## 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.