Biomedical Engineering Reference
In-Depth Information
in phosphoinositide levels at the plasma membrane (33). This suggests that in
neutrophils the time scale of adaptation is so slow that the cells polarize sponta-
neously before there is significant adaptation.
In short, the phosphoinositide distributions display four distinct types of
spatiotemporal dynamics in response to various chemoattractant gradients: 1)
amplification and threshold, 2) polarized sensitivity, 3) spontaneous polariza-
tion, and 4) adaptation. Several mathematical models have been developed to
capture one or more of these dynamics. Iglesias and coworkers have formulated
a series of models that focus on adaptation and amplification (36). Although
these models display perfect adaptation, they capture neither polarized sensitiv-
ity nor spontaneous polarization, and amplification is quite weak compared to
experiments. The model in (37) is concerned primarily with amplification. The
remaining models are formally similar insofar as they contain a
short-range ac-
tivator
that is synthesized autocatalytically, and a
long-range inhibitor
that in-
hibits the synthesis of the activator (38-41). These models differ only with
respect to the postulated mechanisms of activation and inhibition. In the next
section, we present our model and show that it captures the dynamics of amplifi-
cation, threshold, polarized sensitivity and spontaneous polarization. Although
the other two activator-inhibitor models developed by Meinhardt and Postma et
al. have not been studied in detail, the fact that they are formally similar to our
model suggests that they possess similar dynamic properties. The discussion of
adaptation is deferred to §3.
2.
MODEL AND SIMULATION
2.1. Model
The model is an abstraction of the phosphoinositide cycle (Figure 2a). It
contains three variables corresponding to three "lumped" pools (Figure 3a)—
namely, membrane phosphoinositides (
P
), cytosolic and its phosphates (
I
), and
phosphoinositides in the endoplasmic reticulum (
P
s
). The concentrations of these
variables are denoted by
p
,
i
, and
p
s
, respectively. It is assumed that
1. The cell is two-dimensional and disk-shaped. Thus,
p
and
p
s
are based on
the length of the plasma membrane and
i
is based on the area of the cytosol.
2. Radial gradients of the cytosolic inositol phosphate pool are negligibly
small. This is reasonable because inositol phosphates diffuse rapidly.
2
It follows
from assumptions 1 and 2 that the angle, R is the only spatial variable.
3. In the absence of receptor activation, there is
basal
synthesis and degra-
dation of
P
and
I
. The basal synthesis rates of
P
and
I
, denoted
c
p
and
c
i
, follow
zero-order kinetics. The basal degradation rates, denoted
r
p,d
and
r
i,d
, obey first-
order kinetics with rate constants
k
p
and
k
i
, respectively, i.e.,
r
p,d
k
p
p
,
r
i,d
k
i
i
