A scalable approach for solving chemical master equations based on modularization and filtering

Abstract

Solving the chemical master equation (CME) that characterizes the probability evolution of stochastically reacting processes is greatly important for analyzing intracellular reaction systems. Conventional methods for solving CMEs include the simulation-based Monte-Carlo methods, the direct approach (e.g., the finite state projection), and so on; however, they usually do not scale very well with the system dimension either in terms of accuracy or efficiency. To mitigate this problem, we propose a new computational method based on modularization and filtering. Our method first divides the whole system into a leader system and several conditionally independent follower subsystems. Then, we solve the CME by applying the Monte Carlo method to the leader system and the direct approach to the filtered CMEs that characterize the conditional probabilities of the follower subsystems. The system decomposition involved in our method is optimized so that all the subproblems above are low dimensional, and, therefore, our approach scales more favorably with the system dimension. Finally, we demonstrate the efficiency and accuracy of our approach in high-dimensional estimation and inference problems using several biologically relevant examples.Significance StatementDue to the inherent randomness in microscale systems, an intracellular reaction system is often stochastic and modeled by a chemical master equation, which characterizes the probability evolution of this system. Solving this equation for high-dimensional problems is challenging: conventional methods, including the Monte-Carlo methods and the direct approach, are usually inaccurate or inefficient in these cases. To mitigate this problem, we propose a new method for solving CMEs based on modularization and filtering. Our method works by an optimized decomposition of the system, which transforms the high-dimensional problem into several low-dimensional ones; consequently, our approach scales more favorably with the system dimension. Finally, we use several biologically relevant examples to illustrate our approach in high-dimensional estimation and inference problems.