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Keywords:
biochemical networks; gene regulatory networks; oscillating systems; periodic solutions; model reduction; accurate approximation
biochemical networks; gene regulatory networks; oscillating systems; periodic solutions; model reduction; accurate approximation
Summary:
Quasi-steady state assumptions are often used to simplify complex systems of ordinary differential equations in the modelling of biochemical processes. The simplified system is designed to have the same qualitative properties as the original system and to have a small number of variables. This enables to use the stability and bifurcation analysis to reveal a deeper structure in the dynamics of the original system. This contribution shows that introducing delays to quasi-steady state assumptions yields a simplified system that accurately agrees with the original system not only qualitatively but also quantitatively. We derive the proper size of the delays for a particular model of circadian rhythms and present numerical results showing the accuracy of this approach.
References:
[1] Chen, L., Aihara, K.: Stability of genetic regulatory networks with time delay. IEEE Trans. Circuits Syst., I, Fundam. Theory Appl. 49 (2002), 602-608. DOI 10.1109/TCSI.2002.1001949 | MR 1909315
[2] Cotter, S. L., Vejchodský, T., Erban, R.: Adaptive finite element method assisted by stochastic simulation of chemical systems. SIAM J. Sci. Comput. 35 (2013), B107--B131. DOI 10.1137/120877374 | MR 3033062 | Zbl 1264.65158
[3] Erban, R., Chapman, S. J., Kevrekidis, I. G., Vejchodský, T.: Analysis of a stochastic chemical system close to a sniper bifurcation of its mean-field model. SIAM J. Appl. Math. 70 (2009), 984-1016. DOI 10.1137/080731360 | MR 2538635 | Zbl 1200.80010
[4] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering 191 Academic Press, Boston, MA (1993). MR 1218880 | Zbl 0777.34002
[5] Murray, J. D.: Mathematical Biology. Vol. 1: An Introduction. 3rd Interdisciplinary Applied Mathematics 17 Springer, New York (2002). MR 1908418 | Zbl 1006.92001
[6] Savageau, M.: Biochemical systems analysis: I. some mathematical properties of the rate law for the component enzymatic reactions. J. Theor. Biol. 25 (1969), 365-369. DOI 10.1016/S0022-5193(69)80026-3
[7] Segel, L. A., Slemrod, M.: The quasi-steady-state assumption: a case study in perturbation. SIAM Rev. 31 (1989), 446-477. DOI 10.1137/1031091 | MR 1012300 | Zbl 0679.34066
[8] Verdugo, A., Rand, R.: Hopf bifurcation in a DDE model of gene expression. Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 235-242. DOI 10.1016/j.cnsns.2006.05.001 | MR 2360687 | Zbl 1134.34325
[9] Vilar, J. M. G., Kueh, H. Y., Barkai, N., Leibler, S.: Mechanisms of noise-resistance in genetic oscillators. PNAS 99 (2002), 5988-5992. DOI 10.1073/pnas.092133899
[10] Xie, Z., Kulasiri, D.: Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops. J. Theor. Biol. 245 (2007), 290-304. DOI 10.1016/j.jtbi.2006.10.028 | MR 2306447
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