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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