Fig. 2

Substrate cycle. Reaction network (top) as a complex-reaction graph, involving substrate S, product P, enzymes E, F, and complexes ES, FP, and stoichiometric matrix \(\mathbf {S}\) (middle). In addition to the futile cycles \((1,1,0,0,0,0)^\top\) and \((0,0,0,1,1,0)^\top\), corresponding to the two (pairs of) reversible reactions, there is a non-trivial futile cycle \(\mathbf {v}= (1,0,1,1,0,1)^\top\), involving both reversible and irreversible reactions. (Note that this futile cycle is not an actual cycle of the graph.) As a result, the network is thermodynamically sound, but not strictly thermodynamically sound. In a metabolically relevant example from glycolysis/gluconeogenesis, the compounds are S = fructose-6-phosphate, P = fructose-1,6-bisphosphate, E = phosphofructokinase 1, and F = fructose-1,6-bisphosphatase, and reactions 2 and 4 involve additional compounds (bottom). As a consequence, there is no non-trivial futile cycle (in the strict sense of this work). In fact, the vector \(\mathbf {v}\) above then represents the net reaction \(\text{ATP} + \text{H}_{2}\text{O} \rightarrow \text{ADP} + \text{P}_\mathrm {i}\). Still, it is called a futile cycle or substrate cycle in the literature on metabolic networks. (In our approach, reactions producing/consuming the additional compounds \(\text{P}_\mathrm {i}\) must be added to the network to obtain a futile cycle. Such a futile cycle involves the active reactions in \(\mathbf {v}\), and hence the extended network cannot be strictly thermodynamically sound.)