$k$-noncrossing RNA structures with arc-length $ge 3$

In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length $\ge 3$, stack-length $\ge \sigma$ and in which there are at most $k-1$ mutually crossing bonds, denoted by ${\sf T}{k,\sigma}^{[3]}(n)$. In particular we prove that the numbers of 3, 4 and 5-noncrossing RNA structures with arc-length $\ge 3$ and stack-length $\ge 2$ satisfy ${\sf T}{3,2}^{[3]}(n)^{}\sim K_3 n^{-5} 2.5723^n$, ${\sf T}^{[3]}{4,2}(n)\sim K_4 n^{-{21/2}} 3.0306^n$, and ${\sf T}^{[3]}{5,2}(n)\sim K_5 n^{-18} 3.4092^n$, respectively, where $K_3,K_4,K_5$ are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.

💡 Research Summary

The paper addresses the combinatorial enumeration of RNA pseudoknot structures under three simultaneous constraints: (i) a minimum arc length of at least three nucleotides, (ii) a minimum stack length σ (consecutive base‑pair stacks) and (iii) a bound on the number of mutually crossing arcs, expressed as “k‑noncrossing” (at most k‑1 arcs may cross). The authors denote the number of such structures of length n by ${\sf T}_{k,\sigma}^{