Negative memory capacitance and ionic filtering effects in asymmetric nanopores

Abstract

The pervasive model for a solvated, ion-filled nanopore is often a resistor in parallel with a capacitor. For conical nanopore geometries, here we propose the inclusion of a Warburg-like element, which is necessary to explain otherwise anomalous observations such as negative capacitance and low-pass filtering of translocation events (we term this phenomenon as Warburg filtering). The negative capacitance observed here has long equilibration times and memory (that is, mem-capacitance) at negative voltages. We used the transient occlusion of the pore using λ-DNA and 10 kbp DNA to test whether events are being attenuated by purely ionic phenomena when there is sufficient amplifier bandwidth. We argue here that both phenomena can be explained by the inclusion of the Warburg-like element, which is mechanistically linked to concentration polarization and activation energy to generate and maintain localized concentration gradients. We conclude the study with insights into the transduction of molecular translocations into electrical signals, which is not simply based on pulse-like resistance changes but instead on the complex and nonlinear storage of ions that enter dis-equilibrium during molecular transit.

Fig. 1 |. CP and charge storage model.

a, Mean concentration of ions inside a nanopipette under +200 mV and –200 mV as obtained through numerically solving the Poisson–Nernst–Planck and Navier–Stokes equations. Bulk ionic concentration used in the simulation was 10 mM and ionic properties were that of K⁺ and Cl^–. The surface charge on the pore was −10 mC m^–2. b, CP factor as a function of ionic concentration as predicted by numerical simulations (Supplementary Table 1 and Supplementary Fig. 1). c,d, Schematic of a traditional capacitor in which charge is separated in space and can discharge during a voltage change. e,f, Schematic of an ionic negative capacitor in which charge is co-localized but can still discharge with a voltage change. The negative slope of the $Q$ versus $V$ curve is a characteristic of negative capacitance.

Fig. 2 |. Warburg impedance model and its effects on nanopore current measurements.

a, Proposed equivalent circuit model for a nanopipette that contains a Warburg element in series with the nanopore resistance. The Warburg element is modelled as a finite series of RC elements. b, Experimental data showing the raw data of sequential voltage pulses used to generate the $I-V$ curve (left; black arrows mark the observation of negative capacitance). The pore diameter is approximated to be ~30 nm and was filled with 10 mM KCl (pH 7.4). COMSOL simulation results are shown for a nanopipette switching from +500 mV to −500 mV and −500 mV to +500 mV at $t=0.5\text{ms}$ (right). A potential point probe was used for plotting the voltage transients (probe location was 250 nm from the pore). c, Frequency-domain simulation of the voltage across $R_1$, which represents a Warburg element with $n=1$. The current through $R_1$ also exhibits the same attenuation properties as the voltage across $R_1$. d, Comparative analysis of experimental data and the circuit simulation results for a voltage step and transient pulse. The voltage step was acquired from an ~30 nm pore symmetrically filled with 10 mM KCl (pH 7.4) and after −1,000 mV was applied to the pore (zero-voltage bias before the voltage pulse). The transient pulse was obtained by translocating $\lambda$-DNA through an ~20 nm pore at 10 mM KCl ($V=+500\text{mV}$, pH 7.4). Circuit simulations were performed using CircuitLab (www.circuitlab.com).

Fig. 3 |. Current transients depend on pore’s voltage stimulus history.

a, Negative capacitive current response from a voltage pulse (0 V to −1,000 mV) recorded from the same pore (~30 nm) at different salt concentrations: 10, 30, 100 and 300 mM KCl. b, Extracted decay rates for negative capacitance under low-salt (10 mM) and high-salt (1 M) conditions as a function of voltage. c, Negative capacitive current spike for the 0 V to −1,000 mV voltage switch when protocol 1 was executed starting from –1,000 mV versus starting at +1,000 mV. Although the voltage pulse was the same, the history of the voltage pulses changes the capacitive current response. d, Protocol 2 data in which the voltage was maintained at +500 mV for various durations and then switched to −500 mV. e, Peak capacitive current (for example, that shown in d using black arrows) as a function of ion depletion time (duration spent at +500 mV before voltage switch) for different pore sizes. All the recordings were repeated with a minimum of three devices, and the representative traces are shown here.

Fig. 4 |. Ionic filtering attenuates DNA translocation events.

a,b, Schematic (a) of the experimental conditions (that is, DNA location) for data shown in b. $\lambda$-DNA translocation data at different voltages (+500 mV to −500 mV in increments of 100 mV) for the following condition: 500 pM $\lambda$-DNA, 10 mM KCl (pH 7). c, Schematic illustrating the concept of Warburg filtering; specifically, the ionic filter frequency is lower than that of the instrument low-pass filter. d, Event signatures for the following voltages: +500 mV, −500 mV, −400 mV, −300 mV and −200 mV. Unless otherwise noted, the instrument filter was a 10 kHz low-pass filter. e, Current trace and representative DNA events. The analyte, $\lambda$-DNA, was only located inside the nanopipette with 10 mM KCl (pH 7.4) on both sides of the nanopipette and events were observed at negative voltages. The data were recorded at −400 mV. Event signatures, each labelled with their dwell time, illustrate the attenuation of the current enhancements. f, Scatter plot of event properties (peak current enhancement and dwell time) for $\lambda$-DNA inside the pipette and with 10 mM KCl (pH 7.4) on both sides of the nanopipette. g, Representative event signatures for three DNA configurations (linear, partially folded and fully folded) in the attenuated and unattenuated scenarios. For all the experiments, the pore diameter was approximately 20 nm (Supplementary Fig. 22).

Fig. 5 |. Ionic filter characterization and correction.

a, Averages of the rise time and fall time of the sub-regions of $\lambda$-DNA current signatures (event rise and event fall averaged over 625 events). b, Histogram of the time constant extracted from each event’s rise and fall. Event rise data include 625 events and event fall data include 781 events. See Methods for fitting details. c, Schematic showing how the position of DNA inside the nanopipette seems to impact the measured capacitance (that is, RC constant) during the event rise and event fall. d, Current enhancement histograms for events that have dwell times $\left(t_{\text{d}}\right)$ above (top) and below (bottom) the critical event duration $\left(T_{\text{critical}}=T_{\text{r}}+T_{\text{f}}\right)$. Below $T_{\text{critical}}$, attenuation of the current signature is expected. e, Illustration of how DNA events comprise two components: a resistive component and a capacitive component. f, Plot of conductance change versus dwell time showing an attenuation profile that is consistent with a 500 Hz low-pass filter. Note that all the signals were recorded using a 10 kHz low-pass filter. g,h, Extracted and corrected current enhancement histograms for the voltage bias of −700 mV. Gaussian peak fitting was performed on h and shows three peaks corresponding to linear (1), partially folded (2) and fully folded (3).

Fig. 6 |. Ionic filtering depends on the magnitude of ionic perturbation and salt conditions.

a, Current traces showing 10 kbp DNA and $\lambda$-DNA events collected under asymmetric salt conditions (1 M/4 M KCl, pH 7.4). All events were collected at a negative-voltage bias (−400 mV) and events were conductive. b, Decay-rate histogram for both event rise and event fall that correspond to DNA entering and exiting the pore, respectively. c, Scatter plot of $\lambda$-DNA event properties and a theoretical low-pass filter with a cut-off frequency of 146 Hz. d, Histogram of the decay rate of event fall (DNA exit) for $\lambda$-DNA and 10 kbp DNA. All data were collected at a voltage bias of −400 mV. e, Schematic showing that DNA size is a critical factor in ionic dis-equilibrium and therefore (1) ionic filtering effects and (2) translocation memory effects.