  The quasi-steady state assumption (QSSA) forms the basis for rigorous
mathematical justification of the Michaelis-Menten formalism commonly used in
modeling a broad range of intracellular phenomena. A critical supposition of
QSSA-based analyses is that the underlying biochemical reaction is
enzymatically "closed," so that free enzyme is neither added to nor removed
from the reaction over the relevant time period. Yet there are multiple
circumstances in living cells under which this assumption may not hold, e.g.
during translation of genetic elements or metabolic regulatory events. Here we
consider a modified version of the most basic enzyme-catalyzed reaction which
incorporates enzyme input and removal. We extend the QSSA to this enzymatically
"open" system, computing inner approximations to its dynamics, and we compare
the behavior of the full open system, our approximations, and the closed system
under broad range of kinetic parameters. We also derive conditions under which
our new approximations are provably valid; numerical simulations demonstrate
that our approximations remain quite accurate even when these conditions are
not satisfied. Finally, we investigate the possibility of damped oscillatory
behavior in the enzymatically open reaction.
