Statistical physics of a model binary genetic switch with linear feedback

We study the statistical properties of a simple genetic regulatory network that provides heterogeneity within a population of cells. This network consists of a binary genetic switch in which stochastic flipping between the two switch states is mediated by a flipping enzyme. Feedback between the switch state and the flipping rate is provided by a linear feedback mechanism: the flipping enzyme is only produced in the on switch state and the switching rate depends linearly on the copy number of the enzyme. This work generalizes the model of Visco [Phys. Rev. Lett. 101, 118104 (2008)] to a broader class of linear feedback systems. We present a complete analytical solution for the steady-state statistics of the number of enzyme molecules in the on and off states, for the general case where the enzyme can mediate flipping in either direction. For this general case we also solve for the flip time distribution, making a connection to first passage and persistence problems in statistical physics. We show that the statistics are non-Poissonian, leading to a peak in the flip time distribution. The occurrence of such a peak is analyzed as a function of the parameter space. We present a relation between the flip time distributions measured for two relevant choices of initial condition. We also introduce a correlation measure and use this to show that this model can exhibit long-lived temporal correlations, thus providing a primitive form of cellular memory. Motivated by DNA replication as well as by evolutionary mechanisms involving gene duplication, we study the case of two switches in the same cell. This results in correlations between the two switches; these can be either positive or negative depending on the parameter regime.