A constructive approach to quasi-steady state reductions

We present a method to determine the reduction of a (polynomial or rational) ordinary differential equation that models a chemically reacting system, under the assumption that this system admits quasi-steady state (QSS) behavior for certain variables or reactions. We interpret QSS mathematically as a singular perturbation setting to which the classical theorems of Tikhonov and Fenichel apply. Based on a special decomposition of the fast part of the equation, we obtain an explicit formula for a reduced system, defined on the slow manifold (which is a subset of an algebraic variety). Moreover we determine appropriate initial values for the reduced system, which correspond to first integrals of the fast subsystem. These first integrals may not be obtainable in closed form, but locally Taylor expansions are available. We give several examples and applications, and we discuss in detail the separation of a system into fast and slow reactions. It turns out that methods and results from (algorithmic) commutative algebra and algebraic geometry are useful tools for QSS reduction.