Biomedical Engineering Reference
In-Depth Information
prefactor, and thus the primary contributor to the escape rate of all
N
bonds in concert
arises from the deeper potential barrier (Evans & Williams, 2002):
k
0
exp
Fx
β
k
B
T
−
(
N
−
1
)
Δ
U
k
N
→
0
(
F
)=
+
(3.67)
k
B
T
√
κ
0
κ
η
e
−
where
k
0
Δ
U
/
k
B
T
=
is the single-bond escape rate.
3.5.2 S
ERIES
B
ONDS AND
D
OMAINS
There are two primary cases to consider when the molecule under manipulation con-
tains multiple bonds arranged in series (Figure 3.7). The first case is simply bonds
that hold the chain together. For example, a polymer such as poly(ethylene-glycol)
(PEG) consists of a number of ethylene-glycol monomers connected in parallel.
When under a large tensile load, the breakage of any one bond breaks the entire
polymer in half irreversibly. The second case is domains within the polymer chain
that change configuration under force. PEG in aqueous solution fits in this cate-
gory as well because the conformation of each monomer can switch between helical
or planar isomers depending on the level of force (Oesterhelt et al. 1999). Upon
decreasing force, the domains can switch back to their helical form. Another com-
mon example of domains are the folded immunoglobulin domains of Titin, which
unfold under force (Rief et al. 1997). The basic difference between series bonds
and series domains is that domains are generally reversible upon decreasing force,
whereas series bonds typically are not.
In either the series bond or series domain case, the rate of breaking or configura-
tional switch under increasing force is given by the rate for a single sub-unit times
the number of subunits available (Evans & Williams, 2002; Williams, 2003):
Nk
0
exp
Fx
β
k
B
T
k
N
→
N
−
1
(
F
)=
(3.68)
where
N
is the number of closed bonds or domains in the chain. The mean rupture
force of the first bond in the series is (see Equation 3.53)
x
β
exp
Nk
0
k
B
T
E
1
Nk
0
k
B
T
r
f
x
β
k
B
T
F
=
(3.69)
r
f
x
β
or in the fast loading rate limit (
r
f
→
∞)
ln
r
f
x
β
e
−
ln
N
γ
k
B
T
x
β
=
F
−
(3.70)
k
0
k
B
T
Therefore, at large loading rates, the rupture force of one bond in a chain is less than
one lone bond
=
k
B
T
x
β
ln
N
.
When the chain consists of domains that switch configuration upon increasing or
decreasing force, one must account for both forward and reverse processes in the
F
1
by the log of the number of bonds in series
F
F
−
1
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