Canonical RNA Pseudoknot Structures

In this paper, we study k-noncrossing, sigma-canonical RNA pseudoknot structures with minimum arc-length greater or equal to four. Let T(k, sigma)([4])(n) denote the number of these structures. We derive exact enumeration results by computing the generating function T(k, sigma)([4])(z) = summation operator(n) T(k, sigma)([4])(n)z(n) and derive the asymptotic formulas T(k, 3)([4])(n) approximately c(k) n(-(k-1)(2)-(k-1/2)) (gamma(k, 3)([4]))(-n) for k = 3, ..., 9. In particular, we have for k = 3, T(3, 3)([4])(n) approximately c(3)n(-5)2.0348(n). Our results show that the set of biophysically relevant RNA pseudoknot structures is surprisingly small and suggest a new structure class as target for prediction algorithms.