Electrodiffusion Models of Neurons and Extracellular Space Using the Poisson-Nernst-Planck Equations—Numerical Simulation of the Intra- and?Extracellular Potential for an Axon Model

In neurophysiology, extracellular signals—as measured by local field potentials (LFP) or electroencephalography—are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Planck equations of electrodiffusion. The propagation of an action potential along the axonal membrane is investigated by means of numerical simulations. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. We focus on the evolution of the extracellular electric potential. A characteristic up-down-up LFP waveform in the far-field is found. Close to the membrane, the potential shows a more intricate shape. A comparison with the widely used line source approximation reveals similarities and demonstrates the strong influence of membrane currents. However, the electrodiffusion model shows another signal component stemming directly from the intracellular electric field, called the action potential echo. Depending on the neuronal configuration, this might have a significant effect on the LFP. In these situations, electrodiffusion models should be used for quantitative comparisons with experimental data.     

Comparison of jumping and electrodiffusion mechanisms of particle movement in thin membranes. III. Potential clamping in a uniform membrane]

Electric relaxation of a non-uniform membrane was considered during potential clamp. It is shown that in the case of non-uniformity as a high narrow potential barrier in the middle and deep holes along the membrane edges the electrodiffusion model is adequate to the three-barrier hopping one.     

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.

     

Modification of the Kedem-Katchalsky equations

The presented modification of the transport equations of Kedem-Katchalsky resulted in the introduction of (omega s/omega) and omega/(omega-Lp sigma[(1-sigma)C1-(1-sigma s)C2]) factors into the Kedem-Katchalsky equations. The above factors determine the influence of boundary layers on transport across the membrane. The modified Kedem-Katchalsky equations were verified for synthetic membranes and it was shown that the value of the (omega s/omega) factor depended on the type of membrane and the membrane configuration system. This modification facilitated a wider range of application of the Kedem-Katchalsky equations to systems in which the solutions were stirred or unstirred.     

Transport across homoporous and heteroporous membranes in nonideal, nondilute solutions. II. Inequality of phenomenological and tracer solute permeabilities.

The phenomenological solute permeability (omega p) of a membrane measures the flux of solute across it when the concentrations of the solutions on the two sides of the membrane differ. The relationship between omega p and the the conventionally measured tracer permeability (omega T) is examined for homoporous and heteroporous (parallel path) membranes in nonideal, nondilute solutions and in the presence of boundary layers. In general, omega p and omega T are not equal; therefore, predictions of transmembrane solute flux based on omega T are always subject to error. For a homoporous membrane, the two permeabilities become equal as the solutions become ideal and dilute. For heteroporous membranes, omega p is always greater than omega T. An upper bound on omega p- omega T is derived to provide an estimate of the maximum error in predicted solute flux. This bound is also used to show that the difference between omega P and omega T demonstrated earlier for the sucrose-Cuprophan system can be explained if the membrane is heteroporous. The expressions for omega P developed here support the use of a modified osmotic driving force to describe membrane transport in nonideal, nondilute solutions.     

The problem of nonstationary ion fluxes in excitable membranes.

Time-dependent electrodiffusion through a membrane is analysed within a simple model treating the boundary-layers in a consistent manner. It is shown that time-independent reversal potentials for the ion fluxes exist only under steady-state conditions. We argue that this result holds very generally. Therefore nonstationary effect like ion storage and depletion inside the membrane should not contribute to the phenomena of excitability.     

Electrodiffusion model simulation of rectangular current pulses in a voltage-biased biological channel

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employed for the advection-diffusion equation. The rectangular wave shape of previously simulated traveling wave current pulses is preserved by the full set of partial differential equations for electrodiffusion.     

Voltage clamp simulations for multifiber bundles in a double sucrose gap: radial vs. longitudinal resistance effects

Voltage clamp responses of a single excitable fiber were simulated using a core conductor model including a high external resistance (Rs) in series to the fiber membrane to allow for intercellular clefts in a multifiber preparation. In terms of specific resistance, Rs was between 68 and 264 omega cm2. Internal resistivity (Ri) was taken to be zero or 200 omega cm. The aim of the study was to quantify the expected antagonistic effects of external and internal resistances on Na current measurements. With Ri = 0, the external resistance was found to cause a strong depression of fast inward current compared to an ideal space clamp at command potentials between -30 and 30 mV. With Ri = 200 omega cm, the depression of inward current was partially removed. The effects of Rs and Ri on membrane current measurement were illustrated by cable analysis assuming a quasi-steady state of the fiber at peak time of inward current.     

Numerical Solution of Steady-State Electrodiffusion Equations for a Simple Membrane

A technique is given for obtaining numerical solutions to the steady-state electrodiffusion equations for a simple membrane. Solutions are given for several membrane boundary conditions in terms of ratios of current density to mobility for each ion type.     

Parametric resonance and amplification of periodic disturbances in membranes containing ion channels with inactivation

Parametric resonance and amplification of periodic perturbations in the membrane transport of ions through channels with inactivation was studied in computational experiments. It has been shown that a periodic change in the membrane capacitance or in the applied electric current with a frequency approximately 2 omega 0 (omega 0--the own angular frequency of the membrane) may excite stable self-oscillations in the membrane with a frequency of approximately omega 0. For this to occur, the degree of the capacitance modulation m or the amplitude of the applied current i0 must exceed some critical values mcr and i0cr. Excitation of self-oscillations by alternating electric current of the frequency approximately 2 omega 0 has the characteristics of parametric resonance. This can be explained by the fact that the equivalent membrane inductance depends on ionic current and displays periodic changes with a frequency approximately 2 omega 0, as also does the current. Small-amplitude periodic changes in the capacitance (m less than mcr) with frequencies approximately 2 omega 0 may result in significant amplification of periodic perturbations with frequencies approximately omega 0.     

Contribution of electrodiffusion to the dynamics of electrically stimulated changes in mechanical properties of collagen membranes

We have studied the kinetics of oscillatory tensile forces in collagen membranes. These forces were generated by sinusoidal electric fields applied across the membrane. Both the magnitude and phase of the measured force changed with frequency over a three-decade range. The membrane-separated electrolyte baths had different ionic strength but identical non-isoelectric pH. Changes in intramembrane ionic strength due to the electric field were calculated over the same frequency range via an electrodiffusion model that was generalized to include convection and electrokinetic coupling. A comparison of the experimental and theoretical phases and amplitudes versus frequency suggests that electrodiffusion is the dominant rate-limiting process in this electromechanochemical transduction. These results are relevant to electrostatic interactions in connective tissues and to membrane-based filtration devices in which membrane permeability may be actively varied and controlled by an applied electric field.     

Predicted permeability parameters of human ovarian tissue cells to various cryoprotectants and water

This study presents a generic numerical model to simulate the coupled solute and solvent transport in human ovarian tissue sections during addition and removal of chemical additives or cryoprotective agents (CPA). The model accounts for the axial and radial diffusion of the solute (CPA) as well as axial convection of the CPA, and a variable vascular surface area (A) during the transport process. In addition, the model also accounts for the radial movement of the solvent (water) into and out of the vascular spaces. Osmotic responses of various cells within an human ovarian tissue section are predicted by the numerical model with three model parameters: permeability of the tissue cell membrane to water (L(p)), permeability of the tissue cell membrane to the solute or CPA (omega) and the diffusion coefficient of the solute or CPA in the vascular space (D). By fitting the model results with published experimental data on solute/water concentrations within an human ovarian tissue section, I was able to determine the permeability parameters of ovarian tissue cells in the presence of 1.5M solutions of each of the following: dimethyl sulphoxide (DMSO), propylene glycol (PROH), ethylene glycol (EG), and glycerol (GLY), at two temperatures (4 degrees C and 27 degrees C). Modeling Approach 1: Assuming a constant value of solute diffusivity (D = 1.0 x 10(-9) m(2)/sec), the best fit values of L(p) ranged from 0.35 x 10(-14) to 1.43 x 10(-14) m(3)/N-sec while omega ranged from 2.57 x 10(-14) to 70.5 x 10(-14) mol/N-sec. Based on these values of L(p) and omega, the solute reflection coefficient, sigma defined as sigma = 1-omega v(CPA)/L(P) ranged from 0.9961 to 0.9996. Modeling Approach 2: The relative values of omega and sigma from our initial modeling suggest that the embedded ovarian tissue cells are relatively impermeable to all the CPAs investigated (or omega approximately 0 and sigma approximately 1.0). Consequently the model was modified and used to predict the values of L(p) and D assuming omega = 0 and sigma = 1.0. The best fit values of L(p) ranged from 0.44 x 10(-14) to 1.2 x 10(-14) m(3)/N-sec while D ranged from 0.85 x 10(-9) to 2.08 x 10(-9) m(2)/sec. Modeling Approach 3: Finally, the best fit values of D from modeling approach 2 were incorporated into model 1 to re-predict the values of L(p) and omega. It is hoped that the ovarian tissue cell parameters reported here will help to optimize chemical loading and unloading procedures for whole ovarian tissue sections and consequently, tissue cryopreservation procedures.     

Effects of dietary omega-6 and omega-3 fatty acids on levels of long chain polyunsaturated fatty acids in erythrocytes]

C18:2 omega 6/C18:3 omega 3 ratio was lowered in the diet of Elderly subjects. This was done by the replacement of usual sunflower oil by rapeseed oil or by supplementing soybean oil. This diet modification induced an increase of EPA (C20:5 omega 3) and DHA (C22:6 omega 3) in red cell phospholipids. The omega 6 fatty acids (C18:2 and C20:4) were slightly modified. Therefore, dietary C18:2 omega 6/C18:3 omega 3 ratio, seems to play an important role in the determination of membrane highly unsaturated fatty acid levels.     

A gas chromatography-mass spectrometry method for the measurement of fatty acid omega and omega(-1) hydroxylation kinetics by CYP4A1 using an artificial membrane system

A gas chromatography-mass spectrometry assay method for the analysis of lauric, myristic, and palmitic acids and their omega and omega(-1) hydroxylated metabolites from in vitro incubations of cytochrome P450 CYP4A1, involving solid-phase extraction and trimethysilyl derivatization, was developed. The assay was linear, precise, and accurate over the range 0.5 to 50microM for all the analytes. It has the advantages of a more rapid analysis time, an improved sensitivity, and a wider range of analytes compared with other methods. An artificial membrane system was optimized for application to purified CYP4A1 enzyme by investigating the molar ratios of cytochrome b(5) and cytochrome P450 reductase present in the incubation mixture. Using this method, the kinetics of omega and omega(-1) oxidation of lauric, myristic, and palmitic acids by CYP4A enzymes were measured and compared in rat liver microsomes and an artificial membrane system.     

The effect of electric field on the spatial-time patterns in the reaction-diffusion system

A model of electrodiffusion processes in the vicinity of cell membrane was developed. The model takes into account chemical reactions, Coulomb interactions between charged particles and the effect of external electric field. It was concluded that the applied electric field can change the characteristics of space-time patterns in the system. Dissipative structures slowly move and this is accompanied by a change in the number of structure elements. The characteristic equation includes odd powers of the wavenumber, which can lead to the appearance of soliton-like structures. The dissipative structures can appear not only due to the Turing diffusion instability but due to the disperse instability under electric field the applied.     

Charge selectivity of the designed uncharged peptide ion channel Ac-(LSSLLSL)3-CONH2.

Charge selectivity in ion channel proteins is not fully understood. We have studied charge selectivity in a simple model system without charged groups, in which an amphiphilic helical peptide, Ac-(Leu-Ser-Ser-Leu-Leu-Ser-Leu)3-CONH2, forms ion channels across an uncharged phospholipid membrane. We find these channels to conduct both K+ and Cl-, with a permeability ratio (based on reversal potentials) that depends on the direction of the KCl concentration gradient across the membrane. The channel shows high selectivity for K+ when [KCl] is lowered on the side of the membrane that is held at a positive potential (the putative C-terminal side), but only modest K+ selectivity when [KCl] is lowered on the opposite side (the putative N-terminal side). Neither a simple Nernst-Planck electrodiffusion model including screening of the helix dipole potential, nor a multi-ion, state transition model allowing simultaneous cation and anion occupancy of the channel can satisfactorily fit the current-voltage curves over the full range of experimental conditions. However, the C-side/N-side dilution asymmetry in reversal potentials can be simulated with either type of model.     

Generalization of the Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions.

The Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for systems containing n-component, non-ionic solutions is presented. The frictional interpretation of the phenomenological coefficients of membrane and the expressions connecting the practical coefficients (Lp, sigma i, omega ij) with frictional coefficients (fij) are presented.