In this paper, we derive the generating function of RNA structures with pseudoknots. We enumerate all k-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition, we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore, we enumerate for arbitrary k all k-noncrossing, restricted RNA structures i.e. k-noncrossing RNA structures without 2-arcs i.e. arcs of the form (i,i+2), for 1< or =i< or =n-2.
In this paper we enumerate k-noncrossing RNA pseudoknot structures with given minimum stack-length. We show that the numbers of k-noncrossing structures without isolated base pairs are significantly smaller than the number of all k-noncrossing structures. In particular we prove that the number of 3-and 4-noncrossing RNA structures with stack-length ≥ 2 is for large n given by 311.2470 4! n(n−1)...(n−4) 2.5881 n and 1.217 · 10 7 n − 21 2 3.0382 n , respectively. We furthermore show that for k-noncrossing RNA structures the drop in exponential growth rates between the number of all structures and the number of all structures with stack-size ≥ 2 increases significantly. Our results are of importance for prediction algorithms for pseudoknot-RNA and provide evidence that there exist neutral networks of RNA pseudoknot structures.
Abstract. In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures, S k (n), derived in [17], where k − 1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function P n≥0 S k (n)z n and obtain for k = 2 and k = 3 the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2-and 3-noncrossing RNA structures. Our main result is the derivation of the formula S 3 (n) ∼
The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley value) and the fair proportion index. He gave an explanation of this correlation by showing that the contribution of both indices to an edge of the tree becomes identical as the number of taxa tends to infinity. In this note, we show that the Shapley value and the fair proportion index are in fact the same. Moreover, we also consider the modified Shapley value and show that its covariance with the fair proportion index in random phylogenetic trees under the Yule-Harding model and uniform model is indeed close to one.
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