Analysis of the most probable exit path in the synthetic gene network with genetic toggle

In this paper, the most probable exit path is applied to understand the switching of a bistable synthetic gene network with genetic toggle in Escherichia coli from a new perspective. It is worth mentioning that the most probable exit path is a deterministic indicator for researching stochastic system, and its role is analogous to the phase diagram and time history diagram of deterministic system. Firstly, the large deviation theory proves that when a dynamic system has a stable equilibrium point, the cumulative effect of its long-term action can make its sample trajectory reach another region with a probability of 1 from the attractive region under weak random disturbance. The concept of the most probable exit path is given by defining Freidlin-Wentzell action functional. Moreover, considering the noise intensity, Onsager-Machlup action functional is defined to obtain the most probable exit path of the system subject to random disturbance. Based on the Onsager-Machlup action functional, it can be observed that the impact of two types of noise intensities on the most probable exit path for the transition of the system. In addition, based on vector field decomposition, energy landscape is used to analyze the global stability of the system. Finally, this article intuitively demonstrates the switching process of the system with genetic toggle. The analysis of Onsager-Machlup action functional reveals that two independent noises have opposite effects on the system switching. The increase in the intensity of the second Gaussian noise is conductive to more accurately observing the most probable exit path of the system. According to the energy landscape, the reduction of noise intensity is not conductive to switching of the system with genetic toggle.