**BIOCHEMISTRY TOPICS**

### Michaelis-Menten equation

Assmptions underlying the mechanistic model that leads to the Michaelis-Menten equation.

Consider the simple model for the enzymatic catalysis of a unimolecular reaction converting a substrate S to a product P, shown below. We will derive an expression for the initial rate of the enzyme catalyzed reaction represented by this model, which we will call the Michaelis-Menten equation.

Three rate constants are defined in this model: the forward rate constant *k*_{1}
for the formation of the enzyme-substrate (ES) complex from free enzyme and unbound substrate;
the reverse rate constant *k*_{−1} for the dissociation of ES back to E + S;
and the rate constant for the catalytic step, *k*_{2}, *i.e*.
the unimolecular rate constant for the breakdown of ES to E + P.
Note that in the absence of a catalytic step, ES and E + S would reach an equilibrium,
with an equilibrium constant *K*_{eq} = *k*_{1}/*k*_{−1}.
We define a dissociation constant, *K*_{d}, which is the inverse of *K*_{eq}.
Binding affinities are often measured or reported in terms of *K*_{d}.
A lower *K*_{d} means a tighter complex; a higher *K*_{d} means a weaker complex.
The initial rate *V*_{0} of the forward reaction is fundamentally dependent on the concentration of ES,
and is given by the equation *V*0 = *k*_{2}[ES].
We can make two alternative assumptions to obtain an expression to substitute for [ES].

To continue with the derivation of the
rate equation for our simple kinetic model, we note the relation
between the concentrations of free enzyme, the enzyme-substrate complex,
and the total enzyme. In the case where substrate is present in significant
excess over enzyme - a reasonable condition since enzymes accelerate
reactions in amounts substoichiometric to substrate - the analysis
is simplified. We can then solve explicitly for [ES] in terms of
experimental
"knowns" - the total enzyme and total substrate concentrations
- and the parameter *K*_{M}. Then we use the expression
*V*_{0} = *k*_{cat}[ES] to express
the measurable initial reaction rate in these terms, plus the additional
parameter *V*_{max}.

**An important point**: Although
this equation was derived using a particular simple mechanism, it
is valid for many more complex mechanisms. It
reproduces the saturation kinetics characteristic of enzyme-catalyzed
reactions.

That is, *V*_{0} approaches
a limit of *V*_{max} as substrate concentration
increases. We also see that *K*_{M} represents
a substrate concentration at which *V*_{0} reaches
one-half *V*_{max}.
(Substitution of ½*V*_{max} for
*V*_{0} in the above equation and solving for [S] shows
that [S] = *K*_{M}. ) Note that [S] must be quite
large before *V*_{0} becomes close to *V*_{max} .
For instance, at [S] = 5*K*_{M}, the initial rate
is only 83% of *V*_{max}.