Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations

Ion transport through membrane channels is a fundamental process in cell biology [1]. However, atomistic simulations of this process are expensive, and an alternative continuum approach is based on the Poisson-Nernst-Planck (PNP) equations, where the Poisson equation for the electric potential is coupled to the Nernst-Planck equation for the ion concentrations [2]. The continuum PNP model accounts for two effects, (1) molecular diffusion due to ion concentration gradients and (2) ion drift in the applied and self-consistent electric fields. The PNP equations have been employed to study many aspects of ion channel selectivity [3], including the effect of fixed charge [4], ion size [5], and membrane surface charge [6].

Despite the relative efficiency in comparison with atomistic simulations, numerical solution of the PNP equations is still challenging due to several factors, (1) nonlinear coupling of the electric potential and ion concentrations, (2) the need to resolve the channel geometry, as well as boundary layers and internal layers at material interfaces, and (3) maintaining stability in convection-dominated cases [7]. Many PNP solvers have been developed using finite-difference methods [8‐10], finite-element methods [6, 11‐13], and finite-volume methods [14‐16]. It is also important for numerical PNP solvers to preserve positivity of the ion concentrations and conserve mass in time-dependent simulations, and several schemes have been developed for this purpose [10, 17‐20].

The boundary element method (BEM) is another approach to solving linear elliptic boundary value problems, where the differential equation is recast as an integral equation on the domain boundary by convolution with an appropriate Green’s function [21]. While the BEM requires discretization of singular integrals, it has the advantage of reduced spatial dimension. Closely related to the PNP equations is the Poisson-Boltzmann (PB) equation for the electrostatic potential of a solvated protein. The BEM has been applied to solve the 3D linear PB equation on the molecular surface of a protein (see [22, 23] and references therein), but aside from some early work using a Green’s function to solve the PB equation for an ion channel in 1D [24], as far as we know this approach has not been further developed for 2D and 3D ion channel problems.

In fact there are two challenges to applying an integral equation approach to the PNP system, (1) nonlinear coupling of the electric potential and ion concentrations, and (2) the ions are distributed throughout the channel and solvent domain requiring costly volume integrals in 2D and 3D. Nonetheless, we believe that these obstacles can be overcome using iteration to handle the nonlinearity and a fast summation method with adaptive quadrature for the volume integrals. As a step in that direction, the present work develops a novel integral equation method for the 1D steady-state PNP equations.

The paper is organized as follows. Section 2 introduces the 1D steady-state PNP equations. Section 3 uses Green’s 3rd identity to recast the PNP equations as a fixed-point problem involving integral operators for the potential gradient and ion concentrations. Section 4 presents the numerical implementation. Section 5 presents tests for electroneutral and non-electroneutral systems with Robin boundary conditions on the potential and no-flux boundary conditions on the ion concentrations. Section 6 presents a K\(^+\) ion channel simulation following [25], and the work is summarized in Section 7.

This work focuses on the 1D steady-state PNP equations with two ion species. The electrostatic potential \(\phi (x)\) satisfies the Poisson equation,where \(\epsilon\) is the permittivity, the valence is \(z_1 = -1\) for anions and \(z_2 = 1\) for cations, e is the proton charge, \(c_1(x), c_2(x)\) are the ion concentrations, and the domain is \([-L,L]\). This is supplemented with the Nernst-Planck equation,for \(i = 1,2\), where \(D_i, k_B, T\) are the diffusion coefficient of ion species i, Boltzmann constant, and absolute temperature.

Following [10], the PNP equations are non-dimensionalized with respect to the bulk concentration \(c_0\) [ions/Å\(^3\)], channel half-length L [Å], characteristic diffusion coefficient \(D_0\) [Å\(^2\)/s], characteristic potential \(\phi _0\) [V], and water permittivity \(\epsilon _t\) [F/Å]. The resulting non-dimensional equations are for \(i = 1,2\) with domain \([-1,1]\). The coefficients arewhere \(\chi _{1}\) is the ratio of electric potential energy to thermal energy, and \(\chi _{2}/\epsilon\) is the inverse square Debye length. Except as otherwise noted, the non-dimensional diffusion coefficient is \(D_i=1\).

This section recasts the 1D steady-state PNP differential equations as a set of integral equations in the form where \(\textbf{P}, \textbf{NP}\) are operators obtained by integrating the Poisson and Nernst-Planck equations while incorporating the Robin and no-flux boundary conditions and the total concentration condition. Note that (8) defines a fixed-point problem, where \(\textbf{P}\) maps the concentrations \(c_1, c_2\) into the potential gradient \(\phi ^\prime\), while \(\textbf{NP}\) maps the concentration \(c_i\) and potential gradient \(\phi ^\prime\) into the concentration \(c_i\) for \(i=1,2\). The integral form of the PNP equations in (8) is solved by Gummel iteration [26] for \(\phi ^\prime , c_1, c_2\), and once it converges, further processing yields the potential \(\phi\), and the concentration gradients \(c_1^\prime , c_2^\prime\) needed to compute the current. Note that a fixed-point formulation and Gummel iteration were also used in the analysis of semiconductor drift-diffusion equations, although without recasting the differential equations as integral equations [27].

The derivation of the \(\textbf{P}\) and \(\textbf{NP}\) operators relies on Green’s 3rd identity,where f(x) is twice continuously differentiable on \([-1,1]\) and g(x, y) is the 1D free-space Laplace Green’s function,Note that among the two terms on the right side of (9), the first term (or bracket term) involves the boundary values \(f(\pm 1), f^\prime (\pm 1)\), and the second term (or volume integral term) involves \(f^{\prime \prime }(x)\) over the entire interval. One can think of (9) as defining an operator, where \(f(\pm 1),f^\prime (\pm 1),f^{\prime \prime }(x)\) are the input, and f(x) is the output. This point of view is taken in reformulating the PNP equations.

This section presents the numerical method used to solve the integral form of the PNP equations with the Robin and no-flux boundary conditions and total concentration constraint. The domain \([-1,1]\) is divided into N subintervals with endpoints \(x_k, k=0:N\), where \(x_0 = -1, x_N = 1\), and the grid spacing is \(h_k = x_{k+1}-x_k\). We consider two point sets, uniform and Chebyshev, The potential and concentration values and their gradients are denoted as

Two types of integrals appear in the \(\textbf{P}, \textbf{NP}\) operators depending on whether or not the Green’s function is involved. When the Green’s function is not involved, the integral is discretized by the trapezoid rule, where \(w_k\) are the trapezoid weights. Next, we explain how the integrals involving the Green’s function are computed.

In this section, the integral method is applied to compute several examples from prior work in which the PNP equations were solved by finite-difference or finite-element methods [10, 30]. In these examples, there are two ion species with valence \(z_1=-1\) (anions), \(z_2=1\) (cations), and two types of systems depending on the total ion concentration, electroneutral (\(a_1 = a_2\)) and non-electroneutral (\(a_1 \ne a_2\)). Table 1 gives the parameter values. The calculations used either Chebyshev or uniform grid points as noted, and the relaxation parameter \(\omega\) was chosen by trial and error. The method was programmed in Python, and the calculations were done on an iMac computer with the Apple M1 chip.

Finally, we consider a potassium (K\(^+\)) ion channel model following [25]. Figure 5 shows a cross-section of the model comprising two conical baths with KCl solvent and a cylindrical channel with four subregions (buffer, nonpolar, central cavity, and selectivity filter). In this example, the PNP equations are solved in dimensional form with the 1D approximation from [25], where \(A(x)= \pi r(x)^2\) is the cross-sectional area in terms of the radius r(x). Note that the Poisson equation (43a) now includes a fixed negative charge density \(\rho _n(x)\). The model extends over the interval \([-5,8.5]\) (the spatial unit is nm), and Dirichlet boundary conditions are imposed, Table 4 gives the parameter values with units in the caption. The material parameters are constant in each bath and each channel subregion, where the ion mobility is \(\mu _i = eD_i/k_BT\) in terms of the diffusion coefficient \(D_i\). The four channel subregions have constant radius \(r(x) = 0.5\), and while [25] took r(x) to be linear in the baths with \(r(-5) = r(8.5) = 5.5\); here, we consider a piecewise constant approximation indicated schematically by the red dotted lines in Fig. 5. In practice, we used 20 steps in each bath, so altogether there are 43 interfaces (19 in each bath, three in the channel interior, and two at the bath/channel interfaces). At each interface, the potential and ion concentrations satisfy continuity conditions, where the bracket indicates the jump across the interface.

The integral method previously described for systems with one domain was extended to this example with multiple subdomains, and the scheme has the same Gummel iteration form as (39). The unknown boundary and interface variables \(\phi , \phi ^\prime , c_i, c_i^\prime\) at the domain boundary and 43 interfaces satisfy a coupled linear system which is solved in each iteration after which the interior variables are computed using Green’s 3rd identity.

To illustrate the idea in a simple case, consider the Poisson equation (43a) on the interval \([x_L,x_R]\) with Dirichlet BC \(\phi _L,\phi _R\) and one interface at \(x_I\). Here, there are two subdomains, \([x_L,x_I], [x_I,x_R]\), and six unknown boundary/interface values of the potential and potential gradient, \(\phi ^\prime (x_L), \phi ^\prime (x_R), \phi (x_I^{\pm }), \phi ^\prime (x_I^{\pm })\), which satisfy six linear equations comprising two interface conditions (46a),(46b), two equations from integrating once over each subdomain (46c),(46d), and two equations from integrating twice (46e),(46f), where the integrations follow the process described in Section 3.1 leading to (20). In the equations (46), the subscripts and superscripts L and R correspond to the two subdomains, and on the right side of the equal sign, we have the following definitions for the left subdomain, with corresponding definitions for the right subdomain. The Nernst-Planck equation (43b) is treated similarly.

Returning to the model of the K\(^+\) channel with 43 interfaces, the linear system has dimension 174 (four unknowns at each interface and two unknowns at the boundary). The computations used uniform grid spacing \(h = 0.01\) and the same stopping criterion as before (41). In this problem, the mobility/diffusion coefficient ratio is \(\mu _i/D_i = 40\)/V, and the resulting potential gradient \(\phi ^\prime (x)\) is large in some places; hence, the model is drift-dominated, and we used the following continuation scheme; the problem is solved for an increasing sequence of ratios \(\mu _i/D_i\) = 1/V, 10/V, 20/V, and 40/V, with relaxation parameters \(\omega = 0.9, 0.4, 0.26, 0.18\), requiring 12, 69, 148, and 218 iterations, respectively, and the converged solution for one ratio \(\mu _i/D_i\) is the initial guess for the next ratio; the total run time for all four ratios was 44 seconds. The continuation scheme was not needed for the electroneutral and non-electroneutral systems considered earlier since they were not drift-dominated.

Figure 6 shows the computed potential and ion concentration profiles which agree well with the previous results [25]. The negative fixed charge density \(\rho _n\), indicated by green dashed lines, has substantial impact on the potential and ion concentrations. In particular, the polar subregions with negative fixed charge have local potential minima below the boundary potential (−133.7 mV in the buffer and −155.8 mV in the filter), while the potential rises to a local maximum in the nonpolar subregion (−34.3 mV). As a result, the K\(^+\) cations are attracted to the buffer and filter and are expelled from the nonpolar region, while the opposite is true for the Cl\(^-\) anions; this demonstrates the cation selectivity property of the K\(^+\) channel.

We also computed the current–voltage curve for the K\(^+\) channel model, where the current is in terms of the applied voltage \(V_{\textrm{app}}\) across the channel. All other parameters are the same as in Table 4. It follows from the NP equation (43b) that the current (48a) is constant throughout the domain, and in the present calculations, this was satisfied to machine precision.

Figure 7 shows the computed I–V curve which closely matches the expected linear relation. For \(V_{\textrm{app}}\) = 100 mV, the current is 19.4 pA comprising 19.1 pA for K\(^+\) cations and 0.3 pA for Cl\(^-\) anions, which again shows the cation selectivity of the K\(^+\) channel. We checked that the computed current is not sensitive to the number of steps in the bath, and Table 5 shows that the computed current converges to 20.9 pA as the grid spacing h is reduced with convergence rate \(p \approx 2\). For comparison, [25] reported current 22.5 pA comprising 22.2 pA for cations and 0.3 pA for anions. Hence, the two calculations agree for the anion current, but there is a small discrepancy in the cation current.

An integral equation method was presented for the 1D steady-state PNP equations modeling ion transport through membrane channels. Green’s 3rd identity was applied to recast the PNP differential equations as a fixed-point problem involving integral operators for the potential gradient and ion concentrations. The integrals were computed by midpoint and trapezoid rules at either uniform or Chebyshev grid points, and the resulting nonlinear equations were solved by Gummel iteration with relaxation. Each iteration requires solving a small linear system for the boundary and interface variables.

The integral equation method was applied to electroneutral and non-electroneutral systems with Robin boundary conditions on the potential and no-flux boundary conditions on the ion concentrations. The method is 2nd order accurate even in cases with sharp boundary layers. The method was also applied to a drift-dominated K\(^+\) ion channel model following [25]. In this case, the fixed charge density in the channel ensures the selectivity of K\(^+\) ions and exclusion of Cl\(^-\) ions. In these tests, the integral method results agree well with published finite-difference and finite-element results [10, 25, 30].

Various techniques have been used in numerical PNP solvers to improve stability such as positivity-preserving schemes, Slotboom variables, and streamwise upstream differencing. The integral equation method presented here did not employ these techniques, although it did use relaxation to accelerate the iteration and continuation to compute a good initial guess for drift-dominated systems. It is noteworthy that in these calculations, the integral equation method preserved the positivity of the ion concentrations.

In the future work, we aim to extend the integral equation PNP solver to time-dependent problems as well as 2D and 3D ion channel models, where the volume integral can be efficiently computed using adaptive quadrature and a fast summation method [31‐33].