Biology Reference
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landscape along the pulling coordinate, by lowering the energy barrier
(
Fig. 11.6a
)
. Assuming the barrier is sharp enough so the application of a force
does not change the position and shape of the transition state, we obtain
G
(
F
)
F
γ
. Thus, when force is applied, the ligand can escape more easily
from the well, which leads to a faster dissociation rate or, equivalently, a
shorter lifetime. Therefore, the lifetime depends on the applied force and on
the distance to the transition state (
γ
).
$
G
‡
=
1
______
k
τ
(
)
=
τ
0
e
-
F
γ
/
k
B
T
F
) =
(11.2)
(
F
Eq. (11.2)
is equivalent to Bell's equation, except for the deinition and
interpretation of
τ
0
(or
a irst description of how the bond resists the application of force. The wider
the potential, the more the force will affect the dissociation kinetics.
As already mentioned, biological bonds are the result of a combination
of different interactions. This leads to energy landscapes more complex than
11.6b
). In this case, the outermost energy barrier will dominate the lifetime
of the bond, until force drives the outer barrier below the inner one. This will
happen when force exceeds a critical level (
F
c
), determined by the difference
$$
G
‡
1,2
). When force reaches
between the relative barrier heights (
F
c
=
$$
G
‡
1,2
/
γ
1
the inner barrier will dominate dissociation, leading to a different
force dependence of lifetime. The lifetime will then present consecutive
exponential regimes with characteristic lifetimes
τ
01
,
τ
02
and widths
γ
1
,
γ
2
…
(
Fig. 11.6b
)
. This behaviour was irst observed on the streptavidin-biotin
complex by Merkel and coworkers using the biomembrane force probe.
47
It
appeared later to be a common signature of biological bonds.
5,7,33,37,62,63
The model described by
Eq. (11.2)
has been recently reformulated into
a uniied form by some authors,
58,64,65
assuming not a sharp barrier, but a
certain potential shape in which not only the height of the barrier changes
when force is applied but also the position along the reaction coordinate.
Using Dudko, Hummer and Szabo approach,
66
the force dependence of the
lifetime can be described by
(
1 -
(
1 - (1 -
)
1-b
e
Δ
G
‡
)
F
γ
F
γ
_____
k
B
T
_______
_______
τ
(
) =
τ
0
F
b
Δ
G
‡
b
Δ
G
‡
)
b
(11.3),
where
b
is a parameter that selects the particular model of the potential well.
Being
b
= 3/2 for a linear-cubic potential and
b
= 2 for a cusp potential. Notice
that choosing
= 1 returns
Eq. (11.2)
.
The advantage of this approach is that
it also provides an estimate of the barrier height and the approximate shape
of the energy landscape, thus being less phenomenological. As we described
in the previous section, force clamp measurements of receptor-ligand bonds
b
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