Model reduction: When singular perturbation analysis simplifies to partial equilibrium approximation

A geometrical interpretation is provided for the validity of the model reduction methodology based on Partial Equilibrium Approximation, by comparing its algorithm to that of the methodology based on Singular Perturbation Analysis. The cases where the former methodology fails to provide a valid reduced model, while the latter one succeeds, are explored. It is shown that the failure of the Partial Equilibrium Approximation is due to the inability of (i) the stoichiometric vectors of the reactions considered in partial equilibrium to provide a leading order approximation of the fast directions in phase space or/and (ii) the equilibration of the related forward and backward rates to provide a leading order approximation of the equilibrations that develop under the action of the fast time scales. These issues are illustrated geometrically, by analyzing a number of simple examples.