Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph @C(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components. Then, we establish an upper bound on the number of fixed points for F, and more generally on the number of attractors in @C(F), which only depends on X and on the topology of the positive circuits of G(F). This result generalizes the following discrete version of Thomas' conjecture recently proved by Richard and Comet: If G(F) has no positive circuit, then @C(F) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.

[1]  J. Demongeot,et al.  Positive and negative feedback: striking a balance between necessary antagonists. , 2002, Journal of theoretical biology.

[2]  R. Thomas,et al.  Boolean formalization of genetic control circuits. , 1973, Journal of theoretical biology.

[3]  René Thomas,et al.  Logical identification of all steady states: The concept of feedback loop characteristic states , 1993 .

[4]  El Houssine Snoussi.  Necessary Conditions for Multistationarity and Stable Periodicity , 1998 .

[5]  Denis Thieffry,et al.  Segmenting the fly embryo: a logical analysis of the pair-rule cross-regulatory module. , 2003, Journal of theoretical biology.

[6]  Denis Thieffry,et al.  Positive or negative regulatory circuit inference from multilevel dynamics , 2006 .

[7]  Denis Thieffry,et al.  Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework , 2008, Adv. Appl. Math..

[8]  François Robert.  Les systèmes dynamiques discrets , 1995 .

[9]  É. Remy,et al.  From elementary signed circuits to the dynamics of Boolean regulatory networks , 2008 .

[10]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[11]  C. Soulé.  Graphic Requirements for Multistationarity , 2004, Complexus.

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Adrien Richard,et al.  Necessary conditions for multistationarity in discrete dynamical systems , 2007, Discret. Appl. Math..

[14]  D Thieffry,et al.  Toxicity and neuroendocrine regulation of the immune response: a model analysis. , 1996, Journal of theoretical biology.

[15]  René Thomas.  Regulatory networks seen as asynchronous automata: A logical description , 1991 .

[16]  Christophe Soulé,et al.  Mathematical approaches to differentiation and gene regulation. , 2005, Comptes rendus biologies.

[17]  L. Glass,et al.  The logical analysis of continuous, non-linear biochemical control networks. , 1973, Journal of theoretical biology.

[18]  F. Robert.  Dérivée Discrète et Comportement Local d'Une Itération Discrbte , 1983 .

[19]  R. Thomas,et al.  Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior. , 2001, Chaos.

[20]  Eric Goles Ch.,et al.  Positive and negative circuits in discrete neural networks , 2004, IEEE Transactions on Neural Networks.

[21]  J. Aracena.  Maximum Number of Fixed Points in Regulatory Boolean Networks , 2008, Bulletin of mathematical biology.

[22]  Thomas Mestl,et al.  FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS , 1995 .

[23]  El Houssine Snoussi.  Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach , 1989 .

[24]  Adrien Richard,et al.  Application of formal methods to biological regulatory networks: extending Thomas' asynchronous logical approach with temporal logic. , 2004, Journal of theoretical biology.

[25]  H. D. Jong,et al.  Qualitative simulation of genetic regulatory networks using piecewise-linear models , 2004, Bulletin of mathematical biology.

[26]  Denis Thieffry,et al.  Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis , 1999, Bioinform..

[27]  D Thieffry,et al.  GINsim: a software suite for the qualitative modelling, simulation and analysis of regulatory networks. , 2006, Bio Systems.

[28]  Adrien Richard.  Modèle formel pour les réseaux de régulation génétique et influence des circuits de rétroaction , 2006 .

[29]  D. Thieffry,et al.  A logical analysis of the Drosophila gap-gene system. , 2001, Journal of theoretical biology.

[30]  François Robert,et al.  Discrete iterations - a metric study , 1986, Springer series in computational mathematics.

[31]  R. Thomas,et al.  Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits. , 2001, Chaos.

[32]  Elisabeth Remy,et al.  From minimal signed circuits to the dynamics of Boolean regulatory networks , 2008, ECCB.

[33]  R. Thomas,et al.  Laws for the dynamics of regulatory networks. , 1998, The International journal of developmental biology.

[34]  J. Gouzé.  Positive and Negative Circuits in Dynamical Systems , 1998 .

[35]  D. Thieffry,et al.  Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state , 1995 .

[36]  René Thomas.  On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations , 1981 .

[37]  J. Helden,et al.  Establishement of the dorso-ventral pattern during embryonic development of drosophila melanogasater: a logical analysis , 1997, Journal of theoretical biology.

[38]  Andrew P. Sage,et al.  Feedback Loops , 1992, Concise Encyclopedia of Modelling & Simulation.

[39]  Denis Thieffry,et al.  Dynamical behaviour of biological regulatory networks—II. Immunity control in bacteriophage lambda , 1995 .