Biomedical Engineering Reference
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description. Let us assume there are
N
domains total in the chain, and at any point
in time,
N
off
are unfolded and
N
on
are folded, such that
N
=
N
on
+
N
off
.Theratesof
observing unfolding and folding follow as
N
on
k
off
exp
Fx
β
k
N
on
→
N
on
−
1
(
F
)=
N
on
k
off
(
F
)=
(3.71)
k
B
T
N
off
k
on
exp
−
Fx
α
k
B
T
k
N
off
→
(
F
)=
N
off
k
on
(
F
)=
(3.72)
N
off
−
1
where
k
off
and
k
on
are, respectively, the unfolding and folding rates for individual
domains. The simple form of the rates in Equations 3.71 and 3.72 imply that the
total distance
(
between the minima of the folded and unfolded states does
not change appreciably under force. This is not true in general, however, we assume
that for slow to intermediate loading-rates, thermal activation over the barrier should
occur before appreciable distortion of the underlying energy potential takes place.
Under the simple assumption that force is linear with time d
F
x
β
+
x
α
)
=
r
f
d
t
, the equation
of motion for finding
N
on
domains folded at force
F
is given by
r
f
d
N
on
d
F
=
−
(
)+
(
)
k
N
on
→
N
on
−
1
F
k
N
off
→
N
off
−
1
F
(3.73)
=
−
N
on
k
off
(
F
)+
N
off
k
on
(
F
)
(3.74)
Dividing both sides by the total number of domains
N
produces the rate equation
for the probability
p
on
of finding a domain folded at force
F
. In this model, each
domain constitutes a two-level system. Each domain is coupled to one another by
the intervening polymer linkage. When both
k
on
and
k
off
are slow compared with
the rate of the driving force, the domains cannot keep up with the changing force,
and the force at which they fold and unfold will lag behind the force they would
unfold at when driven quasi-statically. That is, each domain will be driven away from
equilibrium. Rief, Fernandez, and Gaub presented a simple Monte Carlo method of
analyzing this two-level system while also accounting for the coupling of the folded
and unfolded states with the nonlinear polymer extension under force (Rief et al.
1998). On the other hand, when either
k
on
or
k
off
are fast enough such that the longest
characteristic relaxation time τ
rel
(
F
)
)]
−
1
is small compared with
the timescale of the measurement, then the system will remain close to equilib-
rium throughout. In this quasi-equilibrium case, a solution for the relative number
of domains in the folded and unfolded states is found directly from the Boltzmann
distribution:
(
F
)=[
k
on
(
F
)+
k
off
(
F
N
on
(
F
)
)
=
exp
[
−
Δ
G
(
F
)
/
k
B
T
]
(3.75)
(
N
off
F
k
on
(
F
)
=
(3.76)
k
off
(
F
)
exp
−
Δ
G
0
)
/
k
B
T
=
+
F
(
x
β
+
x
α
(3.77)
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