Biomedical Engineering Reference
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which implies that without accounting for the spring constant the apparent force-free
unbinding rate is reduced by a factor exp
1
2
k
cant
x
2
β
(
−
/
k
B
T
)
(Walton et al. 2008):
1
2
k
cant
x
2
β
k
eff
k
0
exp
=
(
−
/
k
B
T
)
(3.28)
The reason the Bell model has been so effective at modeling forced rupture may
stem from the temperatures commonly explored in the laboratory. At large
T
ther-
mally activated escape will readily occur before the barrier height is significantly
close to vanishing under force. This will enable passage over energy barriers before
appreciable perturbation of the location of
x
β
occurs. Applying large enough forces
to perturb the underlying bond potential will require fast loading rates to compete
with spontaneous unbinding. Such fast pulling speeds will incur additional hydrody-
namic forces that are not accounted for in the models presented here, which assume
a conservative force field acts on the bond potential. Nevertheless, small probes can
be used that may diminish hydrodynamic effects at large pulling speeds, and low
temperatures may be explored that will enable the observation of perturbation of the
underlying potential by force. In the context of force spectroscopy, Evans explored
this topic for a variety of model potentials (Evans & Ritchie, 1997). The approach
taken by Garg (1995), which was later adopted to model force spectroscopy data
(Dudko et al. 2003), is to utilize a cubic potential
U
(
x
)
to approximate the bond
potential, which under a force field
−
Fx
can be expressed as (Dudko et al. 2006):
2Δ
U
0
x
x
β
3
3
2
x
x
β
−
U
(
x
)
−
Fx
=
Δ
U
0
−
Fx
(3.29)
where Δ
U
0
and
x
β
are the barrier height and minimum-to-barrier distance of the
unperturbed bond. The force-dependent barrier and force constants can be derived
through the first and second derivatives of Equation 3.29 at the extrema
x
β
2
±
[
1
−
2
Fx
β
/
(
3Δ
U
)]
(Garg, 1995):
3
/
2
Δ
U
(
F
)=
Δ
U
0
(
1
−
F
/
F
c
)
(3.30)
1
/
2
κ
0
(
F
)=
κ
0
(
1
−
F
/
F
c
)
(3.31)
1
/
2
κ
b
(
F
)=
κ
b
(
1
−
F
/
F
c
)
(3.32)
3
where
F
c
x
β
is the critical force at which the barrier vanishes due to the
applied force, which coincides with the maximum gradient of the force-free potential
U
(
=
2
Δ
U
0
/
. These functions are then entered into the overdamped limit of Kramers
escape rate (Equation 3.5) in which the minimum and maximum of the potential are
approximated as parabolic:
x
=
0
)
κ
0
(
F
)
κ
b
(
F
)
e
−
Δ
U
(
F
)
/
k
B
T
k
(
F
)=
(3.33)
2πη
√
κ
0
κ
b
2πη
(
3
/
2
1
/
2
e
−
Δ
U
0
(
1
−
F
/
F
c
)
/
k
B
T
=
1
−
F
/
F
c
)
(3.34)
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