Size-independent differences between the mean of discrete stochastic systems and the corresponding continuous deterministic systems

TL;DR

This paper demonstrates that for certain open reaction networks, the mean of stochastic simulations differs fundamentally from deterministic predictions, regardless of molecule count, especially regarding stable zero states.

Contribution

It establishes that stochastic and deterministic models can diverge in predicting system behavior, independent of molecule numbers, for a class of reaction networks with specific stability properties.

Findings

Stochastic systems have a stable zero state unlike deterministic models.

Differences between stochastic and deterministic means are size-independent.

Results generalize previous studies and apply to biological systems.

Abstract

In this paper I show that, for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates systems. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is the only stable stationary state for the discrete stochastic system. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.