Canonical RNA pseudoknot structures

In this paper we study $k$-noncrossing, canonical RNA pseudoknot structures with minimum arc-length $\ge 4$. Let ${\sf T}_{k,\sigma}^{[4]} (n)$ denote the number of these structures. We derive exact enumeration results by computing the generating function ${\bf T}_{k,\sigma}^{[4]}(z)= \sum_n{\sf T}_{k,\sigma}^{[4]}(n)z^n$ and derive the asymptotic formulas ${\sf T}_{k,3}^{[4]}(n)^{}\sim c_k n^{-(k-1)^2-\frac{k-1}{2}} (\gamma_{k,3}^{[4]})^{-n}$ for $k=3,...,9$. In particular we have for $k=3$, ${\sf T}_{3,3}^{[4]}(n)^{}\sim c_3 n^{-5} 2.0348^n$. Our results prove that the set of biophysically relevant RNA pseudoknot structures is surprisingly small and suggest a new structure class as target for prediction algorithms.