18,708 research outputs found

    Shapes of interacting RNA complexes

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    Shapes of interacting RNA complexes are studied using a filtration via their topological genus. A shape of an RNA complex is obtained by (iteratively) collapsing stacks and eliminating hairpin loops. This shape-projection preserves the topological core of the RNA complex and for fixed topological genus there are only finitely many such shapes.Our main result is a new bijection that relates the shapes of RNA complexes with shapes of RNA structures.This allows to compute the shape polynomial of RNA complexes via the shape polynomial of RNA structures. We furthermore present a linear time uniform sampling algorithm for shapes of RNA complexes of fixed topological genus.Comment: 38 pages 24 figure

    Topology of RNA-RNA interaction structures

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    The topological filtration of interacting RNA complexes is studied and the role is analyzed of certain diagrams called irreducible shadows, which form suitable building blocks for more general structures. We prove that for two interacting RNAs, called interaction structures, there exist for fixed genus only finitely many irreducible shadows. This implies that for fixed genus there are only finitely many classes of interaction structures. In particular the simplest case of genus zero already provides the formalism for certain types of structures that occur in nature and are not covered by other filtrations. This case of genus zero interaction structures is already of practical interest, is studied here in detail and found to be expressed by a multiple context-free grammar extending the usual one for RNA secondary structures. We show that in O(n6)O(n^6) time and O(n4)O(n^4) space complexity, this grammar for genus zero interaction structures provides not only minimum free energy solutions but also the complete partition function and base pairing probabilities.Comment: 40 pages 15 figure

    A bijection between unicellular and bicellular maps

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    In this paper we present a combinatorial proof of a relation between the generating functions of unicellular and bicellular maps. This relation is a consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it can be proved using representation theory of the symmetric group. Here we give a bijective proof by rewiring unicellular maps of topological genus (g+1)(g+1) into bicellular maps of genus gg and pairs of unicellular maps of lower topological genera. Our result has immediate consequences for the folding of RNA interaction structures, since the time complexity of folding the transformed structure is O((n+m)5)O((n+m)^5), where n,mn,m are the lengths of the respective backbones, while the folding of the original structure has O(n6)O(n^6) time complexity, where nn is the length of the longer sequence.Comment: 18 pages, 13 figure

    Thermodynamics of a model for RNA folding

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    We analyze the thermodynamic properties of a simplified model for folded RNA molecules recently studied by G. Vernizzi, H. Orland, A. Zee (in {\it Phys. Rev. Lett.} {\bf 94} (2005) 168103). The model consists of a chain of one-flavor base molecules with a flexible backbone and all possible pairing interactions equally allowed. The spatial pseudoknot structure of the model can be efficiently studied by introducing a N×NN \times N hermitian random matrix model at each chain site, and associating Feynman diagrams of these models to spatial configurations of the molecules. We obtain an exact expression for the topological expansion of the partition function of the system. We calculate exact and asymptotic expressions for the free energy, specific heat, entanglement and chemical potential and study their behavior as a function of temperature. Our results are consistent with the interpretation of 1/N1/N as being a measure of the concentration of Mg++\rm{Mg}^{++} in solution.Comment: 11 pages, 4 figure

    Hierarchy and assortativity as new tools for affinity investigation: the case of the TBA aptamer-ligand complex

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    Aptamers are single stranded DNA, RNA or peptide sequences having the ability to bind a variety of specific targets (proteins, molecules as well as ions). Therefore, aptamer production and selection for therapeutic and diagnostic applications is very challenging. Usually they are in vitro generated, but, recently, computational approaches have been developed for the in silico selection, with a higher affinity for the specific target. Anyway, the mechanism of aptamer-ligand formation is not completely clear, and not obvious to predict. This paper aims to develop a computational model able to describe aptamer-ligand affinity performance by using the topological structure of the corresponding graphs, assessed by means of numerical tools such as the conventional degree distribution, but also the rank-degree distribution (hierarchy) and the node assortativity. Calculations are applied to the thrombin binding aptamer (TBA), and the TBA-thrombin complex, produced in the presence of Na+ or K+. The topological analysis reveals different affinity performances between the macromolecules in the presence of the two cations, as expected by previous investigations in literature. These results nominate the graph topological analysis as a novel theoretical tool for testing affinity. Otherwise, starting from the graphs, an electrical network can be obtained by using the specific electrical properties of amino acids and nucleobases. Therefore, a further analysis concerns with the electrical response, which reveals that the resistance sensitively depends on the presence of sodium or potassium thus posing resistance as a crucial physical parameter for testing affinity.Comment: 12 pages, 5 figure

    Enumeration of RNA structures by Matrix Models

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    We enumerate the number of RNA contact structures according to their genus, i.e. the topological character of their pseudoknots. By using a recently proposed matrix model formulation for the RNA folding problem, we obtain exact results for the simple case of an RNA molecule with an infinitely flexible backbone, in which any arbitrary pair of bases is allowed. We analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate-sugar backbone.Comment: RevTeX, 4 pages, 2 figure

    On the combinatorics of sparsification

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    Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, Λ∗\Lambda^*, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of Λ∗\Lambda^*-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of Λ∗\Lambda^*-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the "full" loop-energy model there is a reduction of 98% (experiment).Comment: 27 pages, 12 figure
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