Double perturbation series in the differential equations of enzyme kinetics

The connection between combined singular and ordinary perturbation methods and slow-manifold theory is discussed using the Michaelis-Menten model of enzyme catalysis as an example. This two-step mechanism is described by a planar system of ordinary differential equations (ODEs) with a fast transient and a slow “steady-state” decay mode. The systems of scaled nonlinear ODEs for this mechanism contain a singular (η) and an ordinary (ε) perturbation parameter: η multiplies the velocity component of the fast variable and dominates the fast-mode perturbation series; ε controls the decay toward equilibrium and dominates the slow-mode perturbation series. However, higher order terms in both series contain η and ε. Finite series expansions partially decouple the system of ODEs into fast-mode and slow-mode ODEs; infinite series expansions completely decouple these ODEs. Correspondingly, any slow-mode ODE approximately describes motion on ℳ, the linelike slow manifold of the system, and in the infinite series limit this description is exact. Thus the perturbation treatment and the slow-manifold picture of the system are closely related. The functional equation for ℳ is solved automatically with the manipulative language MAPLE. The formal η and ε single perturbation expansions for the slow mode yield the same double (η,ε) perturbation series expressions to given order. Generalizations of this procedure are discussed.