Biomedical Engineering Reference
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and unbinding rate:
r
f
r
f
1
r
f
d
k
(
F
)
0
k
1
r
f
0
k
1
F
)
d
F
1
F
)
d
F
e
−
(
2
e
−
(
0
=
−
k
(
F
)
d
F
1
r
f
k
1
d
k
(
F
)
(
F
)=
k
(
F
)
d
F
F
∗
F
∗
=
(
)
r
f
dln
k
F
k
(
F
)
(3.55)
d
F
Inserting the Bell transition rate of Equation 3.24 into Equation 3.55, the most prob-
able rupture force under the first-passage model is
x
β
ln
r
f
x
β
k
B
T
F
∗
=
(3.56)
k
0
k
B
T
When comparing with the fast loading rate expansion of Equation 3.54, the mean
is shifted lower than the mode
k
B
T
x
β
F
∗
−
γbecause the distribution is skewed
to low forces due to the finite probability of bond rupture down to zero force (see
Figure 3.5). As the loading rate decreases, the difference between the mode and
mean decreases as well. It is also important to note that the first-passage distribution
does not have a peak when the argument in the logarithm of Equation 3.56 is less
F
=
10
1.4
1.2
10
-1
8
1.0
0.8
0.6
6
F*
/
F (mode)
1
4
r
f
x
k
0
k
B
T
10
10
3
=
10
2
0.4
0.2
0.0
2
〈
F
〉
/
F
(r
f
→ ∞
)
0
0
2
4
6
8
10
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
F
/
F
r
f
x
k
0
k
B
T
FIGURE 3.5
First-passage models of bonds rupture. The distribution of rupture forces
(Equation 3.48) are shown calculated at various normalized loading rates. The mean rupture
force, Equation 3.53, as a function of normalized loading rate is also shown in units of the ther-
mal force scale
F
β
x
β
. The mean crosses between a shallow, linearly increasing regime
into a strongly nonlinear regime when the normalized loading rate is unity. Shown for compar-
ison are the high loading rate limit, Equation 3.54, and most probable (mode), Equation 3.56,
rupture force functions.
=
k
B
T
/
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