Biomedical Engineering Reference
In-Depth Information
of initial and final states is also required to define changes of important thermody-
namic functions, such as the free energy. As a starting point, we model the system as
a Markovian two-state process with time-dependent rates, which is described by the
following coupled equations:
d
d
t
p
on
(
t
)=
−
k
off
(
t
)
p
on
(
t
)+
k
on
(
t
)
p
off
(
t
)
(3.40)
d
d
t
p
off
(
)=
(
)
(
)
−
(
)
(
)
t
k
off
t
p
on
t
k
on
t
p
off
t
(3.41)
where
k
off
(
t
)
and
k
on
(
t
)
are the unbinding and binding transition rates, while
p
on
(
t
)
and
p
off
are, respectively, the probabilities of finding the system in the bound and
unbound states at an observation time
t
. These probabilities have initial conditions
(
t
)
p
on
(
0
)=
1
,
p
off
(
0
)=
0
(3.42)
and
p
on
(
t
)+
p
off
(
t
)=
1.
3.4.1 F
IRST-
P
ASSAGE
A
PPROXIMATION
When the system is driven quickly, the rebinding rate of the system can be neglected.
This is due to the fact that the waiting time for the system to rebind becomes longer
with increasing force, and the speed with which large forces are reached is fast. In
this case, the master equation in Equation 3.40 reduces to the first-order rate process
(Evans & Ritchie, 1997),
d
d
t
p
on
(
t
)=
−
k
off
(
t
)
p
on
(
t
)
(3.43)
(
)=
Beginning with the entire population of states bound
p
on
0
1 and making the sub-
1
r
f
d
F
=
stitution of force for time
d
t
, we must solve for the probability of remaining
bound up to a force
F
:
p
on
F
dp
on
p
on
=
−
1
r
f
F
)
d
F
k
off
(
(3.44)
1
0
Carrying out the integration of the left side of Equation 3.44 yields the probability of
remaining bound for general unbinding rates:
exp
d
F
F
1
r
f
F
)
p
on
(
F
)=
−
k
off
(
(3.45)
0
We see that, due to neglecting reentry (
k
on
(
t
)
≈
0), the probability
p
on
(
t
)
is exactly
[
−
s
k
off
the waiting time probability
P
on
(
t
|
s
)=
exp
(
u
)
d
u
]
of residing
uninterrupted
in the bound state over the time interval from
s
=
0to
t
.
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