Detailed Balance $$=$$ = Complex Balance $$+$$ + Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Mark
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Detailed Balance = Complex Balance + Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains Stefan Müller1
· Badal Joshi2
Received: 12 March 2020 / Accepted: 13 August 2020 © The Author(s) 2020
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. Keywords Chemical reaction network · Arbitrary kinetics · Graph theory · Induced graph · Mixed graph
1 Introduction Detailed balance and complex balance are important concepts in chemical reaction network theory (CRNT). Both principles have been proposed already in the 1870s and 1880s by Ludwig Boltzmann in the kinetic theory of gases (where complex balance is called semi-detailed balance) (Boltzmann 1872, 1887). Around 1900, Rudolf Wegscheider introduced the principle of detailed balance in the field of chemical kinetics (and obtained the necessary conditions on the rate constants named after him) (Wegscheider 1901). Only in the 1970s, Horn and Jackson developed the concept
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Stefan Müller [email protected] Badal Joshi [email protected]
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Faculty of Mathematics, University of Vienna, Vienna, Austria
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Department of Mathematics, California State University San Marcos, San Marcos, CA, USA 0123456789().: V,-vol
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of complex balance (as a generalization of detailed balance) in modern CRNT (Horn and Jackson 1972). Complex-balanced (CB) mass-action systems display remarkably robust dynamics. If one positive equilibrium is CB, then so is every other equilibrium, which justifies calling the entire system CB. Moreover, there is exactly one positive equilibrium in every stoichiometric class (invariant set), and this equilibrium is asymptotically stable (implied by a strict Lyapunov function) (Horn and Jackson 1972). 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). 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 also (Johnston 2014; Müller and Regensburger 2014). Detailed balance has been characterized by Feinberg (1989) and Schuster and Schuster (1989). Fe
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