Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel.

The synthetic channel [acetyl-(LeuSerSerLeuLeuSerLeu)3-CONH2]6 (pore diameter approximately 8 A, length approximately 30 A) is a bundle of six alpha-helices with blocked termini. This simple channel has complex properties, which are difficult to explain, even qualitatively, by traditional theories: its single-channel currents rectify in symmetrical solutions and its selectivity (defined by reversal potential) is a sensitive function of bathing solution. These complex properties can be fit quantitatively if the channel has fixed charge at its ends, forming a kind of macrodipole, bracketing a central charged region, and the shielding of the fixed charges is described by the Poisson-Nernst-Planck (PNP) equations. PNP fits current voltage relations measured in 15 solutions with an r.m.s. error of 3.6% using four adjustable parameters: the diffusion coefficients in the channel's pore DK = 2.1 x 10(-6) and DCl = 2.6 x 10(-7) cm2/s; and the fixed charge at the ends of the channel of +/- 0.12e (with unequal densities 0.71 M = 0.021e/A on the N-side and -1.9 M = -0.058e/A on the C-side). The fixed charge in the central region is 0.31e (with density P2 = 0.47 M = 0.014e/A). In contrast to traditional theories, PNP computes the electric field in the open channel from all of the charges in the system, by a rapid and accurate numerical procedure. In essence, PNP is a theory of the shielding of fixed (i.e., permanent) charge of the channel by mobile charge and by the ionic atmosphere in and near the channel's pore. The theory fits a wide range of data because the ionic contents and potential profile in the channel change significantly with experimental conditions, as they must, if the channel simultaneously satisfies the Poisson and Nernst-Planck equations and boundary conditions. Qualitatively speaking, the theory shows that small changes in the ionic atmosphere of the channel (i.e., shielding) make big changes in the potential profile and even bigger changes in flux, because potential is a sensitive function of charge and shielding, and flux is an exponential function of potential.     

Exact continuum solution for a channel that can be occupied by two ions

The classical Nernst-Planck continuum equation is extended to the case where the channel can be occupied simultaneously by two ions. A two-dimensional partial differential equation is derived to describe the steady-state channel. This differential equation is of the form of the generalized Laplace equation, but it has the novel feature that the boundary conditions are periodic. The finite difference solution takes approximately 8 s on a large computer. The equations are solved for the special case of a cylindrical channel with a fixed charge in the center. It is assumed that the forces on the ions result entirely from the sum of the Born image potential, the fixed charge potential, the interaction potential between the two ions, and the applied voltage. Approximate simple analytical expressions are derived for these potential terms, based on the assumption that the electric field perpendicular to the channel wall is zero. The potentials include the contribution from a diffuse charge (Debye-Huckel) reaction field in the bulk solution for the monovalent cation flux was obtained for channels with a radius of 4 A and lengths of 16 and 32 A and a fixed charge valence of -1 and -1.5. For these channels, a significant fraction (up to 90%) of the total resistance is contributed by the bulk solution and results were obtained for the case where the "channel" included 8 A of bulk solution at each channel end. These results for the two-ion channel were compared with the analytical solution for a one-ion channel. The one-ion channel is a fair approximation to the two-ion channel for a fixed charge of -1, underestimating the flux at high concentrations by approximately 30%. However, for a fixed charge of -1.5, the one-ion model is a poor approximation, with the two-ion flux about seven times that of the one-ion model at high concentrations. The absolute conductance and concentration dependence of these channels (with a fixed charge of -1) mimic the behavior of the large conductance K+ channel and the acetylcholine receptor channel.     

A channel model with fluctuating barrier structures.

Following the theory 'Fluctuations of barrier structure in ionic channels' (L?uger, P., Stephan, W. and Frehland, E. (1980) Biochim. Biophys. Acta 602, 167-180), we constructed a model of a channels with several conformational states. The origin of these conformational states and the source for the transitions from one to the other are given explicitly for the presented model. In this work the effect of multiple conformational states on the ion transport process is analyzed. We considered a channel protein with two main barriers and one binding site. The site is surrounded by dipolar groups. The dipole moment of these groups can be reoriented by thermal activity and also by electrical interaction with the transported ions. Differently polarized states generate different activation energy barriers for the ions. The set of conformational states of the channel is constituted by all the possible polarized states of the binding site. Using the rate-theory analysis of ion transport (Gl?sstone, S., Laider, K.J. and Eyring, H. (1941) The theory of rate processes, McGraw-Hill, New York), the possible coupling between ion flux and the channel conformational transitions has been incorporated into the model by considering the dependence of the rate constants on the heights of the energy barriers. The resulting multistate kinetic equations have been solved numerically. It was shown that the simple saturation characteristic of the flux-concentration curve was obtained. For certain values of the model parameters, the channel shows a strongly different conductance for anions compared to cations. In fact, the model contains an interesting mechanism that exhibits selectivity with respect to the charge of the ions.     

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).     

Ca channel gating during cardiac action potentials.

How do Ca channels conduct Ca ions during the cardiac action potential? We attempt to answer this question by applying a two-microelectrode technique, previously used for Na and K currents, in which we record the patch current and the action potential at the same time (Mazzanti, M., and L. J. DeFelice. 1987. Biophys. J. 12:95-100, and 1988. Biophys. J. 54:1139-1148; Wellis, D., L. J. DeFelice, and M. Mazzanti. 1990. Biophys. J. 57:41-48). In this paper, we also compare the action currents obtained by the technique with the step-protocol currents obtained during standard voltage-clamp experiments. Individual Ca channels were measured in 10 mM Ca/1 Ba and 10 mM Ba. To describe part of our results, we use the nomenclature introduced by Hess, P., J. B. Lansman, and R. W. Tsien (1984. Nature (Lond.). 311:538-544). With Ba as the charge carrier, Ca channel kinetics convert rapidly from long to short open times as the patch voltage changes from 20 to -20 mV. This voltage-dependent conversion occurs during action potentials and in step-protocol experiments. With Ca as the charge carrier, the currents are brief at all voltages, and it is difficult to define either the number of channels in the patch or the conductance of the individual channels. Occasionally, however, Ca-conducting channels spontaneously convert to long-open-time kinetics (in Hess et al., 1984, notation, mode 2).(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Open channel noise. V. Fluctuating barriers to ion entry in gramicidin A channels.

We have measured the fluctuations in the current through gramicidin A (GA) channels in symmetrical solutions of monovalent cations of various concentrations, and compared the spectral density values with those computed using E. Frehland's theory for noise in discrete transport systems (Frehland, E. 1978. Biophys. Chem. 8:255-265). The noise for the transport of NH4+ and Na+ ions in glycerol-monooleate/squalene membranes could be accounted for entirely by "shot noise" in the process of transport through a single-filing pore with two ion binding sites. However, in confirmation of results in a previous paper (Sigworth, F. J., D. W. Urry, and K. U. Prasad. 1987. Biophys. J. 52:1055-1064) currents of Cs+ showed a substantial excess noise at low ion concentrations, as did currents of K+ and Rb+. The excess noise was increased in thicker membranes. The observations are accounted for by a theory that postulates fluctuations of the entry rates of ions into the channel on a time scale of approximately 1 microsecond. These fluctuations occur preferentially when the channel is empty; the presence of bound ions stabilizes the "high conductance" conformation of the channel. The fluctuations are sensed to different degrees by the various ion species, and their kinetics depend on membrane thickness.     

Modeling of the pore domain of the GLUR1 channel: homology with K+ channel and binding of channel blockers

Molecular models of the M2 segments of the GluR1 channel have been elaborated using a molecular mechanics approach. The models are based on the homology between pore-lining segments of AMPA receptor channels and the KcsA K+ channel and on cyclic H bonds at the Q/R site of the AMPA receptor channel. The N-terminal region of an M2 segment of the channel is assumed, like that of the K+ channel, to adopt a helical conformation. Due to a deletion, the C-terminal end of the M2 segment of the AMPA receptor is more stretched than that of the K+ channel. As a result, only a single oxygen ring may be exposed to the AMPA receptor channel pore. Data on the block of AMPA receptor channels by dicationic adamantane derivatives have been used to select the most relevant model. The model with the oxygen of a Gly residue (position +2 from the Q/R site) exposed to the pore best fits the experimental data. This model also fits experimental data for another class of AMPA receptor antagonists, the polyamine amides. According to the model, the side-chains of the C-terminal residues are involved in intra-receptor interactions that stabilize the structure of the channel rather than in interactions with ions in the pore.     

Quantum mechanical calculations of charge effects on gating the KcsA channel

A series of ab initio (density functional) calculations were carried out on side chains of a set of amino acids, plus water, from the (intracellular) gating region of the KcsA K(+) channel. Their atomic coordinates, except hydrogen, are known from X-ray structures [D.A. Doyle, J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, R. MacKinnon, The structure of the potassium channel: molecular basis of K(+) conduction and selectivity, Science 280 (1998) 69-77; R. MacKinnon, S.L. Cohen, A. Kuo, A. Lee, B.T. Chait, Structural conservation in prokaryotic and eukaryotic potassium channels, Science 280 (1998) 106-109; Y. Jiang, A. Lee, J. Chen, M. Cadene, B.T. Chait, R. MacKinnon, The open pore conformation of potassium channels. Nature 417 (2001) 523-526], as are the coordinates of some water oxygen atoms. The 1k4c structure is used for the starting coordinates. Quantum mechanical optimization, in spite of the starting configuration, places the atoms in positions much closer to the 1j95, more tightly closed, configuration. This state shows four water molecules forming a "basket" under the Q119 side chains, blocking the channel. When a hydrated K(+) approaches this "basket", the optimized system shows a strong set of hydrogen bonds with the K(+) at defined positions, preventing further approach of the K(+) to the basket. This optimized structure with hydrated K(+) added shows an ice-like 12 molecule nanocrystal of water. If the water molecules exchange, unless they do it as a group, the channel will remain blocked. The "basket" itself appears to be very stable, although it is possible that the K(+) with its hydrating water molecules may be more mobile, capable of withdrawing from the gate. It is also not surprising that water essentially freezes, or forms a kind of glue, in a nanometer space; this agrees with experimental results on a rather different, but similarly sized (nm dimensions) system [K.B. Jinesh, J.W.M. Frenken, Capillary condensation in atomic scale friction: how water acts like a glue, Phys. Rev. Lett. 96 (2006) 166103/1-4]. It also agrees qualitatively with simulations on channels [A. Anishkin, S. Sukharev, Water dynamics and dewetting transitions in the small mechanosensitive channel MscS, Biophys. J. 86 (2004) 2883-2895; O. Beckstein, M.S.P. Sansom, Liquid-vapor oscillations of water in hydrophobic nanopores, Proc. Natl Acad. Sci. U. S. A. 100 (2003) 7063-7068] and on featureless channel-like systems [J. Lu, M.E. Green, Simulation of water in a pore with charges: application to a gating mechanism for ion channels, Prog. Colloid Polym. Sci. 103 (1997) 121-129], in that it forms a boundary on water that is not obvious from the liquid state. The idea that a structure is stable, even if individual molecules exchange, is well known, for example from the hydration shell of ions. We show that when charges are added in the form of protons to the domains (one proton per domain), the optimized structure is open. No stable water hydrogen bonds hold it together; an opening of 11.0 A appears, measured diagonally between non-neighboring domains as glutamine 119 carbonyl O-O distance. This is comparable to the opening in the MthK potassium channel structure that is generally agreed to be open. The appearance of the opening is in rather good agreement with that found by Perozo and coworkers. In contrast, in the uncharged structure this diagonal distance is 6.5 A, and the water "basket" constricts the uncharged opening still further, with the ice-like structure that couples the K(+) ion to the gating region freezing the entrance to the channel. Comparison with our earlier model for voltage gated channels suggests that a similar mechanism may apply in those channels.     

Charges, currents, and potentials in ionic channels of one conformation.

Flux through an open ionic channel is analyzed with Poisson-Nernst-Planck (PNP) theory. The channel protein is described as an unchanging but nonuniform distribution of permanent charge, the charge distribution observed (in principle) in x-ray diffraction. Appropriate boundary conditions are derived and presented in some generality. Three kinds of charge are present: (a) permanent charge on the atoms of the protein, the charge independent of the electric field; (b) free or mobile charge, carried by ions in the pore as they flux through the channel; and (c) induced (sometimes called polarization) charge, in the pore and protein, created by the electric field, zero when the electric field is zero. The permanent charge produces an offset in potential, a built-in Donnan potential at both ends of the channel pore. The system is completely solved for bathing solutions of two ions. Graphs describe the distribution of potential, concentration, free (i.e., mobile) and induced charge, and the potential energy associated with the concentration of charge, as well as the unidirectional flux as a function of concentration of ions in the bath, for a distribution of permanent charge that is uniform. The model shows surprising complexity, exhibiting some (but not all) of the properties usually attributed to single filing and exchange diffusion. The complexity arises because the arrangement of free and induced charge, and thus of potential and potential energy, varies, sometimes substantially, as conditions change, even though the channel structure and conformation (of permanent charge) is strictly constant. Energy barriers and wells, and the concomitant binding sites and binding phenomena, are outputs of the PNP theory: they are computed, not assumed. They vary in size and location as experimental conditions change, while the conformation of permanent charge remains constant, thus giving the model much of its interesting behavior.     

Negative surface charge near sodium channels of nerve: divalent ions, monovalent ions, and pH.

Evidence is given for a high density of negative surface charge near the sodium channel of myelinated nerve fibres. The voltage dependence of peak sodium permeability is measured in a voltage clamp. The object is to measure voltage shifts in sodium activation as the following external variables are varied: divalent cation concentration and type, monovalent concentration, and pH. With equimolar substitution of divalent ions the order of effectiveness for giving a positive shift is: Ba equals Sr less than Mg less than Ca less than Co approximately equal to Mn less than Ni less than Zn. A tenfold increase of concentration of any of these ions gives a shift of +20 to +25 mV. At low pH, the shift with a tenfold increase in Ca-2+ is much less than at normal pH, and conversely for high pH. Soulutions with no added divalent ions give a shift of minus 18 mV relative to 2 mM Ca-2+. Removal of 7/8 of the cations from the calcium-free solution gives a further shift of minue 35 mV. All shifts are explained quantitatively by assuming that changes in an external surface potential set up by fixed charges near the sodium channel produce the shifts. The model involves a diffuse double layer of counterions at the nerve surface and some binding of H+ions and divalent ions to the fixed charges. Three types of surface groups are postulated: (1) an acid pKa equals 2.88 charge density minus 0.9 nm- minus 2; (i) an acid pKa equals 4.58, charge density minus 0.58 nm- minus 2; (3) a base pKa equals 6.28, charge density +0.33 nm- minus 2. The two acid groups also bind Ca-2+ ions with a dissociation constant K equals 28 M. Reasonable agreement can also be obtained with a lower net surface charge density and stronger binding of divalent ions and H+ ions.     

Energy barrier presented to ions by the vestibule of the biological membrane channel.

The role of the vestibule in influencing the permeation of ions through biological ion channels is investigated. We derive analytical expressions for the electric potential satisfying Poisson's equation with prolate spheroidal boundary conditions. To allow more realistic geometries we devise an iterative method to calculate the electric potential arising from a fixed charge and an arbitrary dielectric boundary, and confirm that the analytical expressions and iterative method give similar potential values. We then investigate the size of the potential barrier presented to an ion by model vestibules of conical and catenary shapes. The height of the potential barrier increases steeply as an ion enters the vestibule and moves toward the constricted region of the channel. We show that the barrier presented by, for example, a 15 degrees conical vestibule can be canceled by placing dipoles with a total moment of about 50 Debyes near the constricted region of the pore. The selectivity of cations and anions can result from the polarity of charge groups or the orientation of dipoles located near the constricted region of the channel.     

Selectivity and permeation in calcium release channel of cardiac muscle: alkali metal ions

Current was measured from single open channels of the calcium release channel (CRC) of cardiac sarcoplasmic reticulum (over the range +/-180 mV) in pure and mixed solutions (e.g., biionic conditions) of the alkali metal ions Li+, K+, Na+, Rb+, Cs+, ranging in concentration from 25 mM to 2 M. The current-voltage (I-V) relations were analyzed by an extension of the Poisson-Nernst-Planck (PNP) formulation of electrodiffusion, which includes local chemical interaction described by an offset in chemical potential, which likely reflects the difference in dehydration/solvation/rehydration energies in the entry/exit steps of permeation. The theory fits all of the data with few adjustable parameters: the diffusion coefficient of each ion species, the average effective charge distribution on the wall of the pore, and an offset in chemical potential for lithium and sodium ions. In particular, the theory explains the discrepancy between "selectivities" defined by conductance sequence and "selectivities" determined by the permeability ratios (i.e., reversal potentials) in biionic conditions. The extended PNP formulation seems to offer a successful combined treatment of selectivity and permeation. Conductance selectivity in this channel arises mostly from friction: different species of ions have different diffusion coefficients in the channel. Permeability selectivity of an ion is determined by its electrochemical potential gradient and local chemical interaction with the channel. Neither selectivity (in CRC) seems to involve different electrostatic interaction of different ions with the channel protein, even though the ions have widely varying diameters.     

Sodium ions as blocking agents and charge carriers in the potassium channel of the squid giant axon

Instantaneous K channel current-voltage (I-V) relations were determined by using internally perfused squid axons. When K was the only internal cation, the I-V relation was linear for outward currents at membrane potentials up to +240 mV inside. With 25-200 mM Na plus 300 mM K in the internal solution, an N-shaped I-V curve was seen. Voltage-dependent blocking of the K channels by Na produces a region of negative slope in the I-V plot (F. Bezanilla and C. M. Armstrong. 1972. J. Gen Physiol, 60: 588). At higher voltages (greater than or equal to 160 mV) we observed a second region of increasing current and a decrease in the fraction of the K conductance blocked by Na. Internal tetraethylammonium (TEA) ions blocked currents over the whole voltage range. In a second series of experiments with K-free, Na-containing internal solutions, the I-V curve turned sharply upward about +160 mV. The current at high voltages increased with increasing internal Na concentration was largely blocked by internal TEA. These data suggest that the K channel becomes substantially more permeable to Na at high voltages. This change is apparently responsible for the relief, at high transmembrane voltages, of the blocking effect seen in axons perfused with Na plus K mixtures. Each time a Na ion passed through, vacating the blocking site, the channel would transiently allow K ions to pass through freely.     

Stochastic theory of singly occupied ion channels. II. Effects of access resistance and potential gradients extending into the bath.

In a previous paper (Jakobsson, E., and S. W. Chiu. 1987. Biophys. J. 52:33-46), we presented the stochastic theory of the singly occupied ion channel as applied to sodium permeation of gramicidin channels, with the assumption of perfect equilibration between the bathing solutions and the ends of the ion channel. In the present paper we couple the previous theory to electrodiffusion of ions from the bulk of the bathing solution to the channel mouth. Our electrodiffusion calculations incorporate estimates of the potential gradients near the channel mouth due to image forces and due to the fraction of the applied potential that falls beyond the ends of the channel. To keep the diffusion calculation one-dimensional, we make the assumption that the electrical potentials in the bath exhibit hemispherical symmetry. As in the previous paper, the flux equations are fit to data on sodium permeation of normal gramicidin A, and gramicidins modified by the fluorination of the valine at the No. 1 position (Barrett Russell, E. W., L. B. Weiss, F. I. Navetta, R. E. Koeppe II, and O. S. Anderson. 1986. Biophys. J. 49:673-686). The conclusions of our previous paper with respect to the effect of fluorination on the mobility, surface potential well depth, and central barrier, are confirmed. However the absolute values of these quantities are somewhat changed when diffusive resistance to the mouth is taken into account, as in the present paper. Future possibilities for more accurate calculations by other methods are outlined.     

Multiple products of the Drosophila Shaker gene may contribute to potassium channel diversity

K+ channels are known through electrophysiology and pharmacology to be an exceptionally diverse group of channels. Molecular studies of the Shaker (Sh) locus in Drosophila have provided the first glimpse of K+ channel structure. The sequences of several Sh cDNA clones have been reported; none are identical. We have isolated and examined 18 additional Sh cDNAs in an attempt to understand the origin, extent, and significance of the variability. The diversity is extensive: we have already identified cDNAs representing at least nine distinct types, and Sh could potentially encode 24 or more products. This diversity, however, fits a simple pattern in which variable 3' and 5' ends are spliced onto a central constant region to yield different cDNA types. These different Sh cDNAs encode proteins with distinct structural features.     

Localization of the K+ lock-In and the Ba2+ binding sites in a voltage-gated calcium-modulated channel. Implications for survival of K+ permeability.

Using Ba2+ as a probe, we performed a detailed characterization of an external K+ binding site located in the pore of a large conductance Ca2+-activated K+ (BKCa) channel from skeletal muscle incorporated into planar lipid bilayers. Internal Ba2+ blocks BKCa channels and decreasing external K+ using a K+ chelator, (+)-18-Crown-6-tetracarboxylic acid, dramatically reduces the duration of the Ba2+-blocked events. Average Ba2+ dwell time changes from 10 s at 10 mM external K+ to 100 ms in the limit of very low [K+]. Using a model where external K+ binds to a site hindering the exit of Ba2+ toward the external side (Neyton, J., and C. Miller. 1988. J. Gen. Physiol. 92:549-568), we calculated a dissociation constant of 2.7 mircoM for K) at this lock-in site. We also found that BK(Ca) channels enter into a long-lasting nonconductive state when the external [K+] is reduced below 4 microM using the crown ether. Channel activity can be recovered by adding K+, Rb+, Cs+, or NH4+ to the external solution. These results suggest that the BK(Ca) channel stability in solutions of very low [K+] is due to K+ binding to a site having a very high affinity. Occupancy of this site by K+ avoids the channel conductance collapse and the exit of Ba2+ toward the external side. External tetraethylammonium also reduced the Ba2+ off rate and impeded the channel from entering into the long-lasting nonconductive state. This effect requires the presence of external K+. It is explained in terms of a model in which the conduction pore contains Ba2+, K+, and tetraethylammonium simultaneously, with the K+ binding site located internal to the tetraethylammonium site. Altogether, these results and the known potassium channel structure (Doyle, D.A., J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, and R. MacKinnon. 1998. Science. 280:69-77) imply that the lock-in site and the Ba2+ sites are the external and internal ion sites of the selectivity filter, respectively.     

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.