Stability analysis under intrinsic fluctuations: a second-moment perspective of gene regulatory networks

Gene regulatory networks with negative feedback play a crucial role in conferring robustness and evolutionary resilience to biological systems. However, the discrete nature of molecular components and probabilistic interactions in these networks are inherently subject to fluctuations, which pose challenges for stability analysis. Traditional analysis methods for stochastic systems, like the Langevin equation and the Fokker-Planck equation, are widely used. However, these methods primarily provide approximations of system behavior and may not be suitable for systems that exhibit non-mass-action kinetics, such as those described by Hill functions. In this study, we employed a second-moment approach to analyze the stability of a gene regulatory network with negative feedback under intrinsic fluctuations. By transforming the stochastic system into a set of ordinary differential equations for the mean concentration and second central moment, we performed a stability analysis similar to that used in deterministic models, where there are no fluctuations. Our results show that the incorporation of the second central moment introduces two additional negative eigenvalues, indicating that the system remains stable under intrinsic fluctuations. Furthermore, the stability of the second central moment suggests that the fluctuations do not induce instability in the system. The stationary values of the mean concentrations were found to be the same as those in the deterministic case, indicating that fluctuations did not influence stationary mean concentrations. This framework provides a practical and insightful method for analyzing the stability of stochastic systems and can be extended to other biochemical networks with regulatory feedback and intrinsic fluctuations through a framework of ordinary differential equations.