ar
X
iv
:0
70
9.
38
50
v1
[
ph
ys
ics
.ch
em
-p
h]
2
4 S
ep
20
07
Physical Review Letters (2007) Vol. 98, 238104
The effect of salt concentration on the electrophoretic speed of a polyelectrolyte
through a nanopore
Sandip Ghosal
Northwestern University,
Department of Mechanical Engineering
2145 Sheridan Road, Evanston, IL 60208
(Dated: February 5, 2008)
In a previous paper [1] a hydrodynamic model for determining the electrophoretic speed of a
polyelectrolyte through an axially symmetric slowly varying nanopore was presented in the limit of
a vanishingly small Debye length. Here the case of a finite Debye layer thickness is considered while
restricting the pore geometry to that of a cylinder of length much larger than the diameter. Further,
the possibility of a uniform surface charge on the walls of the nanopore is taken into account. It
is thereby shown that for fixed ζ-potentials on the surface of the polyelectrolyte and on the pore
wall, the electrophoretic speed is independent of the Debye length. The translocation speed depends
on the salt concentration only to the extent that the ζ-potentials depend on it, and further, this
dependence is very weak. It is shown that the calculated transit times are consistent with recent
measurements in silicon nanopores that reveal this insensitivity to salt concentration.
PACS numbers: 87.15.Tt
The translocation of polymers across nanometer scale
apertures in cell membranes is a common phenomenon
in biological systems [2]. If the polymer carries a charge,
an applied electric potential can drive the translocation.
The change in electrical conductance of a single nanopore
as a polymer transits the pore can be reliably detected
and used to characterize the polymer [3]. A number of
experimental studies [4, 5, 6, 7] as well as a few theoret-
ical ones [1, 8] on the electrically driven translocation of
polymers across nanopores have appeared recently. In-
terest in the phenomenon is to a large extent motivated
by the possibility of refining it to the point where the
base sequence of a DNA strand can be read with single
base resolution as the DNA transits the pore [9]. This
would provide a sequencing method that is faster and
cheaper than existing ones by many orders of magnitude.
A technological challenge is the trade off between noise
and resolution. In typical experiments with solid state
nanopores a single base pair transits the pore in about
∼ 10−8 sec – much too short to be resolved. On the other
hand the voltage across the pore cannot be sufficiently re-
duced to slow down the DNA because then the change
in current would not be detectable above the noise. A
theoretical analysis of the problem to determine how the
translocation speed depends on the controllable param-
eters is therefore of value in guiding the experimental
work.
In an earlier paper [1] (henceforth Paper A) a hydro-
dynamic model was proposed for describing the process
of electrically driven translocation across the nanopore.
The speed of translocation is determined by a balance
of electrical and viscous forces arising from within the
pore with proper accounting for the co- and counter-ions
in the electrolyte. The underlying physics is not unlike
that of electrophoresis of small charged particles in an
applied electric field except that here the proximity of
the pore walls play an important role. The translocation
speed was explicitly calculated for cylindrically symmet-
ric pores by assuming an infinitely thin Debye layer and
slowly varying pore radius. The calculated translocation
speed was shown to be in close agreement with experi-
mental measurements [7] in solid state nanopores. The
assumption of infinitely thin Debye layers was justified
because of the high concentration of salt (1 M KCl) in the
electrolyte used in the experimental work. More recently
Smeets et al. [10] have published experimental data on
a solid state nanopore for an electrolyte with KCl con-
centration varying from 50 mM to 1.0M. Remarkably, it
was found that the most probable translocation time ei-
ther did not vary at all with salt concentration or the
variation was too small to be detected. In this paper
the translocation speed is calculated based on the mech-
anism proposed in Paper A but allowing for a finite Debye
Layer thickness while restricting the geometry to a long
cylindrical pore. The objective is to determine whether
the proposed hydrodynamic model is consistent with the
observed experimental dependance of the translocation
speed on salt concentration.
Figure 1 shows the geometry for our simplified calcu-
lation. The pore shape in the experiment actually re-
sembles a hyperboloid with a smallest diameter of 10.2
nm. Smeets et al. [10] report that the bulk conductance
of the pore is equivalent to that of a cylindrical nanopore
of identical diameter and length L = 34 nm. For the
purpose of comparing our calculation with experiments,
we will consider a cylindrical pore with these dimensions.
Moreover, we will assume the flow field to be uniform in
the axial direction, an assumption that is strictly valid
only for an infinitely long cylinder. Let us model the part
of the polyelectrolyte inside the pore by a straight rigid
cylindrical rod (of radius a = 1 nm) that is co-axial with
the cylindrical pore (of radius R = 5.1 nm) and translo-
cating at a velocity v in the axial direction (x). Such a
model is reasonable since the persistence length of double

30 0.2 0.4 0.6 0.8 1
KCl concentration (M)
0
0.5
1
1.5
2
Tr
an
slo
ca
tio
n
Ti
m
e
(m
s)
FIG. 2: Translocation times for 16.5 µm long ds-DNA through
a 10.2 nm diameter solid state nanopore. Solid line is calcu-
lated from equations (15), (16) and (17), the symbols are
replotted from the data presented in Figure 4(b) (inset) of
Smeets et al. [10]
from equation (12) for the translocation speed in Paper A
if one assumes a uniform cylinder for the pore shape. In
addition, it has the following very simple interpretation in
the limit of thin Debye layers: the part −ǫE0ζw/µ is the
electro-osmotic flow through the pore generated by the
applied field and ǫE0ζp/µ is simply the electrophoretic
speed of an object of arbitrary shape in a reference frame
fixed to the moving fluid in the nanopore [? ].
Equation (15) will now be compared to experimental
data due to Smeets et al. referenced earlier [10]. In
the Debye-Huckel approximation the ζ potential of the
polyelectrolyte, ζp is related to its linear charge density
λ through the formula (see Paper A)
ζp =
λλD
2πaǫ
K0(a/λD)
K1(a/λD)
, (16)
where λD is the Debye length and Kn are the modi-
fied Bessel functions of order n. For a univalent salt
like KCl the Debye length (in nm) is given by [11]
λD = 0.303/
√c where c is the Molar concentration of
the salt. The experimental data is in the range 0.05 to
1.0 M so that λD ranges from 0.30 nm to 1.36 nm. Since
R = 5.1 nm and a = 1.0 nm, there is no significant
overlap between the Debye layers at the polyelectrolyte
and the wall for concentrations above 0.05 M, though
for even smaller concentrations such effects may be ex-
pected. Thus, it is reasonable to use the expression (16)
which is strictly true only for an isolated infinite rigid
rod in an unbounded electrolyte. The dielectric constant
ǫ/ǫ0 = 80 and dynamic viscosity µ = 8.91 × 10−4 Pa
s for the electrolyte are taken as those of water. For
the linear charge density on the DNA we take 5.9 elec-
tronic charges per nm reduced by the Manning factor of
4.2, thus λ = −2.25 × 10−10 C/m. This assumption is
supported by recent force measurement experiments [12]
that show that polyelectrolyte charge is reduced by the
classical Manning factor when the DNA is inside the pore
over a wide range of salt concentrations. The electric field
intensity is obtained by assuming that the entire voltage
drop of 120 mV occurs over the length of the equivalent
cylinder which is L = 34 nm, thus, E0 = −3.53 × 106
V/m. The ζ-potential at the SiO2 wall may be obtained
from the expression
ζw = a0 − a1 log10 c (17)
where c is the molar concentration of K+ ions. The func-
tional form of the dependence on concentration follows
in the low counter-ion concentration limit from the non-
linear Gouy-Chapman model of the Debye layer in case
of symmetric electrolytes. However, it has been shown
to provide a good empirical fit to experimental data for
counter-ion concentrations up to 1.0M [13]. For KCl on
silica a0 ≈ 0 and a1 ≈ −30 mV.
The translocation velocity, v is calculated from equa-
tion (15) for a range of concentrations from 0.01 to
1.01M. The corresponding translocation time for a Lp =
16.5 µm long DNA, t = Lp/v is shown as the solid line in
Figure 2. A notable feature is the lack of sensitivity of the
translocation time to the salt concentration: it changes
by at most a factor of three when the salt concentration
ranges over two orders of magnitude. Taking into ac-
count the considerable scatter in the experimental data
and the various approximations made in the theory, the
agreement between the two is quite reasonable, pointing
to the adequacy of the underlying hydrodynamic model.
The existence of a maximum in the translocation time
at a concentration of about 0.1 M KCl seems to be sup-
ported by the data, although one cannot be completely
certain of this on account of the uncertainty in the data.
The principal uncertainties involved in applying the hy-
drodynamic model to nanopores were discussed in Pa-
per A. Those same considerations apply to the current
calculations as well and need not be repeated here. It
should also be kept in mind that although the motion of
the polymer is treated as a unidirectional translation at
constant speed, the actual translocation takes place via a
drift diffusion process as described by Lubensky and Nel-
son [8]. Here it is assumed, as is done in the classical the-
ory of Brownian motion of particles, that, the mean part
of the motion of the polymer may be obtained through
the solution of a classical hydrodynamics problem that
ignores the fluctuating forces. The hydrodynamic model
or indeed any model that localizes the entire resistive
force at the pore region would predict a translocation
speed that is independent of polymer length. This is
valid only for polymers that are not too long (see Paper
A). For very long polymers the resistive force has an en-
tropic part as discussed by various authors [14, 15, 16].
Sto¨rm et al. [17] have suggested that the viscous drag on
the randomly coiled part of the polymer lying outside the
pore could also be significant.
DNA translocation experiments that have been per-
formed to date can be divided into two classes; those
that use a natural protein nanopore (α-hemolysin) on a

4lipid membrane [3, 4], and those that use a mechanical
nanopore on a solid substrate made by specialized tech-
niques [6, 18]. Although the principle is similar, these
two types of nanopores differ with respect to some im-
portant details. One essential physical difference is that
the narrowest part of the α-hemolysin pore is about 1.5-
2.0 nm in diameter so that only single stranded DNA
or RNA is able to pass through it. When the pore is
blockaded by such a single strand the blockade is al-
most complete in that very few ions and probably none
of the water is able to pass through the blocked pore.
Although solid state pores can be made with pore sizes
approaching 1 nm, most of the experiments to date have
been done with 5 − 10 nm diameter pores which can
be made in a more reliable and reproducible manner.
These larger diameter pores admit both single and dou-
ble stranded DNA, and furthermore dsDNA can enter
the pore in a folded fashion, notwithstanding the relative
rigidity of these polymers [7]. The main observable dif-
ference in terms of translocations across the two kinds of
pores is that the polymer passes through the solid state
pores about two orders of magnitude faster than it does
through α-hemolysin pores. It is important to stress that
the analysis presented here applies to only the 5 − 10
nm solid state nanopores. Although a similar hydrody-
namic model could be constructed to model the viscous
force arising out of the water in the vestibular part of
the α-hemolysin pore, such a model must of necessity
differ from the current one in the details of its formu-
lation. Furthermore, the applicability of the continuum
equations for electrostatics and hydrodynamics would be
questionable to a much greater degree than in the anal-
ysis presented in this paper. It has been suggested that
in order to explain the much slower translocation speed
in protein pores, something other than hydrodynamics
is needed: perhaps an atomic level pore-polymer interac-
tion, an electrostatic self-energy barrier [19] or the energy
cost associated with stripping hydration layers from the
polymer as it enters the pore. The results derived in this
paper neither supports nor refutes the validity of these
alternate mechanisms for the 1.5− 2.0 nm protein pores.
It does however show that for the 5 − 10 nm solid state
pores hydrodynamic resistance can explain the experi-
mental data in the absence of any of the other mecha-
nisms.
In conclusion, the hydrodynamic model introduced in
Paper A to calculate the average transition time of a
polyelectrolyte across a nanopore under an applied elec-
tric field was extended to treat the case of a finite Debye
layer thickness, though the geometry was restricted to the
simple case of a cylindrical pore. The predicted translo-
cation times are found to be consistent with available
experimental data to within the uncertainties inherent in
the experiment and the theory. As a final remark, it is
worth noting a few practical implications of the simple
model presented here in relation to the problem of how
one needs to tune the available parameters to make the
translocation time as large as possible. First, Figure 2
shows that an optimal salt concentration exists for which
the translocation speed is a maximum, though the gain
here is no more than a factor of 3. A better strategy is
suggested by equation (15) which shows that v vanishes
if ζw = ζp. Physically this essentially amounts to bal-
ancing the electrophoretic migration of the DNA against
an opposing electroosmotic flow generated at the wall.
In principle this could be achieved by using an alternate
substrate, a coating on the existing substrate or a physi-
cal or chemical treatment of it that alters its ζ-potential.
The object is to select a substrate such that ζw ≈ ζp
and then “fine tune” the salt concentration to achieve a
closer match. As an example, Poly(methyl methacrylate)
(PMMA) is a commonly used substrate in microfluidic
application for which a0 = −4.06 mV and a1 = −12.57
mV [20]. Using these values in (17) and plotting the re-
sult together with equation (16) it is easily seen that the
two curves intersect at a salt concentration of about 0.6
M. Operating near this molarity with a PMMA substrate
should result in significantly slower translocations.
[1] S. Ghosal, Phys. Rev. E 74, 041901 (2006).
[2] B. Alberts et al., Molecular Biology of the Cell (Garland
Publishing, Taylor & Francis Group, New York, U.S.A.,
1994).
[3] J. Kasianowicz, E. Brandin, D. Branton, and D. Deamer,
Proc. Natl. Acad. Sci. 93, 13770 (1996).
[4] A. Meller et al., Proc. Natl. Acad. Sci. 97, 1079 (2000).
[5] W. Vercoutere et al., Nature Biotechnology 19, 248
(2001).
[6] A. Storm et al., Nature Materials 2, 537 (2003).
[7] A. Storm, J. Chen, H. Zandbergen, and C. Dekker, Phys.
Rev. E 71, 051903 (2005).
[8] D. Lubensky and D. Nelson, Biophys. J. 77, 1824 (1999).
[9] D. Deamer and M. Akeson, Trends in Biotech. 18, 147
(2000).
[10] M. Smeets et al., Nano Letters 6, 89 (2006).
[11] R. Probstein, Physicochemical Hydrodynamics (John Wi-
ley and Sons, Inc., New York, U.S.A., 1994).
[12] U. Keyser et al., Nature Physics 2, 473 (2006).
[13] B. Kirby and E. Hasselbrink, Electrophoresis 25, 187
(2004).
[14] W. Sung and P. Park, Phys. Rev. Lett. 77, 783 (1996).
[15] R. Boehm, Macromolecules 32, 7645 (1999).
[16] M. Muthukumar, J. Chem. Phys. 111, 10371 (1999).
[17] A. Storm et al., Nano Letters 5, 1193 (2005).
[18] J. Li et al., Nature 412, 166 (2001).
[19] D. Bonthuis et al., Phys. Rev. Lett. 97, (2006).
[20] B. Kirby and E. Hasselbrink, Electrophoresis 25, 203
(2004).