Building on the derivation of Michaelis-Menten kinetics, we now turn to enzymes with multiple substrate-binding sites.

Early work in this regard was
carried out by Adair and Pauling, operating under the **rapid equilibrium
approximation**. Remember that this assumption states that all substrate
binding and dissociation steps happen much more rapidly than catalytically
productive steps. In 1954, King and Altman showed how to solve any kinetic
system without resorting to this approximation, but this is unnecessarily
complicated for our purposes. Even if the rapid equilibrium approximation does
not hold for our system, the forms of the equations we derive will not change –
instead, the significance of the various *K _{M}* values will be
different. This is analogous to the difference between

We begin with the simplest model
of multiple binding: a two-site sequential model. Here, an enzyme *E* can
bind a single molecule of substrate *S* to form a singly-occupied complex *ES*
with equilibrium dissociation constant *K _{D1}*.

In order to calculate the
reaction velocity for this system, we need to know the relative concentrations
of the active species *ES* and *ESS*. We will describe a
straightforward derivation of these quantities, and then a shortcut that allows
one to quickly solve more complicated kinetic systems.

From the definitions of the two dissociation constants, we have:

Relative to the total enzyme
concentration, *E _{T}*:

The total reaction velocity for
this system is given by (where *V*_{max1} = *k _{cat1}*[

This graph shows the relative concentrations of the
two active species as a function of substrate concentration, when both *K _{D}*s
are set to 10 μM. Note the initial increase and subsequent decrease in the
level of

There is a quicker way to calculate reaction velocity equations of this type. To return to the scheme of a two-site sequential model:

Each species in the system has a
relative concentration term that appears in the denominator of the velocity
equation, while terms from all active species appear in the numerator. To find
the relative concentration term of any species, simply work your way backwards
from that species to the free enzyme; the specific term is equal to the product
of all the substrate molecules that bind, divided by all the equilibrium
constants in the chain. So the specific term for *ES* is [*S*]/*K _{D1}*
and the term for

The denominator of the velocity
equation is the sum of all specific terms: 1 + [*S*]/*K _{D1}*
+ [