Biomedical Engineering Reference
In-Depth Information
This must be compared with the electrophoretic speed
mEðrÞ
. Biased diffusion
dominates when
mEðrÞ>Dr
=
; or more simply, when
mVðrÞ = >
1. Substituting this
into (
10.1
)weobtain:
d
2
m
8
lD
DV:
r
¼
(10.2)
Note that
r*
grows with DNA length because the diffusion constant
D
decreases with
length. In other words, longer DNA molecules begin to sense the nanopore farther
away from the pore than shorter DNA molecules. However, the rate
R
diff
at which
DNA coils randomly and enters a hemisphere of radius
r*
is proportional to their
diffusion constant, and is given by Smoluchowski theory as
R
diff
¼
2
pDr
. Combining
this with (
10.2
), we find:
pDr
¼
pd
2
m
4
l
DV:
R
diff
¼
2
(10.3)
Thus, in the regime where diffusion to the pore is the rate-limiting step, the capture
rate should be
independent of DNA length
. This result may seem counterintuitive,
since free diffusion for longer DNA is inherently slower than diffusion for shorter
DNA, and the Smoluchowski equation dictates that
R
diff
is proportional to
D
.
However, the effective capture radius
r*
grows with
D
1
, which cancels out this
length dependence. This result predicts the diffusion-limited “current density” of
DNA translocating through the pore to be
J
diff
¼ R
diff
c
, where
c
is the bulk DNA
concentration [
50
].
The DNA translocation rate is only diffusion-limited for a certain range of DNA
lengths. For other lengths, the rate-limiting step occurs when the polymer is
threaded into the pore (step (iii) in Fig.
10.3
)[
50
]. The confinement of the DNA
end, as well as possible unfavorable interactions of the highly charged DNA with
the pore itself, creates a free energy barrier to capture. This barrier was experimen-
tally observed for DNA transport through the 1.5 nm protein pore
-HL [
17
,
36
],
characterized by an exponential dependence of the capture rate on voltage that
could be explained theoretically: When DNA capture is governed by an energy
barrier, its rate, according to classical Kramers theory, can be written in the form
J ¼ Rc ¼ o
a
[
52
], where
U
is the height of the threading
barrier in the absence of applied voltage and
q
is the effective charge of a DNA
end segment, which is DNA length-independent. The pre-factor
exp
½
ðqDV UÞ=k
B
T
in the expression
above is usually interpreted as the threading attempt rate. Since in stage (iii) the
DNA coil is at or very near to the mouth of the pore, the biasing effect of the local
potential
VðrÞ
o
on this attempt rate must be considered. Specifically, we find that the
electrical bias leads to two capture enhancement mechanisms [
50
]:
1. An exponential attempt rate enhancement: The potential well
VðrÞ
traps the DNA
a distance
r
g
from the pore mouth, where threading is repeatedly attempted.
Due to the energy barrier, DNA molecules are delivered to the pore mouth
multiple times before a successful translocation occurs. Therefore, while the
probability of finding a DNA coil within a distance
r
g
from the pore mouth
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