleotides in an RNA sequence) or $\alpha$, $\bar{\alpha}$, $\beta$ and $\bar{\beta}$ (i.e., generators of the free group $\langle\alpha,\beta\rangle$ and their inverses). We show that the expected fraction $\rho(n)$ of unpaired bases in an optimal RNA secondary structure (with only Watson-Crick bonds and no pseudo-knots) converges to a constant $\lambda_2$ with $0<\lambda_2<1$ as $n\to\infty$. Thus, a positive proportion of the bases of a random RNA string do not form hydrogen bonds. We do not know the exact value of $\lambda_2$, but we derive upper and lower bounds for it. In terms of free groups, $\rho(n)$ is the ratio of the length of the shortest word representing $X$ in the generating set consisting of conjugates of generators and their inverses to the word length of $X$ with respect to the standard generators and their inverses. Thus for a typical word the word length in the (infinite) generating set consisting of the conjugates of standard generators grows linearly with the word length in the standard generators. In fact, we show that a similar result holds for all non-abelian finitely generated free groups $\langle\alpha_1,\dots,\alpha_k\rangle$, $k\geq 2$.