Biomedical Engineering Reference
In-Depth Information
Table 1 . Parameter values used in the simulations (from (39) and (42))
Z i = 0.01
dimensionless rate of basal synthesis of inositol phosphates
Z p = 0.01
dimensionless rate of basal synthesis of phosphoinositides
E i = 1
dimensionless angular diffusivity of inositol phosphates
E p = 0.001
dimensionless angular diffusivity of phosphoinositides in plasma
membrane
E s p = 0.001
dimensionless angular diffusivity of phosphoinositides in endoplasmic
reticulum
L f = 2.4
dimensionless rate constant for receptor mediated phosphoinositide
formation
L p = 0.1
dimensionless rate constant for basal degradation of phosphoinositides
L i = 0.1
dimensionless rate constant for basal degradation of inositol phosphates
L - = 6.7 x 10 -3
dimensionless rate constant for receptor-ligand dissociation
r t = 50,000
total number of receptors
2.2.1. Amplification and Threshold
In the following simulation, the cell is assumed to be at a uniform steady
state (Q - , Q s - , t - ) corresponding to the uniform active receptor profile S = S - . At
time t 0, the cell is exposed to a steady chemoattractant gradient and the active
receptor distribution instantly inherits the chemoattractant concentration profile
(Figure 4a). Despite the mild gradient of S, a pronounced phosphoinositide peak
ultimately develops at the leading edge, Y = 1/2 (Figure 4b). Compared to the
polarized distribution of membrane phosphoinositides, the concentration profile
of inositol phosphates is virtually flat (Figure 4c).
In terms of the model, formation of the phosphoinositide peak can be ex-
plained as follows. Because of their autocatalytic and cooperative kinetics,
membrane phosphoinositides (P) are strongly amplified beyond a certain thresh-
old. To see this, observe that immediately after receptor activation, Q s Q s - = 1 -
Q, t t - , and diffusion is negligibly small compared to the reaction. Hence, the
initial dynamics of the membrane phosphoinositides at any point of the plasma
membrane is approximated by the equation
s
Q
(
)
2
=
LSQ Q QJ Z LQ
1
+
.
[11]
f
p
p
s
U
Figure 3b shows that if S is small at a point, the membrane phosphoinositi-
des display bistable dynamics at that point, i.e., there are two stable steady states
separated by an unstable steady state, which acts as a threshold because Q moves
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