Complete mathematical characterization of two simple toggle-switch biological systems

Prokaryotic gene expression is dynamic and noisy, and can lead to phenotypic variations. Previous work has shown that the selection of such variation can be modeled through basic reaction networks described by O.D.E. displaying a bistable behavior. While previous mathematical studies have shown that mono- or bistable behavior depends on the rate of the reactions in the system, no analytical solution of the curves delineating the actual parameter conditions that result in mono or bistability has so far been provided. In this work we provide the first explicit analytical solution for the boundary curve that separates the parameter space defining domains where double positive and double negative feedback loops become bistable.