Global behaviors analysis for tryptophan operon system with bounded noise

Global behavior for stochastic system of regulation of bacterial tryptophan (trp) operon is analyzed in this paper. This bioprocess cannot avoid the internal and external disturbances reflected on the cell growth rate within the biosynthesis of trp. These factors can affect the integrated results of repression and attenuation, leading to genetic circuit or switch. The equilibria and bifurcations of the deterministic system are analyzed. Then, a stochastic model is presented by considering the effect of a bounded Markov diffusion process on critical parameters. In order to analyze the global behavior, we compute the control sets through solving the associated control system by using parallel control set algorithm. In particular, we compute the bifurcation as discontinuous topology changes in the support of a stationary measure. The probability distributions of relative supports are also provided. The results provide indication on how the disturbances affect the bioprocess with sustained oscillations.