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Balasubramaniam, P.; Rakkiyappan, R.; Krishnasamy, R.

Stochastic stability of Markovian jumping uncertain stochastic genetic regulatory networks with interval time-varying delays. (English) Zbl 1194.92030

Math. Biosci. 226, No. 2, 97-108 (2010).
Summary: This paper investigates the robust stability problem of stochastic genetic regulatory networks with interval time-varying delays and Markovian jumping parameters. The structure variations at discrete time instances during the process of gene regulations, known as hybrid genetic regulatory networks based on Markov processes, are proposed. The jumping parameters considered are generated from a continuous-time discrete-state homogeneous Markov process, which is governed by a Markov process with discrete and finite state space. A new type of Markovian jumping matrices \(P_i\) and \(Q_i\) is introduced in this paper. The parameter uncertainties are assumed to be norm bounded and the discrete delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are unavoidable. Based on the Lyapunov-Krasovskii functional and stochastic stability theory, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities. Some numerical examples are given to illustrate the effectiveness of our theoretical results.

Cited in 18 Documents

MSC:

92C42 Systems biology, networks
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
15A45 Miscellaneous inequalities involving matrices
60J25 Continuous-time Markov processes on general state spaces

Keywords:

asymptotic stability; genetic regulatory networks; linear matrix inequality; Lyapunov-Krasovskii functional; Markovian jumping parameters; uncertain systems
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\textit{P. Balasubramaniam} et al., Math. Biosci. 226, No. 2, 97--108 (2010; Zbl 1194.92030)
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References:

[1] Bower, U.; Balouri, H., Computational Modelling of Genetic and Biochemical Networks (2001), MIT: MIT Cambridge, MA
[2] Davidson, E., Genomic Regulatory Systems (2001), Academic Press: Academic Press San Diego, CA
[3] Kitano, H., Foundations of Systems Biology (2001), MIT: MIT Cambridge, MA
[4] Hood, L.; Galas, D., The digital code of DNA, Nature, 421, 444 (2003)
[5] Becskei, A.; Serrano, L., Engineering stability in gene networks by autoregulation, Nature, 405, 590 (2000)
[6] Chen, L.; Aihara, K., Stability of genetic regulatory networks with time delay, IEEE Trans. Circuits Syst. I, 49, 602 (2002) · Zbl 1368.92117
[7] Gardner, T.; Cantor, C.; Collins, J., Construction of a genetic toggle switch in Escherichia coli, Nature, 403, 339 (2000)
[8] Li, C.; Chen, L.; Aihara, K., Synchronization of coupled nonidentical genetic oscillators, Phys. Biol., 3, 37 (2006)
[9] Li, C.; Chen, L.; Aihara, K., Stability of genetic networks with sum regulatory logic: Lur’s system and LMI approach, IEEE Trans. Circuits Syst. I, 53, 2451 (2006) · Zbl 1374.92045
[10] Wang, Y.; Ma, Z.; Shen, J.; Liu, Z.; Chen, L., Periodic oscillation in delayed gene networks with SUM regulatory logic and small perturbations, Math. Biosci., 220, 34 (2009) · Zbl 1169.92020
[11] Ren, F.; Cao, J., Asymptotic and robust stability of genetic regulatory networks with time-varying delays, Neurocomputing, 71, 834 (2008)
[12] Wu, M.; Liu, F.; Shi, P.; He, Y.; Yokoyama, R., Exponential stability analysis for neural networks with time-varying delay, IEEE Trans. Syst. Man Cybern. B, 38, 1152 (2008)
[13] Mou, S.; Gao, H.; Qiang, W.; Chen, K., New delay-dependent exponential stability for neural networks with time delay, IEEE Trans. Syst. Man Cybern. B, 38, 571 (2008)
[14] Wang, Z.; Gao, H.; Cao, J.; Liu, X., On delayed genetic regulatory networks with polytopic uncertainties: robust stability analysis, IEEE Trans. Nanobiosci., 7, 154 (2008)
[15] Hirata, H.; Yoshiura, S.; Ohtsuka, T.; Bessho, Y.; Harada, T.; Yoshikawa, K.; Kageyama, R., Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science, 298, 840 (2002)
[16] Lewis, J., Autoinhibition with transcriptional delay: a simple mechanism for the zebra fish somitogenesis oscillator, Curr. Biol., 13, 1398 (2003)
[17] Kauffman, S. A., Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol., 22, 437 (1969)
[18] Thomas, R., Boolean formalization of genetic control circuits, J. Theor. Biol., 42, 3, 563 (1973)
[19] Chen, T.; He, H.; Church, G., Modelling gene expression with differential equations, Proc. Pacific Sympos. Biocomput., 4, 29 (1999)
[20] D’haeseleer, P.; Wen, X.; Fuhrman, S.; Somogyi, R., Linear modelling of mRNA expression levels during CNS development and injury, Proc. Pacific Sympos. Biocomput., 4, 41 (1999)
[21] de Hoon, M.; Imoto, S.; Kobayashi, K.; Ogasawara, N.; Miyano, S., Infering gene regulatory networks from time-ordered gene expression data of bacillus subtilis using differential equations, Proc. Pacific Sympos. Biocomput., 8, 17 (2003) · Zbl 1219.92032
[22] Monk, N. A.M., Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays, Curr. Biol., 13, 1409 (2003)
[23] Smolen, P.; Baxter, D.; Byrne, J., Mathematical modelling of gene networks, Neuron, 26, 567 (2000)
[24] Smolen, P.; Baxter, D.; Byrne, J., Modelling circadian oscillators with interlocking positive and negative feedback loops, J. Neurosci., 21, 6644 (2001)
[25] Tian, T.; Burragea, K.; Burragea, P. M.; Carlettib, M., Stochastic delay differential equations for genetic regulatory networks, J. Comput. Appl. Math., 205, 696 (2007)
[26] McAdams, H. H.; Arkin, A., Stochastic mechanisms in gene expression, Proc. Nat. Acad. Sci. USA, 94, 814 (1997)
[27] Hasty, J.; Pradlines, J.; Dolnik, M.; Collins, J. J., Noise-based switches and amplifiers for gene expression, Proc. Nat. Acad. Sci. USA, 97, 2075 (2000)
[28] Raser, J.; O’Shea, E., Noise in gene expression: origins, consequences, and control, Science, 309, 2010 (2005)
[29] Elowitz, M.; Levine, A.; Siggia, E.; Swain, P., Stochastic gene expression in a single cell, Science, 297, 1183 (2002)
[30] Kim, S., Can Markov chain models mimic biological regulation?, J. Biol. Syst., 10, 337 (2002) · Zbl 1113.92309
[31] Isaacs, F. J.; Hasty, J.; Cantor, C. R.; Collins, J. J., Prediction and measurement of an autoregulatory genetic module, Proc. Nat. Acad. Sci. USA, 100, 7714 (2003)
[33] Li, C.; Chen, L.; Aihara, K., Stochastic stability of genetic networks with disturbance attenuation, IEEE Trans. Circuits Syst. I, 54, 892 (2007)
[36] Wei, G.; Wang, Z.; Lam, J.; Fraser, K.; Prasada Rao, G.; Liu, X., Robust filtering for stochastic genetic regulatory networks with time-varying delay, Math. Biosci., 220, 73 (2009) · Zbl 1168.92020
[37] Chen, B.-S.; Wu, W.-S., Robust filtering circuit design for stochastic gene networks under intrinsic and extrinsic molecular noises, Math. Biosci., 211, 342 (2008) · Zbl 1130.92023
[38] Sun, Y.; Feng, G.; Cao, J., Stochastic stability of Markovian switching genetic regulatory networks, Phys. Lett. A, 373, 1646 (2009) · Zbl 1229.92041
[39] Bolouri, J.; Davidson, H., Modelling transcriptional regulatory networks, BioEssay, 24, 1118 (2002)
[40] Jong, H. D., Modelling and simulation of genetic regulatory systems: a literature review, J. Comput. Biol., 9, 67 (2002)
[41] Kalir, S.; Mangan, S.; Alon, U., A coherent feed-forward loop with a SUM input function prolongs flagells expression in Escherichia coli, Mol. Syst. Biol. (2005)
[42] Yuh, C. H.; Bolouri, H.; Davidson, E. H., Genomic cis-regulatory logic: experimental and computational analysis of a sea urchin gene, Science, 279, 1896 (1998)
[43] Elowitz, M. B.; Leibler, S., A synthetic oscillatory network of transcriptional regulators, Nature, 403, 335 (2000)
[44] Cao, Y.; Lam, J., Robust \(H_∞\) control of uncertain Markovian jump systems with time-delay, IEEE Trans. Automat. Control, 45, 77 (2000) · Zbl 0983.93075
[45] Wu, J.; Chen, T.; Wang, L., Delay-dependent robust stability and \(H_∞\) control for jump linear systems with delays, Syst. Control Lett., 55, 939 (2006) · Zbl 1117.93072
[46] Shu, Z.; Lam, J.; Xu, S., Robust stabilization of Markovian delay systems with delay-dependent exponential estimates, Automatica, 42, 2001 (2006) · Zbl 1113.60079
[47] Chen, W.; Lu, X., Mean square exponential stability of uncertain stochastic delayed neural networks, Phys. Lett. A, 372, 1061 (2008) · Zbl 1217.92005
[48] Arnold, L., Stochastic Differential Equations: Theory and Appplicatons (1974), Wiley: Wiley London
[49] Chen, W.; Lu, X., Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach, Syst. Control Lett., 54, 547 (2005) · Zbl 1129.93547
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