Nonlinear Electrical Effects in Lipid Bilayer Membranes: III. The Dissociation Field Effect

In the course of an analysis of nonlinear electrical effects in lipid bilayer membranes, the influence of the dissociation field (or Wien) effect on the membrane conductivity is investigated. It is shown that the theory of Onsager for the Wien effect in a macroscopic phase can be applied to a thin membrane when the proper boundary conditions at the membrane-solution interface are introduced. It is assumed that an activation energy is associated with the passage of the ion across the interface. The mathematical treatment of the model is restricted to the case for which cations and anions have identical properties except for the charge sign. The resulting differential equations for the ion concentration within the membrane are integrated numerically. The analysis shows that the influence of the Wien effect on the membrane conductivity is appreciable only if the energy barrier at the interface is sufficiently high, i.e. if the rate limiting step for the ion transport is the passage of the ion across the interface.     

Semiconductor theory of ion transport in thin lipid membranes: I. Potential and field distributions

In the theory as presented in this paper and the following one, we shall attempt to apply the semiconductor principles and methods to the study of ion transport in thin lipid membranes. Detailed formulations are given on the potential energy barriers at the interfaces, voltage drops in the polar and non-polar regions, and potential and field distributions in the diffuse double layer and within a charged membrane. These results will be used mainly as the boundary conditions for the solution of ion flow as to be given in the following paper. The analysis clearly indicates that the ion transport is interface-limited and is profoundly influenced by the presence of surface charges. An explanation of Na⁺ extrusion in nerve membrane is given based on the field distribution analysis. The theory also suggests that the “membrane potential” depends mainly on surface charges but not necessarily on ion permeation through the membrane.     

Ion transport through thin lipid films

Currents through thin lipid films were measured versus temperature, applied voltage, pH value, ionic strength and ionic species. All experimental data tend to fit a two-component current theory, that is, the film current composed of a surface barrier jumping current and a surface recombination current. A good agreement was obtained in surface barrier height between theory and experiment. Breakdown phenomenon was observed and is interpreted as possibly arising from avalanche multiplication of ions, most probably from surface traps.     

Electrical noise from lipid bilayer membranes in the presence of hydrophobic ions

In the presence of the hydrophobic ion dipicrylamine, lipid bilayer membranes exhibit a characteristic type of noise spectrum which is different from other forms of noise described so far. The spectral density of current noise measured in zero voltage increases in proportion to the square of frequency at low frequencies and becomes constant at high frequencies. The observed form of the noise spectrum can be interpreted on the basis of a transport model for hydrophobic ions in which it is assumed that the ions are adsorbed in potential-energy minima at either membrane surface and are able to cross the central energy barrier by thermal activation. Accordingly, current-noise results from random fluctuations in the number of ions jumping over the barrier from right to left and from left to right. On the basis of this model the rate constant ki for the translocation of the hydrophobic ion across the barrier, as well as the mean surface concentration Nt of adsorbed ions may be calculated from the observed spectral intensity of current noise. The values of ki obtained in this way closely agree with the results of previous relaxation experiments. A similar, although less quantitative, agreement is also found for the surface concentration Nt.     

Electrical capacitances of thin lipid films

Based on the considerations of surface charge and surface recombination (or ion binding), the capacitances of the diffused double layer and the polar region are calculated as functions of ionic strength, pH value and external bias. Electrical capacitances of thin lipid films were measured under various conditions. In all cases, the change in capacitance is not more than 5%. Agreement between theory and experiment is fairly good except for the capacitance-voltage relation. It is concluded that though the magnitude of the observed membrane capacitance is determined by the center hydrocarbon region, its constancy and stability are attributed to the largeness and guarding action of the interface capacitances.     

A three-barrier model for the hemocyanin channel

Keyhole limpet hemocyanin forms ion-conducting channels in planar lipid bilayer membranes. Ionic current through the open hemocyanin channel presents the following characteristics: (a) it is carried mainly by cations; (b) it is a nonlinear function of membrane potential; (c) channel conductance is a saturating function of ion activity; (d) it shows ionic competition. A model for the open hemocyanin channel is developed from absolute reaction rate theory. The model calls for three energy barriers in the channel. Two energy barriers represent the entrance and exit of the ion into and out of the channel. The third barrier separates two energy minima that represent two binding sites. Furthermore, only one ion is allowed inside the channel at a given time. This model is able to recreate all the hemocyanin characteristics found experimentally in negatively charged and neutral membranes.     

Thickness dependence of monoglyceride bilayer membrane conductance.

We have studied the conductance properties of unmodified monoglyceride membranes as a function of monoglyceride chain length. As membrane thickness decreases from 31 to 20 nm, the steepness of the current-voltage (I-V) curve increases from 80 mV per e-fold current increase to 52 mV per e-fold current increase. The zero-voltage conductance increases more than 1,000-fold and the apparent activation energy of conductance decreases from 18.4 to 14.2 kcal/mol. We have analyzed our results using both the Nernst-Planck equation and absolute rate theory. Both approaches are consistent with our results and give consistent values for the parameters describing the I-V curves. We conclude that both the surface ion concentration and the distance from the surface of the membrane at which the energy of an ion rises appreciably above its value in solution (position of the barrier) are invariant with thickness.     

A theory of ion permeation through membranes with fixed neutral sites

Summary Some model membranes and biological membranes behave as if ion permeation were controlled by fixed neutral sites, i.e., by groups that are polar but lack net charge. By solving the boundary conditions and Nernst-Planck flux equations, this paper derives the expected properties of four types of membranes with fixed neutral sites: model 1, a membrane thick enough that microscopic electroneutrality is obeyed; model 2, same as model 1 but with a free-solution shunt in parallel; model 3, a membrane thin enough that microscopic electroneutrality is violated; and model 4, same as model 3 but with a free-solution shunt in parallel. The conductance-concentration relation and the current-voltage relation in symmetrical solutions are approximately linear for all four models. Partial ionic conductances are independent of each other for a thin membrane but not for a thick membrane. Sets of permeability ratios derived from conductances, dilution potentials, or biionic potentials agree with each other in a thin membrane but not in a thick membrane. The current-voltage relation in asymmetrical single-salt solutions is linear for a thick membrane but nonlinear for a thin membrane. Examples of potential and concentration profiles in a thin membrane are calculated to illustrate the meaning of space charge and the electroneutrality condition. The experimentally determined properties (by A. Cass, A. Finkelstein & V. Krespi) of thin lipid membranes containing “pores” of the anion-selective antibiotic nystatin are in reasonable agreement with model 3. Tests are suggested for deciding if a membrane of unknown structure has neutral sites, whether it is thick or thin, and whether the sites are fixed or mobile.     

A laser-temperature-jump method for the study of the rate of transfer of hydrophobic ions and carriers across the interface of thin lipid membranes

The first application of a laser-temperature-jump apparatus for the study of ion transport through planar (artificial) lipid membranes is described. The relaxation of the electric current is detected, either continuously at a constant applied voltage or discontinuously by a series of short voltage pulses. The second technique, a combined voltage- and temperature-jump method, is especially appropriate to investigate the kinetics of the adsorption/desorption process of hydrophobic ions and neutral carriers of cations at the membrane interface and to separate this phenomenon from the diffusion process through the unstirred aqueous layers adjacent to the membrane. The aim is to determine the rate-limiting step of transport. The permeation rate of the hydrophobic anion 2,4,6-trinitrophenolate is limited by the inner membrane barrier. For tetraphenylberate the rate constant of translocation across the inner barrier and that of desorption from the membrane into water are found to be of comparable magnitude. The membrane permeability of the neutral macrocyclic ion carrier enniatin B is strongly interface limited by its comparatively small rate of desorption into water. These results show that the frequently used a priori assumption of partition equilibrium at the membrane interfaces during transport is not justified.     

Electrochemical and NMR spectroscopic investigations of the influence of the probe molecule Eu(fod)3 on the permeability of lipid membranes to ions

Investigating the action of the fluorinated europium complex Eu(fod)3 on lipid membranes we found that the complex facilitates the ion transfer through the membrane. Electric measurements on planar lipid membranes showed that the membrane conductivity increases considerably by insertion of the complex into the membrane. The increase in the conductivity was only obtained if both layers of the membrane were modified with the complex. 1H NMR spectroscopic studies using DOPC liposomes gave information about the location of the modifier complex in the lipid membrane. From chemical shift effects we concluded that the complex resides in the choline head group region of the membrane and also in the membrane interior near the -C =C- lipid double bond, but not in the center of the bilayer. For understanding of the mentioned conductivity effect we assume that the europium complex induces defects of yet unknown structure in the lipid matrix which provide paths for the ion transfer through the membrane. As appropriate measurements revealed, these paths seem to conduct cations predominantly. Investigating the current voltage behavior of the modified lipid membranes in dependence on the ion concentration we obtained different shaped current-voltage curves. Calculation showed that a model with only one energy barrier inside the membrane is unable to describe these curves kinetically. However, by assuming two energy barriers--one barrier in each membrane lipid layer--the observed curve can be described satisfactorily.     

Energetics of permeation of thin lipid membranes by ions

The energy of an ion in a thin hydrocarbon membrane relative to its energy in a bulk aqueous phase is considered in terms of the electrostatic and surface components that may be expected to be involved. Except when diffusion activation energies are large compared to partition free energies, the latter will control permeation rate and the state of an ion having the lowest partition energy will be critical for its permeability. This minimum is found when an ion is surrounded with a thin layer of water. All ions of the same charge will tend to be at their lowest state in a sphere of water of the same size. It is concluded, therefore, that all ions of a given charge will have about the same permeability in lipid membranes.     

Theory for carrier-mediated zero-current conductance of bilayers extended to allow for nonequilibrium of interfacial reactions,spatially dependent mobilities and barrier shape

Summary A generalized form of the electrodiffusion equation, allowing for any shape of symmetrical energy barrier and any spatial dependence of the diffusion coefficient, is used to deduce theoretically the carrier-mediated conductance for thin (e.g., bilayer) membranes in the limit of low applied current. Both the Nernst-Planck and the Eyring single-barrier treatments are special cases of this more general approach, which allows for the effect of non-uniform properties of the lipid and non-uniform profiles of the forces acting within the membrane interior. Two independent mechanisms for ions to cross the membrane-solution interfaces are considered; namely, (1) the reaction at the interface between ions from solution and carriers from the membrane, and (2) the partition across the interfaces of complexes already formed in the solution. The rates of these reactions are taken into account using the rate equations of chemical kinetics; and the Poisson-Boltzmann equation is integrated in the aqueous solutions to evaluate the effect of charged polar head groups of the lipid. The analysis leads to an expression for the conductance, which, in the approximation of constant field, is an explicit function of such experimentally variable parameters as the concentrations and types of permeant ions and carriers in the aqueous phases, the total ionic strength and the nature of the polar head groups of the lipid. The functional relationship observable in an unknown membrane can, in principle, enable one to deduce such information as the mechanism of ion permeation across the interfaces, the magnitude of the surface charge, and the degree of ion-carrier complexation in the aqueous solutions.     

The inner membrane barrier of lipid membranes experienced by the valinomycin/Rb+ complex: charge pulse experiments at high membrane voltages.

The kinetic analysis of charge pulse experiments at planar lipid membranes in the presence of macrocyclic ion carriers has been limited so far to the low voltage range, where, under certain simplifying conditions, an analytical solution is available. In the present study, initial voltages of up to 300 mV were applied to the membrane, and the voltage decay through the conductive pathways of the membrane was followed as a function of time. The system of differential equations derived from the transport model was solved numerically and was compared with the experimental data. The generalized kinetic analysis of charge pulse experiments and of steady-state current-voltage curves was used to study the voltage dependence of the individual transport steps and to obtain information on the shape of the inner membrane barrier. The data were found to be consistent with a comparatively broad inner barrier such as a trapezoidal barrier or an image force barrier. The inner barrier was found to sense 70-76% of the voltage applied to the membrane. As a consequence, 24-30% of the voltage acts on the two interfacial barriers between membrane and water. The data refer to membranes formed from monoolein, monoeicosenoin, or monoerucin in n-decane.     

Electrical noise from lipid bilayer membranes in the presence of hydrophobic ions

Summary In the presence of the hydrophobic ion dipicrylamine, lipid bilayer membranes exhibit a characteristic type of noise spectrum which is different from other forms of noise described so far. The spectral density of current noise measured at zero voltage increases in proportion to the square of frequency at low frequencies and becomes constant at high frequencies. The observed form of the noise spectrum can be interpreted on the basis of a transport model for hydrophobic ions in which it is assumed that the ions are adsorbed in potential-energy minima at either membrane surface and are able to cross the central energy barrier by thermal activation. Accordingly, current-noise results from random fluctuations in the number of ions jumping over the barrier from right to left and from left to right. On the basis of this model the rate constantk _i for the translocation of the hydrophobic ion across the barrier, as well as the mean surface concentrationN _t of adsorbed ions may be caluculated from the observed spectral intensity of current noise. The values ofk _i obtained in this way closely agree with the results of previous relaxation experiments. A similar, although less quantitative, agreement is also found for the surface concentrationN _t .     

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.     

Curvature-electric effect in black lipid membranes

Upon periodical bending of a BLM, by means of oscillating hydrostatic pressure with sound frequency, the generation of an a.c. electric current with the same frequency can be observed under short circuit conditions. Previously, this phenomenon was attributed by us to a displacement current due to the oscillating flexoelectric polarization. The latter is proportional to the membrane curvature and depends on the lipid dipole moment and surface charge.The theory of this effect is outlined here. Earlier results concerning dipolar and quadrupolar contributions to the total current are presented and new expressions about charge contributions are derived for the two basic regimes of free and blocked lateral lipid exchange.Further, a systematic study of the frequency dependence of the amplitude and phase of the curvature-electric signal from a bacterial phosphatidylethanolamine/n-decane BLM is reported. Constant membrane curvature at each vibration frequency was assured by a calibration of the capacitance current observed with a small transmembrane voltage.The frequency dependence of the curvature-electric current amplitude was characterized by two regions: low frequency plateau and high frequency slope, the boundary between them being about 160 Hz. Such behaviour suggested a switching of the mechanism of membrane polarization from free to blocked lateral lipid exchange. Frequency dependence of the phase shift was characterized by low frequency and high frequency plateaus and a gradual transition between them. From phase measurements on initially curved membranes the sign of the membrane flexo-coefficient was found to be negative.The influence of some modifiers of the surface charge and surface dipole, as well as of the membrane conductivity, upon the value of the effect was studied. Surface charge was separately measured by the internal field compensation method under an ionic strength gradient. The membrane flexoelectric coefficient was evaluated and compared to the theoretical predictions. A conclusion was drawn that under the present experimental conditions the main contribution to the effect comes from the curvatureinduced shift of the surface charge equilibrium.Presented at the Tenth International Liquid Crystal Conference, 15–21 July 1984, York, UK     

Functional Consequences of Oxidative Membrane Damage

The interaction of reactive oxygen species with biological membranes is known to produce a great variety of different functional modifications. Part of these modifications may be classified as direct effects. They are due to direct interaction of the reactive species with the molecular machinery under study with a subsequent chemical and functional modification of these molecules. An important part of the observed functional modifications are, however, indirect effects. They are the consequence of an oxidative modification of the environment of biological macromolecules. Lipid peroxidation—via its generation of chemically reactive products—contributes to the loss of cellular functions through the inactivation of membrane enzymes and even of cytoplasmic (i.e., water soluble) proteins. Oxidation of membrane lipids may, however, also increase the efficiency of membrane functions. This was observed for a series of transport systems. Lipid peroxidation was accompanied by activation of certain types of ion channels and ion carriers. The effect is due to an increase of the polarity of the membrane interior by accumulation of polar oxidation products. The concomitant change of the dielectric constant, which may be detected via the increase of the membrane capacitance, facilitates the opening of membrane channels and lowers the inner membrane barrier for the movement of ions across the membrane. The predominant effect, however, at least at a greater extent of lipid peroxidation, is the inhibition of membrane functions. The strong increase of the leak conductance contributes to the depolarization of the membrane potential, it destroys the barrier properties of the membrane and it may finally lead, via an increase of cytoplasmic Ca²⁺ concentration, to cell death. The conclusions were derived from experiments performed with different systems: model systems in planar lipid membranes, native ion channels either reconstituted in lipid membranes or investigated in their natural environment by the patch-clamp method, and two important ion pumps, the Na/K-ATPase and the sarcoplasmic reticulum (SR) Ca-ATPase.     

Influence of surface potentials on the mitochondrial H+ pump and on lipid-phase transitions.

It has been shown that the surface potential of lipid membranes, as well as of mitochondria, can be shifted more positive by absorption of alkylbiguanides. Both phospholipid vesicles and natural membranes respond in an analogous way to this shift. Ion activities at the immediate membrane surface are influenced by sign and magnitude of the surface charge. Corresponding effects on ion transport and on fluorescence-probe binding can be observed. The mitochondrial H+ pump is inhibited when the surface charge is shifted more positive. In contrast,the absolute charge density determines the temperature of the ordered-fluid transition. The latter is increased by biguanides, suggesting that the membrane is rendered more rigid. The experiments make obvious that physical relations derived from model systems apply equally well to lipid-containing natural membranes.     

Calculation of deformation energies and conformations in lipid membranes containing gramicidin channels.

In this paper we calculate surface conformation and deformation free energy associated with the incorporation of gramicidin channels into phospholipid bilayer membranes. Two types of membranes are considered. One is a relatively thin solvent-free membrane. The other is a thicker solvent-containing membrane. We follow the approach used for the thin membrane case by Huang (1986) in that we use smectic liquid crystal theory to evaluate the free energy associated with distorting the membrane to other than a flat configuration. Our approach is different from Huang, however, in two ways. One is that we include a term for surface tension, which Huang did not. The second is that one of our four boundary conditions for solving the fourth-order differential equation describing the free energy of the surface is different from Huang's. The details of the difference are described in the text. Our results confirm that for thin membranes Huang's neglect of surface tension is appropriate. However, the precise geometrical form that we calculate for the surface of the thin membrane in the region of the gramicidin channel is somewhat different from his. For thicker membranes that have to deform to a greater extent to accommodate the channel, we find that the contribution of surface tension to the total energy in the deformed surface is significant. Computed results for the shape of the deformed surface, the total energy in the deformed surface, and the contributions of different components to the total energy, are presented for the two types of membranes considered. These results may be significant for understanding the mechanisms of dimer formation and breakup, and the access resistance for ions entering gramicidin channels.