Single-variable reaction systems: Deterministic and stochastic models

Biochemical reaction networks are often described by deterministic models based on macroscopic rateequations. However, for small numbers of molecules, intrinsic noise can play a significant role and stochasticmethods may thus be required. In this work, we analyze the differences and similarities betweena class of macroscopic deterministic models and corresponding mesoscopic stochastic models. We deriveexpressions that provide a clear and intuitive view upon the behavior of the stochastic model. In particular,these expressions show the dependence of both the dynamics and the stationary distribution of thestochastic model on the number of molecules in the system. As expected, most properties of the stochasticmodel correspond well with those in the deterministic model if the number of molecules is largeenough. However, for some properties, both models are inconsistent, even if the number of moleculesin the stochastic model tends to infinity. Throughout this paper, we use a bistable autophosphorylationcycle as a running example. For such a bistable system, we give an explicit proof that the rate of convergenceto the stationary distribution (or the second eigenvalue of the transition matrix) depends exponentiallyon the number of molecules.