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.     

Comparison of jumping and electrodiffusion mechanisms of particle movement in thin membranes. II. Potential clamping. Jumping and continuous uniform mechanism]

Electric relaxation of the membrane in three-barrier hopping and electrodiffusion uniform models has been considered. Both models are shown to have different relaxation characteristics which do not coiside with each other.     

Simple model of ion transport through alamethicin channels in lipid membranes

The electrical characteristics of wide membrane channels such as those induced in lipid membranes by alamethicin have been analyzed using an electrodiffusion model. The channel is considered to be a water filled cylinder in which the potential energy barrier is a result of the difference in polarization energy of the ion environment when the ion is located inside as compared to outside of the channel. In addition, an electric field related to the channel structure is assumed. It is shown that without postulating any specific chemical ion-channel interaction one can reproduce experimental membrane potentials for NaCl, KCl, and CaCl₂ concentration gradients with a single set of channel parameters. The calculations also yield experimental J-V characteristics of discrete conduction states. In addition, a simple mechanism of interchannel coupling based on the above model is discussed. The model suggests a unifying approach to the problem of the origin of interionic selectivity of membrane channels induced by polyene antibiotics.     

Effect of ionic polarizability on electrodiffusion in lipid bilayer membranes.

Ion-carrier complexes and organic ions of similar size and shape have mobilities in lipid bilayer membranes which span several orders of magnitude. In this communication, an examination is made of the hypothesis that the basis for this unusually wide range of ionic mobilities is the potential energy barrier arising from image forces which selectively act on ions according to their polarizability. Using Poisson's equation to evaluate the electrostatic interaction between an ion and its surroundings, the potential energy barrier to ion transport due to image effects is computed, with the result that the potential energy barrier height depends strongly on ionic polarizability. Theoretical membrane potential energy profile calculations are used in conjunction with Nernst-Planck electrodiffusion equation to analyze the available mobility data for several ion-carrier complexes and lipid-soluble ions in lipid bilayer membranes. The variation among the mobilities of different ions is shown to be in agreement with theoretical predictions based on ionic polarizability and size. Furthermore, the important influence exerted by image forces on ion transport in lipid bilayer membranes compared to the frictional effect of membrane viscosity is established by contrasting available data on the activation energy of ionic conductivity with that for membrane fluidity.     

Anomalous Reactances in Electrodiffusion Systems

A frequency response analysis of a constrained diffusion boundary has been made by linearizing the Nernst-Planck equations for a small applied AC current. The number of time constants and their dependence on ionic concentrations and electric field as well as membrane parameters such as dielectric constant, thickness, etc. have been evaluated by this method. Numerical solutions have been carried out for cases when the Planck charging time can be neglected and the results are presented in the form of impedance loci. These impedance loci show that if the membrane separates two univalent electrolytes with a common anion it will exhibit a combined capacitative inductive response with a 90° phase angle. The dependence of these anomalous reactances on ionic concentrations and the electric field is consistent with the behavior of the Hodgkin-Huxley axon suggesting that a homogeneous electrodiffusion regime could be adequate as a basic model for the kinetic behavior of biological membranes.     

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.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

The role of proteins in a dipole model for steady-state ionic transport through biological membranes.

The steady-state current-voltage characteristics of biological membranes are analyzed for means of an application of the electrodiffusion theory to the passage of ions through "dielectric pores", with orientable dipoles at the pore-water interfaces. A detailed evaluation of the electrostatic potential barrier shows, indeed, that the ions have practically no chance to penetrate into the phospholipid bilayer, but that they can cross the membrane through local protein inclusions, of high dielectric constant. A "gating mechanism" can be provided, moreover, by a change of the potential barrier, resulting from a dipole reorientation at the pore-water interface. Dipole-dipole interactions are opposed to the orienting effect of an applied field, but they can be neglected when the separation between the dipoles exceeds a certain critical value. The high polarizability of the pore material leads to an amplification of the effect of an applied field on the orientable dipoles. It is therefore possible to achieve a satisfactory agreement with the experimental results of Gilbert and Ehrenstein (Biophys. J., 9: 447, 1969) for the squid axon, and, in particular, to account for the width of the negative resistance regions with a relatively small value for the length of the orientable dipoles.     

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.     

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.     

The independence principle. A reconsideration

The electrodiffusion model presented in the previous paper, which specifically excludes ion-ion interactions, is analyzed for the ratio of one-way fluxes (flux ratio) as a function of the ionic driving force across the membrane. Significant deviations from the behavior expected on the basis of the Ussing relation are found. These are sufficient to explain the “nonindependent” ion movement noted in some biological flux ratio data. One-way fluxes are dependent on the ionic concentration on both sides of the membrane. The coupling of these fluxes to ionic concentrations comes from the dependence of ionic mobility and the diffusion coefficient on the equilibrium potential. It is concluded that nonindependent behavior in experimental data is not sufficient to implicate ion-ion interaction as the source of the discrepancy.     

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.     

Potassium channels as multi-ion single-file pores

A literature review reveals many lines of evidence that both delayed rectifier and inward rectifier potassium channels are multi-ion pores. These include unidirectional flux ratios given by the 2--2.5 power of the electrochemical activity ratio, very steeply voltage-dependent block with monovalent blocking ions, relief of block by permeant ions added to the side opposite from the blocking ion, rectification depending on E--EK, and a minimum in the reversal potential or conductance as external K+ ions are replaced by an equivalent concentration of T1+ ions. We consider a channel with a linear sequence of energy barriers and binding sites. The channel can be occupied by more than one ion at a time, and ions hop in single file into vacant sites with rate constants that depend on barrier heights, membrane potential, and interionic repulsion. Such multi-ion models reproduce qualitatively the special flux properties of potassium channels when the barriers for hopping out of the pore are larger than for hopping between sites within the pore and when there is repulsion between ions. These conditions also produce multiple maxima in the conductance-ion activity relationship. In agreement with Armstrong's hypothesis (1969. J. Gen. Physiol. 54:553--575), inward rectification may be understood in terms of block by an internal blocking cation. Potassium channels must have at least three sites and often contain at least two ions at a time.     

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. I. Solution in transform space

Transient mass transfer in a Krogh tissue cylinder is described by a model taking into account axial diffusion in both blood and tissue, a localized permeability barrier at the capillary membrane and a diffusion barrier on the outer surface and at the ends of the cylinder. Radial diffusion in both blood and tissue is assumed to be infinitely fast. In contrast to previous work, which has usually relied on numerical methods for solving the equations, an exact solution is presented here in Laplace transform space. This allows calculation of the moments of the concentration at any point in the cylinder. Numerical results indicate that the moments of the residence time distribution are affected by the boundary conditions used, and that the discrepancies between the predictions using different conditions may be large in some physiological situations. Order-of-magnitude calculations are used to estimate when the use of simpler models may be feasible. The transform space solution may also be useful for parameter estimation, but it seems preferable to extend the present results to a time-domain solution for this purpose.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.