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Modeling and simulation of genetic regulatory systems: A literature review
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 2002
"... In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between ..."
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Cited by 738 (14 self)
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In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rulebased formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 114 (8 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
Genetic network analyzer: qualitative simulation of genetic regulatory networks
 Bioinformatics
, 2003
"... Motivation: The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for the development of appropriate computer ..."
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Cited by 93 (14 self)
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Motivation: The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for the development of appropriate computer tools for the modeling and simulation of gene regulation processes. Results: We present Genetic Network Analyzer (GNA), a computer tool for the modeling and simulation of genetic regulatory networks. The tool is based on a qualitative simulation method that employs coarsegrained models of regulatory networks. The use of GNA is illustrated by a case study of the network of genes and interactions regulating the initiation of sporulation in Bacillus subtilis. Availability: GNA and the model of the sporulation network are available at
Validation of qualitative models of genetic regulatory networks by model checking: Analysis of the nutritional stress response in Escherichia coli
 Bioinformatics
, 2005
"... The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their mutual interactions, socalled genetic regulatory networks. In order to gain an understanding of how the behavior of an organism – e.g., the response of a ..."
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Cited by 85 (18 self)
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The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their mutual interactions, socalled genetic regulatory networks. In order to gain an understanding of how the behavior of an organism – e.g., the response of a
Robustness and fragility of boolean models for genetic regulatory networks
 J. Theoretical Biology
, 2005
"... Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene ..."
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Cited by 71 (10 self)
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Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre and posttranslational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.
A Class of Piecewise Linear Differential Equations Arising In Biological Models
, 2003
"... We investigate the properties of the solutions of a class of piecewiselinear differential equations. The equations are appropriate to model biological systems (e.g., genetic networks) in which there are switchlike interactions between the elements. The analysis uses the concept of Filippov solutio ..."
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Cited by 56 (18 self)
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We investigate the properties of the solutions of a class of piecewiselinear differential equations. The equations are appropriate to model biological systems (e.g., genetic networks) in which there are switchlike interactions between the elements. The analysis uses the concept of Filippov solutions of differential equations with a discontinuous righthand side. It gives an insight into the socalled singular solutions which lie on the surfaces of discontinuity. We show that this notion clarifies the study of several examples studied in the literature.
R (2006) Methods of robustness analysis for boolean models of gene control networks
 IET Systems Biology
"... As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network ..."
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Cited by 47 (17 self)
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As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuoustime character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model’s gene pattern prediction as a function of the timescales of the various processes. 1
Piecewiselinear models of genetic regulatory networks: equilibria and . . .
 J. MATH. BIOL.
, 2005
"... A formalism based on piecewiselinear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be wellsuited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutio ..."
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Cited by 46 (20 self)
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A formalism based on piecewiselinear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be wellsuited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.
Model Checking Genetic Regulatory Networks using GNA and CADP
 In: Proceedings of the 11th International SPIN Workshop on Model Checking of Software SPIN’2004
, 2004
"... who are interested in the interdisciplinary methods and applications relevant to the analysis, design and management of complex systems. 15 St. Mary’s St. Brookline MA 02446 l 617.358.1295 l www.bu.edu/systems ..."
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Cited by 45 (6 self)
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who are interested in the interdisciplinary methods and applications relevant to the analysis, design and management of complex systems. 15 St. Mary’s St. Brookline MA 02446 l 617.358.1295 l www.bu.edu/systems
Qualitative simulation of the initiation of sporulation in Bacillus subtilis
 Bulletin of Mathematical Biology
, 2004
"... Under conditions of nutrient deprivation, the Gram positive soil bacterium Bacillus subtilis can abandon vegetative growth and form a dormant, environmentallyresistant spore instead. The decision to either divide or sporulate is controlled by a ∗Corresponding address: Institut National de Recherche ..."
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Cited by 36 (1 self)
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Under conditions of nutrient deprivation, the Gram positive soil bacterium Bacillus subtilis can abandon vegetative growth and form a dormant, environmentallyresistant spore instead. The decision to either divide or sporulate is controlled by a ∗Corresponding address: Institut National de Recherche en Informatique et en Automatique (INRIA), Unite ́ de recherche RhôneAlpes, 655 avenue de l’Europe, Montbonnot, 38334 Saint Ismier