In this paper we analyse stochastic expression of a single gene with its dynamics given by the classical Goodwin model with mRNA and protein contribution. We compare the effect of the presence of positive and negative feedback on the transcription regulation. In such cases we observe two qualitatively different types of asymptotic behaviour. In the case of a negative feedback loop, under sufficient conditions, one can find a stationary density for mRNA and protein molecules. In the case of a positive feedback loop we observe extinction of both types of molecules with time.

中文翻译：

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Goodwin模型在随机基因表达的自动调节反馈中的应用。

在本文中，我们分析了单个基因的随机表达及其动力学，该动力学是由经典的Goodwin模型给出的，具有mRNA和蛋白质作用。我们比较了正面和负面反馈的存在对转录调控的影响。在这种情况下，我们观察到两种性质上不同的渐近行为。在负反馈回路的情况下，在足够的条件下，可以找到mRNA和蛋白质分子的固定密度。在正反馈回路的情况下，我们观察到两种类型的分子都随时间消失。