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.     

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.     

Free energy of a potassium ion in a model of the channel formed by an amphipathic leucine-serine peptide

We use molecular dynamics simulations to investigate the position-dependent free energy of a potassium ion in a model of an ion channel formed by the synthetic amphipathic leucine-serine peptide, LS3. The channel model is a parallel bundle of six LS3 helices around which are packed 146 methane-like spheres in order to mimic a membrane. At either end of and within the channel are 1051 water molecules, plus four ions (two potassium and two chloride). The free energy of a potassium ion in the channel was estimated using the weighted histogram analysis (WHAM) method. This is the first time to our knowledge that such a calculation has been carried out as a function of the position of an ion in three dimensions within a channel. The results indicate that for this channel, which is lined by hydrophilic serine sidechains, there is a relatively weak dependence of the free energy on the axial/off-axial position of the ion. There are some off-axis local minima, especially in the C-terminal half of the channel. Using the free energy results, a single channel current-voltage curve was estimated using a one-dimensional Nernst-Planck equation. Although reasonable agreement with experiment is achieved for K(+) ions flowing from the N-terminal to the C-terminal mouth, in the opposite direction the current is underestimated. This underestimation may be a consequence of under-sampling of the conformational dynamics of the channel. We suggest that our simulations may have captured, for example, a sub-conductance level (i.e. an incompletely open state) of the LS3 channel.     

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

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

Electrostatic calculations for an ion channel. II. Kinetic behavior of the gramicidin A channel.

A theoretical model of the gramicidin A channel is presented and the kinetic behavior of the model is derived and compared with previous experimental results. The major assumption of the model is that the only interaction between ions in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate that there will be potential wells at each end of the channel and, at high concentrations, that both wells can be occupied. The kinetics are based on two reaction steps: movement of the ion from the bulk solution to the well and movement between the two wells. The kinetics for this reaction rate approach are identical to those based on the Nernst-Planck equation in the limit where the movement between the two wells is rate limiting. The experimental results for sodium and potassium are consistent with a maximum of two ions per channel. To explain the thallium results it is necessary to allow three ions per channel. It is shown that this case is compatible with the electrostatic calculations if the presence of an anion is included. The theoretical kinetics are in reasonable quantitative agreement with the following experimental measurements: single channel conductance of sodium, potassium, and thallium; bi-ionic potential and permeability ratio between sodium-potassium and potassium-thallium; the limiting conductance of potassium and thallium at high applied voltages; current-voltage curves for sodium and potassium at low (but not high) concentrations; and the inhibition of sodium conductance by thallium. The results suggest that the potential well is located close to the channel mouth and that the conductance is partially limited by the rate going from the bulk solution to the well. For thallium, this entrance rate is probably diffusion limited.     

The gramicidin a channel: A review of its permeability characteristics with special reference to the single-file aspect of transport

Summary Gramicidin A forms univalent cation-selective channels of 4 Å diameter in phospholipid bilayer membranes. The transport of ions and water throughout most of the channel length is by a singlefile process; that is, cations and water molecules cannot pass each other within the channel. The implications of this single-file mode of transport for ion movement are considered. In particular, we show that there is no significant electrostatic barrier to ion movement between the energy wells at the two ends of the channel. The rate of ion translocation (e.g., Na⁺ or Cs⁺) through the channel between these wells is limited by the necessity for an ion to move six water molecules in single file along with it; this also limits the maximum possible value for channel conductance. At all attainable concentrations of NaCl, the gramicidin A channel never contains more than one sodium ion, whereas even at 0.1M CsCl, some channels contain two cesium ions. There is no necessity to postulate more than two ion-binding sites in the channel or occupancy of the channel by more than two ions at any time.     

A semi-microscopic Monte Carlo study of permeation energetics in a gramicidin-like channel: the origin of cation selectivity.

The influence of a gramicidin-like channel former on ion free energy barriers is studied using Monte Carlo simulation. The model explicitly describes the ion, the water dipoles, and the peptide carbonyls; the remaining degrees of freedom, bulk electrolyte, non-polar lipid and peptide regions, and electronic (high frequency) permittivity, are treated in continuum terms. Contributions of the channel waters and peptide COs are studied both separately and collectively. We found that if constrained to their original orientations, the COs substantially increase the cationic permeation free energy; with or without water present, CO reorientation is crucial for ion-CO interaction to lower cation free energy barriers; the translocation free energy profiles for potassium-, rubidium-, and cesium-like cations exhibit no broad barriers; the lipid-bound peptide interacts more effectively with anions than cations; anionic translocation free energy profiles exhibit well defined maxima. Using experimental data to estimate transfer free energies of ions and water from bulk electrolyte to a non-polar dielectric (continuum lipid), we found reasonable ion permeation profiles; cations bind and permeate, whereas anions cannot enter the channel. Cation selectivity arises because, for ions of the same size and charge, anions bind hydration water more strongly.     

Fluctuations of barrier structure in ionic channels

In rate-theory analysis of ion transport in channels, the energy of binding sites and the height of activation barriers are usually considered to be time-independent and not influenced by the movement of the ion. The assumption of a fixed barrier structure seems questionable, however, in view of the fact that proteins may exist in a large number of conformational states and may rapidly move from one state to the other. In this study, some of the effects of multiple conformational states of a channel on ion transport are analyzed. In the first part of the paper, the ion permeability of a channel with n binding sites is treated on the assumption that interconversion of channel states is much faster than ion transfer between binding sites. Under this condition, the form of the flux equation remains the same as for a channel with fixed barriers, provided that the rate constants for ion jumps are replaced by weighted averages over the rate constants for the individual conformational states. In the second part, a channel with two (main) barriers and a single (main) binding site is considered, with the rates of conformational transitions being arbitrary. This case, in particular, includes the situation where a jump of the ion is followed by a slow transition to a more polarized state of the binding site. Under this condition, the conductance of the channel exhibits a nonlinear dependence on ion concentration which is different from a simple saturation behavior. Under non-stationary conditions damped oscillations may occur.     

Ion flux across lipid bilayer membranes with charged surfaces

Summary In this paper we derive expressions for the ion flux across lipid bilayer membranes with charged surfaces treating the membrane as a continuous phase interposed between two electrolyte solutions and calculating the ion flux with the Nernst-Planck equations. The theoretical results are compared with experiments of Seufert and Hashimoto on lipid bilayer membranes with charged surface active agents added to the membranes. If the charge of both membrane surfaces has the same sign the flux of the gegenions is greatly increased whereas the flux of the coions decreases to a small amount. For oppositely charged membrane surfaces the membrane behaves like a np semiconductor and typical rectification voltage-current characteristics are obtained.     

Electrodiffusion of ions approaching the mouth of a conducting membrane channel.

The movement of ions in the aqueous medium as they approach the mouth (radius a) of a conducting membrane channel is analyzed. Starting with the Nernst-Planck and Poisson equations, we derive a nonlinear integrodifferential equation for the electric potential, phi(r), a less than or equal to r less than infinity. The formulation allows deviations from charge neutrality and dependence of phi(r) on ion flux. A numerical solution is obtained by converting the equation to an integral equation that is solved by an iterative method for an assumed mouth potential, combined with a shooting method to adjust the mouth potential until the numerical solution agrees with an asymptotic expansion of the potential at r-a much greater than lambda (lambda = Debye length). Approximate analytic solutions are obtained by assuming charge neutrality (Läuger, 1976) and by linearizing. The linear approximation agrees with the exact solution under most physiological conditions, but the charge-neutrality solution is only valid for r much greater than lambda and thus cannot be used unless a much greater than lambda. Families of curves of ion flux vs. potential drop across the electrolyte, phi(infinity)-phi (a), and of permeant ion density at the channel mouth, n1(a), vs. flux are obtained for different values of a/lambda and S = a d phi/dr(a). If a much greater than lambda and S = O, the maximum flux (which is approached when n1(a)----0) is reduced by 50% compared to the value predicted by the charge-neutrality solution. Access resistance is shown to be a factor a/[2 (a + lambda)] times the published formula (Hille, 1968), which was derived without including deviations from charge neutrality and ion density gradients and hence does not apply when there is no counter-ion current. The results are applied to an idealized diffusion-limited channel with symmetric electrolytes. For S = O, the current/voltage curves saturate at a value dependent on a/lambda; for S greater than O, they increase linearly for large voltage.     

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.     

Electrostatic radius of the gramicidin channel determined from voltage dependence of H+ ion conductance.

The results of Decker and Levitt (1987) suggest that the conductance of H+ ion through the gramicidin channel is limited primarily by diffusion in the bulk solution at the channel mouth. It is assumed in this paper that the H+ conductance is 100% diffusion limited. This means that all the factors that influence the H+ flux are external to the channel and are presumed to be known. In particular, the diffusion coefficient of H+ in this region is assumed to be equal to the bulk solution value and the only force acting on the ion is that due to the applied voltage. A model of the H+ flux is derived, based on the Nernst-Planck equation. It has three adjustable parameters: the electrostatic radius, the capture distance, and the radius of the H+ ion. The acceptable range of the parameters was determined by comparing the predictions of the model with the experimental measurements of the H+ conductance at pH 3.75. The best fit was obtained for an electrostatic radius in the range 2.3-2.7 A. This is in good agreement with earlier predictions (2.5 A) based on the assumption that the dielectric constant of the channel water is equal to that of bulk water. The addition of 1 M choline Cl- (an impermeant) increases the H+ current at low voltage and decreases it at high voltage. The increase can be explained by the small surface charge that results from the separation of charge produced by exclusion of the large choline cation (relative to Cl-) from the membrane surface. The decrease at high voltages can be accounted for by the change in the profile of the applied potential produced by the increase in ionic strength.     

Flux, coupling, and selectivity in ionic channels of one conformation.

Ions crossing biological membranes are described as a concentration of charge flowing through a selective open channel of one conformation and analyzed by a combination of Poisson and Nernst-Planck equations and boundary conditions, called the PNP theory for short. The ion fluxes in this theory interact much as ion fluxes interact in biological channels and mediated transporters, provided the theoretical channel contains permanent charge and has selectivity created by (electro-chemical) resistance at its ends. Interaction occurs because the flux of different ionic species depends on the same electric field. That electric field is a variable, changing with experimental conditions because the screening (i.e., shielding) of the permanent charge within the channel changes with experimental conditions. For example, the screening of charge and the shape of the electric field depend on the concentration of all ionic species on both sides of the channel. As experimental interventions vary the screening, the electric field varies, and thus the flux of each ionic species varies conjointly, and is, in that sense, coupled. Interdependence and interaction are the rule, independence is the exception, in this channel.     

Microscopic model for selective permeation in ion channels.

Ionic permeation in the selectivity filter of ion channels is analyzed by a microscopic model based on molecular kinetic theory. The energy and flux equations are derived by assuming that: (a) the selectivity filter is formed by a symmetrical array of carbonyl groups; (b) ion movement is near the axis of the channel; (c) a fraction of water molecules is separated from the ion while it moves across the selectivity filter; (d) the applied voltage drops linearly across the selectivity filter; (e) ions move independently. Energy profiles, single channel conductances, and the degree of hydration of K+ in a hypothetical K+ channel are examined by varying the following microscopic parameters: ion radius and mass, channel radius, number of effective water dipoles, and number of carbonyl groups. The i-V curve is linear up to +/- 170 mV. If the positions of energy maxima and minima are fixed, this linear range is reduced to +/- 50 mV. Channel radius and ion-water interactions are found to be two major channel structural determinants for selectivity sequences. Both radius and mass of an ion are important in selectivity mediated by these interactions. The theory predicts a total of 15 possible kinetic selectivity sequences for alkali cations in ion channels with a single selectivity filter.     

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.     

Ion coordination in the amphotericin B channel.

The antifungal polyene antibiotic amphotericin B forms channels in lipid membranes that are permeable to ions, water, and nonelectrolytes. Anion, cation, and ion pair coordination in the water-filled pore of the "barrel" unit of the channels was studied by molecular dynamics simulations. Unlike the case of the gramicidin A channel, the water molecules do not create a single-file configuration in the pore, and some cross sections of the channel contain three or four water molecules. Both the anion and cation are strongly bound to ligand groups and water molecules located in the channel. The coordination number of the ions is about six. The chloride has two binding sites in the pore. The binding with water is dominant; more than four water molecules are localized in the anion coordination sphere. Three motifs of the ion coordination were monitored. The dominant motif occurs when the anion is bound to one ligand group. The ion is bound to two or three ligand groups in the less favorable configurations. The strong affinity of cations to the channel is determined by the negatively charged ligand oxygens, whose electrostatic field dominates over the field of the hydrogens. The ligand contribution to the coordination number of the sodium ion is noticeably higher than in the case of the anion. As in the case of the anion, there are three motifs of the cation coordination. The favorable one occurs when the cation is bound to two ligand oxygens. In the less favorable cases, the cation is bound to three or four oxygens. In the contact ion pair, the cation and anion are bound to two ligand oxygens and one ligand hydrogen, respectively. There exist intermediate solvent-shared states of the ion pair. The average distances between ions in these states are twice as large as that of the contact ion pair. The stability of the solvent-shared state is defined by the water molecule oriented along the electrostatic field of both ions.