Phase-Plane Geometries in Coupled Enzyme Assays

The determination of a substrate or enzyme activity by coupling of one enzymatic reaction with another easily detectable (indicator) reaction is a common practice in the biochemical sciences. Usually, the kinetics of enzyme reactions is simplified with singular perturbation analysis to derive rate or time course expressions valid under the quasi-steady-state and reactant stationary state assumptions. In this paper, the dynamical behavior of coupled enzyme catalyzed assays is studied by analysis in the phase plane. We analyze two types of time-dependent slow manifolds - Sisyphus and Laelaps manifolds - that occur in asymptotically autonomous vector fields that arise from enzyme coupled assays. Projection onto slow manifolds yields various reduced models, and we develop a mathematical framework from which to analyze coupled enzyme assays through scaling and phase-plane analysis. We demonstrate the necessity of fast indicator reactions to derive mathematical expressions which could be used for the accurate estimation of enzyme kinetic parameters through experimental assays in the laboratory.