Biology Reference
In-Depth Information
where square brackets [ ] are used to denote concentration. Thus, [Drug]
implies the concentration, or more rigorously, the activity of the drug
that is free in solution (i.e., unbound, or not bound to the receptor). Here
a reversible reaction is one in which the drug-receptor complex can
easily dissociate into the unbound forms [shown by the double-headed
arrow in Eq. (7-1)]. The drug and receptor can also easily reform
the drug-receptor complex. The pharmacological effect is assumed
to be proportional to the concentration of the drug-receptor complex.
This is the simplest of all possible drug-receptor interaction
mechanisms.
The binding of the drug and receptor follows the law of mass action,
which states that the concentration of the drug-receptor complex is
proportional to the product of the molar concentrations of the unbound
drug and unoccupied receptors. The law of mass action was
experimentally derived and reported by Waage and Guldberg in 1864.
Once a conceptual model is rigorously stated, it can be translated into
a mathematical model. Here, the concentration of the drug-receptor
complex can be formulated in terms of the unbound (free) concentrations
of the drug and receptor as:
½
Drug
Receptor
¼
K
a
½
Drug
½
Receptor
:
(7-2)
The proportionality constant K
a
is called association equilibrium constant.
These reactions are also commonly written in terms of a dissociation
constant, K
d
, where:
K
a
¼
1
=
K
d
:
We shall derive Eq. (7-2) from the differential equations governing the
reaction. Assume that the reaction has the following general form:
A
þ
B
$
C
:
Then, the rates of change in the concentrations of A, B, and C will be
given by the following differential equations:
d
½
C
¼
k
1
½
A
½
B
k
2
½
C
dt
½
d
A
¼
k
2
½
C
k
1
½
A
½
B
(7-3)
dt
d
dt
¼
½
B
k
2
½
C
k
1
½
A
½
B
:
Here, k
1
and k
2
are the respective reaction rate constants for the
reactions A
þ
!
!
þ
B. These equations are derived
using considerations similar to those discussed in Chapters 1 and 2 for
B
C and C
A
