Biomedical Engineering Reference
In-Depth Information
we have taken into account an instantaneous VEGF-receptor complex inter-
nalization rate of 4.3 10 4 per second [152], and 45 kDa as the molecular
weight for a VEGF molecule [64].
For the construction of a mathematical model of VEGF-mediated intracel-
lular events, we use a PDE-type approach based on the model by L. Munaron
for the proangiogenic calcium signals in a static EC [283]. However, with re-
spect to that model, we here explicitly describe the VEGF-induced production
of second messengers AA and NO, as well as the calcium feedback mechanism
for AA biosynthesis. Moreover, we use slightly different flux distributions for
both calcium entry and extrusion (see below for details).
Summing up, the current levels of AA and NO at cell location x (i.e.,
( ( x ) ) 2 fN;Cg) are controlled by standard reaction-diusion equations
consisting in terms for their production, homogeneous diffusion and natural
decay:
@a
@t = D a r 2 a
k a R
K a + R
v c c
c a + c
a a
+
+
;
| {z }
diffusion
|{z}
decay
| {z }
VEGFinduced production
| {z }
Cainduced production
(6.7)
@n
@t = D n r 2 n
k n R
K n + R
c
c n + c
a
a a + a
n n
+
+
v ca
:
| {z }
diffusion
|{z}
decay
| {z }
VEGFinduced production
| {z }
AA and Cainduced prod:
(6.8)
The diffusion parameters have been estimated with laboratory-based tech-
niques [226, 276]. The degradation of both the intracellular messengers (and
thus the following production, respectively, of eicosanoids [209] and perox-
ynitrites [300]) is considered constant, at rates a , n > 0 and measured in
[283], where it has been estimated fitting experimental measurements made
by the same group on bovine aortic endothelial cells [274]. The third terms in
Equations (6.7) and (6.8) describe the VEGF-induced production rate of AA
and NO at the cell submembrane region, as
R(x;t) = X
x 0 2 0 x
B(x 0 ;t);
(6.9)
where ( ( x ) ) = C, ( ( x 0 ) ) = M and B(x 0 ;t) is dened in (6.6). We apply
Michaelis{Menten laws supposing that the intermediate receptor-ligand com-
plexes are in a pseudo-steady state. In particular, the Michaelis{Menten coef-
ficients of the VEGF receptor-mediated production of AA have been chosen
for the best fit with experimental measures made for the same theoretical pa-
per [283]. Note that such single simplified reactions, we bundle multiple signal
transduction cascades that are known intermediates in AA and NO genera-
tion, such as the activity of PLA2 and eNOS enzymes. The last term in Eq.
(6.7) implements the calcium-dependent feedback mechanism in AA biosyn-
thesis, while the analogous term in Equation (6.8) accounts for the double
 
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