Comparison of Dynamic Lattice Monte Carlo Simulations and the Dielectric Self-Energy Poisson−Nernst−Planck Continuum Theory for Model Ion Channels

A detailed study of ion transport through the SARS-CoV-2 E protein ion channel

The envelope (E) protein encoded in the genome of an RNA virus is crucial for the replication, budding and pathophysiology of the virus. In the light of the ongoing pandemic, we explored similarities/differences between SARS-CoV-1 and SARS-CoV-2 E protein ion channels in terms of their selectivity. Further, we also examined the impact of variation of the bath concentration and introduction of potential and concentration gradients across the channel on the binding ratios of sodium and chloride ions for the SARS-CoV-2 E protein. Ion transport is described through the fourth-order Poisson-Nernst-Planck-Bikerman (4PNPBik) model which generalizes the traditional model by including ionic interactions between ions and their surrounding medium and non-ionic interactions between particles due to their finite size. Governing equations are solved numerically using the immersed boundary-lattice Boltzmann method (IB-LBM). The mathematical model has been validated by comparing analytical and experimental ion activity. The SARS-CoV-1 E protein ion channel is found to be more permeable to cationic ions, while the SARS-CoV-2 E protein has similar selectivity for both cationic and anionic species. For SARS-CoV-2, an increase in the bath concentration results in an increase in the binding ratio for sodium ions. Furthermore, the chloride binding ratio increases as the concentration gradient increases. A potential gradient has a minimal effect on the binding ratio. The SARS-CoV-2 E protein was found to support higher ionic currents than the SARS-CoV-1 E protein. Furthermore, the ionic current increased with increasing bath concentrations.

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.

Investigating ion transport inside the pentameric ion channel encoded in COVID-19 E protein

Ion flow inside an ion channel can be described through continuum based Born-Poisson-Nernst-Planck (BPNP) equations in conjunction with the Lennard-Jones potential. Keeping in mind the ongoing pandemic, in this study, an attempt has been made to understand the selectivity and the current voltage relation of the COVID-19 E protein pentameric ion channel. Two ionic species, namely Na^{+} and Cl^{-}, have been considered here. E protein is one of the smallest structural protein which is embedded in the outer membrane of the virus. Once the virus is inside the host cell, this protein is expressed abundantly and is responsible for activities such as replication and budding of the virus. In the literature, we can find a few experimental studies focusing on understanding the activity of the channel formed by E proteins of different viruses. Here, we attempt the same study for the COVID-19 E protein ion channel through mathematical modeling. The channel geometry is calculated from the protein data bank file which was provided by NARLabs, Taiwan, using the hole program. Further, it was used to obtain the charge distribution using the pdbtopqr online program. The immersed boundary-lattice Boltzmann method (IB-LBM) has been implemented to numerically solve the system of equations in the channel generated by the protein data bank file. Further, an in-house code which operates on multiple GPUs and uses the cuda platform has been developed to achieve the goal of performing the current investigation.

A Focus on Two Electrokinetics Issues

This review article intends to communicate the new understanding and viewpoints on two fundamental electrokinetics topics that have only become available recently. The first is on the holistic approach to the Poisson–Boltzmann equation that can account for the effects arising from the interaction between the mobile ions in the Debye layer and the surface charge. The second is on the physical picture of the inner electro-hydrodynamic flow field of an electrophoretic particle and its drag coefficient. For the first issue, the traditional Poisson–Boltzmann equation focuses only on the mobile ions in the Debye layer; effects such as charge regulation and the isoelectronic point arising from the interaction between the mobile ions in the Debye layer and the surface charge are left to supplemental measures. However, a holistic treatment is entirely possible in which the whole electrical double layer—the Debye layer and the surface charge—is treated consistently from the beginning. While the derived form of the Poisson–Boltzmann equation remains unchanged, the zeta potential boundary condition becomes a calculated quantity that can reflect the various effects due to the interaction between the surface charges and the mobile ions in the liquid. The second issue, regarding the drag coefficient of a spherical electrophoretic particle, has existed ever since the breakthrough by Smoluchowski a century ago that linked the zeta potential of the particle to its mobility. Due to the highly nonlinear mathematics involved in the electro-hydrodynamics inside the Debye layer, there has been a lack of an exact solution for the electrophoretic flow field. Recent numerical simulation results show that the flow field comprises an inner region and an outer region, separated by a rather sharp interface. As the inner flow field is carried along by the particle, the measured drag is that at the inner/outer interface rather than at the solid/liquid interface. This identification and its associated physical picture of the inner flow field resolves a long-standing puzzle regarding the electrophoretic drag coefficient.

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.

Separation Selectivity of CH4/CO2 Gas Mixtures in the ZIF-8 Membrane Explored by Dynamic Monte Carlo Simulations

Here we report a series of nonequilibrium dynamic Monte Carlo simulations combined with dual control volume (DCV-DMC) to explore the separation selectivity of CH₄/CO₂ gas mixtures in the ZIF-8 membrane with a thickness of up to about 20 nm. Meanwhile, an improved DCV-DMC approach coupled with the corresponding potential map (PM-DCV-DMC) is further developed to speed up the computational efficiency of conventional DCV-DMC simulations. Our simulation results provide the molecular-level density and selectivity profiles along the permeation direction of both CH₄ and CO₂ molecules in the ZIF-8 membrane, indicating that the parts near membrane surfaces at both ends play a key role in determining the separation selectivity. All densities initially show a sharp increase in the individual maximum within the first outermost unit cell at the feed side and follow a long fluctuating decrease process. Accordingly, the corresponding selectivity profiles initially display a long fluctuating increase in the individual maximum and follow a sharp decrease near the membrane surface at the permeation side. Furthermore, the effects of feed composition, temperature, and pressure on the relevant separation selectivity are also discussed in detail, where the temperature has a greater influence on the separation selectivity than the feed composition and pressure. More importantly, the predicted separation selectivities from our PM-DCV-DMC simulations are well consistent with previous experimental results.

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.

Anomalies in the equilibrium and nonequilibrium properties of correlated ions in complex molecular environments

Emergent statistical attributes, and therefore the equations of state, of an assembly of interacting charge carriers embedded within a complex molecular environment frequently exhibit a variety of anomalies, particularly in the high-density (equivalently, the concentration) regime, which are not well understood, because they do not fall under the low-concentration phenomenologies of Debye-Hückel-Onsager and Poisson-Nernst-Planck, including their variants. To go beyond, we here use physical concepts and mathematical tools from quantum scattering theory, transport theory with the Stosszahlansatz of Boltzmann, and classical electrodynamics (Lorentz gauge) and obtain analytical expressions both for the average and the frequency-wave vector-dependent longitudinal and transverse current densities, diffusion coefficient, and the charge density, and therefore the analytical expressions for (a) the chemical potential, activity coefficient, and the equivalent conductivity for strong electrolytes and (b) the current-voltage characteristics for ion-transport processes in complex molecular environments. Using a method analogous to the notion of Debye length and thence the electrical double layer, we here identify a pair of characteristic length scales (longitudinal and the transverse), which, being wave vector and frequency dependent, manifestly exhibit nontrivial fluctuations in space-time. As a unifying theme, we advance a quantity (inverse length dimension), g_{scat}^{(a)}, which embodies all dynamical interactions, through various quantum scattering lengths, relevant to molecular species a, and the analytical behavior which helps us to rationalize the properties of strong electrolytes, including anomalies, in all concentration regimes. As an example, the behavior of g_{scat}^{(a)} in the high-concentration regime explains the anomalous increase of the Debye length with concentration, as seen in a recent experiment on electrolyte solutions. We also put forth an extension of the standard diffusion equation, which manifestly incorporates the effects arising from the underlying microscopic collisions among constituent molecular species. Furthermore, we show a nontrivial connection between the current-voltage characteristics of electrolyte solutions and the Landauer's approach to electrical conduction in mesoscopic solids and thereby establish a definite conceptual bridge between the two disjoint subjects. For numerical insight, we present results on the aqueous solution of KCl as an example of strong electrolyte, and the transport (conduction as well as diffusion) of K^{+} ions in water, as an example of ion transport across the voltage-gated channels in biological cells.

Membrane Position Dependency of the pKa and Conductivity of the Protein Ion Channel

The dependency of current-voltage characteristics of the α-hemolysin channel on the channel position within the membrane was studied using Poisson-Nernst-Planck theory of ion conductivity with soft repulsion between mobile ions and protein atoms (SP-PNP). The presence of the membrane environment also influences the protonation state of the residues at the boundary of the water-lipid interface. In this work, we predict that Asp and Lys residues at the protein rim change their protonation state upon penetration to the lipid environment. Free energies of protein insertion in the membrane for different penetration depths were estimated using the Poisson-Boltzmann/solvent-accessible surface area (PB/SASA) model. The results show that rectification and reversal potentials are very sensitive to the relative position of channel in the membrane, which in turn contributes to alternative protonation states of lipid-penetrating ionizable groups. The prediction of channel position based on the matching of calculated rectification with experimentally determined rectification is in good agreement with recent neutron reflection experiments. Based on the results, we conclude that α-hemolysin membrane position is determined by a combination of factors and not only by the pattern of the surface hydrophobicity as is typically assumed.

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.

Computational simulations of asymmetric fluxes of large molecules through gap junction channel pores

Gap junction channels are formed out of connexin isoforms, which enable molecule and ion selective diffusion amongst neighboring cells. HeLa cells expressing distinct connexins (Cx) allow the formation of heterotypic channels, where we observed a molecular charge-independent preferential flux of large fluorescent molecules in the Cx45 to Cx43 direction. We hypothesize that the pore's shape is a significant factor along-side charge and transjunctional voltages for this asymmetric flux. To test this hypothesis, we developed a 3D computational model simulating Brownian diffusion of large molecules in a gap junction channel pore. The basic pore contour was derived from x-ray crystallographic structures of Cx43 and Cx26 and approximated using basic geometric shapes. Lucifer yellow dye molecules and cesium counter-ions were modeled as spheres using their respective Stokes radii. Our simulation results from simple diffusion and constant concentration gradient experiments showed that only charged particles yield asymmetric fluxes in heterotypic pores. While increasing the inner mouth size resulted in a near-quadratic rise in flux, the rise was asymptotic for outer mouth radii increase. Probability maps and average force per particle per pore section explain the asymmetric flux with variation in pore shape. Furthermore, the simulation results are in agreement with our in vitro experimental results with HeLa cells in Cx43-Cx45 heterotypic configurations. The presence of asymmetric fluxes can help us to understand effects of the molecular structure of the pore and predict potential differences in vivo.

Self-Optimized Biological Channels in Facilitating the Transmembrane Movement of Charged Molecules

We consider an anisotropically two-dimensional diffusion of a charged molecule (particle) through a large biological channel under an external voltage. The channel is modeled as a cylinder of three structure parameters: radius, length, and surface density of negative charges located at the channel interior-lining. These charges induce inside the channel a potential that plays a key role in controlling the particle current through the channel. It was shown that to facilitate the transmembrane particle movement the channel should be reasonably self-optimized so that its potential coincides with the resonant one, resulting in a large particle current across the channel. Observed facilitation appears to be an intrinsic property of biological channels, regardless of the external voltage or the particle concentration gradient. This facilitation is very selective in the sense that a channel of definite structure parameters can facilitate the transmembrane movement of only particles of proper valence at corresponding temperatures. Calculations also show that the modeled channel is nonohmic with the ion conductance which exhibits a resonance at the same channel potential as that identified in the current.

A comprehensive QSPR model for dielectric constants of binary solvent mixtures

The dielectric constant is a key physicochemical property in solubility, chemical equilibrium and the synthesis of compounds in pharmaceutical/chemical sciences. In this context, a quantitative structure-property relationship (QSPR) model was designed from 3207 binary solvent mixtures by using 23 calculated experimental-theoretical descriptors including solvent fractions (f₁ and f₂), individual dielectric constants of solvents (dc₁ and dc₂), temperature, and Abraham and Hansen solvation parameters. The QSPR model was developed using a genetic algorithm based multiple linear regression (GA-MLR) and robust regression. Jackknifing was implemented for internal-external validation of the selected descriptors by GA containing f₁, f₂, dc₁ and dc₂. Implementation of jackknifing on the selected descriptors revealed that p values were close to zero. Consequently, the significance of selected descriptors was confirmed through the sign change point of view and their validity was verified. The model was evaluated using the r² and Q²_(F3) parameters as criteria of model prediction ability. The r² values were equal to 0.925 and 0.922, and Q²_(F3) were reported as 0.873 and 0.862 for the cross-validation and prediction steps, respectively. Finally, model performance was clearly acceptable to anticipate the modelling of dielectric constants for a wide range of binary solvent mixtures.

Numerical methods for a Poisson-Nernst-Planck-Fermi model of biological ion channels

Numerical methods are proposed for an advanced Poisson-Nernst-Planck-Fermi (PNPF) model for studying ion transport through biological ion channels. PNPF contains many more correlations than most models and simulations of channels, because it includes water and calculates dielectric properties consistently as outputs. This model accounts for the steric effect of ions and water molecules with different sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of polarized water molecules in an inhomogeneous aqueous electrolyte. The steric energy is shown to be comparable to the electrical energy under physiological conditions, demonstrating the crucial role of the excluded volume of particles and the voids in the natural function of channel proteins. Water is shown to play a critical role in both correlation and steric effects in the model. We extend the classical Scharfetter-Gummel (SG) method for semiconductor devices to include the steric potential for ion channels, which is a fundamental physical property not present in semiconductors. Together with a simplified matched interface and boundary (SMIB) method for treating molecular surfaces and singular charges of channel proteins, the extended SG method is shown to exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. The generalized SG stability condition shows why the standard discretization (without SG exponential fitting) of NP equations may fail and that divalent Ca(2+) may cause more unstable discrete Ca(2+) fluxes than that of monovalent Na(+). Two different methods-called the SMIB and multiscale methods-are proposed for two different types of channels, namely, the gramicidin A channel and an L-type calcium channel, depending on whether water is allowed to pass through the channel. Numerical methods are first validated with constructed models whose exact solutions are known. The experimental data of both channels are then used to verify and explain novel features of PNPF as compared with previous PNP models. The PNPF currents are in accord with the experimental I-V (V for applied voltages) data of the gramicidin A channel and I-C (C for bath concentrations) data of the calcium channel with 10(-8)-fold bath concentrations that pose severe challenges in theoretical simulations.

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.

Control of flux by narrow passages and hidden targets in cellular biology

Critical biological processes, such as synaptic plasticity and transmission, activation of genes by transcription factors, or double-strained DNA break repair, are controlled by diffusion in structures that have both large and small spatial scales. These may be small binding sites inside or on the surface of the cell, or narrow passages between subcellular compartments. The great disparity in spatial scales is the key to controlling cell function by structure. We report here recent progress on resolving analytical and numerical difficulties in extracting properties from experimental data, from biophysical models, and from Brownian dynamics simulations of diffusion in multi-scale structures. This progress is achieved by developing an analytical approximation methodology for solving the model equations. The reported results are applied to analysis and simulations of subcellular processes and to the quantification of their biological functions.

One-dimensional transport of interacting particles: currents, density profiles, phase diagrams, and symmetries

Driven lattice gases serve as canonical models for investigating collective transport phenomena and properties of nonequilibrium steady states. Here we study one-dimensional transport with nearest-neighbor interactions both in closed bulk systems and in open channels coupled to two particle reservoirs at the ends of the channel. For the widely employed Glauber rates we derive an exact current-density relation in the bulk for unidirectional hopping. An approach based on time-dependent density functional theory provides a good description of the kinetics. For open systems, the system-reservoir couplings are shown to have a striking influence on boundary-induced phase diagrams. The role of particle-hole symmetry is discussed, and its consequence for the topology of the phase diagrams. It is furthermore demonstrated that systems with weak bias can be mapped onto systems with unidirectional hopping.

Detecting and characterizing individual molecules with single nanopores

Single-nanometer-scale pores have demonstrated the capability for the detection, identification, and characterization of individual molecules. This measurement method could soon extend the existing commercial instrumentation or provide solutions to niche applications in many fields, including health care and the basic sciences. However, that paradigm shift requires a significantly better understanding of the physics and chemistry that govern the interactions between nanopores and analytes. We describe herein some of our methods and approaches to address this issue.

Web interface for Brownian dynamics simulation of ion transport and its applications to beta-barrel pores

Brownian dynamics (BD) based on accurate potential of mean force is an efficient and accurate method for simulating ion transport through wide ion channels. Here, a web-based graphical user interface (GUI) is presented for carrying out grand canonical Monte Carlo (GCMC) BD simulations of channel proteins: http://www.charmm-gui.org/input/gcmcbd. The webserver is designed to help users avoid most of the technical difficulties and issues encountered in setting up and simulating complex pore systems. GCMC/BD simulation results for three proteins, the voltage dependent anion channel (VDAC), α-Hemolysin (α-HL), and the protective antigen pore of the anthrax toxin (PA), are presented to illustrate the system setup, input preparation, and typical output (conductance, ion density profile, ion selectivity, and ion asymmetry). Two models for the input diffusion constants for potassium and chloride ions in the pore are compared: scaling of the bulk diffusion constants by 0.5, as deduced from previous all-atom molecular dynamics simulations of VDAC, and a hydrodynamics based model (HD) of diffusion through a tube. The HD model yields excellent agreement with experimental conductances for VDAC and α-HL, while scaling bulk diffusion constants by 0.5 leads to underestimates of 10-20%. For PA, simulated ion conduction values overestimate experimental values by a factor of 1.5-7 (depending on His protonation state and the transmembrane potential), implying that the currently available computational model of this protein requires further structural refinement.

Variational multiscale models for charge transport

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field.

Voltage dependence of a stochastic model of activation of an alpha helical S4 sensor in a K channel membrane

The voltage dependence of the ionic and gating currents of a K channel is dependent on the activation barriers of a voltage sensor with a potential function which may be derived from the principal electrostatic forces on an S4 segment in an inhomogeneous dielectric medium. By variation of the parameters of a voltage-sensing domain model, consistent with x-ray structures and biophysical data, the lowest frequency of the survival probability of each stationary state derived from a solution of the Smoluchowski equation provides a good fit to the voltage dependence of the slowest time constant of the ionic current in a depolarized membrane, and the gating current exhibits a rising phase that precedes an exponential relaxation. For each depolarizing potential, the calculated time dependence of the survival probabilities of the closed states of an alpha helical S4 sensor are in accord with an empirical model of the ionic and gating currents recorded during the activation process.

A method for treating the passage of a charged hard sphere ion as it passes through a sharp dielectric boundary

In the implicit solvent models of electrolytes (such as the primitive model (PM)), the ions are modeled as point charges in the centers of spheres (hard spheres in the case of the PM). The surfaces of the spheres are not polarizable which makes these models appropriate to use in computer simulations of electrolyte systems where these ions do not leave their host dielectrics. The same assumption makes them inappropriate in simulations where these ions cross dielectric boundaries because the interaction energy of the point charge with the polarization charge induced on the dielectric boundary diverges. In this paper, we propose a procedure to treat the passage of such ions through dielectric interfaces with an interpolation method. Inspired by the "bubble ion" model (in which the ion's surface is polarizable), we define a space-dependent effective dielectric coefficient, ε(eff)(r), for the ion that overlaps with the dielectric boundary. Then, we replace the "bubble ion" with a point charge that has an effective charge q/ε(eff)(r) and remove the portion of the dielectric boundary where the ion overlaps with it. We implement the interpolation procedure using the induced charge computation method [D. Boda, D. Gillespie, W. Nonner, D. Henderson, and B. Eisenberg, Phys. Rev. E 69, 046702 (2004)]. We analyze the various energy terms using a spherical ion passing through an infinite flat dielectric boundary as an example.

Insights on the permeability of wide protein channels: measurement and interpretation of ion selectivity

Ion channels are hollow proteins that have evolved to exhibit discrimination between charged solutes. This property, known as ion selectivity is critical for several biological functions. By using the bacterial porin OmpF as a model system of wide protein channels, we demonstrate that significant insights can be gained when selectivity measurements are combined with electrodiffusion continuum models and simulations based on the atomic structure. A correct interpretation of the mechanisms ruling the many sources of channel discrimination is a first, indispensable step for the understanding of the controlled movement of ions into or out of cells characteristic of many physiological processes. We conclude that the scattered information gathered from several independent approaches should be appropriately merged to provide a unified and coherent picture of the channel selectivity.

Molecular dynamics and brownian dynamics investigation of ion permeation and anesthetic halothane effects on a proton-gated ion channel

Bacterial Gloeobacter violaceus pentameric ligand-gated ion channel (GLIC) is activated to cation permeation upon lowering the solution pH. Its function can be modulated by anesthetic halothane. In the present work, we integrate molecular dynamics (MD) and Brownian dynamics (BD) simulations to elucidate the ion conduction, charge selectivity, and halothane modulation mechanisms in GLIC, based on recently resolved X-ray crystal structures of the open-channel GLIC. MD calculations of the potential of mean force (PMF) for a Na(+) revealed two energy barriers in the extracellular domain (R109 and K38) and at the hydrophobic gate of transmembrane domain (I233), respectively. An energy well for Na(+) was near the intracellular entrance: the depth of this energy well was modulated strongly by the protonation state of E222. The energy barrier for Cl(-) was found to be 3-4 times higher than that for Na(+). Ion permeation characteristics were determined through BD simulations using a hybrid MD/continuum electrostatics approach to evaluate the energy profiles governing the ion movement. The resultant channel conductance and a near-zero permeability ratio (P(Cl)/P(Na)) were comparable to experimental data. On the basis of these calculations, we suggest that a ring of five E222 residues may act as an electrostatic gate. In addition, the hydrophobic gate region may play a role in charge selectivity due to a higher dehydration energy barrier for Cl(-) ions. The effect of halothane on the Na(+) PMF was also evaluated. Halothane was found to perturb salt bridges in GLIC that may be crucial for channel gating and open-channel stability, but had no significant impact on the single ion PMF profiles.

Soft wall ion channel in continuum representation with application to modeling ion currents in α-hemolysin

A soft repulsion (SR) model of short-range interactions between mobile ions and protein atoms is introduced in the framework of continuum representation of the protein and solvent. The Poisson-Nernst-Plank (PNP) theory of ion transport through biological channels is modified to incorporate this soft wall protein model. Two sets of SR parameters are introduced. The first is parametrized for all essential amino acid residues using all atom molecular dynamic simulations; the second is a truncated Lennard-Jones potential. We have further designed an energy-based algorithm for the determination of the ion accessible volume, which is appropriate for a particular system discretization. The effects of these models of short-range interactions were tested by computing current-voltage characteristics of the α-hemolysin channel. The introduced SR potentials significantly improve prediction of channel selectivity. In addition, we studied the effect of the choice of some space-dependent diffusion coefficient distributions on the predicted current-voltage properties. We conclude that the diffusion coefficient distributions largely affect total currents and have little effect on rectifications, selectivity, or reversal potential. The PNP-SR algorithm is implemented in a new efficient parallel Poisson, Poisson-Boltzmann, and PNP equation solver, also incorporated in a graphical molecular modeling package HARLEM.

Discrete-state representation of ion permeation coupled to fast gating in a model of ClC chloride channels: comparison to multi-ion continuous space Brownian dynamics simulations

A discrete-state model of chloride ion motion in a ClC chloride channel is constructed, following a previously developed multi-ion continuous space model of the same system (Cheng, M. H.; Mamonov, A. B.; Dukes, J. W.; Coalson, R. D. J. Phys. Chem. B 2007, 111, 5956) that included a simplistic representation of the fast gate in this channel. The reducibility of the many-body continuous space to the eight discrete-state model considered in the present work is examined in detail by performing three-dimensional Brownian dynamics simulations of each allowed state-to-state transition in order to extract the appropriate rate constant for this process, and then inserting the pairwise rate constants thereby obtained into an appropriate set of kinetic master equations. Experimental properties of interest, including the rate of Cl(-) ion permeation through the open channel and the average rate of closing of the fast gate as a function of bulk Cl(-) ion concentrations in the intracellular and extracellular electrolyte reservoirs are computed. Good agreement is found between the results obtained via the eight discrete-state model versus the multi-ion continuous space model, thereby encouraging continued development of the discrete-state model to include more complex behaviors observed experimentally in these channels.

Nonlinear hopping transport in ring systems and open channels

We study the nonlinear hopping transport in one-dimensional rings and open channels. Analytical results are derived for the stationary current response to a constant bias without assuming any specific coupling of the rates to the external fields. It is shown that anomalous large effective jump lengths, as observed in recent experiments by taking the ratio of the third-order nonlinear and the linear conductivity, can occur already in ordered systems. Rectification effects due to site energy disorder in ring systems are expected to become irrelevant for large system sizes. In open channels, in contrast, rectification effects occur already for disorder in the jump barriers and do not vanish in the thermodynamic limit. Numerical solutions for a sinusoidal bias show that the ring system provides a good description for the transport behavior in the open channel for intermediate and high frequencies. For low frequencies temporal variations in the mean particle number have to be taken into account in the open channel, which cannot be captured in the more simple ring model.

The electrostatics of VDAC: implications for selectivity and gating

The voltage-dependent anion channel (VDAC) is the major pathway mediating the transfer of metabolites and ions across the mitochondrial outer membrane. Two hallmarks of the channel in the open state are high metabolite flux and anion selectivity, while the partially closed state blocks metabolites and is cation selective. Here we report the results from electrostatics calculations carried out on the recently determined high-resolution structure of murine VDAC1 (mVDAC1). Poisson-Boltzmann calculations show that the ion transfer free energy through the channel is favorable for anions, suggesting that mVDAC1 represents the open state. This claim is buttressed by Poisson-Nernst-Planck calculations that predict a high single-channel conductance indicative of the open state and an anion selectivity of 1.75--nearly a twofold selectivity for anions over cations. These calculations were repeated on mutant channels and gave selectivity changes in accord with experimental observations. We were then able to engineer an in silico mutant channel with three point mutations that converted mVDAC1 into a channel with a preference for cations. Finally, we investigated two proposals for how the channel gates between the open and the closed state. Both models involve the movement of the N-terminal helix, but neither motion produced the observed voltage sensitivity, nor did either model result in a cation-selective channel, which is observed experimentally. Thus, we were able to rule out certain models for channel gating, but the true motion has yet to be determined.

Discretization of the induced-charge boundary integral equation

Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

Molecular basis for cation selectivity in claudin-2-based paracellular pores: identification of an electrostatic interaction site

Paracellular ion transport in epithelia is mediated by pores formed by members of the claudin family. The degree of selectivity and the molecular mechanism of ion permeation through claudin pores are poorly understood. By expressing a high-conductance claudin isoform, claudin-2, in high-resistance Madin-Darby canine kidney cells under the control of an inducible promoter, we were able to quantitate claudin pore permeability. Claudin-2 pores were found to be narrow, fluid filled, and cation selective. Charge selectivity was mediated by the electrostatic interaction of partially dehydrated permeating cations with a negatively charged site within the pore that is formed by the side chain carboxyl group of aspartate-65. Thus, paracellular pores use intrapore electrostatic binding sites to achieve a high conductance with a high degree of charge selectivity.

Computational prediction of ion permeation characteristics in the glycine receptor modified by photo-sensitive compounds

We conduct computational analyses of ion permeation characteristics in a model glycine receptor (GlyR) modified by photo-sensitive compounds. In particular, we consider hypothetical attachment to the channel of charge-neutral chemical groups which can be photo-activated by shining light of an appropriate wavelength on the system. After illumination, the attached molecules become charged via a photodissociation process or excited into a charge-separated state (thus generating a significant electric dipole). We carry out Brownian Dynamics simulations of ion flow through the channel in the presence of the additional charges generated in this fashion. Based on these calculations, we predict that photo-activation of appropriately positioned photo-sensitive compounds near the channel mouth can significantly modify the rate of ion permeation and the current rectification ratio. Possible implications for GlyR-based device designs are briefly discussed.

Nanoscopic porous sensors

There are thousands of different nanometer-scale pores in biology, many of which act as sensors for specific chemical agents. Recent work suggests that protein and solid-state nanopores have many potential uses in a wide variety of analytical applications. In this review we survey this field of research and discuss the prospects for advances that could be made in the near future.

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

Kinetic lattice grand canonical Monte Carlo simulation for ion current calculations in a model ion channel system

An algorithm in which kinetic lattice grand canonical Monte Carlo simulations are combined with mean field theory (KLGCMC/MF) is presented to calculate ion currents in a model ion channel system. In this simulation, the relevant region of the system is treated by KLGCMC simulations, while the rest of the system is described by modified Poisson-Boltzmann mean field theory. Calculation of reaction field due to induced charges on the channel/water and membrane/water boundaries is carried out using a basis-set expansion method [Im and Roux, J. Chem. Phys. 115, 4850 (2001)]. Calculation of ion currents, electrostatic potentials, and ion concentrations, as obtained from the KLGCMC/MF simulations, shows good agreement with Poisson-Nernst-Planck (PNP) theory predictions when the channel and membrane have the same dielectric constant as water. If the channel and membrane have a lower dielectric constant than water, however, there is a considerable difference between the KLGCMC/MF and PNP predictions. This difference is attributed to the reaction field, which is missing in PNP theory. It is demonstrated that the reaction field as well as fixed charges in the channel play key roles in selective ion transport. Limitations and further development of the current KLGCMC/MF approach are also discussed.

An accurate and efficient empirical approach for calculating the dielectric self-energy and ion-ion pair potential in continuum models of biological ion channels

This paper presents empirical formulas for calculating the dielectric self-energy and ion-ion pair interactions in cylindrical ion channels. The proposed approach can be extended to more complex channel structures, for example, (i) a "straight" channel with variable radius and (ii) a "curved" channel with constant radius. For calibration purposes, we compare results obtained based on the approximate effective potentials developed herein to exact electrostatic calculations obtained via the algorithm of Graf et al.: the agreement is satisfactory. A dynamic lattice Monte Carlo (DLMC) technique is used to further assess the accuracy and efficiency of the proposed empirical potentials. The concentration profiles and current-voltage curves produced with our simple empirical energy formulas are in excellent agreement with numerical results obtained using the algorithm of Graf et al., which calculates all relevant electrostatic forces exactly. The use of effective ion-ion potentials greatly reduces the computer memory required to perform DLMC ion permeation simulations in dielectrically inhomogeneous environments, thus enabling treatment of larger systems than can be handled by numerically exact techniques.

Modeling the fast gating mechanism in the ClC-0 chloride channel

A simplified three-dimensional model ClC-0 chloride channel is constructed to couple the permeation of Cl- ions to the motion of a glutamate side chain that acts as the putative fast gate in the ClC-0 channel. The gate is treated as a single spherical particle attached by a rod to a pivot point. This particle moves in a one-dimensional arc under the influence of a bistable potential, which mimics the isomerization process by which the glutamate side chain moves from an open state (not blocking the channel pore) to a closed state (blocking the channel pore, at a position which also acts as a binding site for Cl- ions moving through the channel). A dynamic Monte Carlo (DMC) technique is utilized to perform Brownian dynamics simulations to investigate the dependence of the gate closing rate on both internal and external chloride concentration and the gate charge as well. To accelerate the simulation of gate closing to a time scale that can be accommodated with current methodology and computer power, namely, microseconds, parameters that govern the motion of the bare gate (i.e., in the absence of coupling to the permeating ions) are chosen appropriately. Our simulation results are in qualitative agreement with experimental observations and consistent with the "foot-in-the-door" mechanism (Chen et al. J. Gen. Physiol. 2003, 122, 641; Chen and Miller J. Gen. Physiol. 1996, 108, 237), although the absolute time scale of gate closing in the real channel is much longer (millisecond time scale). A simple model based on the fractional occupation probability of the Cl- binding site that is ultimately blocked by the fast gate suggests straightforward scalability of simulation results for the model channel considered herein to experimentally realistic time scales.

Simulation of biological ion channels with technology computer-aided design

Computer simulations of realistic ion channel structures have always been challenging and a subject of rigorous study. Simulations based on continuum electrostatics have proven to be computationally cheap and reasonably accurate in predicting a channel's behavior. In this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for KcsA channel and study its steady-state response. SILVACO is a well-established program, typically used by electrical engineers to simulate the process flow and electrical characteristics of solid-state devices. By employing this simulation program, we have presented an alternative computing platform for performing ion channel simulations, besides the known methods of writing codes in programming languages. With the ease of varying the different parameters in the channel's vestibule and the ability of incorporating surface charges, we have shown the wide-ranging possibilities of using a device simulator for ion channel simulations. Our simulated results closely agree with the experimental data, validating our model.

A rigid-body Newtonian propagation scheme based on instantaneous decomposition into rotation and translation blocks

The rotation and translation block (RTB) method of Durand et al. [Biopolymers 34, 759 (1994)] and Tama et al. [Proteins 41, 1 (2000)] provides an appealing way to calculate low-frequency normal modes of large biomolecules by restricting the space of motions to exclude internal motions of preselected rigid fragments within the molecule. These fragments are modeled essentially as rigid bodies and the need to calculate high-frequency relative motions of the atoms that form them is obviated in a natural way. Here we extend the RTB approach into a method for computing the classical (Newtonian) dynamics of a biomolecule, or any large molecule, with effective rigid-body constraints applied to a prechosen set of internal molecular fragments. This method, to be termed RTB dynamics, is easy to implement, conserves the total energy of the system, does not require the construction of the matrix of second spatial derivatives of the potential-energy function (Hessian matrix), and can be used to compute the classical dynamics of a system moving in an arbitrary anharmonic force field. An elementary numerical application to signal propagation in the small membrane-bound polypeptide gramicidin-A is presented for illustration purposes.