Biomedical Engineering Reference
In-Depth Information
the same d
x
, the number of cross-bridges in subensembles is the same and constant
for any
x
due to the lack of register between myosin and actin. Assuming that the
cross-bridge can interact only with the closest actin binding site, the state of the
cross-bridges can be described by fractions
n
j
(x,t)
giving the fraction of cross-
bridges in state
j
(
j
is one of
W
,or
S
for two state model) at time
t
in subensemble
at
x
to
x
+
d
x
. Taking that the distance between actin binding sites is
d
, fractions
n
j
(x,t)
are defined for
x
in the interval
(
−
d/
2
,d/
2
)
. At any time
t
, all cross-
bridges are in one of the two states,
j
n
j
(x,t)
1. Changes in cross-bridge states
are induced by chemical transition from one state to another or sliding of actin and
myosin filaments relative to each other with the velocity
v
of sarcomere lengthening.
For example, for state
W
, this would result in the following governing equation
=
∂n
W
∂t
+
∂n
W
∂x
v(t)
=
k
SW
n
S
−
k
WS
n
W
,
(7.2)
where
k
WS
and
k
SW
are the first order kinetic rate constants for transition from state
W
to state
S
.
The integral properties of the muscle, such as developed stress and ATPase rate
could be found from integration over subensembles (Hill,
1974
). The Cauchy stress
σ
a
developed by the cross-bridges in a half-sarcomere is, according to Zahalak and
Ma (
1990
)
n
S
(x,t)F
S
(x)
d
x
,
d
2
d
2
ml
s
d
σ
a
=
n
W
(x,t)F
W
d
x
+
(7.3)
2
2
−
−
where
m
is the number of cross-bridges in the unit volume and
l
s
is the length of the
half-sarcomere. According to our assumptions
F
W
is zero because only the strong
binding state generates force. Assuming that
F
S
is proportional to
x
with Hooke
constant
K
, the stress equation will have the following form
2
ml
s
K
d
σ
a
=
n
S
(x,t)x
d
x.
(7.4)
d
2
−
The average cross-bridge ATP consumption rate is
d
2
k
SW
n
S
(x,t)
k
WS
n
W
(x,t)
d
x,
1
d
V
AT P
=
−
(7.5)
2
−
leading to the total ATP consumption per cross-bridge during a beat
T
c
d
2
k
SW
n
S
(x,t)
k
WS
n
W
(x,t)
d
x
d
t,
1
d
V
beat
AT P
=
−
(7.6)
2
0
−
where
T
c
is the period of a beat.
