Systematic approximation methods for stochastic biochemical kinetics

Abstract

Experimental studies have shown that the protein abundance in living cells varies from few tens to several thousands molecules per species. Molecular fluctuations roughly scale as the inverse square root of the number of molecules due to the random timing of reactions. It is hence expected that intrinsic noise plays an important role in the dynamics of biochemical networks. The Chemical Master Equation is the accepted description of these systems under well-mixed conditions. Because analytical solutions to this equation are available only for simple systems, one often has to resort to approximation methods. A popular technique is an expansion in the inverse volume to which the reactants are confined, called van Kampen's system size expansion. Its leading order terms are given by the phenomenological rate equations and the linear noise approximation that quantify the mean concentrations and the Gaussian fluctuations about them, respectively. While these approximations are valid in the limit of large molecule numbers, it is known that physiological conditions often imply low molecule numbers. We here develop systematic approximation methods based on higher terms in the system size expansion for general biochemical networks. We present an asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion. We then derive an analytical approximation of the corresponding time-dependent probability distribution. Finally, we devise a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions. We show through the use of biological examples that the first few terms of the expansion yield accurate approximations even for low number of molecules. The theory is hence expected to closely resemble the outcomes of single cell experiments.