New alphabet-dependent morphological transition in a random RNA alignment

TL;DR

This paper investigates how the secondary structure of random RNA molecules changes with the number of nucleotide types, revealing a sharp transition at a critical alphabet size that affects the structure's perfection.

Contribution

It identifies and proves bounds for a morphological transition in RNA secondary structures depending on nucleotide alphabet size, supported by numerical evidence.

Findings

Transition occurs at critical alphabet size c_cr between 2 and 4.

For c ≤ c_cr, structures are nearly perfect and gapless.

For c > c_cr, structures contain gaps and are less perfect.

Abstract

We study the fraction $f$ of nucleotides involved in the formation of a cactus--like secondary structure of random heteropolymer RNA--like molecules. In the low--temperature limit we study this fraction as a function of the number $c$ of different nucleotide species. We show, that with changing $c$, the secondary structures of random RNAs undergo a morphological transition: $f(c)\to 1$ for $c \le c_{\rm cr}$ as the chain length $n$ goes to infinity, signaling the formation of a virtually "perfect" gapless secondary structure; while $f(c)<1$ for $c>c_{\rm cr}$, what means that a non-perfect structure with gaps is formed. The strict upper and lower bounds $2 \le c_{\rm cr} \le 4$ are proven, and the numerical evidence for $c_{\rm cr}$ is presented. The relevance of the transition from the evolutional point of view is discussed.