DOUBLE PERTURBATION-SERIES IN THE DIFFERENTIAL-EQUATIONS OF ENZYME-KINETICS

The connection between combined singular and ordinary perturbation met hods and slow-manifold theory is discussed using the Michaelis-Menten model of enzyme catalysis as an example. This two-step mechanism is de scribed by a planar system of ordinary differential equations (ODEs) w ith a fast transient and a slow ''steady-state'' decay mode. The syste ms of scaled nonlinear ODEs for this mechanism contain a singular (eta ) and an ordinary (epsilon) perturbation parameter: eta multiplies the velocity component of the fast variable and dominates the fast-mode p erturbation series; epsilon controls the decay toward equilibrium and dominates the slow-mode perturbation series. However, higher order ter ms in both series contain eta and epsilon. Finite series expansions pa rtially decouple the system of ODEs into fast-mode and slow-mode ODEs; infinite series expansions completely decouple these ODEs. Correspond ingly, any slow-mode ODE approximately describes motion on M, the line like slow manifold of the system, and in the infinite series limit thi s description is exact. Thus the perturbation treatment and the slow-m anifold picture of the system are closely related. The functional equa tion for M is solved automatically with the manipulative language MAPL E. The formal eta and epsilon Single perturbation expansions for the s low mode yield the same double (eta,epsilon) perturbation series expre ssions to given order. Generalizations of this procedure are discussed . (C) 1998 American Institute of Physics.