Using singular perturbation theory to determine kinetic parameters in a non-standard coupled enzyme assay (Q2192670)

Enzyme assays are an important tool for characterizing enzymes. In the classic assay, a single reaction converts a substrate into a product, using an enzyme as a catalyst, and the product is measured over time to estimate the initial reaction velocity. When characterizing an enzyme for which the reaction product is difficult to observe, an auxiliary enzyme can be introduced to convert the product of the primary enzyme reaction into a chemical that can more easily be measured. This is called a coupled enzyme assay. However, the problem is more complicated if the reaction catalysed by the auxiliary enzyme has a product which is also a substrate for the primary enzyme. In this paper, the authors interested in understanding how to characterize a primary enzyme in a non-standard coupled enzyme assay, where one of the products of the auxiliary enzyme is a substrate for the primary enzyme and where can be explicitly accounted for the reversibility of the auxiliary enzyme reaction. In the analysis, the authors assume that the time-scales of enzyme complex formation are much shorter than the other reaction timescales in the model, and that the effective reaction velocities are governed by Michaelis-Menten-type laws. For these reasons, they use singular perturbation theory to analyse the mathematical model consisting of the following dimensionless governing ODEs \begin{align*} \frac{\mathrm{d}s_1}{\mathrm{d}t}&=-\epsilon a(v_1-v_2+v_{-2}),\\ \frac{\mathrm{d}s_2}{\mathrm{d}t}&=-\epsilon bv_1,\\ \frac{\mathrm{d}s_4}{\mathrm{d}t}&=v_1-v_2+v_{-2},\\ \frac{\mathrm{d}s_5}{\mathrm{d}t}&=-\epsilon c(v_2-v_{-2}),\\ \frac{\mathrm{d}s_6}{\mathrm{d}t}&=v_2-v_{-2}, \end{align*} where \(s_i,\) \(i=1,2,4,5,6\) denote the scaled substrates, \(v_1=v_1(s_1,s_2),\) \(v_2=v_1(s_4,s_5),\) \(v_{-2}=v_{-2}(s_1,s_6)\) are dimensionless reaction velocities and the parameter \(\epsilon\) can always be made to be small by suitable choices of the relative concentrations of the primary and auxiliary enzyme. The authors discussed and demonstrated that the results can be used effectively in characterizing enzymes by performing the coupled enzyme assay modeled in the paper and applying achieved theoretical results.