Biomedical Engineering Reference
In-Depth Information
3.4.1.1 Distribution
The distribution of first-passage times, or first rupture force, is straightforward.
The probability density of first-passage times ρ
fp
(
t
)
coincides with the probabil-
ity of unbinding events during the interval
divided by the duration of the
interval d
t
. This density is equivalent to the negative derivative of the waiting time
probability of the bound state, ρ
fp
[
t
,
t
+
d
t
]
(
t
)=
−
d
p
on
(
t
)
/
d
t
. In terms of the force, the den-
sity is given by
1
r
f
k
ρ
fp
(
F
)=
(
F
)
p
on
(
F
)
(3.46)
exp
d
F
F
1
r
f
1
r
f
F
)
=
k
(
F
)
−
k
(
(3.47)
0
To understand the behavior of Equation 3.46, consider the initial moment when
F
=
0.
In this case, the probability of finding the system bound is
p
on
(
0
)=
1 and the density
of transition times is equivalent to the kinetic transition rate ρ
fp
.As
the force on the bound state increases, the rate of unbinding increases, while the
probability of bound states decreases. Thus, the probability
p
on
(
0
)=
k
off
(
0
)
(
F
)
acts to modulate
(
)
the unbinding transition rate
k
off
in proportion to the fraction of the system still
occupying the bound state. Under the Bell rate, Equation 3.24, the probability density
of first-passage rupture forces is (Evans & Ritchie, 1997):
F
exp
1
r
f
k
k
B
T
r
f
x
β
(
ρ
fp
(
F
)=
(
F
)
−
k
(
F
)
−
k
(
0
))
(3.48)
3.4.1.2 Mean and Most Probable Rupture Force
A useful data set for analyzing the characteristics of a bond under force is the trend
of the rupture force with loading rate. The unbinding transition rate (Equation 3.24)
depends only on the instantaneous value of the force, not the history of the force with
time. Therefore,
k
is independent of loading rate. However, from the first-passage
distribution in Equation 3.46, we see that to describe the frequency of rupture events
we must scale
k
(
F
)
that the bond is still formed at force
F
. This probability is loading-rate-dependent because the fraction of bonds remain-
ing at a force
F
will depend on the length of time
F
(
F
)
by the probability
p
on
(
F
)
r
f
spent to reach that force.
This can be inferred directly from the first-order rate process in Equation 3.43, when
expressed in terms of the force, d
p
on
/
1
r
f
k
off
, which shows that
the rate of losing bound states with force is inversely proportional to the loading
rate,
r
f
. Hence for faster loading rates, the bound state tends to persist to higher
forces.
Two common, and very similar, statistics of the rupture force distribution are
the mean
(
F
)
/
d
F
=
−
(
F
)
p
on
(
F
)
and most probable (or mode)
F
∗
. The mean of the first-passage rup-
ture force distribution (Equation 3.46) can be determined by any of the following
F
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