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Stability analysis for delayed genetic regulatory networks with reaction--diffusion terms

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Abstract

This paper deals with the problem of asymptotic stability for delayed genetic regulatory networks with reaction--diffusion terms. For three types of boundary conditions, delay-dependent criteria are established by the Lyapunov--Krasovskii functional approach, respectively. The obtained results are expressed in terms of linear matrix inequalities. Numerical examples illustrating the effectiveness of the proposed approach are provided.

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References

  1. Cao J, Ren F (2008) Exponential stability of discrete-time genetic regulatory networks with delays. IEEE Trans Neural Netw 19:520–523

    Article  Google Scholar 

  2. Chen L, Aihara K (2002) Stability of genetic regulatory networks with time delay. IEEE Trans Circuits Syst I 49:602–608

    Article  MathSciNet  Google Scholar 

  3. De Jong H (2002) Modelling and simulation of genetic regulatory systems: a literature review. J Comput Biol 9:67–113

    Article  Google Scholar 

  4. Elf J, Doncic A, Ehrenberg M (2003) Mesoscopic reaction-diffusion in intracellular signaling. Proc SPIE 5110:114–124

    Article  Google Scholar 

  5. Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 406:335–338

    Article  Google Scholar 

  6. Elowitz MB, Surette MG, Wolf PE, Stock JB, Leibler S (1999) Protein mobility in the cytoplasm of Escherichia coli. J Bacteriol 181:197–203

    Google Scholar 

  7. Evans LC (1998) Partial differential equations. American Mathematical Society, Providence

    MATH  Google Scholar 

  8. Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403:339–342

    Article  Google Scholar 

  9. Grammaticos B, Carstea AS, Ramani A (2006) On the dynamics of a gene regulatory network. J Phys A Math Gen 39:2965–2971

    Article  MathSciNet  MATH  Google Scholar 

  10. Ji DH, Lee DW, Koo JH, Won SC, Lee SM, Park JH Synchronization of neutral complex dynamical networks with coupling time varying delays. Nonlinear Dyn (in press)

  11. Ji DH, Park JH, Yoo WJ, Won SC, Lee SM (2010) Synchronization criterion for Lur’e type complex dynamical networks with time-varying delay. Phys Lett A 374:1218–1227

    Article  Google Scholar 

  12. Kobayashi T, Chen L, Aihara K (2003) Modeling genetic switches with positive feedback loops. J Theor Biol 221:379–399

    Article  MathSciNet  Google Scholar 

  13. Li C, Chen L, Aihara K (2006) Stability of genetic networks with SUM regulatory logic: Lur’e system and LMI approach. IEEE Trans Circuits Syst I 53:2451–2458

    Article  MathSciNet  Google Scholar 

  14. Li C, Chen L, Aihara K (2006) Synchronization of coupled nonidentical genetic oscillators. Phys Biol 3:37–44

    Article  Google Scholar 

  15. Li J, Zhang F, Yan J (2009) Global exponential stability of nonautonomous neural networks with time-varying delays and reaction-diffusion terms. IEEE Trans Circuits Syst II 233:241–247

    MathSciNet  MATH  Google Scholar 

  16. Liu Z, Peng J (2010) Delay-independent stability of stochastic reaction-diffusion neural networks with Dirichlet boundary conditions. Neural Comput Appl 19:151–158

    Article  Google Scholar 

  17. Lou X, Ye Q, Cui B (2010) Exponential stability of genetic regulatory networks with random delays. Neurocomputing 73:759–769

    Article  Google Scholar 

  18. Lu J (2007) Robust global exponential stability for interval reaction-diffusion hopfield neural networks with distributed delays. IEEE Trans Circuits Syst II 54:1115–1119

    Article  Google Scholar 

  19. Meng TC, Somani S, Dhar P (2004) Modeling and simulation of biological systems with stochasticity. Silico Biol 4:293–309

    Google Scholar 

  20. Park JH, Lee SM, Jung HY (2009) LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks. J Optim Theory Appl 143:357–367

    Article  MathSciNet  MATH  Google Scholar 

  21. Perkins TJ, Jaeger J, Reinitz J, Glass L (2006) Reverse engineering the gap gene network of Drosophila melanogaster. PLoS Comput Biol 2:0417–0428

    Article  Google Scholar 

  22. Qiu Z (2010) The asymptotical behavior of cyclic genetic regulatory networks. Nonlinear Anal Real World Appl 11:1067–1086

    Article  MathSciNet  MATH  Google Scholar 

  23. Ren F, Cao J (2008) Robust stability of genetic regulatory networks with interval time-varying delays. Neurocomputing 71:834–842

    Article  Google Scholar 

  24. Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31:64–68

    Article  Google Scholar 

  25. Smolen P, Baxter DA, Byrne JH (2000) Mathematical modeling of gene networks. Neuron 26:567–580

    Article  Google Scholar 

  26. Stetter M, Deco G, Dejori M (2003) Large-scale computational modeling of genetic regulatory networks. Artific Intell Rev 20:75–93

    Article  Google Scholar 

  27. Sun Y, Feng G, Cao J (2009) Stochastic stability of markovian switching genetic regulatory networks. Phys Lett A 373:1646–1652

    Article  MathSciNet  Google Scholar 

  28. Van Zon JS, Morelli MJ, Tanase-Nicola S, Ten Wolde PR (2006) Diffusion of transcription factors can drastically enhance the noise in gene expression. Biophys J 91:4350–4367

    Article  Google Scholar 

  29. Van Zon JS, Ten Wolde PR (2005) Simulating biochemical networks at the particle level and in time and space: greens function reaction dynamics. Phys Rev Lett 94:1–4

    Article  Google Scholar 

  30. Wang G, Cao J, Liang J (2009) Robust stability of stochastic genetic network with Markovian jumping parameters. Proc Inst Mech Eng Part I J Syst Control Eng 223:797–807

    Article  Google Scholar 

  31. Wang L, Zhang F, Wang Y (2008) Stochastic exponential stability of the delayed reaction-diffusion recurren neural networks with Markovian jumping parameters. Phys Lett A 372:3201–3209

    Article  MathSciNet  Google Scholar 

  32. Wang L, Zhang Y, Zhang Z, Wang Y (2009) LMI-based approach for global exponential robust stability for reaction-diffusion uncertain neural networks with time-varying delay. Chaos Solitons Fractals 41:900–905

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang Z, Gao H, Cao J, Liu X (2008) On delayed genetic regulatory networks with polytypic uncertainties: robust stability analysis. IEEE Trans Nano Biosci 7:154–163

    Google Scholar 

  34. Wang ZS, Zhang H, Li P (2010) An LMI approach to stability analysis of reaction-diffusion Cohen- Grossberg neural networks concerning Dirichlet boundary conditions and distributed delays. IEEE Trans Syst Man Cyber Part B 40:1596–1606

    Article  Google Scholar 

  35. Xu S, Chen T, Lam J (2003) Robust \(H_{\infty}\) filtering for uncertain Markovian jump systems with modedependent time delays. IEEE Trans Automat Cont 48:900–907

    Article  MathSciNet  Google Scholar 

  36. Xu S, Lam J (2007) On equivalence and efficiency of certain stability criteria for time-delay systems. IEEE Trans Automat Cont 52:95–101

    Article  MathSciNet  Google Scholar 

  37. Zhang B, Xu S, Zong G, Zou Y (2009) Delay-dependent exponential stability for uncertain stochastic hopfield neural networks with time-varying delays. IEEE Trans Circuits Syst I 56:1241–1247

    Article  MathSciNet  Google Scholar 

  38. Zhou Q, Xu S, Chen B, Li H, Chu Y (2009) Stability analysis of delayed genetic regulatory networks with stochastic disturbances. Phys Lett A 373:3715–3723

    Article  Google Scholar 

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Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province under Grant BK2008047, NSFC 61074043, and Qing Lan Project.

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Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, 210094, Nanjing, People’s Republic of China

    Qian Ma, Shengyuan Xu & Yun Zou

  2. School of Information Science and Engineering, Changzhou University, 213164, Changzhou, People’s Republic of China

    Guodong Shi

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  1. Qian Ma
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  2. Guodong Shi
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  4. Yun Zou
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Correspondence to Shengyuan Xu.

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Ma, Q., Shi, G., Xu, S. et al. Stability analysis for delayed genetic regulatory networks with reaction--diffusion terms. Neural Comput & Applic 20, 507–516 (2011). https://doi.org/10.1007/s00521-011-0575-9

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