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Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays. (English) Zbl 1156.92031
Summary: To understand how a gene regulatory network functioning as an oscillator is built, a genetic regulatory network with two transcriptional delays is investigated. We show by mathematical analysis and simulation that autorepression of mRNA and protein can provide a mechanism for the intracellular oscillator. Based on the linear stability approach and bifurcation theory, sufficient conditions for the oscillation of the genetic networks are derived, and critical values of Hopf bifurcations are assessed. In particular, the genetic network can exhibit Hopf bifurcations (oscillation appears) as the sum of delays or transcriptional rates passes through some critical values. Moreover, the robustness of amplitudes against change in delay can also be obtained from the delayed genetic network; the period of oscillations increases with the total time delay in an almost linear way. While it is exactly opposite for transcriptional rates, the amplitude of oscillations always increases as the transcriptional rate increases; the robustness of the period against change in the transcriptional rate occurs. Some simple genetic regulatory networks are used to study the impact of delays and transcriptional rates on the system dynamics when there are delays.
##### MSC:
 92D10 Genetics 92C40 Biochemistry, molecular biology 34K60 Qualitative investigation and simulation of models 92C37 Cell biology 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional differential equations
##### References:
 [1] Barbai, N.; Leibler, S.: Biological rhythms: circadian clocks limited by noise, Nature 403, 267 (2000) [2] Becskei, A.; Boselli, M. G.; Van Oudenaarden, A.: Amplitude control of cell-cycle waves by nuclear import, Nat. cell biol. 6, 451 (2004) [3] Becskei, A.; Serrano, L.: Engineering stability in gene networks by autoregulation, Nature 405, 590 (2000) [4] Belair, J.; Campbell, S. A.; Driessche, P.: Frustration, stability, and delay-induced oscillations in a neural network model, SIAM J. Appl. math. 56, 245 (1996) · Zbl 0840.92003 · doi:10.1137/S0036139994274526 [5] Bessho, Y.; Hirata, H.; Masamizu, Y.; Kageyama, R.: Periodic repression by the bhlh factor hes7 is an essential mechanism for the somite segmentation clock, Genes dev. 17, 1451 (2003) [6] Campbell, S. A.; Ruan, S.; Wei, J.: Qualitative analysis of a neural network model with multiple time delays, Int. J. Bifurcation chaos appl. Sci. eng. 9, 1585 (1999) · Zbl 1192.37115 · doi:10.1142/S0218127499001103 [7] Chen, L.; Aihara, K.: Stability of genetic regulatory networks with time delay, IEEE trans. CAS-I 49, 602 (2002) [8] Chen, L.; Wang, R.; Kobayashi, T.; Aihara, K.: Dynamics of gene regulatory networks with cell division cycle, Phys. rev. E 70, 011909 (2004) [9] Cooke, K.; Grossman, Z.: Discrete delay, distributed delay and stability switches, J. math. Anal. appl. 86, 592 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8 [10] Dunlap, J. C.: Molecular bases for circadian clocks, Cell 96, 271 (1999) [11] Elowitz, M. B.; Leibler, S.: A synthetic oscillatory network of transcriptional regulators, Nature 403, 335 (2000) [12] Erneux, T.; Large, L.; Lee, M.; Goedgebuer, J. -P.: Ikeda Hopf bifurcation revisited, Physica D 194, 49 (2004) · Zbl 1099.34065 · doi:10.1016/j.physd.2004.01.038 [13] Gardner, T. S.; Cantor, C. R.; Collins, J. J.: Construction of a genetic toggle dwitch inescherichia coli, Nature 403, 339 (2000) [14] Glass, L.; Kauffman, S. A.: The logical analysis of continuous, nonlinear biochemical control networks, J. theor. Biol. 39, 103 (1973) [15] Glass, L.; Mackey, M. C.: From clocks to chaos: the rhythms of life, (1988) [16] Goldbeter, A.: A model for circadian oscillations in the drosophila period $protein\left(PER\right)$, Proc. R. Soc. lond. B 261, 319 (1995) [17] Gonze, D.; Leloup, J. -C.; Goldbeter, A.: Theoretical models for circadian rhythms in neurospora and drosophila, C. R. Hebd. acad. Sci.$\left(Paris\right)$ III 323, 57 (2000) [18] Goodwin, B. C.: Oscillatory behavior in enzymatic control process, Adv. enzyme reg. 3, 425 (1965) [19] Guo, S.; Chen, Y.; Wu, J.: Two-parameter bifurcations in a network of two neurons with multiple delays, J. differential equations 244, No. 2, 444 (2008) · Zbl 1136.34058 · doi:10.1016/j.jde.2007.09.008 [20] Hasty, J.; Dolnik, M.; Rottschäfer, V.; Collins, J. J.: Synthetic gene network for entraining and amplifying cellular oscillations, Phys. rev. Lett. 88 (2002) [21] 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) [22] Hunding, A.: Limit cycles in enzyme systems with nonlinear negative feedback, Biophys. struct. Mech. 1, 47 (1974) [23] Jiang, Y. J.; Aerne, B. L.; Smithers, L.; Haddon, C.; Ish-Horowicz, D.; Lewis, J.: Notch signalling and the synchronization of the somite segmentation clock, Nature 408, 475 (2000) [24] Jordan, D. W.; Smith, P.: Nonlinear ordinary differential equations, (1999) [25] Kippert, F.; Hunt, P.: Ultradian clocks in eukaryotic microbes: from behavioural observation to functional genomics, Bioessays 22, 16 (2000) [26] Kobayashi, T.; Chen, L.; Aihara, K.: Design of genetic switches with only positive feedback loops, Proc. IEEE comput. Soc. conf. Bioinf., 151 (2002) [27] Kobayashi, T.; Chen, L.; Aihara, K.: Modelling genetic switches with positive feedback loops, J. theor. Biol. 221, 379 (2003) [28] Mallet-Paret, J.; Sell, G. R.: The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. differential equations 125, 441 (1996) · Zbl 0849.34056 · doi:10.1006/jdeq.1996.0037 [29] Mallet-Paret, J.; Sell, G. R.: Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. differential equations 125, 385 (1996) · Zbl 0849.34055 · doi:10.1006/jdeq.1996.0036 [30] Mcmillen, D.; Kopell, N.; Hasty, J.; Collins, J. J.: Synchronizing genetic relaxation oscillators by intercell signaling, Proc. natl. Acad. sci. USA 99, 679 (2002) [31] Ozbudak, E. M.; Thattai, M.; Lim, H. N.; Shraiman, B.; Oudenaardeb, A. V.: Multi-stability in the lactose utilization network ofescherichia coli, Nature 427, 737 (2004) [32] Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator – prey systems with discrete delays, Quart. appl. Math. 59, 159 (2001) · Zbl 1035.34084 [33] Smith, H.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems 41 (1995) · Zbl 0821.34003 [34] Stelling, J.; Sauer, U.; Szallasi, Z.; Iii, F. J. Doyle; Doyle, J.: Robustness of cellular functions, Cell 118, 675 (2004) [35] Wang, R.; Chen, L.; Aihara, K.: Construction of genetic oscillators with interlocked feedback networks, J. theor. Biol. 242, 454 (2006) [36] Wang, R.; Jing, Z.; Chen, L.: Periodic oscillators in genetic networks with negative feedback loops, WSEAS trans. Math. 3, 150 (2004) [37] Wu, J.; Zou, X.: Patterns of sustained oscillations in neural networks with delayed interactions, Appl. math. Comput. 73, 55 (1995) · Zbl 0857.92003 · doi:10.1016/0096-3003(94)00203-G