Main

Voltage-driven transport of macromolecules through biological6,7 and artificial8,9,10,11,12 nanopores provides an ideal system for studying the physics of the translocation process13. Electrophoretic migration is the main driving force for DNA translocation through nanopores and is a consequence of the force exerted by an externally applied electric field on the charges on the polyelectrolyte chain. As DNA in solution is negatively charged, it is screened by a layer of more or less mobile, positively charged counterions that also experience the electric field. It has long been recognized that the force exerted on the counterions induces a hydrodynamic drag force on the DNA that locally balances the electrical force, leading to much slower migration than would be expected from Stokes drag alone (see, for example, ref. 1). The electrophoretic translocation of a polymer through a small pore obeys similar principles, but here the geometry of the pore (or the local structure of the gel) is expected to profoundly influence the hydrodynamic flow profile around the DNA and thereby the drag force opposing movement1,5,14. Despite increasingly sophisticated experiments3,12,15,16, an unambiguous manifestation of hydrodynamic interactions in DNA translocation has not yet been reported. Here, we address this issue using our recently developed apparatus combining solid-state nanopores with optical tweezers3 to arrest DNA translocation, and subsequently measure the stalling force on a single DNA strand (see the Methods section). A schematic diagram of the measurement apparatus is shown in Fig. 1a.

Figure 1: Experimental configuration.
figure 1

a, Schematic diagram of the measurement apparatus. Using optical tweezers, DNA translocation is stalled and the force on the DNA is measured. The nanopore ionic conductance is also measured simultaneously. b, Cylindrical model of DNA in a nanopore. The blue arrows on either side of the DNA schematically depict the velocity of the moving fluid due to electro-osmosis.

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In contrast with a gel electrophoresis experiment, the electric field in a nanopore experiment is confined to the immediate vicinity of the pore and electrical forces act locally on a short segment of the DNA. The electric field strength in the pore typically reaches ∼106 V m−1 owing to the very thin (∼60 nm) free-standing membranes. The large persistence length of λ-DNA, about 50 nm, ensures that the DNA segment inside the pore is practically fully extended. The situation is depicted schematically in Fig. 1b, where L and R represent the length and radius of the pore and ΔV is the applied potential. DNA has a bare line charge density λbare=−0.96 nC m−1 (2 electrons per base pair) and is modelled here as a uniformly charged cylinder with radius a=1.1 nm. The bare electrostatic force on the DNA backbone is represented by Fbare=λbareΔV (ref. 3). It is opposed by a drag force Fdrag caused by the electrophoretic motion of its counterions in the opposite direction: Fdrag is thus an intrinsic component of the translocation process that is also ultimately traceable to electrical forces5. The resulting net force is the electrophoretic force driving the translocation, given by Felec=Fbare−Fdrag. In our case, Felec is balanced by an opposing mechanical force Fmech=−Felec from the optical tweezers that arrests the DNA inside the pore.

The origin of Fdrag lies in the spatial distribution of the ions and the corresponding fluid flow profile. A mean-field description of the ion distributions is given by the Poisson–Boltzmann formalism, in which the electrostatic potential is given by and the corresponding ion distributions as . Here, λD is the Debye length, is the reduced electrostatic potential, e is the elementary charge, Φ is the potential, kB is the Boltzmann constant, T is the temperature, n is the number density of ions and z is the valency of the ionic species. Figure 2a shows calculated potentials for two pore sizes obtained by numerical evaluation of the Poisson–Boltzmann equation in a cylindrical geometry (see the Methods section). The corresponding ion distributions are shown in Fig. 2b. The Debye layer of the DNA is characterized by a depletion of coions, whereas counterions pile up to very high densities owing to the high DNA charge. As these distributions are solutions to the full (nonlinearized) Poisson–Boltzmann equation, the effect of Manning condensation17 is included.

Figure 2: Mean-field calculations.
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a, Reduced electrostatic potential for a ‘large’ pore (green, R=10 nm) and a ‘small’ pore (blue, R=3 nm). The grey line indicates the location of the pore wall for the small pore. The salt concentration is cbulk=20 mM (λD=2.2 nm) and the pore walls are taken as uncharged in this particular calculation for simplicity. b, Corresponding ion distributions. In ‘large’ pores (R≫λD), the distribution of ions around the DNA is not significantly affected by the pore wall, as indicated by the observation that the green curves reach the bulk value. However, in ‘small’ pores (R≲λD), the counterion cloud (solid curves) is compressed by the pore wall and the pore can be depleted of coions (dashed curves), leaving mainly positive counterions to balance the negative charges on the DNA. This situation is exemplified by the blue curves. c, Fluid velocity profiles calculated by combining results from a and b with the Stokes equation. d, Normalized surface integral of the shear stress at each position r, calculated as , with τ being the shear stress in the fluid. At the DNA surface, . e, Calculated reduced surface potentials for DNA (red) and nanopore wall (grey) as a function of the pore radius. The grey area qualitatively depicts the ‘small pore’ region, in which the ion distributions are influenced by the confinement of the nanopore.

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In our DNA stalling experiment, the distribution of the counterions largely determines the velocity profile of the induced electro-osmotic flow. The flow velocity profile can be calculated by solving the Stokes equation , where η is the dynamic viscosity of the fluid, vz is the fluid velocity, Ez is the applied electric field and ρ(r) is the ionic charge distribution. Calculated fluid flow profiles are shown in Fig. 2c. Subsequently, to relate the flow profile to viscous drag, the shear stress in the fluid is calculated as τ(r)=−η(dvz/dr). Figure 2d shows , which is equal to the ratio Fdrag/Fbare when evaluated at the DNA surface (r=a). It shows that Fdrag is of the same order of magnitude as the bare electrostatic force acting on the DNA that it opposes. Fdrag is larger for the larger pore, corresponding to a smaller stalling force.

For comparison with our experiments, it is convenient to express the above model in a form that directly relates the electrophoretic force to the properties of the nanopore. As was shown previously5, the Poisson–Boltzmann and Stokes equations can be combined to yield an expression for the electrophoretic force on a stationary molecule,

Φ(a) and Φ(R) are the surface potentials of the DNA and the nanopore, respectively, and ε is the dielectric constant of water. and deduced from the Poisson–Boltzmann equation are plotted in Fig. 2e as a function of R. In large pores, the surface potentials are independent of R. Equation (1) then predicts that the measured force Fmech has a simple ln−1(R/a) dependence on the size of the pore. In smaller pores, on the other hand, the Debye layer is compressed by the pore wall (Fig. 2). This results in a dependence of the surface potentials on the pore size and a correspondingly more complicated dependence of Fmech on R.

A way to directly test the above model is to measure the DNA stalling force Fmech as a function of the pore radius R. As previous work has focused on smaller pores, however, we first demonstrate that it is possible to detect the presence of a single DNA molecule in very large (R≫a) nanopores. The detection of DNA in the pore is based on measuring the step in ionic conductance ΔG when DNA enters the pore. ΔG was previously shown to be caused by competition between two contributions: volume exclusion, lowering the number of ions available for conductance, and an excess of DNA counterions, effectively increasing the number of ions available for conductance12. Which of these two effects dominates depends on the bulk concentration of electrolyte, causing ΔG to be positive (conductance increase) in the present experiments with 20–50 mM salt. Under these optimized salt conditions, the signal-to-noise ratio of DNA capture is predicted to be high even in very large pores18. A typical step in the current together with the corresponding change in position Z of the bead is shown in Fig. 3a. We observe a clear step in the current even though the DNA changes the conductance of the nanopore by less than 1%. A typical histogram of capture events is shown in Fig. 3b. These data show that we are able to controllably insert and detect single molecules even in nanopores with R=45 nm, where the DNA covers only 1/2000th of the nanopore cross-sectional area.

Figure 3: Conductance change due to DNA capture.
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a, Typical example of pore current I (upper panel, red averaged) and bead position Z (lower panel) during DNA capture in an R=39 nm pore in 33 mM KCl at 80 mV. Current measurements were low-pass filtered at 1 kHz. b, Histogram of ΔG of 88 DNA captures in the nanopore under the same conditions as in a. c, ΔG in pores of different radii. Capture events typically occurred at voltages of 60–100 mV. The horizontal dashed line is a guide to the eye. Triangles: 20 mM KCl, stars: 33 mM KCl, diamonds: 50 mM KCl, green star: data from b, error bars: standard deviation evaluated from ΔG histograms. The open symbols represent data from free translocation experiments without optical tweezers.

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Experimentally determined ΔG values as a function of nanopore radius are shown in Fig. 3c for 10 nanopores with radii ranging from R=3 to 45 nm. The grey area indicates the ‘small pore’ region in which, according to the calculations in Fig. 2b, the ion distributions near the DNA are influenced by the presence of the nanopore wall. In apparent concurrence with the simulations, ΔG is approximately constant in large nanopores. The larger ΔG in small nanopores is consistent with the compression of the diffuse screening layer, ensuring charge neutrality in the pore. A more quantitative comparison of the measured ΔG with theory is difficult, however: the access resistance Racc becomes an increasingly important contribution to the total resistance of the system in large pores19, and it is currently unknown how the presence of DNA threading through the pore affects Racc.

Having established the feasibility of arresting and detecting DNA molecules in large as well as small pores, we now turn to the main result of this letter, namely, the experimentally determined magnitude of the stalling force Fmech as function of the nanopore radius R. Figure 4a shows Fmech as a function of applied voltage ΔV in nanopores with R=4 nm (left panel) and R=39 nm (right panel). This relationship is approximately linear in both cases. However, Fmech/ΔV in the small pore is a factor of two larger than in the large pore. Figure 4b shows the ratio Fmech/ΔV as a function of pore size: Fmech/ΔV decreases gradually with increasing size, as expected on the basis of equation (1). The dashed curve results from combining the surface potentials for the full DNA charge (Fig. 2e) with equation (1), without any extra adjustments. In spite of the inherent simplifications of a one-dimensional model, the theoretical result captures the trend in the data quite well, although it quantitatively overestimates the stalling force by ∼50%. This difference can be attributed to (a combination of) several factors, including a reduction of counterion mobility at the DNA surface12,20, or an extra opposing electro-osmotic fluid flow resulting from fixed charges on the nanopore surface5. In terms of equation (1), both effects can be represented by lowering the magnitude of the surface potential difference ΔΦ=Φ(a)−Φ(R), thereby reducing the magnitude of Fmech. Empirically reducing ΔΦ by 33% results in the solid curve in Fig. 4b, an excellent fit to the experimental data. This reduction of ΔΦ is equivalent to approximately 50% of the DNA’s bare charge being screened by ions that are effectively immobilized on its surface (as shown in Supplementary Fig. S1) or to the presence of a 15 mC m−2 surface charge on the pore wall (Supplementary Fig. S2). Although our experiment cannot directly distinguish between these two mechanisms, the inferred surface charge density is typical for SiO2 surfaces21. This suggests that the surface charge density in our nanopores is not strongly influenced by the fabrication process and that the pore charge is responsible for a substantial part of the observed correction.

Figure 4: Pore-size dependence of the DNA stalling force.
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a, Measurements of the stalling force as a function of the applied potential in a small (left panel) and a large nanopore (right panel). In each case, two measurements obtained at different distances between the bead and the nanopore are shown. b, Measured force versus pore radius. The blue and green symbols correspond to data in a. The data in the small-pore area were taken from ref. 3. The curves represent the theoretical result for the bare DNA charge (dashed curve) and for a reduced ΔΦ (solid line, see text). The inset shows schematically how the force on the DNA counterions is divided between the DNA and the pore walls through viscous drag in a small and a large pore; the yellow arrow represents the bare electrostatic force acting on the DNA. Triangles: 20 mM KCl, stars: 33 mM KCl, diamonds: 50 mM KCl. Error bars stem from the uncertainties in the calibration of the optical trap, which are estimated at 10–30% of the calibrated trap stiffness.

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External factors not taken into account in our simple model might affect the absolute value of the stalling force as well. For example, it might be biased by electrostatic and/or hydrodynamic forces on the bead or on the portion of the DNA residing in the electric field just outside the nanopore. However, our experiments show no evidence for such effects, as we detected no change in the stalling force when the bead was positioned at increasing distances from the nanopore3. In addition, experimental force–voltage curves are linear in our voltage range (Fig. 4a), indicating that entropic forces are small compared with the electrostatic forces, consistent with independent measurements of the entropic forces in DNA stretching22.

The dependence of the measured force on pore radius in Fig. 4b directly demonstrates that the electrophoretic force in DNA translocation is in part determined by hydrodynamic coupling between the DNA’s counterions and the nanopore wall. The nanopore geometry determines the magnitude of the drag force exerted on the DNA by these counterions. This coupling has important consequences for the interpretation of experimentally determined forces because these forces are manifestly not determined by the intrinsic properties of the DNA alone1,23.

Variations of the force can also be expected with increasing salt concentration because this changes the Debye length and thereby both the DNA surface potential and presumably the stalling force Fmech. This expectation is however complicated by the facts that the pore surface charge density also changes considerably with salt concentration12 and that the mean-field theory used here becomes unreliable at high ionic strength. In previous experiments, Fmech was in fact found to be independent of salt concentration up to 1 M (ref. 3), and an interpretation of those data based on a model in which Φ(R) depends on salt concentration was suggested by Ghosal5.

In free DNA translocation, the restoring force Fmech is absent and effectively replaced by an extra Stokes drag FStokes proportional to the velocity of the molecule. The molecule translocates at constant velocity, the electrophoretic force being balanced by FStokes. The relation between the stalling force Fmech, as measured with the optical tweezers, and the translocation velocity vtrans, as determined from free DNA translocation experiments, is given by4,5,14

This result is equivalent to the Stokes drag that would be experienced by an uncharged cylinder translocating through a cylindrical pore at constant velocity vtrans. Thus, Fmech and vtrans are proportional quantities, related by a geometry-dependent factor. This expectation is qualitatively confirmed by experimental data3,12, which show that Fmech and vtrans are indeed roughly proportional. Calculating vtrans as the λ-DNA contour length divided by the experimentally determined translocation time gives vtrans=(16 μm/1.1 ms)≈15 mm s−1, resulting in Fmech/vtrans=1,750 pN s m−1 (data for 5-nm-radius pores). Equation (2) however gives a value of 249 pN s m−1 for this factor, using η=1×10−3 N s m−2, L=60 nm, R=5 nm and a=1.1 nm. This is seven times lower than the experimentally obtained value. Translocation of DNA through these pores therefore occurs at a much lower velocity than expected from the electrophoretic forces measured with the optical tweezers and the mean-field description. As similar nanopores were used in all of these experiments, this conclusion points to a more fundamental difference between the static situation of a stalled molecule and the dynamic situation of a translocating molecule. In free DNA translocation, the conformation of the DNA outside the pore results in extra drag on the moving molecule24, which may account for the discrepancy.

In conclusion, we have shown that the electrophoretic force on a DNA molecule in a nanopore depends on the geometry of the pore. This is not expected from simple electrostatics but can be understood straightforwardly by considering the hydrodynamic drag induced by the counterions screening the DNA in solution. Numerical calculations based on mean-field equations yield a good description of the pore-size dependence of the measured forces.

Methods

Experimental procedures.

The nanopores are fabricated by drilling and subsequently expanding holes with ∼1 nm accuracy in a 60-nm-thick SiO2/SiN/SiO2 membrane using a transmission electron microscope25. The experimental apparatus has been described elsewhere26 and is shown schematically in Fig. 1a. In brief, a microfluidic sample cell containing the nanopore is mounted on an inverted microscope. A standard current amplifier (Axon 200b, Molecular Devices) is used to detect the presence of single DNA molecules in the nanopore. An optical trap required for the force measurements is created by coupling an infrared laser (wavelength 1,064 nm) into the microscope objective. In the trap, micrometre-sized beads coated with DNA molecules (λ-DNA, length=48.6 kb) are held in close proximity to the nanopore until a single DNA molecule is captured electrophoretically by the nanopore3,27. The translocation is subsequently stalled by the restoring force of the optical trap. The force measurements are carried out with a typical distance of 5 μm between the bead and the nanopore. The optical trap is aligned above the centre of the nanopore with accuracy of a few tens of nanometres28. In all experiments, the solution has pH=8.0 set by adding 1 mM of TrisHCl-EDTA.

Modelling.

Converging numerical solutions to the nonlinear Poisson–Boltzmann equation in one-dimensional geometries were obtained using the COMSOL Multiphysics package (COMSOL, Inc.) and a custom MATLAB script (The Mathworks, Inc.). One boundary condition was determined by the charge density on the DNA, where dΦ/dr=−λbare/2πa ε. A second boundary condition depended on the properties of the nanopore surface. Nanopores composed of SiO2 are typically negatively charged, leading to an extra liquid flow opposite to the direction of motion of the DNA. To simplify the discussion, the nanopore walls were assumed uncharged in the calculations of Fig. 2. The situation including charged walls is however described in Supplementary Fig. S2. The boundary condition on the insulating pore wall was thus taken as dΦ/dr=0. All of the calculations were carried out for monovalent symmetric electrolyte with bulk concentration cbulk=20 mM.