Mass fluctuation kinetics: Capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations

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Abstract
The intrinsic stochastic effects in chemical reactions, and particularly in biochemical networks, may result in behaviors significantly different from those predicted by deterministic mass action kinetics (MAK). Analyzing stochastic effects, however, is often computationally taxing and complex. The authors describe here the derivation and application of what they term the mass fluctuation kinetics (MFK), a set of deterministic equations to track the means, variances, and covariances of the concentrations of the chemical species in the system. These equations are obtained by approximating the dynamics of the first and second moments of the chemical master equation. Apart from needing knowledge of the system volume, the MFK description requires only the same information used to specify the MAK model, and is not significantly harder to write down or apply. When the effects of fluctuations are negligible, the MFK description typically reduces to MAK. The MFK equations are capable of describing the average behavior of the network substantially better than MAK, because they incorporate the effects of fluctuations on the evolution of the means. They also account for the effects of the means on the evolution of the variances and covariances, to produce quite accurate uncertainty bands around the average behavior. The MFK computations, although approximate, are significantly faster than Monte Carlo methods for computing first and second moments in systems of chemical reactions. They may therefore be used, perhaps along with a few Monte Carlo simulations of sample state trajectories, to efficiently provide a detailed picture of the behavior of a chemical system.