Concentration fluctuations in a mesoscopic oscillating chemical reaction system

Under sustained pumping, kinetics of macroscopic nonlinear biochemical reaction systems far from equilibrium either can be in a stationary steady state or can execute sustained oscillations about a fixed mean. For a system of two dynamic species X and Y, the concentrations n(x) and n(y) will be constant or will repetitively trace a closed loop in the (n(x), n(y)) phase plane, respectively. We study a mesciscopic system with n(x) and n(y) very small; hence the occurrence of random fluctuations modifies the deterministic behavior and the law of mass action is replaced by a stochastic model. We show that nx and ny execute cyclic random walks in the (n(x), n(y)) plane whether or not the deterministic kinetics for the corresponding macroscopic system represents a steady or an oscillating state. Probability distributions and correlation functions for n(x)(t) and n(y)(t) show quantitative but not qualitative differences between states that would appear as either oscillating or steady in the corresponding macroscopic systems. A diffusion-like equation for probability P(n(x), n(y), t) is obtained for the two-dimensional Brownian motion in the (n(x), n(y)) phase plane. In the limit of large n(x), n(y), the deterministic nonlinear kinetics derived from mass action is recovered. The nature of large fluctuations in an oscillating nonequilibrium system and the conceptual difference between "thermal stochasticity" and "temporal complexity" are clarified by this analysis. This result is relevant to fluorescence correlation spectroscopy and metabolic reaction networks.