Concentration Polarization in Translocation of DNA through Nanopores and Nanochannels

In this Letter we provide a theory to show that high-field electrokinetic translocation of DNA through nanopores or nanochannels causes large transient variations of the ionic concentrations in front and at the back of the DNA due to concentration polarization (CP). The CP causes strong local conductivity variations, which can successfully explain the nontrivial current transients and ionic distributions observed in molecular dynamics simulations of nanopore DNA translocations as well as the transient current dips and spikes measured for translocating hairpin DNA. Most importantly, as the future of sequencing of DNA by nanopore translocation will be based on time-varying electrical conductance, CP, must be considered in experimental design and interpretation—currently these studies are mostly based on the incomplete pore conductance models that ignore CP and transients in the electrical conductance.

Figure 1

(a) Schematic of DNA (in red) with a surrounding EDL (in blue) in a nanochannel, and the location of the CP-induced ion-depletion (between planes 1 and 2) and ion-enrichment (between planes 3 and 4) zones. The arrows (in blue) depicting the cationic current are located inside the nanochannel and are directed from left to right, whereas the arrows (in red) depicting the anionic current are located inside the nanochannel and are directed from right to left. (b) The different dimensions corresponding to the DNA transport in a nanochannel. The channel height is $2H$ and width $w$ ($2H=w$), $a$ is the DNA radius of cross section and ${\lambda }_D$ is the DNA-EDL thickness. (c) (left) Schematic of the event of a DNA entering a nanopore (radius $R$) [also see Figs. 6(a),(b) in 9], and the locations of the planes 1 and 2 in between which ion depletion will occur. (c) (right) Schematic of the event of a DNA (shown in red) exiting a nanopore [also see Figs. 6(d),(e) in 9], and the locations of the planes 3 and 4 in between which ion enrichment will occur. (d) The different dimensions corresponding to the DNA translocation in a nanopore, having radius $R$. In each figure, the purple arrow (located outside the nanochannel or nanopore, and directed from right to left) denotes the direction of movement of the DNA molecule and the brown arrow (located outside the nanochannel or nanopore, and directed from left to right) denotes the direction of the applied electric field (denoted as $E$). Not shown for sake of clarity are EDLs at the pore or channel walls of thicknesses identical to the DNA-EDL thickness.

Figure 2

Variation of ${\tau }_S$ and ${\tau }_{\mathrm{DNA}}^{c/p}$ [we denote ${\tau }_{\mathrm{DNA}}^c$ as ${\tau }^c$ and ${\tau }_{\mathrm{DNA}}^p$ as ${\tau }^p$ and take $l_{\mathrm{DNA}}=10.0\mu \mathrm{m}$ ($l_{\mathrm{pore}}=20.0\mathrm{nm}$) and ${\mu }_{\mathrm{DNA}}={10}^{-9}{\mathrm{m}}^2/\mathrm{Vs}$] with applied average current density $j_0$ (defined as $j_0=2\times {10}^3\mu {\mathrm{FEc}}_{\propto }$) for different ionic concentrations and nanochannel or nanopore dimensions (here NC refers to nanochannel and NP refers to nanopore). Also, except where it is mentioned we take $c_{\propto }=1\mathrm{M}$ and $R(\mathrm{\text{or}}H)=1.25\mathrm{nm}$. The different constant parameters are $D_{+}=D_{-}=D=2\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$, ${\mu }_{+}={\mu }_{-}=\mu =8\times {10}^{-8}{\mathrm{m}}^2/\mathrm{V}\mathrm{s}$, ${\zeta }_w=-10\mathrm{mV}$, $q_B=4$, and $\sigma =(2.4\times {10}^4/1.8\times {10}^{-5})\mathrm{e}/\mathrm{m}$.

Figure 3

(a),(c): Analytical and numerical simulation results for electrolyte depletion just outside the axial DNA EDL (at $x=-5{\lambda }_D$) in a $30\times 30\mathrm{nm}$ nanochannel with 10 mM KCl ($K^{+}$ are the cations and ${\mathrm{Cl}}^{-}$ are the anions) for (a) $j_0=1.508\times {10}^4\mathrm{A}/{\mathrm{m}}^2$ and (c) $j_0=1.508\times {10}^6\mathrm{A}/{\mathrm{m}}^2$. (b)(,d): Numerical simulation results for anionic (red dotted line) and cationic (blue bold line) concentration distributions for (b) $j_0=1.508\times {10}^4\mathrm{A}/{\mathrm{m}}^2$ and $t=2.5\sim \mathrm{ms}$ and (d) $j_0=1.508\times {10}^6\mathrm{A}/{\mathrm{m}}^2$ and $t=10$, 20, $30\mu \mathrm{s}$ (the numbers adjacent to the profiles denote the corresponding time in $\mu \mathrm{s}$). For the corresponding average electric fields see caption of Fig. 2. In (b) and (d), the purple arrow indicates DNA movement direction and the brown arrow applied field direction. Also in these figures we denote the location of DNA by a black line. The different parameters used for the analytical calculations are as in Fig. 2. For the numerical calculations, in addition to these parameters, we consider an immobile (negative) charge concentration of 2.628 mM over a length of $20\mu \mathrm{m}$ (typical for $\lambda$-DNA stretched in a $30\times 30\mathrm{nm}$ channel).

Figure 4

Numerical simulation results for cation and anion distribution in a nanopore for (a) an entering DNA at $t=10\mathrm{ns}$ after current application and (c) an exiting DNA at $t=100\mathrm{ns}$ after current application. Blue bold lines and red dashed lines depict the cation and anion concentrations, respectively, at $t=10$, 100 ns and dash-dotted lines the cation (blue) and anion (red) concentrations at $t=0$ (i.e., before current application). Numerical simulation results for temporal variation of cation (blue bold lines) and anion (red dashed lines) concentrations at $x=10\mathrm{nm}$ are shown in (b) and (d). In (b) this location is in front of the DNA and in (d) at the back of the DNA. In figs. (a) and (c), the purple arrow indicates DNA movement direction and the brown arrow applied field direction. The location of the DNA is indicated by a black line. We also use $R=1.25\mathrm{nm}$, $j_0=1.508\times {10}^9\mathrm{A}/{\mathrm{m}}^2$ (initial field $\sim {10}^8\mathrm{V}/\mathrm{m}$), DNA $\mathrm{\text{length}}=10\mathrm{nm}$, $\mathrm{\text{pore}}\mathrm{\text{length}}=20\mathrm{nm}$ (the corresponding bias voltage will therefore be 2 V). At the DNA section the immobile (negative) charge concentration is equal to 1232.4 mM, whereas that at the bare channel section is equal to 330.836 mM. At the pore ends bulk concentrations (1000 mM) exist.