An exact moment-based approach for high-order chemical reaction–diffusion networks: From mass action to Hill functions

Biochemical systems are inherently stochastic, particularly those with small-molecule populations. The spatial distribution of molecules plays a critical role and requires the inclusion of spatial coordinates in their analysis. Stochastic models such as the chemical master equation are commonly used to study these systems. However, analytical solutions are limited to specific cases, and stochastic simulations require significant computational resources. To mitigate these challenges, approximation methods, such as the moment approach, reduce the system to a set of ordinary differential equations, thereby lowering the computational requirements. This study investigates the conditions under which the second-moment approach yields exact results during the dynamic evolution of chemical reaction-diffusion networks. The analysis encompasses second-order or higher-order reactions and Hill functions without relying on higher-order moment estimations or closure approximations. Starting with stationary states, we extended the analysis to a dynamic evolution. An enzymatic process and an antithetic feedback system were examined for purely reactive systems, demonstrating the approach's accuracy in capturing system behavior and quantifying errors. The study was further extended to gene regulatory networks governed by Hill functions, including both purely reactive and reaction-diffusion systems, validating the method in spatially distributed contexts. This framework enables precise characterization of biochemical systems, avoiding information loss typically associated with approximations and allowing for stability analysis under fluctuations. These findings optimize computational strategies while providing insights into intracellular signaling and regulatory processes, paving the way for efficient and accurate stochastic modeling in biochemical systems.