Biomedical Engineering Reference
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where in the case of bond rupture, the Hamiltonian is the sum of the time-invariant
intermolecular potential of the bond
U
(
x
)
and the time-dependent pulling potential
applied to the bond
V
(
x
,
t
)
:
H
(
x
(
t
)
,
t
)=
U
(
x
(
t
))+
V
(
x
(
t
)
,
t
)
(3.111)
As illustrated in Figure 3.1, two common scenarios for
U
arise: (1) a unimolec-
ular system, such as a protein or RNA, described by a bistable potential; or (2) a
bimolecular system, such as a ligand-receptor bond, which is described by a sin-
gle metastable state within the potential. In the second case, the total Hamiltonian
becomes bistable due to the pulling potential
V
(
x
)
. However, the resulting solu-
tions for the work done on these two systems differs significantly.
(
x
,
t
)
Unimolecular system with bistable U(x).
We will assume the bistable potential
has two minima separated by a constant distanceΔ
x
, with the initial state at
x
0.
We will also take the simple assumption that a single transition from state 1 to state
2 occurs in the observation window. While under the influence of a pulling potential
V
(
0
)=
1
2
k
cant
2
, the system will switch states from
x
(
,
)=
x
t
(
x
−
vt
)
(
t
)=
0to
x
(
t
)=
Δ
x
at
an arbitrary time
t
s
(or switching force
F
s
=
k
cant
vt
s
). Referring to Figure 3.11, the
work then follows as
t
s
t
f
k
cant
v
2
t
d
t
W
=
+
k
cant
v
(
vt
−
Δ
x
)
d
t
(3.112)
0
t
s
1
2
k
cant
Δ
x
2
1
2
k
cant
2
=
k
cant
vt
s
Δ
x
−
+
(
vt
f
−
Δ
x
)
(3.113)
1
2
k
cant
Δ
x
2
1
2
k
cant
2
=
F
s
Δ
x
−
+
(
F
f
/
k
cant
−
Δ
x
)
(3.114)
W
molecule
W
spring
(
t
f
)
where the work is separated into energy used to switch the molecule between states
and the energy finally stored in the spring
W
spring
(
)=
(
,
)
t
f
V
Δ
x
t
f
after stretching it
k
cant
vt
f
. Equation 3.114 states
that, although the driven transition occurs at force
F
s
, the total work done on the
system depends on how long you pull on it. This is true because the thermodynamic
work accounts for the entire system, including the pulling device (Jarzynski, 2006,
2007).
Note, however, that we assume the pulling potential to move slowly enough
through the solution to contribute a conservative force to the system. Therefore,
W
spring
up to the final time
t
f
, or equivalently, final force
F
f
=
is the reversible work done on the spring, independent of pulling speed.
The total reversible work, or free energy, for the bistable system will then be a com-
bination of the minimal work to switch states of the moleculeΔ
G
0
plus the reversible
work to stretch the spring:
(
t
f
)
Δ
G
=
Δ
G
0
+
W
spring
(
t
f
)
(3.115)
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