RNA Secondary Structures Having a Compatible Sequence of Certain Nucleotide Ratios.

Given a random RNA secondary structure, S, we study RNA sequences having fixed ratios of nucleotides 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 nucleotide ratios, there exists a convex region, in which, in the limit of long sequences, a random structure asymptotically almost surely (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 GC-sequences (GC denoting the nucleotides guanine and cytosine, respectively) having a ratio of 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 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.