Complex and detailed balancing of chemical reaction networks revisited

Abstract

The characterization of the notions of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff's Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively.

Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics

“…which demonstrates the role of G as a Lyapunov function. The relative entropy L ({Z σx }|{Z σx }) was known to be a Lyapunov function for detailed-balanced networks [76,81], but we provided its clear connection to the transformed nonequilibrium Gibbs free energy. To summarize, instead of minimizing the nonequilibrium Gibbs free energy G (70) as in closed CRNs, the dynamics minimizes the transformed nonequilibrium Gibbs free energy G in open nondriven detailed-balanced CRNs.…”

Bond Graph Representation of Chemical Reaction Networks

“…The complex approach to modelling chemical reaction networks as introduced by Feinberg [13], Horn and Jackson [14] and Feinberg and Horn [15] and expanded by van der Schaft et al [18,19,20,28] has been given a bond graph interpretation thus enabling results from the complex approach to be applied to the bond graph approach and vice versa. In particular, the decomposition of the stoichiometric matrix N into the complex composition matrix [20] Z and the complex graph incidence matrix D (where N = ZD) is given a bond graph interpretation.…”

“…In parallel with the seminal work of Oster et al [7,8], the mathematical foundations of chemical reaction networks (CRN) were being laid by Feinberg [13], Horn and Jackson [14] and Feinberg and Horn [15]. This approach to chemical reaction network theory was further developed by Sontag [16], Angeli [17], and van der Schaft et al [18,19,20]. General results on stability of both closed and open systems of chemical reactions have been derived and applied to reveal dynamic features of complex (bio)chemical networks [21], dissipation in noisy chemical networks Polettini et al [22], metabolic networks [23] and multistability in interferon signalling Otero-Muras et al [24].…”

Semistability of complex balanced kinetic systems with arbitrary time delays

“…Finally, an ordinary or delayed kinetic system is called complex balanced if it has a positive complexed balanced equilibrium. It is well-known [24] that if Eq. (1) and hence (2) has a positive complex balanced equilibrium x, then any other positive equilibrium is complex balanced and the set of all positive equilibria E can be characterized by…”