Biomedical Engineering Reference
In-Depth Information
7.2 Theoretical Background
According to the sliding filament theory, shortening in sarcomere length during con-
traction is caused by thick and thin filaments sliding along each other. Thick and thin
filaments consist primarily of the protein myosin and actin, respectively. To slide
along, the energy from ATP hydrolysis is used in the interaction between myosin
heads and actin sites. One possibility to connect the energetic and mechanical be-
havior for the contraction process is to use a Huxley-type model and the Hill for-
malism to describe the actin-myosin interaction in a thermodynamically consistent
way. Cross-bridges, as defined by Hill, represent the projection from thick filament,
despite if it is attached or not to actin site (Hill,
1974
).
The initial Huxley model involves two biochemical states for cross-bridges: one
state where actin and myosin are not attached (
W
) and one where they are attached
(
S
). The rate constants to describe the transition from one state to the other are func-
tions of the relative distance between the nearest cross-bridge equilibrium position
and actin binding site. In this formalism several assumptions have been made: the
cross-bridge is considered to have only one head and this head has ability to bind
to only one actin site with significant probability. Cross-bridge behavior is assumed
not to depend on other cross-bridge behavior and each cross-bridge can be in differ-
ent biochemical states. The force
F
n
produced by an attached cross-bridge in bound
state
S
is assumed to be elastic and depends linearly on the axial distance
x
along
the myosin and actin filaments between the equilibrium position of the myosin head
and the nearest actin binding site.
According to Hill (
1974
), the force produced by the cross-bridge at position
x
is
related to the free energy
G
in the corresponding state
n
:
F
n
=
∂G
n
/∂x
. Hence the
linear dependency of force on
x
, leads to a parabolic dependence of the free energy
on
x
. Such a relationship between mechanical force and free energy links the chem-
ical reactions with mechanics. Namely, the transition between states is described by
forward and reverse rate constants
k
forward
and
k
reverse
, respectively. The ratio be-
tween rate constants is determined by the difference in free energies of biochemical
states. If we consider a reaction between state
W
and
S
then the ratio between the
rate constants of the reaction is defined as
exp
,
k
forward
k
reverse
=
G
S
(x)
−
G
W
(x)
RT
−
(7.1)
where
R
and
T
are the universal gas constant and absolute temperature, respectively.
It is clear from this relationship that only one of the rate constants can be given at
each
x
with another one determined by the free energy difference. As there is no
force associated with state
W
, the free energy for it is not dependent on
x
.For
values of
x
, where
G
S
<G
W
the strongly bound state is thermodynamically more
stable; otherwise the weakly bounded state is favorable.
To describe muscle contraction, we use a kinetic formalism developed by Hill
(
1974
). In short, it is possible to divide cross-bridges into subensembles according
to the distance
x
between cross-bridge and the closest actin binding site. The cross-
bridges are in the same subensemble, if the distance is between
x
and
x
+
d
x
.For
