Biomedical Engineering Reference
In-Depth Information
Fig. 7.3
Kinetic scheme of
actin and myosin interaction
used in three-state
Huxley-type cross-bridge
model:
W
is a weakly bound
biochemical state,
S
1
and
S
2
are strongly bound states,
where myosin head is able to
generate force
7.3 Methods
7.3.1 Model Description
Our mathematical model of actomyosin interaction is a further development of the
cross-bridge model of Vendelin et al. (
2000
). This model consists of a three state
Huxley-type model with two strong binding states (
S
1
,
S
2
) and one weak bind-
ing state (
W
) for cross-bridge interaction and a model of Ca
2
+
induced activation
(Fig.
7.3
). According to Eq. (
7.3
), the Cauchy stress
σ
a
developed by the cross-
bridges in a half-sarcomere for the considered three state model is
x
2
)
d
x
,
2
2
ml
s
K
d
σ
a
=
n
S
1
(x,t)(x
−
x
1
)
d
x
+
n
S
2
(x,t)(x
−
(7.7)
d
2
d
2
−
−
where
x
1
and
x
2
are the minimum positions of free energy profiles for biochemical
states
S
1
and
S
2
. The cross-bridge attachment and detachment in the muscle fiber
are governed by the following system of equations
∂n
S
1
v
∂n
S
1
∂t
=
k
WS
1
n
W
+
k
S
2
S
1
n
S
2
−
(k
S
1
W
+
k
S
1
S
2
)n
S
1
−
∂x
,
(7.8)
∂n
S
2
∂t
=
k
s
1
S
2
n
S
1
+
k
WS
1
n
W
−
(k
S
2
S
1
+
k
S
2
W
)n
S
2
−
v
∂n
S
2
∂x
,
(7.9)
n
W
=
A
−
n
S
1
−
n
S
2
,
(7.10)
where
n
W
,
n
S
1
,
n
S
2
are the fractions of the cross bridges in states
W
,
S
1
,
S
2
, respec-
tively,
A
is the relative amount of activated cross bridges and
v
is the velocity of
half sarcomere lengthening
d
l
s
d
t
.
v
=
(7.11)
The rate constants are constrained as follows:
exp
,
k
WS
1
k
S
1
W
=
G
S
1
(x)
−
G
W
(x)
−
(7.12)
RT
exp
,
k
S
1
S
2
G
S
2
(x)
−
G
S
1
(x)
RT
k
S
2
S
1
=
−
(7.13)
