Membrane Potentials at Zero Current: The Significance of a Constant Ionic Permeability Ratio

The possibility has been examined that the Goldman-Hodgkin-Katz equation for V(0), the total membrane potential at zero current, can be derived with constant permeability ratios from a thermodynamic treatment. The flux equations have been integrated under zero current conditions subject only to the restriction that the total membrane potential should be independent of internal concentration profiles, which is the requirement for the premeability ratios to be phenomenological constants, independent of solution conditions. No assumptions have been made concerning the electric potential profile. It was found that a constant permeability ratio can only be characteristic of systems satisfying certain relationships between ionic conductances and chemical potentials. From these relationships it was possible to define the permeability ratio in terms of the thermodynamic properties of the membrane quite generally and to identify the permeability ratio as the product of mobility ratio and ratio of partition coefficients. Moreover, the ionic conductance ratio at any point in the membrane has been shown to be expressable explicitly in terms of the permeability ratio and the activities of an external solution which would be in equilibrium with the point under consideration. Lastly, a number of conclusions have been reached regarding the range of applicability of the Goldman-Hodgkin-Katz equation with constant permeability ratios.     

The Double Fixed Charge Membrane: Solution-Membrane Ion Partition Effects and Membrane Potentials

An analysis is made of the effect of solution-membrane partition of ions on the electrostatic potential and ion concentration profiles in fixed charge membranes. It is shown that the inclusion of partition effects gives rise to large solution-membrane “Donnan” potentials even when the concentration of fixed charges is of the same order as the concentration of the external solution. This effect renders the system and the simplified analysis of the double fixed charge membrane (FCM) previously given more applicable to biological membranes. An analysis is also given of the voltage dependence of the fluxes of individual ion species in the double FCM when it separates different ionic solutions and an expression is deduced for the membrane resting potential. Although the latter is similar in form to the Goldman-Hodgkin-Katz (GHK) equation the corresponding value of the permeability ratio P_C1/P_K is under certain specified conditions both concentration and potential dependent.     

Membrane transport of polysialic acid chains: modulation of transmembrane potential

Polysialic acid (polySia) is a long polyanionic polymer (with the degree of polymerization, DP, up to 200) of negatively charged sialic acid monomers. PolySia chains are bound to the external surface of some neuroinvasive bacterial cells and neural cells. PolySia serves as a potent regulator of cell interactions via its unusual biophysical properties. In the present paper, the analysis, based on the Goldman-Hodgkin-Katz equation, of transmembrane potential changes resulting from transmembrane translocation of polySia is performed. The relationships between the transmembrane potential and the polySia DP (up to 200), the temperature, the cation/ anion permeability ratio, and the inner/outer concentration ratio of polySia has been plotted and discussed. The maximal membrane potential changes, up to 118 mV, were found for a permeability ratio greater than one. The increase of the polySia chain length resulted in the diminution of this effect. The temperature-dependent changes in membrane potential were less than 7 mV in the range 0-50 degrees C. The change in the concentration ratio (into its reciprocal) resulted in a mirror reflection of the membrane potential curves. The results show that the expression of polySia chains in bacterial cells can be responsible for the modulation of the transmembrane potential of the bacterial inner membrane. We suggest that the polySia chains can influence the transmembrane potential of neural cell membranes in a similar way. This analysis also describes the effect of the transmembrane translocation of negatively charged polyanionic polynucleotydes on the cell membrane potential.     

Dependence of cellular potential on ionic concentrations. Data supporting a modification of the constant field equation.

The resting potential in the squid axon has been measured at various concentrations of Cl, K, Na, and Ca ions. The results of these measurements are compared with the Goldman-Hodgkin-Katz (GHK) equation and a modified constant field equation. This modified equation was derived by including currents carried by divalent ions and the effects of the unstirred layer and the periaxonal space. It is shown that, although the GHK equation can fit the V vs. [K]o data well, it has difficulty explaining the observed dependence of V on [Na]o when the axon is bathed in K-free artificial sea water. The use of the modified constant field equation removes this difficulty.     

Potassium current activated by depolarization of dissociated neurons from adult guinea pig hippocampus

Currents were generated by depolarizing pulses in voltage-clamped, dissociated neurons from the CA1 region of adult guinea pig hippocampus in solutions containing 1 microm tetrodotoxin. When the extracellular potassium concentration was 100 mM, the currents reversed at -8.1 +/- 1.6 mV (n = 5), close to the calculated potassium equilibrium potential of -7 mV. The currents were depressed by 30 mM tetraethylammonium in the extracellular solution but were unaffected by 4-aminopyridine at concentrations of 0.5 or 1 mM. It was concluded that the currents were depolarization-activated potassium currents. Instantaneous current-voltage curves were nonlinear but could be fitted by a Goldman-Hodgkin-Katz equation with PNa/PK = 0.04. Conductance-voltage curves could be described by a Boltzmann-type equation: the average maximum conductance was 65.2 +/- 15.7 nS (n = 9) and the potential at which gK was half-maximal was -4.8 +/- 3.9 mV (mean +/- 1 SEM, n = 10). The relationship between the null potential and the extracellular potassium concentration was nonlinear and could be fitted by a Goldman-Hodgkin-Katz equation with PNa/PK = 0.04. The rising phase of potassium currents and the decay of tail currents could be fitted with exponentials with single time constants that varied with membrane potential. Potassium currents inactivated to a steady level with a time constant of approximately 450 ms that did not vary with potential. The currents were depressed by substituting cobalt or cadmium for extracellular calcium but similar effects were not obtained by substituting magnesium for calcium.     

[K+] dependence of open-channel conductance in cloned inward rectifier potassium channels (IRK1, Kir2.1).

Potassium conduction through unblocked inwardly rectifying (IRK1, Kir2.1) potassium channels was measured in inside-out-patches from Xenopus oocytes, after removal of polyamine-induced strong inward rectification. Unblocked IRK1 channel current-voltage (I-V) relations show very mild inward rectification in symmetrical solutions, are linearized in nonsymmetrical solutions that bring the K+ reversal potential to extreme negative values, and follow Goldman-Hodgkin-Katz constant field equation at extreme positive E alpha. When intracellular K+ concentration (KIN) was varied, at constant extracellular K+ concentration (KOUT) the conductance at the reversal potential (GREV) followed closely the predictions of the Goldman-Hodgkin-Katz constant field equation at low concentrations and saturated sharply at concentrations of > 150 mM. Similarly, when KOUT was varied, at constant KIN, GREV saturated at concentrations of > 150 mM. A square-root dependence of conductance on KOUT is a well-known property of inward rectifier potassium channels and is a property of the open channel. A nonsymmetrical two-site three-barrier model can qualitatively explain both the I-V relations and the [K+] dependence of conductance of open IRK1 (Kir2.1) channels.     

A physical model of nerve axon—I. Ionic distribution,potential profile,and resting potential

A theory which is based on a set of assumptions different from those of the sodium theory is developed. Here the mobile ions are considered to be distributed at Donnan equilibrium and the axoplasm is regarded as an analog of a cation-exchanger. Following the spirit of the Debye-Hückel theory, some important features of the ionic distribution and electrical potential of the nerve fiber have been calculated. The results appear to be in better agreement with the experimental observations than the Goldman-Hodgkin-Katz equation. A summary of this work has been reported in the 1974 March meeting of the American Physical Society and the 1976 annual meeting of the Biophysical Society.     

The conductance of single cardiac sodium channels from guinea pig depends on the intracellular sodium concentration.

Currents through DPI 201-106 modified single sodium channels have been measured in cell-free inside-out patches from guinea-pig ventricular myocytes. Single-channel conductance and reversal potential of the sodium channel have been calculated at different intracellular sodium concentrations [( Na+]i) from microscopic I-V curves, which were obtained by application of linear voltage ramps. The relation between the reversal potential and [Na+]i could be fitted with a modified Goldman-Hodgkin-Katz equation with a relative permeability for K+ over Na+ ions of 0.054. The zero-current conductance of the Na channel as a function of [Na+]i shows a plateau value at low Na concentrations, and increases in a sigmoidal manner at higher concentrations. It is concluded that the Na channel can carry outward currents and that its conductance depends on [Na+]i.     

Membrane potential dependency of glutamic acid transport in rabbit jejunal brush-border membrane vesicles: K+ and H+ effects

We have applied our recently developed approach for quantitative generation and estimation of membrane potential differences (Berteloot, A. (1986) Biochim. Biophys. Acta 857, 180-188) to the reevaluation of glutamic acid transport rheogenicity in rabbit jejunal brush-border membrane vesicles. Membrane diffusion-potentials were created by altering iodide concentrations in the intra- and extravesicular compartments while keeping isosmolarity, isotonicity and ionic strength constant by chloride replacement. The known value of ion permeabilities relative to sodium in this preparation also allows calculation of membrane potential differences using the Goldman-Hodgkin-Katz equation. This strategy appears superior to more classical methods involving ionophore-induced membrane diffusion-potentials of protons or potassium as both cations have been shown to participate in the transport mechanism. In this paper, we demonstrate that this approach is perfectly suitable for the investigation of membrane potential dependency of glutamic acid transport as our results showed that chloride replacement by iodide did not affect uptake in vesicles with membrane potential clamped to zero by gramicidin D (sodium conditions) or by gramicidin D plus valimonycin (sodium + potassium conditions). The method thus allows to dissociate membrane potential effects from possible effects that might be introduced by altering the anion species. In these conditions, our studies clearly demonstrate that glutamic acid uptake, whether analyzed over a 1 min time scale or under initial rate conditions, was sensitive to membrane potential differences. However, our results also show that the electrogenicity of the transport system varied depending upon the intravesicular presence or absence of potassium, its presence stimulating the membrane potential dependency of uptake. This effect is modulated by the internal pH and it is concluded that inside H+ and K+ are not equivalent as countertransported cations. The external pH also seems to modulate the response to potential by acting on the fully loaded form(s) of the transporter. The possibility that outside H+ competes for (an) external Na+ binding site(s) and/or precludes the attachment of (an) extra sodium ion(s) should be considered.     

The causal theory of the resting potential of cells

In this pedagogical article the causal theory of the resting potential of cells is presented, which for given extracellular ion concentrations predicts the intracellular ones simultaneously with the resting potential. In addition to the Na, K-pump, fixed charges on the membrane surfaces are taken into account. The equation determining the resting potential in the causal theory suggests a new explanation of the genesis of the resting potential. The usual criterion for an ion pump to be electrogenic is not relevant for the whole of the resting potential, and may therefore be misleading. The physical meaning of the Goldman-Hodgkin-Katz formula for the membrane potential as a diffusion potential is also explained and tested with numbers for the giant axon of the squid. A significant discrepancy between theory and experiment is found which calls for an experimental re-examination of the constitutive equations for passive potassium and sodium currents.     

Ionic selectivity of the sodium channel of frog skeletal muscle

The ionic selectivity of the Na channel to a variety of metal and organic cations is studied in frog semitendinosus muscle. Na channel currents are measured under voltage clamp conditions in fibers bathed in solutions with all Na+ replaced by a test ion. Permeability ratios are calculated from measured reversal potentials using the Goldman-Hodgkin-Katz equation. The permeability sequence was Na+ approximately Li+ approximately hydroxylammonium greater than hydrazinium greater than ammonium greater than guanidinium greater than K+ greater than aminoguanidinium in the ratios 1:0.96:0.94:0.31:0.11:0.093:0.048:0.031. No inward currents were observed for Ca++, methylammonium, methylguanidinium, tetraethylammonium, and tetramethylammonium. The results are consistent with the Hille model of the Na channel selectivity filter of the node of Ranvier and suggest that the selectivity filter of the two channels is the same.     

Mode of action of amiloride in toad urinary bladder

Summary The effect of amiloride on the sensitivity to Na of the mucosal border of toad urinary bladder was investigated by recording Na concentration-dependent transepithelial potential difference (V _t ) and the intracellular potential. When mucosal Na concentration was normal, amiloride added to the mucosal solution at 10^–4 m markedly reduced the mucosal membrane potential (V _m ) and altered the potential profile from a two-step type to a well type. Similar changes were observed when Na was totally eliminated from the mucosal medium. The serosal membrane potential was insensitive to amiloride and elimination of mucosal Na. In the absence of amiloride, theV _t could be described by the Goldman-Hodgkin-Katz equation in the range of mucosal Na concentration from 0 to 16mm, and amiloride extended this concentration range. By using the Goldman-Hodgkin-Katz equation, Na permeability was calculated from the data ofV _t 's obtained in the allowed ranges of Na concentration and compared before and after the addition of amiloride. The results show that Na permeability decreases to 1/600 of control when the maximum dose of amiloride (10^–4 m) is applied. The relationship between Na permeability and amiloride concentration is well explained on the basis of assumptions that amiloride binds to the Na site of the mucosal border in one-to-one fashion and in a competitive manner with Na and that Na permeability reduces in proportion to increase in number of the sites bound with amiloride.     

Voltage dependent membrane conductances in cultured renal distal cells.

Cultured Na(+)-transporting epithelia from amphibian renal distal tubule (A6) were impaled with microelectrodes and analyzed at short-circuit and after transepithelial voltage perturbation to evaluate the influence of voltage on apical and basolateral membrane conductances. For equivalent circuit analysis, amiloride was applied at each setting of transepithelial potential. At short-circuit, apical and basolateral membrane conductances averaged 88 and 497 microS/cm2, respectively (n = 10). Apical membrane conductance, essentially due to Na(+)-specific pathways, decreased after depolarization of the apical membrane. The drop was considerably larger than predicted by the Goldman-Hodgkin-Katz (GHK) constant-field equation. This suggests decrease in permeability of the apical Na+ channels upon depolarization. Basolateral membrane conductance, preferentially determined by K+ channels, increased after hyperpolarization of the basolateral membrane. This behavior is contrary to the prediction of the GHK constant field equation and reflects inward rectification of the K+ channels. The observed rectification patterns can be valuable for maintenance of cellular homeostasis.     

Application of the Goldman-Hodgkin-Katz current equation to membrane current-voltage data

A new approach for the analysis of current-voltage (IV) data is presented and applied to a variety of published data collected from various systems. Our analysis of published results shows that our method of analysis can account for the observed IV data. The calculated permeability coefficients are in reasonable agreement with those calculated from ion fluxes. In those cases where two ions are assumed to carry the current, the ratios of the calculated permeability coefficients are in agreement with those ratios determined from the Goldman-Hodgkin-Katz voltage equation. In most cases, the entire IV curve can be accounted for by using our method of analysis. In several examples, only a portion of the IV curve is in agreement with the predictions. We attribute the failure to account for the IV data to reflect the failure of one or more of the assumptions used by the GHK current equation. In other cases, assuming that an additional ion carries the current, the treatment can account for the IV data. However, the identity of the extra ion cannot be established from these published data without additional studies (e.g., ionic replacement studies).     

Anion and cation permeability of a chloride channel in rat hippocampal neurons

The ionic permeability of a voltage-dependent Cl channel of rat hippocampal neurons was studied with the patch-clamp method. The unitary conductance of this channel was approximately 30 pS in symmetrical 150 mM NaCl saline. Reversal potentials interpreted in terms of the Goldman-Hodgkin-Katz voltage equation indicate a Cl:Na permeability ratio of approximately 5:1 for conditions where there is a salt gradient. Many anions are permeant; permeability generally follows a lyotropic sequence. Permeant cations include Li, Na, K, and Cs. The unitary conductance does not saturate for NaCl concentrations up to 1 M. No Na current is observed when the anion Cl is replaced by the impermeant anion SO4. Unitary conductance depends on the cation species present. The channel is reversibly blocked by extracellular Zn or 9-anthracene carboxylic acid. Physiological concentrations of Ca or Mg do not affect the Na:Cl permeability ratio. The permeability properties of the channel are consistent with a permeation mechanism that involves an activated complex of an anionic site, an extrinsic cation, and an extrinsic anion.     

Effects of Fusicoccin in 'Isolated' Guard Cells of Commelina communis L

The effects of 3 ? 10^–2 mol m^–3 FC on rubidium fluxesand contents in isolated guard cells of Commelina communis L.have been studied using ⁸⁶RbCl. Fusicoccin causes a marked stimulationof influx and an immediate, apparently irreversible, decreasein efflux of ⁸⁶Rb. The effect on influx is usually more importantin determining the new net flux of Rb. Observed fluxes differmarkedly from those predicted by the Goldman-Hodgkin-Katz equation,suggesting that FC does not act solely via an effect upon theplasmalemma potential. Fusicoccin appears to have a more directeffect upon the ion movements associated with changes in stomatalaperture than either ABA or transfer to the dark. Observed changesin Rb content cannot account fully for the osmotic changes associatedwith aperture increase. Key words: Fusicoccin, Guard cells, Ion fluxes, Commelina communis     

Generation of membrane potential beyond the conceptual range of Donnan theory and Goldman-Hodgkin-Katz equation

Donnan theory and Goldman-Hodgkin-Katz equation (GHK eq.) state that the nonzero membrane potential is generated by the asymmetric ion distribution between two solutions separated by a semipermeable membrane and/or by the continuous ion transport across the semipermeable membrane. However, there have been a number of reports of the membrane potential generation behaviors in conflict with those theories. The authors of this paper performed the experimental and theoretical investigation of membrane potential and found that (1) Donnan theory is valid only when the macroscopic electroneutrality is sufficed and (2) Potential behavior across a certain type of membrane appears to be inexplicable on the concept of GHK eq. Consequently, the authors derived a conclusion that the existing theories have some limitations for predicting the membrane potential behavior and we need to find a theory to overcome those limitations. The authors suggest that the ion adsorption theory named Ling’s adsorption theory, which attributes the membrane potential generation to the mobile ion adsorption onto the adsorption sites, could overcome those problems.     

EDHF is not K+ but may be due to spread of current from the endothelium in guinea pig arterioles

Endothelium-derived hyperpolarizing factor (EDHF)-attributed hyperpolarizations and relaxations were recorded simultaneously from submucosal arterioles of guinea pigs with the use of intracellular microelectrodes and a video-based system, respectively. Membrane currents were recorded from electrically short segments of arterioles under single-electrode voltage clamp. Substance P evoked an outward current with a current-voltage relationship that was well described by the Goldman-Hodgkin-Katz equation for a K+ current, consistent with the involvement of intermediate- and small-conductance Ca2+-activated K+ channels. 1-Ethyl-2-benzimidazolinone relaxed the arterioles and evoked hyperpolarizations that were blocked by charybdotoxin, but not by iberiotoxin. Application of K+ induced depolarization under conditions in which EDHF evoked hyperpolarization. The Ba2+-sensitive component of the K+-induced current was inwardly rectifying, in contrast to the outwardly rectifying current evoked by substance P. EDHF-attributed hyperpolarizations in dye-identified smooth muscle cells were indistinguishable from those recorded from dye-identified endothelial cells in the same arterioles. These results provide evidence that EDHF is not K+ but may involve electrotonic spread of hyperpolarization from the endothelial cells to the smooth muscle cells.     

The low conductance K(+) channel in human colonic crypt cells has a voltage-dependent permeability not affected by Mg(++)

The low conductance K(+) channel found in human colonocytes was investigated using the patch-clamp technique. The channel is Ca(++)-dependent and is blocked by Ba(++) (5 mM) with a decrease in open probability from 0.42 to 0.19. At -40 mV the slope conductance was 29 pS (using intracellular solution in the pipette). In inside-out patches, inward rectification was seen both with KCl (pipette)/NaCl (bath) solutions as well as KCl/KCl solutions. The rectification could not be affected by omitting Mg(++) from the pipette or the bath solution, neither by exposing the patches to the polyamine spermine (1 mM). Using the Goldman-Hodgkin-Katz equation we show that the permeability decreased in a linear fashion from approximately 5.2 x 10(-14) cm(3)/s to 1.8 x 10(-14) cm(3)/s (-100 to +100 mV), both with and without Mg(++) in the solutions. There was no significant difference in the nominal values of permeability. This property of the K(+) channel may facilitate the hyperpolarization needed to sustain a chloride secretion.     

The reliability of relative anion-cation permeabilities deduced from reversal (Dilution) potential measurements in ion channel studies

Measurements of anion-cation permeability ratios (e.g., P _Cl/P _Na) are most readily made by measuring changes in zero-current reversal potential when the salt concentration on one side of the membrane (e.g., external NaCl) is decreased. This is particularly useful for measuring changes in ion selectivity in wild-type and mutant channels, such as those of the ligand-gated ion channel superfamily, and has shown that many of these channels have a significant permeability to counter-ions. One Brownian dynamics study of ion permeation through such narrow ion channels failed to observe such counter-ion movement, although later, another Brownian dynamics study did observe counter-ion movement through simulations of the same channels. The question has been raised as to the reliability of such reversal potential measurements for determining permeability ratios, particularly given the use of an equation such as the Goldman-Hodgkin-Katz (GHK) equation, which is often used to calculate such ratios. A new derivation of the GHK equation in terms of activity coefficients is also included. The application of irreversible thermodynamics will be shown to qualitatively support the reliability of such experimental anion-cation permeability values derived from reversal potential measurements. It will then be shown that for such zero-current situations, different electrodiffusion models, with very different underlying assumptions, produce almost identical relative permeabilities (and reversal potentials). Finally, the results of the two Brownian dynamics simulation studies and the relationship between reversal potentials and relative permeability will be discussed.