Dual chemical reaction networks and implications for Lyapunov-based structural stability

Given a class of (bio)Chemical Reaction Networks (CRNs) identified by a stoichiometric matrix S, we define as dual reaction network, CRN*, the class of (bio)Chemical Reaction Networks identified by the transpose stoichiometric matrix S. We consider both the dynamical systems describing the time evolution of the species concentrations and of the reaction rates. First, based on the analysis of the Jacobian matrix, we show that the structural (i.e., parameter-independent) local stability properties are equivalent for a CRN and its dual CRN*. We also assess the structural global stability properties of the two dual networks, analysing both concentration and rate representations. We prove that the existence of a polyhedral (or piecewise-linear) Lyapunov function in concentrations for a CRN is equivalent to the existence of a piecewise-linear in rates Lyapunov function for the dual CRN*; in fact, if V is a polyhedral Lyapunov function for a CRN, the dual polyhedral function V* is a piecewise-linear in rates Lyapunov function for the dual network. We finally show how duality can be exploited to gain additional insight into biochemical reaction networks.