Rescue of the Quasi-Steady-State Approximation in a Model for Oscillations in an Enzymatic Cascade

A three-variable model describing the oscillatory activity of a cascade of enzyme reactions is analyzed. A quasi-steady-state approximation reduces the three equations to a system of two equations which admits only a stable steady state. This apparent failure of the quasi-steadystate approximation to describe the limit-cycle oscillations observed in the full, three-variable system is analyzed in detail. We first show that the oscillations occur in the full system provided the Michaelis constants are sufficiently small. We then develop a method for determining the correct limit for application of the quasi-steady-state approximation. The leading problem consists of two equations for a conservative oscillator, and a higher order analysis is required in order to determine the amplitude of the limit-cycle oscillations. Finally, we observe a good agreement when comparing exact numerical and approximate bifurcation diagrams.