Reaction networks are common in many fields of science such as chemistry,
biology and ecology. In a chemical network, for instance, several molecular species
form a web of reactions, that produce more complex molecules. In order to
characterize the functionality of these networks one seeks parameters such as the
average population sizes and reaction rates of the different reactive species.
This is commonly done using rate equations, which are based on the mean field
approximation. However, if the system is small, and the average population sizes
are low, the system becomes dominated by fluctuations, the mean field
approximation no longer applies, and stochastic methods are called upon.
The problem is that existing methods, such as Monte Carlo simulations, or the
direct integration of the master equation, scale very badly with the complexity of the
network, and thus cannot efficiently treat elaborate networks which include many
reactive species. Here I will present a new method based on moment equations,
which enables the simulation of reaction networks far beyond the feasibility limit
of the commonly used methods. In its most greedy version the number of equations
is just one equation for each reactive species and one equation for each reaction,
which in terms of efficiency is comparable to that of the rate equations. The accuracy,
on the other hand, is, in many cases, indistinguishable from that of the master equation.
The application fields range from the interstellar chemistry to the metabolic networks
within the living cell.
