Steady state of an auto-regulatory gene network with partial binding and transcription leakage

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A positive feedback loop, where the gene product directly activates its own production, is the simplest form of a gene regulatory network that exhibits a bistable behavior. An analytic solution for the steady state can be obtained only under various simplifying assumptions, such as simultaneous binding of the activator molecules to the gene or complete inactivation of the gene in the absence of protein binding. Here, I consider a feedback loop without such unrealistic simplification and analyze the fixed point structure of the resulting rate equation. The current model is the most general deterministic model that has been solved analytically. I find that the bifurcation diagram is qualitatively different from that obtained in the absence of transcription leakage.