Mathematical analysis of a biochemical oscillator with delay. (English) Zbl 1329.34131
Summary: We analyze a nonlinear delay differential equation (DDE) model with negative feedback and constant time delay. The model is constructed from a previously studied biochemical reaction network of gene transcription and protein synthesis. Linear analysis of the associated DDE model gives a critical time delay beyond which a periodic motion is born in a Hopf bifurcation. The method of multiple scales is then used to analyze the nonlinear system to obtain expressions for the amplitude and frequency of oscillation as a function of the system parameters. We use our closed form analytical expressions to study the importance of a well-balanced ratio between synthesis and degradation rates in the existence of periodic solutions. We show that our theoretical results are in agreement with numerical simulations and with experimental evidence found in the biological literature.
MSC:
34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
92C40 | Biochemistry, molecular biology |
34K13 | Periodic solutions to functional-differential equations |
34K18 | Bifurcation theory of functional-differential equations |
34K20 | Stability theory of functional-differential equations |
Keywords:
biochemical oscillator; delay differential equation; method of multiple scales; Hopf bifurcationReferences:
[1] | Goodwin, B. C., Oscillatory behavior in enzymatic control processes, Adv. Enzyme Regul., 3, 0, 425-437 (1965) |
[2] | Higgins, J., The theory of oscillating reactions, Ind. Eng. Chem., 59, 5, 18-62 (1967) |
[3] | Mackey, M.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 4300, 287-289 (1977) · Zbl 1383.92036 |
[4] | Goldbeter, A., Biochemical Oscillations and Cellular Rhythms (1997), Cambridge University Press |
[5] | Monk, N., Oscillatory expression of Hes1, p53, and NF-kB driven by transcriptional time delays, Curr. Biol., 59, 16, 1409-1413 (1967) |
[6] | Scheper, T.o.; Klinkenberg, D.; Pennartz, C.; van Pelt, J., A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19, 1, 40-47 (1999) |
[7] | Verdugo, A.; Rand, R., Center manifold analysis of a DDE model of gene expression, Commun. Nonlinear Sci. Numer. Simul., 13, 1112-1120 (2008) · Zbl 1221.37197 |
[8] | Das, S.; Chatterjee, A., Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynam., 30, 4, 323-335 (2002) · Zbl 1038.34075 |
[9] | Verdugo, A.; Rand, R., Hopf bifurcation in a DDE model of gene expression, Commun. Nonlinear Sci. Numer. Simul., 13, 235-242 (2008) · Zbl 1134.34325 |
[10] | Kalmár-Nagy, T.; Stépán, G.; Moon, F. C., Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynam., 2, 121-142 (2001) · Zbl 1005.70019 |
[11] | Rand, R.; Verdugo, A., Hopf bifurcation formula for first order differential-delay equations, Commun. Nonlinear Sci. Numer. Simul., 12, 859-864 (2007) · Zbl 1123.34334 |
[12] | Bernard, S.; Cajavec, B.; Pujo-Menjouet, L.; Mackey, M.; Herzel, H., Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations, Philos. Trans. A Math. Phys. Eng. Sci., 364, 1155-1170 (2006) · Zbl 1152.92316 |
[13] | Wei, J.; Yu, C., Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proc. Roy. Soc. Edinburgh, 139, 879-895 (2009) · Zbl 1185.34124 |
[14] | Zhang, T.; Song, Y.; Zang, H., The stability and Hopf bifurcation analysis of a gene expression model, J. Math. Anal. Appl., 395, 103-113 (2012) · Zbl 1263.34118 |
[15] | Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press · Zbl 0474.34002 |
[16] | Novak, B.; Tyson, J., Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol., 9, 981-991 (2008) |
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