Asymptotic Enumeration of RNA Structures with Pseudoknots

In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for $k$-noncrossing RNA structures. Our results are based on the generating function for the number of $k$-noncrossing RNA pseudoknot structures, ${\sf S}k(n)$, derived in \cite{Reidys:07pseu}, where $k-1$ denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function $\sum{n\ge 0}{\sf S}_k(n)z^n$ and obtain for $k=2$ and $k=3$ the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary $k$ singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula ${\sf S}_3(n) \sim \frac{10.4724\cdot 4!}{n(n-1)…(n-4)} (\frac{5+\sqrt{21}}{2})^n$.

💡 Research Summary

The paper addresses the long‑standing combinatorial problem of counting RNA secondary structures that contain pseudoknots, i.e., configurations where base‑pair arcs may cross. The authors work within the framework of k‑noncrossing RNA structures, where at most (k-1) arcs are allowed to mutually intersect. For a fixed (k) they denote by ({\sf S}_k(n)) the number of such structures on a sequence of length (n) and study the generating function
\