Abstract
Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves slower than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, well known in biochemistry. We discuss both two timescale reductions of fast-slow systems and multiple timescale reductions of multiscale networks. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyberbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with linear, monomial or polynomial approximate conservation laws. Keywords: Model order reduction, chemical reaction networks, singular perturbations, multiple timescales, tropical geometry.