Biology Reference
In-Depth Information
provide direct data of lifetime versus force. Thus,
Eqs. (11.2)
a
nd
(11.3)
can
be directly itted to determine the interaction parameters.
11.3.3.1 Dynamic force spectra
In practice, the application of the force is not instantaneous but force is
ramped up at a constant rate. In this case of a bond pulled by a constantly
increasing force, constant loading rate, the height of the energy barrier
diminishes with time. Thus, the rupture force will depend on the loading rate.
Three loading regimes can be differentiated in the plots of most probable
rupture force versus loading rate.
10,48,57
At very low loading rates, i.e. near
equilibrium conditions, the attempt rate is much faster that the loading
rate and dissociation is governed by the activation energy and the thermal
energy available. Thus, the most probable rupture force is not affected by
the loading rate.
67
At very high loading rates, above the adiabatic limit, only
accessible in molecular dynamics simulations, the applied increasing force
reaches the maximum rupture force (
$
G
‡
/
γ
) much faster than the
intrinsic dissociation rate. In this regime, the rupture force will have this
constant value with a linear contribution due to luid damping and viscous
friction.
48,57
At intermediate loading rates, at which most AFM measurements
are carried out, the loading rate is comparable with the forced dissociation
rate. Thus, the most probable rupture force will strongly depend on the
loading rate. In addition, given the wide difference between the timescales
for protein relaxation (
F
max
=
t
D
~ 10
10
to 10
9
seconds) and AFM measurements
(>10
seconds), the dissociation times during forced rupture of the bonds
become continuous functions that lead to probability distributions of rupture
forces,
5
p
(
F
) (
Fig. 11.5b
)
. In this regime, the probability
S
(
t
) that the bond is
y
still intact at time
t
can be described by a irst order rate equation
S
Y
dS
/
dt
= −
S
(
t
)/
τ
(
F
(
t
)), assuming negligible rebinding.
p
(
F
) is related to the survival
probability by -
S
Y
dt
=
p
(
F
)
dF
and is given by
F
0
[
)
τ
(
) =
e
-∫
F
(
f
f
)]
-1
df
_____________
r
f
τ
(
p
(
F
)
(11.4)
F
At constant loading rate, we can derive analytically the distribution of
rupture forces using
Eqs. (11.2)
and
(11.4)
*
(
k
B
T
)
) =
e
F
γ
/
k
B
T
_______
______
p
(
F
τ
0
exp
r
f
[1 - e
γ
/
]
(11.5)
F
k
B
T
τ
0
γ
F
*
) is located at the maximum of the
probability distribution
[
Eq. (11.5)
]
. Thus, imposing
The most probable rupture force (
dp
(
F
)/
dF
= 0 leads to
* Using
Eq. 11.3
it is also possible to determine the distribution of rupture forces. We refer the
reader to the original articles.
64, 66
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