Robust stability of stochastic genetic network with Markovian jumping parameters

Abstract In this paper, the robust exponential stability problem is investigated for a class of Markovian jumping genetic networks which involve both uncertain parameters and stochastic disturbances. Under the assumption that the jumping parameters are generated from a continuous-time discrete-state homogeneous Markov process, the stability problem is first studied for a deterministic genetic model. By constructing suitable Lyapunov functionals and conducting some stochastic analysis, the stability criteria are derived in the form of linear matrix inequalities (LMIs), which can be easily checked in practice. Then, based on the derived results, sufficient LMI conditions are obtained explicitly for an indeterministic genetic system where the parameter uncertainties are norm-bounded. An illustrative example is presented to demonstrate the effectiveness and usefulness of the proposed stability criteria.

[1]  Jinde Cao,et al.  Robust stability of genetic regulatory networks with distributed delay , 2008, Cognitive Neurodynamics.

[2]  Xuerong Mao,et al.  Exponential stability of stochastic delay interval systems with Markovian switching , 2002, IEEE Trans. Autom. Control..

[3]  Kazuyuki Aihara,et al.  Stochastic Stability of Genetic Networks With Disturbance Attenuation , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[4]  Yeung Sam Hung,et al.  Stability analysis of uncertain genetic sum regulatory networks , 2008, Autom..

[5]  K. Aihara,et al.  Stability of genetic regulatory networks with time delay , 2002 .

[6]  S Zeiser,et al.  Simulation of genetic networks modelled by piecewise deterministic Markov processes. , 2008, IET systems biology.

[7]  Kazuyuki Aihara,et al.  Multivariate analysis of noise in genetic regulatory networks. , 2004, Journal of theoretical biology.

[8]  X. Mao,et al.  Robust stability of uncertain stochastic differential delay equations , 1998 .

[9]  K. Burrage,et al.  Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage lambda. , 2004, Journal of theoretical biology.

[10]  Zidong Wang,et al.  Exponential stability of delayed recurrent neural networks with Markovian jumping parameters , 2006 .

[11]  Jinde Cao,et al.  Asymptotic and robust stability of genetic regulatory networks with time-varying delays , 2008, Neurocomputing.

[12]  Lihua Xie,et al.  H/sub infinity / control and quadratic stabilization of systems with parameter uncertainty via output feedback , 1992 .

[13]  A. Ninfa,et al.  Development of Genetic Circuitry Exhibiting Toggle Switch or Oscillatory Behavior in Escherichia coli , 2003, Cell.

[14]  Ofer Biham,et al.  Stochastic simulations of genetic switch systems. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Kazuyuki Aihara,et al.  Stability of Genetic Networks With SUM Regulatory Logic: Lur'e System and LMI Approach , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[16]  Zidong Wang,et al.  Exponential stability of uncertain stochastic neural networks with mixed time-delays , 2007 .

[17]  K. Burrage,et al.  Stochastic delay differential equations for genetic regulatory networks , 2007 .

[18]  Jinde Cao,et al.  On Delayed Genetic Regulatory Networks With Polytopic Uncertainties: Robust Stability Analysis , 2008, IEEE Transactions on NanoBioscience.

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

[20]  P. Swain,et al.  Stochastic Gene Expression in a Single Cell , 2002, Science.

[21]  Jinde Cao,et al.  Robust exponential stability analysis for stochastic genetic networks with uncertain parameters , 2009 .

[22]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[23]  Jinde Cao,et al.  Global asymptotic and robust stability of recurrent neural networks with time delays , 2005, IEEE Trans. Circuits Syst. I Regul. Pap..

[24]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[25]  M. Lewis,et al.  The lac repressor. , 2005, Comptes rendus biologies.

[26]  J. Paulsson.  Summing up the noise in gene networks , 2004, Nature.

[27]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[28]  Ting Chen,et al.  Modeling Gene Expression with Differential Equations , 1998, Pacific Symposium on Biocomputing.

[29]  Jinde Cao,et al.  Exponential Stability of Discrete-Time Genetic Regulatory Networks With Delays , 2008, IEEE Transactions on Neural Networks.