The adaptation property in non-equilibrium chemical systems

The goal of this paper is to understand the relationship between the property of adaptation (i.e. the insensitivity of the concentration of some substances on the change of the concentration of some chemical signals) and the absence of thermal equilibrium of the system. We prove that, unless the conserved quantities of a signalling system satisfy a very specific factorization assumption, adaptation cannot be achieved in a robust manner by a general class of systems that satisfy the detailed balance property and that do not exchange substances or energy with the environment. We also prove that robust adaptation can be achieved by systems that satisfy the detailed balance property, but exchange substances with the environment. We also recover classical adaptation mechanisms freezing the concentrations of some substances in systems with mass conservation and with detailed balance in some suitable limit regimes.