On algebraic properties of extreme pathways in metabolic networks

We give a concise development of some of the major algebraic properties of extreme pathways (pathways that cannot be the result of combining other pathways) of metabolic networks, contrasting them to those of elementary flux modes (pathways involving a minimal set of reactions). In particular, we show that an extreme pathway can be recognized by a rank test as simple as the existing rank test for elementary flux modes, without computing all the modes. We make the observation that, unlike elementary flux modes, the property of being an extreme pathway depends on the presence or absence of reactions beyond those involved in the pathway itself. Hence, the property of being an extreme pathway is not a local property. As a consequence, we find that the set of all elementary flux modes for a network includes all the elementary flux modes for all its subnetworks, but that this property does not hold for the set of all extreme pathways.