Biomedical Engineering Reference
In-Depth Information
[
]
where
is the concentration of the calcium-calmodulin complex,
K
CaCaM
is the half-activation constant,
α
is a positive constant and
CaCaM
Ca
2
+
]
e
is the external
[
Ca
2
+
]
e
is not a function of time and the MLCK
calcium concentration. In (
4.2
),
[
reaction rates are constant values.
The MLCK-activity can also be related to the intracellular
Ca
2
+
]
[
by using the
first and fourth equations from (
4.1
) and setting
k
1
=
k
6
and
k
2
=
k
5
. Thus (Murtada
et al.,
2010a
),
(
d
t
n
M
+
d
k
2
(n
Mp
+
n
AMp
)
−
d
t
n
AM
)
k
1
=
k
6
=
,
(4.3)
n
M
+
n
AM
which in steady-state reduces to
k
2
Phos
1
k
1
=
k
6
=
Phos
,
(4.4)
−
where Phos
n
AMp
)
is the fraction of phosphorylated cross-bridges (Rem-
bold and Murphy,
1990a
). The relationship between intracellular
=
(n
Mp
+
Ca
2
+
]
and Phos
was estimated in swine carotid SM by measuring aequorin light signal into a sig-
moidal function, i.e.
[
0
.
686
Phos
=−
0
.
04
+
,
(4.5)
10
−[
3
.
645
(
0
.
004
[
Ca
2
+
]
i
−
6
.
018
)
+
18
.
92
]
1
+
Ca
2
+
]
i
is the intracellular calcium concentration (Rembold and Murphy,
1990a
). In a similar approach as for the external calcium concentration
[
where
Ca
2
+
]
e
in
(
4.2
), the MLCK-activity can be related to the intracellular calcium concentration
[
[
Ca
2
+
]
i
according to
Ca
2
+
]
h
i
[
k
1
=
k
6
=
ε
(
ED
50
)
h
,
(4.6)
Ca
2
+
]
i
[
+
where
ε
is a fitting parameter describing the maximal MLCK activity,
h
is a pa-
rameter related to the steepness of the relationship and ED
50
is the half-activation
constant for
Ca
2
+
]
i
to MLCK.
Through these approaches the external and intracellular calcium concentrations
can be coupled to the fraction of the attached cross-bridges
n
AMp
+
[
n
AM
by fitting
the chemical parameters against dose-response relationships (Murtada et al.,
2010a
)
or by comparing to myosin phosphorylation data (Murtada et al.,
2010b
,
2012
).
4.3.2 Mechanical Model of the Smooth Muscle Contractile Unit
To introduce a description of the average elastic elongation of the attached cross-
bridges in a smooth muscle contractile unit, and a related framework of the filament
sliding evolution law to simulate filament sliding during contraction and relaxation,
a mechanical model is necessary.
