  Biochemical processes typically involve many chemical species, some in
abundance and some in low molecule numbers. Here we first identify the rate
constant limits under which the concentrations of a given set of species will
tend to infinity (the abundant species) while the concentrations of all other
species remains constant (the non-abundant species). Subsequently we prove that
in this limit, the fluctuations in the molecule numbers of non-abundant species
are accurately described by a hybrid stochastic description consisting of a
chemical master equation coupled to deterministic rate equations. This is a
reduced description when compared to the conventional chemical master equation
which describes the fluctuations in both abundant and non-abundant species. We
show that the reduced master equation can be solved exactly for a number of
biochemical networks involving gene expression and enzyme catalysis, whose
conventional chemical master equation description is analytically impenetrable.
We use the linear noise approximation to obtain approximate expressions for the
difference between the variance of fluctuations in the non-abundant species as
predicted by the hybrid approach and by the conventional chemical master
equation. Furthermore we show that surprisingly, irrespective of any separation
in the mean molecule numbers of various species, the conventional and hybrid
master equations exactly agree for a class of chemical systems.
