  In recent years moment-closure approximations (MA) of the chemical master
equation have become a popular method for the study of stochastic effects in
chemical reaction systems. Several different MA methods have been proposed and
applied in the literature, but it remains unclear how they perform with respect
to each other. In this paper we study the normal, Poisson, log-normal and
central-moment-neglect MAs by applying them to understand the stochastic
properties of chemical systems whose deterministic rate equations show the
properties of bistability, ultrasensitivity and oscillatory behaviour. Our
results suggest that the normal MA is favourable over the other studied MAs. In
particular we found that (i) the size of the region of parameter space where a
closure gives physically meaningful results, e.g. positive mean and variance,
is considerably larger for the normal closure than for the other three
closures; (ii) the accuracy of the predictions of the four closures (relative
to simulations using the stochastic simulation algorithm) is comparable in
those regions of parameter space where all closures give physically meaningful
results; (iii) the Poisson and log-normal MAs are not uniquely defined for
systems involving conservation laws in molecule numbers. We also describe the
new software package MOCA which enables the automated numerical analysis of
various MA methods in a graphical user interface and which was used to perform
the comparative analysis presented in this paper. MOCA allows the user to
develop novel closure methods and can treat polynomial, non-polynomial, as well
as time-dependent propensity functions, thus being applicable to virtually any
chemical reaction system.
