  Detailed balance of a chemical reaction network can be defined in several
different ways. Here we investigate the relationship among four types of
detailed balance conditions: deterministic, stochastic, local, and zero-order
local detailed balance. We show that the four types of detailed balance are
equivalent when different reactions lead to different species changes and are
not equivalent when some different reactions lead to the same species change.
Under the condition of local detailed balance, we further show that the system
has a global potential defined over the whole space, which plays the central
role in the large deviation theory and the Freidlin-Wentzell-type metastability
theory of chemical reaction networks. Finally, we provide a new sufficient
condition for stochastic detailed balance, which is applied to construct a
class of high-dimensional chemical reaction networks that both satisfies
stochastic detailed balance and displays multistability.
