Counting RNA pseudoknotted structures

International audienceIn 2004, Condon and coauthors gave a hierarchical classification of exact RNA structure prediction algorithms according to the generality of structure classes that they handle. We complete this classification by adding two recent prediction algo- rithms. More importantly, we precisely quantify the hierarchy by giving closed or asymptotic formulas for the theoretical number of structures of given size n in all the classes but one. This allows to assess the tradeoff between the expressiveness and the computational complexity of RNA structure prediction algorithms

A Combinatorial Framework for Designing (Pseudoknotted) RNA Algorithms

We extend an hypergraph representation, introduced by Finkelstein and Roytberg, to unify dynamic programming algorithms in the context of RNA folding with pseudoknots. Classic applications of RNA dynamic programming energy minimization, partition function, base-pair probabilities...) are reformulated within this framework, giving rise to very simple algorithms. This reformulation allows one to conceptually detach the conformation space/energy model -- captured by the hypergraph model -- from the specific application, assuming unambiguity of the decomposition. To ensure the latter property, we propose a new combinatorial methodology based on generating functions. We extend the set of generic applications by proposing an exact algorithm for extracting generalized moments in weighted distribution, generalizing a prior contribution by Miklos and al. Finally, we illustrate our full-fledged programme on three exemplary conformation spaces (secondary structures, Akutsu's simple type pseudoknots and kissing hairpins). This readily gives sets of algorithms that are either novel or have complexity comparable to classic implementations for minimization and Boltzmann ensemble applications of dynamic programming

Combinatorics of locally optimal RNA secondary structures

It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$. Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes -1 towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07427\cdot n^{-3/2} \cdot 2.35467^n$ many saturated structures for a sequence of length $n$. In this paper, we consider the base stacking energy model, a mild variant of the Nussinov model, where each stacked base pair contributes -1 toward the energy of the structure. Locally optimal structures with respect to the base stacking energy model are exactly those secondary structures, whose stems cannot be extended. Such structures were first considered by Evers and Giegerich, who described a dynamic programming algorithm to enumerate all locally optimal structures. In this paper, we apply methods from enumerative combinatorics to compute the asymptotic number of such structures. Additionally, we consider analogous combinatorial problems for secondary structures with annotated single-stranded, stacking nucleotides (dangles).Comment: 27 page

Counting RNA pseudoknotted structures (extended abstract)

In 2004, Condon and coauthors gave a hierarchical classification of exact RNA structure prediction algorithms according to the generality of structure classes that they handle. We complete this classification by adding two recent prediction algorithms. More importantly, we precisely quantify the hierarchy by giving closed or asymptotic formulas for the theoretical number of structures of given size n in all the classes but one. This allows to assess the tradeoff between the expressiveness and the computational complexity of RNA structure prediction algorithms. \parEn 2004, Condon et ses coauteurs ont défini une classification des algorithmes exacts de prédiction de structure d'ARN, selon le degré de généralité des classes de structures qu'ils sont capables de prédire. Nous complétons cette classification en y ajoutant deux algorithmes récents. Chose plus importante, nous quantifions la hiérarchie des algorithmes, en donnant des formules closes ou asymptotiques pour le nombre théorique de structures de taille donnée n dans chacune des classes, sauf une. Ceci fournit un moyen d'évaluer, pour chaque algorithme, le compromis entre son degré de généralité et sa complexité

Counting RNA pseudoknotted structures (extended abstract)

Abstract. In 2004, Condon and coauthors gave a hierarchical classification of exact RNA structure prediction algorithms according to the generality of structure classes that they handle. We complete this classification by adding two recent prediction algorithms. More importantly, we precisely quantify the hierarchy by giving closed or asymptotic formulas for the theoretical number of structures of given size n in all the classes but one. This allows to assess the tradeoff between the expressiveness and the computational complexity of RNA structure prediction algorithms. Résumé. En 2004, Condon et ses coauteurs ont défini une classification des algorithmes exacts de prédiction de structure d’ARN, selon le degré de généralité des classes de structures qu’ils sont capables de prédire. Nous complétons cette classification en y ajoutant deux algorithmes récents. Chose plus importante, nous quantifions la hiérarchie des algorithmes, en donnant des formules closes ou asymptotiques pour le nombre théorique de structures de taille donnée n dans chacune des classes, sauf une. Ceci fournit un moyen d’évaluer, pour chaque algorithme, le compromis entre son degré de généralité et sa complexité