Abstract
We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, “formal balance,” and the new notion of “cycle balance” are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.
Highlights
Detailed balance and complex balance are important concepts in chemical reaction network theory (CRNT)
We show that complex balance plus a condition significantly weaker than formal balance, namely the absence of directed cycles in an induced graph, is equivalent to detailed balance
We introduce two other variants of cycle balance which are defined by inequalities and which are weaker than formal balance
Summary
Detailed balance and complex balance are important concepts in chemical reaction network theory (CRNT). For mass-action kinetics, complex balance has been characterized by Horn (1972), and explicit conditions on the “tree constants” of the underlying graph have been provided by Craciun et al (2009); see (Johnston 2014; Müller and Regensburger 2014). Schuster and Schuster consider “generalized mass-action kinetics” in the sense that the net reaction rate contains a mass-action factor (as for enzyme kinetics) They provide “generalized Wegscheider’s conditions” on the equilibrium constants; they obtain r − s (= γ + δ) independent conditions. We provide new conditions on a complex-balanced equilibrium of a reversible chemical reaction network to be detailed-balanced. We give an elementary graph-theoretic (non-algebraic) proof of the previous result (without using the conditions on the tree/equilibrium constants for complex/detailed balance). We apply it to different types of networks (balance in reaction networks and balance in Markov chains)
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