On the stability of hybrid limit cycles and isolated equilibria in a genetic network with binary hysteresis

A mathematical model for a two-gene regulatory network is derived and several of its properties are analyzed. Due to the presence of continuous dynamics and binary hysteresis, we propose a hybrid system model. Binary hysteresis with different thresholds captures the interaction between the genes. We analyze properties of the solutions and asymptotic stability of equilibria in the system as a function of its parameters. Our analysis reveals the presence of limit cycles for a certain range of parameters, a behavior that is associated with the presence of binary hysteresis. The set of points defining the limit cycle is characterized and its asymptotic stability properties are studied. Numerical simulations are presented to illustrate some of the results.

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

[2]  T. Mestl,et al.  A mathematical framework for describing and analysing gene regulatory networks. , 1995, Journal of theoretical biology.

[3]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[4]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[5]  Sui Huang.  Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery , 1999, Journal of Molecular Medicine.

[6]  Madalena Chaves,et al.  Robustness and fragility of Boolean models for genetic regulatory networks. , 2005, Journal of theoretical biology.

[7]  J. Gouzé,et al.  A class of piecewise linear differential equations arising in biological models , 2002 .

[8]  Martin Fussenegger,et al.  Hysteresis in a synthetic mammalian gene network. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Jayajit Das,et al.  Digital Signaling and Hysteresis Characterize Ras Activation in Lymphoid Cells , 2009, Cell.

[10]  Jean-Luc Gouzé,et al.  Hybrid Modeling and Simulation of Genetic Regulatory Networks: A Qualitative Approach , 2003, HSCC.

[11]  Claire J. Tomlin,et al.  Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model , 2001, HSCC.

[12]  J. Hu,et al.  Modeling of Hysteresis in a Mammalian Gene Regulatory Network , 2010 .

[13]  Ashish Tiwari,et al.  Symbolic Systems Biology: Hybrid Modeling and Analysis of Biological Networks , 2004, HSCC.

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

[15]  Liang Qiao,et al.  Bistability and Oscillations in the Huang-Ferrell Model of MAPK Signaling , 2007, PLoS Comput. Biol..

[16]  Nancy Kopell,et al.  Synchrony in a Population of Hysteresis-Based Genetic Oscillators , 2004, SIAM J. Appl. Math..

[17]  M Chaves,et al.  Methods of robustness analysis for Boolean models of gene control networks. , 2006, Systems biology.

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