Short relaxation times but long transient times in both simple and complex reaction networks

TL;DR

This paper reveals that in reaction networks, the standard relaxation time often underestimates the duration of transient behaviors, with perturbation lifetimes potentially being much longer, even in simple systems.

Contribution

The study demonstrates through mathematical analysis that characteristic times for perturbation lifetimes can vastly exceed relaxation times in both simple and complex reaction networks.

Findings

Perturbation lifetimes can be much longer than relaxation times.

Standard relaxation time may not reflect true transient durations.

Long transient times are observed in various reaction network models.

Abstract

When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time $\tau$, i.e., the inverse of that decay rate, is the longest characteristic time in the system. However that need not be true, and in particular other times such as the lifetime of an infinitesimal perturbation can be much longer. In the present work we demonstrate that this paradoxical property can arise even in quite simple systems such as a chain of reactions obeying mass action kinetics. By mathematical analysis of simple reaction networks, we pin-point the reason why the standard relaxation time does not provide relevant information on the potentially long transient times of typical infinitesimal perturbations. Overall, we consider four characteristic times and study their behavior in both simple chains and in more complex reaction networks taken from the publicly available database "Biomodels." In all these systems involving mass action rates, Michaelis-Menten reversible kinetics, or phenomenological laws for reaction rates, we find that the characteristic times corresponding to lifetimes of tracers and of concentration perturbations can be much longer than $\tau$.