  Mixed feedback loops combining transcriptional and post-transcriptional
regulations are common in cellular regulatory networks. They consist of two
genes, encoding a transcription factor and a small non-coding 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 {\it Escherichia coli}.
