Abstract
When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time τ, i.e. the inverse of the decay rate, is the longest characteristic time in the system. However, that need not be true, 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 linear chain of reactions obeying mass action (MA) 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 behaviour in both simple linear chains and in more complex reaction networks taken from the publicly available database ‘Biomodels’. In all these systems, whether involving MA 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 significantly longer than τ.
Highlights
Networks have been used to model systems involving large numbers of components, agents or species [1]
Relaxation is often driven by the local dynamics, and as a result characteristic times of the system are comparable to that of the individual processes
Pure diffusion provides a simple example of this effect: on a linear lattice of N nodes, the characteristic times of the whole system grow as N2
Summary
Networks have been used to model systems involving large numbers of components, agents or species [1]. Our focus is on the emergence of large characteristic times in reaction networks close to their steady state. One usually takes for granted the rule of thumb that the steady states will pretty much be reached within two or three times t and that any study can focus on just determining t [4]. In metabolic networks, this rule of thumb is often used to classify different timescales [3,5,6]. We will study more general systems using reaction networks published by other authors. We compare the behaviours of four characteristic times in these systems, investigating the causes that can render them non-informative or make their ratios diverge
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