Biomedical Engineering Reference
In-Depth Information
where the dependence on the nucleotide concentrations is embedded in the zero
force rate
ˉ
ij
,
0
and the force dependence is described by the factor
ij
(
)
[
49
,
61
].
These factors are subject to constraints to fulfill detailed balance: For all chemical
transitions, the force factors
F
; for the mechanical
stepping transitions between states (
DT
) and (
TD
) [states 2 and 5 in Fig.
3.1
b],
25
(
ij
(
F
)
satisfy
ij
(
F
)
=
ji
(
F
)
[
65
].
Compared to a mechanical forward step, which is completed within
F
)
=
52
(
F
)
×
exp
(
−
F
/
k
B
T
)
with the step size
s, the chem-
ical transitions are rather slow and take several ms. Thus, the chemical reaction paths
are explicitly accessible to experiments and experimentally obtained reaction rates
can be implemented as the transition rates of the network model [
61
]. One important
property of these networks is that they involve several motor cycles, which provide
the free energy transduction between ATP hydrolysis and mechanical work. As one
varies the nucleotide concentrations and the external load force, the fluxes on these
cycles change and different cycle fluxes dominate for different parameter regimes.
In this sense, the chemomechanical networks of a single motor as introduced in [
61
]
contain several competing motor cycles. Imposing cyclic balance conditions [
62
]
on all motor cycles ensures that the network description satisfies both the first and
second law of thermodynamics.
µ
3.3.1.2 Motor Pair Network
To extend the network description to coupled motors, we consider a pair of two
kinesin-1 dimers that are attached to the same cargo and walk on the same fila-
ment. We refer to the two motors in the pair as the leading and the trailing motor,
respectively, according to their relative positions in the direction of motion.
The modeling so far points to two key questions: What happens if one of the
motors dissociates from the filament? And second, how does the translocation of one
of the motors influence the motor pair system? For single motors, the answers are
rather simple because the single motor run is terminated when the motor dissociates
from the filament and the unbound motor is not spatially restricted. For the motor
pair, unbinding from and rebinding to the filament provides an alternating sequence
of 1-motor runs, where the cargo is pulled by one active motor, and 2-motor runs,
where the cargo is actively pulled by both motors, as outlined in the upper row of
Fig.
3.2
. During 1-motor runs, the dynamics of the bound or active motor can be
described by a random walk on the single motor network as discussed above. The
only difference to the case of a single motor is that the average dwell time in any state
i
now also involves the rebinding rate of the second (unbound or inactive) motor.
A 1-motor run is terminated either by unbinding of the remaining motor, which
corresponds to the termination of the motor pair walk, or by rebinding of the inactive
motor, which initiates a 2-motor run. During 2-motor runs, the state-space consists of
combinations of the 7 chemical states of the individual motors, i.e., 7
2
49 states.
Concerning the coupling of the motor pair, we consider the flexible stalks of the
kinesin motors as linear springs. Since both springs are only coupled via the cargo,
which is taken to be rigid, we can effectively describe the system by one linear spring
=
Search WWH ::
Custom Search