A theory for the effects of diffusion-reaction coupling on the carrier-mediated ionic permeability in lipid bilayers

The zero-current membrane potential and the current-voltage relations are discussed theoretically for the case in which ionic transport is mediated by carriers that form complexes with ions in the aqueous phase (‘solution complexation’ mechanism). Interest for this topic originated partly from the finding that gradients of the neutral cyclic peptide PV, cyclo (dVal-lPro-lVal-dPro)₃, commonly thought to act as a carrier via ‘solution complexation’, generate Nernstian potentials across lipid bilayers separating solutions of identical ion composition. It is shown that the general expression for the potential in a gradient of carriers reduces to the Nernst equation under any of the following conditions: slow aqueous reaction; impermeability of the membrane to the neutral carriers; high concentration of the complexing ions in solution; finite permeability of the membrane to the neutral carrier, but faster rate of movement from the membrane surface into the torus than across the middle or out of the membrane. In symmetrical solutions, the conductance is most typically characterized by a quantity that we designate by δ*, which has the dimensions of a length and is generally a complex function of ion activity. Comparing the thory with previous data on dioleoylphosphatidylcholine membranes in the presence of PV and K⁺, the order of magnitude of the rates of the aqueous reaction and of the membrane permeability to the neutral carriers is tentatively estimated.     

Effects of unstirred layers on the steady-state zero-current conductance of bilayer membranes mediated by neutral carriers of ions

Some effects of diffusion polarization and chemical reactions on the steady-state zero-current conductance of lipid bilayers mediated by neutral carriers of ions have been studied theoretically and experimentally. Assuming that ion permeation across the interfaces occurs via a heterogeneous reaction between ions in the solution and carriers in the membrane, the relationship between the conductance and the aqueous concentration of carriers is shown to be linear only in a limited range of sufficiently low concentrations. At higher carrier concentrations, which for the most strongly bound cations are within the range of the experimentally accessible values, the conductance is expected to become limited by diffusion of the carried ion in the unstirred layers and therefore reach an upper limiting value independent of the membrane properties. This expectation has been successfully verified for glyceryl-monooleate membranes in the presence of the ions K+, Rb+ and NH+4 and carriers such as valinomycin and trinactin. The experimental results support, at least for the present system, the generally accepted view that complexation between ions and the macrocyclic antibiotics occurs at the membrane surface; it is shown, in fact, that for a different mechanism, such as that by which the complexes would form in the aqueous solutions and cross the interfaces as lipid-soluble ions, the same type of saturation would be expected to be observable only for unrealistically high values of the rate constants of the ion-carrier association. A previously proposed criterion to distinguish between these two mechanisms, based on the dependence of the conductance on the ion concentration, is discussed from the viewpoint of this more comprehensive model.     

Electrostatic calculations for an ion channel. I. Energy and potential profiles and interactions between ions

The electrostatic energy profile of one, two, or three ions in an aqueous channel through a lipid membrane is calculated. It is shown that the previous solution to this problem (based on the assumption that the channel is infinitely long) significantly overestimates the electrostatic energy barrier. For example, for a 3-A radius pore, the energy is 16 kT for the infinite channel and 6.7 kT for an ion in the center of a channel 25 A long. The energy as a function of the position of the ion is also determined. With this energy profile, the rate of crossing the membrane (using the Nernst-Planck equation) was estimated and found to be compatible with the maximum conductance observed for the gramicidin A channel. The total electrostatic energy (as a function of position) required to place two or three ions in the channel is also calculated. The electrostatic interaction is small for two ions at opposite ends of the channel and large for any positioning of the three ions. Finally, the gradient through the channel of an applied potential is calculated. The solution to these problems is based on solving an equivalent problem in which an appropriate surface charge is placed on the boundary between the lipid and aqueous regions. The magnitude of the surface charge is obtained from the numerical solution for a system of coupled integral equations.     

Surface charge, surface dipoles and membrane conductance

The conductance of lipid membranes in the presence of nonactin is changed by the adsorption of small amounts of ionic and zwitterionic surfactants. The conductance changes are, in many instances, not accounted for by the variation in surface charge or diffuse double layer potential as calculated from Gouy-Chapman theory. The changes are, however, all accurately accounted for by the variation in total potential across the membrane interface. This potential includes contributions from surface dipoles and specific adsorption, as well as any diffuse double layer effects not included in the Gouy-Chapman theory.The total potential changes were inferred from Volta or compensational potential changes at bulk oil (and monolayer)/aqueous solution interfaces. Surface charge densities were found by standard thermodynamic methods involving the use of the Gibbs equation. Electrokinetic potentials for the appropriate surfaces were also measured and, in general, agreed well with the diffuse double layer potentials calculated from the Gouy-Chapman theory.     

Photogating of ionic currents across lipid bilayers. Electrostatics of ions and dipoles inside the membrane.

The conductances of the lipophilic ions tetraphenylboride and tetraphenylphosphonium across a lipid bilayer can be increased or decreased, i.e., gated, by the photoformation of closed-shell metalloporphyrin cations within the bilayer. The gating can be effected by pulsed or continuous light or by chemical oxidants. At high concentrations of lipophilic anions where the dark conductance is saturated due to space charge in the bilayer, the photogated conductance can increase 15-fold. The formation of porphyrin cations allows the conductance to increase to its nonspace charge limited value. Conversely, the decrease of conductance in the light of phosphonium cations diminishes toward zero as the dark conductance becomes space charge limited. We present electrostatic models of the space charge limited conductance that accurately fit the data. One model includes an exponentially varying dielectric constant for the polar regions of the bilayer that allows an analytical solution to the electrostatic problem. The exponential variation of the dielectric constant effectively screens the potential and implies that the inside and outside of real dielectric interfaces can be electrically isolated from one another. The charge density, the distance into the membrane of the ions, about one-quarter of its thickness, and the dielectric constant at that position are determined by these models. These calculations indicate that there is insufficient porphyrin charge density to cancel the boride ion space charge and the following article proposes a novel ion chain mechanism to explain these effects. These models indicate that the positive potential arising from oriented carbonyl ester groups, previously used to explain the 10(3)-fold larger conductance of hydrophobic anions over cations, is smaller than previously estimated. However, the synergistic movement of the positive choline group into the membrane can account for the large positive potential.     

Mechanism of the electric response of lipid bilayers to bitter substances.

In order to clarify by what mechanism the lipid bilayer membrane changes its potential under the stimulation of bitter substances, a microscopic model for the effects of the substances on the membrane is presented and studied theoretically. It is assumed that the substances are adsorbed on the membrane and change the partition coefficients of ions between the membrane and the stimulation solution, the dipole orientation in the polar head, and the diffusion constants of ions in the membrane. It is shown, based on the comparison of the calculated results with the experimental ones, that the response arises mainly from a change in the partition coefficients. Protons play an essential role in the membrane potential variation due to the change in their partition coefficients. The present model reproduces the following observed unique properties in the response of lipid bilayers to bitter substances, which cannot be accounted for by the usual channel model for the membrane potential: 1) the response of the membrane potential appears even under the condition that there is no ion gradient across the membrane, 2) the response remains even when the salt in the stimulating solution is replaced with the salt made of an impermeable cation, and 3) the direction of the polarization of the potential is not reversed, even when the ion gradient across the bilayer is reversed.     

Surface dipole potential at the interface between water and self-assembled monolayers of phosphatidylserine and phosphatidic acid.

The nature and magnitude of the surface dipole potential chi at a membrane/water interface still remain open to discussion. By combining measurements of differential capacity C and charge density sigma at the interface between self-assembled monolayers of phosphatidylserine and phosphatidic acid supported by mercury and aqueous electrolytes of different concentration and pH, a sigmoidal dependence of chi upon sigma is revealed, with the inflection at sigma = 0. This behavior is strongly reminiscent of the surface dipole potential due to reorientation of adsorbed water molecules at electrified interfaces. The small increase in C with a decrease in the frequency of the AC signal below approximately 80 Hz, as observed with phospholipid monolayers with partially protonated polar groups, is explained either by a sluggish collective reorientation of some polar groups of the lipid or by a sluggish movement of protons across the polar head region.     

Membrane surface-charge titration probed by gramicidin A channel conductance.

We manipulate lipid bilayer surface charge and gauge its influence on gramicidin A channel conductance by two strategies: titration of the lipid charge through bulk solution pH and dilution of a charged lipid by neutral. Using diphytanoyl phosphatidylserine (PS) bilayers with CsCl aqueous solutions, we show that the effects of lipid charge titration on channel conductance are masked 1) by conductance saturation with Cs+ ions in the neutral pH range and 2) by increased proton concentration when the bathing solution pH is less than 3. A smeared charge model permits us to separate different contributions to the channel conductance and to introduce a new method for "bilayer pKa" determination. We use the Gouy-Chapman expression for the charged surface potential to obtain equilibria of protons and cations with lipid charges. To calculate cation concentration at the channel mouth, we compare different models for the ion distribution, exact and linearized forms of the planar Poisson-Boltzmann equation, as well as the construction of a "Gibbs dividing surface" between salt bath and charged membrane. All approximations yield the intrinsic pKain of PS lipid in 0.1 M CsCl to be in the range 2.5-3.0. By diluting PS surface charge at a fixed pH with admixed neutral diphytanoyl phosphatidylcholine (PC), we obtain a conductance decrease in magnitude greater than expected from the electrostatic model. This observation is in accord with the different conductance saturation values for PS and PC lipids reported earlier (, Biochim. Biophys. Acta. 552:369-378) and verified in the present work for solvent-free membranes. In addition to electrostatic effects of surface charge, gramicidin A channel conductance is also influenced by lipid-dependent structural factors.     

Effect of ultrasound on a bilayer lipid membrane.

The effects of continuous wave ultrasound at a frequency of 1 MHz in the intensity range of 0-1.4 W/cm2 on an oxidized cholesterol bilayer lipid membrane (BLM) were observed. Ultrasound at 1.5 W/cm2 broke the membrane; in the range from 0.5 to 1.4 W/cm2, it accelerated the draining of the bulk lipid solution from the annulus to the Teflon support. At all intensities it has no effect on the conductance, the capacitance, or the dependence of each on the voltage applied across the membrane. Electrical parameters were measured in the presence of aqueous solutions of NaCl, KCl, and distilled water. The motivation and results of this project are explained in relation to an overall objective of determining the specific effects of ultrasound on biological membranes.     

Permeation of ions across the potassium channel: brownian dynamics studies

The physical mechanisms underlying the transport of ions across a model potassium channel are described. The shape of the model channel corresponds closely to that deduced from crystallography. From electrostatic calculations, we show that an ion permeating the channel, in the absence of any residual charges, encounters an insurmountable energy barrier arising from induced surface charges. Carbonyl groups along the selectivity filter, helix dipoles near the oval chamber, and mouth dipoles near the channel entrances together transform the energy barrier into a deep energy well. Two ions are attracted to this well, and their presence in the channel permits ions to diffuse across it under the influence of an electric field. Using Brownian dynamics simulations, we determine the magnitude of currents flowing across the channel under various conditions. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength causes a steady decrease in the channel conductance. The current also decreases systematically when the effective dielectric constant of the channel is lowered. The conductance with the optimal choice of dipoles reproduces the experimental value when the dielectric constant of the channel is assumed to be 60. The current-voltage relationship obtained with symmetrical solutions is linear when the applied potential is less than approximately 100 mV but deviates from Ohm's law at a higher applied potential. The reversal potentials obtained with asymmetrical solutions are in agreement with those predicted by the Nernst equation. The conductance exhibits the saturation property observed experimentally. We discuss the implications of these findings for the transport of ions across the potassium channels and membrane channels in general.     

Divalent Ions and the Surface Potential of Charged Phospholipid Membranes

Phospholipid bilayer membranes were bathed in a decimolar solution of monovalent ions, and the conductance produced by neutral carriers of these monovalent cations and anions was used to assess the electric potential at the surface of the membrane. When the bilayers were formed from a neutral lipid, phosphatidylethanolamine, the addition of alkaline earth cations produced no detectable surface potential, indicating that little or no binding occurs to the polar head group with these ions. When the bilayers were formed from a negatively charged lipid, phosphatidylserine, the addition of Sr and Ba decreased the magnitude of the surface potential as predicted by the theory of the diffuse double layer. In particular, the potential decreased 27 mv for a 10-fold increase in concentration in the millimolar-decimolar range. A 10-fold increase in the Ca or Mg concentration also produced a 27 mv decrease in potential in this region, which was again due to screening, but it was necessary to invoke some specific binding to account for the observation that these cations were effective at a lower concentration than Ba or Sr. It is suggested that the ability of the alkaline earth cations to shift the conductance-voltage curves of a nerve along the voltage axis by 20–26 mv for a 10-fold increase in concentration may be due to essentially a screening rather than a binding phenomenon.     

Modification of ion transport in lipid bilayer membranes in the presence of 2,4-dichlorophenoxyacetic acid. II. Suppression of tetraphenylborate conductance and changes of interfacial potentials.

It has been shown that the blocking of negatively charged tetraphenylborate ion transport in phosphatidylcholine (PC)-cholesterol membranes by the herbicide 2,4-dichlorophenoxyacetic acid (2,4-D) is dominated by suppression of TPhB- diffusion across the membrane interior, rather than by the decrease of adsorption of TPhB- ions at the membrane surface. The blocking effect can be associated with the decrease of electric potential inside the membrane with respect to that of the aqueous medium, this decreases being proportional to the concentration of 2,4-D in the aqueous solution. It has been estimated that 25 - 30% of the total 2,4-D-induced change of the potential difference is between the plane of absorption of TPhB- and the aqueous solution, and the remaining fraction is between the membrane interior and the absorption plane. The results of this study support the dipolar hypothesis of 2,4-D action in lipid membranes. These conclusions are further supported by measurements changes of electric potential difference across air/water and air/lipid monolayer/water interfaces. It has been found that the electric potential of the nonpolar side of the interface decreases in the presence of neutral molecules of 2,4-D and that this effect becomes more prominent in presence of electrolyte. We have confirmed that PC-cholesterol monolayer cannot be considered as a model for half of the bilayer membrane because of the disagreement between the changes of the interfacial potential difference of PC-cholesterol monolayers and those determined from studied of transport of positive and negative ions across bilayer membranes. In contract, we have found close agreement between the 2,4-D-induced changes of electric potential of the lipid hydrocarbon region in glycerolmonooleate (GMO) membranes and GMO monolayers. We suggest that the action of 2,4-D in lipid membranes is not associated with the changes of orientation of dipoles of lipids constituting the membranes, but rather with a layer of 2,4-D molecules absorbed at the nonpolar/polar membrane boundary.     

A theory for the effects of neutral carriers such as the macrotetralide actin antibiotics on the electric properties of bilayer membranes

Summary To develop a quantitiative theoretical treatment for the effects of neutral macrocyclic antibiotics on the electrical properties of phospholipid bilayer membranes, this paper proceeds from the known ability of such molecules to form stoichiometric, lipid-soluble complexes with cations and deduces the electrical properties that a simple organic solvent phase would have if it were made into a membrane of the thinness of the phospholipid bilayer. In effect, we postulate that the essential barrier to ion movement across a bilayer membrane is its liquid-like hydrocarbon interior and that the neutral macrocyclic antibiotics bind monovalent cations and solubilize them in the membrane as mobile positively charged complexes. Using the Poisson-Boltzmann equation to describe the equilibrium profile of the electrical potential, it is shown that an excess of the positive complexes over all the other ions is expected in the membrane as a net space charge for appropriate conditions of membrane thickness and values of the partition coefficients of the various ionic species and without requiring the presence of fixed charges. Describing the fluxes of these complexes by the Nernst-Planck equation and neglecting the contribution to the electric current of uncomplexed ions, theoretical expressions are derived for the membrane potential in ionic mixtures, as well as for the limiting value of the membrane conductance at zero current when the membrane is interposed between identical solutions. The expressions are given in terms of the ionic activities and antibiotic concentrations in the aqueous solutions so as to be accessible to direct experimental test. Under suitable experimental conditions, the membrane potential is described by an equation recognizible as the Goldman-Hodgkin-Katz equation, in which the permeability ratios are combinations of parameters predicted from the present theory to be independently determinable from the ratio of membrane conductances in single salt solutions. Since this identity between permeability and conductance ratios is expected also for systems obeying the Independence Principle of Hodgkin and Huxley, the applicability of this principle to membranes exposed to antibiotics is discussed, and it is shown that this principle is compatible with the permeation mechanism proposed here.     

Phloretin-induced changes in ion transport across lipid bilayer membranes

Phloretin, the aglucone derivative of phlorizin, increases cation conductance and decreases anion conductance in lipid bilayer membranes. In this paper we present evidence that phloretin acts almost exclusively by altering the permeability of the membrane interior and not by modifying the partition of the permanent species between the membrane and the bulk aqueous phases. We base our conclusion on an analysis of the current responses to a senylborate, and the cation complex, peptide PV-K+. These results are consistent with the hypothesis that phloretin decreases the intrinsic positive internal membrane potential but does not modify to a great extent the potential energy minima at the membrane interfaces. Phloretin increases the conductance for the nonactin-K+ complex, but above 10(-5) M the steady- state nonactin-K+ voltage-current curve changes from superlinear to sublinear. These results imply that, above 10(-5) M phloretin, the nonactin-5+ transport across the membrane becomes interfacially limited.     

Properties of the porin of Haemophilus influenzae type b in planar lipid bilayer membranes

The major outer membrane protein (40 kDa) of the bacterium Haemophilus influenzae type b is a porin which forms transmembrane permeability channels. It has an exclusion limit for oligosaccharides of about 1.4 kDa. When this protein was added to the aqueous phase which was bathing a planar lipid bilayer, it caused the conductance of the membrane to increase by several orders of magnitude. At low protein concentrations (2-10 pM), the conductance of the membrane increased in a stepwise fashion with an average single-channel conductance of 1.1 nS in 1 M KCl. Single-channel experiments were performed with a variety of different salts. The conductance of single channels was proportional to the specific conductance of the aqueous solution which was bathing the membrane. Current through the pores was proportional to the applied voltage, indicating that these pores are not voltage-controlled. The 40 kDa porin was very slightly cation-selective: the pores were about 1.6-times more permeable to potassium ions than to chloride ions. These properties of the 40 kDa porin are those of large water-filled channels and are characteristic of most bacterial porins. The single-channel conductance of the porin is, however, much smaller than might be expected from its exclusion limit. A model is proposed which could explain the differences in apparent pore size.     

Strong electrolyte continuum theory solution for equilibrium profiles, diffusion limitation, and conductance in charged ion channels.

The solution for the ion flux through a membrane channel that incorporates the electrolyte nature of the aqueous solution is a difficult theoretical problem that, until now, has not been properly formulated. The difficulty arises from the complicated electrostatic problem presented by a high dielectric aqueous channel piercing a low dielectric lipid membrane. The problem is greatly simplified by assuming that the ratio of the dielectric constant of the water to that of the lipid is infinite. It is shown that this is a good approximation for most channels of biological interest. This assumption allows one to derive simple analytical expressions for the Born image potential and the potential from a fixed charge in the channel, and it leads to a differential equation for the potential from the background electrolyte. This leads to a rigorous solution for the ion flux or the equilibrium potential based on a combination of the Nernst-Planck equation and strong electrolyte theory (i.e., Gouy-Chapman or Debye-Huckel). This approach is illustrated by solving the system of equations for the specific case of a large channel containing fixed negative charges. The following characteristics of this channels are discussed: anion and mono- and divalent cation conductance, saturation of current with increasing concentration, current-voltage relationship, influence of location and valence of fixed charge, and interaction between ions. The qualitative behavior of this channel is similar to that of the acetylcholine receptor channel.     

Discrete conductance fluctuations in lipid bilayer protein membranes

Discrete fluctuations in conductance of lipid bilayer membranes may be observed during the initial stages of membrane interaction with EIM ("excitability inducing material"), during destruction of the EIM conductance by proteolysis, and during the potential-dependent transitions between low and high conductance states in the "excitable" membranes. The discrete conductance steps observed during the initial reaction of EIM with the lipid membranes are remarkably uniform, even in membranes of widely varying lipid composition. They range only from 2 to 6 x 10_-10 ohm_-1 and average 4 x 10_-10 ohm_-1. Steps found during destruction of the EIM conductance by proteolysis are somewhat smaller. The transition between high conductance and low conductance states may involve steps as small as 0.5 x 10_-10 ohm_-1. These phenomena are consistent with the formation of a stable protein bridge across the lipid membrane to provide a polar channel for the transport of cations. T6he uniform conductance fluctuations observed during the formation of these macromolecular channels may indicate that the ions in a conductive channel, in its open state, are largely protected from the influence of the polar groups of the membrane lipids. Potential-dependent changes in conductance may be due to configurational or positional changes in the protein channel. Differences in lipid-lipid and lipid-macromolecule interactions may account for the variations in switching kinetics in various membrane systems.     

Mechanism of anion-cation selectivity of amphotericin B channels

Summary Zero current potential and conductance of ionic channels formed by polyene antibiotic amphotericin B in a lipid bilayer were studied in various electrolyte solutions. Nonpermeant magnesium and sulphate ions were used to independently vary the concentration of monovalent anions and cations as well as to maintain the high ionic strength of the two solutions separated by the membrane. Under certain conditions the channels select very strongly for anions over cations. They are permeable to small inorganic anions. However, in the absence of these anions the channels are practically impermeable to any cation. In the presence of a permeant anion the contribution of monovalent cations to channel conductance grows with an increase in the anion concentration. The ratio of cation-to-anion permeability coefficients is independent of the membrane potential and cation concentration, but it does depend linearly on the sum of concentrations of a permeant anion in the two solutions. These results are accounted for on the assumption that a cation can enter only an anion-occupied channel to form an ionic pair at the center of the channel. The cation is also assumed to slip past the anion and then to leave the channel for the opposite solution. This model with only few parameters can quantitatively describe the concentration dependences of conductance and zero current potential under various conditions.     

Effect of 3-phenylindole on lipophilic ion and carrier-mediated ion transport across bilayer lipid membranes

The physical effects of 3-phenylindole, an antimicrobial compound which interacts with phospholipids, on ion transport across phosphatidylcholine-cholesterol bilayers have been investigated using three lipophilic ions and one ion-carrier complex. It was found that 3-phenylindole increased membrane electrical conductance of positively charged membrane probes and decreased electrical conductance of negatively charged probes. The enhancement of conductance detected by nonactin-K+ complex and tetraphenylarsonium+ was several orders of magnitude, whereas the suppression of conductance due to tetraphenylborate- and dipicrylamine- was less than a factor of ten. Presence of 3-phenylindole in aqueous phase slightly decreased adsorption of tetraphenylborate- and dipicrylamine- at the membrane surface. From the voltage dependence of the steady-state conductance it was shown that 3-phenylindole induced kinetic limitation of membrane transport of potassium mediated by nonactin. No such limitation was found in the case of tetraphenylarsonium+ transport. These results are shown to be consistent with the present concept of ion diffusion in membranes and the assumption that 3-phenylindole decreases the electric potential in the membrane interior. The asymmetry of the effect of 3-phenylindole on the magnitude of conductance changes for positively and negatively charged membrane permeable ions is also discussed as a reflection of the discreteness of both the absorbed 3-phenylindole and lipid dipoles.     

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.