Secondary structures of polynucleotides can be view as a certain class of planar vertex-labeled graphs. We construct recursion formulae enumerating various sub-classes of these graphs as well as certain structural elements (sub-graphs). First order asymptotics are derived and their dependence on the logic of base pairing is computed and discussed. Key words. Planar Graphs, Generating Functions, Asymptotic Enumeration, Secondary Structure AMS subject classifications. 05A15, 05A16, 05C30, 92C40 1. Introduction. Presumably the most important problem and the greatest challenge in present day theoretical biophysics deals with deciphering the code that transforms sequences of biopolymers into spatial molecular structures. A sequence is properly visualized as a string of symbols which together with the environment encodes the molecular architecture of the biopolymer. In case of one particular class of biopolymers, the ribonucleic acid (RNA) molecules, decoding of information stored in th...

Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures that incorporates "non-nested" pseudoknots. We also introduce a measure for the complexity of more general contact structures in terms of the chromatic number of their intersection graph. We show that RNA structures without nested pseudoknots form a special class of planar graphs. Upper bounds on their number are derived, showing that there are fewer different structures than sequences. 1. Introduction Presumably the most important problem and the greatest challenge in present day theoretical biophysics is deciphering the code that transforms sequences of biopolymers into spatial molecular structures. A sequence is properly visualized as a string of symbols which together with the environment encodes the molecul...

this paper we consider the problem of computing two similar recurrences: the one-dimensional case

i Abstract Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules,however contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures that incorporates "non-nested" pseudoknots. We also introduce a measure for the complexity of more general contact structures in terms of the chromatic number of their intersection graph. We show that RNA structures without nested pseudoknots form a special class of planar graphs, the so called "bi-secondary structures". Upper bounds on their number are derived, showing that there are fewer different structures than sequences. An energy function capable of dealing with bi-secondary structures was implemented into a generalized kinetic folding algorithm. Sterical hindrances involved in pseudoknot formation are taken into account with the help of two simplifications: stacked regions are viewed ...

Abstract. 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 ≤ i ≤ n − 2. 1.

Discretized models of biopolymer structures can be used not only as approximations of the actual spatial structures but also as a computationally feasible approach to the generic features of the sequence-structure relationships. We review the combinatorics of nucleic acid secondary structures as well as lattice models of proteins, and show how properties such as the existence of extended neutral networks or shape space covering can be explained on this basis.

As one of the earliest problems in computational biology, RNA secondary structure prediction (sometimes referred to as “RNA folding”) problem has attracted attention again, thanks to the recent discoveries of many novel non-coding RNA molecules. The two common approaches to this problem are de novo prediction of RNA secondary structure based on energy minimization and the consensus folding approach (computing the common secondary structure for a set of unaligned RNA sequences). Consensus folding algorithms work well when the correct seed alignment is part of the input to the problem. However, seed alignment itself is a challenging problem for diverged RNA families. In this paper, we propose a novel framework to predict the common secondary structure for unaligned RNA sequences. By matching putative stacks in RNA sequences, we make use of both primary sequence information and thermodynamic stability for prediction at the same time. We show that our method can predict the correct common RNA secondary structures even when we are given only a limited number of unaligned RNA sequences, and it outperforms current algorithms in sensitivity and accuracy. Key words: RNA secondary structure prediction, RNA consensus folding, RNA stack configuration, dynamic programming. 1.

In previous publications ( J. Geom.Phys.38 (2001) 81-139 and references therein) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics. MCS:83C45 Subj.Class.:Quantum gravity Keywords: Surface automorphisms; dynamical systems; gravity; Grassmannians; Schubert calculus; enumerative combinatorics.

Introduction. (0.1) The Stasheff polytope, or associahedron, Kn, is a convex polytope of dimension n − 2 whose vertices correspond to complete parenthesizings of the product of n factors x1,..., xn. It was introduced by J. Stasheff [St] in his study of homotopy associativity for binary multiplications on topological spaces. In this paper we describe a surprising appearance of Stasheff

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, Sk(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 Sk(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