A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced

Abstract

Certain chemical reaction networks (CRNs) when modeled as a deterministic dynamical system taken with mass-action kinetics have the property of reaction network detailed balance (RNDB) which is achieved by imposing network-related constraints on the reaction rate constants. Markov chains (whether arising as models of CRNs or otherwise) have their own notion of detailed balance, imposed by the network structure of the graph of the transition matrix of the Markov chain. When considering Markov chains arising from chemical reaction networks with mass-action kinetics, we will refer to this property as Markov chain detailed balance (MCDB). Finally, we refer to the stochastic analog of RNDB as Whittle stochastic detailed balance (WSDB). It is known that RNDB and WSDB are equivalent. We prove that WSDB and MCDB are also intimately related but are not equivalent. While RNDB implies MCDB, the converse is not true. The conditions on rate constants that result in networks with MCDB but without RNDB are stringent, and thus examples of this phenomenon are rare, a notable exception is a network whose Markov chain is a birth and death process. We give a new algorithm to find conditions on the rate constants that are required for MCDB.

Graphically Balanced Equilibria and Stationary Measures of Reaction Networks

“…As an immediate consequence of Corollary 4.10 and Theorem 5.6, the following can be stated, in the spirit of Theorem 5.2 (due to [15]):…”

“…In [1] it is further proven that such models are non-explosive. In [15] it is proven that if an ODE model with mass action kinetics has a positive detailed balanced equilibrium, then the stationary distributions of the corresponding Markov chain model are detailed balanced, in the classical probabilistic sense [21]. Finally, in [7] it is shown that if an ODE model with mass action kinetics is complex balanced, then the stochastic counterpart has a so called complex balanced stationary distribution, and the converse holds as well.…”

Chemical Boltzmann Machines

“…Such systems must be coupled to implicit reservoirs of fuel molecules that can drive the species of interest into a non-equilibrium steady state [41,42,43]. Usually -but not always [44,45] -the resultant Markov chain violates detailed balance. In Section 3.1, we shall consider a system that exhibits detailed balance at the level of the Markov chain, but is necessarily non-equilibrium and violates detailed balance at the detailed chemical level.…”

Detailed balance in micro- and macrokinetics and micro-distinguishability of macro-processes

“…For example, in chemical mass action law kinetics we can consider the reaction mechanism A ⇋ B (rate constants k ±1 ), A + B ⇋ 2B (rate constants k ±1 ) [22]. We can also create a stochastic model for this system with the states (xA, yB) (x, y are nonnegative integers) and the elementary transitions (xA, yB) ⇋ ((x − 1)A, (y + 1)B) (rate constants κ + = k +1 x + k +2 x 2 , κ − = k −1 (y + 1) + k −2 (x − 1)(y + 1)).…”

“…Thus, macroscopic detailed balance may be violated in this example when microscopic detailed balance holds. (For more examples and theoretic consideration of the relations between detailed balance in mass action law chemical kinetics and stochastic models of these systems see [22].) Indeed, both of the macroscopic elementary processes A ⇋ B and A + B ⇋ 2B correspond to the same set of microscopic elementary processes (xA, yB) ⇋ ((x − 1)A, (y + 1)B).…”