RNA secondary structures having a compatible sequence of certain nucleotide ratios

TL;DR

This paper investigates the conditions under which random RNA secondary structures are compatible with sequences having fixed nucleotide ratios, identifying a convex region in the ratio space where compatibility is almost certain for long sequences.

Contribution

It characterizes the convex region of nucleotide ratios that almost surely yield compatible sequences in random RNA structures, considering various base pairing rules and restrictions.

Findings

A convex region in nucleotide ratio space guarantees compatibility in long sequences.

Sequences with G ratio less than 1/3 are almost surely incompatible.

Sequences with G ratio greater than 1/3 are almost surely compatible.

Abstract

Given a random RNA secondary structure, $S$, we study RNA sequences having fixed ratios of nuclotides that are compatible with $S$. We perform this analysis for RNA secondary structures subject to various base pairing rules and minimum arc- and stack-length restrictions. Our main result reads as follows: in the simplex of the nucleotide ratios there exists a convex region in which, in the limit of long sequences, a random structure a.a.s.~has compatible sequence with these ratios and outside of which a.a.s.~a random structure has no such compatible sequence. We localize this region for RNA secondary structures subject to various base pairing rules and minimum arc- and stack-length restrictions. In particular, for {\bf GC}-sequences having a ratio of {\bf G} nucleotides smaller than $1/3$, a random RNA secondary structure without any minimum arc- and stack-length restrictions has a.a.s.~no such compatible sequence. For sequences having a ratio of {\bf G} nucleotides larger than $1/3$, a random RNA secondary structure has a.a.s. such compatible sequences. We discuss our results in the context of various families of RNA structures.