Biomedical Engineering Reference
In-Depth Information
an arbitrary time
t
s
(or unbinding force
F
s
=
k
cant
vt
s
). The thermodynamic work is
t
s
t
f
k
cant
v
2
t
d
t
W
=
+
k
cant
v
(
vt
−
vt
)
d
t
(3.119)
0
t
s
1
2
k
cant
v
2
t
s
=
(3.120)
F
s
2
k
cant
=
(3.121)
Unlike the unimolecular system above, the thermodynamic work done on a bimolec-
ular system is independent of the final observation time
t
f
because no work is done
on the molecule when it breaks away and resides around the minimum of the pulling
potential (see Figure 3.11). While this is a sufficient approximation, it is not neces-
sarily true in general, as hydrodynamic effects will contribute further dissipation as
the probe is dragged through the fluid. Therefore, Equation 3.121 is valid at slow to
moderate pulling speeds when viscous damping is negligible.
The dissipated energy in this case is given by
W
d
=
W
−
Δ
G
(3.122)
F
s
2
k
cant
−
Δ
G
0
=
(3.123)
Therefore, the work done on bond rupture (dissociation of a bimolecular complex)
can be derived from the integral over the force-extension trajectory or, in the case of
a bond loaded by a spring of stiffness
k
cant
, the work is simply one-half the square of
the rupture force over the spring constant.
3.7 SUMMARY
Single-molecule manipulation techniques are arguably the only method available to
study the fundamental energy landscapes that govern inter- and intramolecular bond-
ing and mechanics. Initial developments provided access to a previously inaccessible
parameter of a bond—the transition state. Early work has also been concerned pri-
marily with kinetics, under the assumption that driven molecular systems are too far
from equilibrium to permit exploration of equilibrium properties. But recent devel-
opments in nonequilibrium thermodynamics have provided amazing tools, such as
fluctuation theorems, which link nonequilibrium measurements to their equilibrium
roots. In addition, recent findings have also shown that bond rupture experiments
can be performed very close to equilibrium conditions, allowing estimates of bond-
free energies directly. Therefore, force spectroscopy should no longer be thought of
as a purely irreversible, kinetic regime technique. The dynamics can be reversible
under appropriate conditions and in principle, the full breadth of information rang-
ing from equilibrium free energies to activation barriers to kinetics should be acces-
sible. Some exciting future developments of the technique will most likely be found
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