Reduced models of networks of coupled enzymatic reactions

TL;DR

This paper extends the Michaelis-Menten approximation to complex enzymatic networks, deriving reduced models that simplify network dynamics while rigorously validating the approximation conditions.

Contribution

It generalizes the total quasi steady state assumption (tQSSA) to large enzymatic networks and provides a systematic method for model reduction using geometric singular perturbation theory.

Findings

Extended tQSSA conditions to enzymatic networks.

Derived reduced equations involving only protein concentrations.

Proved validity of the approximation rigorously.

Abstract

The Michaelis-Menten equation has played a central role in our understanding of biochemical processes. It has long been understood how this equation approximates the dynamics of irreversible enzymatic reactions. However, a similar approximation in the case of networks, where the product of one reaction can act as an enzyme in another, has not been fully developed. Here we rigorously derive such an approximation in a class of coupled enzymatic networks where the individual interactions are of Michaelis-Menten type. We show that the sufficient conditions for the validity of the total quasi steady state assumption (tQSSA), obtained in a single protein case by Borghans, de Boer and Segel can be extended to sufficient conditions for the validity of the tQSSA in a large class of enzymatic networks. Secondly, we derive reduced equations that approximate the network's dynamics and involve only protein concentrations. This significantly reduces the number of equations necessary to model such systems. We prove the validity of this approximation using geometric singular perturbation theory and results about matrix differentiation. The ideas used in deriving the approximating equations are quite general, and can be used to systematize other model reductions.