Approaches for the estimation of timescales in nonlinear dynamical systems: Timescale separation in enzyme kinetics as a case study.

The derivation of timescales is frequently introduced as an art form in papers and textbooks. The best scaling techniques require the application of physical intuition to identify dimensionless variables that are one unit order of magnitude and small parameters, which can simplify nonlinear differential equations. However, physical intuition requires prior knowledge of the solution to the dynamical systems under investigation. There are problems where the application of physical intuition is not straightforward. Therefore, it is necessary to apply mathematical techniques to estimate scales for the separation of timescales and simplification. In this review, we present three mathematical techniques - determination of pairwise balances, principle of minimum simplification and scaling by inverse rates - to scale dynamical systems with limited prior knowledge of model behavior. We illustrate the application of these techniques with the Michaelis-Menten reaction, which is widely studied to introduce scaling and simplification techniques in textbooks. We show that the pairwise balance approach, though commonly introduced as a method for nondimensionalization, can fail to derive a separation between timescales. The other techniques we review here can be applied to a number of dynamical systems, where the separation of timescales can lead to the simplification of a complex nonlinear problem.