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64
Reverseengineering of polynomial dynamical systems,
 Adv. Appl. Math.
, 2007
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Identification of genetic network dynamics with unate structure
 in "Bioinformatics", 2010
"... Motivation: Modern experimental techniques for timecourse measurement of gene expression enable the identification of dynamical models of genetic regulatory networks. In general, identification involves fitting appropriate network structures and parameters to the data. For a given set of genes, exp ..."
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Motivation: Modern experimental techniques for timecourse measurement of gene expression enable the identification of dynamical models of genetic regulatory networks. In general, identification involves fitting appropriate network structures and parameters to the data. For a given set of genes, exploring all possible network structures is clearly prohibitive. Modelling and identification methods for the a priori selection of network structures compatible with biological knowledge and experimental data are necessary to make the identification problem tractable. Results: We propose a differential equation modelling framework where the regulatory interactions among genes are expressed in terms of unate functions, a class of gene activation rules commonly encountered in Boolean network modelling. We establish analytical properties of the models in the class and exploit them to devise a twostep procedure for gene network reconstruction from product concentration and synthesis rate time series. The first step isolates a family of model structures compatible with the data from a set of most relevant biological hypotheses. The second step explores this family and returns a pool of best fitting models along with estimates of their parameters. The method is tested on a simulated network and compared to stateoftheart network inference methods on the benchmark synthetic network IRMA. Contact:
Complex qualitative models in biology: a new approach
 In: European Conference on Complex Systems
, 2005
"... Abstract. We advocate the use of qualitative models in the analysis of large biological systems. We show how qualitative models are linked to theoretical differential models and practical graphical models of biological networks. A new technique for analyzing qualitative models is introduced, which i ..."
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Abstract. We advocate the use of qualitative models in the analysis of large biological systems. We show how qualitative models are linked to theoretical differential models and practical graphical models of biological networks. A new technique for analyzing qualitative models is introduced, which is based on an efficient representation of qualitative systems. As shown through several applications, this representation is a relevant tool for the understanding and testing of large and complex biological networks. 1
Reverse engineering discrete dynamical systems from data sets with random input vectors
 Journal of Computational Biology
, 2006
"... Recently a new algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is based on methods from computational algebra and finds most parsimonious models for a given data set. We derive mathematically rigorous estimates for the expected amount of data n ..."
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Recently a new algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is based on methods from computational algebra and finds most parsimonious models for a given data set. We derive mathematically rigorous estimates for the expected amount of data needed by this algorithm to find the correct model. In particular, we demonstrate that for one type of input parameter (graded term orders), the expected data requirements scale polynomially with the number n of chemicals in the network, while for another type of input parameters (randomly chosen lex orders) this number scales exponentially in n. We also show that for a modification of the algorithm, the expected data requirements scale as the logarithm of n. 1
On minimality and equivalence of Petri nets
 PROCEEDINGS OF CONCURRENCY, SPECIFICATION AND PROGRAMMING CS&P’2012 WORKSHOP, 2:382–393, 2012. PROC. BIOPPN 2013, A SATELLITE EVENT OF PETRI NETS 2013
"... The context of this work is the reconstruction of Petri net models for biological systems from experimental data. Such methods aim at generating all network alternatives fitting the given data. To keep the solution set small while guaranteeing its completeness, the idea is to generate only Petri ne ..."
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The context of this work is the reconstruction of Petri net models for biological systems from experimental data. Such methods aim at generating all network alternatives fitting the given data. To keep the solution set small while guaranteeing its completeness, the idea is to generate only Petri nets being minimal in the sense that all other networks fitting the data contain the reconstructed ones. In this paper, we consider Petri nets with extensions in two directions: priority relations among the transitions of a network in order to allow the modelization of deterministic systems, and controlarcs in order to represent catalytic or inhibitory dependencies. We de ne a containment relation for Petri nets taking both concepts, priority relations and controlarcs, into account. We discuss the consequences for extended Petri nets differing in their sets of controlarcs and priority relations, and the impact of our results towards the reconstruction of such Petri nets.
An algebraic algorithm for the identification of Glass networks with periodic orbits along cyclic attractors
 IN PROCEEDINGS OF ALGEBRAIC BIOLOGY (AB) (LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... Glass piecewise linear ODE models are frequently used for simulation of neural and gene regulatory networks. Efficient computational tools for automatic synthesis of such models are highly desirable. However, the existing algorithms for the identification of desired models are limited to fourdimens ..."
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Glass piecewise linear ODE models are frequently used for simulation of neural and gene regulatory networks. Efficient computational tools for automatic synthesis of such models are highly desirable. However, the existing algorithms for the identification of desired models are limited to fourdimensional networks, and rely on numerical solutions of eigenvalue problems. We suggest a novel algebraic criterion to detect the type of the phase flow along network cyclic attractors that is based on a corollary of the PerronFrobenius theorem. We show an application of the criterion to the analysis of bifurcations in the networks. We propose to encode the identification of models with periodic orbits along cyclic attractors as a propositional formula, and solving it using stateoftheart SATbased tools for real linear arithmetic. New lower bounds for the number of equivalence classes are calculated for cyclic attractors in sixdimensional networks. Experimental results indicate that the runtime of our algorithm increases slower than the size of the search space of the problem.
Computing Gröbner Bases of Ideals of Few Points in High Dimensions
"... A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is pot ..."
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A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Gröbner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Gröbner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Gröbner bases of zerodimensional ideals that is optimized for the case when the number m of points is much smaller than the number n of indeterminates. The algorithm identifies those variables that are essential, that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Gröbner basis in terms of these variables. When n is much larger than m, the complexity is dominated by nm 3. The algorithm has been implemented and tested in the computer algebra system Macaulay 2. We provide a comparison of its performance to the BuchbergerMöller algorithm, as built into the system. Keywords: Gröbner bases, BuchbergerMöller algorithm, essential variables, runtime complexity, computational biology applications
Fixed Points in Discrete Models for Regulatory Genetic Networks
, 2007
"... It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that ..."
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Cited by 3 (0 self)
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It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that enable the detection of steady states that are modeled by fixed points in discrete finite dynamical systems. We discuss two algebraic models, a univariate model and a multivariate model. We show that these two models are equivalent and that one can be converted to the other by means of a discrete Fourier transform. We give a new, more general definition of a linear finite dynamical system and we give a necessary and sufficient condition for such a system to be a fixed point system, that is, all cycles are of length one. We show how this result for generalized linear systems can be used to determine when certain nonlinear systems (monomial dynamical systems over finite fields) are fixed point systems. We also show how it is possible to determine in polynomial time when an ordinary linear system (defined over a finite field) is a fixed point system. We conclude with a necessary condition for a univariate finite dynamical system to be a fixed point system.