25 research outputs found
Distribution of graph-distances in Boltzmann ensembles of RNA secondary structures
Large RNA molecules often carry multiple functional domains whose spatial
arrangement is an important determinant of their function. Pre-mRNA splicing,
furthermore, relies on the spatial proximity of the splice junctions that can
be separated by very long introns. Similar effects appear in the processing of
RNA virus genomes. Albeit a crude measure, the distribution of spatial
distances in thermodynamic equilibrium therefore provides useful information on
the overall shape of the molecule can provide insights into the interplay of
its functional domains. Spatial distance can be approximated by the
graph-distance in RNA secondary structure. We show here that the equilibrium
distribution of graph-distances between arbitrary nucleotides can be computed
in polynomial time by means of dynamic programming. A naive implementation
would yield recursions with a very high time complexity of O(n^11). Although we
were able to reduce this to O(n^6) for many practical applications a further
reduction seems difficult. We conclude, therefore, that sampling approaches,
which are much easier to implement, are also theoretically favorable for most
real-life applications, in particular since these primarily concern long-range
interactions in very large RNA molecules.Comment: Peer-reviewed and presented as part of the 13th Workshop on
Algorithms in Bioinformatics (WABI2013
Impact Of The Energy Model On The Complexity Of RNA Folding With Pseudoknots
International audiencePredicting the folding of an RNA sequence, while allowing general pseudoknots (PK), consists in finding a minimal free-energy matching of its positions. Assuming independently contributing base-pairs, the problem can be solved in -time using a variant of the maximal weighted matching. By contrast, the problem was previously proven NP-Hard in the more realistic nearest-neighbor energy model. In this work, we consider an intermediate model, called the stacking-pairs energy model. We extend a result by Lyngs\o, showing that RNA folding with PK is NP-Hard within a large class of parametrization for the model. We also show the approximability of the problem, by giving a practical algorithm that achieves at least a -approximation for any parametrization of the stacking model. This contrasts nicely with the nearest-neighbor version of the problem, which we prove cannot be approximated within any positive ratio, unless .La prédiction du repliement, avec pseudonoeuds généraux, d'une séquence d'ARN de taille est équivalent à la recherche d'un couplage d'énergie libre minimale. Dans un modèle d'énergie simple, où chaque paire de base contribue indépendamment à l'énergie, ce problème peut être résolu en temps grâce à une variante d'un algorithme de couplage pondéré maximal. Cependant, le même problème a été démontré NP-difficile dans le modèle d'énergie dit des plus proches voisins. Dans ce travail, nous étudions les propriétés du problème sous un modèle d'empilements, constituant un modèle intermédiaire entre ceux d'appariement et des plus proches voisins. Nous démontrons tout d'abord que le repliement avec pseudo-noeuds de l'ARN reste NP-difficile dans de nombreuses valuations du modèle d'énergie. . Par ailleurs, nous montrons que ce problème est approximable, en proposant un algorithme polynomial garantissant une -approximation. Ce résultat illustre une différence essentielle entre ce modèle et celui des plus proches voisins, pour lequel nous montrons qu'il ne peut être approché à aucun ratio positif par un algorithme en temps polynomial sauf si
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
Dynamics of Local Search Trajectory in Traveling Salesman Problem
This paper investigates dynamics of a local search trajectory generated by running the Or-opt heuristic on the traveling salesman problem. This study evaluates the dynamics of the local search heuristic by estimating the correlation dimension for the search trajectory, and finds that the local heuristic search process exhibits the transition from high-dimensional stochastic to low-dimensional chaotic behavior. The detection of dynamical complexity for a heuristic search process has both practical as well as theoretical relevance. The revealed dynamics may cast new light on design and analysis of heuristics and result in the potential for improved search process.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45818/1/10732_2005_Article_3604.pd
On Acyclic Orientations and Sequential Dynamical Systems
AbstractWe study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1,…,n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (Fi,Y: F2n→F2n)i, and (c) a permutation π∈Sn. The function Fi,Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions Fi,Y are applied. By composing the functions Fi,Y in the order given by π we obtain the sequential dynamical system (SDS):[FY,π]=Fπ(n),Y∘⋯∘Fπ(1),Y: F2n→F2n.In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (Fi,Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs
Evolution of Random Catalytic Networks
In this paper we investigate the evolution of populations of sequences on a random catalytic network. Sequences are mapped into structures, between which are catalytic interactions that determine their instantaneous fitness. The catalytic network is constructed as a random directed graph. We prove that at certain parameter values, the probability of some relevant subgraphs of this graph, for example cycles without outgoing edges, is maximized. Populations evolving under point mutations realise a comparatively small induced subgraph of the complete catalytic network. We present results which show that populations reliably discover and persist on directed cycles in the catalytic graph, though these may be lost because of stochastic effects, and study the effect of population size on this behavior. 1 Introduction Understanding evolutionary dynamics requires an understanding of the dynamics at a number of different levels--- the population dynamics which result from selection, the mechan..