Application of the Goodwin model to autoregulatory feedback for stochastic gene expression

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.