  The application of the quasi-steady-state approximation to the
Michaelis-Menten reaction embedded in large open chemical reaction networks is
a popular model reduction technique in deterministic and stochastic simulations
of biochemical reactions inside cells. It is frequently assumed that the
predictions of the reduced master equations obtained using the stochastic
quasi-steady-state approach are in very good agreement with the predictions of
the full master equations, provided the conditions for the validity of the
deterministic quasi-steady-state approximation are fulfilled. We here use the
linear-noise approximation to show that this assumption is not generally
justified for the Michaelis-Menten reaction with substrate input, the simplest
example of an open embedded enzyme reaction. The reduced master equation
approach is found to considerably overestimate the size of intrinsic noise at
low copy numbers of molecules. A simple formula is obtained for the relative
error between the predictions of the reduced and full master equations for the
variance of the substrate concentration fluctuations. The maximum error is
reached when modeling moderately or highly efficient enzymes, in which case the
error is approximately 30%. The theoretical predictions are validated by
stochastic simulations using experimental parameter values for enzymes involved
in proteolysis, gluconeogenesis and fermentation.
