RNA pseudoknot structures with arc-length $\ge 3$ and stack-length $\ge \sigma$

Abstract

In this paper, we enumerate $k$-noncrossing RNA pseudoknot structures with a 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 $T_{k,\sigma }^{\left[3\right]}\left(n\right)$. We prove that the numbers of$k$-noncrossing RNA structures with arc-length $\ge 3$ and stack-length $\ge 2$ satisfy $T_{k,2}^{\left[3\right]}{\left(n\right)}^{}\sim C_k{n}^{-{(k-1)}^2-\frac{k-1}{2}}{\left({\gamma }_{k,2}^{\left[3\right]}\right)}^{-n}$. In the case $k=3,4,5$, we derive $T_{3,2}^{\left[3\right]}{\left(n\right)}^{}\sim C_3{n}^{-5}2.5721^n$, $T_{4,2}^{\left[3\right]}\left(n\right)\sim C_4{n}^{-\frac{21}{2}}3.0306^n$, and $T_{5,2}^{\left[3\right]}\left(n\right)\sim C_5{n}^{-18}3.4092^n$, respectively, where $C_3,C_4,C_5$ are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.