An Efficient Characterization of Complex-Balanced, Detailed-Balanced, and Weakly Reversible Systems

Abstract

Very often, models in biology, chemistry, physics and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or powerlaw dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems. redirect the reaction z → y * . Instead of the reaction z → y * with flux J z→y * , we have M reactions z → y j with fluxes J ′ z→y j = J y * →y j . Let (G ′ , J ′ ) denote this newest flux system. Recall that flux equivalence means Equation (11) holds at each vertex of G and G ′ . Here we only need to look at the vertex z to show that (G ′ , J ′ ) ∼ (G, J ). Note that y j − z = w j − w 0 . From P (G,J) (y * ) = 0, we also have M j=1 J y * →y j = J z→y * . Thus, the weighted sum of vectors coming out of z isFinally, we prove that the potentials are unchanged. Trivially P (G,J ) (y * ) = P (G ′ ,J ′ ) (y * ) = 0. Also P (G,J) (y j ) = J y * →y j = J ′ z→y j = P (G ′ ,J ′ ) (y j ) for j = 1, 2, . . . , M . Last but not least,J y * →y j = J z→y * = −P (G,J) (z).We have shown that the resulting flux system (G ′ , J ′ ) is flux equivalent to the original flux system (G, J ), and the potential at each vertex is preserved.Remark. In Lemma 4.3, the source vertex z may not be distinct from y j .

Single-target networks

“…Another method, staying within the realm of mass-action systems, is that of dynamical equivalence. The associated dynamical system (1) of a mass-action system G κ is uniquely defined; however, different reaction networks can give rise to the same system of differential equations under mass-action kinetics [14,34,11]. We say that a system of differential equationsẋ = f (x) can be realized by a network G if there exists a vector of rate constants κ > 0 such that the associated dynamical system of G κ isẋ = f (x).…”

“…Indeed, two massaction systems are dynamically equivalent if and only if the corresponding fluxes at an arbitrary state satisfy (6) for every vertex. See [11] for the correspondence between mass-action systems and fluxes on a network.…”

“…An implementation of some of these algorithms is available as a MATLAB toolbox [35]. Although these algorithms require a predetermined set of vertices as input, in the case of detailed-balanced, complex-balanced, reversible or weakly reversible realizations, it suffices to use the exponents of the monomials in the differential equations [11].…”

Detailed Balance = Complex Balance + Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains

“…In various important cases, it has been shown that positive CB equilibria are globally stable (Anderson 2011;Craciun et al 2013), a property that is conjectured to hold for all CB systems (Horn 1974;Craciun 2015). Finally, mass-action systems that are not CB may be dynamically equivalent to CB systems and have all their strong properties (Craciun et al 2020).…”

“…Finally, mass-action systems that are not CB may be dynamically equivalent to CB systems and have all their strong properties (Craciun et al. 2020 ).…”