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. The dynamical behavior of couple enzyme catalyzed assays is studied by analysis in the phase plane. 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, we analyze two types of time-dependent slow manifolds that occur in asymptotically autonomous vector fields that arise from enzyme coupled assays. We show that the motion of the slow manifolds relative to the motion of the solution must be taken into account in order to formulate accurate leading order asymptotic solutions. We also develop a rigorous mathematical framework from which to analyze enzyme catalyzed indicator reaction from couple enzyme assays.