Critical Parameters for Singular Perturbation Reductions of Chemical Reaction Networks

Abstract

We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path towards such reductions. In the present paper, we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov–Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here, we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero) or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex-balanced reaction networks (of deficiency zero) and first-order reaction networks.

A proof of unlimited multistability for phosphorylation cycles

“…For the specific reduction of the system to be applicable, we need the transverse eigenvalues to have negative real parts. This is true not only for this case but in general for reduction of systems by (so-called) removal of non-interacting species [9][10][11]25]. So this part of the proof could potentially be generalised.…”