Biomedical Engineering Reference
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(a)
(b)
(c)
assisting
force
opposing
force
force
force
Fig. 3.3
Force-dependent dynamics of a single motor:
a
Schematic setup of a typical single mole-
cule experiment, in which a single kinesin is held with an optical trap that exerts the force
F
on the
motor in the direction opposite to its walking direction. The motor steps forward with the force-
dependent stepping rate
ʱ(
F
)
and the step size
. Force-dependent unbinding of the motor from
the filament is described by the rate
ʵ
1
(
F
)
.
b
Piecewise-linear parametrization of the force velocity
relation
v
si
.
c
Parameteriza-
tion of the force-dependent unbinding rate of the single motor. The stall force
F
s
in (
b
)andthe
detachment force
F
d
in (
c
) provide the basic force scales for the single motor behavior
(
F
)
, from which the stepping rate is determined via
ʱ(
F
)
=
v
si
(
F
)/
to include backward steps or functional dependencies on other parameters such as
the nucleotide concentrations [
7
]. Incorporating additional features of single motors
usually requires additional parameters that need to be determined either directly from
experiments or calculated from more microscopic models such as the chemomechan-
ical networks described above. In the case of backward stepping, the forward stepping
rate and the backward stepping rate can be determined from the force-velocity rela-
tion and the force-dependent ratio of forward to backward steps [
8
,
61
]. The latter
quantity has been measured for kinesin-1 [
19
].
3.3.2.2 Two Elastically Coupled Molecular Motors
In the following, we use the coarse-grained single motor description that we intro-
duced above to study two elastically coupled molecular motors. As a generic case, we
focus on two identical motors coupled via their stalks to a common cargo. Below, we
use this model to determine the time
t
2
that two motors stay simultaneously attached
to the filament and the resulting velocity
v
2
of the cargo, two key quantities for an
even more coarse-grained description of transport by a motor pair as described at
the end of this section. In general, these two quantities are expected to depend on
the single motor dynamics and on the coupling. Because of the stochastic stepping
of the motors, the elastic elements between them are stretched (or compressed) and
relaxed. Thus strain forces are generated that in turn influence the stepping of the
motors [
7
].
Assuming a linear force-extension relation of the elastic coupling, the only para-
meter, in addition to the single motor parameters, is again the coupling strength
K
.
Since the motors step in a discrete manner, the induced strain forces have discrete
values
F
i
=
i
K
,
(3.8)
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