  The Michaelis-Menten equation has played a central role in our understanding
of biochemical processes. It has long been understood how this equation
approximates the dynamics of irreversible enzymatic reactions. However, a
similar approximation in the case of networks, where the product of one
reaction can act as an enzyme in another, has not been fully developed. Here we
rigorously derive such an approximation in a class of coupled enzymatic
networks where the individual interactions are of Michaelis-Menten type. We
show that the sufficient conditions for the validity of the total quasi steady
state assumption (tQSSA), obtained in a single protein case by Borghans, de
Boer and Segel can be extended to sufficient conditions for the validity of the
tQSSA in a large class of enzymatic networks. Secondly, we derive reduced
equations that approximate the network's dynamics and involve only protein
concentrations. This significantly reduces the number of equations necessary to
model such systems. We prove the validity of this approximation using geometric
singular perturbation theory and results about matrix differentiation. The
ideas used in deriving the approximating equations are quite general, and can
be used to systematize other model reductions.
