Bifurcation- and noise-induced tipping in two-parametric gene transcriptional regulatory system

A regime shift from one state to another often takes place during the gene transcriptional regulations process. Changes in parameters and presence of noise can trigger critical transitions that could have a significant and irreversible effect. It is thus necessary to utilize early warning indicators to predict this phenomenon under noisy conditions as the parameters are slowly raised or lowered over time. However, in most papers, early-warning indicators are related to the concept of local stability. So, this paper intends to study the global effects of control parameters and noise intensity on critical transitions. The results indicate that the stability of the low concentration state decreases with increasing noise intensity. The degradation rate can inhibit protein synthesis, thus reducing the concentration level. Contrastingly, changing the transcription rate has an opposite effect, as the noise intensity increases. Furthermore, under the influence of noise, once the time-varying system parameters enter the parameter-dependent basin of the unsafe regime (PDBUR), it is an indication that a critical transition imminent, implying that effective measures need to be taken to prevent the occurrence of critical jumps. Numerical results show that the proposed method is in approximating the parameter range, and provides theoretical guidance for studies and predictive of bifurcation- and noise-induced tipping.