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We study the statistical properties of a simple genetic regulatory networkthat provides heterogeneity within a population of cells. This network consistsof a binary genetic switch in which stochastic flipping between the two switchstates is mediated by a "flipping" enzyme. Feedback between the switch stateand the flipping rate is provided by a linear feedback mechanism: the flippingenzyme is only produced in the on switch state and the switching rate dependslinearly on the copy number of the enzyme. This work generalises the model of[Phys. Rev. Lett., 101, 118104] to a broader class of linear feedback systems.We present a complete analytical solution for the steady-state statistics ofthe number of enzyme molecules in the on and off states, for the general casewhere the enzyme can mediate flipping in either direction. For this generalcase we also solve for the flip time distribution, making a connection to firstpassage and persistence problems in statistical physics. We show that thestatistics of the model are non-Poissonian, leading to a peak in the flip timedistribution. The occurrence of such a peak is analysed as a function of theparameter space. We present a new relation between the flip time distributionsmeasured for two relevant choices of initial condition. We also introduce a newcorrelation measure to show that this model can exhibit long-lived temporalcorrelations, thus providing a primitive form of cellular memory. Motivated byDNA replication as well as by evolutionary mechanisms involving geneduplication, we study the case of two switches in the same cell. This resultsin correlations between the two switches; these can either positive or negativedepending on the parameter regime.

Statistical physics of a model binary genetic switch with linear feedback