Biomedical Engineering Reference
In-Depth Information
experiments. It is therefore useful to consider the way in which these effects vary
with experimentally accessible parameters such as DNA length, voltage and tem-
perature, all of which may substantially affect these two processes. Before we
consider these steps in detail, we must obtain a description of the spatial distribution
of the electrical potential near the pore, as it plays a central role in both mechanisms.
Under the high ionic strength conditions typically used in nanopore experiments
(e.g.
>
50 mM of monovalent salt), the characteristic Debye screening length is of
the order of ~1 nm or smaller. In the absence of an external electric field and with the
system in thermal equilibrium, this implies that static electrical charges in the system
are
effectively screened
. Therefore, a highly negatively charged object (such as a
DNA molecule) will not be electrically attracted to the area around the pore, even if
this area is positively charged. However, when an external electrical potential
DV
is
applied far away from the nanopore, a finite current density of ions flows through the
pore,
drastically
changing the electrical profile of the system. Ohm's law states that
at every point in space the electric field lines must follow the electric current lines,
i.e.
j ¼ sE
, where
j
is the current density,
E
is the electric field, and
is the
conductivity (proportional to the local salt concentration). The steady-state ion-
current flow through a cylindrical pore of diameter
d
and length
l
dictates a long-
range power-law dependence of the electric potential in and around the pore surface
(see the Supporting Information section in Wanunu et al.,
2010
for derivation) [
50
]:
s
d
2
8
lr
DV:
VðrÞ¼
(10.1)
This profile decays as 1
=
with the distance
r
from the pore.
In the negatively-biased chamber, (
10.1
) describes an attractive, funnel-shaped
potential landscape for the negatively-charged DNA coil. Far away from the
nanopore the DNA motion is purely diffusive, because the electrostatic forces
pulling the DNA towards the pore are negligible in comparison with the thermal
forces randomizing its motion. Closer to the pore, DNA diffusion begins to be
biased along the potential gradient lines by the increasingly strong electric field.
The resulting motion may be described by the space-dependent drift velocity
vðrÞ¼mrVðrÞ
, where
m
is the DNA electrophoretic mobility. It is well known
that the electrophoretic mobility of DNA does not depend on its length for
molecules longer than a few persistence lengths (
l
p
50 nm) [
38
,
39
]. This is the
result of an opposing electrophoretic drag force; while the charge of the DNA does
scale linearly with the number of base pairs
N
, the corresponding electrostatic force
Q
E
is almost entirely canceled by the opposing force transmitted to the DNA
through hydrodynamic drag of K
+
counterions flowing in the other direction
through the DNA coil [
27
]. Therefore, there is a regime in the vicinity of the pore
where short as well as long DNA molecules are expected to be equally mobilized by
the long-range potential
VðrÞ
[
50
].
To estimate the effective range where DNA motion is biased towards the pore, we
evaluate the critical radius
r*
from the nanopore where biased diffusion (for
r<r
)
dominates over unbiased diffusion (for
r>r
), as shown in Fig.
10.3
[
50
]. If the
DNA coil is positioned at a distance
r
from the pore, then pure diffusion over this
distance would require time
r
2
=
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