Detailed balance in reversible chemical reaction networks (CRNs) is a property possessed by certain chemical reaction networks (CRNs) when modeled as a deterministic dynamical system taken with mass-action kinetics whose reaction rate parameters are appropriately constrained, the constraints being imposed by the network structure of the CRN. We will refer to this property as reaction network detailed balance (RNDB)  . 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 , in the sense that they require an identical set of conditions on the rate constants . We prove that WSDB and MCDB are also intimately related but are not equivalent , although they are sometimes confused for each other . 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. Using the fact that RNDB implies MCDB, we give a new algorithm to find conditions on the rate constants that are required for MCDB and we obtain an explicit formula for the stationary distribution of networks with RNDB .
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 .
R
Detailed balance in reversible
Certain
D
is a property possessed by certain chemical reaction networks (CRNs)
R
whose reaction rate parameters are appropriately constrained, the constraints being imposed by the network structure of the CRN. We will refer to this property as
have the property of
A
which is achieved by imposing network-related constraints on the reaction rate constants
D
, in the sense that they require an identical set of conditions on the rate constants
D
, although they are sometimes confused for each other
R
Using the fact that RNDB implies MCDB, we
We
D
and we obtain an explicit formula for the stationary distribution of networks with RNDB