Approximation of fast dynamics in kinetic networks using non-negative polynomials

Kinetic models of reaction networks often feature sets of fast-reacting species. If the slow timescale is of interest, these species can be assumed to be in equilibrium and a singular perturbation approximation can be used to render the dynamics of the slow reacting subsystem. However, to obtain a reduction in the number of equations that represent the dynamics of the reaction network, an explicit representation of the equilibrium species is required. In most cases the equilibrium relations are given in implicit form, and algebraic manipulations to obtain an explicit form are prohibitive. This paper examines the use of constrained polynomial fitting techniques to obtain an approximation of the explicit form that is consistent with the physical constraints of the reaction network. This approximation can be combined with the slow reacting subsystem to form a reduced-order representation of the reaction network which is physically consistent. This reduced-order representation can then be used for analysis and control of the kinetic network.