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# Breakdown of the Nernst–Einstein relation in carbon nanotube porins

## Abstract

For over 100 years, the Nernst–Einstein relation has linked a charged particle’s electrophoretic mobility and diffusion coefficient. Here we report experimental measurements of diffusion and electromigration of K+ ions in narrow 0.8-nm-diameter single-walled carbon nanotube porins (CNTPs) and demonstrate that the Nernst–Einstein relation in these channels breaks down by more than three orders of magnitude. Molecular dynamics simulations using polarizable force fields show that K+ ion diffusion in CNTPs in the presence of a single-file water chain is three orders of magnitude slower than bulk diffusion. Intriguingly, the simulations also reveal a disintegration of the water chain upon application of electric fields, resulting in the formation of distinct K+–water clusters, which then traverse the CNTP at high velocity. Finally, we show that although individual ion–water clusters still obey the Nernst–Einstein relation, the overall relation breaks down because of two distinct mechanisms for ion diffusion and electromigration.

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## Data availability

All data (experimental and theory/computational) reported in this paper have been deposited in FigShare (https://figshare.com) and are available at https://doi.org/10.6084/m9.figshare.20728327.

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## Acknowledgements

This work was supported as part of the Center for Enhanced Nanofluidic Transport (CENT), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under award number DE-SC0019112. Work at the Lawrence Livermore National Laboratory was performed under the auspices of the US Department of Energy under Contract DE-AC52-07NA27344. Z.L. was supported by the Scientific Research Foundation of Graduate School of Southeast University (grant number YBJJ1802), Postgraduate Research and Practice Innovation Program of Jiangsu Province (grant number KYCX18_0067) and China Scholarship Council (CSC 201806090020). R.P.M. and D.B. acknowledge the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant number ACI1548562, for computational resources used to develop polarizable force fields, and the Lawrence Livermore National Laboratory for computational resources used to carry out MD simulations of ion transport. The authors also thank Dr. Tuan Anh Pham for facilitating access to LLNL computing resources, and Professor Martin Z. Bazant, Professor Michael S. Strano, and Dr. Satish K. Iyemperumal for fruitful discussions.

## Author information

Authors

### Contributions

Z.L. and A.N. designed the experiments and R.P.M. and D.B. designed the modelling component involving molecular simulations. Z.L. and Y.L. performed ion diffusion measurements, and S.Z. and Y.-C.Y. performed electrophoretic transport measurements. R.P.M. formulated the theoretical framework involving all-atomistic polarizable force fields and carried out the MD simulations and theoretical analysis. Z.L., R.P.M., Y.L., S.Z., Y.Z., Y.C., D.B. and A.N. analysed the data. A.N. and D.B. directed the project. All authors contributed to the writing of the manuscript.

### Corresponding authors

Correspondence to Daniel Blankschtein or Aleksandr Noy.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Nanotechnology thanks Chih-Jen Shih and Nikita Kavokine for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Potassium ions transport into LUVs in the absence of CNTPs.

A representative stopped-flow kinetic trace recorded for LUVs after exposure to a 25 mM K+ ion concentration gradient. No K+ ion transport was observed on the timescale of the experiment.

### Extended Data Fig. 2 Calibration of the fluorescence of the PBFI dye.

a. Changes in the dye fluorescence at a given K+ ion concentration. b. Calibration curve of the PBFI dye in accordance to the calibration equation. The derived value of Fmax from the linear fitting is 2.399 and the dissociation constant Kd is 11.41 mM. Data are shown as mean ± s.d. (n = 3 measurements).

### Extended Data Fig. 3 Simultaneous determinations of K+ ion concentrations inside the liposomes.

a. K+ ion concentration kinetics after exposure of CNTP–LUVs to different K+ ion concentration gradients. b. K+ ion concentration kinetics after exposure of control LUVs to a K+ ion concentration gradient of 25 mM.

### Extended Data Fig. 4 Ionic conductance of individual carbon nanotube porins.

Example trace showing jumps that correspond to individual CNTPs incorporating and leaving the lipid bilayer formed by the painting method. Data was taken in a buffer containing 100 mM KCl, 10 mM HEPES at pH 7.5 with an applied voltage of 100 mV. Traces were analysed using a custom Python CUSUM software to extract stepwise conductance increases.

### Extended Data Fig. 5 I–V characteristics of individual carbon nanotube porins.

a. Schematics of the experimental set-up where a small area lipid bilayer with a single CNTP was formed over a SiNx nanopore fabricated on a silicon chip. An external voltage was applied across the bilayer to drive the ion transport through the CNTP channel. b. Current–voltage (I–V) curves recorded for an individual CNTP channel at different KCl concentrations, where the conductance obtained from the slopes of the linear fitting lines is 6.0 ± 0.7 pS at 5 mM KCl, 12.5 ± 3.5 pS at 20 mM KCl, 24.9 ± 5.8 pS at 50 mM KCl, and 35.8 ± 3.7 pS at 100 mM KCl. The conductance value recorded in a 100 mM KCl solution is similar to the value obtained in the CNTP insertion measurement.

### Extended Data Fig. 6 Schematic of the multiscale model used to calculate the effective diffusion coefficient.

As discussed in the Methods section, the system is divided into 3 parts: (i) the bulk solution (highlighted in grey), (ii) the CNT entrance region (highlighted in purple), and (iii) the CNT interior (highlighted in yellow). The K+ ion diffusion in the CNT entrance and interior regions corresponds to anomalous subdiffusion, which is described using the generalized time-fractional diffusion equation with fractional diffusion coefficients and exponents.

### Extended Data Fig. 7 Quantifying the role of entrance effects on the diffusion coefficient of K+ ions in the presence of the single-file water chain.

a. MSD of the K+ ion at the CNT entrance for the CNTP with 3 COO groups. b. MSD of the K+ ion at the CNT entrance for the CNTP with 6 COO groups. The MSD profiles in a and b have been obtained from 80 independent MD simulations, and are utilized in the multiscale model to calculate the effective diffusion coefficient, Deff, for CNTPs with 3 COO and 6 COO groups, respectively.

### Extended Data Fig. 8 Formation of ion–water clusters during a 15 ns MD simulation run for the CNTP with 3 COO– groups.

a. Plot showing the variation of the number of water molecules inside the CNT, Nwater, during the first 7.5 ns of the MD simulation run in the presence of a 0.4 V·nm-1 external electric field applied along the z direction. Events corresponding to the translocation of K+ ions in the form of ion–water clusters are highlighted by the magenta solid boxes on the plot. The snapshots correspond to the time steps at which the K+ ions in the form of ion–water clusters cross the central region of the CNT (that is, z = 27 Å). b. Plot showing the variation of Nwater during the 7.5–15 ns MD simulation run. As shown in a and b, there are 5 distinct ion translocation events during the 15 ns MD simulation run, which are indicated by the magenta arrows.

### Extended Data Fig. 9 Formation of ion–water clusters during a 15 ns MD simulation run for the CNTP with 6 COO– groups.

a. Plot showing the variation of the number of water molecules inside the CNT, Nwater during the first 7.5 ns of the MD simulation run. Similar to Extended Data Fig. 8, events corresponding to the translocation of K+ ions in the form of ion–water clusters are highlighted by the magenta solid boxes. b. Plot showing the variation of Nwater during the 7.5–15 ns MD simulation run. As shown in a and b, there are 10 distinct ion translocation events during the 15 ns MD simulation run, which are indicated by the magenta arrows.

### Extended Data Fig. 10 Calculation of the electrophoretic mobilities and diffusion coefficients of ion–water clusters.

a. Variation of the z position of the K+ ion in cluster i (see Fig. 3e) as a function of the simulation time obtained in the presence of a 0.25 V·nm-1 electric field applied along the z direction. The simulation data (solid curve) is obtained from 5 independent MD simulations with different random velocities of the molecules in the system. The dashed curve represents the fit to the data obtained using the terminal velocity of the K+ ion, $$v_z = \frac{{dz}}{{dt}}$$, which is estimated to be 24.7 ± 3.9 m·s-1. b. Variation of the z position of the K+ ion in cluster iv (see Fig. 3e) as a function of the simulation time obtained in the presence of a 0.25 V·nm-1 electric field applied along the z direction. Similar to a, the simulation data is obtained from 5 independent MD simulations. The calculated terminal velocity in this case is: vz = 54.7 ± 5.6 m·s-1. The terminal velocities obtained at three different electric fields have been used to obtain the data shown in Fig. 3g, h. c. Plot showing the variation of the MSD of cluster i in the absence of any electric field, that is, where Ez = 0. d. Plot showing the variation of the MSD of cluster iv in the absence of any electric field. Additional data on the variation of the terminal velocity and MSD of cluster v is presented in Supplementary Discussion S6.

## Supplementary information

### Supplementary Information

Supplementary Discussion 1–7, which covers: (1) the classical MD simulations carried out using all-atomistic polarizable force fields, (2) estimation of the number of COO groups at the CNT entrance, (3) alignment of the CNT with respect to the lipid bilayer, (4) analysis of the ion-induced electronic polarization of the carbon atoms in the CNT, (5) calculation of the diffusion coefficient of K+ ions in the presence of the single-file water chain, (6) investigation of the electrophoretic transport of K+ ions; (7) experimental data for carbon nanotube porin characterization.

### Supplementary Video 1

A video of a 2 ns duration all-atomistic MD simulation using polarizable force fields showing the formation of ion–water clusters inside a CNTP with 6 COO groups in the presence of a 0.4 V nm−1 electric field applied along the z direction. The colour code used to represent the various atoms in the system is the same as that used in Fig. 3. The lipid bilayer surrounding the CNT is not shown for clarity.

### Supplementary Video 2

A video of a 2 ns duration all-atomistic MD simulation using non-polarizable force fields showing the absence of any K+ ion translocation through the CNTP with 6 COO groups in the presence of a 0.4 V nm−1 electric field applied along the z direction. All other settings are kept identical to those used to generate Supplementary Video 1. K+ ion translocation was also not observed for the CNTP with 3 COO groups.

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Li, Z., Misra, R.P., Li, Y. et al. Breakdown of the Nernst–Einstein relation in carbon nanotube porins. Nat. Nanotechnol. 18, 177–183 (2023). https://doi.org/10.1038/s41565-022-01276-0

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