Periaxonal surface calcium binding and distribution of charge on the faces of squid axon potassium channel molecules

Summary The calcium binding constant associated with external surface charge in a position to influence the voltage sensing charges for potassium channel gating appears to be 30 molar^–1, a value much larger than previously thought and in approximate agreement with that found for artificial membranes composed of the lipid brain phosphatidylserene. Fixed charge on the periaxonal membrane surface is distributed in such a way that much larger charges occur at a distance of at least 8 angstroms from the channel pore openings. The separation between the ion pathway and the channel gating charge appears to be greater than or equal to 8 angstroms. Periaxonal surface charge which is in a position to determine the surface potential for gating has a magnitude greater than or equal to one (negative) electronic charge per 182 square angstrom before calcium binding, which is reduced to –e/625 Å in a normal divalent ionic environment. With the normal divalent ionic composition of seawater the surface potential at a position to influence the gating voltage sensor is –15 millivolts relative to the bulk external potential. The external surface potential is –3 mV at the pore mouth. There appears to be a negligible amount of fixed charge on the axoplasmic surface in the vicinity of the ion channel opening. Further, our results confirm earlier measurements that have given a negligible amount of axoplasmic surface fixed charge whose field components would be in a position to influence the channel gating charges.     

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.     

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 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.     

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).     

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.     

How does vestibule surface charge affect ion conduction and toxin binding in a sodium channel?

We describe various models for the dielectric geometry and pore mouth charge distribution of a Na channel. The electric potential due to the vestibule charges is then computed on the basis of the nonlinear Possion-Boltzmann equation. The results are used to account for the effect of permeant ion concentration and ionic strength on channel conductance and on toxin association rate constants for Na channels. We find that a single negatively charged group near the entrance to the channel constriction is adequate to account for deviations from Michaelis-Menten conductance kinetics and for the concentration dependence of toxin-binding coefficients. We find further that only a limited range of vestibule geometries and pore mouth charge distributions are consistent with experiment.     

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.     

Rapid calculation of the solution scattering profile from a macromolecule of known structure

If one expands the structure factor equation in spherical coordinates, rotational averaging of the molecular Fourier transform, which leads directly to the solution scattering profile, is greatly simplified. It becomes a projection in the polar and azimuthal angular variables. The profile is given by I(R) = 1/2 infinity sigma n = 0 n sigma m = 0 epsilon mNm,n magnitude of Gm,n(R) 2 where Gm,n(R) = sigma jfjYm,n(theta j, phi j)jn(2 pi rjR) The index j runs over all atoms; r, theta, phi are atomic coordinates and epsilon and N are constants; the Ym,n are complex spherical harmonics, and jn are spherical Bessel functions; R = 2 sin theta/lambda. The effects of solvent have been modeled by subtracting from each protein atom a properly weighted water. Hydrogens have been included by using scattering curves fj derived from the spherical averaging of protein atoms with their attached hydrogens. This approach may also be satisfactory for neutron scattering. Published scattering profiles for lysozyme and BPTI have been accurately matched in less than one-tenth the time required by other methods. Separate, adjustable temperature factors for the protein, solvent waters, and bound waters are used, and appear to be needed. In the case of BPTI, as suggested by NMR observations, the observed diffraction pattern was much better accounted for by including only 4 tightly bound waters rather than the roughly 60 seen by crystallography.     

Effects of external protons on single cardiac sodium channels from guinea pig ventricular myocytes.

The effects of external protons on single sodium channel currents recorded from cell-attached patches on guinea pig ventricular myocytes were investigated. Extracellular protons reduce single channel current amplitude in a dose-dependent manner, consistent with a simple rapid channel block model where protons bind to a site within the channel with an apparent pKH of 5.10. The reduction in single channel current amplitude by protons is voltage independent between -70 and -20 mV. Increasing external proton concentration also shifts channel gating parameters to more positive voltages, consistent with previous macroscopic results. Similar voltage shifts are seen in the steady-state inactivation (h infinity) curve, the time constant for macroscopic current inactivation (tau h), and the first latency function describing channel activation. As pHo decreases from 7.4 to 5.5 the midpoint of the h infinity curve shifts from -107.6 +/- 2.6 mV (mean +/- SD, n = 16) to -94.3 +/- 1.9 mV (n = 3, P less than 0.001). These effects on channel gating are consistent with a reduction in negative surface potential due to titration of negative external surface charge. The Gouy-Chapman-Stern surface charge model incorporating specific proton binding provides an excellent fit to the dose-response curve for the shift in the midpoint of the h infinity curve with protons, yielding an estimate for total negative surface charge density of -1e/490 A2 and a pKH for proton binding of 5.16. By reducing external surface Na+ concentration, titration of negative surface charge can also quantitatively account for the reduction in single Na+ channel current amplitude, although we cannot rule out a potential role for channel block. Thus, titration by protons of a single class of negatively charged sites may account for effects on both single channel current amplitude and gating.     

Cation-selective mutations in the M2 domain of the inhibitory glycine receptor channel reveal determinants of ion-charge selectivity

Ligand-gated ion channel receptors mediate neuronal inhibition or excitation depending on their ion charge selectivity. An investigation into the determinants of ion charge selectivity of the anion-selective alpha1 homomeric glycine receptor (alpha1 glycine receptor [GlyR]) was undertaken using point mutations to residues lining the extra- and intracellular ends of the ion channel. Five mutant GlyRs were studied. A single substitution at the intracellular mouth of the channel (A-1'E GlyR) was sufficient to convert the channels to select cations over anions with P(Cl)/P(Na) = 0.34. This result delimits the selectivity filter and provides evidence that electrostatic interactions between permeating ions and pore residues are a critical factor in ion charge selectivity. The P-2'Delta mutant GlyR retained its anion selectivity (P(Cl)/P(Na) = 3.81), but it was much reduced compared with the wild-type (WT) GlyR (P(Cl)/P(Na) = 27.9). When the A-1'E and the P-2'Delta mutations were combined (selectivity double mutant [SDM] GlyR), the relative cation permeability was enhanced (P(Cl)/P(Na) = 0.13). The SDM GlyR was also Ca(2+) permeable (P(Ca)/P(Na) = 0.29). Neutralizing the extracellular mouth of the SDM GlyR ion channel (SDM+R19'A GlyR) produced a more Ca(2+)-permeable channel (P(Ca)/P(Na) = 0.73), without drastically altering monovalent charge selectivity (P(Cl)/P(Na) = 0.23). The SDM+R19'E GlyR, which introduces a negatively charged ring at the extracellular mouth of the channel, further enhanced Ca(2+) permeability (P(Ca)/P(Na) = 0.92), with little effect on monovalent selectivity (P(Cl)/P(Na) = 0.19). Estimates of the minimum pore diameter of the A-1'E, SDM, SDM+R19'A, and SDM+R19'E GlyRs revealed that these pores are larger than the alpha1 GlyR, with the SDM-based GlyRs being comparable in diameter to the cation-selective nicotinic acetylcholine receptors. This result provides evidence that the diameter of the ion channel is also an important factor in ion charge selectivity.     

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero.     

How pore mouth charge distributions alter the permeability of transmembrane ionic channels.

This paper investigates the effects that surface dipole layers and surface charge layers along the pore mouth-water interface can have on the electrical properties of a transmembrane channel. Three specific molecular sources are considered: dipole layers formed by membrane phospholipids, dipole layers lining the mouth of a channel-forming protein, and charged groups in the mouth of a channel-forming protein. We find, consistent with previous work, that changing the lipid-water potential difference only influences channel conduction if the rate-limiting step takes place well inside the channel constriction. We find that either mouth dipoles or mouth charges can act as powerful ion attractors increasing either cation or anion concentration near the channel entrance to many times its bulk value, especially at low ionic strengths. The effects are sufficient to reconcile the apparently contradictory properties of high selectivity and high conductivity, observed for a number of K+ channel systems. We find that localizing the electrical sources closer to the constriction entrance substantially increases their effectiveness as ion attractors; this phenomenon is especially marked for dipolar distributions. An approximate treatment of electrolyte shielding is used to discriminate between the various mechanisms for increasing ionic concentration near the constriction entrance. Dipolar potentials are far less sensitive to ionic strength variation than potentials due to fixed charges. We suggest that the K+ channel from sarcoplasmic reticulum does not have a fixed negative charge near the constriction entrance; we suggest further that the Ca+2-activated K+ channel from transverse tubule does have such a charge.     

Constant fields and constant gradients in open ionic channels.

Ions enter cells through pores in proteins that are holes in dielectrics. The energy of interaction between ion and charge induced on the dielectric is many kT, and so the dielectric properties of channel and pore are important. We describe ionic movement by (three-dimensional) Nemst-Planck equations (including flux and net charge). Potential is described by Poisson's equation in the pore and Laplace's equation in the channel wall, allowing induced but not permanent charge. Asymptotic expansions are constructed exploiting the long narrow shape of the pore and the relatively high dielectric constant of the pore's contents. The resulting one-dimensional equations can be integrated numerically; they can be analyzed when channels are short or long (compared with the Debye length). Traditional constant field equations are derived if the induced charge is small, e.g., if the channel is short or if the total concentration gradient is zero. A constant gradient of concentration is derived if the channel is long. Plots directly comparable to experiments are given of current vs voltage, reversal potential vs. concentration, and slope conductance vs. concentration. This dielectric theory can easily be tested: its parameters can be determined by traditional constant field measurements. The dielectric theory then predicts current-voltage relations quite different from constant field, usually more linear, when gradients of total concentration are imposed. Numerical analysis shows that the interaction of ion and channel can be described by a mean potential if, but only if, the induced charge is negligible, that is to say, the electric field is spatially constant.     

Hindered diffusion in agarose gels: test of effective medium model.

The diffusivities of uncharged macromolecules in gels (D) are typically lower than in free solution (D infinity), because of a combination of hydrodynamic and steric factors. To examine these factors, we measured D and D infinity for dilute solutions of several fluorescein-labeled macromolecules, using an image-based fluorescence recovery after photobleaching technique. Test macromolecules with Stokes-Einstein radii (rs) of 2.1-6.2 nm, including three globular proteins (bovine serum albumin, ovalbumin, lactalbumin) and four narrow fractions of Ficoll, were studied in agarose gels with agarose volume fractions (phi) of 0.038-0.073. The gels were characterized by measuring the hydraulic permeability of supported agarose membranes, allowing calculation of the Darcy permeability (kappa) for each gel sample. It was found that kappa, which is a measure of the intrinsic hydraulic conductance of the gel, decreased by an order of magnitude as phi was increased over the range indicated. The diffusivity ratio D/D infinity, which varied from 0.20 to 0.63, decreased with increases in rs or phi. Thus as expected, diffusional hindrances were the most severe for large macromolecules and/or relatively concentrated gels. According to a recently proposed theory for hindered diffusion through fibrous media, the diffusivity ratio is given by the product of a hydrodynamic factor (F) and a steric factor (S). The functional form is D/D infinity = F(rs/k1/2) S(f), where f = [(rs+rf)/rf]2 phi and rf is the fiber radius. Values of D/D infinity calculated from this effective medium theory, without use of adjustable parameters, were in much better agreement with the measured values than were predictions based on other approaches. The strengths and limitations of the effective medium theory for predicting diffusivities in gels are discussed.     

Heparin influence on alpha-staphylotoxin formed channel

The effects of heparin on ion channels formed by Staphylococcus aureus alpha-toxin (ST channel) in lipid bilayers were studied under voltage clamp conditions. Heparin concentrations as small as 100 pM induced a sharp dose-dependent increase in channel voltage sensitivity. This was only observed when heparin was added to the negative-potential side of lipid bilayers in the presence of divalent cations. Divalent cations differ in their efficiency: Zn2+>Ca2+>Mg2+. The apparent positive gating charge increased 2-3-fold with heparin addition as well as with acidification of the bathing solution. 'Free' carboxyl groups and carboxyl groups in ion pairs of the protein moiety are hypothesized to interact with sulfated groups of heparin through divalent cation bridges. The cis mouth of the channel (that protrudes beyond the membrane plane on the side of ST addition and to which voltage was applied) is less sensitive to heparin than the trans-mouth. It is suggested that charged residues which interact with heparin at the cis mouth of ST channels and which contribute to the effective gating charge at negative voltage may be physically different from those at the trans mouth and at positive voltage.     

Modeling the electrophoresis of lysozyme. II. Inclusion of ion relaxation.

In this work, boundary element methods are used to model the electrophoretic mobility of lysozyme over the pH range 2-6. The model treats the protein as a rigid body of arbitrary shape and charge distribution derived from the crystal structure. Extending earlier studies, the present work treats the equilibrium electrostatic potential at the level of the full Poisson-Boltzmann (PB) equation and accounts for ion relaxation. This is achieved by solving simultaneously the Poisson, ion transport, and Navier-Stokes equations by an iterative boundary element procedure. Treating the equilibrium electrostatics at the level of the full rather than the linear PB equation, but leaving relaxation out, does improve agreement between experimental and simulated mobilities, including ion relaxation improves it even more. The effects of nonlinear electrostatics and ion relaxation are greatest at low pH, where the net charge on lysozyme is greatest. In the absence of relaxation, a linear dependence of mobility and average polyion surface potential, (lambda zero)s, is observed, and the mobility is well described by the equation [formula: see text] where epsilon 0 is the dielectric constant of the solvent, and eta is the solvent viscosity. This breaks down, however, when ion relaxation is included and the mobility is less than predicted by the above equation. Whether or not ion relaxation is included, the mobility is found to be fairly insensitive to the charge distribution within the lysozyme model or the internal dielectric constant.     

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.