Biomedical Engineering Reference
In-Depth Information
10.4.1.5 Smooth Muscle Cells
In the undamaged state, the energy of the smooth muscle cells
Ψ
smc
can be ex-
pressed as follows (Murtada et al.,
2010
):
n
IV
)
I
smc
1
2
1
2
μ
smc
(n
III
+
Ψ
smc
=
+
u
rs
−
,
(10.11)
4
where
μ
smc
characterizes the stiffness of the actin-myosin filament apparatus (in
kPa). The kinetics of the actin-myosin powerstroke are modeled through a four-
state model described by Hai and Murphy (
1988
) and adopted by Murtada et al.
(
2010
) and Stålhand et al. (
2011
). This model describes the transitions between the
four states
n
I
,
n
II
,
n
III
and
n
IV
of the myosin heads as a function of the calcium
concentration as follows:
⎡
⎤
⎡
⎤
⎡
⎤
n
I
˙
−
κ
1
κ
2
0
κ
7
n
I
n
II
n
III
n
IV
⎣
⎦
=
⎣
⎦
⎣
⎦
n
II
˙
κ
1
−
(κ
2
+
κ
3
)
κ
4
0
.
(10.12)
n
III
˙
0
κ
3
−
(κ
4
+
κ
5
)
κ
6
n
IV
0
0
κ
5
−
(κ
6
+
κ
7
)
Here,
n
are the fractions of the four states, which sum up to one,
n
i
=
1. The
κ
i
(in s
−
1
) are the rate constants of the model, where
κ
1
and
κ
6
are a function of the cal-
cium concentration. In particular,
n
I
and
n
II
are the fractions of dephosphorylated
and phosphorylated myosin heads that are not attached to the actin filament, and
thus not mechanically contributing, while
n
III
and
n
IV
are the fractions of phospho-
rylated and dephosphorylated myosin heads, or cross-bridges, attached to the actin
filaments, and thus contributing to the stiffness. The power-stroke occurs through a
conformational change in state III, after which the myosin heads transform back into
state II. As long as the myosin heads remain phosphorylated, they cycle back and
forth between states II and III, thus generating contraction. In state IV, the myosin
heads are still attached to the actin filament but dephosphorylated and thus unable
to perform a power stroke.
In Eq. (
10.11
),
u
rs
is the average normalized relative sliding between the myosin
and the actin filaments. It follows a viscous evolution law:
η
P
smc
P
mat
,
1
u
rs
=
˙
−
(10.13)
where
η
is a viscosity parameter (in MPa),
P
smc
denotes the active stress exerted by
the attached myosin heads and
P
mat
denotes the stress from the surrounding matrix.
The active stress
P
smc
can be approximated by the following step function
⎧
⎨
for
P
mat
<κ
c
n
III
,
κ
c
n
III
P
smc
P
mat
=
else
,
(10.14)
⎩
n
IV
)<P
mat
,
κ
c
(n
III
+
n
IV
)
for
κ
c
/(n
III
+
where
κ
c
is a material parameter (in MPa) related to the driving force per myosin
head, see Murtada et al. (
2010
) for details.
