Current Flow in a Cylindrical Nanopore with an Object–Implications for Virus Sensing

Abstract

Interest is growing in nanopores as real-time, low-cost, label-free virus size sensors. To optimize their performance, we evaluate how external electric field and ion concentrations and pore wall charges influence currents and object (disk) radius-current relationship using simulations. The physics was described using the Poisson-Nernst-Planck and Navier–Stokes equations. In a charged cylindrical nanopore with a charged disk, elevated external electric field produces higher (and polarity independent) ion concentrations and greater ion current (largely migratory). Elevated external ion concentrations also lead to higher concentrations (mainly away from the pore wall), greater axial electric field especially in the disk-pore wall space, and finally larger current. At low concentrations, current is disk radius independent. The current rises as concentrations increase. Interestingly, the rise is greater for larger disks (except when the pore is blocked mechanically). Smaller cross-sectional area for current flow or volume exclusion of electrolyte by object thus cannot be universally accepted as explanations of current blockage. Ion current rises when pore wall charge density increases, but its direction is independent of charge sign. Current-disk radius relationship is also independent of pore wall charge sign. If the pore wall and disk charges have the same sign, larger current with bigger disk is due to higher counter-ion accumulation in the object-pore wall space. However, if their signs are opposite, it is largely due to elevated axial electric field in the object-pore wall space. Finally in uncharged nanopores, current diminishes when disk radius increases making them better sensors of virus size.

Introduction

Micropores have been used for a long time for rapid estimation of cell numbers, size, and “dwell time” (Coulter counter; [1, 2]). Interest in using pores as low-cost, rapid, and label-free estimators of properties of pathogens (such as viruses) is large and growing. However, viruses are smaller than cells and vary in size greatly (from several to several hundred nanometers [3]). Estimating their size–an important descriptor in their classification [4]–thus often requires nanopores. We classify the pore size on a logarithmic scale (with base 10; [5]). If the pore size ranges from 0.1 to 100 nm, it is considered a nanopore, and if it ranges from 0.1 to 100 µm, it is considered a micropore [5]. Nanopores are already used as sensors of single molecules–DNA [6–9], proteins [10–12], peptides [13, 14], and amino acids [15, 16], and solid-state nanopores as diagnostic tools for infectious diseases [17].

In all cases, the “object” size is estimated from a transient and discrete ionic current blockage during its translocation through the pore–the larger the object (and the smaller the cross-sectional area for current flow) the larger the current blockage. This “resistive-pulse method” [18] based on the size-exclusion principle [2] can also yield the shape of objects if pores with irregularities (a pore is considered irregular if its radius changes axially) are used [19, 20]. When an object passes through a pore, it often rotates. If the object is non-spherical, the cross-sectional area for the current flow changes and this leads to current fluctuations that contain information about how non-spherical the object is [21]. When a spherical object rotates, the cross-sectional area for the current flow does not change. Although this method is now used for size and shape characterization of larger viruses [4, 20, 22, 23], it is unclear whether it would be suitable for characterization of smaller viruses using nanopores. However, shape is a powerful descriptor of viruses that vary in their shape greatly and can be globular (coronaviruses), oval shaped (poxviruses), rod-like (as many plant viruses), or head and tail like (bacteriophages) [24].

What is critically important is to understand the relationship between the cross-sectional area for current flow and current. Presently, the “size-exclusion principle” is the most widely held view. Expanding on this notion, it has been argued that the volume exclusion of electrolyte solution by the object also contributes to the current decrease during its translocation [25, 26]. In this study, we challenge these notions as a general explanation of current blockage because ions are as a rule not uniformly distributed within charged pores. Screening charges formed to counterbalance surface charges are concentrated near the pore wall [27–29]. Given such charges’ distribution, the current-object size dependence cannot be one of progressive linear decrease, but it is unclear what the relationship is, and what influences it.

The object if charged also induces screening charges within nanopore. It has been suggested that this leads to larger current, partially counterbalancing the current decrease caused by smaller cross-sectional area for current flow and volume exclusion of electrolyte solution by the object, finally producing multi-level current changes [26]. This is an interesting, but a simplistic proposition because charged disk surfaces that may induce such currents will also largely block them mechanically. However, a charged object will affect the electric field (both its radial and axial components) within the nanopore (especially if the disk and the pore wall are charged differently). The radial electric field controls the radial migratory current around the disk and may change the overall axial migratory current in the disk-pore wall space, but it is unclear how and how much. Finally, even an uncharged object changes the electrostatics (and thus electro-kinetics) within the nanopore because it is a low permittivity material (ε_r,d = 2; [30]) within a highly polarizable core of the nanopore (ε_r,w = 80; [30]).

The presence of two charged surfaces in close proximity can change ion currents and conductivity greatly. Even a “simple” charge non-uniformity at the pore walls can lead to pronounced changes of ion fluxes [31], pore conductance [32], and current rectification [27, 32–35]. In engineering, such nanopores are used as ionic diodes and transistors [36–39] or serve as ion filters [29]. A systematic evaluation of how external factors and pore parameters influence the current in a charged cylindrical nanopore with a charged disk is clearly needed.

We evaluate how external electric field and ion concentrations and pore wall charges influence ion currents and disk radius-current relationship. The transport of K⁺ and Cl⁻ was simulated using the Poisson–Nernst–Planck (PNP) equations [40] and was coupled to the transport of water using the Navier–Stokes (NS) equation [41, 42]. The electrostatic-electro-kinetic-hydrodynamic variables (potential, mobile charge density and ion currents) were subsequently evaluated.

Methods

Geometry of Simulation Domain, Parameters, Constants, and Boundary Conditions

Figure 1 gives the simulation space which consists of a cylindrical nanopore, two compartments flanking the nanopore, and a piece of membrane separating the compartments. All subdomains and boundaries are enumerated, and their dimensions are given (Fig. 1A). Figure 1B depicts meshing of the simulation space and Fig. 1C shows its 3D representation. Table 1 lists the electrostatics, electro-kinetics, and fluidics boundary conditions implemented in this study and Table 2 details the parameters and physical constants used. Note that the Poisson equation (i.e., electrostatics) was implemented in all subdomains, whereas the Nernst-Planck equation (i.e., electro-kinetics) and the Navier–Stokes equation (i.e., hydrodynamics) were only defined in subdomain 1 (i.e., electrolyte solution). The finite element method system was used to solve coupled PNP-NS equations (see below). In all simulations, diffusion constants of ions (K⁺ and Cl⁻) are known [56], but it is not so clear what the values may be in the confined space of the nanopore, where the electrostatic and non-electrostatic interactions with its walls [54] restrict their movement. To simplify the computation, we assumed that both diffusion constants (but also the dielectric constant of water and its viscosity) are homogeneous, isotropic, and the same as in the aqueous solution.

Fig. 1.

Model geometry. a Hemi-section of the simulation space consisting of 3 subdomains–a cylindrical nanopore, one compartment on each side and a membrane patch separating them. In total, the model has 14 boundaries. Disk was positioned 1.5 nm above the nanopore center; its thickness was 0.5 nm, whereas its radius (r_o; subdomain S3) ranged from 0.5 to 3.25 nm. Radius of cylindrical nanopore was 3.5 nm. Other dimensions are given on the figure. The 3D-model is generated by the rotation of the hemi-section about central axis by 360°. All boundary conditions are given in Table 1. b Simulation space meshing. c 3D representation of the simulation space

Table 1.

Boundary conditions

Boundary	Hydrodynamics	Electrostatics (Poisson’s eqn.)	Electrostatics (Laplace’s eqn.)	Electro-kinetics
B1, B10	Axial symmetry	Axial symmetry	Axial symmetry	Axial symmetry
B2 and B3	Pressure (pu = 0 Pa); no viscous stress	Electric potential Vu (0 V)	Electric potential Vu (as specified)	K+-Cl− (400 mM or as specified)
B4	Not applicable (not in contact with water; NS equation does not apply)	Zero charge symmetry (i.e., n·D = 0, D is an electrical displacement vector)a	Zero charge symmetry (i.e., n·D = 0, D is an electrical displacement vector)a	Not applicable (not in contact with liquid electrolyte; PNP equations do not apply)
B5 and B6	Pressure (pd = 0 Pa); No viscous stress	Electric potential Vd (0 V)	Electric potential Vd (0 V)	K+-Cl− (400 mM or as specified)
B7 and B9	No slip	Surface charge density σ = 0 C/m2	Not applicableb	No flux
B8	No slip	Surface charge density σw	Not applicableb	No flux
B11	Not applicable (not in contact with water; NS equation does not apply)	Axial symmetry	Axial symmetry	Not applicable (not in contact with liquid electrolyte; PNP equations do not apply)
B12-B14	No slip	Surface charge density σo	Not applicableb	No flux

^aZero charge symmetry (i.e., n·D = 0) signifies that the normal component of the electric displacement on a given boundary is zero and that there is only a tangential component. Note that D = ε0E + P, where ε0 is permittivity of vacuum, E is electric field and P is polarization

^bLaplace’s equation (∇²ϕ = 0) calculates the electric potential (ϕ) due to external electric field. The electric field due to charges (mobile or fixed) is not considered, and relative permittivities play no role. To solve this equation for ϕ, the external electric fields at the upper and lower boundaries (i.e., B2, B3, B5, and B6) are defined, whereas the electric potentials at the internal boundaries (i.e., B7, B8, B9, and B12-B14) are not required. No boundaries effectively exist there when solving Laplace’s equation

Table 2.

Model parameters and physical constants

Params	Values	Unit	Description	Refs
T	300.0	K	Temperature
R	8.314	J/(mol·K)	Universal gas constant	[30]
e	1.602 × 10−19	C	Elementary charge	[30]
DK	1.960 × 10−9	m2/s	Diffusion coefficient of K+ ions	[56]
DCl	2.030 × 10−9	m2/s	Diffusion coefficient of Cl− ions	[56]
ρ	1.0 × 103	kg/m3	Fluid density	[30]
μ	1.0 × 10−3	Pa s	Fluid viscosity	[30]
ε0	8.854 × 10−12	F/m	Permittivity of vacuum	[30]
εr,w	80.0	Dimensionless	Relative permittivity of water	[30]
εr,m	2.0	Dimensionless	Relative permittivity of membrane	[30]
εr,d	2.0	Dimensionless	Relative permittivity of object (disk)	[30]

General Formulation of Mathematical Simulations

We used the Nernst-Planck equation to calculate ionic fluxes within a nanopore for both K⁺ and Cl⁻ ions. The Poisson equation was used to compute the electric potential (ψ) due to mobile charges (inside the nanopore and in two compartments flanking it) and fixed charges (on the pore and disk walls; Eq. (1)), while the Laplace’s equation (Eq. (2) calculates the electric potential (ϕ) due to applied external electric field (i.e., the potential difference between the upper and lower controlling edges) as follows:

$$-\mathrm{\nabla }\bullet {\epsilon }_0{\epsilon }_r\mathrm{\nabla }\psi ={\rho }_e$$ 1

$${\mathrm{\nabla }}^2\varphi =0$$ 2

where ε₀ is the permittivity of vacuum, ε_r is the relative dielectric constant of a chosen subdomain (as shown in Fig. 1 there are three subdomains in the model; their ε_rs are given in Table 2), and $\mathrm{\nabla }$ is the gradient operator. The charge density ρ_e (mobile) was calculated as follows:

$${\rho }_{\mathrm{e}}=F\sum z_a{c}_a\left(=,e,\sum ,z_a,n_a\right)$$ 3

Note that c_a is the molar concentration of each ion in [mol/m³], z_a is the valence of ion a, n_a is the number density of ion a, and F is the Faraday constant (9.648 × 10⁴ C/mol). The contribution of fixed charges (σ_w–pore wall charge density and σ_o–disk charge density) is included through electrostatic boundary conditions (Table 1). The International System of Units (SI) is used in all simulations, and mol/m³ thus equals to mmol/liter (or simply mM).

The movement (convection–diffusion-migration) of ions in the electrolytic fluid was defined by the Nernst-Planck equation as follows:

$${\mathbf{J}}_{\mathbf{a}}=\mathbf{u}{c}_a-D_a\mathrm{\nabla }{c}_a-m_a{z}_a{\mathit{Fc}}_a\mathrm{\nabla }\mathrm{\Phi }$$ 4

where J_a is the molar flux in mol/(m²·s), whereas u denotes the fluid velocity calculated by the NS equation. D_a and m_a represent diffusivity and mobility of ion a, which are related by m_a = D_a/(RT). R and T account for the universal gas constant (R = 8.314 J/(mol·K)) and absolute temperature in Kelvin, respectively. Finally, Φ denotes the total electric potential, which is the sum of ϕ and ψ.

Equation (5) gives the conservation of ionic mass in a steady-state situation.

$$\mathrm{\nabla }\bullet {\mathit{J}}_{\mathit{a}}=0$$ 5

The Navier–Stokes equation (at steady-state condition), in the presence of external forces, was applied to model fluid velocity as follows:

$$\rho \left(\mathit{u},\bullet ,\mathrm{\nabla },\mathit{u}\right)=-\mathrm{\nabla }p+\mathrm{\nabla }\bullet \left[\mu ,\left[\mathrm{\nabla },\mathit{u},+,{\left(\mathrm{\nabla },\mathit{u}\right)}^T\right]\right]+{\mathit{F}}_{\mathit{e}}$$ 6

$$\mathrm{\nabla }\bullet \mathit{u}=0$$ 7

Equation (6) describes the conservation of momentum, whereas Eq. (7) accounts for the conservation of mass. The ρ, μ, and p stand for fluid density, viscosity, and pressure (defined as the isotropic part of the fluidic stress tensor), respectively. The vector u denotes the fluid velocity. Finally, F_e accounts for the electric force per unit volume, calculated as ${\mathbf{F}}_{\mathbf{e}}=-{\rho }_e\mathrm{\nabla }\mathrm{\Phi }$.

At nanoscales, the Reynolds number is expected to be very low and fluid flow laminar with negligible inertial effects [43]. Indeed, we find that the Reynolds number is < 0.01, even for the highest water velocity within the nanopore (i.e., very low ensuring that the flow is laminar). Moreover, the nanopore size and the Debye length [44] are comparable. We thus used no-slip condition in all simulations, but a slip condition may be a better choice when the surfaces are strongly hydrophobic. Finally, our choice of a no-slip boundary condition enables us to resolve the flow inside the electric double layer.

Mathematical Model in the Cylindrical Coordinate System

This study uses the cylindrical coordinate system (the 2D r-z plane). The simplified Navier–Stokes equation (note that the velocity and its gradient in the θ−direction are ignored, and there is an axial symmetry in the z−direction) in terms of components of the stress tensor τ are thus as follows:

$$r-\mathrm{component}:\rho \left(u,\frac{\partial u}{\partial r},+,v,\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial r}-\left[\frac{1}{r},\frac{\partial }{\partial r},\left({r\tau }_{\mathit{rr}}\right),+,\frac{\partial {\tau }_{\mathit{rz}}}{\partial z}\right]+F_{e-r}$$ 8

$$z-\mathrm{component}:\rho \left(u,\frac{\partial v}{\partial r},+,v,\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial z}-\left[\frac{1}{r},\frac{\partial }{\partial r},\left({r\tau }_{\mathit{rz}}\right),+,\frac{\partial {\tau }_{\mathit{zz}}}{\partial z}\right]+F_{e-z}$$ 9

The u and v are the r- and z-components of the fluid velocity, respectively. The τ_ij are the ij-th components of the viscous stress tensor τ, which for the Newtonian fluids in 2D r-z plane (in the cylindrical coordinate system) are defined as:

$${\tau }_{\mathit{rr}}=-\mu \left[2,\frac{\partial u}{\partial r},-,\frac{2}{3},\left(\mathrm{\nabla },\bullet ,\mathit{u}\right)\right]$$ 10

$${\tau }_{\mathit{zz}}=-\mu \left[2,\frac{\partial v}{\partial z},-,\frac{2}{3},\left(\mathrm{\nabla },\bullet ,\mathit{u}\right)\right]$$ 11

$${\tau }_{\mathit{rz}}=-\mu \left[\frac{\partial v}{\partial r},+,\frac{\partial u}{\partial z}\right]$$ 12

$$\mathrm{where}\mathrm{\nabla }\bullet \mathit{u}=\frac{1}{r}\frac{\partial }{\partial r}\left(r,u\right)+\frac{\partial v}{\partial z}$$ 13

The electrostatic force is defined as ${\mathbf{F}}_{\mathbf{e}}=-{\rho }_e\mathrm{\nabla }\mathrm{\Phi }$, whereas the gradient operator in the cylindrical coordinate system is given by:

$$\begin{array}{cc}{\left[\mathrm{\nabla },\mathrm{\Phi }\right]}_r=\frac{\partial \mathrm{\Phi }}{\partial r},& {\left[\mathrm{\nabla },\mathrm{\Phi }\right]}_z=\frac{\partial \mathrm{\Phi }}{\partial z}\end{array}$$ 14

When the components of the stress tensor (given in Eqs. (10)–(14)) are substituted into Eqs. (8) and (9) with constants ρ and μ, we obtain Eqs. (15) and (16) describing the Navier–Stokes equation in terms of velocity gradients.

$$r-\mathrm{component}:\rho \left(u,\frac{\partial u}{\partial r},+,v,\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial r}+\mu \left[\frac{\partial }{\partial r},\left(\frac{1}{r},\frac{\partial }{\partial r},\left(r,u\right)\right),+,\frac{{\partial }^{2}u}{\partial z^2}\right]-{\rho }_e\frac{\partial \mathrm{\Phi }}{\partial r}$$ 15

$$z-\mathrm{component}:\rho \left(u,\frac{\partial v}{\partial r},+,v,\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu \left[\frac{1}{r},\frac{\partial }{\partial r},\left(r,\frac{\partial v}{\partial r}\right),+,\frac{{\partial }^{2}v}{\partial z^2}\right]-{\rho }_e\frac{\partial \mathrm{\Phi }}{\partial z}$$ 16

Finally, note that the divergence operator in Eqs. (1) and (5) is defined as:

$$\mathrm{\nabla }\bullet \mathit{J}=\frac{1}{r}\frac{\partial }{\partial r}\left(r,J_r\right)+\frac{\partial J_z}{\partial z}$$ 17

The J_r and J_z are the r- and z- components of vector J.

The system of coupled PNP and NS equations was solved using a finite element method based on a commercial software package COMSOL Multiphysics 4.3 (COMSOL, Burlington, MA, USA), whereas the postprocessing was performed using a MATLAB software package for scientific and engineering computation (MathWorks, Natick, MA, USA).

Results

Effect of External Electric Field on Ion Currents in a Charged Nanopore with a Charged Disk

The color-coded 2D distributions shown on top depict the effect of external axial electric field (E_ex) on the axial electric field (E_ax), K⁺ and Cl⁻ concentrations (co_K and co_Cl) within the nanopore (Fig. 2). To test whether the mesh sizes are adequately small (note that the mesh size differs depending on the location and is smaller near the disk and close to the edges; Fig. 1), we made some simulations where size was 100% bigger. In each set of 2D distributions, the results with double size mesh are shown as third panels (their V_u was +1 V). This change did not affect the 2D distributions of the E_ax, co_K, and co_Cl (Fig. 2). Mesh size is thus adequately fine.

Fig. 2.

Effect of external electric field on electric field and ion concentrations within a nanopore. External axial electric field (E_ex) alters axial electric field (E_ax) within a cylindrical nanopore with charged walls and an object (a disk), but also K⁺ and Cl⁻ concentrations (co_K and co_Cl). Disk radius (r_o) was 3 nm, charge density at the pore wall (σ_w) was −160 mC/m² (this is equivalent to 1 e/nm²), and at the disk (σ_o), it was +160 mC/m². Ion concentrations (K⁺ and Cl⁻) at the controlling edges of the upper (co_u) and lower (co_d) compartment were 400 mM. The color coded 2D distributions of E_ax, co_K and co_Cl are given on top for the voltage at upper controlling edges (V_u) with V_u = −1 V (left panels), +1 V (middle panels) and +1 V but with mesh doubled in size (right panels). In this case and throughout the text the potential (V_d) at the lower controlling edges was 0 V, whereas the pressure at both upper and lower controlling edges was 0 Pa (see Table 1). The calibration bars are as indicated. A, B Radial profiles of K⁺ concentrations at z = 4 nm (i.e., 2.25 nm above the disk upper surface and from pore center to the wall; co_K,u), or at z = 1.5 nm (i.e., from the axial center of the disk to the pore wall; co_K,o) and corresponding Cl⁻ concentrations (co_Cl,u and co_Cl,o). C Radial profiles of E_ax,u and E_ax,o. D, E V_u dependence of co_K,u (and co_K,o) and co_Cl,u (and co_Cl,o) near the pore wall and in the pore center. F V_u dependence of E_ax,u and E_ax,o

Radial profiles of the E_ax, co_K, and co_Cl above the object-disk (at z = 4 nm which is indicated by the upper horizontal line; E_ax,u, co_K,u and co_Cl,u) and in the disk-pore wall space (at the disk’s axial mid-point; i.e., at z = 1.5 nm; E_ax,o, co_K,o and co_Cl,o), which are critical for understanding how current is generated and controlled, are also shown. Lower horizontal line depicts the axial midpoint (i.e., z = 0 nm). The co_K,u is high in a very narrow space near the (negatively) charged pore wall. In the pore center it is low, but significantly above the value at the controlling edges (upper or lower). The co_K,o is similar in value near the pore wall, but is near zero close to the (positively charged) disk. Given that in between the pore and disk walls it differs significantly from the co_K,u, there is a significant co_K axial gradient in small space near the disk tip. The co_Cl,u is low near the pore wall and rises towards its center, whereas the co_Cl,o (also low near the wall) rises to high values near the (positively) charged disk.

Neither the co_K,u (and co_K,o) nor co_Cl,u (and co_Cl,o) radial profiles change significantly, if the E_ex is reversed (Fig. 2A, B). However, as can be seen from 2D distributions the panels of c_K (and c_Cl) are not identical if V_u is reversed. The co_K,u (and co_K,o) and co_Cl,u (and co_Cl,o) radial profiles should reveal some differences if different axial positions are chosen. The E_ax,u and E_ax,o radial profiles (which are quite uniform) change with the E_ex reversal. The E_ax,o profiles change too but more so (Fig. 2C). Complete V_u dependence reveals that the co_K and co_Cl (near the pore wall and in the pore center) are both qualitatively similar–high at very positive and negative V_us, and low at zero V_u (Fig. 2D, E). The E_ax,us however depend on the V_u linearly, but the E_ax,os are higher and more V_u-dependent than the E_ax,us (Fig. 2F).

All axial current densities–migratory (σ_I,mig), diffusive (σ_I,_diff) and convective (σ_I,conv)–are affected by the E_ex (Fig. 3). Figure 3A-D show their radial profiles and those of the total current density (σ_I,_tot) for two extreme V_us. The σ_I,conv,u is confined largely to the spaces near the pore wall (being zero near pore center)–a consequence of the presence of E_ax and elevated ion (counter-ion) concentration there that drive the water flow. When water flow changes direction with the E_ax reversal the σ_I,conv,u changes direction too. The σ_I,_diff,u is also prominent near the pore wall and this is due to large co_K and co_Cl axial gradients there (see the difference of co_K,u and co_K,o (and co_Cl,u and co_Cl,o) radial profiles (Fig. 2A, B)). The σ_I,_diff,u profiles do not change significantly if the E_ex is reversed, and this is not surprising because the co_K and co_Cl radial profiles do not change either. The σ_I,mig,u (the largest of all) is however non-zero at any radial distance, and its radial profiles are reversed with E_ax reversal. Two factors explain σ_I,mig,u profiles–non-zero and constant E_ax at any radial distance, whose values reverse with E_ax reversal, and co_K and co_Cl radial profiles that are essentially insensitive to the E_ex reversal (Fig. 2A, B).

Fig. 3.

Effect of external electric field on ion currents. The color-coded 2D distributions of ion current densities (convective–σ_I,conv, diffusive–σ_I,diff, migratory–σ_I,mig, and total–σ_I,tot) are given on top for the V_u = −1 V (a; upper panels) or +1 V (b; lower panels; the same simulations as in Fig. 2). The K⁺ and Cl⁻ contributions and their sum are shown separately. The σ_I,mig is clearly affected and to a lesser extent the σ_I,conv by V_u change. A–D Radial profiles of σ_I,conv,_u, σ_I,_diff,_u, σ_I,mig,_u, and σ_I,_tot,_u, and σ_I,conv,_o, σ_I,_diff,_o, σ_I,mig,_o, and σ_I,_tot for the V_u as indicated. The σ_I,mig clearly dominates and largely determines the amplitude and the shape of the σ_I,_tot. The V_u-dependence of I_conv,u and I_conv,o (E), I_diff,u and I_diff,o (F), and I_mig,u and I_mig,o (together with their K⁺ and Cl⁻ contributions; G), and I_tot (H). V_u-dependence of the I_conv,u, I_diff,u, I_mig,u, and I_tot,u in the absence of a disk is also given (as indicated)

The V_u-dependence plots of the Is summarize these findings. The migratory currents (I_mig,u and I_mig,o), which are the greatest at all V_us, depend linearly on the V_u (Fig. 3E-H). Given that neither the co_Ks nor co_Cls change greatly with the V_u (and thus the E_ax,u and E_ax,o) reversal, and the fact that the E_ax changes linearly with the E_ex, linear dependence of I_migs is as expected. The K⁺ contributes the most to the I_mig, and this is also as expected–K⁺ is a counter-ion to the negatively charged pore wall. Note that the I_conv,u and I_conv,o also depend linearly on V_u, unlike the I_diff,u and I_diff,o that do not. Finally, note that if there is no disk the I_conv,u becomes more V_u-dependent whereas the I_mig,u and I_tot,u become less so. Small and largely V_u-independent I_diff,u becomes even smaller and less V_u-dependent.

Effect of External Ion Concentrations on Currents

If the external K⁺ and Cl⁻ concentrations (they are identical at both sides of the nanopore throughout this study) change, the co_K and co_Cl will also change. How large the change is and where, is important as it influences the E_ax and ultimately the current–its all three components. In the presence of an object the change may be difficult to predict. DNA translocation reduces the current compared to the baseline current at high concentration, but at low concentration it increases it [45]. As Fig. 4 shows the effect on the co_Ks and co_Cls is qualitatively as expected. The co_K,u near (negatively charged) pore wall is high, but changes only moderately. In the pore center it is low but depends more on the external concentrations. Interestingly, it is above the external value (Fig. 4A, D). The co_K,o is also low (but not as low as the co_K,u) near the pore wall (the disk is positively charged), but rises more as external concentrations increase.

Fig. 4.

Effect of external ion concentrations on concentrations and electric field within nanopore. Color-coded 2D distributions depict the effect of external ion concentrations on the co_K, co_Cl and E_ex within nanopore. The V_u was 1 V, the σ_w was −160 mC/m², and the σ_o was +160 mC/m². Radial profiles of co_K,u and co_K,o (A), co_Cl,u and co_Cl,o (B), and E_ax (C) at external co_K and co_Cl as indicated. Concentration dependence of co_K,u and co_K,o (D) co_Cl,u and co_Cl,o (E) and E_ax,u and E_ax,o (F) near the pore wall, and in the pore center (or near the disk wall)

The co_Cl,u and co_Cl,o are also as expected near zero close to the pore wall, especially when the external concentrations are low. In the pore center, the co_Cl,u is above the external values (the disk is positively charged) and higher at high external concentrations. Finally, the co_Cl,o is quite high near the disk and very dependent on external concentration (Fig. 4B, E). In general thus the co_Cls also rise as the external concentrations increase but those that are the largest (the co_Cl,o near the disk and the co_Cl,u near the pore center) rise the most (Fig. 4E). It is intuitively not very obvious how the E_ax,_u or E_ax,_o radial profiles may be influenced by the external co_K and co_Cl. As these simulations show they both become more negative at high external K⁺ and Cl⁻ concentration especially in the pore wall-disk space (Fig. 4C, F).

What should σ_I profiles be given such co_K, co_Cl, and E_ax radial profiles? The σ_I,conv,_u which is small and negative near the pore center (but zero at the pore or disk walls; see Sect. 2) has a negative peak near the pore wall. The σ_I,conv,_o peak is also negative but is larger. They both become more negative at higher external concentrations (Fig. 5A). Elevated K⁺ concentration near positively charged pore wall and positive V_u lead to negative I_tot, that drives negative water flow (not shown) and thus negative I_conv, which becomes more negative at high external concentrations, and this remains the case regardless of axial position. The σ_I,diff,u is positive and elevated near the pore wall but very insensitive to the external concentration change, whereas the σ_I,diff,o is negative and small but depends on external concentration (Fig. 5B). Positive σ_I,diff,u is not surprising due to an axial K⁺ concentration gradient induced by the positive V_u, but it is more difficult to predict what the σ_I,diff,o would be because of complex c_K and c_Cl distributions in disk-pore wall space. Finally, the σ_I,mig,u is near zero in the pore center and negative near the pore wall at low concentrations. It becomes negative even in the pore center but especially near the pore wall at high concentrations. This is due to the significantly greater co_Cl,u throughout nanopore and negative and larger E_ax (Fig. 4). The σ_I,mig,o is negative in the disk-pore wall space at low or high concentrations. This becomes much more so at elevated external concentrations near both the disk and pore wall. Given that the σ_I,mig is the largest it determines the σ_I,tot profile (Fig. 5C, D). Finally, note that with no disk the I_conv,u becomes more concentration dependent whereas the I_mig,u and I_tot,u become less so. Small I_diff,u becomes smaller and essentially concentration independent.

Fig. 5.

Effect of external ion concentrations on current densities within nanopore. The σ_I,conv and σ_I,mig depend significantly on external ion concentrations, but changing the relative error needed to terminate simulations led to no visible change (see text;. see 2D color-coded distributions on top with external concentrations as indicated). Third panels indicate results with relative error reduced by ten times (the same simulations as in Fig. 4). Radial profiles of σ_I,conv,u, σ_I,conv,o (A), σ_I,diff,u and σ_I,diff,o (B), σ_I,mig,u and σ_I,mig,o (C) and σ_I,tot,u and σ_I,tot,o (D). Concentration dependence of I_conv,u and I_conv,o (E), I_diff,u and I_diff,o (F) and I_mig,u and I_mig,o (G) and I_tot (H) with K⁺ and Cl⁻ contributions for the selected loci as indicated. Concentration dependence of the I_conv,u, I_diff,u, I_mig,u and I_tot,u with no disk within nanopore is also shown (as indicated)

How Is depend on external concentrations is sometimes easy to predict, but not always. The total I_conv,u and I_conv,o are small but increasing as the external concentrations rise. They are carried largely by K⁺ ions concentrated near the negatively charged pore wall (Fig. 5E). The total I_diff,u is small, positive (carried by K⁺ ions and increasing modestly as the external concentration rises). Interestingly, the total I_diff,o which is even smaller is negative and carried by the Cl⁻ ions (Fig. 5F). The I_mig,u and I_mig,o (the largest at any concentration and which determine the I_tot,u and I_tot,o), which are negative and almost identical, increase as external concentrations rise (Fig. 5G, H). Finally, we show the 2D distributions with relative error needed to terminate simulations reduced by ten times (third panel in each case). This did not lead to any visible change demonstrating that the chosen relative error is adequately small.

Effect of Pore Wall Charges on Ion Currents

How much do the pore wall charges affect co_K and co_Cl within the nanopore with a charged disk, and what are the consequences for ion currents passing through? If the σ_w changes from −160 to 0 mC/m² to +160 mC/m², the co_K and co_Cl change greatly throughout the nanopore. The changes near the pore wall are as expected. Away from the pore wall, the co_K changes little above the disk when σ_w changes from −160 to 0 mC/m² but increases significantly when the σ_w rises to +160 mC/m². Below the disk the co_K decreases to very low levels before rising, but very modestly. Interestingly, the co_Cl changes similarly both above and below the disk. The σ_w influences the E_ax too but the effect is confined to or near the disk (Fig. 6).

Fig. 6.

Effect of pore wall charges on concentrations and electric field within nanopore. The co_K, co_Cl, and E_ax are all σ_w dependent (see 2D color-coded distributions on top). The V_u was 1 V, the external co_K and co_Cl were 400 mM, and the σ_o was +160 mC/m². Radial profiles of co_K,u and co_K,o (A), co_Cl,u and co_Cl,o (B). C Radial profiles of E_ax,u and E_ax,o. C₁ Radial profiles of E_rad with the σ_w being −160 mC/m², 0 mC/m², and +160 mC/m². The σ_w dependence of co_K,u and co_K,o (D), co_Cl,u and co_Cl,o (E) and E_ax,u and E_ax,o at selected loci as indicated

The radial profiles confirm visual impressions. Near the pore wall, the co_K,u and co_K,o are either high (negative σ_w), the same as the co_K,u in the pore center (zero σ_w) or low (positive σ_w). In the pore center, the co_K,u is in all cases above the external value but is especially elevated for positive σ_w–probably an effect of positive V_u (Fig. 6A). The co_Cl,u near the pore wall is also easy to predict–high near positive wall, zero if it is negative and the same as in the pore center if the σ_w is zero. In the pore center, it is always above its external value, but higher for positive σ_w. It is high near the (positively charged) disk but is still influenced by the σ_w (Fig. 6B). The effect of σ_w on the E_ax is interesting. Whereas the E_ax,u is always negative (with flat radial profile), it is more negative for negative σ_w. Finally, the E_ax,o is both much more negative, and much more σ_w dependent (Fig. 6C). We also evaluated the E_rad. If σ_w changes the E_rad,u should also change, and it does from being very near zero at radial distances below 2 nm to very significantly positive beyond (if the σ_w is negative) or similarly significantly negative (positive σ_w). If σ_w is zero, the E_rad,u rises to positive but small values (Fig. 6C₁).

The σ_w dependence of co_K, co_Cl, and E_ax provides some additional insights. As the σ_w changes from negative to positive, the co_K,u and co_K,o decrease from very high to low values near the pore wall, whereas in the pore center the co_K,u rises from low values, and near the disk the co_K,o remains low (Fig. 6D). In contrast, the co_Cl,o rises greatly near the disk and pore walls, whereas the co_Cl,u rises similarly near the pore wall but its rise in the pore center is less pronounced (Fig. 6E). Finally, the E_ax,u is negative and not very σ_w dependent, whereas the E_ax,o is much more negative and its σ_w dependence is complex (Fig. 6F).

The σ_w effect on σ_Is is pronounced but some changes are surprising (Fig. 7). The σ_I,conv,u (zero in the pore center) is negative near the pore wall regardless of σ_w sign, but more so if the σ_w is negative. The σ_I,conv,o is more negative than σ_I,conv,u for similarly charged pore wall and disk. However, it is positive near the disk and negative near the wall if the σ_o and σ_w differ in sign (Fig. 7A). This is not surprising because water movement drives positive and negative ions in the same direction. What the σ_I,_diff radial profiles should be is intuitively less obvious, owing to the spatial complexity of co_K and co_Cl near the disk and pore walls. The σ_I,_diff,u is zero in the pore center, negative near positively charged pore wall, but positive if it is negative. As already discussed this is likely due to the V_u that affects K⁺ and Cl⁻ differently. The σ_I,_diff,o is positive near positively charged pore wall, but negative and small if it is negatively charged (in both cases the disk was positively charged; Fig. 7B). The σ_I,mig,u is negative at any radial distance but more so near the pore wall (especially if the σ_w is negative), as is the σ_I,mig,o which is larger and negative near both the disk and the pore walls regardless of the sign of their charges (Fig. 7C). Negative σ_I,migs are not surprising given the positive V_u. As expected the σ_I,_tot is similar to that of σ_I,_mig (the σ_I,conv and σ_I,diff are much smaller; Fig. 7D).

Fig. 7.

Effect of pore wall surface charge density (σ_w) on migratory and convective current densities is significant. See 2D color-coded distributions on top (the same simulations as in Fig. 6). Radial profiles of σ_I,conv,u and σ_I,conv,o (A), σ_I,_diff,u and σ_I,_diff,o (B), σ_I,mig,u and σ_I,mig,o (C), and σ_I,tot,u and σ_I,tot,o (D). The σ_w and axial distance z were as indicated. Radial profiles of σ_I,_rad,diff (B₁), σ_I,rad,mig (C₁) and σ_I,rad,tot (D₁). The σ_w and axial distance z were as indicated. The σ_w-dependence of I_conv,u and I_conv,o (E), I_diff,u and I_diff,o(F), I_mig,u and I_mig,o(G), and I_tot (H) at selected loci as indicated. Dependence of the I_conv,u, I_diff,u, I_mig,u, and I_tot,u on σ_w in the absence of a disk is also depicted (as indicated)

The radial current densities (σ_I,rads) just above the upper disk surface (0.25 nm above) were also estimated. If the radial currents are significant, they may contribute to the axial currents in the disk-pore wall space. The σ_I,rad,convs are small (not shown). The σ_I,rad,diffs are indeed comparable to σ_I,diffs. If the σ_w is negative, the σ_I,rad,diff is present even at short radial distances and gradually increases near the disk’s tip. If the σ_w is positive, the σ_I,rad,diff is smaller and negative up to the disk’s tip, but then it rises to significant and positive values. Interestingly, if the σ_w is zero, the σ_I,rad,diff remains significant and negative near the tip (Fig. 7 B1). The σ_I,rad,migs are also elevated and high near the tip, but they oppose the σ_I,rad,diffs (Fig. 7 C1). Finally, the σ_I,rad,tots are moderate, present at any radial distance, rise near the tip, and are insensitive of the σ_w sign (Fig. 7 D1).

The I_mig,u and I_mig,o are the largest of all Is and very similar–negative (especially for high σ_w) and essentially independent of σ_w sign (Fig. 7G). Both the I_conv,u and I_conv,o are small and depend differently on σ_w (Fig. 7E), and the same is true for I_diff,u and I_diff,o (Fig. 7F). The I_tot,u and I_tot,o overlap completely (Fig. 7H). If no disk is present within nanopore, the I_conv,u becomes more σ_w-dependent. The I_mig,u and I_tot,u become not only less σ_w-dependent, but also smaller in value at higher σ_w regardless of its sign. Finally, the I_diff,u becomes near zero at any σ_w.

In an Uncharged Nanopore, the Current Magnitude Diminishes as the Disk Size Increases

As already stated, the I_tot is the smallest if the σ_w is near zero. Interestingly in such a case, the r_o dependence is qualitatively different. The 2D distributions shown on top compare the σ_Is, E_ax, and co_Cl for a short and long disk when the σ_w is zero or positive and significant (i.e., 0 or +160 mC/m²; Fig. 8). It is not evident based on visual observations whether or how I_diffs will differ, when the σ_w is +160 mC/m² instead of 0 mC/m² regardless of disk radius. Both the I_conv and I_mig are clearly more negative when the σ_w is +160 mC/m² instead of 0 mC/m² especially near the pore wall. The co_Cl and E_ax distributions are also shown as they illustrate complex co_Cl (especially near the pore wall) and E_ax (in the pore interior) changes that influence different currents. The Cl⁻ is depicted because it is a dominant ion within the nanopore (either both the σ_w and σ_o are positive or the σ_o is positive and the σ_w is zero; Fig. 8).

Fig. 8.

In a nanopore with an uncharged wall greater disk radius (r_o) leads to smaller total ion current. The color-coded 2D distributions of ion current densities (r_o was either 0.5 or 3.0 nm) and Cl⁻ concentration and E_ax (for a disk whose r_o was 3.0 nm) are shown on top. The V_u was +1 V, the σ_o was +160 mC/m², the external co_K and co_Cl were 400 mM, and the σ_w was as indicated. A Axial profiles of E_ax at 0.25 nm distance from the pore wall. B Corresponding axial profiles of Cl⁻ concentration. The r_o-dependence of I_conv (C), I_diff (D), I_mig (E), and I_tot (F) for the σ_w as indicated

Figure 8A and B give the E_ax and co_Cl axial profiles at the radial mid-point of the disk-pore wall space (i.e., with the r_d of 2 nm or 3.25 nm and from z = 1.25 nm to z = 1.75 nm). If the disk is small (r_o = 0.5 nm) the axial profiles of both E_ax and co_Cl are constant. Interestingly even in this case their values depend on the σ_w. If the σ_w is 0 mC/m², the E_ax is clearly more negative and the co_Cl is lower. With large disk (r_o = 3.0 nm), the E_ax is more negative at the lower end especially when the σ_w is 0 mC/m²but is almost zero at the higher end regardless of the σ_w. The co_Cl also changes. It is lower when the σ_w is 0 mC/m² especially at the lower end. The E_ax and co_Cl are thus both affected by the σ_w changes even when the disk is small, but if it is large, they also change significantly axially.

The currents depend also on σ_w and at all disk radii, but there are important differences. If the σ_w is zero, the I_conv is also near zero and is largely r_o-independent. If the σ_w is positive, it is negative (having the most negative value for r_o of 2 nm). There is very little ion accumulation near uncharged pore wall, but if the wall is positively charged Cl⁻ accumulates. Given that the V_u is positive, water will move upwards (not shown) producing a negative I_conv. The I_conv decreases eventually when very large disk blocks almost completely mechanically the water flow and I_conv (Fig. 8C). The I_diffr_o-dependence is influenced differently by the σ_w. The I_diff is near zero for small r_o regardless of the σ_w. It rises as the r_o increases but more when the σ_w is 0 mC/m² (Fig. 8D). Although the co_Cl is higher when the σ_w is large and positive, it changes more axially when the σ_w is 0 mC/m². When the r_o is small, the co_Cl is not only low but changes very little axially (Fig. 8B). The I_mig is negative (and the largest of all Is) if the σ_w is 0 mC/m²and diminishes in amplitude (as does the I_tot) as r_o increases (Fig. 8E). It is difficult to give a simple explanation for this change–the E_ax does not change overall but becomes highly non-uniform axially and the co_Cl in fact increases. When the σ_w is positive (+ 160 mC/m²), the I_mig and I_tot become more negative with greater r_o (except when the disk comes very near the wall; Fig. 8E, F). The most likely explanation is the rise of Cl⁻ concentration in the disk-pore wall space (Fig. 8B).

At High External Ion Concentrations, Ion Current is More r_o-Dependent

As external concentrations increase the I_tot rises (Fig. 5H), but it is unclear whether, and if so how, its r_o-dependence may change. The V_u remained +1 V but the σ_o(+ 160 mC/m²) and σ_w(− 160 mC/m²) differed in sign (unlike in Fig. 8). At two comparatively low but different external concentrations (20 and 100 mM), the I_conv diminishes as r_o increases regardless of external concentrations, but its r_o dependence is greater at high concentrations (Fig. 9A). In contrast, the I_diff (also small) rises as r_o increases but is concentration independent (Fig. 9B). This is most likely because the axial concentration gradients in the disk-pore wall space–the K⁺ (near the pore wall) and Cl⁻ (near the disk)–increased. Finally, the I_mig (the largest current at any r_o; Fig. 9D) is not r_o-dependent at low concentration but at high concentration it is not only larger, but becomes moderately r_o-dependent, rising as r_o increases except when the disk gets very near the pore wall (Fig. 9C).

Fig. 9.

Ion current is more r_o-dependent at high ion concentrations. The color-coded 2D distributions of ion current densities (the disk whose radius r_o was either 0.5 nm or 3.0 nm) are shown on top. A–D The r_o dependence of I_conv, I_diff, I_mig, and I_tot for co_K,u and co_Cl,u of 20 mM or 100 mM. The V_u was +1 V, σ_o was +160 mC/m² and σ_w was −160 mC/m²

Discussion

Background

The size of cells and generally micro-particles traversing through a micro-pore is often evaluated by measuring how much their transit reduces the current [2, 46, 47]. A basic idea is that a bigger cell (which leaves smaller cross-sectional area for the passage of ions) leads to smaller current. The cell shape can also be assessed (with irregular micro-pores) from current fluctuations produced by cell rotation [21]. However, it is still not clear why axial irregularity may be critical in inducing object rotation, whether the object would rotate in nanopores, and whether its rotation would be detectable [19–21]. Given that many viruses are at the nanoscale and that thus nanopores will have to be used for their detection a systematic evaluation of what controls the current and current-object size relationship in nanopores is clearly needed.

In this study, we determine using simulations how external factors (electric field and ion concentrations), and pore properties (pore wall charge density) affect the current and object (disk) radius-current relationship in cylindrical nanopores. Three components of axial ionic current are dissected above the disk and in the disk-pore wall space–diffusive (due to concentration gradient; I_diff), convective (due to water movement; I_conv), and migratory (due to electric field; I_mig), as each component is controlled differently. Finally, we dissected the corresponding radial currents and estimate them just above the disk. Note that we do not consider osmotic gradient term in the interaction between ions and the fluid. This will be a part of our future study.

Effect of External Electric Field on the Current Flow

The effect of external electric field (E_ex) on ion currents was tested in a negatively charged nanopore with a positively charged large disk. The voltages applied at the controlling edges resemble those considered previously (ranging from −1 to +1 V; [48]). Greater E_ex leads to a proportionally greater I_mig, but much smaller I_conv and I_diff also depend on the E_ex. Note that not only at high voltages, but over a wide range of voltages (and thus over a wide range of E_ex values), the I_mig is the largest current and determines the I_tot. At any point within the nanopore, the I_mig is determined by two factors: (a) E_ax and (b) ion concentrations. We tested the E_ex effect above the disk and in the disk-pore wall space. The E_ax radial profiles are uniform. The E_ax depends (linearly) on the E_ex, but its dependence is greater in the pore wall-disk space. As in conically shaped nanopores, the E_ex influences the co_K and co_Cl [25, 49, 50], which rise with greater E_ex regardless of tested location, and independently of E_ex polarity. Finally note that the disk presence within nanopore leads to greater V_u-dependence of the I_tot,u (and I_mig,u), but that of the I_conv,u diminishes.

Effect of External Ion Concentrations on Ion Currents and Their r_o Dependence

External K⁺ and Cl⁻ concentrations influence those within nanopore, but unequally. The concentrations varied over a significant range (from 10 to 400 mM)–i.e., beyond values used experimentally (100–300 mM; [20]), and if they rise, the nanopore concentrations increase. Given that in these simulations the pore wall is negatively charged, and disk is positively charged K⁺ ions are counter-ions for the pore wall and co-ions for the disk surface. Opposite is the case for Cl⁻ ions. As expected near the pore wall, the co_K is high (but more so above the disk than in the disk-pore wall space), whereas in the pore center it is low but even lower near the disk in the disk-pore wall space. The co_Cl is opposite–low near the pore wall above the disk and in the disk-pore wall space. In the pore center, it is high (moderately) but higher near the disk in the disk-pore wall space. As the external concentrations rise, both co_K and co_Cl rise regardless of location tested. The co_K increases little near the pore wall above the disk, but more so in the disk-pore wall space. The co_Cl remains low near the pore wall, but in the pore center and especially near the disk in the disk-pore wall space, it rises significantly.

The E_ax was negative (V_u was +1 V and V_d 0 V) except at low external concentrations in the center of the nanopore. It becomes progressively more negative as external concentrations increased, especially in the pore wall-disk space (near the pore wall or the disk). This is not entirely surprising because in that space the ion concentrations change greatly both radially and axially, especially when the disk almost reaches the pore wall. All current components increase–the I_mig (the greatest), the I_conv (that adds to the I_mig), and the I_diff (that opposes it) owing to elevate charges near the pore wall and greater E_ax. The I_tot, which is r_o independent at low concentration, becomes r_o dependent at high concentration, but only modestly. It is also worth pointing that if the disk is present within the nanopore the concentration dependence of the I_tot,u and I_mig,u rises but that of the I_conv,u diminishes.

Effect of Pore Wall Charges on Ion Concentrations and Currents and Current r_o Dependence

In this study, the σ_w changed from −160 to +160 mC/m². These values are equivalent to 1 e/nm²–thus equal or beyond those observed biologically or used experimentally. At the surface of biological membranes, the σ_w has been estimated at 40–160 mC/m² or 0.25 to 1 e/nm² [51, 52], and at the surface layer of SiO₂ membrane at 26 mC/m² [53]. As expected the σ_w influences the ion concentrations greatly near the charged pore wall or disk surfaces. We find it surprising how large the σ_w influence is throughout the nanopore including in the pore center. It is also notable how different the counter-ion and co-ion concentrations could be above and below the disk. The effect on the E_ax is also difficult to predict–pronounced and complex in the disk-pore wall space, but also present though limited above the disk.

How σ_I radial profiles and corresponding currents should depend on the σ_w is sometimes possible to assess but not always. As expected the σ_I,diff and σ_I,conv are confined near the pore wall regardless of the σ_w value or its sign. However, the σ_I,mig, though greater near the pore wall, is not insignificant near the pore center. The σ_I,mig rises greatly as σ_w increases but its direction is independent of the σ_w sign. The I_mig-σ_w relationship is thus almost parabolic. Greater concentration of counter-ions near the pore wall contributes to greater I_mig at higher σ_w. It however does not affect its direction (K⁺ movement in one direction is the same as Cl⁻ movement in the opposite direction). The E_ax also changes (especially in the pore wall-disk space) influencing the I_mig, but the E_ax-σ_w relationship is complex.

Given that the nanopore does not have current sources or sinks, the I_tot has to be independent of where axially it is estimated and it is. The I_mig which clearly makes the largest contribution to the I_tot should thus depend very little on where they are axially estimated, and that is also the case. On the other hand much smaller I_diff and I_conv may differ significantly depending on where axially they are estimated, and they clearly do. It has been suggested that the (a) presence of an object within nanopore leads to volume exclusion of electrolyte solution thus reducing the axial current flow and (b) charged object induces screening charges leading to larger axial currents, finally producing multi-level current changes [26]. However, this study clearly demonstrates that the I_tot,u (and I_mig,u) become larger and not smaller (regardless of σ_w sign) at high σ_w if disk is present within the nanopore. However, if the σ_w is low (or zero), the I_tot,u (and I_mig,u) become smaller.

To show how the σ_w influences the I-r_o relationship we chose two σ_ws (0 and +160 mC/m²). In an uncharged nanopore, the I_mig decreases in value (rather than increasing) with greater r_o but remains the largest current contribution. The I_conv (negative and small for positive σ_w) becomes almost zero for all r_os. The I_diff is positive and near zero for small r_o regardless of σ_w. It rises in value with greater r_o but more so in an uncharged nanopore. Should the I_diff be positive or negative and how should it change in value as the r_o rises? Consider the Cl⁻ (counter-ion) concentration near the disk in an uncharged pore. It is elevated near the disk but diminishes upwards. It thus produces a positive I_diff. Greater I_diff at larger r_o is also not unexpected. Larger r_o leads to more fixed (and thus also more mobile) charges. Whereas a positively charged pore would lead to more charges (Cl⁻ ions) the axial gradient is likely smaller. Overall, in an uncharged nanopore, the I_tot decreases significantly in value with greater r_o, instead of increasing modestly as observed when σ_w is positive and high. A comparison of I-r_o relationships with σ_ws that are either negative or positive but high (the σ_o was +160 mC/m² in both cases) is also interesting. The I_tot-r_o relationship depends on the σ_w value but is largely independent of its sign (i.e., whether the σ_w and σ_o have the same or opposite signs). However, if the σ_w and σ_o have the same sign, larger I_tot at greater r_o is due to higher counter-ion accumulation in the object-pore wall space as suggested before [26]. In contrast, if the σ_w and σ_o are of opposite sign, similarly greater I_tot is largely due to an elevated E_ax in that space.

It is clear from the above that the uncharged nanopores would make better sensors of size and shape of nano-size objects (such as viruses). Note also, that in a charged nanopore (and to a lesser extent in an uncharged nanopore too), the I_tot is almost r_o-independent for r_os ranging from 0 to 2–2.5 nm in a 3.5 nm radius nanopore. Suggestions that the I_tot block is due to the volume exclusion of electrolyte solution by the object thus cannot be universally accepted. If the object does not get very close (< 1 nm) to the pore wall, it will not reduce the I_tot significantly, because the ion current flows largely in 1 nm space near the pore wall.

Our simulations are three-dimensional based on axial symmetry. True three-dimensional simulations can evaluate radially asymmetrical situations whereby one side of the disk is closer to the pore wall than the opposite side. Given that the I_tot depends relatively little on the r_o the I_tot should not depend greatly on how asymmetrical the radial position of the disk is. The cross-sectional area for the current flow remains constant. However, if the disk is positioned non-centrally (i.e., if it is axially asymmetrical), the I_tot changes. It diminishes linearly if the disk is positioned more upwardly (15 nm change in axial position leads to I_tot reduction of 39%; not shown).

Comment on Continuum Poisson-Nernst-Planck and Navier–Stokes Simulations

Continuum PNP-NS simulations do not capture all features of the nanopore ion and water system. As shown by the molecular dynamics simulations water density and viscosity are greater near the pore wall due to non-electrostatic (described by Lennard–Jones potentials) and electrostatic interactions [54], whereas the Navier–Stokes equations assume that the water is incompressible. The ions are also layered near the pore wall and show a peak near, but not at its surface, because the ions cannot get closer to the surface than an ionic radius [54]. This is also not predicted by the continuum simulations. Nevertheless, most properties of the nanopore ion and water system are well described by the continuum PNP-NS equations for water and ion transport [55].

Conclusions

In a charged cylindrical nanopore with a charged disk elevated E_ex leads to higher E_ax, co_K, co_Cl, and I_tot (which is largely determined by the I_mig regardless of the E_ex value). The co_K and co_Cl rise (estimated above the disk or in the disk-pore wall space) is largely E_ex polarity independent. The I_tot also rises with greater σ_w and interestingly it is σ_w sign independent. If the σ_w and σ_o have the same sign, larger I_tot is due to higher counter-ion accumulation in the object-pore wall space, but if their signs are opposite similarly larger I_tot is mainly due to greater E_ax in the object-pore wall space. Elevated external ion concentrations lead to greater co_K and co_Cl (mainly away from the pore wall), raise the E_ax (especially in the disk-pore wall space), and lead to larger I_tot. The I_tot is not r_o-dependent at low concentrations, but at high concentrations it is, though only modestly. Surprisingly, at high concentrations the I_tot rises modestly when disk becomes bigger. Regardless of external concentrations the I_tot depends very little on the r_o if it is < 2 nm. The r_o-independence of I_tot for small disks is not surprising because the current flow is largely confined to the space near the pore wall. Smaller cross-sectional area for current flow (size-exclusion principle) or volume exclusion of electrolyte solution by the object thus cannot be universally accepted as explanations of current blockage. However, the radial currents near the charged disk surface due to screening charges and E_rad can contribute to axial currents. Finally, if the pore wall is uncharged, the I_tot diminishes significantly as r_o increases due to progressively smaller I_mig, and larger I_diff, which opposes it. Though smaller than the I_mig, the I_diff is not insignificant.

Author Contribution

MIG: conceptualization, simulations, analysis, and writing. MT: model development and writing.

Funding

This work was supported by the grant from the Natural Sciences and Engineering Research Council of Canada to M.I.G (Grant no. 24776).

Declarations

Ethics Approval

Not applicable.

Competing Interests

The authors declare no competing interests.