Biomedical Engineering Reference
In-Depth Information
L
is the free ligand concentration
K
d
is the apparent dissociation constant derived from the law of mass
action
K
A
is the ligand concentration producing half occupation that is also the
microscopic dissociation constant
n
is the Hill coefficient
A coefficient of
n
= 1 indicates completely independent binding, regard-
less of how many additional ligands are already bound. Numbers of
n
> 1
indicate positive cooperativeness, meaning that once one ligand molecule
is bound to the enzyme, its affinity for other ligand molecules increases.
Conversely, numbers of
n
< 1 indicate negative cooperativeness, such that
once one ligand molecule is bound to the enzyme, its affinity for other ligand
molecules decreases.
In cell biology, cell responses such as differentiation, proliferation, and
apoptosis are all related to various ligand-receptor reactions, of which some
are stimulatory and others are inhibitory [3]. The same observations are true
for the production of molecules due to receptor-ligand interactions. Pivonka
et al. [3] assumed that a cellular process is governed by a single factor
x,
which is a ligand that governs the production rate of a cell or molecule
z
through binding to its receptor on the cell. They then expressed the rate of
production
z
per unit time as a function of the concentration
x
in its active
form
x
*
as
z
=
f
(
x
*
)
( 7. 2)
The input function
f
is, in general, a monotonic, S-shaped function. In
modeling the cell responses governed by Equation (7.2), the Hill equation is
often used to describe the molecular input function. The activation (“act” for
short) and repression (“rep” for short) forms of the Hill equation [51] for the
production rate of a new cell or molecule are [3]
∗
x
Kx
β
(
)
∗
fx
=β⋅Π =
( 7. 3)
act
∗
+
1
β
(
)
∗
fx
=β⋅Π =
( 7. 4)
rep
x
K
∗
1
+
2
where β is the maximal production rate of molecule
z,
Π
i
(
i
=
act,
rep
) is the
input function, and
K
1
and
K
2
are activation and repression coefficients. Note
here that we have already assumed that the Hill coefficient equals one. In
the model described in this chapter, Qin and Wang [2] extended the Hill
equation to the case where two ligands,
x
and
y,
both affect the production
