Biomedical Engineering Reference
In-Depth Information
relationships:
∞
F
=
F
ρ
fp
(
F
)
d
F
(3.49)
0
∞
1
r
f
=
(
)
(
)
Fk
off
F
p
on
F
d
F
(3.50)
0
∞
=
p
on
(
F
)
d
F
(3.51)
0
1
=
F
(
p
on
)
d
p
on
(3.52)
0
where
F
is the inverse of the probability of finding the system in the bound state,
Equation 3.45, which can be solved analytically for simple transition rates such as
Equation 3.24.
Under the Bell transition rate in Equation 3.24, the mean rupture force under the
first-passage model is
(
p
on
)
∞
F
=
p
on
(
F
)
d
F
0
exp
d
F
d
F
F
∞
1
r
f
F
)
=
−
k
off
(
0
0
exp
k
0
k
B
T
r
f
x
β
exp
r
f
x
β
e
Fx
β
/
k
B
T
d
F
k
0
k
B
T
∞
=
−
0
x
β
exp
k
0
k
B
T
e
−
u
u
k
B
T
∞
k
0
k
B
T
r
f
x
β
=
d
u
r
f
x
β
x
β
exp
k
0
k
B
T
E
1
k
0
k
B
T
r
f
x
β
k
B
T
=
(3.53)
r
f
x
β
(
)
where E
1
is the exponential integral. Equation 3.53 increases from zero force
linearly with loading rate
r
f
, then follows a nonlinear trend, and asymptotically
approaches the commonly used form at large loading rate (Williams, 2003),
z
x
β
ln
r
f
x
β
e
−
γ
k
B
T
=
lim
r
f
→
∞
F
,
(3.54)
k
off
k
B
T
where γ
is the Euler constant. Note that the exponential integral follows
the simple interpolation e
z
E
1
=
0
.
577
...
) =
e
−
γ
over all values of the argument
z
.
The most probable rupture force (or mode,
F
∗
) coincides with the maximum
of the rupture force distribution. Using the first-passage distribution in Equa-
tion 3.46, Evans & Ritchie (1997) derived a useful expression for the mode by taking
dρ
fp
(
z
ln
(
1
+
/
z
)
F
∗
)
/
(
d
F
=
0 and expressing the following relationship between loading rate
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