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.     

Induction of conductance heterogeneity in gramicidin channels

In previous work from our laboratory, 5-10% of the channels formed by [Val1]gramicidin A have conductances that fall outside the narrow range that conventionally has defined the standard gramicidin channel [e.g., see Russell et al. (1986) Biophys. J. 49, 673]. Reports from other laboratories, however, show that up to 50% of [Val1]gramicidin channels have conductances that fall outside the range for standard channels [e.g., see Prasad et al. (1986) Biochemistry 25, 456]. This laboratory-to-laboratory variation in the distribution of gramicidin single-channel conductances suggests that the conductance variants are induced by some environmental factor(s) [Busath et al. (1987) Biophys. J. 51, 79]. In order to test whether extrinsic agents can induce such conductance heterogeneity, we examined the effects of nonionic or zwitterionic detergents upon gramicidin channel behavior. In phospholipid bilayers, detergent addition induces many changes in gramicidin channel behavior: all detergents tested increase the channel appearance rate and average duration; most detergents decrease the conductance of the standard channel; and all but one of the detergents increase the conductance heterogeneity. These results show that the conductance heterogeneity can result from environmental perturbations, thus providing a possible explanation for the laboratory-to-laboratory variation in the heterogeneity of gramicidin channels. In addition, the differential detergent effects suggest possible mechanisms by which detergents can induce the conformational perturbations that result in gramicidin single-channel conductance variations.     

Nonlinear relationship between V+max and h infinity in frog skeletal muscle.

The relationship between the maximum velocity of action potential upstroke (V+max) and steady-state Na+ channel inactivation (h infinity) was studied in frog skeletal muscle during repetitive discharges evoked in the presence of cevadine (1 mumol/l). Conventional microelectrodes and vaseline-gap voltage-clamp techniques were used. A severe degree of nonlinearity was found between (h infinity) and (V+max) especially when the Na+ conductance (gNa) was small. The observed nonlinearity could be explained as a property of the normal Na+ channel gating in skeletal muscle rather than that of cevadine-modified channels. Part of this work has been published in abstract form in Biophys. J. 57: 105A, 1990.     

Multiple-state model of ionic channel in reproducing the time course of electrophysiological events.

BACKGROUND: The predictions of the Hodgkin-Huxley model do not accurately fit all the measurements of voltage-clamp currents, gating charge and single-channel currents. There are many quantitative differences between the predicted and measured characteristics of the sodium and potassium channels. For example, the two-state gate model has exponential onset kinetics, whereas the sodium and potassium conductances show S-shaped activation and the sodium conductance shows an exponential inactivation. In this paper we shall examine a more general channel model that can more faithfully represent the measured properties of ionic channels in the membrane of the excitable cell. METHODS: The model is based on the generalisation of the notion of a channel with a discrete set of states. Each state has state attributes such as the state conductance, state ionic current and state gating charge. These variables can have quite different waveforms in time, in contrast with a two-state gate channel model, in which all have the same waveforms. RESULTS: The kinetics of all variables are equivalent: gating and ionic currents give equivalent information about channel kinetics; both the equilibrium values of the current and the time constants are functions of membrane potential. The results are in almost perfect concordance with the experimental data regarding the characteristics of nerve impulse. CONCLUSIONS: The expected values of the gating charge and the ionic conductance are weighted sums of the state occupancy probabilities, but the weights differ: for the expected value of the gating charge the weights are the state gating charges and for the expected value of the ionic conductance the weights are the state conductances. Since these weights are different, the expected values of the gating charge and the ionic conductance will differ.     

Dissecting lidocaine action: diethylamide and phenol mimic separate modes of lidocaine block of sodium channels from heart and skeletal muscle.

We have investigated block of sodium channels by diethylamide and phenol, which resemble the hydrophilic tertiary amine head and the hydrophobic aromatic tail of the lidocaine molecule, respectively. Diethylamide and phenol separately mimicked the fast and slow modes of block caused by lidocaine. Experiments were performed using single batrachotoxin-activated bovine cardiac and rat skeletal muscle sodium channels incorporated into neutral planar lipid bilayers. Diethylamide, only from the intracellular side, caused a voltage-dependent reduction in apparent single channel amplitude ('fast' block). Block was similar for cardiac and skeletal muscle channels, and increased in potency when extracellular sodium was replaced by N-methylglucamine, consistent with an intrapore blocking site. Thus, although occurring at 15-fold higher concentrations, block by diethylamide closely resembles the fast mode of block by lidocaine (Zamponi, G. W., D. D. Doyle, and R. J. French. 1993. Biophys. J. 65:80-90). For cardiac sodium channels, phenol bound to a closed state causing the appearance of long blocked events whose duration increased with phenol concentration. This slow block depended neither on voltage nor on the side of application, and disappeared upon treatment of the channel with trypsin. For skeletal muscle channels, slow phenol block occurred with only very low probability. Thus, phenol block resembles the slow mode of block observed for lidocaine (Zamponi, G. W., D. D. Doyle, and R. J. French. 1993. Biophys. J. 65:91-100). Our data suggest that there are separate sites for fast lidocaine block of the open channel and slow block of the "inactivated" channel. Fast block by diethylamide inhibited the long, spontaneous, trypsin-sensitive (inactivation-like) closures of cardiac channels, and hence secondarily antagonized slow block by phenol or lidocaine. This antagonism would potentiate shifts in the balance between the two modes of action of a tertiary amine drug caused by changes in the relative concentrations of the charged (fast blocking) and neutral (slow blocking) forms of the drug.     

Proton Inhibition of Sodium Channels: Mechanism of Gating Shifts and Reduced Conductance

Extracellular acidosis affects both permeation and gating of the expressed rat skeletal muscle Na⁺ channel (μ1). Reduction of the extracellular pH produced a progressive decrease in the maximal whole-cell conductance and a depolarizing shift in the whole-cell current-voltage relationship. A smaller depolarizing shift in the steady-state inactivation curve was observed. The pK of the reduction of maximal conductance was 6.1 over the pH range studied. An upper limit estimate of the pK of the shift of the half-activation voltage was 6.1. The relative reduction in the maximal whole-cell conductance did not change with higher [Na⁺] _o . The conductance of single fenvalerate-modified Na⁺ channels was reduced by extracellular protons. Although the single-channel conductance increased with higher [Na⁺] _o , the maximal conductances at pH 7.6, 7.0 and 6.0 did not converge at [Na⁺] _o up to 280 mm, inconsistent with a simple electrostatic effect. A model incorporating both Na⁺ and H⁺ binding in the pore and cation binding to a Gouy-Chapman surface charge provided a robust fit to the single-channel conductance data with an estimated surface charge density of 1e⁻/439?². Neither surface charge nor proton block alone suffices to explain the effects of extracellular acidosis on Na⁺ channel permeation; both effects play major roles in mediating the response to extracellular pH. Received: 14 May 1996/Revised: 19 September 1996     

Effect of temperature on the inward rectifier and gramicidin A-induced channels in the membrane of frog skeletal muscle fibres

The conductance of frog skeletal muscle fibres in isotonic K2SO4 solution has been measured. Experiments were carried out under current-clamp conditions using a double sucrose-gap technique. The potassium conductances of the inward rectifier and the gramicidin channel in the same muscle fibre were compared. Potassium conductance of the inward rectifier increased with the temperature, with a value of Q10 1.55 +/- 0.09 (n = 8) under hyperpolarization, and Q10 2.38 +/- 0.23 (n = 6) for the depolarizing stimulus, the difference between Q10 of potassium and gramicidin channels being statistically insignificant.     

Density and sub-cellular distribution of cardiac and neuronal sodium channel isoforms in rat ventricular myocytes

In cardiac ventricular myocytes, Na current is generated mainly by the cardiac NaV1.5 isoform, but the presence of "neuronal" Na channel isoforms in the heart has been demonstrated recently. In this study, we quantified the density and sub-cellular distribution of cardiac and neuronal channel isoforms in rat ventricular myocytes. INa was recorded using the patch clamp technique in control and detubulated myocytes. Detubulation reduced cell capacitance (by approximately 29%) but maximum conductance was not altered (1.94+/-0.15, 14 control vs 1.98+/-0.19 nS/pF, 17 detubulated myocytes). The kinetic properties of INa were similar in both cell types suggesting good voltage control of surface and t-tubule membranes. We calculated Na channel densities assuming the sub-cellular current localization we recently provided (neuronal isoform: approximately 11% of total sarcolemmal current, approximately 3% of cell surface, and approximately 31% of t-tubule current). Single channel conductances were assumed to be 2.2 and 2.5 pS for the cardiac and neuronal isoforms, respectively, after accounting for the use of low Na concentration. We calculated that the density of the cardiac Na channel isoform is relatively constant (in channels/microm2: approximately 11 in total sarcolemma, approximately 13 at the cell surface, approximately 10 at the t-tubules). In contrast, neuronal Na channel isoforms are concentrated at the t-tubules (in channels/microm2: approximately 1 in total sarcolemma, approximately 0.3 at the cell surface, approximately 2.5 at the t-tubules). We conclude that, in contrast to skeletal muscle in which Na channel density is higher at the cell surface than the t-tubules, in ventricular cardiac myocytes the sub-cellular distribution of Na channel density is relatively homogeneous (approximately 13 channels/microm2).     

Heterogeneity of calcium channels from a purified dihydropyridine receptor preparation.

Dihydropyridine receptors were purified from rabbit skeletal muscle transverse tubule membranes and incorporated into planar lipid bilayers. Calcium channels from both the purified dihydropyridine receptor preparation and the intact transverse tubule membranes exhibited two sizes of unitary currents, corresponding to conductances of 7 +/- 1 pS and 16 +/- 3 pS in 80 mM BaCl2. Both conductance levels were selective for divalent cations over monovalent cations and anions. Cadmium, an inorganic calcium channel blocker, reduced the single channel conductance of calcium channels from the purified preparation. The organic calcium channel antagonist nifedipine reduced the probability of a single channel being open with little effect on the single channel conductance. The presence of two conductance levels in both the intact transverse tubule membranes and the purified dihydropyridine receptor preparation suggests that the calcium channel may have multiple conductance levels or that multiple types of calcium channels with closely related structures are present in transverse tubule membranes.     

Asymmetric block of a monovalent cation-selective channel of rabbit cardiac sarcoplasmic reticulum by succinyl choline

Summary We have investigated the effect of the skeletal muscle relaxant succinyl choline (SC) on the conduction of potassium ions through a monovalent cation-selective channel present in the cardiac muscle sarcoplasmic reticulum membrane (CSR). This channel has been studied under voltage-clamp conditions following the fusion of purified CSR membrane vesicles with preformed planar phospholipid bilayers. The channel assumes a fixed orientation in the bilayer and displays two conducting states (B. Tomlins, A.J. Williams & R.A.P. Montgomery, 1984,J. Membrane Biol. 80: 191–199). SC blocks potassium conductance through the channel in a voltage-dependent manner. Block occurs from both sides of the channel, in both conducting states and is resolved as discrete flickering events. Although SC is capable of blocking potassium conductance from both sides of the membrane, block is asymmetric. The zero-voltage dissociation constant for block from the cis side of the membrane is approximately threefold lower than that from thetrans side. Block from thecis side displays a linear dependence on SC concentration for both open states and is competitive with potassium ions at saturating potassium activities, consistent with a singlesite blocking model. The degree of SC-induced block is also influenced by membrane surface charge. SC block differs from that previously described for bis quaternary ammonium (bis Qn) compounds such as decamethonium in that SC blocks preferentially from thecis side of the channel.     

Characterization of ion channels on the surface membrane of adult rat skeletal muscle.

The channels present on the surface membrane of isolated rat flexor digitorum brevis muscle fibers were surveyed using the patch clamp technique. 85 out of 139 fibers had a novel channel which excluded the anions chloride, sulfate, and isethionate with a permeability ratio of chloride to sodium of less than 0.05. The selectivity sequence for cations was Na+ = K+ = Cs+ greater than Ca++ = Mg++ greater than N-Methyl-D-Glucamine. The channel remained closed for long periods, and had a large conductance of approximately 320 pS with several subconductance states at approximately 34 pS levels. Channel activity was not voltage dependent and the reversal potential for cations in muscle fibers of approximately 0 mV results in the channel's behaving as a physiological leakage conductance. Voltage activated potassium channels were present in 65 of the cell attached patches and had conductances of mostly 6, 12, and 25 pS. The voltage sensitivity of the potassium channels was consistent with that of the delayed rectifier current. Only three patches contained chloride channels. The scarcity of chloride channels despite the known high chloride conductance of skeletal muscle suggests that most of the chloride channels must be located in the transverse tubular system.     

Stilbene derivatives inhibit the activity of the inner mitochondrial membrane chloride channels

Ion channels selective for chloride ions are present in all biological membranes, where they regulate the cell volume or membrane potential. Various chloride channels from mitochondrial membranes have been described in recent years. The aim of our study was to characterize the effect of stilbene derivatives on single-chloride channel activity in the inner mitochondrial membrane. The measurements were performed after the reconstitution into a planar lipid bilayer of the inner mitochondrial membranes from rat skeletal muscle (SMM), rat brain (BM) and heart (HM) mitochondria. After incorporation in a symmetric 450/450 mM KCl solution (cis/trans), the chloride channels were recorded with a mean conductance of 155 ± 5 pS (rat skeletal muscle) and 120 ± 16 pS (rat brain). The conductances of the chloride channels from the rat heart mitochondria in 250/50 mM KCl (cis/trans) gradient solutions were within the 70–130 pS range. The chloride channels were inhibited by these two stilbene derivatives: 4,4′-diisothiocyanostilbene-2,2′-disulfonic acid (DIDS) and 4-acetamido-4′-isothiocyanostilbene-2,2′-disulfonic acid (SITS). The skeletal muscle mitochondrial chloride channel was blocked after the addition of 1 mM DIDS or SITS, whereas the brain mitochondrial channel was blocked by 300 μM DIDS or SITS. The chloride channel from the rat heart mitochondria was inhibited by 50–100 μM DIDS. The inhibitory effect of DIDS was irreversible. Our results confirm the presence of chloride channels sensitive to stilbene derivatives in the inner mitochondrial membrane from rat skeletal muscle, brain and heart cells.     

Effects of phospholipid surface charge on ion conduction in the K+ channel of sarcoplasmic reticulum.

Single-channel K+ currents through sarcoplasmic reticulum K+ channels were compared after reconstitution into planar bilayers formed from neutral or negatively charged phospholipids. In neutral bilayers, the channel conductance saturates with K+ concentration according to a rectangular hyperbola, with half-saturation at 40 mM K+, and maximum conductance of 220 pS. In negatively charged bilayers (70% phosphatidylserine/30% phosphatidylethanolamine), the conductance is, at a given K+ concentration, higher than in neutral bilayers. This effect of negative surface charge is increasingly pronounced at lower ionic strength. The maximum conductance at high K+ approaches 220 pS in negative bilayers, and the channel's ionic selectivity is unaffected by lipid charge. The divalent channel blocker " bisQ11 " causes discrete blocking events in both neutral and negatively charged bilayers; the apparent rate constant of blocking is sensitive to surface charge, while the unblocking rate is largely unaffected. Bilayers containing a positively charged phosphatidylcholine analogue led to K+ conductances lower than those seen in neutral bilayers. The results are consistent with a simple mechanism in which the local K+ concentration sensed by the channel's entryway is determined by both the bulk K+ concentration and the bulk lipid surface potential, as given by the Gouy-Chapman model of the electrified interface. To be described by this approach, the channel's entryway must be assumed to be located 1-2 nm away from the lipid surface, on both sides of the membrane.     

Single channel recordings from calcium-activated potassium channels in cultured rat muscle

Single channel recordings from cultured rat skeletal muscle have revealed a large conductance (230 pS) channel with a high selectivity for K⁺ over Na⁺. In excised patches of membrane, the probability of channel opening is sensitive to micromolar concentrations of calcium ions at the intracellular surface of the patch. Channel openings appear grouped together into bursts whose duration increases with Ca²⁺ and membrane depolarization. Statistical analysis of the individual open times during each burst showed that there are two distinct open states of similar conductance but dissimilar average lifetimes. These channels might contribute to a macroscopic calcium-activated potassium conductance in rat skeletal muscle and other preparations.     

Single chloride-selective channels active at resting membrane potentials in cultured rat skeletal muscle.

The patch-clamp technique was used to characterize channels that could contribute to the resting Cl-conductance in the surface membrane of cultured rat skeletal muscle. Two Cl- -selective channels, in addition to the Cl- -selective channel of large conductance described previously (Blatz and Magleby, 1983), were observed. One of these channels had fast kinetics and a conductance of 45 +/- 1.8 pS (SE) in symmetrical 100 mM KCl. The other had slow kinetics and a conductance of 61 +/- 2.4 pS. The channel with fast kinetics typically closed within 1 ms after opening and flickered between the open and shut states. The channel with slow kinetics typically closed within 10 ms after opening and displayed less flickering. Both channels were active in excised patches of membrane held at potentials similar to resting membrane potentials in intact cells, and both were open a greater percentage of time with depolarization. Under conditions of high ion concentrations, both channels exhibited nonideal selectivity for Cl- over K+ with the permeability ratio PK/PCl of 0.15-0.2. Additional experiments on the fast Cl- channel indicated that its activity decreased with lowered pHi and that SO2-4 and CH3SO-4 were ineffective charge carriers. These findings, plus the observation that the fast Cl- channel was also active in membrane patches on intact cells, suggest that the fast Cl- channel provides a molecular basis for at least some of the resting Cl- conductance. The extent to which the slow Cl- channel contributes is less clear as it was typically active only after excised patches of membrane had been exposed to high concentrations of KCl at the inner membrane surface.     

K channel kinetics during the spontaneous heart beat in embryonic chick ventricle cells.

By averaging the current that passes through cell-attached patches on beating heart cells, while measuring action potentials with a whole-cell electrode, we were able to study K channels during beating. In 7-d chick ventricle in 1.3 mM K physiological solutions at room temperature, delayed-rectifier channels have three linear conductance states: 60, 30, and 15 pS. The 60 and 15 pS conductances can exist alone, but all three states may appear in the same patch as interconverting conductance levels. The delayed-rectifier conductance states have low densities (less than 10 channels per 10-microns diam cell), and all have a reversal potential near -75 mV and the same average kinetics. Outward K current through delayed-rectifier channels follows the upstroke without appreciable delay and lasts throughout the action potential. No inward current flows through delayed-rectifier channels during beating. The early outward channel has a nonlinear conductance of 18-9 pS depending on the potential. It also turns on immediately after the upstroke of the action potential and lasts on average only 50 ms. The early outward channel has an extrapolated reversal potential near -30 mV; no inward current flows during beating. The inward-rectifier has an extrapolated conductance and reversal potential of 2-3 pS and -80 mV in 1.3 mM K. Channel kinetics are independent of external K between 10 and 120 mM, and the channel conducts current only during the late repolarization and diastolic phases of the action potential. No outward current flows through inward-rectifier channels during beating. This work parallels a previous study of Na channels using similar techniques (Mazzanti, M., and L. J. DeFelice. 1987, Biophys. J. 52:95-100).     

General continuum theory for multiion channel. II. Application to acetylcholine channel.

The general theory (Levitt, D. G. 1990. Biophys. J. 59:271-277) is applied to a model channel that resembles the acetylcholine receptor channel (ACH). The model incorporates the known features of the ACH geometry and fixed charge locations. The channel has a wide mouth facing the outer solution, tapering to a narrow region facing the interior of the cell. Rings of fixed negative charge are placed at the two surfaces where the bilayer begins, corresponding to the known charges at the ends of the M2 segment. It is assumed that the forces acting on the ion are electrostatic: ion-channel wall, ion-ion, Born image and applied voltage. Analytical expressions for these forces are derived that take account of the low dielectric lipid region. In addition, there is a local hard sphere repulsive force that prevents ions from piling up on each other in regions of the channel with a high fixed charge density. A classical continuum theory is used to obtain an expression for the diffusion coefficient in the channel. The model can mimic the major qualitative and, in many cases, quantitative experimental features of the ACH channel: current-voltage relation, conductance versus concentration and interaction between monovalent and divalent ions. The model calculations were also compared with the site directed mutagenesis experiments of Imoto, K., C. Busch, B. Sakmann, M. Mishina, T. Konno, J. Nakai, H. Bujo, Y. Mori, K. Fukuda, and S. Numa. (1988. Nature (Lond.). 335:645-648) in which the charge at the ends of the channel was systematically varied.     

Transmembrane peptide NB of influenza B: a simulation, structure, and conductance study

The putative transmembrane segment of the ion channel forming peptide NB from influenza B was synthesized by standard solid-phase peptide synthesis. Insertion into the planar lipid bilayer revealed ion channel activity with conductance levels of 20, 61, 107, and 142 pS in a 0.5 M KCl buffer solution. In addition, levels at -100 mV show conductances of 251 and 413 pS. A linear current-voltage relation reveals a voltage-independent channel formation. In methanol and in vesicles the peptide appears to adopt an alpha-helical-like structure. Computational models of alpha-helix bundles using N = 4, 5, and 6 NB peptides per bundle revealed water-filled pores after 1 ns of MD simulation in a solvated lipid bilayer. Calculated conductance values [using HOLE (Smart et al. (1997) Biophys. J. 72, 1109-1126)] of ca. 20, 60, and 90 pS, respectively, suggested that the multiple conductance levels seen experimentally must correspond to different degrees of oligomerization of the peptide to form channels.     

Ion selectivity of porcine skeletal muscle Ca2+ release channels is unaffected by the Arg615 to Cys615 mutation.

The Arg615 to Cys615 mutation of the sarcoplasmic reticulum (SR) Ca2+ release channel of malignant hyperthermia susceptible (MHS) pigs results in a decreased sensitivity of the channel to inhibitory Ca2+ concentrations. To investigate whether this mutation also affects the ion selectivity filter of the channel, the monovalent cation conductances and ion permeability ratios of single Ca2+ release channels incorporated into planar lipid bilayers were compared. Monovalent cation conductances in symmetrical solutions were: Li+, 183 pS +/- 3 (n = 21); Na+, 474 pS +/- 6 (n = 29); K+, 771 pS +/- 7 (n = 29); Rb+, 502 pS +/- 10 (n = 22); and Cs+, 527 pS +/- 5 (n = 16). The single-channel conductances of MHS and normal Ca2+ release channel were not significantly different for any of the monovalent cations tested. Permeability ratios measured under biionic conditions had the permeability sequence Ca2+ >> Li+ > Na+ > K+ > or Rb+ > Cs+, with no significant difference noted between MHS and normal channels. This systematic examination of the conduction properties of the pig skeletal muscle Ca2+ release channel indicated a higher Ca2+ selectivity (PCa2+:Pk+ approximately 15.5) than the sixfold Ca2+ selectivity previously reported for rabbit skeletal (Smith et al., 1988) or sheep cardiac muscle (Tinker et al., 1992) Ca2+ release channels. These results also indicate that although Ca2+ regulation of Ca2+ release channel activity is altered, the Arg615 to Cys615 mutation of the porcine Ca2+ release channel does not affect the conductance or ion selectivity properties of the channel.     

Far-field analysis of coupled bulk and boundary layer diffusion toward an ion channel entrance.

We present a far-field analysis of ion diffusion toward a channel embedded in a membrane with a fixed charge density. The Smoluchowski equation, which represents the 3D problem, is approximated by a system of coupled three- and two-dimensional diffusions. The 2D diffusion models the quasi-two-dimensional diffusion of ions in a boundary layer in which the electrical potential interaction with the membrane surface charge is important. The 3D diffusion models ion transport in the bulk region outside the boundary layer. Analytical expressions for concentration and flux are developed that are accurate far from the channel entrance. These provide boundary conditions for a numerical solution of the problem. Our results are used to calculate far-field ion flows corresponding to experiments of Bell and Miller (Biophys. J. 45:279, 1984).