Biomedical Engineering Reference
In-Depth Information
regulation of NO production (both AA- and Ca
2+
-mediated). They are de-
scribed by others saturating functions of Michaelis{Menten type, work within
the entire cell cytosol and agree with electrophysiological measurements [283],
which have proved that they happen at comparable rates and time scales. In
particular, the maximal rate of AA- and Ca-dependent NO release has been
estimated by fitting the experimental measures made using fluorescent probes
and selective electrodes in [37, 274].
The local intracellular level of calcium c(x;t), where (
(
x
)
) 2 fN;Cg,
is determined by a balance between the influxes through either AA- and NO-
activated PM channels, respectively denoted by F
AA
and F
NO
, extrusion of
the ion from the cytosol, F
out
, and its buffering. It is thus controlled by the
following diffusion equation:
8
<
:
@c
@t
= K
buff
D
c
r
2
c
in ;
| {z }
diffusion
n
@
x
rc(@x;t) =
F
AA
(@x;t)
+
F
NO
(@x;t
)
|
F
out
(@
x;t
)
| {z }
efflux
at @;
{z
}
influxes
(6.10)
where n
@
x
is the unit outward normal to @x 2 @. The coecient of diusion,
D
c
, is assumed to be homogeneous throughout the cell (the diffusion of calcium
across the nuclear envelope is taken to occur through non selective pores,
whose permeability is proportional to the diffusion constant of the ion in the
cytosol [89, 283]). The scaling factor
K
buff
=
K
off
K
off
+ K
on
b
T
models the activity of intracellular endogenous buffers (proteins including
calmodulin and others, cytoskeleton, mitochondria), which have a significant
impact on the overall calcium dynamics, influencing its effective diffusion and
contributing to decrease its level [24, 33, 212]. bT
T
is the total concentration
of buffer sites (considered constant and experimentally estimated in different
cell types [132, 283]), K
on
is the rate of calcium uptake, K
off
the rate of its
release. This approximation works under the assumption that the buffering
reactions take place on a faster timescale than the flux dynamics described
above, and thus they can be considered in a quasi-steady state. This is the case
of immobile buffers, characterized by low anity and fast kinetics [371, 372].
F
out
represents the overall rate of calcium eux from the cell at time t
and incorporates the extrusion of the ion both via PM ATP-ase and Ca
2+
-
Na
2+
exchangers, which is given in several models as a sum of Hill functions
[283, 399]. However, in order to avoid over-complications while remaining close
to the curves set in literature in the considered agonist concentration range,
we here approximate F
out
as a single Michaelis{Menten form:
c(x;t)
Ca
out
+ c(x;t)
F
out
(@x;t) = k
Ca
;
(6.11)
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