Biomedical Engineering Reference
In-Depth Information
where @x 2 @, k Ca is the maximal rate of calcium extrusion and Ca out the
calcium concentration at which the rate of eux is half maximal.
The Ca 2+ influx distributions from the extracellular environment through
independent and either AA- and NO-activated channels, respectively given by
F AA and F NO , are assumed to be saturably dependent on the relative second
messenger concentration:
a(x;t)
q a + a(x;t)
m
F AA (@x;t) = F AA;max
;
(6.12)
n(x;t)
q n + n(x;t)
m
F NO (@x;t) = F NO;max
;
(6.13)
where, as usual @x 2 @, and a(x;t) and n(x;t) are derived by Eqs. (6.7) and
(6.8). We select a quadratic m = 2 sigmoidal activation reflecting the fact
that two AA or NO molecules are supposed to be bound to the respective
Ca 2+ channel in order to open it. In particular, the flux parameters have been
chosen to fit the currents obtained by path clamp experiments in [274, 283].
The level of calcium in the extracellular environment (i.e., at site x :
( ( x ) ) = M) evolves according to:
8
<
@c
@t = D c r 2 c
outside ;
| {z }
diffusion
n @ x rc(@x;t) = F AA (@x;t) F NO (@x;t)
|
+ F out (@x;t)
| {z }
flux from cells
at @;
:
{z
}
fluxes to cells
n @ x rc(@x;t) = 0 at @;
(6.14)
where, @ is the boundary of the domain and, as usually, n @ x is the unit
outward normal to @x, and the uxes F AA , F NO , and F out are specified in
Equations (6.11), (6.12), and (6.13).
The model of VEGF-induced calcium signals, although very simple, in
the absence of more specific evidence [134, 282], is acceptable for low agonist
concentrations and it is experimentally validated [351, 372]. Indeed, it is based
on the hypothesis of two independent calcium channels (AA- or NO-activated),
but we cannot exclude either the existence of channels co-modulated by AA
and NO or potential cross regulations between different channel types: for
example, several member of the TRP family (in particular TRPC and TRPV
sub-families) can be also involved in the proangiogenic signals [221, 282, 292].
There would be also two additional fluxes that influence the level of cy-
tosolic calcium: the rate of its release from the ER and the rate of its re-
sequestration back in the endoplasmic reticulum [8, 330]. We neglect both
contributions, since the former is not stimulated by low concentrations of AA
and NO [133, 274, 390], and the latter is typically involved only in the re-
covery of calcium response after agonist removal [265]. Finally, it is useful to
 
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