  We recall the perturbation expansion for Michaelis-Menten kinetics, beyond
the standard quasi-steady-state approximation (sQSSA). Against this background,
we are able to appropriately apply the alternative approach to the study of
singularly perturbed differential equations that is based on the
renormalization group (SPDERG), by clarifying similarities and differences. In
the present demanding situation, we directly renormalize the bare initial
condition value for the substrate. Our main results are: i) the 2nd order
SPDERG uniform approximations to the correct solutions contain, up to 1st
order, the same outer components as the known perturbation expansion ones; ii)
the differential equation to be solved for the derivation of the 1st order
outer substrate component is simpler within the SPDERG approach; iii) the
approximations better reproduce the numerical solutions of the original problem
in a region encompassing the matching one, because of the 2nd order terms in
the inner components, calculated here for the first time to our knowledge: iv)
the refined SPDERG uniform approximations, that we propose, give the correct
asymptotically vanishing solutions, too, and allow to obtain results nearly
indistinguishable from the solutions of the original problem in a large part of
the whole relevant time window, even in the studied unfavourable kinetic
constant case, for an expansion parameter value as large as {\epsilon} = 0.5.
