- Email: [email protected]
Dynamical modeling of biological regulatory networks
BioSystems 84 (2006) 77–80
Editorial
Dynamical modeling of biological regulatory networks This special issue of BioSystems has grown out of a workshop on the dynamical modeling of biological regulatory networks held in Montreal in the summer of 2004. Most of the articles in this issue were presented at that workshop, organized in the framework of the Journ´ees Ouvertes de Biologie, Informatique et Math´ematiques (JOBIM), the annual French-speaking bioinformatics conference. While regulatory networks are found on all levels in biology, from the molecular level up to the ecological level, this special issue is mainly concerned with the lowest level of this hierarchy. That is, it focuses on the networks of interactions between molecular components that allow the cells to grow, differentiate, and adapt to their environment. These regulatory networks are basically information-processing systems. On the one hand, they sense and integrate signals indicating the state of the cell and its environment. On the other hand, they control cellular processes influencing the state of the cell and its environment. Genetic regulatory networks are a prime example of such networks, and in fact most (but not all) articles below deal with these networks of genes, proteins, and their mutual interactions. The concerted efforts of genetics, molecular biology, biochemistry, and physiology have led to the accumulation of enormous amounts of data on the molecular components of regulatory networks and their interactions, reported in the literature or stored in databases accessible through the internet. In addition, a variety of high-throughput experimental techniques have become available for studying the structure and dynamics of regulatory networks on a genome-wide scale. However, it is clear that, in addition to powerful experimental tools, the study of the dynamics of biological regulatory networks also requires the support of mathematical and computational tools. Since most networks of interest consist of a large number of molecular components involved in
complex feedback loops, predicting the behavior of the system by intuition alone quickly becomes unfeasible. The use of mathematical models in combination with computer tools allows for an unambiguous description of a network and the systematic prediction of its dynamics. The development of mathematical methods and computer tools for the study of biological regulatory networks has to meet a number of challenges, in relation to specific network features, or to the nature of currently available data. First of all, molecular regulatory networks are large and complex, which means that the scalability of the methods and tools becomes an issue. Second, the components and their interactions are quite heterogeneous, making it far from straightforward to characterize the relation between structure and dynamics of the system. Third, quantitative information on the molecular concentrations and kinetic parameters is mostly absent, while current knowledge of the molecular details of interactions is usually incomplete. Fourth, regulatory networks are often tightly integrated with the systems they control as, for example, in the case of genetic regulatory networks and the metabolism of the cell. Fifth, the questions relevant to the dynamics of regulatory networks are quite diverse, ranging from the stochastic nature of individual interactions to the evolutionary conservation of the network structure, so that a plurality of methods need to be available and applied in combination. Although research aiming at coping with these challenges has become very popular – as witnessed by the exponential increase in publications, specialized conferences and workshops – it is important to bear in mind that the current efforts have much profited from a solid tradition in mathematical and theoretical biology. Indeed, the modeling of metabolic reactions can be traced back to beginning of the 20th century (Michaelis and Menten, 1913), while kinetic models of large metabolic networks
0303-2647/$ – see front matter © 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2005.10.002
78
Editorial / BioSystems 84 (2006) 77–80
have been developed since the late 1960s (e.g., Garfinkel et al., 1970). Regulatory aspects of metabolism have received attention in the theory of metabolic control (for a review, see Heinrich and Schuster, 1996). The dynamic analysis of genetic regulatory networks has been triggered by the work of Jacob and Monod. In particular, in a paper presented at the Cold Spring Harbor Symposium on Quantitative Biology in 1961, Monod and Jacob have tentatively associated simple regulatory structures with specific dynamical properties, generalizing their previous work on the regulation of the lac operon (Monod and Jacob, 1961). Two years later, Goodwin (1963) presented a model of autogenic regulation, which was soon followed by a series of studies on the mathematical characterization of simple positive and negative feedback loops (for a review, see Rosen, 1968). In parallel, Sugita (1963) interpreted gene regulation in logical terms, opening up the road to the development of Boolean and related logical models of genetic regulatory networks by Kauffman (1969), Glass and Kauffman (1973) and Thomas (1973). Contributions to the understanding of the relation between the structure and dynamics of genetic regulatory networks were further provided by Savageau and Thomas, who related specific patterns of gene regulation to biological functions (Savageau, 1974, 1977; Thomas, 1981; Thomas and d’Ari, 1990). The articles in this special issue implicitly or explicitly draw on such early ideas, while carrying them further in several directions by proposing novel methods, computer tools, and applications. The article by Grefenstette and coworkers develops a model for gene regulation in the context of synchronous Boolean networks (Kauffman, 1993). The authors first define an ensemble of regulatory networks based on an explicit, parametric representation of protein–protein and protein–DNA interactions. Given the molecular interactions involved in the regulation of a particular gene, they show how the corresponding Boolean regulation function can be inferred. By randomly sampling the ensemble of regulatory networks, statistics on the network topology and the distribution of Boolean regulation functions can be obtained. An intriguing result of this analysis is the observation that the distribution of Boolean functions is not uniform, but highly biased towards what the authors call monotonic regulatory functions. According to such functions, the regulator of a gene either occurs in inhibitory or activatory complexes, but not in both. Some empirical evidence exists that monotonic regulatory functions are also over-represented in real biological networks. Logical models are also central in the contribution of Larrinaga and coworkers. The authors describe the
Java software suite GINsim, which implements a logical method for the qualitative modeling, simulation, and analysis pioneered by Thomas and d’Ari (1990). In comparison with the Boolean network models used by Grefenstette et al., the logical models introduced by Thomas allow asynchronous transitions and employ multivalued instead of Boolean variables. GINsim assists the user in all stages of the modeling process, from the graph-based specification of the model to the visual interpretation of the simulation results. The tool has been applied to the analysis of a preliminary model of the inter-cellular regulatory network involved in the early development of the Drosophila wing. The simulations of this network, consisting of five genes that interact within and between cells, qualitatively reproduce the wild-type developmental pathway and several mutant phenotypes (for examples of other applications dealing with Drosophila early development, see S´anchez and Thieffry, 2001, 2003). In the next article, Mendoza also employs the logical method of Thomas to model the network of interactions controlling T helper lymphocyte differentiation, from a precursor Th0 into Th1 and Th2 cells. The latter two cell types are involved in the control of the immune responses: cell-mediated and inflammatory immune responses for Th1, humoral immune responses for Th2. Deviations in the responses are known to cause diverse pathologies. The logical model of Mendoza synthesizes published experimental data on almost twenty different molecules and their interactions, involved in signal transduction and gene regulation. The author analyzes the feedback loops and the attractors of the system, and identifies stable states that can be related to the Th0, Th1, and Th2 phenotypes. The differentiation process takes the form of a switch from the Th0 steady state to the Th1 or Th2 steady state, depending on external stimulation. The author further shows how the inclusion of molecular details in the model presented here, as compared to previous minimal models of the immune response, allows the prediction of mutant phenotypes, many of which still await experimental verification. Whereas the previous articles assume the regulatory structure of a genetic network to be given, and compare the results of simulations with experimental data, Perkins and coworkers study the inverse problem. Indeed, these authors are concerned with the inference of genetic regulatory networks from time-series measurements of gene expression obtained with highthroughput functional genomic techniques, such as DNA microarrays. More specifically, comparing Boolean and piecewise-linear differential equation models (originally introduced in (Glass and Kauffman, 1973)), Perkins and
Editorial / BioSystems 84 (2006) 77–80
coworkers provide an estimate of the amount of data required for model inference, and its scaling with the number of regulators per gene (see also Perkins et al., 2004). Interestingly, more precise knowledge on the distribution of the regulation functions, as provided by the analysis of Grefenstette et al., may lower the complexity estimates. Piecewise-linear differential equation models are also used in the article of Ropers et al. Here, they are applied to the analysis of the genetic regulatory network controlling the carbon starvation response in Escherichia coli. Under carbon starvation conditions, E. coli cells abandon their exponential-growth state to enter a more resistant, non-growth state called stationary phase. This growth–phase transition is controlled by a regulatory network whose functioning is still little understood, despite the fact that E. coli is a paradigm of the bacterial world. In order to analyze the carbon starvation response, the authors use the computer tool GNA, which allows the qualitative simulation of genetic regulatory networks described by piecewise-linear differential equation models (de Jong et al., 2003, 2004). The simulations have identified crucial features of the transition between exponential and stationary phase, and have led to new predictions of the qualitative dynamics of the system that are currently under experimental validation. The final article of this special issue by Siegel et al. provides an integrated analysis of genetic regulatory networks and metabolic networks. The authors represent the interactions between mRNA, proteins, and metabolites in a graph-based formalism and model the dynamics of this system in terms of differential equations. Within this formal framework, they aim at predicting qualitative changes in the asymptotic state of the system in response to (external) perturbations, as well as at identifying conflicts with available measurements. To this end, the differential equation model is transformed into a system of qualitative linear equations (of the kind that has been studied in the field of qualitative reasoning (Kuipers, 1994; Trav´e-Massuy`es and Dague, 2003)). The authors then propose algorithms to solve the resulting systems and to identify opposing qualitative influences upon a variable. This approach is illustrated by the construction of a model of the regulation of fatty acid synthesis in liver, analyzed under fasting conditions. Acknowledgements We wish to thank S´ebastien Provencher, co-organizer of the JOBIM satellite workshop in Montreal, as well
79
as David Fogel and Roland Somogyi for editorial support. References de Jong, H., Geiselmann, J., Hernandez, C., Page, M., 2003. Genetic network analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19, 336–344. de Jong, H., Gouz´e, J.-L., Hernandez, C., Page, M., Sari, T., Geiselmann, J., 2004. Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull. Math. Biol. 66, 301– 340. Garfinkel, D., Garfinkel, L., Pring, M., Green, S.B., Chance, B., 1970. Computer applications to biochemical kinetics. Ann. Rev. Biochem. 39, 473–498. Glass, L., Kauffman, S.A., 1973. The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103– 129. Goodwin, B.C., 1963. Temporal Organization in Cells. Academic Press, New York. Heinrich, R., Schuster, S., 1996. The Regulation of Cellular Systems. Chapman & Hall, New York. Kauffman, S.A., 1969. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437– 467. Kauffman, S.A., 1993. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York. Kuipers, B.J., 1994. Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. MIT Press, Cambridge, MA. Michaelis, L., Menten, M., 1913. Die Kinetik der Invertinwirkung. Biochem. Z. 49, 333–369. Monod, J., Jacob, F., 1961. General conclusions: teleonomic mechanisms in cellular metabolism, growth, and differentiation. CSH Symp. Quant. Biol. 26, 389–401. Perkins, T.J., Hallett, M., Glass, L., 2004. Inferring models of gene expression dynamics. J. Theor. Biol. 230, 289–299. Rosen, R., 1968. Recent developments in the theory of control and regulation of cellular processes. Int. Rev. Cytol. 23, 25– 88. S´anchez, L., Thieffry, D., 2001. A logical analysis of the Drosophila gap-gene system. J. Theor. Biol. 211, 115–141. S´anchez, L., Thieffry, D., 2003. Segmenting the fly embryo: a logical analysis of the pair-rule cross-regulatory module. J. Theor. Biol. 224, 517–537. Savageau, M.A., 1974. Genetic regulatory mechanisms and the ecological niche of Escherichia coli. Proc. Natl. Acad. Sci. U.S.A. 71, 2453–2455. Savageau, M.A., 1977. Design of molecular control mechanisms and the demand for gene expression. Proc. Natl. Acad. Sci. U.S.A. 74, 5647–5651. Sugita, M., 1963. Functional analysis of chemical systems in vivo using a logical circuit equivalent. II. The idea of a molecular automation. J. Theor. Biol. 4, 179–192. Thomas, R., 1973. Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585. Thomas, R., 1981. On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Della Dora, J., Demongeot, J., Lacolle, B. (Eds.), Numerical Methods in the Study of Critical Phenomena. Springer–Verlag, Berlin, pp. 180–193.
80
Editorial / BioSystems 84 (2006) 77–80
Thomas, R., d’Ari, R., 1990. Biological Feedback. CRC Press, Boca Raton, FL. Trav´e-Massuy`es, L., Dague, P., 2003. Mod`eles et raisonnements qualitatifs. Herm`es, Paris.
Hidde de Jong Institut National de Recherche en Informatique et en Automatique, Unit´e de recherche Rhˆone-Alpes 655 Avenue de l’Europe, Montbonnot 38334 Saint Ismier Cedex, France E-mail address: [email protected] (H. de Jong)
Claudine Chaouiya Denis Thieffry∗ Institut de Biologie du D´eveloppement de Marseille CNRS/INSERM/Universit´e de la M´editerran´ee, Campus de Luminy, Case 907 13288 Marseille Cedex 9, France ∗ Corresponding author. E-mail addresses: [email protected] (C.Chaouiya) [email protected] (D. Thieffry)
Editorial
Dynamical modeling of biological regulatory networks This special issue of BioSystems has grown out of a workshop on the dynamical modeling of biological regulatory networks held in Montreal in the summer of 2004. Most of the articles in this issue were presented at that workshop, organized in the framework of the Journ´ees Ouvertes de Biologie, Informatique et Math´ematiques (JOBIM), the annual French-speaking bioinformatics conference. While regulatory networks are found on all levels in biology, from the molecular level up to the ecological level, this special issue is mainly concerned with the lowest level of this hierarchy. That is, it focuses on the networks of interactions between molecular components that allow the cells to grow, differentiate, and adapt to their environment. These regulatory networks are basically information-processing systems. On the one hand, they sense and integrate signals indicating the state of the cell and its environment. On the other hand, they control cellular processes influencing the state of the cell and its environment. Genetic regulatory networks are a prime example of such networks, and in fact most (but not all) articles below deal with these networks of genes, proteins, and their mutual interactions. The concerted efforts of genetics, molecular biology, biochemistry, and physiology have led to the accumulation of enormous amounts of data on the molecular components of regulatory networks and their interactions, reported in the literature or stored in databases accessible through the internet. In addition, a variety of high-throughput experimental techniques have become available for studying the structure and dynamics of regulatory networks on a genome-wide scale. However, it is clear that, in addition to powerful experimental tools, the study of the dynamics of biological regulatory networks also requires the support of mathematical and computational tools. Since most networks of interest consist of a large number of molecular components involved in
complex feedback loops, predicting the behavior of the system by intuition alone quickly becomes unfeasible. The use of mathematical models in combination with computer tools allows for an unambiguous description of a network and the systematic prediction of its dynamics. The development of mathematical methods and computer tools for the study of biological regulatory networks has to meet a number of challenges, in relation to specific network features, or to the nature of currently available data. First of all, molecular regulatory networks are large and complex, which means that the scalability of the methods and tools becomes an issue. Second, the components and their interactions are quite heterogeneous, making it far from straightforward to characterize the relation between structure and dynamics of the system. Third, quantitative information on the molecular concentrations and kinetic parameters is mostly absent, while current knowledge of the molecular details of interactions is usually incomplete. Fourth, regulatory networks are often tightly integrated with the systems they control as, for example, in the case of genetic regulatory networks and the metabolism of the cell. Fifth, the questions relevant to the dynamics of regulatory networks are quite diverse, ranging from the stochastic nature of individual interactions to the evolutionary conservation of the network structure, so that a plurality of methods need to be available and applied in combination. Although research aiming at coping with these challenges has become very popular – as witnessed by the exponential increase in publications, specialized conferences and workshops – it is important to bear in mind that the current efforts have much profited from a solid tradition in mathematical and theoretical biology. Indeed, the modeling of metabolic reactions can be traced back to beginning of the 20th century (Michaelis and Menten, 1913), while kinetic models of large metabolic networks
0303-2647/$ – see front matter © 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2005.10.002
78
Editorial / BioSystems 84 (2006) 77–80
have been developed since the late 1960s (e.g., Garfinkel et al., 1970). Regulatory aspects of metabolism have received attention in the theory of metabolic control (for a review, see Heinrich and Schuster, 1996). The dynamic analysis of genetic regulatory networks has been triggered by the work of Jacob and Monod. In particular, in a paper presented at the Cold Spring Harbor Symposium on Quantitative Biology in 1961, Monod and Jacob have tentatively associated simple regulatory structures with specific dynamical properties, generalizing their previous work on the regulation of the lac operon (Monod and Jacob, 1961). Two years later, Goodwin (1963) presented a model of autogenic regulation, which was soon followed by a series of studies on the mathematical characterization of simple positive and negative feedback loops (for a review, see Rosen, 1968). In parallel, Sugita (1963) interpreted gene regulation in logical terms, opening up the road to the development of Boolean and related logical models of genetic regulatory networks by Kauffman (1969), Glass and Kauffman (1973) and Thomas (1973). Contributions to the understanding of the relation between the structure and dynamics of genetic regulatory networks were further provided by Savageau and Thomas, who related specific patterns of gene regulation to biological functions (Savageau, 1974, 1977; Thomas, 1981; Thomas and d’Ari, 1990). The articles in this special issue implicitly or explicitly draw on such early ideas, while carrying them further in several directions by proposing novel methods, computer tools, and applications. The article by Grefenstette and coworkers develops a model for gene regulation in the context of synchronous Boolean networks (Kauffman, 1993). The authors first define an ensemble of regulatory networks based on an explicit, parametric representation of protein–protein and protein–DNA interactions. Given the molecular interactions involved in the regulation of a particular gene, they show how the corresponding Boolean regulation function can be inferred. By randomly sampling the ensemble of regulatory networks, statistics on the network topology and the distribution of Boolean regulation functions can be obtained. An intriguing result of this analysis is the observation that the distribution of Boolean functions is not uniform, but highly biased towards what the authors call monotonic regulatory functions. According to such functions, the regulator of a gene either occurs in inhibitory or activatory complexes, but not in both. Some empirical evidence exists that monotonic regulatory functions are also over-represented in real biological networks. Logical models are also central in the contribution of Larrinaga and coworkers. The authors describe the
Java software suite GINsim, which implements a logical method for the qualitative modeling, simulation, and analysis pioneered by Thomas and d’Ari (1990). In comparison with the Boolean network models used by Grefenstette et al., the logical models introduced by Thomas allow asynchronous transitions and employ multivalued instead of Boolean variables. GINsim assists the user in all stages of the modeling process, from the graph-based specification of the model to the visual interpretation of the simulation results. The tool has been applied to the analysis of a preliminary model of the inter-cellular regulatory network involved in the early development of the Drosophila wing. The simulations of this network, consisting of five genes that interact within and between cells, qualitatively reproduce the wild-type developmental pathway and several mutant phenotypes (for examples of other applications dealing with Drosophila early development, see S´anchez and Thieffry, 2001, 2003). In the next article, Mendoza also employs the logical method of Thomas to model the network of interactions controlling T helper lymphocyte differentiation, from a precursor Th0 into Th1 and Th2 cells. The latter two cell types are involved in the control of the immune responses: cell-mediated and inflammatory immune responses for Th1, humoral immune responses for Th2. Deviations in the responses are known to cause diverse pathologies. The logical model of Mendoza synthesizes published experimental data on almost twenty different molecules and their interactions, involved in signal transduction and gene regulation. The author analyzes the feedback loops and the attractors of the system, and identifies stable states that can be related to the Th0, Th1, and Th2 phenotypes. The differentiation process takes the form of a switch from the Th0 steady state to the Th1 or Th2 steady state, depending on external stimulation. The author further shows how the inclusion of molecular details in the model presented here, as compared to previous minimal models of the immune response, allows the prediction of mutant phenotypes, many of which still await experimental verification. Whereas the previous articles assume the regulatory structure of a genetic network to be given, and compare the results of simulations with experimental data, Perkins and coworkers study the inverse problem. Indeed, these authors are concerned with the inference of genetic regulatory networks from time-series measurements of gene expression obtained with highthroughput functional genomic techniques, such as DNA microarrays. More specifically, comparing Boolean and piecewise-linear differential equation models (originally introduced in (Glass and Kauffman, 1973)), Perkins and
Editorial / BioSystems 84 (2006) 77–80
coworkers provide an estimate of the amount of data required for model inference, and its scaling with the number of regulators per gene (see also Perkins et al., 2004). Interestingly, more precise knowledge on the distribution of the regulation functions, as provided by the analysis of Grefenstette et al., may lower the complexity estimates. Piecewise-linear differential equation models are also used in the article of Ropers et al. Here, they are applied to the analysis of the genetic regulatory network controlling the carbon starvation response in Escherichia coli. Under carbon starvation conditions, E. coli cells abandon their exponential-growth state to enter a more resistant, non-growth state called stationary phase. This growth–phase transition is controlled by a regulatory network whose functioning is still little understood, despite the fact that E. coli is a paradigm of the bacterial world. In order to analyze the carbon starvation response, the authors use the computer tool GNA, which allows the qualitative simulation of genetic regulatory networks described by piecewise-linear differential equation models (de Jong et al., 2003, 2004). The simulations have identified crucial features of the transition between exponential and stationary phase, and have led to new predictions of the qualitative dynamics of the system that are currently under experimental validation. The final article of this special issue by Siegel et al. provides an integrated analysis of genetic regulatory networks and metabolic networks. The authors represent the interactions between mRNA, proteins, and metabolites in a graph-based formalism and model the dynamics of this system in terms of differential equations. Within this formal framework, they aim at predicting qualitative changes in the asymptotic state of the system in response to (external) perturbations, as well as at identifying conflicts with available measurements. To this end, the differential equation model is transformed into a system of qualitative linear equations (of the kind that has been studied in the field of qualitative reasoning (Kuipers, 1994; Trav´e-Massuy`es and Dague, 2003)). The authors then propose algorithms to solve the resulting systems and to identify opposing qualitative influences upon a variable. This approach is illustrated by the construction of a model of the regulation of fatty acid synthesis in liver, analyzed under fasting conditions. Acknowledgements We wish to thank S´ebastien Provencher, co-organizer of the JOBIM satellite workshop in Montreal, as well
79
as David Fogel and Roland Somogyi for editorial support. References de Jong, H., Geiselmann, J., Hernandez, C., Page, M., 2003. Genetic network analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19, 336–344. de Jong, H., Gouz´e, J.-L., Hernandez, C., Page, M., Sari, T., Geiselmann, J., 2004. Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull. Math. Biol. 66, 301– 340. Garfinkel, D., Garfinkel, L., Pring, M., Green, S.B., Chance, B., 1970. Computer applications to biochemical kinetics. Ann. Rev. Biochem. 39, 473–498. Glass, L., Kauffman, S.A., 1973. The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103– 129. Goodwin, B.C., 1963. Temporal Organization in Cells. Academic Press, New York. Heinrich, R., Schuster, S., 1996. The Regulation of Cellular Systems. Chapman & Hall, New York. Kauffman, S.A., 1969. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437– 467. Kauffman, S.A., 1993. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York. Kuipers, B.J., 1994. Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. MIT Press, Cambridge, MA. Michaelis, L., Menten, M., 1913. Die Kinetik der Invertinwirkung. Biochem. Z. 49, 333–369. Monod, J., Jacob, F., 1961. General conclusions: teleonomic mechanisms in cellular metabolism, growth, and differentiation. CSH Symp. Quant. Biol. 26, 389–401. Perkins, T.J., Hallett, M., Glass, L., 2004. Inferring models of gene expression dynamics. J. Theor. Biol. 230, 289–299. Rosen, R., 1968. Recent developments in the theory of control and regulation of cellular processes. Int. Rev. Cytol. 23, 25– 88. S´anchez, L., Thieffry, D., 2001. A logical analysis of the Drosophila gap-gene system. J. Theor. Biol. 211, 115–141. S´anchez, L., Thieffry, D., 2003. Segmenting the fly embryo: a logical analysis of the pair-rule cross-regulatory module. J. Theor. Biol. 224, 517–537. Savageau, M.A., 1974. Genetic regulatory mechanisms and the ecological niche of Escherichia coli. Proc. Natl. Acad. Sci. U.S.A. 71, 2453–2455. Savageau, M.A., 1977. Design of molecular control mechanisms and the demand for gene expression. Proc. Natl. Acad. Sci. U.S.A. 74, 5647–5651. Sugita, M., 1963. Functional analysis of chemical systems in vivo using a logical circuit equivalent. II. The idea of a molecular automation. J. Theor. Biol. 4, 179–192. Thomas, R., 1973. Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585. Thomas, R., 1981. On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Della Dora, J., Demongeot, J., Lacolle, B. (Eds.), Numerical Methods in the Study of Critical Phenomena. Springer–Verlag, Berlin, pp. 180–193.
80
Editorial / BioSystems 84 (2006) 77–80
Thomas, R., d’Ari, R., 1990. Biological Feedback. CRC Press, Boca Raton, FL. Trav´e-Massuy`es, L., Dague, P., 2003. Mod`eles et raisonnements qualitatifs. Herm`es, Paris.
Hidde de Jong Institut National de Recherche en Informatique et en Automatique, Unit´e de recherche Rhˆone-Alpes 655 Avenue de l’Europe, Montbonnot 38334 Saint Ismier Cedex, France E-mail address: [email protected] (H. de Jong)
Claudine Chaouiya Denis Thieffry∗ Institut de Biologie du D´eveloppement de Marseille CNRS/INSERM/Universit´e de la M´editerran´ee, Campus de Luminy, Case 907 13288 Marseille Cedex 9, France ∗ Corresponding author. E-mail addresses: [email protected] (C.Chaouiya) [email protected] (D. Thieffry)