  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.
