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Reduction of Chemical Reaction Networks with Approximate Conservation Laws

Aur\'elien DesoeuvresAlexandru IosifChristoph L\"uders ...+3 Thomas Sturm
Dec 2022
Model reduction of fast-slow chemical reaction networks based on thequasi-steady state approximation fails when the fast subsystem has firstintegrals. We call these first integrals approximate conservation laws. Inorder to define fast subsystems and identify approximate conservation laws, weuse ideas from tropical geometry. We prove that any approximate conservationlaw evolves slower than all the species involved in it and therefore representsa supplementary slow variable in an extended system. By elimination of somevariables of the extended system, we obtain networks without approximateconservation laws, which can be reduced by standard singular perturbationmethods. The field of applications of approximate conservation laws covers thequasi-equilibrium approximation, well known in biochemistry. We discuss bothtwo timescale reductions of fast-slow systems and multiple timescale reductionsof multiscale networks. Networks with multiple timescales have hierarchicalrelaxation. At a given timescale, our multiple timescale reduction methoddefines three subsystems composed of (i) slaved fast variables satisfyingalgebraic equations, (ii) slow driving variables satisfying reduced ordinarydifferential equations, and (iii) quenched much slower variables that areconstant. The algebraic equations satisfied by fast variables define chains ofnested normally hyberbolic invariant manifolds. In such chains, fastermanifolds are of higher dimension and contain the slower manifolds. Ourreduction methods are introduced algorithmically for networks with linear,monomial or polynomial approximate conservation laws. Keywords: Model order reduction, chemical reaction networks, singularperturbations, multiple timescales, tropical geometry.