Biomedical Engineering Reference
In-Depth Information
state, (
p
-
,
p
s
-
,
i
-
), corresponding to a uniform chemoattractant concentration;
hence,
r
=
r
-
= constant. At time
t
= 0, the cell is perturbed by exposing it to a
chemoattractant profile that is instantly mirrored by the active receptor profile,
r
(
t
,R). The dynamics of
P
,
P
s
, and
I
are then governed by the equations
2
D
s
p
s
p
p
=
krpp kpi c kp
2
+
+
,
[1]
f
s
r
p
p
s
t
R
2
s
R
2
D
s
p
s
2
p
(
)
p
2
s
=
krpp kpi c kp
+
+
s
,
[2]
f
s
r
p
p
s
t
R
2
s
R
2
D
2
s
i
s
i
(
)
2
=
skrpp kpi c ki
+
+
i
.
[3]
f
s
r
i
i
s
t
R
2
s
R
2
Here,
D
p
,
D
,
D
i
denote the lateral diffusivities of
P
,
P
s
, and
I
, respectively, and
R
denotes the cell radius. The factor
s
, denoting membrane length per unit cell
area, is required since synthesis and removal rates of
P
are based on the length
of the plasma membrane. Since the cell is circular, all concentrations and fluxes
must be equal at R = 0 and R = 2Q. Thus, we impose the
periodic
boundary con-
ditions
s
()( )
() (
)
xt
x t
s
0,
s
2 ,
Q
xt x t
0,
=
2 ,
Q
,
=
,
t
>
0
,
[4]
s
R
s
R
where
x
p
,
p
s
,
i
. The initial conditions for the reduced equations are
p
(0,R) =
p
-
,
p
s
(0,R) =
p
t
-
p
-
,
i
(0,R) =
i
-
, 0 < R < 2Q.
[5]
Here, the initial condition for
p
s
reflects the assumption that the total amount of
phosphoinositide in the membrane and the endoplasmic reticulum is conserved,
so that the average phosphoinositide concentration, denoted
p
t
, is constant (39).
It is convenient to define the dimensionless variables
p
p
i
R
t
Q
wwww
,
Q
s
,
J
,
Y
,
U
w
,
s
(
)
p
p
sp
2 3.1416
×
1/
k sp
t
t
t
r
t
and dimensionless parameters
2
2
krp
c p
/
k
D C
/
r
r
p
ft t
p t
p
S
ww w w w
s
s
L
Z
L
E
,
f
p
p
p
k sp
k sp
k sp
k sp
t
r
t
r
t
r
t
r
t
