  Approximations based on moment-closure (MA) are commonly used to obtain
estimates of the mean molecule numbers and of the variance of fluctuations in
the number of molecules of chemical systems. The advantage of this approach is
that it can be far less computationally expensive than exact stochastic
simulations of the chemical master equation. Here we numerically study the
conditions under which the MA equations yield results reflecting the true
stochastic dynamics of the system. We show that for bistable and oscillatory
chemical systems with deterministic initial conditions, the solution of the MA
equations can be interpreted as a valid approximation to the true moments of
the CME, only when the steady-state mean molecule numbers obtained from the
chemical master equation fall within a certain finite range. The same validity
criterion for monostable systems implies that the steady-state mean molecule
numbers obtained from the chemical master equation must be above a certain
threshold. For mean molecule numbers outside of this range of validity, the MA
equations lead to either qualitatively wrong oscillatory dynamics or to
unphysical predictions such as negative variances in the molecule numbers or
multiple steady-state moments of the stationary distribution as the initial
conditions are varied. Our results clarify the range of validity of the MA
approach and show that pitfalls in the interpretation of the results can only
be overcome through the systematic comparison of the solutions of the MA
equations of a certain order with those of higher orders.
