We propose a simple model of lac operon that describes the expression of B-galactosidase from lac Z gene in Escherichia coli, through the interaction among several identical mRNA. Our goal is to explore the complex dynamics (i.e. the oscillation phenomenon) of this architecture mediated by this interaction. This model was theoretically and numerically investigated using distributed time delay. We considered the average delay as a bifurcation parameter and the nonlinear degradation rate as a control parameter. Sufficient conditions for local stability were gained by using the Routh-Hurwitz criterion in the case of a weak delay kernel. Then we proved that Hopf bifurcation happened and the direction of the periodic solution was determined using multiple time scale technique. Our results suggest that the interaction among several identical mRNA plays the main role in gene regulation.

中文翻译：

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具有分布时滞和非线性退化率的lac Operon模型的稳定性和Hopf分支分析。

我们提出了一个简单的lac操纵子模型，该模型描述了通过几个相同的mRNA相互作用从lac Z基因在大肠杆菌中表达的B-半乳糖苷酶。我们的目标是探索这种相互作用介导的这种体系结构的复杂动力学（即振荡现象）。使用分布式时延对模型进行了理论和数值研究。我们将平均延迟视为分叉参数，并将非线性退化率视为控制参数。在弱延迟内核的情况下，通过使用Routh-Hurwitz准则获得了足够的局部稳定性条件。然后我们证明了发生霍普夫分叉，并使用多时标技术确定了周期解的方向。