Pseudoknot RNA Structures with Arc-Length ≥4

Abstract

In this article, we study k-noncrossing RNA structures with minimum arc-length 4 and at most k - 1 mutually crossing bonds. Let T(k)([4])(n) denote the number of k-noncrossing RNA structures with arc-length > or =4 over n vertices. We (a) prove a functional equation for the generating function summation operator(n> or =0) T(k)([4])(n)z(n) and (b) derive for 4 < or = k < or = 9 the asymptotic formula T(k)([4])(n) approximately c(k) n(-((k-1)(2)+(k-1)/2)) gamma(k)(-n). Furthermore, we explicitly compute the exponential growth rates gamma(k)(-1) and asymptotic formulas for 4 < or = k < or = 9.