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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.
92C40Biochemistry, molecular biology
34K60Qualitative investigation and simulation of models
92C37Cell biology
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
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