Enzyme Kinetics and Metabolic Control Analysis
193
r =
k2eas
s
+
K.„
(6.8)
Derivation of eq (6.8) by Michaelis and Menten in 1913 signified virtually the beginning of
quantitative enzymology. Their equilibrium assumption for step 1 of (6.2) was relaxed by Briggs
and Haldane (1925), who used a similarly speculative hypothesis of pseudo-steady state for
ES.
Both models - which typically for mechanistically based models can assign a definite physical
meaning to the parameters - lead to eq. (6.1) where
Km
is called the Michaelis constant in honor
of the recognized “father” of enzymology1. The picture of
Km
as an equilibrium constant for a
dissociation reaction according to (6.8) is illustrative: a substrate that is easily captured by the
enzyme (the enzyme has a high affinity for the substrate) will have a small
K m
value. If
k2
is large
- i.e. if the rate of the second reaction in (6.2) is high then the rate of conversion of the substrate
by the enzyme is high, also at low values of
5.
Note 6.1 Assumptions in the mechanistic models for enzyme kinetics
The simplicity of the derivation of the two mechanistically based models for the rate of an enzyme
reaction could lead the reader to believe that here, for once, there is a trustworthy piece of modeling in
biotechnology. Indeed, the rate expression (6.1) is very robust - much more so than the apparently
equivalent Monod model for cell kinetics, eq. (7.16). It allows for a significant extrapolation from the
data that are used to determine the two parameters
k
and
and the parameters are clearly related to
stringently defined parameters of elementary reactions. Still, it may be useful to give a few comments to
illustrate the effect of the approximations that lurk behind both the Michaelis Menten and the Briggs
Haldane derivation of eq. (6.1). First of all the comments given below may help the user to avoid
mistakes in an experimental set up which is aimed to calculate the two kinetic parameters.
First one can easily see that the derivation of eq. (6.6) leads to the same result as (6.8) if, indeed the first
step of (6.2) is an equilibrium reaction. Then both
kA
and k, in (6.2) are infinitely large and the ratio k.i/'k,
=
Km
is equal to the equilibrium constant
Kcq
of (6.7) and (6.8). Also the assumption of (6.3) that (es) is
constant is true. The amount of
ES
consumed by the second step of (6.2) is immediately replenished by
the fast equilibrium reaction in step 1.
But (6.1) is also derived by (6.2) to (6.6) without the apparently unnecessary assumption of a fast
equilibrium - the value of
Km
must only be interpreted differently. How trustworthy is the assumption of
quasi-stationarity of (es)? Clearly if all three rate constants
k,, kA
and
k2
are small it may take a very long
time until (es) becomes constant. In fact eq. (6.3) can easily be integrated (by separation of variables) for
a constant s, and (es) can therefore be found as a function of time. When es(t) is inserted in the overall
rate expression (6.6) one obtains
r = b - -—
(1 - exp[—
(kts +
+
k 2
)?] )
s + K.
a )
It is well established that
k.h k_,
and
k,s
are very large, 100 to 1000 s'1
for normal enzymes, and the
Menten had a sad fate, discriminated as a Jew and ending his life in London in the early 1930’s