Biomedical Engineering Reference
In-Depth Information
Fig. 6.1
The mechanical
element for the smooth
muscle comprising a
contractile unit and a parallel
spring. The reference length
of the element is
L
(
top
). The
filaments are first translated a
distance
−
u
ft
in the figure by
the friction clutch (
middle
)
followed by an extension of
the cross-bridges a distance
u
cd
(
bottom
)
and unattached myosin
(
B
)
, phosphorylated and attached myosin
(
C
)
, and dephos-
phorylated and attached myosin
(
D
)
. Because myosin must be attached to actin to
generate force, only the states C and D are associated with force generation. The
difference is that the myosin heads in state C undergo the cross-bridge cycle and
generate force through the power stroke while myosin heads in state D are believed
to be non-cycling and work as passive springs resisting extension. The latter state is,
therefore, often referred to as the 'latch state'. The transformation between the four
myosin states is given by a first-order kinetic model (Hai and Murphy,
1988
),
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
n
A
n
B
n
C
n
D
−
k
1
k
2
0
k
7
n
A
n
B
n
C
n
D
d
d
t
−
k
2
−
k
1
k
3
k
4
0
,
(6.1)
−
k
4
−
0
k
3
k
5
k
6
0
0
k
5
−
k
6
−
k
7
where
n
A
,
n
B
,
n
C
, and
n
D
are the fractions of myosin in the states A, B, C, and D,
respectively, and
k
1
,...,k
7
are reaction rates. Since the myosin states are given as
fractions, their sum must equal one, i.e.,
n
A
+
n
B
+
n
C
+
n
D
=
1
.
(6.2)
The reaction rates
k
1
and
k
6
in Eq. (
6.1
) control the phosphorylation of myosin. The
phosphorylation is governed by a complex chain of events (Alberts et al.,
2008
)but
is ultimately dependent on the intracellular calcium ion concentration in a sigmoid-
shaped manner, see Arner (
1982
). This behavior can be modeled by taking the rate
