Goldman-Hodgkin-Katz Equation Research Articles

Electrogenic transport by lipophilic guanidinium salts as anion carriers in bilayer membranes

Anion transport by four isomers of tetradecylguanidinium in vesicle bilayer membranes was investigated by ion-selective electrode and pH-sensitive fluorescence methods. On a macroscopic scale, these carriers mediate hydroxide-anion antiport. In the absence of an externally imposed ionic gradient, the carriers produce a pH gradient due to rapid partition-deprotonation of the carrier and diffusion of the guanidine conjugate base. The formally coupled anion transport is slow relative to this process, and the system exhibits a corresponding membrane polarisation detected with potential-sensitive fluorescent dyes. Selective anion transport in competition with the efficient hydroxide transport is detected only with relatively easily extracted anions and N,N′-disubstituted isomers. Membrane polarisation changes can be deduced from simple equivalent circuits derived from the Goldman–Hodgkin–Katz voltage equation.

Nitric oxide decreases the permselectivity of the paracellular pathway in thick ascending limbs.

Thick ascending limbs reabsorb 25% to 30% of the filtered NaCl. About 50% to 70% is reabsorbed via the transcellular pathway and 30% to 50% is reabsorbed through the Na-selective paracellular pathway. Nitric oxide (NO) inhibits transepithelial Na reabsorption, but its effects on the paracellular pathway are unknown. We hypothesized that NO decreases the selectivity of the paracellular pathway in thick ascending limbs via cGMP-dependent protein kinase. To assess relative Na/Cl permeability ratios (PNa/PCl), we perfused rat thick ascending limbs and measured the effect of reducing bath NaCl on transepithelial voltage, creating dilution potentials, with vehicle, NO donors, and endogenous NO. PNa/PCl was calculated using the Goldman-Hodgkin-Katz equation. Reducing bath Na/Cl to 16/8, 32/24, and 64/56 mmol/L created dilution potentials of -13.6±2.2, -10.8±3.0, and -6.1±0.9 mV, respectively. Calculated PNa/PCls were 2.0±0.2, 2.2±0.5, and 1.9±0.2. The NO donor spermine NONOate (200 µmol/L) blunted the dilution potential caused by 32/24 mmol/L Na/Cl from -11.1±2.1 to -6.5±1.6 mV (P<0.004) and PNa/PCl from 2.2±0.4 to 1.5±0.2. Nitroglycerin (200 µmol/L), another NO donor, also reduced PNa/PCl. Controls showed no significant changes. Dibutyryl-cGMP decreased dilution potentials from -13.4±2.9 to -7.5±1.8 mV (n=6; P<0.01). cGMP-dependent protein kinase inhibition with KT5823 (4 µmol/L) blocked the effect of spermine NONOate, whereas phosphodiesterase 2 inhibition did not. Endogenously produced NO mimicked the effect of the NO donors. In conclusion, NO reduces the selectivity of the paracellular pathway in thick ascending limbs via cGMP and cGMP-dependent protein kinase.

The Warburg Effect as a VDAC‐Hexokinase‐Mediated Electrical Suppression of Mitochondrial Energy Metabolism

It is known that nearly half of tumor energy metabolism is provided by glycolysis under normal aerobic conditions (the Warburg Effect) that has been attributed to a respiratory impairment in cancer cells. In the last years, the mitochondrial impairment in cancer cells has been explained by the closure of mitochondrial outer membrane VDACs (porins) by free tubulin (as molecular corks) leading to a suppression of mitochondrial metabolism. Here, we further developed a concept that mitochondria in cancer cells are suppressed by the outer membrane potential (OMP) generated by the metabolically‐dependent mechanism involving the functioning of the VDAC‐hexokinase and/or the ANT(adenine nucleotide translocator)‐VDAC‐hexokinase complexes.These complexes and generated OMP allow initiation of glycolysis preferentially by ATP generated in mitochondria. High selectivity of channeling of mitochondrial energy into the cytosol of cancer cells, to only initiate glycolysis, is achieved due to the electrical closure of free VDACs (non‐bound to hexokinase) by the metabolically‐dependent OMP, by a mode preventing turbo acceleration of glycolysis by cytosolic ATP. Even at low values of OMP, not high enough to completely close VDACs, the ATP flux from the mitochondrial intermembrane space should be almost completely suppressed according to Goldman equation. The value of OMP directly generatedby the VDAC‐hexokinase complex, for example, depends on the Gibbs free energy of the hexokinase reaction. It means that VDAC might function as a physiologically important electrical switcher of tumor energy metabolism, allowing a clear explanation of the Warburg Effect mechanism. (Research grants #111852128625 and #520154531565, Colciencias).

Membrane potential generated by ion adsorption.

It has been widely acknowledged that the Goldman-Hodgkin-Katz (GHK) equation fully explains membrane potential behavior. The fundamental facet of the GHK equation lies in its consideration of permeability of membrane to ions, when the membrane serves as a separator for separating two electrolytic solutions. The GHK equation describes that: variation of membrane permeability to ion in accordance with ion species results in the variation of the membrane potential. However, nonzero potential was observed even across the impermeable membrane (or separator) separating two electrolytic solutions. It gave rise to a question concerning the validity of the GHK equation for explaining the membrane potential generation. In this work, an alternative theory was proposed. It is the adsorption theory. The adsorption theory attributes the membrane potential generation to the ion adsorption onto the membrane (or separator) surface not to the ion passage through the membrane (or separator). The computationally obtained potential behavior based on the adsorption theory was in good agreement with the experimentally observed potential whether the membrane (or separator) was permeable to ions or not. It was strongly speculated that the membrane potential origin could lie primarily in the ion adsorption on the membrane (or separator) rather than the membrane permeability to ions. It might be necessary to reconsider the origin of membrane potential which has been so far believed explicable by the GHK equation.

INFLUENCE OF DIFFERENT MODELING PROTOCOLS FOR CALCIUM CURRENTS IN THE AHP FEATURES OF A TYPE-S MOTONEURON

INFLUENCE OF DIFFERENT MODELING PROTOCOLS FOR CALCIUM CURRENTS IN THE AHP FEATURES OF A TYPE-S MOTONEURON

K+ Channels of Squid Giant Axons Open by an Osmotic Stress in Hypertonic Solutions Containing Nonelectrolytes

In hypertonic solutions made by adding nonelectrolytes, K(+) channels of squid giant axons opened at usual asymmetrical K(+) concentrations in two different time courses; an initial instantaneous activation (I (IN)) and a sigmoidal activation typical of a delayed rectifier K(+) channel (I (D)). The current-voltage relation curve for I (IN) was fitted well with Goldman equation described with a periaxonal K(+) concentration at the membrane potential above -10mV. Using the activation-voltage curve obtained from tail currents, K(+) channels for I (IN) are confirmed to activate at the membrane potential that is lower by 50mV than those for I (D). Both I (IN) and I (D) closed similarly at the holding potential below -100mV. The logarithm of I (IN)/I (D) was linearly related with the osmolarity for various nonelectrolytes. Solute inaccessible volumes obtained from the slope increased with the nonelectrolyte size from 15 to 85 water molecules. K(+) channels representing I (D) were blocked by open channel blocker tetra-butyl ammonium (TBA) more efficiently than in the absence of I (IN), which was explained by the mechanism that K(+) channels for I (D) were first converted to those for I (IN) by the osmotic pressure and then blocked. So K(+) channels for I (IN) were suggested to be derived from the delayed rectifier K(+) channels. Therefore, the osmotic pressure is suggested to exert delayed-rectifier K(+) channels to open in shrinking rather hydrophilic flexible parts outside the pore than the pore itself, which is compatible with the recent structure of open K(+) channel pore.

Components of neuronal chloride transport in rat and human neocortex

Considerable evidence indicates disturbances in the ionic gradient of GABAA receptor-mediated inhibition of neurones in human epileptogenic tissues. Two contending mechanisms have been proposed, reduced outward and increased inward Cl⁻ transporters. We investigated the properties of Cl⁻ transport in human and rat neocortical neurones (layer II/III) using intracellular recordings in slices of cortical tissue. We measured the alterations in reversal potential of the pharmacologically isolated inhibitory postsynaptic potential mediated by GABAA receptors (IPSPA) to estimate the ionic gradient and kinetics of Cl⁻ efflux after Cl⁻ injections before and during application of selected blockers of Cl⁻ routes (furosemide, bumetanide, 9-anthracene carboxylic acid and Cs+). Neurones from human epileptogenic cortex exhibited a fairly depolarized reversal potential of GABAA receptor-mediated inhibition (EIPSP-A) of -61.9 ± 8.3 mV. In about half of the neurones, the EIPSP-A averaged -55.2 ± 5.7 mV, in the other half, 68.6 ± 2.3 mV, similar to rat neurones (-68.9 ± 2.6 mV). After injections of Cl⁻, IPSPA recovered in human neurones with an average time constant (τ) of 19.0 ± 9.6 s (rat neurones: 7.2 ± 2.4 s). We calculated Cl⁻ extrusion rates (1/τ) via individual routes from the τ values obtained in different experimental conditions, revealing that, for example, the K+-coupled Cl⁻ transporter KCC2 comprises 45.3% of the total rate in rat neurones. In human neurones, the total rate of Cl⁻ extrusion was 63.9% smaller, and rates via KCC2, the Na+-K+-2Cl⁻ transporter NKCC1 and the voltage-gatedCl− channelClCwere smaller than in rat neurones by 80.0%, 61.7% and 79.9%, respectively. The rate via anion exchangers conversely was 14.4% larger in human than in rat neurones. We propose that (i) KCC2 is the major route of Cl⁻ extrusion in cortical neurones, (ii) reduced KCC2 is the initial step of disturbed Cl⁻ regulation and (iii) reductions in KCC2 contribute to depolarizing EIPSP-A of neurones in human epileptogenic neocortex.

Study on Rapid Determination Method of Alkali-Aggregate Reaction by Electrochemical Technique

This study was conducted to grasp the possibilities of a method for determining alkali-aggregate reaction rapidly, through the use of potential difference between two aqueous solutions separated by a cation exchange membrane. The rhyolite powder (75∼150μm) was used in the test samples. The membrane potential according to the equation for theory (Goldman equation) was calculated by substitution of the analysis value of Na, K and Si concentrations at aqueous electrolyte solutions.The results obtained were as follows : (1) The potential difference was measured with “Positive (+)” and “Negative (-)” values at testing periods of 3∼15minutes, by the use of “Innocuous” and “Deleterious” powder interpreted according to ASR chemical method (JIS A 1145), respectively. (2) The potential difference was highly correlated with Rc/Sc ratio tested to ASR chemical method. (3) The membrane potential according to the theory equation was calculated for “Positive (+)” and “Negative (-)” value, respectively, as in the case of the potential difference by the membrane. (4) From these results, it was recognized that the method of test for alkali-aggregate reaction by electrochemical technique was able to judge rapidly, and held great promise at the site of mine.

Early biophysics of the NMDA receptor channel

The voltage-dependent block of NMDA receptor channels by external Mg2+ (Mgo) was first reported in 1984 in two papers (Mayer et al. 1984; Nowak et al. 1984) that have had a great influence on our understanding of the functional role of NMDA receptors. However, the observations also offered a new way to explain the ‘gating’ of a voltage-dependent conductance, and our paper of 1988 published in The Journal of Physiology aimed primarily at a biophysical description of the Mgo block based on single channel records (Ascher & Nowak, 1988). To characterize the Mgo block we followed the approach used by Neher & Steinbach (1978) to study how QX222 transforms into ‘bursts’ the single openings of nicotinic ACh receptor channels. Assuming that Mgo enters the NMDA receptor channel, binds to a blocking site situated deep in the membrane and can only leave to the outside after unbinding, we evaluated the rates of Mg2+ binding and unbinding in various Mg2+ concentrations and at various potentials. We then deduced from the voltage dependence of these rates the depth of the blocking site in the membrane. This depth was evaluated by a coefficient ‘δ’ that could vary between 0 and 1. Our value of δ was close to 1, suggesting that the blocking site was actually very close to the inner limit of the membrane. This value was somewhat higher than the values obtained by analysis of the I–V relations of whole cell currents by Mayer & Westbrook (1985). We also characterized at the single channel level the Ca2+ permeability of the NMDA receptor channel. We measured the shifts of the reversal potential in different external Ca2+ concentrations and deduced the ratio of the permeabilities of Ca2+ and monovalent cations from the Goldman–Hodgkin–Katz voltage equation. Our results agreed with the values obtained by Mayer & Westbrook (1987) using I–V relations for whole cell current. We also observed that an increase in external Ca2+ reduced the single channel conductance, indicating that Ca2+ permeates the channel more slowly than monovalent cations. Our evaluation of the depth of the Mgo blocking site was soon put in doubt by the observation that the value of δ we deduced for Mgo block was not easily reconciled with the voltage dependence of the block by internal Mg2+ (Mgi) (Johnson & Ascher, 1990). The ‘crossing of the deltas’ paradox was solved by Jon Johnson and his collaborators, who showed that access of Mg2+ to the channel is prevented when permeant ions bind at the outer surface of the membrane. In the model of Antonov & Johnson (1999) the δ for Mgo is now equal to 0.5. We should also acknowledge that our single channel recordings made us miss the ‘slow Mgo unblock’ which was later described by Spruston et al. (1995) on whole cell current relaxations following voltage jumps, modelled by Vargas-Caballero & Robinson (2004) and by Kampa et al. (2004), and shown to be NR2 subunit dependent by Clarke & Johnson (2006). Recently the same authors (Clarke & Johnson, 2008) have shown that the slow block is the consequence of a voltage dependent gating which does not require Mgo. From 1991 onward, the cloning of the NMDA receptor subunits radically renewed the study of Mg2+ block and Ca2+ permeability. It rapidly led to the identification of the key amino acids involved in the binding of Mg2+ (reviewed by Dingledine et al. 1999), and it also revealed the heterogeneity of Mg block among NMDA receptors subtypes (not yet well understood). Concerning the Ca2+ permeability, the use of calcium indicators has allowed direct comparison of the Ca2+ influx and the total current and thus evaluation of the ‘fractional Ca2+ current’ (Pf). When appropriate corrections are made, the value of Pf agrees very well with the predictions of the GHK equation (Schneggenburger 1996). The molecular structures responsible for the Ca2+ permeability have been partially identified and comprise both a deep site, the N site of the NR1 subunit, and a superficial site at the entrance of the channel, the DRPEER motif, also specific to the NR1 subunit (Watanabe et al. 2002). Despite all these advances, one cannot yet say that either Mg2+ block or Ca2+ permeation are understood at the molecular level. We still lack a structural model of the NMDA receptor channel, but it may not be too far away.

Anomalous Mole Fraction Effect in ClC-2 Chloride Channel Pore

The Cl− pathway of E. Coli ClC H+/Cl− exchanger has three binding sites that can be occupied by Cl−, however SeCN− (an anion similar to SCN−) occupies only one site. This pathway serves as a model for pore structure of ClC channels whose gating is coupled to permeation. Thus ClC channels may gate differently in the presence of SCN−. We analyzed the relationship between gating and pore properties by performing SCN− mole-fraction experiments in ClC-2 channels expressed in HEK cells. Internal and external solutions with different SCN− fractions were prepared mixing [Cl−] + [SCN−] = 140 mM. Replacing 100% Cl− with SCN− on both sides of the membrane shifted to the left the voltage (Vm) vs channel P0 curve by ∼-18 mV without changing the slope. In contrast, the protopore gate PP vs Vm curve was shallower and shifted rightward (+5 mV). Extracellular SCN− mole-fractions produced negative shifts on reversal potential (ER) values which are not described by GHK equation with a constant PSCN/PCl ratio. The PSCN/PCl ratios increased from 1.4 to 2.5 when the mole fraction increased, suggesting extracellular SCN− enters the pore better than Cl−. In addition, the slope of P0 vs Vm curve was steeper and the mid-point voltage (V0.5) did not change. Intracellular SCN− mole fractions produced nonlinear negative shifts on ER suggesting that from this side Cl− enters the pore better than SCN−. The slope of the Po vs Vm curve was shallower and V0.5 and slope conductance vs mole fraction displayed the classical anomalous behaviour. Interestingly, the protopore gate located in the Cl− pathway also displays the same behaviour. Our data show that the ClC-2 channel has indeed a multi-ion pore and that SCN− enters this pore preferably from the extracellular side.Supported by CONACyT.

Determining K+ channel activation curves from K+ channel currents often requires the Goldman–Hodgkin–Katz equation

Potassium ion current in nerve membrane, IK, has traditionally been described by IK = gK(V − EK), where gK is the K ion conductance, V is membrane potential and EK is the K+ Nernst potential. This description has been unchallenged by most investigators in neuroscience since its introduction almost 60 years ago. The problem with the IK ∼ (V − EK) proportionality is that it is inconsistent with the unequal distribution of K ions in the intra- and extracellular bathing media. Under physiological conditions the intracellular K+ concentration is significantly higher than the extracellular concentration. Consequently, the slope conductance at potentials positive to EK cannot be the same as that for potentials negative to EK, as the linear proportionality between IK and (V − EK) requires. Instead IK has a non-linear dependence on (V − EK) which is well described by the Goldman–Hodgkin–Katz equation. The implications of this result for K+ channel gating and membrane excitability are reviewed in this report.

Identification of Pore Residues Engaged in Determining Divalent Cationic Permeation in Transient Receptor Potential Melastatin Subtype Channel 2

The molecular basis for divalent cationic permeability in transient receptor potential melastatin subtype (TRPM) channels is not fully understood. Here we studied the roles of all eight acidic residues, glutamate or aspartate, and also the glutamine residue between pore helix and selectivity filter in the pore of TRPM2 channel. Mutants with alanine substitution in each of the acidic residues, except Glu-960 and Asp-987, formed functional channels. These channels exhibited similar Ca2+ and Mg2+ permeability to wild type channel, with the exception of the E1022A mutant, which displayed increased Mg2+ permeability. More conservative E960Q, E960D, and D987N mutations also led to loss of function. The D987E mutant was functional and showed greater Ca2+ permeability along with concentration-dependent inhibition of Na+-carrying currents by Ca2+. Incorporation of negative charge in place of Gln-981 between the pore helix and selectivity filter by changing it to glutamate, which is present in the more Ca2+-permeable TRPM channels, substantially increased Ca2+ permeability. Expression of concatemers linking wild type and E960D mutant subunits resulted in functional channels that exhibited reduced Ca2+ permeability. These data taken together suggest that Glu-960, Gln-981, Asp-987, and Glu-1022 residues are engaged in determining divalent cationic permeation properties of the TRPM2 channel.

A computational neuron model based on Poisson-Nernst-Planck theory

Modelling of the nerve impulse is often simplified to one spatial dimension, for example by using cable theory. In reality signals propagate in a complex three dimensional environment, and neurons may electrostatically affect cells in close proximity. To investigate this, the electrochemistry of the neuron environment is modelled using the Poisson--Nernst--Planck theory of electrodiffusion. An accurate numerical solver for the Poisson--Nernst--Planck equations in three dimensions is developed. The solver, integrated with a simple computational model of ion channels, is capable of simulating the dynamics of multiple electrical charge species for an arbitrary configuration of membranes and ion channels. Preliminary simulations of simplified neurons show resting membrane potentials broadly consistent with the Goldman equation of electrochemistry, but with interesting differences in some cases. The model can be applied to the detailed study of the nerve impulse. References Hundsdorfer, W., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer--Verlag, Berlin, 2003. Kurnikova, M. G., Coalson, R. D., Graf, P. and Nitzan, A., A Lattice Relaxation Algorithm for Three-Dimensional Poisson--Nernst--Planck Theory with Application to Ion Transport through the Gramicidin A Channel, Biophysical Journal , 76 , 642--656, 1999. http://www.biophysj.org/cgi/content/full/76/2/642 Nicholls, J. G., Martin, A. R., Wallace B. G. and Fuchs, P. A., From Neuron to Brain , Sinauer Associates, Sunderland, MA., 2001. Nicholson, C., Extracellular space as the pathway for neuron-glial cell interaction, In Kettenmann, H. and Ransom, B.R., editors, Neuroglia , Oxford University Press, New York, 387--397, 1995. Qian, N., and Sejnowski, T. J., Electrodiffusion Model of Electrical Conduction in Neuronal Processes, In Woody, C. D., and McGaugh, J. L., editors, Cellular Mechanisms of Conditioning and Behavioral Plasticity , Plenum Press, New York, 237--244, 1988. Visscher, P.B., Fields and Electrodynamics , John Wiley and sons, USA, 1988. Chung, S. H., Andersen, O. S. and Krishnamurthy, V., Biological Membrane Ion Channels: Dynamics, Structure and Applications , Springer Science+Business Media, New York, 2007. Crank, J. and Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Advances in Computational Mathematics , 6 , 207--226, 1996. doi:10.1007/BF02127704 Goldman, D. E., Potential, Impedance, and Rectification in Membranes, Journal of General Physiology , 27 , 37--60, 1943. http://www.jgp.org/cgi/reprint/27/1/37 Hille, B., Ion Channels of Excitable Membranes , Sinauer Associates, Sunderland, MA., 2001.

A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

The Hodgkin and Huxley (HH) model predicts sustained repetitive firing of nerve action potentials for a suprathreshold depolarizing current pulse for as long as the pulse is applied (type 2 excitability). Squid giant axons, the preparation for which the model was intended, fire only once at the beginning of the pulse (type 3 behaviour). This discrepancy between the theory and experiments can be removed by modifying a single parameter in the HH equations for the K+ current as determined from the analysis in this paper. K+ currents in general have been described by IK=gK(V-EK), where gK is the membrane's K+ current conductance and EK is the K+ Nernst potential. However, IK has a nonlinear dependence on (V-EK) well described by the Goldman-Hodgkin-Katz equation that determines the voltage dependence of gK. This experimental finding is the basis for the modification in the HH equations describing type 3 behaviour. Our analysis may have broad significance given the use of IK=gK(V-EK) to describe K+ currents in a wide variety of biological preparations.

Simultaneous Monitoring of Ionophore- and Inhibitor-mediated Plasma and Mitochondrial Membrane Potential Changes in Cultured Neurons

Although natural and synthetic ionophores are widely exploited in cell studies, for example, to influence cytoplasmic free calcium concentrations and to depolarize in situ mitochondria, their inherent lack of membrane selectivity means that they affect the ion permeability of both plasma and mitochondrial membranes. A similar ambiguity affects the interpretation of signals from fluorescent membrane-permeant cations (usually termed "mitochondrial membrane potential indicators"), because the accumulation of these probes is influenced by both plasma and mitochondrial membrane potentials. To resolve some of these problems a technique is developed to allow simultaneous monitoring of plasma and mitochondrial membrane potentials at single-cell resolution using a cationic and anionic fluorescent probe. A computer program is described that transforms the fluorescence changes into dynamic estimates of changes in plasma and mitochondrial potentials. Exploiting this technique, primary cultures of rat cerebellar granule neurons display a concentration-dependent response to ionomycin: low concentrations mimic nigericin by hyperpolarizing the mitochondria while slowly depolarizing the plasma membrane and maintaining a stable elevated cytoplasmic calcium. Higher ionomycin concentrations induce a stochastic failure of calcium homeostasis that precedes both mitochondrial depolarization and an enhanced rate of plasma membrane depolarization. In addition, the protonophore carbonyl cyanide p-trifluoromethoxyphenylhydrazone only selectively depolarizes mitochondria at submicromolar concentrations. ATP synthase reversal following respiratory chain inhibition depolarizes the mitochondria by 26 mV.

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., PCl/PNa) 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.

On the nature of liquid junction and membrane potentials

Whenever a spatially inhomogeneous electrolyte, composed of ions with different mobilities, is allowed to diffuse, charge separation and an electric potential difference is created. Such potential differences across very thin membranes (e.g. biomembranes) are often interpreted using the steady state Goldman equation, which is usually derived by assuming a spatially constant electric field. Through the fundamental Poisson equation of electrostatics, this implies the absence of free charge density that must provide the source of any such field. A similarly paradoxical situation is encountered for thick membranes (e.g. in ion-selective electrodes) for which the diffusion potential is normally interpreted using the Henderson equation. Standard derivations of the Henderson equation appeal to local electroneutrality, which is also incompatible with sources of electric fields, as these require separated charges. We analyse self-consistent solutions of the Nernst-Planck-Poisson equations for a 1 : 1-univalent electrolyte to show that the Goldman and Henderson steady-state membrane potentials are artefacts of extraneous charges created in the reservoirs of electrolyte solution on either side of the membrane, due to the unphysical nature of the usual (Dirichlet) boundary conditions assumed to apply at the membrane-electrolyte interfaces. We also show, with the aid of numerical simulations, that a transient electric potential difference develops in any confined, but initially non-uniform, electrolyte solution. This potential difference ultimately decays to zero in the real steady state of the electrolyte, which corresponds to thermodynamic equilibrium. We explain the surprising fact that such transient potential differences are well described by the Henderson equation by using a computer algebra system to extend previous steady-state singular perturbation theories to the time-dependent case. Our work therefore accounts for the success of the Henderson equation in analysing experimental liquid-junction potentials.

Basal Nonselective Cation Permeability in Rat Cardiac Microvascular Endothelial Cells

The presence of a basal nonselective cation permeability was mainly investigated in primary cultures of rat cardiac microvascular endothelial cells (CMEC) by applying both the patch-clamp technique and Fura-2 microfluorimetry. With low EGTA in the pipette solution, the resting membrane potential of CMEC was −21.2 ± 1.1 mV, and a Ca 2+-activated Cl − conductance was present. When the intracellular Ca 2+ was buffered with high EGTA, the membrane potential decreased to 5.5 ± 1.2 mV. In this condition, full or partial substitution of external Na + by NMDG + proportionally reduced the inward component of the basal I–V relationship. This current was dependent on extracellular monovalent cations with a permeability sequence of K + > Cs + > Na + > Li + and was inhibited by Ca 2+, La 3+, Gd 3+, and amiloride. The K +/Na + permeability ratio, determined using the Goldman–Hodgkin–Katz equation, was 2.01. The outward component of the basal I–V relationship was reduced when intracellular K + was replaced by NMDG +, but was not sensitive to substitution by Cs +. Finally, microfluorimetric experiments indicated the existence of a basal Ca 2+ entry pathway, inhibited by La 3+ and Gd 3+. The basal nonselective cation permeability in CMEC could be involved both in the control of myocardial ionic homeostasis, according to the model of the blood–heart barrier, and in the modulation of Ca 2+-dependent processes.

Uncovering misconceptions about the resting membrane potential.

Resting membrane potential is one of the most difficult physiological concepts that students must master. Although the upper-division undergraduates in my physiology course have completed a semester of neurophysiology in which they worked problems using the Nernst and Goldman equations, they often

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 × 10 −14 cm 3/s to 1.8 × 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.