Comparison of Approximate Kinetics for Unireactant Enzymes: Michaelis-Menten against the Equivalent Server

Mathematical models are widely used to create complex biochemical models. Model reduction in order to limit the complexity of a system is an important topic in the analysis of the model. A way to lower the complexity is to identify simple and recurrent sets of reactions and to substitute them with one or more reactions in such a way that the important properties are preserved but the analysis is easier. In this paper we consider the typical recurrent reaction scheme E + S −−⇀ ↽− ES −−→ E + P which describes the mechanism that an enzyme, E, binds a substrate, S, and the resulting substrate-bound enzyme, ES, gives rise to the generation of the product, P . If the initial quantities and the reaction rates are known, the temporal behaviour of all the quantities involved in the above reactions can be described exactly by a set of differential equations. It is often the case however that, as not all necessary information is available, only approximate analysis can be carried out. The most well-known approximate approach for the enzyme mechanism is provided by the kinetics of Michaelis-Menten. We propose, based on the concept of the flow-equivalent server which is used in Petri nets to model reduction, an alternative approximate kinetics for the analysis of enzymatic reactions. We evaluate the goodness of the proposed approximation with respect to both the exact analysis and the approximate kinetics of Michaelis and Menten. We show that the proposed new approximate kinetics can be used and gives satisfactory approximation not only in the standard deterministic setting but also in the case when the behaviour is modeled by a stochastic process.