Nernst Planck Poisson System Research Articles

Mathematical analysis of Poisson–Nernst–Planck models with permanent charges and boundary layers: studies on individual fluxes

This work focuses on a one-dimensional Poisson–Nernst–Planck system including small permanent charges for ionic flows with one cation and one anion through a membrane channel. Our main interest is to examine the qualitative properties of the individual fluxes with boundary layers that is more realistic for ion channel problem study. Our result shows that the individual fluxes depend sensitively on multiple system parameters such as permanent charges, channel geometry, boundary conditions (concentrations and potentials) and boundary layers. For the relatively simple setting and assumptions of the model in this work, we are able to characterize the different effects of the nonlinear interaction among these system parameters in detail and gain a better understanding of the internal dynamics of ionic flows through membrane channels. Our analysis indicates that the small positive permanent charge cannot strengthen the flux of cation while reduce that of anion. Critical electric potentials which play crucial roles in studying ionic flow properties are identified. Some can be estimated experimentally. Numerical simulations are further performed and numerical results are consistent with our analytical ones.

Efficient, Positive, and Energy Stable Schemes for Multi-D Poisson–Nernst–Planck Systems

In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson–Nernst–Planck equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme is also shown to preserve the steady-states. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady-states.

Small Permanent Charge Effects on Individual Fluxes via Poisson–Nernst–Planck Models with Multiple Cations

A quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail. Of particular interest is the competition between two cations, which is related to the selectivity phenomena of ion channels. Critical potentials are identified and their roles in characterizing ionic flow properties are studied. Some critical potentials can be estimated experimentally, and this provides an efficient way to adjust/control boundary conditions (electric potential and concentrations) to observe distinct qualitative properties of ionic flows. Mathematical analysis further indicates that to optimize the effect of permanent charges, a short and narrow filter, within which the permanent charge is confined, is expected, which is consistent with the typical structure of an ion channel.

Numerical analysis of locally conservative weak Galerkin dual-mixed finite element method for the time-dependent Poisson–Nernst–Planck system

In this study, a linearized locally conservative scheme, based on using a weak Galerkin (WG)-mixed finite element method (MFEM), is developed for the Poisson–Nernst–Planck (PNP) system. In the dual-mixed formulation of the PNP equation, in addition to the three unknowns of concentrations p, n and the potential ψ, their fluxes, namely, σp=∇p+pσψ and σn=∇n−nσψ and σψ=∇ψ are introduced. These fluxes have an essential role in specifying the Debye layer and computing the electric current. The WG-MFEM considered here uses discontinuous functions to construct the approximation space. Also, a linearization scheme is employed to treat nonlinear terms. In the proposed method, the important physical laws of mass conservation and free energy dissipation are preserved without any restriction on the time step. Error estimates are developed and analyzed for both semi- and fully discrete WG-MFEM schemes. Furthermore, optimal error estimates (under adequate regularity assumptions on the solution) are derived. Several numerical results are provided and they demonstrate the efficiency of the proposed method and validate the convergence theorems.

Competition between Cations via Classical Poisson-Nernst-Planck Models with Nonzero but Small Permanent Charges.

We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.

Dynamics of Classical Poisson–Nernst–Planck Systems with Multiple Cations and Boundary Layers

We study a quasi-one-dimensional classical Poisson–Nernst–Planck model for ionic flow through a membrane channel with two positively charged ion species (cations) and one negatively charged, and with zero permanent charges. We treat the model problem as a boundary value problem of a singularly perturbed differential system. Under the framework of the geometric singular perturbation theory, together with specific structures of this concrete model, the existence of solutions to the boundary value problem is established and, for a special case that the two cations have the same valences, we are able to derive approximations of the individual fluxes and the I–V (current–voltage) relation explicitly, from which, our two main focuses in this work, boundary layer effects on ionic flows and competitions between two cations, are analyzed in great details. Critical potentials are identified and their roles in characterizing these effects are studied. Nonlinear interplays among physical parameters, such as boundary concentrations and potentials, diffusion coefficients and ion valences, are characterized, which could potentially provide efficient ways to control and affect some biological functions. Numerical simulations are performed, and numerical results are consistent with our analytical ones.

Model reduction-based initialization methods for solving the Poisson-Nernst-Plank equations in three-dimensional ion channel simulations

The Poisson-Nernst-Planck (PNP) model is frequently-used in simulating ion transport through ion channel systems. Due to the nonlinearity and coupling of PNP model, choosing appropriate initial values is crucial to obtaining a convergent result or accelerating the convergence rate of finite element solution, especially for large channel systems. Continuation is an effective and commonly adopted strategy to provide good initial guesses for the solution procedure. However, this method needs multiple times to solve the whole system at different conditions. We utilize a reduced model describing a near-or partial-equilibrium state as an approximation of the original PNP system (describing a non-equilibrium process in general). Based on the reduced model, we design three initialization methods for the solution of PNP equations under general conditions. These methods provide the initial guess of the PNP system by solving a specifically designed Poisson-Boltzmann-like model, Smoluchowski-Poisson-Boltzmann-like model, and linear approximation model. Simulations of potassium channels 1BL8 and 2JK4 demonstrate that these methods can effectively reduce the number of Gummel iteration steps and the total CPU time in the solution of the PNP equations, and especially do not need the continuation approach anymore. The reason is that these initial guesses can approximate the PNP solution well in the channel region. Besides, our numerical experiments demonstrate that as one of the initialization methods, the linear approximation method can even produce very close results such as current-voltage curves to that from the PNP model when the membrane potential is not high.

Asymptotic analysis of compression sensing in ionic polymer metal composites: The role of interphase regions with variable properties

Ionic Polymer Metal Composites (IPMCs) consist of two noble metal electrodes plating an electroactive polymeric membrane, referred to as ionomer, which is electroneutralised by a solvent including mobile ions. The IPMC manufacturing leads to thin interphase regions next to the electrodes, the so-called Composite Layers (CLs), in which metal atoms occupy interstitial sites within the ionomer. In this work we extend previous efforts of our group on IPMC compression sensing to include the important effect of CLs, where large variations of the electrochemical properties occur. In IPMC compression sensing the application of a through-the-thickness displacement leads to a shortcircuit electric response, here assumed to be governed by a linearised <i>modified</i> Poisson-Nernst-Planck (PNP) system of partial differential equations (PDEs), to be solved for the time-evolving electric potential and mobile ions concentration as functions of the displacement field evaluated through the linear momentum balance. The variation of material properties in the CLs requires the simultaneous integration of the governing system of PDEs in three regions: the membrane and the two CLs. To this purpose, we resort to the perturbative method of matched asymptotic expansions. Except for a numerical inverse Laplace transform, this allows us to obtain an analytical solution through which we establish an equivalent circuit model elucidating the main features of the IPMC sensing behaviour. We validate and discuss the analytical solution through comparison with finite element analyses, whereby we also numerically solve the nonlinear modified PNP systems fully coupled with the linear momentum balance accounting for the electrochemical stresses. We finally provide some insight into the role of CLs in the IPMC sensing behaviour, by assessing its sensitivity to some key parameters. We expect the obtained results to aid the design of optimised IPMC sensors.

Electric discharge of electrocytes: Modelling, analysis and simulation

In this paper, we investigate the electric discharge of electrocytes by extending our previous work on the generation of electric potential. We first give a complete formulation of a single cell unit consisting of an electrocyte and a resistor, based on a Poisson-Nernst-Planck (PNP) system with various membrane currents as interfacial conditions for the electrocyte and a Maxwell’s model for the resistor. Our previous work can be treated as a special case with an infinite resistor (or open circuit). Using asymptotic analysis, we simplify our PNP system and reduce it to an ordinary differential equation (ODE) based model. Unlike the case of an infinite resistor, our numerical simulations of the new model reveal several distinct features. A finite current is generated, which leads to non-constant electric potentials in the bulk of intracellular and extracellular regions. Furthermore, the current induces an additional action potential (AP) at the non-innervated membrane, contrary to the case of an open circuit where an AP is generated only at the innervated membrane. The voltage drop inside the electrocyte is caused by an internal resistance due to mobile ions. We show that our single cell model can be used as the basis for a system with stacked electrocytes and the total current during the discharge of an electric eel can be estimated by using our model.

Transient electrohydrodynamic flow with concentration-dependent fluid properties: Modelling and energy-stable numerical schemes

Transport of electrolytic solutions under influence of electric fields occurs in phenomena ranging from biology to geophysics. Here, we present a continuum model for single-phase electrohydrodynamic flow, which can be derived from fundamental thermodynamic principles. This results in a generalized Navier–Stokes–Poisson–Nernst–Planck system, where fluid properties such as density and permittivity depend on the ion concentration fields. We propose strategies for constructing numerical schemes for this set of equations, where the electrochemical and the hydrodynamic subproblems are decoupled at each time step. We provide time discretizations of the model that suffice to satisfy the same energy dissipation law as the continuous model. In particular, we propose both linear and non-linear discretizations of the electrochemical subproblem, along with a projection scheme for the fluid flow. The efficiency of the approach is demonstrated by numerical simulations using several of the proposed schemes.

Modeling Actuation of Ionomer Cilia in Salt Solution Under an External Electric Field

Abstract A recent experiment by Kim’s group from the University of Nevada, Las Vegas, has shown the possibility of actuating ionomer cilia in salt solution. When these actuators are placed between two external electrodes, across which a small voltage is applied, they move toward the cathode. This is in stark contrast with ionic polymer metal composites, where the same ionomers are plated by metal electrodes but bending occurs toward the anode. Here, we seek to unravel the factors underlying the motion of ionomer cilia in salt solution through a physically based model of actuation. In our model, electrochemistry is described through the Poisson–Nernst–Planck system in terms of concentrations of cations and anions and voltage. Through finite element analysis, we establish that Maxwell stress is the main driving force for the motion of the cilia. This study constitutes a first effort toward understanding the motion of ionomer cilia in salt solution, which, in turn, may help elucidate the physical underpinnings of actuation in ionic polymer metal composites.

Electric potential generation of electrocytes: Modelling, analysis, and computation

In this paper, we developed a one-dimensional model for electric potential generation of electrocytes in electric eels. The model is based on the Poisson–Nernst–Planck system for ion transport coupled with membrane fluxes including the Hodgkin–Huxley type. Using asymptotic analysis, we derived a simplified zero-dimensional model, which we denote as the membrane model in this paper, as a leading order approximation. Our analysis provides justification for the assumption in membrane models that electric potential is constant in the intracellular space. This is essential to explain the superposition of two membrane potentials that leads to a significant transcellular potential. Numerical simulations are also carried out to support our analytical findings.

Solution of Ion Channel Flow Using Immersed Boundary-Lattice Boltzmann Methods.

Poisson-Nernst-Planck (PNP) model has been extensively used for the study of channel flow under the influence of electrochemical gradients. PNP theory is a continuum description of ion flow where ionic distributions are described in terms of concentrations. Nonionic interparticle interactions are not considered in this theory as in continuum framework, their impact on the solution is minimal. This theory holds true for dilute flows or flows where channel radius is significantly larger than ion radius. However, for ion channel flows, where channel dimensions and ionic radius are of similar magnitude, nonionic interactions, particularly related to the size of the ions (steric effect), play an important role in defining the selectivity of the channel, concentration distribution of ionic species, and current across the channel, etc. To account for the effect of size of ions, several modifications to PNP equations have been proposed. One such approach is the introduction of Lennard-Jones potential to the energy variational formulation of PNP system. This study focuses on understanding the role of steric effect on flow properties. To discretize the system, Lattice Boltzmann method has been used. The system is defined by modified PNP equations where the steric effect is described by Lennard-Jones potential. In addition, boundary conditions for the complex channel geometry have been treated using immersed boundary method.

A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson–Nernst–Planck System

In this paper, based on the reformulation of the Nernst–Planck equations, we construct an unconditionally stable semi-implicit linearized difference scheme for the time dependent Poisson–Nernst–Planck system. The method preserves several important physical laws at full discrete level without any constraints on the time step size, including: mass conservation, ion concentration positivity, free energy dissipation. The proposed method is a linearized method, which can be implemented efficiently. Furthermore, the method can be extended to the case of multi-ions without any difficulty. The paper ends with supporting numerical results which confirm well with theoretical findings.

Global existence of solutions for the Poisson–Nernst–Planck system with steric effects

In this work, we study the global existence of a modified Poisson–Nernst–Planck system with steric effects. This model involves cross-diffusion and non-local terms. The main idea of the proof is to approximate the system by truncating the diffusion matrix and apply the Schauder’s fixed-point theorem. Moreover, we obtain L2 uniform in time estimates from the energy inequality (3.5). The global existence then follows.

Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains

In this paper, we consider radial solutions of the Poisson-Nernst-Planck (PNP) system with variable dielectric coefficients $\varepsilon g(x)$ in $N$-dimensional annular domains, $N≥2$. When the parameter $\varepsilon$ tends to zero, the PNP system admits a boundary layer solution as a steady state, which satisfies the charge conserving Poisson-Boltzmann (CCPB) equation. For the stability of the radial boundary layer solutions to the time-dependent radial PNP system, we estimate the radial solution of the perturbed problem with global electroneutrality. We generalize the argument of the one spatial dimension case (cf. [18]) and find a new way to transform the perturbed problem. By choosing a suitable weighted norm, we then derive the associated energy law which can be used to prove that the $H^{-1}_x$ norm of the solution of the perturbed problem decays exponentially in time with the exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound.

Sensitivity analysis of the Poisson Nernst-Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model.

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst-Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

The inviscid limit and well-posedness for the Euler–Nernst–Planck–Poisson system

ABSTRACTIn this paper, we mainly study the Cauchy problem of the Euler–Nernst–Planck–Poisson (ENPP) system. We first establish local well-posedness for the Cauchy problem of the ENPP system in Besov spaces. Then we present a blow-up criterion of solutions to the ENPP system. Moreover, we prove that the solutions of the Navier–Stokes–Nernst–Planck–Poisson system converge to the solutions of the ENPP system as the viscosity ν goes to zero, and the convergence rate is at least of order.

Electroneutral models for dynamic Poisson-Nernst-Planck systems.

The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modeling and computational challenges. In this paper, we derive simplified electroneutral (EN) models where the thin boundary layers are replaced by effective boundary conditions. There are two major advantages of EN models. First, it is much cheaper to solve them numerically. Second, EN models are easier to deal with compared to the original PNP system; therefore, it would also be easier to derive macroscopic models for cellular structures using EN models. Even though the approach used here is applicable to higher-dimensional cases, this paper mainly focuses on the one-dimensional system, including the general multi-ion case. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to the EN system for the bulk region. This EN system can be solved directly and efficiently without computing the solution in the boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back higher-order contributions into the effective boundary conditions. For Dirichlet boundary conditions, the higher-order terms can be neglected and the classical results (continuity of electrochemical potential) are recovered. For flux boundary conditions, higher-order terms account for the accumulation of ions in boundary layer and neglecting them leads to physically incorrect solutions. To validate the EN model, numerical computations are carried out for several examples. Our results show that solving the EN model is much more efficient than the original PNP system. Implemented with the Hodgkin-Huxley model, the computational time for solving the EN model is significantly reduced without sacrificing the accuracy of the solution due to the fact that it allows for relatively large mesh and time-step sizes.

Convergence analysis of a BDF2 / mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system

This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy–Nernst–Planck–Poisson system. The scheme uses the second order backward difference formula (BDF2) in time and the mixed finite element method with Raviart–Thomas elements in space. In the first step, we show that the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an L ∞ bound of the concentrations / masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis works without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions.