Stochastic analysis of bistability in coherent mixed feedback loops combining transcriptional and posttranscriptional regulations

Mixed feedback loops combining transcriptional and posttranscriptional regulations are common in cellular regulatory networks. They consist of two genes, encoding a transcription factor and a small noncoding RNA (sRNA), which mutually regulate each other's expression. We present a theoretical and numerical study of coherent mixed feedback loops of this type, in which both regulations are negative. Under suitable conditions, these feedback loops are expected to exhibit bistability, namely, two stable states, one dominated by the transcriptional repressor and the other dominated by the sRNA. We use deterministic methods based on rate equation models, in order to identify the range of parameters in which bistability takes place. However, the deterministic models do not account for the finite lifetimes of the bistable states and the spontaneous, fluctuation-driven transitions between them. Therefore, we use stochastic methods to calculate the average lifetimes of the two states. It is found that these lifetimes strongly depend on rate coefficients such as the transcription rates of the transcriptional repressor and the sRNA. In particular, we show that the fraction of time the system spends in the sRNA-dominated state follows a monotonically decreasing sigmoid function of the transcriptional repressor transcription rate. The biological relevance of these results is discussed in the context of such mixed feedback loops in Escherichia coli. It is shown that the fluctuation-driven transitions and the dependence of some rate coefficients on the biological conditions enable the cells to switch to the state which is better suited for the existing conditions and to remain in that state as long as these conditions persist.