Characterizing the Detailed Balance Property by Means of Measurements

In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions Rᵢⱼ (t) R ij ( t ) that measure how the system reacts to the injection of the substance j at time $$t=0$$ t = 0 , by measuring the concentration of the substance i ≠ j i ≠ j for $$t >0$$ t > 0 . In particular, we obtain a condition involving two reciprocal measurements (that is Rᵢⱼ(t) R ij ( t ) , Rⱼᵢ(t) R ji ( t ) ) that is necessary, but not sufficient, for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied by the graph associated to the network, as well as a stability property that guarantees that the chemical rates are not fine-tuned.