Golden aspect ratio for ion transport simulation in nanopores

Access resistance indicates how well current carriers from a bulk medium can converge to a pore or opening and is an important concept in nanofluidic devices and in cell physiology. In simplified scenarios, when the bulk dimensions are infinite in all directions, it depends only on the resistivity and pore radius. These conditions are not valid in all-atom molecular dynamics simulations of transport, due to the computational cost of large simulation cells, and can even break down in micro- and nanoscale systems due to strong confinement. Here, we examine a scaling theory for the access resistance that predicts a special simulation cell aspect ratio—the golden aspect ratio—where finite-size effects are eliminated. Using both continuum and all-atom simulations, we demonstrate that this golden aspect ratio exists and that it takes on a universal value in linear response and moderate concentrations. Outside of linear response, it gains an apparent dependence on characteristics of the transport scenario (concentration, voltages, etc.) for small simulation cells, but this dependence vanishes at larger length scales. These results will enable the use of all-atom molecular dynamics simulations to study contextual properties of access resistance—its dependence on protein and molecular-scale fluctuations, the presence of charges, and other functional groups—and yield the opportunity to quantitatively compare computed and measured resistances.

Figure 1

Contour plots of potential from continuum (PNP) simulations of ion transport through a nanopore. (a) Potential drop $V_0$ across the simulation cell with a pore of radius $a$ in a membrane of thickness $h_p$. The equipotential surfaces (thick, shaded red and blue lines shown on top of a heat map) are elliptical near the pore—i.e., an access-like region. The finite size of the simulation cell curtails the access region, forcing the equipotential surfaces to transition into nearly flat profiles closer to the electrodes—i.e., a bulk-like region. The top half shows the rotational elliptical coordinates, $\xi$ and $\eta$, we employ in modeling the access resistance. (b) Equipotential surfaces in simulation cells of increasing $L$ and $H$, with $H>L$, show the finite-size scaling of access-resistance. (c) Potential map in simulation cells with $L>H$ ($L=32$ nm and $H=16$ nm). The equipotential surfaces are quasielliptical almost up to the end of the cell, but with the surfaces somewhat vertically distorted when they approach the electrode surfaces on the top and bottom. (d) As the cell size increases at a constant aspect ratio $\alpha =H/L$, more of the equipotential surfaces become ellipsoidal, i.e., accesslike, and the total resistance converges to its value in an infinite bulk.

Figure 2

Variation of the resistance with changes in one of the cell dimensions. The pore radius is $a=1$ nm, the membrane thickness $h_p=1$ nm, and the resistivity $\gamma =71\mathrm{M}\mathrm{\Omega }\mathrm{nm}$ (taken to match our all-atom MD value of 1 M KCl from prior results [34]). (a) The resistance $R$ versus cross-sectional length $L$ with electrolyte height fixed at $H=14.5$ nm. For $L\gtrsim H/f, R$ is smaller than the actual resistance for an infinite-size electrolyte, $R_{\infty }$ (horizontal dotted line). The scaling form still correctly predicts $R_{\infty }$ when fit for small $L$ (solid lines) but to do so it has to display non-monotonic behavior (extrapolated, dashed lines). The computed data at large $L$, however, converges to a value below $R_{\infty }$ as given by Eq. (11). We note that obtaining $f$ from where $R$ crosses $R_{\infty }$ gives about 1.0 for cylindrical and 1.2 for rectangular cells. This is in close agreement with the golden aspect ratio we obtain later, with the difference from the exact values likely due to the strong non-monotonic effects here. (b) The resistance $R$ versus total electrolyte height $H$ when cross-section is fixed at $L=50$ nm. For $H/f\gtrsim L, R$ increases linearly with increasing $H$ as shown by the fitted solid lines. For $H<L, R$ decreases with decreasing $H$ as the negative correction term $-f^{\prime }\gamma /H$ becomes larger. The factor $f^{\prime }\approx 0.4$ obtained from fitting is similar to simple estimates (about 0.3). The standard error of the fits for $f$ and $R_{\infty }$ are less than 0.2 %. Their fitted values are shown in bold font in the legends.

Figure 3

Resistance versus $L$ for different aspect ratios $\alpha$, with $a=1$ nm, $h_p=1$ nm, and $\gamma =71\mathrm{M}\mathrm{\Omega }\mathrm{nm}$. For small $\alpha , dR/dL>0$ and for large $\alpha , dR/dL<0$. At special value of $\alpha$ in between—the “golden aspect ratio,” $dR/dL=0$ and the resistance is constant (i.e., no finite size effects). Here, the gold line is very close to, but not quite at, the golden aspect ratio. The solid lines show the fits $R=\frac{\gamma }{\mathcal{G}L}\left(\alpha -f\right)+R_{\infty }$ for $\alpha >1$ and $R=-\frac{\gamma f^{\prime }}{\alpha L}+R_{\infty }$ for $\alpha <1$, where $\mathcal{G}=\pi /4$ for cylindrical cells (left panel) and $\mathcal{G}=1$ for rectangular cells (right panel). The fit parameters are shown in bold. The dashed lines extrapolate these fits, which match well with the calculations for large $L$ and yield consistent values for $R_{\infty }$. The performance of the scaling analysis using small $L$ indicates that the simulation cell sizes achievable with all-atom MD should be sufficient to find the total resistance (pore plus access). The error of the fits for the $f$'s and the $R_{\infty }$'s are about 0.5 % and 0.1 %, respectively (except for $\alpha =0.25$ where the respective errors are about about 3 % and 0.5 %).

Figure 4

Universality of the golden aspect ratio with pore size. (a) The resistance versus $L$ for cylindrical cells with $\alpha =1.07, h_p=1$ nm, $\gamma =71\mathrm{M}\mathrm{\Omega }\mathrm{nm}$, and various pore radii. The golden aspect ratio is fairly constant over this range of $a$. The solid lines show the fits $R=\frac{4\gamma }{\pi L}\left(\alpha -f\right)+R_{\infty }$ and the dashed lines show the extrapolations—which make good predictions of $R$ at large $L$. The fit parameters are shown in bold. The errors of the fits for $f$'s and $R_{\infty }$'s are less than 0.5 % and 0.1 %, respectively. As the pore radius increases, we examine fits for larger $L$, to roughly keep proportionality between $L$ and $a$ as there are (nonscaling) finite-size effects that will affect the simulations when the radius starts to become comparable to the cell cross-sectional length. The range of the fitting can influence the extracted $f$ (we expect that ultraprecise calculations at really large $L$ will reveal the exact golden aspect ratio, which will be close to 1.07). (b) $\mathrm{\Delta }R$ from fits to $R=\gamma \left(1/2a+h_{\mathrm{eff}}/\pi a^2\right)+\mathrm{\Delta }R,$ for cylindrical cells of various $\alpha$, with $h_{\mathrm{eff}}$ and $\mathrm{\Delta }R$ as fitting parameters. We get $\mathrm{\Delta }R\approx 0$ for the “golden aspect ratio” ${\alpha }^{★}=1.07$ (dashed lines), the fit of which is shown in the inset. The effective height, $h_{\mathrm{eff}}$, is ($5\pm 1$)% larger than the 1-nm membrane thickness; see Fig. 5. (c) $\mathrm{\Delta }R$ in the fits $R=\gamma /4a+\mathrm{\Delta }R$ for half cylinders of different aspect ratios with the pore mouth set at potential $V=0$. We again get $\mathrm{\Delta }R\approx 0$ for ${\alpha }^{★}=1.07$ (dashed lines), whose fit is shown in the inset. The interpolated lines in the insets are for visual clarity only.

Figure 5

The resistance versus $a$ for cylindrical cells with $L=32$ nm, the golden aspect ratio ${\alpha }^{★}=1.07$, and $h_p=1.0\mathrm{nm}$, 2.0 nm, 3.0 nm, and 4.0 nm. The resistance fits into the model, $R=\gamma \left(1/2a+h_{\mathrm{eff}}/\pi a^2\right)+\mathrm{\Delta }R$ with $h_{\mathrm{eff}}$ and $\mathrm{\Delta }R$ as fitting parameters (shown in bold in the legend). The fitted $h_{\mathrm{eff}}$ is slightly larger than the actual membrane thickness but its relative difference diminishes as the membrane thickness is increased. This is due to the curvature of the potential at the pore mouth and not due to numerical grid-size errors (a ten times smaller grid increases this value to 7%, and thus we expect this height correction to converge to something of this order.). The standard errors of $h_{\mathrm{eff}}$ and $\mathrm{\Delta }R$ are about 0.01 nm and 0.04 $\mathrm{M}\mathrm{\Omega }$ in each case.

Figure 6

PNP solution of the resistance versus $L$ for cylindrical cells with $h_p=1$ nm, $a=1$ nm, and various aspect ratios ($\alpha =1.4$, 1.1, and 0.8 from top to bottom). The labels at the top show the applied potential and the labels at the right show the concentration of KCl. The golden aspect ratio remains the same as in the homogeneous medium calculation for the smaller voltages (10 mV and 100 mV) and for both 1 mol/L and 0.1 mol/L concentrations. For 300 mV applied voltage, the golden aspect ratio seems to take on a lower value due to the increased concentration of ions near the membrane, which is more pronounced for low concentrations due to a diminished ability to screen and generate local fields. The legends show the fits to Eq. (12) with the fit parameters shown in bold. The error of the fits for $f$'s and $R_{\infty }$'s are about 0.5% and 0.1%, respectively.

Figure 7

The resistivity along the $z$ axis for various applied potentials and cell dimensions $L$. The pore radius is $a=1$ nm and the KCl concentration is $0.1\mathrm{mol}/\mathrm{L}$ (${\gamma }_b=710\mathrm{M}\mathrm{\Omega }\mathrm{nm}$). Large applied voltages enhance the density of charge carriers near the pore, decreasing the resistivity. Since the access resistance contribution is largest near the pore, see Eq. (5), this enhancement lowers the overall access resistance and gives an apparent depression to the golden aspect ratio when examining small simulation cells.

Figure 8

PNP solution of the resistance versus $L$ at applied potential of 1 V and KCl concentrations 0.1 mol/L (top panel) and 1.0 mol/L (bottom panel) for cylindrical cells with $h_p=1$ nm, $a=1$ nm, and aspect ratios ($\alpha =1.4$, 1.1 and 0.8). The solid lines show the fit with $f$ and $R_{\infty }$ as fitting parameters (shown in bold) and the dashed lines show the extrapolation. Since the applied field is large, the resistance at smaller $L$ ($<100$ nm) is highly nonlinear and cannot be scaled to extract $R_{\infty }$ with Eq. (12) as is done for the smaller applied voltages in Fig. 6. Hence, a larger $L$ is necessary to employ the scaling form when the electric field is large and ion concentrations are small. Note that the resistance for the 0.1 mol/L solution is substantially smaller than at lower voltages. For 0.1 mol/L solution, the error of the fit for $f$ and $R_{\infty }$ are about 3% and 0.03%, respectively, whereas for 1.0 mol/L solution the respective errors are less than 1% and 0.01%.

Figure 9

MD results of the resistance versus $L$ at an applied bias of 1 V and KCl concentration 1.0 mol/L, $a=1.81$ nm, and aspect ratios 1.9, 1.2, and 0.5 (from top to bottom). The raw resistance from MD is shown as open circle. Due to the equilibration, the final cell aspect ratios are not exactly 1.9, 1.2, and 0.5. The slight change in height can be corrected for to set $H=\alpha L+h_p$. The data with error bars show this slightly corrected data. The error bars represent $\pm 1$ block standard error, see Ref. [68], which reflects error due to finite sampling time. The legends show the fits to Eq. (12) with the fit parameters shown in bold. The golden aspect ratio for MD is $\approx 1.2$, which is the same as that from continuum simulations (for a rectangular cell). The scaling and/or golden aspect ratio simulations yield a resistance in agreement with the classical expression, $2R_{\mathrm{Hall}}+R_{\mathrm{pore}}$, as we demonstrated with the scaling procedure only in Sahu et al. [34].