Robust Stability Criteria for Uncertain Genetic Regulatory Networks with Time-Varying Delays

Authors: Wenqin Wang, Shouming Zhong

Abstract:

This paper presents the robust stability criteria for uncertain genetic regulatory networks with time-varying delays. One key point of the criterion is that the decomposition of the matrix ˜D into ˜D = ˜D1 + ˜D2. This decomposition corresponds to a decomposition of the delayed terms into two groups: the stabilizing ones and the destabilizing ones. This technique enables one to take the stabilizing effect of part of the delayed terms into account. Meanwhile, by choosing an appropriate new Lyapunov functional, a new delay-dependent stability criteria is obtained and formulated in terms of linear matrix inequalities (LMIs). Finally, numerical examples are presented to illustrate the effectiveness of the theoretical results.

Keywords: Genetic regulatory network, Time-varying delay, Uncertain system, Lyapunov-Krasovskii functional

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079200

References:

[1] S. Huang, Gene expression profilling genetic networks and cellular states: an integrating concept for tumorigenesis and drug discovery, J. Mol. Med. 77 (1999) 469.
[2] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol. 22 (3) (1969) 437.
[3] R. Somogyi, C. Sniegoski, Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation, Complexity (1996) 45-63.
[4] R. Thomas, Boolean formalization of genetic control circuits, J. Theor. Biol. 42 (3) (1973) 563.
[5] X. Lou, Q. Ye, B. Cui, Exponential stability of genetic regulatory networks with random delays, Neurocomputing, 73 (2010), 759-769
[6] F. Ren, J. Cao, Asymptotic and robust stability of genetic regulatory networks with time-varying delays, Neurocomputing, 71 (2008), 834- 842
[7] M. de Hoon, S. Imoto, K. Kobayashi, N. Ogasawara, S. Miyano, Infering gene regulatory networks from time-ordered gene expression data of bacillus subtilis using differential equations, in: Proc. Pacific Symposium on Biocomputing, vol. 8, pp. 17-28, 2003.
[8] L. Chen, K. Aihara, Stability of genetic regulatory networks with time delay, IEEE Transactions on Circuits and Systems I 49 (2002)602-608.
[9] H. Iba, A. Mimura, Inference of a gene regulatory network by means of interactive evolutionary computing, Information Sciences 145 (2002)225-236
[10] T. Tian, K. Burragea, P.M. Burragea, M. Carlettib, Stochastic delay differential equations for genetic regulatory networks, Journal of Computational and Applied Mathematics 205 (2007) 696-707
[11] H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa, R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science 298 (2002)840-843.
[12] Y.He, Q.Wang, C.Lin, M.Wu, Delay-range-dependent stability for systems with time-varying delay. Automatica, 43,(2007). 371-376.
[13] L.Wu, C.Wang, Q.Zeng, Observer-based sliding mode control for a class of uncertain nonlinear neutral delay systems, J.Franklin Inst.345 (2008) 233-253.
[14] X. Song, S. Xu, H. Shen, Robust H1 control for uncertain fuzzy systems with distributed delays via output feedback controllers, Information Sciences 178 (2008) 4341-4356.
[15] Tomioka R, Kimurab H, Kobayashib TJ, Aihara K. Multivariate analysis of noise in genetic regulatory networks. J Theor Biol (2004) 229,501-21.
[16] C.-H. Yuh, H. Bolouri, and E. H. Davidson, Genomic cis-regulatory logic: Experimental and computational analysis of a sea urchin gene, Science, 279, (1998). 1896-1902.
[17] N. E. Buchler, U. Gerland, and T. Hwa, On schemes of combinatorial transcription logic, Proc. Natl. Acas. Sci. USA, 100, (2003). 5136-5141 .
[18] Y. Setty, A. E. Mayo, M. G. Surette, and U. Alon, Detailed map of a cis-regulatory input function, Proc. Natl. Acad. Sci. USA, 100,(2003). 7702-7707.
[19] Y. Wang, J. Shen, B. Niu, Z. Liu, L. Chen, Robustness of interval gene networks with multiple time-varying delays and noise, Neurocomputing, 72 (2009), 3303-3310
[20] L. Li, X. Liu, New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays, Information Sciences 179 (2009) 1134-1148.
[21] X. Song, S. Xu, H. Shen, Robust H∞ control for uncertain fuzzy systems with distributed delays via output feedback controllers, Information Sciences 178 (2008) 4341-4356.
[22] G. Wang, J. Cao, Robust exponential stability analysis for stochastic genetic networks with uncertain parameters, Commun Nonlinear Sci Numer Simulat 14 (2009) 3369-3378
[23] C. Li, L. Chen, K. Aihara, Stability of genetic networks with SUM regulatory logic: Lure system and LMI approach, IEEE Transactions on Circuits and Systems I 53 (2006) 2451-2458.
[24] H. Wu, X. Liao, W. Feng , S. Guo , W. Zhang, Robust stability for uncertain genetic regulatory networks with interval time-varying delays,Information Sciences 180 (2010) 3532-3545
[25] L. Xie, Output feedback H∞ control of systems with parameter uncertainty, International Journal of Control, Vol. 63,No.4,741-750,1996.
[26] P. Park, J. W. Ko, C. K. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (2011) 235-238.
[27] D. Yang, C. Hua, Y. Chen, P. Wei, H. Yang, New delay-dependent global asymptotic stability criteria of delayed BAM neural networks, Chaos, Solitons and Fractals 42 (2009) 854-864.
[28] C. Shen, S. Zhong, New delay-dependent robust stability criterion for uncertain neutral systems with time-varying delay and nonlinear uncertainties, Chaos, Solitons and Fractals 40 (2009) 2277-2285.