Uniform Approximation of Solutions by Elimination of Intermediate Species in Deterministic Reaction Networks

Chemical reactions often proceed through the formation and the consumption of intermediate species. An example is the creation and subsequent degradation of the substrate-enzyme complexes in an enzymatic reaction. In this paper we provide a setting, based on ordinary differential equations, in which the presence of intermediate species has little effect on the overall dynamics of a biological system. The result provides a method to perform model reduction by elimination of intermediate species. We study the problem in a multiscale setting, where the species abundances as well a the reaction rates scale to different orders of magnitudes. The different time and concentration scales are parameterised by a single parameter $N$. We show that a solution to the original reaction system is uniformly approximated on compact time intervals to a solution of a reduced reaction system without intermediates and to a solution of a certain limiting reaction systems, which does not depend on $N$. Known approximation techniques such as the theorems by Tikhonov and Fenichel cannot readily be used in this framework.