18,708 research outputs found
Shapes of interacting RNA complexes
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
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
time and 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
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
into bicellular maps of genus 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 , where are the lengths of the
respective backbones, while the folding of the original structure has
time complexity, where is the length of the longer sequence.Comment: 18 pages, 13 figure
Thermodynamics of a model for RNA folding
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 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 as
being a measure of the concentration of in solution.Comment: 11 pages, 4 figure
Hierarchy and assortativity as new tools for affinity investigation: the case of the TBA aptamer-ligand complex
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
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
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, ,
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 -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 -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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