Mass Action Dynamics of Coupled Reactions using Fluctuation Theory

Comprehensive and predictive simulation of coupled reaction networks has long been a goal of biology and other fields. Currently, metabolic network models that utilize enzyme mass action kinetics have predictive power but are limited in scope and application by the fact that the determination of enzyme rate constants is laborious and low throughput. We present a statistical thermodynamic formulation of the law of mass action for coupled reactions at both steady states and non-stationary states. The formulation is based on a fluctuation theorem for coupled reactions and uses chemical potentials instead of rate constants. When used to model deterministic systems, the theory corresponds to a rescaling of the time dependent reactions in such a way that steady states can be reached on the same time scale but with significantly fewer computational steps. The significance for applications in systems biology is discussed. Introduction One hundred and fifty years ago Peter Waage and Cato Maximillian Gulberg published their first article describing the law of mass action, that the rate of a chemical reaction is proportional to the concentration of the reacting species (1-4). For a simple reaction A 1 −1 ! ⇀ !! ↽ ! ! B , the forward rate due to reaction 1 is simply, forward rate =k1[A] , 1 where the brackets indicate the concentration the constant of proportionality k1 is known as the rate constant and a similar relation exists for the reverse reaction -1. The net rate is given by, net rate =k1[A]−k−1[B]. 2 All introductory chemistry texts describe the law of mass action in one form or another. Although the relationship is simple and can easily be applied to many reactions, the application to more complex systems such as biological metabolism is challenging because most rate constants are not available and measuring the missing rate constants is very labor intensive. Thermodynamic (5, 6) and other approaches (7-11) to the law of mass action have been proposed that do not use rate constants, but these approaches are only valid at steady states. Studies linking thermodynamics and kinetics have historically used the concept of chemical affinities where the affinity is defined with respect to the extent of a reaction ξ (12). If ξ varies from 0 (no reaction) to 1 (complete stoichiometric reaction),