Pseudoknot RNA Structures with Arc-Length $\ge 3$

Abstract

In this paper we study $k$-noncrossing RNA structures with arc-length $\ge 3$, i.e. RNA molecules in which for any $i$, the nucleotides labeled $i$ and $i+j$ ($j=1,2$) cannot form a bond and in which there are at most $k-1$ mutually crossing arcs. Let ${\mathsf{S}}_{k,3}(n)$ denote their number. Based on a novel functional equation for the generating function ${\sum }_{n\ge 0}{\mathsf{S}}_{k,3}(n)z^n$, we derive for arbitrary $k\ge 3$ exponential growth factors and for $k=3$ the subexponential factor. Our main result is the derivation of the formula ${\mathsf{S}}_{3,3}(n)\sim \frac{6.11170\cdot 4!}{n(n-1)...(n-4)}4.54920^n$.

18 pages, 4 figures