Stochastic theory of stable and bistable biochemical systems: Effects of noise on enzymatic catalysis

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Abstract
A theory is developed to analyze the effects of noise on the dynamics of stable and bistable biochemical systems where a substrate, continuously provided by a constant external flow, is enzymatically converted into product. Analytical expressions for the ensemble average concentration of substrate at steady state, 〈x〉, are derived. It is found that noise induces a bias in 〈x〉. The bias is always positive in the case of noncooperative enzymatic catalysis, but it can be positive, negative, or zero in the case of cooperative enzymatic catalysis. In the case of bistable systems, 〈x〉 is dramatically affected by noise. A first-order phase transition between two concentration states is approached as the noise becomes asymptotically small. The transition occurs at a critical value of the flow that can be arrived at by means of an equal-area rule. This rule is a generalized form of the Maxwell rule for the van der Waals gas.