The 5'-3' distance of RNA secondary structures

Recently Yoffe {\it et al.} observed that the average distances between $5'-3'$ ends of RNA molecules are very small and largely independent of sequence length. This observation is based on numerical computations as well as theoretical arguments maximizing certain entropy functionals. In this paper we compute the exact distribution of $5'-3'$ distances of RNA secondary structures for any finite $n$. We furthermore compute the limit distribution and show that already for $n=30$ the exact distribution and the limit distribution are very close. Our results show that the distances of random RNA secondary structures are distinctively lower than those of minimum free energy structures of random RNA sequences.