Electrodiffusion Model Research Articles

A quantitative electrodiffusion model of antibacterial heavy metals: Ion influx through hydrophilic pores

Excessive heavy-metal ions in the cytoplasm, being toxic ions, can inactivate bacteria, which has been widely used as an effective antibacterial method. Nevertheless, how such toxic ions can be transported into bacteria is poorly understood. Here, we discovered a new mechanism showing that toxic ions electro-diffuse into the cytoplasm through the inherent hydrophilic pores in the cell membrane of living bacteria. Transmembrane potential induces recoverable defects by enhancing the thermal fluctuation of phospholipid molecules. By integrating electroporation, surface potential, and cell osmotic pressure, we develop a simple model for the influx rate of heavy metal ions into living bacteria. Our modeling results agree well with the experimental ones on the survival rates of Gram-negative and Gram-positive bacteria.

Global Zero-relaxation Limit Problem of the Electro-diffusion Model Arising in Electro-Hydrodynamics

Global Zero-relaxation Limit Problem of the Electro-diffusion Model Arising in Electro-Hydrodynamics

HAZniCS – Software Components for Multiphysics Problems

We introduce the software toolbox HAZniCS for solving interface-coupled multiphysics problems. HAZniCS is a suite of modules that combines the well-known FEniCS framework for finite element discretization with solver and graph library HAZmath. The focus of this article is on the design and implementation of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics. The robustness and efficiency of the numerical algorithms and methods is shown in several numerical examples, namely the Darcy-Stokes equations that model the flow of cerebrospinal fluid in the human brain and the mixed-dimensional model of electrodiffusion in the brain tissue.

On the Global Well-Posedness and Analyticity of Some Electrodiffusion Models in Ideal Fluids and Porous Media

On the Global Well-Posedness and Analyticity of Some Electrodiffusion Models in Ideal Fluids and Porous Media

Multi-oligomeric states of alamethicin ion channel: Assemblies and conductance

Transmembrane assemblies of the peptaibol alamethicin (ALM) are among the most extensively studied ion channels not only because of their antimicrobial activity but also as models for channel structure and aggregation. In this study, several oligomeric states of ALM are investigated with molecular dynamics simulations to establish properties of the channel and obtain free energy profiles for ion transport and the corresponding values of conductance. The hexamer, heptamer, and octamer of ALM in phospholipid membrane are found to be stable but highly dynamic in barrel-stave structures, with calculated conductance equal to 18, 195, and 1270 pS, respectively, in 1 M KCl ion solution. The corresponding free energy profiles, reported for the first time, are reconstructed from simulations at applied voltage of 200 mV with the aid of the electrodiffusion model both with and without the knowledge of diffusivity. The calculated free energy barriers are equal to 2.5, 1.5, and 0.5 kcal/mol for K+ and 4.0, 2.2, and 1.5 kcal/mol for Cl−, for hexamer, heptamer, and octamer, respectively. The calculated conductance and the ratio between conductance in consecutive states are in good agreement with those measured experimentally. This suggests that the hexamer is the lowest conducting state, with measured conductance equal to 19 pS. The selectivity of K+ over Cl− is calculated as 1.5 and 2.3 for the octameric and heptameric channels, close to the selectivity measured for high-conductance states. Selectivity increases to 13 in the hexameric channel in which the narrowest Gln7 site has a pore radius of only ∼1.6 Å, again in accord with experiment. A good agreement found between calculated and measured conductance through a hexamer templated on cyclodextrin lands additional support for the results of our simulations, and the comparison with ALM reveals the dependence of conductance on the nature of phospholipid membrane.

Structure and Computational Electrophysiology of Ac-LS3, a Synthetic Ion Channel.

Computer simulations are reported on Ac-LS3, a synthetic ion channel, containing 21 residues with a Leu-Ser-Ser-Leu-Leu-Ser-Leu heptad repeat, which forms ions channels upon application of voltage. A hexameric, coiled-coil bundle initially positioned perpendicular to the membrane settled into a stable, tilted structure after 1.5 μs, most likely to improve contacts between the non-polar exterior of the channel and the hydrophobic core of the membrane. Once tilted, the bundle remained in this state during subsequent simulations of nearly 10 μs at voltages ranging from 200 to -100 mV. In contrast, attempts to identify a stable pentameric structure failed, thus supporting the hypothesis that the channel is a hexamer. Results at 100 mV were used to reconstruct the free energy profiles for K+ and Cl- in the channel. This was done by way of several methods in which results of molecular dynamics (MD) simulations were combined with the electrodiffusion model. Two of them developed recently do not require knowledge of the diffusivity. Instead, they utilize one-sided density profiles and committor probabilities. The consistency between different methods is very good, supporting the utility of the newly developed methods for reconstructing free energies of ions in channels. The flux of K+, which accounts for most of the current through the channel, calculated directly from MD matches well the total measured current. However, the current of Cl- is somewhat overestimated, possibly due to a slightly unbalanced force field involving chloride. The current-voltage dependence was also reconstructed by way of a recently developed, efficient method that requires simulations only at a single voltage, yielding good agreement with the experiment. Taken together, the results demonstrate that computational electrophysiology has become a reliable tool for studying how channels mediate ion transport through membranes.

Accurate numerical simulation of electrodiffusion and water movement in brain tissue.

Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide a new physiological insight into brain homeostasis. However, in order to provide solid answers and resolve controversies, the accuracy of the predictions is essential. Ionic electrodiffusion models typically comprise non-trivial systems of non-linear and highly coupled partial and ordinary differential equations that govern phenomena on disparate time scales. Here, we study numerical challenges related to approximating these systems. We consider a homogenized model for electrodiffusion and osmosis in brain tissue and present and evaluate different associated finite element-based splitting schemes in terms of their numerical properties, including accuracy, convergence and computational efficiency for both idealized scenarios and for the physiologically relevant setting of cortical spreading depression (CSD). We find that the schemes display optimal convergence rates in space for problems with smooth manufactured solutions. However, the physiological CSD setting is challenging: we find that the accurate computation of CSD wave characteristics (wave speed and wave width) requires a very fine spatial and fine temporal resolution.

A tridomain model for potassium clearance in optic nerve of Necturus

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system are important in a wide range of biological and medical processes. However, the exact driving mechanisms are often not known. In this work, we investigate flows induced by action potentials in an optic nerve as a prototype of the central nervous system. Different from traditional fluid dynamics problems, flows in biological tissues such as the central nervous system are coupled with ion transport. They are driven by osmosis created by concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group Orkand et al. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.

Optic nerve microcirculation: Fluid flow and electrodiffusion

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system (CNS) are important in a wide range of biological and medical processes [C. Nicholson and S. Hrabětová, “Brain extracellular space: The final frontier of neuroscience,” Biophys. J. 113(10), 2133 (2017)]. However, the exact driving mechanisms are often not known. In this paper, we investigate flows induced by action potentials in an optic nerve as a prototype of the CNS. Different from traditional fluid dynamics problems, flows in biological tissues such as the CNS are coupled with ion transport. It is driven by osmosis created by the concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group [R. K. Orkand, J. G. Nicholls, and S. W. Kuffler, “Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia,” J. Neurophysiol. 29(4), 788 (1966)]. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.

Computational Electrophysiology from a Single Molecular Dynamics Simulation and the Electrodiffusion Model.

The availability of high-resolution structures of ion channels opens the doors to reliable computations of electrophysiological properties, such as the dependence of ionic currents and selectivities on applied voltage. We develop two theoretical approaches for calculating these properties from molecular dynamics simulations at a single voltage, or even in the absence of voltage, combined with the electrodiffusion model in which ion motion in the channel is represented as one-dimensional diffusion in the potential of mean force exerted by other components of the system and the applied electric field. No knowledge of diffusivity or ion densities at other voltages is needed. Instead, in one approach, one-sided ion fluxes and density profiles are used to determine the free energy profile. In the other approach, committor probabilities for ions transported at the selected voltage are used for this purpose. Both approaches have been validated in an example of a simple ion channel formed by trichotoxin. The potentials of mean force calculated by way of the proposed approaches and obtained from traditional methods are in excellent agreement. Furthermore, the current-voltage dependence agrees very well with results obtained by way of computationally more demanding methods. We also have readily calculated the reversal potential, a computationally challenging electrophysiological property. The key assumptions of the electrodiffusion model, such as the independence of crossing events or the insensitivity of the potential of mean force to applied voltage, have been found to be satisfied. We also show that the voltage changes linearly in the hydrophobic core of the membrane and is constant elsewhere.

Quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics

In this paper we study quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck–Nernst–Poisson and Navier–Stokes equations. Different from other studies, we consider the physical case that the mobilities of the charges are different. For the generally smooth doping profile and for the ill-prepared initial data, under the assumption that the difference between the mobilities of two kinds of charges is very small, the quasi-neutral limit with an initial layer structure is rigorously proved by using the weighted energy method coupled with multi-scaling asymptotic expansions.

A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model.

Cortical spreading depression (SD) is a spreading disruption of ionic homeostasis in the brain during which neurons experience complete and prolonged depolarizations. SD is the basis of migraine aura and is increasingly associated with many other brain pathologies. Here, we study the role of glutamate and NMDA receptor dynamics in the context of an ionic electrodiffusion model. We perform simulations in one (1D) and two (2D) spatial dimension. Our 1D simulations reproduce the "inverted saddle" shape of the extracellular voltage signal for the first time. Our simulations suggest that SD propagation depends on two overlapping mechanisms; one dependent on extracellular glutamate diffusion and NMDA receptors and the other dependent on extracellular potassium diffusion and persistent sodium channel conductance. In 2D simulations, we study the dynamics of spiral waves. We study the properties of the spiral waves in relation to the planar 1D wave, and also compute the energy expenditure associated with the recurrent SD spirals.

Modeling and numerical analysis of the effect of dissociation/recombination of water molecules on the transport of salt ions in diffusion layer

Introduction. The paper presents a theoretical study on binary salt ion transport considering the water dissociation/recombination reaction. The work objectives are as follows: to build a mathematical model; to develop an algorithm for the numerical solution to the boundary value problem corresponding to the mathematical model; to work out the similarity theory including the transition to a dimensionless form using characteristic quantities; to determine a physical meaning of trivial similarity criteria; to find nontrivial similarity criteria; to build and analyze the volt-ampere characteristic (VAC).Materials and Methods. The theoretical study and numerical analysis of the transport of binary salt ions consider the dissociation/recombination reaction of water. In this case, the heat transfer equation and the mathematical model of electrodiffusion of four types of ions simultaneously (two salt ions, as well as ????+ and ????????−ions) in the diffusion layer of electromembrane systems with a perfectly selective membrane are used. For the first-order differential equations, a singularly perturbed boundary-value problem is set. In the equation for the electric field, the right side is independent of the intensity. In the numerical solution to the digitized system of equations by the Newton-Kantorovich method, this causes the stability of the method. In this regard, the boundary-value problem is reduced for numerical solution: a transition to a system of the second-order equations is provided, and the missing boundary conditions for the electric field strength are calculated.Research Results. A new mathematical model, a numerical algorithm to solve a boundary value problem, and software are developed. A numerical analysis is carried out, and fundamental laws of the transport of salt ions are determined considering the dissociation/recombination reaction of water molecules, temperature effects, and Joule heating. The VAC is built and analyzed.Discussion and Conclusions. The transport of binary salt ions through a diffusion layer near a cation exchange membrane is considered. A mathematical model of this process is proposed. It takes into account the temperature effects due to dissociation/recombination reactions of water molecules and Joule heating in a solution. The basic laws of the transport of salt ions are established considering the dissociation/recombination reaction of water molecules and temperature effects. The temperature effects of the dissociation/recombination reaction and the Joule heating in the electroneutrality region (ENR) are almost imperceptible (with the exception of the recombination region, RR). The Joule heating in the space-charge region (SCR) is by two orders of magnitude larger than the cooling effect of the water dissociation reaction. Upon recombination, approximately the same heat is released in the RR as during Joule heating in the expanded SCR. However, due to the small size of the RR, the effect of this heat is imperceptible. Therefore, we can assume that there is only one heat source at the interface in the SCR, which, due to its noticeable size, causes a significant increase in temperature in the entire diffusion layer. It follows that the emergence and development of gravitational convection is possible. General conclusions, following from the results obtained, open up the possibility of intensifying the process of transport of salt ions in the electrodialysis machines.

Location Specificity of Transcranial Electrical Stimulation on Neuronal Electrodynamics: A Mathematical Model of Ion Channel Gating Dynamics and Ionic Flux Due to Neurostimulation.

Transcranial Electrical Stimulation (TES) continues to demonstrate success as a medical intervention for individuals with neurodegenerative diseases. Despite promising results from these neuromodulation modalities, the cellular level mechanisms by which this neurotherapy operates are not fully comprehended. In particular, the effects of TES on ion channel gating and ion transport are not known. Using the Poisson-Nernst-Planck model of electrodiffusion, coupled with a Hodgkin-Huxley based model of cellular ion transport, we present a model of TES that, for the first time, integrates electric potential energy, individualized ion species, voltage-gated ion channels, and transmembrane ionic flux during TES administration. Computational simulations are executed on a biologically-inspired domain with medically-based TES treatment parameters and quantify neuron-level electrical processes resulting from this form of neurostimulation. Results confirm prior findings that show that TES polarizes the cell membrane, however, these are extended as simulations in this paper show that polarization occurs in a location specific manner, where the type and degree of polarization depends on the position on the membrane within a node of Ranvier. In addition, results demonstrate that TES causes ion channel gating variables to change in a location specific fashion and, as a result, transmembrane current from distinct ion species depends on both time and membrane location. Another simulation finding is that intracellular calcium concentrations increase significantly due to a TES-induced calcium influx. As cytosolic calcium is critical in intracellular signaling pathways that govern proper neurotransmitter secretion as well as support cell viability, this alteration in calcium homeostasis suggests a possible mechanism by which TES operates at the neuronal level to achieve neurotherapeutic success.

Ionic Currents in a Flat Membrane

The ionic currents passing through the membranes forming the surface of a cell play an important role in the behavior of biological objects. The behavior of a flat membrane in an electrolyte is considered in terms of the electrodiffusion model. It is shown that an instability leading to the development of ionic currents in a cell can emerge under the action of membrane protein density fluctuations, an external potential, or electrolyte ion concentrations. A long-wavelength mode describing the change in electrolyte ion concentrations along the membrane has been found. The instability thresholds have been determined. The influence of membrane surface charges on the instability thresholds has been studied.

From electrodiffusion theory to the electrohydrodynamics of leaky dielectrics through the weak electrolyte limit

The Taylor–Melcher (TM) model is the standard model for describing the dynamics of poorly conducting leaky dielectric fluids under an electric field. The TM model treats the fluids as ohmic conductors, without modelling the underlying ion dynamics. On the other hand, electrodiffusion models, which have been successful in describing electrokinetic phenomena, incorporate ionic concentration dynamics. Mathematical reconciliation of the electrodiffusion picture and the TM model has been a major issue for electrohydrodynamic theory. Here, we derive the TM model from an electrodiffusion model in which we explicitly model the electrochemistry of ion dissociation. We introduce salt dissociation reaction terms in the bulk electrodiffusion equations and take the limit in which the salt dissociation is weak; the assumption of weak dissociation corresponds to the fact that the TM model describes poor conductors. Together with the assumption that the Debye length is small, we derive the TM model with or without the surface charge convection term depending upon the scaling of relevant dimensionless parameters. An important quantity that emerges is the Galvani potential (GP), the jump in voltage across the liquid–liquid interface between the two leaky dielectric media; the GP arises as a natural consequence of the interfacial boundary conditions for the ionic concentrations, and is absent under certain parametric conditions. When the GP is absent, we recover the TM model. Our analysis also reveals the structure of the Debye layer at the liquid–liquid interface, which suggests how interfacial singularities may arise under strong imposed electric fields. In the presence of a non-zero GP, our model predicts that the liquid droplet will drift under an imposed electric field, the velocity of which is computed explicitly to leading order.

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

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

Phenomenon of two transition times in chronopotentiometry of electrically inhomogeneous ion exchange membranes

Transition time in chronopotentiometry is an important parameter, which is widely used for electrochemical characterization of various systems. Occurrence of multiple transition times is typical for multicomponent or multilayer electrode and membrane systems. In this paper we show that there may be another cause of multiple transition times. It is electrical heterogeneity of ion exchange membrane surface. It is found that there are two transition times on the chronopotentiograms of a commercial anion-exchange MA-41 (Shchekinoazot) and two specially prepared cation-exchange heterogeneous membranes in dilute electrolyte solutions (0.02 M NaCl in the experiment). It is found that both transition times are determined by diffusion limitations of ion delivery to the membrane surface in the depleted diffusion layer. The value of the first transition time depends on the dimensions of the conductive regions and their surface fraction; this transition time is determined as the time necessary for the depletion of electrolyte concentration near the conductive regions of the surface. The rate of concentration depletion depends on the electromigration through the conductive regions, on the one hand, and on the normal the tangential ion diffusion to the conductive surface regions, which mitigate the concentration decrease, on the other hand. The experimental value of the first transition time for the laboratory-made membranes is in a good agreement with simulation using a 3D electrodiffusion model. The value of the second transition time is in a good agreement with the Sand theory, hence it is conditioned by the normal diffusion delivery of electrolyte from the solution bulk to the entire membrane surface. The tangential diffusion plays a secondary role in this stage of concentration polarization since current-induced convection levels off the concentrations along the heterogeneous membrane surface.

Excitable Membrane, Ion Channel, Electrostatics: A Historical Sketch

The roughly one-tenth of a volt across an excitable membrane may not seem large, but across a membrane two lipid molecules thick it yields an electric field comparable to one causing atmospheric lightning. Early attempts to explain the physical basis of the nerve impulse modeled its ion currents as stemming from two processes: the movement of ionic charges by the electric field and the ion diffusion due to ion-concentration differences across the membrane. Hampered by its assumptions of constant properties, this classical electrodiffusion model couldn’t explain axonal membrane data. An eclectic approach combining an equation from electrodiffusion theory with elements of cable theory, wave theory and circuit theory—capacitor, batteries, variable conductors—transformed biophysical science with its successful fitting of the action potential's time course. The development of glass pipette electrodes that restrict ion flow to microscopic membrane patches led to the isolation of ion channels, glycoprotein macromolecules that selectively conduct ion currents. Because it's easy to visualize water-filled pores across these macromolecules, fitted with gates that block or allow ion currents across the membrane, they were named voltage-gated ion channels. This approach is challenged because of its: simplistic models of everyday devices, including screws and paddles; assumption of constant channel capacitance; and disregard of electrostatic repulsions between same-charge residues. Genetic engineering techniques brought the discovery of structural features of many such ion channels. One universal feature is a membrane-spanning segment with a regular array of positively charged amino-acid residues. All also contain multiple amino acids whose sidechains fork into two branches. An electrostatic model based on the ion channel's strong similarities to ferroelectric liquid crystals containing branched chains helps explain the depolarization-driven activation of voltage-sensitive ion channels. [See Leuchtag H R (2008) Voltage-Sensitive Ion Channels, Springer; (2017) Biophys. J. 112(3):544a; 112(3):543a-544a; 112(3):464a.]

Series solution of Painlevé II in electrodiffusion: conjectured convergence

A perturbation series solution is constructed with the use of Airy functions, for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, with two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlevé II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion for the Painlevé transcendent describing the electric field has been obtained. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.