## Stability of systems with stochastic delays and applications to genetic regulatory networks.(English)Zbl 1353.37106

### MSC:

 37H10 Generation, random and stochastic difference and differential equations 92C42 Systems biology, networks

### Keywords:

stability; stochastic delay; genetic networks

### Software:

Knut; PDDE-CONT; DDE-BIFTOOL
Full Text:

### References:

 [1] R. F. V. Anderson, The relative variance criterion for stability of delay systems, J. Dynam. Differential Equations, 5 (1993), pp. 105–128, https://doi.org/10.1007/BF01063737. · Zbl 0770.34054 [2] A. Arkin, J. Ross, and H. H. McAdams, Stochastic kinetic analysis of developmental pathway bifurcation in Phage $$λ$$-infected Escherichia coli cells, Genetics, 149 (1998), pp. 1633–1648. [3] L. Arnold, H. Crauel, and V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), pp. 451–461, https://doi.org/10.1137/0321027. · Zbl 0514.93069 [4] G. K. Basak, Stabilization of dynamical systems by adding a colored noise, IEEE Trans. Automat. Control, 46 (2001), pp. 1107–1111, https://doi.org/10.1109/9.935065. · Zbl 1005.93047 [5] M. B. G. Cloosterman, N. van de Wouw, W. P. M. H. Heemels, and H. Nijmeijer, Stability of networked control systems with uncertain time-varying delays, IEEE Trans. Automat. Control, 54 (2009), pp. 1575–1580, https://doi.org/10.1109/TAC.2009.2015543. · Zbl 1367.93459 [6] D. Del Vecchio and R. M. Murray, Biomolecular Feedback Systems, Princeton University Press, Princeton, NJ, 2014. · Zbl 1300.92001 [7] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Appl. Math. Sci. 110, Springer, New York, 1995. · Zbl 0826.34002 [8] K. Engelborghs, T. Luzyanina, and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-biftool, ACM Trans. Math. Software, 28 (2002), pp. 1–21, https://doi.org/10.1145/513001.513002. · Zbl 1070.65556 [9] T. Erneux, Applied Delay Differential Equations, Surv. Tutor. Appl. Math. Sci. 3, Springer, New York, 2009. · Zbl 1201.34002 [10] M. Farkas and G. Stépán, On perturbation of the kernel in infinite delay systems, Z. Angew. Math. Mech., 72 (1992), pp. 153–156, https://doi.org/10.1002/zamm.19920720216. [11] R. M. Feldman and C. V. Flores, Applied Probability and Stochastic Processes, Springer, Berlin, Heidelberg, 2010. · Zbl 1201.60001 [12] H. Gao and T. Chen, New results on stability of discrete-time systems with time-varying state delay, IEEE Trans. Automat. Control, 52 (2007), pp. 328–334, https://doi.org/10.1109/TAC.2006.890320. · Zbl 1366.39011 [13] M. Ghasemi, S. Zhao, T. Insperger, and T. Kalmár-Nagy, Act-and-wait control of discrete systems with random delays, in Proceedings of the American Control Conference, 2012, pp. 5440–5443. [14] D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), pp. 2340–2361, https://doi.org/10.1021/j100540a008. [15] C. Gupta, J. M. López, R. Azencott, M. R. Bennett, K. Josić, and W. Ott, Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations, J. Chem. Phys., 140 (2014), 204108, https://doi.org/10.1063/1.4878662. [16] F. Hartung, T. Insperger, G. Stépán, and J. Turi, Approximate stability charts for milling processes using semi-discretization, Appl. Math. Comput., 174 (2006), pp. 51–73, https://doi.org/10.1016/j.amc.2005.05.008. · Zbl 1168.65364 [17] T. Insperger and G. Stépán, Semi-discretization method for delayed systems, Internat. J. Numer. Methods Engrg., 55 (2002), pp. 503–518, https://doi.org/10.1002/nme.505. · Zbl 1032.34071 [18] T. Insperger and G. Stépán, Stability analysis of turning with periodic spindle speed modulation via semidiscretization, J. Vib. Control, 10 (2004), pp. 1835–1855, https://doi.org/10.1177/1077546304044891. · Zbl 1093.70010 [19] T. Insperger and G. Stépán, Semi-discretization for Time-Delay Systems: Stability and Engineering Applications, Appl. Math. Sci. 178, Springer, New York, 2011. [20] T. Insperger, G. Stépán, and Turi, State-dependent delay in regenerative turning processes, Nonlinear Dynam., 47 (2007), pp. 275–283, https://doi.org/10.1007/s11071-006-9068-2. · Zbl 1177.74197 [21] K. Josić, J. M. López, W. Ott, L. Shiau, and M. R. Bennett, Stochastic delay accelerates signaling in gene networks, PLoS Comput. Biol., 7 (2011), e1002264, https://doi.org/10.1371/journal.pcbi.1002264. [22] I. Kats, On the stability of systems with random delay in the first approximation, Prikl. Mat. Meh., 31 (1967), pp. 447–452 (in Russian); J. Appl. Math. Mech., 31 (1967), pp. 478–482 (in English). [23] S. Klumpp, A superresolution census of RNA polymerase, Biophys. J., 105 (2013), pp. 2613–2614, https://doi.org/10.1016/j.bpj.2013.11.018. [24] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Math. Sci. Eng. 180, Academic Press, London, 1986. [25] I. Kolmanovsky and T. L. Maizenberg, Stochastic stability of a class of nonlinear systems with randomly varying time-delay, in Proceedings of the American Control Conference, 2000, pp. 4304–4308. [26] H. J. Kushner, Introduction to Stochastic Control, Holt, Rinehart and Winston, New York, Montreal, London, 1971. · Zbl 0293.93018 [27] A. A. Kwiecińska, Stabilization of partial differential equations by noise, Stochastic Process. Appl., 79 (1999), pp. 179–184, https://doi.org/10.1016/S0304-4149(98)00080-5. [28] J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol., 13 (2003), pp. 1398–1408, https://doi.org/10.1016/S0960-9822(03)00534-7. [29] E. A. Lidskii, On stability of motion of a system with random delays, Differential$$'$$nye Uravnenia, 1 (1965), pp. 96–101 (in Russian). [30] X. Liu and J. Shen, Stability theory of hybrid dynamical systems with time delay, IEEE Trans. Automat. Control, 51 (2006), pp. 620–625, https://doi.org/10.1109/TAC.2006.872751. · Zbl 1366.93441 [31] D. M. Longo, J. Selimkhanov, J. D. Kearns, J. Hasty, A. Hoffmann, and L. S. Tsimring, Dual delayed feedback provides sensitivity and robustness to the nf-$$κ$$b signaling module, PLoS Comput. Biol., 9 (2013), e1003112, https://doi.org/10.1371/journal.pcbi.1003112. [32] M. C. Mackey, A. Longtin, and A. Lasota, Noise-induced global asymptotic stability, J. Statist. Phys., 60 (1990), pp. 735–751, https://doi.org/10.1007/BF01025992. · Zbl 1086.34534 [33] J. Mallet-Paret and H. L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), pp. 367–421, https://doi.org/10.1007/BF01054041. · Zbl 0712.34060 [34] X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Automat. Control, 47 (2002), pp. 1604–1612, https://doi.org/10.1109/TAC.2002.803529. · Zbl 1364.93685 [35] W. Mather, M. R. Bennett, J. Hasty, and L. S. Tsimring, Delay-induced degrade-and-fire oscillations in small genetic circuits, Phys. Rev. Lett., 102 (2009), 068105, https://doi.org/10.1103/PhysRevLett.102.068105. [36] H. H. McAdams and A. Arkin, It’s a noisy business! Genetic regulation at the nanomolar scale, Trends Genet., 15 (1999), pp. 65–69, https://doi.org/10.1016/S0168-9525(98)01659-X. [37] W. Michiels, V. Van Assche, and S.-I. Niculescu, Stabilization of time-delay systems with a controlled time-varying delay, IEEE Trans. Automat. Control, 50 (2005), pp. 493–504, https://doi.org/10.1109/TAC.2005.844723. · Zbl 1365.93411 [38] N. A. Monk, Oscillatory expression of Hes1, p53, and NF-$$κ$$b driven by transcriptional time delays, Curr. Biol., 13 (2003), pp. 1409–1413, https://doi.org/10.1016/S0960-9822(03)00494-9. [39] A. N. Naganathan and V. Mun͂oz, Scaling of folding times with protein size, J. Am. Chem. Soc., 127 (2005), pp. 480–481, https://doi.org/10.1021/ja044449u. [40] G. Orosz, J. Moehlis, and R. M. Murray, Controlling biological networks by time-delayed signals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368 (2010), pp. 439–454, https://doi.org/10.1098/rsta.2009.0242. · Zbl 1205.34109 [41] P. G. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automat. Control, 4 (1999), pp. 876–877. · Zbl 0957.34069 [42] X. Pu and H. Zhao, Stability of a kind of hybrid systems with time delay, in Advances in Electronic Engineering, Communication and Management, Lect. Notes Electr. Eng. 139, Springer, Berlin, Heidelberg, 2012, pp. 159–165, https://doi.org/10.1007/978-3-642-27287-5_26. [43] W. B. Qin, M. M. Gomez, and G. Orosz, Stability and frequency response under stochastic communication delays with applications to connected cruise control design, IEEE Trans. Intell. Transp. Syst., (2016), https://doi.org/10.1109/TITS.2016.2574246. [44] D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., B. Krauskopf, H. M. Osinga, and J. Galan-Vioque, eds., Springer, Dordrecht, 2007, pp. 359–399, https://doi.org/10.1007/978-1-4020-6356-5_12. · Zbl 1132.34001 [45] H. Smith, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts Appl. Math. 57, Springer, New York, 2011. · Zbl 1227.34001 [46] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Pitman Res. Notes Math. Ser. 210, Longman, Harlow/John Wiley and Sons, New York, 1989. [47] J. Stricker, S. Cookson, M. R. Bennett, W. H. Mather, L. S. Tsimring, and J. Hasty, A fast, robust and tunable synthetic gene oscillator, Nature, 456 (2008), pp. 516–519, https://doi.org/10.1038/nature07389. [48] T. Tian, K. Burrage, P. M. Burrage, and M. Carletti, Stochastic delay differential equations for genetic regulatory networks, J. Comput. Appl. Math., 205 (2007), pp. 696–707, https://doi.org/10.1016/j.cam.2006.02.063. · Zbl 1112.92029 [49] M. Tigges, T. T. Marquez-Lago, J. Stelling, and M. Fussenegger, A tunable synthetic mammalian oscillator, Nature, 457 (2009), pp. 309–312, https://doi.org/10.1038/nature07616. [50] O. S. Venturelli, H. El-Samad, and R. M. Murray, Synergistic dual positive feedback loops established by molecular sequestration generate robust bimodal response, Proc. Natl. Acad. Sci. USA, 109 (2012), pp. E3324–E3333, https://doi.org/10.1073/pnas.1211902109. [51] E. I. Verriest and W. Michiels, Stability analysis of systems with stochastically varying delays, Systems Control Lett., 58 (2009), pp. 783–791, https://doi.org/10.1016/j.sysconle.2009.08.009. · Zbl 1181.93088 [52] U. Vogel and K. F. Jensen, The RNA chain elongation rate in Escherichia coli depends on the growth rate, J. Bacteriol., 176 (1994), pp. 2807–2813. [53] P. Xing-Cheng and Y. Wei, Stability of hybrid stochastic systems with time-delay, ISRN Math. Anal., 2014, 423413. · Zbl 1286.93195 [54] C. Yuan and X. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control, 16 (2010), pp. 595–608, https://doi.org/10.3166/ejc.16.595-608. · Zbl 1216.93095 [55] D. Yue, Y. Zhang, E. Tian, and C. Peng, Delay-distribution-dependent exponential stability criteria for discrete-time recurrent neural networks with stochastic delay, IEEE Trans. Neural Netw., 19 (2008), pp. 1299–1306, https://doi.org/10.1109/TNN.2008.2000166.
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