A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived.
Bibliographical noteFunding Information:
This work was supported in part by NIH Grant #29123 to H.G. Othmer. C.J. Gadgil acknowledges funding from the Minnesota Supercomputing Institute (Research Scholar program). Computations were carried out using MSI and Digital Technology Center resources. We thank one of the reviewers for an extremely thorough review and for pointing out the connections to similar results in queuing theory and the theory of branching processes.