An Exact Constant-Field Solution for a Simple Membrane

We show that the exact steady-state solution to the electrodiffusion equations for a simple membrane is the constant electric field solution when the ion environment is electroneutral on both sides of the membrane and the total numbers of ions of the same valence on both sides are equal.     

Diffusion of Ions in Myelinated Nerve Fibers

The diffusion of ions towards or away from the inner side of the nodal membrane in preparations, the cut ends of which are placed in various media, was investigated. The ion concentration changes were calculated by numerical solution of the unidimensional electrodiffusion equation under a variety of media compositions, axoplasmic diffusion coefficients, and internal anionic compositions. The potassium and cesium ion diffusion along the axon towards the node was determined experimentally by two different electrophysiological methods. On the basis of comparison between the experimental data and the computational predictions the axoplasmic potassium ion diffusion coefficient was determined to be almost equal to that in free aqueous solution, while that of cesium ion was close to one half of that in aqueous solution. Utilizing the values of diffusion parameters thus determined, we solved the electrodiffusion equation for a number of common experimental procedures. We found that in short fibers, cut 0.1-0.2 cm at each side of the node, the concentration approached values close to the new steady-state values within 5-30 min. In long fibers (over 1 cm long) steady-state concentrations were obtained only after a few hours. Under some conditions the internal concentrations transiently overshot the steady-state values. The diffusion potentials generated in the system were also evaluated. The ion concentration changes and generation of diffusion potential cannot be prevented by using side pools with cation content identical to that of the axoplasm.     

Frequency relations of capacitance and conductance of the system electrolyte--membrane--electrolyte in a model framework]

An electrodiffusion model of immediate permeability of ions through the lipid membrane was considered. The model suggests the existence of finite immovable layers of the electrolyte near the membrane. A technique is given for linearization and obtaining analytical solutions of the time-dependent electrodiffusion equations for one ion species. The expression for admittence of the system was obtained. The model allowed to obtain the curves of c (omega) and g (omega) which agreed with the empiric ones.     

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.     

The Effect of temperature gradient on the transport phenomenon in roots of maize plants grown under salinity conditions. substance,heat, and ion flows

An analysis of flows through primary root and first node root tissues of plants grown under conditions of salinity and nutrient deficiency induced by temperature gradients was carried out using. a mathematical model. The results obtained show that high KNO₃ concentration in Knop’s nutrient solution (salinity) causes an inhibition of volume and heat flows and that the omission of KNO₃ from Knop’s nutrient solution (deficiency) stimulates these flows. The causes of the inhibition lay in the fact that salinity reduced hydraulic, electric, and osmotic conductivity when compared with the control (Knop’s solution), but relative to nutrient deficiency, it increased osmotic conductivity, electrodiffusion, diffusion, and filtration of heat flow induced by the electric and heat power. The causes of the stimulation were that deficiency partially decreased conductivities, similarly as salinity when compared with the control, and also decreased osmotic abilities of the system. By contrast, it increased heat conductivity and corresponding filtrations (diffusion-thermal, thermoosmotic). In first node root tissues, it increased all conductivities with the exception of electric conductivity, then osmotic, electroosmotic, diffusion, electrodiffusion, and filtration of heat flow and current flow, that is the number of possible ways of solution transport through root tissues increased. Part II.     

Stochastic theory of singly occupied ion channels. II. Effects of access resistance and potential gradients extending into the bath.

In a previous paper (Jakobsson, E., and S. W. Chiu. 1987. Biophys. J. 52:33-46), we presented the stochastic theory of the singly occupied ion channel as applied to sodium permeation of gramicidin channels, with the assumption of perfect equilibration between the bathing solutions and the ends of the ion channel. In the present paper we couple the previous theory to electrodiffusion of ions from the bulk of the bathing solution to the channel mouth. Our electrodiffusion calculations incorporate estimates of the potential gradients near the channel mouth due to image forces and due to the fraction of the applied potential that falls beyond the ends of the channel. To keep the diffusion calculation one-dimensional, we make the assumption that the electrical potentials in the bath exhibit hemispherical symmetry. As in the previous paper, the flux equations are fit to data on sodium permeation of normal gramicidin A, and gramicidins modified by the fluorination of the valine at the No. 1 position (Barrett Russell, E. W., L. B. Weiss, F. I. Navetta, R. E. Koeppe II, and O. S. Anderson. 1986. Biophys. J. 49:673-686). The conclusions of our previous paper with respect to the effect of fluorination on the mobility, surface potential well depth, and central barrier, are confirmed. However the absolute values of these quantities are somewhat changed when diffusive resistance to the mouth is taken into account, as in the present paper. Future possibilities for more accurate calculations by other methods are outlined.     

Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations

We present numerical solutions for the one-dimensional Nernst-Planck and Poisson system of equations for steady-state electrodiffusion. Commonly used approximate solutions to these equations invoke assumptions of local electroneutrality (Planck approximation) or constant electric field (Goldman approximation). Calculations were performed to test the ranges over which these approximate theories are valid. For a dilutional junction of a 1:1 electrolyte, separated from adjoining perfectly stirred solutions by sharp boundaries, the Planck approximation is valid for values of kappa dL greater than 10, where 1/kappa d is the Debye length of the more dilute solution. The Goldman approximation is valid for kappa cL less than 0.1 where 1/kappa c is the Debye length of the more concentrated solution. These results suggest that the modeling of electrodiffusive flows in and near membrane ion channels may require numerical solutions of this set of equations rather than the use of either limiting case.     

A general approach to modeling conduction and concentration dynamics in excitable cells of concentric cylindrical geometry.

This paper discusses mathematical approaches for modeling the propagation of the action potential and ion concentration dynamics in a general class of excitable cells and cell assemblies of concentric cylindrical geometry. Examples include myelinated and unmyelinated axons, single strands of interconnected cardiac cells and outer hair cells. A key feature in some of the cells is the presence of a small working volume such as the periaxonal space between the myelin sheath and the axon in the myelinated axon and the extracisternal space between the plasma membrane and the subsurface cisterna of the outer hair cell. Proper treatment of these cell types requires a modeling approach which can readily address these anatomical properties and the non-uniform biophysical properties of the concentric membranes and the ionic composition of the volumes between the membranes. An electrodiffusion approach is first developed in which the Nernst-Planck equation is used to characterize axial ion fluxes. It is then demonstrated that this "full" model can be stepwise reduced, eventually becoming equivalent to the standard cable equation formulation. This is done in a manner that permits direct comparisons between the full and simplified models by running simulations using a single parameter set. An intermediate approach where the contributions of the axial currents to ion concentration changes and the effect of varying ion concentrations on solution conductivities are ignored is derived and is found adequate in many cases. Two application examples are given: a "cardiac strand" model, for which the intermediate formulation is shown sufficient and a model of the myelinated axon, for which the full electrodiffusion formulation is clearly necessary. The latter finding is due to spatial inhomogeneities in the anatomy and distribution of ion channels and transporters in the myelinated axon and the restricted periaxonal space between the myelin sheath and the axon.     

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.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.     

The effect of temperature gradient on the transport phenomenon in roots of maize plants grown under salinity conditions. conductivity and filtration properties

An analysis of flows through primary root and first node root tissues of plants grown under conditions of salinity and nutrient deficiency induced by temperature gradients was carried out using. a mathematical model. The results obtained show that high KNO₃ concentration in Knop’s nutrient solution (salinity) causes an inhibition of volume and heat flows and that the omission of KNO₃ from Knop’s nutrient solution (deficiency) stimulates these flows. The causes of the inhibition lay in the fact that salinity reduced hydraulic, electric, and osmotic conductivity when compared with the control (Knop’s solution), but relative to nutrient deficiency, it increased osmotic conductivity, electrodiffusion, diffusion, and filtration of heat flow induced by the electric and heat power. The causes of the stimulation were that deficiency partially decreased conductivities, similarly as salinity when compared with the control, and also decreased osmotic abilities of the system. By contrast, it increased heat conductivity and corresponding filtrations (diffusion-thermal, thermoosmotic). In first node root tissues, it increased all conductivities with the exception of electric conductivity, then osmotic, electroosmotic, diffusion, electrodiffusion, and filtration of heat flow and current flow, that is the number of possible ways of solution transport through root tissues increased.     

Linearity, saturation and blocking in a large multiionic channel: divalent cation modulation of the OmpF porin conductance

Measurement of unitary conductance is a fundamental step in the characterization of a protein ion channel permeabilizing a membrane. We study here the effect of salts of divalent cations on the OmpF channel conductance with a particular emphasis in dissecting the role of the electrolyte itself, the role of the counterion accumulation induced by the protein channel charges and other effects not found in salts of monovalent cations. We show that current saturation and blocking are not exclusive properties of narrow (single-file) ion channels but may be observed in large, multiionic channels like bacterial porins. Single-channel conductance measurements performed over a wide range of salt concentrations (up to 3 M) combined with continuum electrodiffusion calculations demonstrate that current saturation cannot be simply ascribed to ion interaction with protein channel residues.     

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.     

Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events.

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc. are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and the shut times, are too short to be detected reliably. If a fixed dead-time tau is assumed then it is possible to define, as an approximation to what is actually observed, an 'extended opening' or e-opening which starts with an opening of duration at least tau followed by any number of openings and shuttings, all the shut times being shorter than tau; the e-opening ends when a shut time longer than tau occurs. A similar definition is used for e-shut times. The probability densities, f(t), of these extended times have previously been obtained as expressions which become progressively more complicated, and numerically unstable to compute, as t-->infinity. In this paper we present, for the two-state model, an alternative representation as an infinite series of which a small number of terms gives a very accurate approximation of f (t) for large t. For the general model we present an asymptotic representation as a mixture of exponentials which is accurate for all except quite small values of t. Some simple model-independent corrections for missed events are discussed in relationship to the exact solutions.     

Phase transitions and ion currents in a model ferroelectric channel unit

The hypothesis of ferroelectric electrodiffusion is examined mathematically. A thermodynamic potential, the elastic Gibbs function, written in polynomial form, provides the dielectric equation of state for the model. The other equations of electrodiffusion theory complete the model. This system reduces to a second-order partial differential equation, which is formally solved by the phase-plane method. This solution, applied to the Na channel, leads to a propagating phase-transition wave accompanied by movement of ionic charge. This may be readily interpreted as a transmembrane wave traveling along a ferroelectric unit within, and transporting ions through, the channel. Comparison of the temperature dependence of axonal conduction velocity with that of the spontaneous polarization of Rochelle salt suggests that the Na channel of squid axon contains a ferroelectric unit having a lower Curie point, but decomposing before reaching its upper Curie point. Comparison with data from reconstitution experiments suggests that the ferroelectric unit is a carbohydrate enclosed in an intrinsic protein structure to form a glycoprotein channel. The value experimentally estimated for the surface charge of the Na channel is within the range of spontaneous polarizations of typical ferroelectric crystals. It is argued that the ferroelectric probably is a single crystal of the order-disorder type, which undergoes a first-order transition between a ferroelectric and a paraelectric state during excitational activity. The hypothesis of ferroelectric channel units is consistent with the existence and directionality of the observed "gating" currents.     

Electrodiffusion of ions approaching the mouth of a conducting membrane channel.

The movement of ions in the aqueous medium as they approach the mouth (radius a) of a conducting membrane channel is analyzed. Starting with the Nernst-Planck and Poisson equations, we derive a nonlinear integrodifferential equation for the electric potential, phi(r), a less than or equal to r less than infinity. The formulation allows deviations from charge neutrality and dependence of phi(r) on ion flux. A numerical solution is obtained by converting the equation to an integral equation that is solved by an iterative method for an assumed mouth potential, combined with a shooting method to adjust the mouth potential until the numerical solution agrees with an asymptotic expansion of the potential at r-a much greater than lambda (lambda = Debye length). Approximate analytic solutions are obtained by assuming charge neutrality (Läuger, 1976) and by linearizing. The linear approximation agrees with the exact solution under most physiological conditions, but the charge-neutrality solution is only valid for r much greater than lambda and thus cannot be used unless a much greater than lambda. Families of curves of ion flux vs. potential drop across the electrolyte, phi(infinity)-phi (a), and of permeant ion density at the channel mouth, n1(a), vs. flux are obtained for different values of a/lambda and S = a d phi/dr(a). If a much greater than lambda and S = O, the maximum flux (which is approached when n1(a)----0) is reduced by 50% compared to the value predicted by the charge-neutrality solution. Access resistance is shown to be a factor a/[2 (a + lambda)] times the published formula (Hille, 1968), which was derived without including deviations from charge neutrality and ion density gradients and hence does not apply when there is no counter-ion current. The results are applied to an idealized diffusion-limited channel with symmetric electrolytes. For S = O, the current/voltage curves saturate at a value dependent on a/lambda; for S greater than O, they increase linearly for large voltage.