Biomedical Engineering Reference
In-Depth Information
phosphoinositide synthesis relative to its rate of diffusion, the transition layer
moves in a wavelike fashion at a velocity that is proportional to the net rate of
phosphoinositide accumulation within the transition layer.
The response to sharp chemoattractant gradients (Figure 5c) can be ex-
plained as follows. After the steady state has developed, the concentration of the
inhibitor,
I
, is high throughout the cell. This tends to increase the threshold at all
points of the plasma membrane. When the gradient is switched, the active recep-
tor concentration decreases at the previous "front" and increases at the current
"front." It follows from the earlier discussion regarding thresholds (see Figure
3b) that at the current "front" the tendency of the threshold to increase due to
elevated inhibitor concentrations is mitigated by the higher active receptor con-
centration. However, at the "previous" front the tendency of the threshold to
increase due to elevated inhibitor concentrations is further exacerbated by the
lower active receptor concentration. The thresholds at the "previous" front be-
come so large that, despite the large concentrations of membrane phosphoinosi-
tides, they fall short of the threshold, and the preexisting peak collapses.
2.2.3. Spontaneous Polarization
The random location of the polarity in
spontaneous
polarization suggests
that some variable that is upstream of the phosphoinositides in the signal trans-
duction pathway undergoes stochastic fluctuations. The most upstream source of
the stochastic fluctuations is receptor-ligand binding (44). Following (5), we
construct a stochastic model of receptor-ligand binding.
To this end, we partition a cell containing
r
t
receptors into
n
equal sections.
We assume that this cell is exposed to some uniform chemoattractant concentra-
tion
l
. If ligand binds instantaneously to the receptors, the mean number of ac-
tive receptors in each section, denoted
r
m
, is (
r
t
/
n
)
l
/(
k
-
/
k
+
+l), where
k
+
and
k
-
are
the rate constants for receptor-ligand association and dissociation, respectively.
The number of active receptors in each section, denoted,
r
i
, is given by the sto-
chastic differential equation
(
)
(
¯
)
(
)
n
.
[12]
dr
=+
k l
+
k
r
r dt
+
k l r n
+
/
r
+
k r dW i
,
=
1, 2...
¡
°
¢
±
i
m
i
t
m
m
i
where the first term on the right denotes the
deterministic
part of receptor ligand
binding, which has the effect bringing
r
i
back to its mean value,
r
m
. The second
term on the right denotes the
stochastic
part of the binding process. Here,
dW
i
denotes the Wiener process, which is a Gaussian random number generator with
zero mean and standard deviation (
dt
)
1/2
. It should be noted that the standard de-
viation of the random process, E
r
i
, is proportional to (
r
t
)
1/2
, but the relative
spread, defined as the ratio, E
r
i
/r
m
, is inversely proportional to (
r
t
)
1/2
. Thus, the
