Stochastic modeling of gene positive autoregulation networks involving signal molecules

Positive autoregulation in gene regulation networks has been shown in the past to exhibit stochastic behavior, including stochastic bistability, in which an initially uniform cell population develops into two distinct subpopulations. However, positive autoregulation is often mediated by signal molecules, which have not been considered in prior stochastic analysis of these networks. Here we propose both a full model of such a network that includes a signal molecule, and a simplified model in which the signal molecules have been eliminated through the use of two simplifications. The simplified model is amenable to direct mathematical analysis that shows that stochastic bistability is possible. We use stochastic Petri networks for simulating both types of models. The simulation results show that 1), the stochastic behavior of the two models is similar; and 2), that the analytical steady-state distribution of the simplified model matches well the transient results at times equal to that of a cell generation. A discussion of the simplifications we used in the context of the results indicates the importance of the signal molecule number as a factor determining the presence of bistability. This is further supported from a deterministic steady-state analysis of the full model that is shown to be a useful indicator of potential stochastic bistability. We use the regulation of SdiA in Escherichia coli as an example, due to the importance of this protein and of the signal molecule, a bacterial autoinducer, that is involved. However, the use of kinetic parameter values representing typical cellular activities make the conclusions applicable to other signal-mediated positive autoregulation networks as well.