Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates

A stabilized finite volume element method for solving Poisson-Nernst-Planck equations

One difficulty in solving the Poisson-Nernst-Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convection-dominant problem in the Nernst-Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady-state PNP equations is proposed and analyzed. Numerical tests on four ion-channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable.

Electrostatic Reaction Inhibition in Nanoparticle Catalysis

Electrostatic reaction inhibition in heterogeneous catalysis emerges if charged reactants and products with similar charges are adsorbed on the catalyst and thus repel the approaching reactants. In this work, we study the effects of electrostatic inhibition on the reaction rate of unimolecular reactions catalyzed on the surface of a spherical model nanoparticle using particle-based reaction-diffusion simulations. Moreover, we derive closed rate equations based on an approximate Debye-Smoluchowski rate theory, valid for diffusion-controlled reactions, and a modified Langmuir adsorption isotherm, relevant for reaction-controlled reactions, to account for electrostatic inhibition in the Debye-Hückel limit. We study the kinetics of reactions ranging from low to high adsorptions on the nanoparticle surface and from the surface- to diffusion-controlled limits for charge valencies 1 and 2. In the diffusion-controlled limit, electrostatic inhibition drastically slows down the reactions for strong adsorption and low ionic concentration, which is well described by our theory. In particular, the rate decreases with adsorption affinity because, in this case, the inhibiting products are generated at a high rate. In the (slow) reaction-controlled limit, the effect of electrostatic inhibition is much weaker, as semiquantitatively reproduced by our electrostatic-modified Langmuir theory. We finally propose and verify a simple interpolation formula that describes electrostatic inhibition for all reaction speeds ("diffusion-influenced" reactions) in general.

Electrodiffusion Phenomena in Neuroscience and the Nernst–Planck–Poisson Equations

This work is aimed to give an electrochemical insight into the ionic transport phenomena in the cellular environment of organized brain tissue. The Nernst–Planck–Poisson (NPP) model is presented, and its applications in the description of electrodiffusion phenomena relevant in nanoscale neurophysiology are reviewed. These phenomena include: the signal propagation in neurons, the liquid junction potential in extracellular space, electrochemical transport in ion channels, the electrical potential distortions invisible to patch-clamp technique, and calcium transport through mitochondrial membrane. The limitations, as well as the extensions of the NPP model that allow us to overcome these limitations, are also discussed.

A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface

Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces.

Hysteresis and linear stability analysis on multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric interactions

In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson-Nernst-Planck equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain S-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the S-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the S-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis transition dynamics, from a steady-state solution on the unstable branch to one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.

Accurate modeling of a biological nanopore with an extended continuum framework

Despite the broad success of biological nanopores as powerful instruments for the analysis of proteins and nucleic acids at the single-molecule level, a fast simulation methodology to accurately model their nanofluidic properties is currently unavailable. This limits the rational engineering of nanopore traits and makes the unambiguous interpretation of experimental results challenging. Here, we present a continuum approach that can faithfully reproduce the experimentally measured ionic conductance of the biological nanopore Cytolysin A (ClyA) over a wide range of ionic strengths and bias potentials. Our model consists of the extended Poisson-Nernst-Planck and Navier-Stokes (ePNP-NS) equations and a computationally efficient 2D-axisymmetric representation for the geometry and charge distribution of the nanopore. Importantly, the ePNP-NS equations achieve this accuracy by self-consistently considering the finite size of the ions and the influence of both the ionic strength and the nanoscopic scale of the pore on the local properties of the electrolyte. These comprise the mobility and diffusivity of the ions, and the density, viscosity and relative permittivity of the solvent. Crucially, by applying our methodology to ClyA, a biological nanopore used for single-molecule enzymology studies, we could directly quantify several nanofluidic characteristics difficult to determine experimentally. These include the ion selectivity, the ion concentration distributions, the electrostatic potential landscape, the magnitude of the electro-osmotic flow field, and the internal pressure distribution. Hence, this work provides a means to obtain fundamental new insights into the nanofluidic properties of biological nanopores and paves the way towards their rational engineering.

Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck–Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

Improved Poisson-Boltzmann Methods for High-Performance Computing

Implicit solvent models based on the Poisson-Boltzmann equation (PBE) have been widely used to study electrostatic interactions in biophysical processes. These models often treat the solvent and solute regions as high and low dielectric continua, leading to a large jump in dielectrics across the molecular surface which is difficult to handle. Higher order interface schemes are often needed to seek higher accuracy for PBE applications. However, these methods are usually very liberal in the use of grid points nearby the molecular surface, making them difficult to use on high-performance computing platforms. Alternatively, the harmonic average (HA) method has been used to approximate dielectric interface conditions near the molecular surface with surprisingly good convergence and is well suited for high-performance computing. By adopting a 7-point stencil, the HA method is advantageous in generating simple 7-banded coefficient matrices, which greatly facilitate linear system solution with dense data parallelism, on high-performance computing platforms such as a graphics processing unit (GPU). However, the HA method is limited due to its lower accuracy. Therefore, it would be of great interest for high-performance applications to develop more accurate methods while retaining the simplicity and effectiveness of the 7-point stencil discretization scheme. In this study, we have developed two new algorithms based on the spirit of the HA method by introducing more physical interface relations and imposing the discretized Poisson's equation to the second order, respectively. Our testing shows that, for typical biomolecules, the new methods significantly improve the numerical accuracy to that comparable to the second-order solvers and with ∼65% overall efficiency gain on widely available high-performance GPU platforms.

Selectivity of the KcsA potassium channel: Analysis and computation

Ion channels regulate the flux of ions through cell membranes and play significant roles in many physiological functions. Most of the existing literature focuses on computational approaches based on molecular dynamics simulation or numerical solution of the modified Poisson-Nernst-Planck (PNP) system. In this paper, we present an analytical and computational study of a mathematical model of the KcsA potassium channel, including the effects of ion size (Bikerman model) and solvation energy (Born model). Under equilibrium conditions, we obtain an analytical solution of our modified PNP system, which is used to explain selectivity of KcsA of various ions (K^{+}, Na^{+}, Cl^{-}, Ca^{2+}, and Ba^{2+}) due to negative permanent charges inside the filter region and the effect of ion sizes. Our results show that K^{+} is always selected over Na^{+}, as smaller Na^{+} ions have larger solvation energy. As the amount of negative charges in the filter exceeds a critical value, divalent ions (Ca^{2+} and Ba^{2+}) can enter the filter region and block the KcsA channel. For the nonequilibrium cases, due to difficulties associated with a pure analytical or numerical approach, we use a hybrid analytical-numerical method to solve the modified PNP system. Our predictions of selectivity of KcsA channels and saturation phenomenon of the current-voltage (I-V) curve agree with experimental observations.

An efficient second-order poisson-boltzmann method

Immersed interface method (IIM) is a promising high-accuracy numerical scheme for the Poisson-Boltzmann model that has been widely used to study electrostatic interactions in biomolecules. However, the IIM suffers from instability and slow convergence for typical applications. In this study, we introduced both analytical interface and surface regulation into IIM to address these issues. The analytical interface setup leads to better accuracy and its convergence closely follows a quadratic manner as predicted by theory. The surface regulation further speeds up the convergence for nontrivial biomolecules. In addition, uncertainties of the numerical energies for tested systems are also reduced by about half. More interestingly, the analytical setup significantly improves the linear solver efficiency and stability by generating more precise and better-conditioned linear systems. Finally, we implemented the bottleneck linear system solver on GPUs to further improve the efficiency of the method, so it can be widely used for practical biomolecular applications. © 2019 Wiley Periodicals, Inc.

Incorporating Born solvation energy into the three-dimensional Poisson-Nernst-Planck model to study ion selectivity in KcsA K^{+} channels

Potassium channels are much more permeable to potassium than sodium ions, although potassium ions are larger and both carry the same positive charge. This puzzle cannot be solved based on the traditional Poisson-Nernst-Planck (PNP) theory of electrodiffusion because the PNP model treats all ions as point charges, does not incorporate ion size information, and therefore cannot discriminate potassium from sodium ions. The PNP model can qualitatively capture some macroscopic properties of certain channel systems such as current-voltage characteristics, conductance rectification, and inverse membrane potential. However, the traditional PNP model is a continuum mean-field model and has no or underestimates the discrete ion effects, in particular the ion solvation or self-energy (which can be described by Born model). It is known that the dehydration effect (closely related to ion size) is crucial to selective permeation in potassium channels. Therefore, we incorporated Born solvation energy into the PNP model to account for ion hydration and dehydration effects when passing through inhomogeneous dielectric channel environments. A variational approach was adopted to derive a Born-energy-modified PNP (BPNP) model. The model was applied to study a cylindrical nanopore and a realistic KcsA channel, and three-dimensional finite element simulations were performed. The BPNP model can distinguish different ion species by ion radius and predict selectivity for K^{+} over Na^{+} in KcsA channels. Furthermore, ion current rectification in the KcsA channel was observed by both the PNP and BPNP models. The I-V curve of the BPNP model for the KcsA channel indicated an inward rectifier effect for K^{+} (rectification ratio of ∼3/2) but indicated an outward rectifier effect for Na^{+} (rectification ratio of ∼1/6).

Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions

We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+.

Interaction between Charged Cylinders in Electrolyte Solution; Excluded Volume Effect

Electrostatic interactions govern the physical properties of charged cylindrical structures in electrolyte solutions. Besides the surface charge on the cylinders, another factor influencing the electrostatic interactions are the mobile ions. The finite size of the mobile ions is included by the excluded volume effect within the lattice statistics, while the electrostatic interactions are considered by means of the mean electrostatic field. In this article we consider charged parallel cylinders embedded into an electrolyte solution of mobile monovalent ions. A modified nonlinear Poisson-Boltzmann equation is proposed via variational procedure, and we implement the finite element method to solve it numerically. Excluded volume effect of the system containing two and multiple charged parallel cylinders are taken into account. Numerical results show that the excluded volume effect decreases the concentration of counterion and increases the electrostatic potential near the charged cylinders. The angular distribution of counterion around the particular cylinder is asymmetric. The study of the electrostatic interaction between two parallel equally charged cylinders reveals that an increase in the free energy is seen when the ionic strength is decreased. The free energy decreases as a function of the cylinders separation distance. On the contrary for two oppositely charged cylinders, the free energy increases with increasing cylinder separation distance, while for two cylinders with different charged density it shows nonmonotonic variation with the increasing cylinders separation distance.

Fractional Poisson-Nernst-Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson-Nernst-Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker-Planck equation. Then, it is generalized to the macroscopic fractional Poisson-Nernst-Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.

Charged Substrate and Product Together Contribute Like a Nonreactive Species to the Overall Electrostatic Steering in Diffusion-Reaction Processes

The Debye-Hückel limiting law is used to study the binding kinetics of substrate-enzyme system as well as to estimate the reaction rate of a electrostatically steered diffusion-controlled reaction process. It is based on a linearized Poisson-Boltzmann model and known for its accurate predictions in dilute solutions. However, the substrate and product particles are in nonequilibrium states and are possibly charged, and their contributions to the total electrostatic field cannot be explicitly studied in the Poisson-Boltzmann model. Hence the influences of substrate and product on reaction rate coefficient were not known. In this work, we consider all the charged species, including the charged substrate, product, and mobile salt ions in a Poisson-Nernst-Planck model, and then compare the results with previous work. The results indicate that both the charged substrate and product can significantly influence the reaction rate coefficient with different behaviors under different setups of computational conditions. It is interesting to find that when substrate and product are both considered, under an overall neutral boundary condition for all the bulk charged species, the computed reaction rate kinetics recovers a similar Debye-Hückel limiting law again. This phenomenon implies that the charged product counteracts the influence of charged substrate on reaction rate coefficient. Our analysis discloses the fact that the total charge concentration of substrate and product, though in a nonequilibrium state individually, obeys an equilibrium Boltzmann distribution, and therefore contributes as a normal charged ion species to ionic strength. This explains why the Debye-Hückel limiting law still works in a considerable range of conditions even though the effects of charged substrate and product particles are not specifically and explicitly considered in the theory.

Nonlocal Poisson-Fermi model for ionic solvent

We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-like kernel function. The Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.

Charge Central Interpretation of the Full Nonlinear PB Equation: Implications for Accurate and Scalable Modeling of Solvation Interactions

Continuum solvation modeling based upon the Poisson-Boltzmann equation (PBE) is widely used in structural and functional analysis of biomolecules. In this work, we propose a charge-central interpretation of the full nonlinear PBE electrostatic interactions. The validity of the charge-central view or simply charge view, as formulated as a vacuum Poisson equation with effective charges, was first demonstrated by reproducing both electrostatic potentials and energies from the original solvated full nonlinear PBE. There are at least two benefits when the charge-central framework is applied. First the convergence analyses show that the use of polarization charges allows a much faster converging numerical procedure for electrostatic energy and forces calculation for the full nonlinear PBE. Second, the formulation of the solvated electrostatic interactions as effective charges in vacuum allows scalable algorithms to be deployed for large biomolecular systems. Here, we exploited the charge-view interpretation and developed a particle-particle particle-mesh (P3M) strategy for the full nonlinear PBE systems. We also studied the accuracy and convergence of solvation forces with the charge-view and the P3M methods. It is interesting to note that the convergence of both the charge-view and the P3M methods is more rapid than the original full nonlinear PBE method. Given the developments and validations documented here, we are working to adapt the P3M treatment of the full nonlinear PBE model to molecular dynamics simulations.

MEAN-FIELD THEORY AND COMPUTATION OF ELECTROSTATICS WITH IONIC CONCENTRATION DEPENDENT DIELECTRICS

We construct a mean-field variational model to study how the dependence of dielectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and the ionic ideal-gas entropy. The electrostatic potential is determined by Poisson's equation in which the dielectric coefficient depends on the sum of concentrations of individual ionic species. This dependence is assumed to be qualitatively the same as that on the salt concentration for which experimental data are available and analytical forms can be obtained by the data fitting. We derive the first and second variations of the free-energy functional, obtain the generalized Boltzmann distributions, and show that the free-energy functional is in general nonconvex. To validate our mathematical analysis, we numerically minimize our electrostatic free-energy functional for a radially symmetric charged system. Our extensive computations reveal several features that are significantly different from a system modeled with a dielectric coefficient independent of ionic concentration. These include the non-monotonicity of ionic concentrations, the ionic depletion near a charged surface that has been previously predicted by a one-dimensional model, and the enhancement of such depletion due to the increase of surface charges or bulk ionic concentrations.

Effect of thermodiffusion on pH-regulated surface charge properties of nanoparticle

Surface properties of nanoparticle are of high importance in the field of biotechnology, drug delivery and micro/nanofabrication. In this article, we developed a comprehensive theoretical model and subsequently solved that numerically to study the effect of thermodiffusion of ions on surface charge properties of nanoparticle. The theoretical study has been done considering silica nanoparticle for two aqueous solutions NaCl and KCl. The effect of solution pH in conjunction with nanoparticle temperature on surface charge density has been obtained for different salt concentrations (1, 10 and 100 mM) and nanoparticle size (diameter of 2 and 100 nm). It is observed from the results that with increasing temperature of the nanoparticle, the negative surface charge density gets higher due to increasing thermodiffusion effect. It is also found out that the magnitude of surface charge density is higher for KCl solution than NaCl solution under same condition which is attributed mostly due to less thermodiffusion of counterions for KCl than NaCl. Present study also shows that magnitude of surface charge density decreases with increasing nanoparticle size until it reaches a limiting value (called critical size) above which the effect of nanoparticle size on surface charge density is insignificant.

Drastic alteration of diffusioosmosis due to steric effects

We demonstrate essential quantitative and qualitative distinctions between the steric effects on classical electrokinetic phenomena like electroosmosis and on diffusioosmosis.

Influence of chemical microstructure of single-ion polymeric electrolyte membranes on performance of lithium-ion batteries

A novel protocol to generate and control porosity in polymeric structures is presented for fabrication of single ion polymer electrolyte (SIPE) membranes for lithium ion batteries. A series of SIPEs with varying ratios of aliphatic and aromatic segments was successfully synthesized and subsequently blended with PVDF-HFP to fabricate membranes of various sizes of pores. The membranes were characterized using techniques including SEM, solvent uptake capacity measurement and ionic conductivity. We demonstrate that appropriate membrane porosity enhances ionic conductivity, reduces interfacial resistance between electrodes and electrolyte and ultimately boosts performance of Li-ion batteries. The implication of the structure-performance relationship for battery design is discussed.

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Recent progress in adapting Poisson–Boltzmann methods to molecular simulations

Electrostatic solvation modeling based upon the Poisson–Boltzmann equation is widely used in studies of biomolecular structures and functions. This manuscript provides a thorough review of published efforts to adapt the numerical Poisson–Boltzmann methods to molecular simulations so that these methods can be extended to biomolecular studies involving conformational fluctuation and/or dynamics. We first review the fundamental works on how to define the electrostatic free energy and the Maxwell stress tensor. These topics are followed by three different strategies in developing algorithms to compute electrostatic forces and how to improve their numerical performance. Finally procedures are also presented in detail on how to discretize these algorithms for numerical calculations. Given the pioneer works reviewed here, further developmental efforts will be on how to balance efficiency and accuracy in these theoretical sound approaches — two important issues in applying any numerical algorithms for routine biomolecular applications. Even if not reviewed here, more advanced numerical solvers are certainly necessary to achieve higher accuracy than the widely used classical methods to improve the overall performance of the numerical Poisson–Boltzmann methods.

Electrostatic forces in the Poisson-Boltzmann systems

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

Electrodiffusion kinetics of ionic transport in a simple membrane channel

We employ numerical techniques for solving time-dependent full Poisson-Nernst-Planck (PNP) equations in 2D to analyze transient behavior of a simple ion channel subject to a sudden electric potential jump across the membrane (voltage clamp). Calculated spatiotemporal profiles of the ionic concentrations and electric potential show that two principal exponential processes can be distinguished in the electrodiffusion kinetics, in agreement with original Planck's predictions. The initial fast process corresponds to the dielectric relaxation, while the steady state is approached in a second slower exponential process attributed to the nonlinear ionic redistribution. Effects of the model parameters such as the channel length, height of the potential step, boundary concentrations, permittivity of the channel interior, and ionic mobilities on electrodiffusion kinetics are studied. Numerical solutions are used to determine spatiotemporal profiles of the electric field, ionic fluxes, and both the conductive and displacement currents. We demonstrate that the displacement current is a significant transient component of the total electric current through the channel. The presented results provide additional information about the classical voltage-clamp problem and offer further physical insights into the mechanism of electrodiffusion. The used numerical approach can be readily extended to multi-ionic models with a more structured domain geometry in 2D or 3D, and it is directly applicable to other systems, such as synthetic nanopores, nanofluidic channels, and nanopipettes.

Exploring accurate Poisson-Boltzmann methods for biomolecular simulations

Accurate and efficient treatment of electrostatics is a crucial step in computational analyses of biomolecular structures and dynamics. In this study, we have explored a second-order finite-difference numerical method to solve the widely used Poisson-Boltzmann equation for electrostatic analyses of realistic bio-molecules. The so-called immersed interface method was first validated and found to be consistent with the classical weighted harmonic averaging method for a diversified set of test biomolecules. The numerical accuracy and convergence behaviors of the new method were next analyzed in its computation of numerical reaction field grid potentials, energies, and atomic solvation forces. Overall similar convergence behaviors were observed as those by the classical method. Interestingly, the new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, though the numerical advantage of the new method is reduced when grid potentials are extrapolated to the molecular surface. Our exploratory study indicates the need for further improving interpolation/extrapolation schemes in addition to the developments of higher-order numerical methods that have attracted most attention in the field.

Ionic Size Effects: Generalized Boltzmann Distributions, Counterion Stratification, and Modified Debye Length

Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Recent studies by a variational mean-field approach that includes ionic size effects and by Monte Carlo simulations both suggest that the counterion stratification is determined by the ionic valence-to-volume ratios. Central in the mean-field approach is a free-energy functional of ionic concentrations in which the ionic size effects are included through the entropic effect of solvent molecules. The corresponding equilibrium conditions define the generalized Boltzmann distributions relating the ionic concentrations to the electrostatic potential. This paper presents a detailed analysis and numerical calculations of such a free-energy functional to understand the dependence of the ionic charge density on the electrostatic potential through the generalized Boltzmann distributions, the role of ionic valence-to-volume ratios in the counterion stratification, and the modification of Debye length due to the effect of ionic sizes.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of the bio-ions Na(+), K(+), Ca(2+), and Cl(-) as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

A molecular level prototype for mechanoelectrical transducer in mammalian hair cells

The mechanoelectrical transducer (MET) is a crucial component of mammalian auditory system. The gating mechanism of the MET channel remains a puzzling issue, though there are many speculations, due to the lack of essential molecular building blocks. To understand the working principle of mammalian MET, we propose a molecular level prototype which constitutes a charged blocker, a realistic ion channel and its surrounding membrane. To validate the proposed prototype, we make use of a well-established ion channel theory, the Poisson-Nernst-Planck equations, for three-dimensional (3D) numerical simulations. A wide variety of model parameters, including bulk ion concentration, applied external voltage, blocker charge and blocker displacement, are explored to understand the basic function of the proposed MET prototype. We show that our prototype prediction of channel open probability in response to blocker relative displacement is in remarkable accordance with experimental observation of rat cochlea outer hair cells. Our results appear to suggest that tip links which connect hair bundles gate MET channels.

Progress in developing Poisson-Boltzmann equation solvers

This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects.

Ionic Interactions Are Everywhere

Ionic solutions are dominated by interactions because they must be electrically neutral, but classical theory assumes no interactions. Biological solutions are rather like seawater, concentrated enough so that the diameter of ions also produces important interactions. In my view, the theory of complex fluids is needed to deal with the interacting reality of biological solutions.

Exploring a charge-central strategy in the solution of Poisson's equation for biomolecular applications

Continuum solvent treatments based on the Poisson-Boltzmann equation have been widely accepted for energetic analysis of biomolecular systems. In these approaches, the molecular solute is treated as a low dielectric region and the solvent is treated as a high dielectric continuum. The existence of a sharp dielectric jump at the solute-solvent interface poses a challenge to model the solvation energetics accurately with such a simple mathematical model. In this study, we explored and evaluated a strategy based on the "induced surface charge" to eliminate the dielectric jump within the finite-difference discretization scheme. In addition to the use of the induced surface charges in solving the equation, the second-order accurate immersed interface method is also incorporated to discretize the equation. The resultant linear system is solved with the GMRES algorithm to explicitly impose the flux conservation condition across the solvent-solute interface. The new strategy was evaluated on both analytical and realistic biomolecular systems. The numerical tests demonstrate the feasibility of utilizing induced surface charge in the finite-difference solution of the Poisson-Boltzmann equation. The analysis data further show that the strategy is consistent with theory and the classical finite-difference method on the tested systems. Limitations of the current implementations and further improvements are also analyzed and discussed to fully bring out its potential of achieving higher numerical accuracy.

Competitive adsorption and ordered packing of counterions near highly charged surfaces: From mean-field theory to Monte Carlo simulations

Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect-included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both the mean-field theory and the MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for further studies on the ionic size effect with application to large-scale biomolecular modeling.

Lateral dynamics of charged lipids and peripheral proteins in spatially heterogeneous membranes: comparison of continuous and Monte Carlo approaches

Biological membranes are complex environments whose physico-chemical properties are of utmost importance for the understanding of many crucial biological processes. Much attention has been given in the literature to the description of membranes along the z-axis perpendicular to the membrane. Here, we instead consider the lateral dynamics of lipids and peripheral proteins due to their electrostatic interaction. Previously, we constructed a Monte Carlo automaton capable of simulating mutual diffusive dynamics of charged lipids and associated positively charged peptides. Here, we derive and numerically analyze a system of Poisson-Boltzmann-Nernst-Planck (PBNP) equations that provide a mean-field approximation compatible with our Monte Carlo model. The thorough comparison between the mean-field PBNP equations and Monte Carlo simulations demonstrates that both the approaches are in a good qualitative agreement in all tested scenarios. We find that the two methods quantitatively deviate when the local charge density is high, presumably because the Poisson-Boltzmann formalism is applicable in the so-called weak coupling limit, whose validity is restricted to low charge densities. Nevertheless, we conclude that the mean-field PBNP approach provides a good approximation for the considerably more detailed Monte Carlo model at only a fraction of the associated computational cost and allows simulation of the membrane lateral dynamics on the space and time scales relevant for the realistic biological problems.