How electrolyte shielding influences the electrical potential in transmembrane ion channels.

The electrical potential due to fixed charge distributions is strongly altered in the vicinity of a membrane and notably dependent on aqueous electrolyte concentration. We present an efficient way to solve the nonlinear Poisson-Boltzmann equation applicable to general cylindrically symmetric dielectric geometries. It generalizes Gouy-Chapman theory to systems containing transmembrane channels. The method is applied to three channel systems: gramicidin, gap junction, and porin. We find that for a long, narrow channel such as gramicidin concentration variation has little influence on the electrical image barrier to ion permeation. However, electrolyte shielding reduces the image induced contribution to the energy required for multiple occupancy. In addition, the presence of electrolyte significantly affects the voltage profile due to an applied potential, substantially compressing the electric field to the immediate vicinity of the pore itself. In the large diameter channels, where bulk electrolyte may be assumed to enter the pore, the electrolyte greatly reduces the image barrier to ion permeation. At physiological ionic strengths this barrier is negligible and the channel may be readily multiply occupied. At all ionic strengths considered (l greater than 0.005 M) the image barrier saturates rapidly and is essentially constant more than one channel radius from the entrance to the pore. At lower ionic strengths (l less than 0.016 M) there are noticeable (greater than 20 mV) energy penalties associated with multiple occupancy.     

Electrostatic modeling of ion pores. Energy barriers and electric field profiles

This paper presents calculations of the image potential for an ion in an aqueous pore through lipid membrane and the electric field produced in such a pore when a transmembrane potential is applied. The method used is one introduced by Levitt (1978, Biophys. J. 22:209), who solved an equivalent problem, in which a surface charge density is placed at the dielectric boundary. It is shown that there are singularities in this surface charge density if the model system has sharp corners. Numerically accurate calculations require exact treatment of these singularities. The major result of this paper is the development of a projection method that explicitly accounts for this behavior. It is shown how this technique can be used to compute, both reliably and efficiently, the electrical potential within a model pore in response to any electrical source. As the length of a channel with fixed radius is increased, the peak in the image potential approaches that of an infinitely long channel more rapidly than previously believed. When a transmembrane potential is applied the electric field within a pore is constant over most of its length. Unless the channel is much longer than its radius, the field extends well into the aqueous domain. For sufficiently dissimilar dielectrics the calculated values for the peak in the image potential and for the field well within the pore can be summarized by simple empirical expressions that are accurate to within 5%.     

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.     

Long-pore Electrostatics in Inward-rectifier Potassium Channels

Inward-rectifier potassium (Kir) channels differ from the canonical K⁺ channel structure in that they possess a long extended pore (~85 Å) for ion conduction that reaches deeply into the cytoplasm. This unique structural feature is presumably involved in regulating functional properties specific to Kir channels, such as conductance, rectification block, and ligand-dependent gating. To elucidate the underpinnings of these functional roles, we examine the electrostatics of an ion along this extended pore. Homology models are constructed based on the open-state model of KirBac1.1 for four mammalian Kir channels: Kir1.1/ROMK, Kir2.1/IRK, Kir3.1/GIRK, and Kir6.2/KATP. By solving the Poisson-Boltzmann equation, the electrostatic free energy of a K⁺ ion is determined along each pore, revealing that mammalian Kir channels provide a favorable environment for cations and suggesting the existence of high-density regions in the cytoplasmic domain and cavity. The contribution from the reaction field (the self-energy arising from the dielectric polarization induced by the ion's charge in the complex geometry of the pore) is unfavorable inside the long pore. However, this is well compensated by the electrostatic interaction with the static field arising from the protein charges and shielded by the dielectric surrounding. Decomposition of the static field provides a list of residues that display remarkable correspondence with existing mutagenesis data identifying amino acids that affect conduction and rectification. Many of these residues demonstrate interactions with the ion over long distances, up to 40 Å, suggesting that mutations potentially affect ion or blocker energetics over the entire pore. These results provide a foundation for understanding ion interactions in Kir channels and extend to the study of ion permeation, block, and gating in long, cation-specific pores.     

Estimating the dielectric constant of the channel protein and pore

When modelling biological ion channels using Brownian dynamics (BD) or Poisson–Nernst–Planck theory, the force encountered by permeant ions is calculated by solving Poisson’s equation. Two free parameters needed to solve this equation are the dielectric constant of water in the pore and the dielectric constant of the protein forming the channel. Although these values can in theory be deduced by various methods, they do not give a reliable answer when applied to channel-like geometries that contain charged particles. To determine the appropriate values of the dielectric constants, here we solve the inverse problem. Given the structure of the MthK channel, we attempt to determine the values of the protein and pore dielectric constants that minimize the discrepancies between the experimentally-determined current–voltage curve and the curve obtained from BD simulations. Two different methods have been applied to determine these values. First, we use all possible pairs of the pore dielectric constant of water, ranging from 20 to 80 in steps of 10, and the protein dielectric constant of 2–10 in steps of 2, and compare the simulated results with the experimental values. We find that the best agreement is obtained with experiment when a protein dielectric constant of 2 and a pore water dielectric constant of 60 is used. Second, we employ a learning-based stochastic optimization algorithm to pick out the optimum combination of the two dielectric constants. From the algorithm we obtain an optimum value of 2 for the protein dielectric constant and 64 for the pore dielectric constant.     

Investigating the dielectric effects of channel pore water on the electrostatic barriers of the permeation ion by the finite difference Poisson-Boltzmann method

In this paper, the finite difference Poisson-Boltzmann (FDPB) method with four dielectric constants is developed to study the effect of dielectric saturation on the electrostatic barriers of the permeation ion. In this method, the inner shape of the channel pore is explicitly represented, and the fact that the dielectric constant inside the channel pore is different from that of bulk water is taken into account. A model channel system which is a right-handed twist bundle with four α-helical segments is provided for this study. From the FDPB calculations, it is found that the difference of the ionic electrostatic solvation energy for wider domains depends strongly on the pore radius in the vicinity of the ion when the pore dielectric constant is changed from 78 to 5. However, the electrostatic solvation energy of the permeation ion can not be significantly affected by the dielectric constant in regions with small pore radii. Our results indicate that the local electrostatic interactions inside the ion channel are of major importance for ion electrostatic solvation energies, and the effect of dielectric saturation on the electrostatic barriers is coupled to the interior channel dimensions. Received: 28 January 1997 / Accepted: 24 September 1997     

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.     

Carrier-like behaviour from a static but electrically responsive model pore.

Because of the low dielectric constant of most proteins and lipids, the electric field of an ion passing through a narrow pore is long range and will interact with neighbouring ionizable residues of the channel protein. The electrical structure of the channel may thus change transiently in response to an ion passing through the pore. Model calculations then reveal that the ratio of the unidirectional ion fluxes may approach 1 as expected for a carrier or shuttling ionophore rather than the Ussing ratio expected for a pore. Saturation behaviour also becomes carrier-like. Computer simulation is reported showing a continuous variation between pore-like and carrier-like behaviour as the parameters of the system are allowed to change smoothly.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.     

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.     

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.     

Energy barriers for passage of ions through channels. Exact solution of two electrostatic problems

Exact solutions are given to two electrostatic problems relevant to ion permeation through pores in membranes. The first assesses the importance of the pore forming molecule as a dielectric shield. It is shown on the basis of structural and dielectric considerations alone (neglecting effects attributable to possible charge distribution at the interior surface of the pre-former) that the minimum electrostatic barrier for monovalent ion passage through a gramicidin-like channel is 11 kT. It is further shown that given favorable circumstances, dielectric shielding might dramatically reduce the barrier to ion passage through potassium channels. The second problem considers the error introduced by treating ions as point charges. It is shown that for structureless pores the point charge approximation introduces no meaningful error, even if the ratio of ion radius to pore radius is as great as 0.95.     

A kinetic role for ionizable sites in membrane channel proteins

Electrically charged residues in a membrane channel protein will certainly have a direct effect upon its gating and selectivity if they are near the channel pore. It is customary to regard the charged state of such residues as a fixed feature of the channel. In this paper it is argued that far from being fixed, the charged state of ionizable residues near the pore will very probably change rapidly in response to the channel opening and to ions passing through it. Calculations are presented using simple models which demonstrate that changes in the dielectric environment and changes in the distances to other charged groups resulting from channel opening can shift the effective pK values of the sites by 3 or 4 units leading to switching of its charged state. Examples are given of how this time dependent charge state of ionizable residues may play an important role in the functioning of channels. Also, by considering the influence of the electric field due to the mobile ion upon the charge state of a residue in the channel wall, it is shown that a channel lined with acid residues may very effectively block the passage of cations while allowing the passage of anions.     

Ion-channel entrances influence permeation. Net charge, size, shape, and binding considerations.

Many ion channels have wide entrances that serve as transition zones to the more selective narrow region of the pore. Here some physical features of these vestibules are explored. They are considered to have a defined size, funnel shape, and net-negative charge. Ion size, ionic screening of the negatively charged residues, cation binding, and blockage of current are analyzed to determine how the vestibules influence transport. These properties are coupled to an Eyring rate theory model for the narrow length of the pore. The results include the following: Wide vestibules allow the pore to have a short narrow region. Therefore, ions encounter a shorter length of restricted diffusion, and the channel conductance can be greater. The potential produced by the net-negative charge in the vestibules attracts cations into the pore. Since this potential varies with electrolyte concentration, the conductance measured at low electrolyte concentrations is larger than expected from measurements at high concentrations. Net charge inside the vestibules creates a local potential that confers some cation vs. anion, and divalent vs. monovalent selectivity. Large cations are less effective at screening (diminishing) the net-charge potential because they cannot enter the pore as well as small cations. Therefore, at an equivalent bulk concentration the attractive negative potential is larger, which causes large cations to saturate sites in the pore at lower concentrations. Small amounts of large or divalent cations can lead to misinterpretation of the permeation properties of a small monovalent cation.     

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.     

Charge formation in microporous membranes by single-acid site dissociation

The electrolyte concentration and pH dependence of the effective charge density of weak ion exchange membranes have been studied by combining solutions of the Poisson-Boltzmann equation in cylindrical pores with a simple dissociation equilibrium of weakly acid groups attached to the pore walls. Analytical expressions for the effective charge density and wall potential are presented which describe these quantities in terms of pH, electrolyte (1 : 1) concentration, acid constant, density of acid groups and the pore size. The concentration dependence of the effective fixed charge density experimentally observed for cellulose membranes and NaCl-solutions agrees quantitatively with the theoretical predictions. For track-etched mica membranes and KCl-solutions the influence of pH and electrolyte concentration on the effective charge density can be qualitatively explained. Also an interpretation of electro-osmotic findings obtained with an asymmetric cellulose acetate membrane and NaCl-solutions is given.     

Dependence of ion flow through channels on the density of fixed charges at the channel opening. Voltage control of inverse titration curves

Model calculations were done to investigate the effect of titratable fixed charges at channel openings on ion flow through open channels. The current titration curves (channel current vs. bulk pH) can assume the shape expected from the change of the ionic surface concentration with pH (c-control), or be inverted, i.e., follow the change of the electrical field within the membrane (V-control). The relationships were explored pars pro toto for Goldman-Hodgkin-Katz channels, two-barrier one-site channels and six-barrier five-site channels. With net current flowing in the direction of the concentration gradient and from the titrated fixed charge layer into the channel, c-control is the sign of low channel occupancy (entrance-step limitation) and V-control the sign of high channel occupancy (exit-step limitation). At intermediate occupancy, the current titration curve can be nearly invariant to pH.     

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.     

Immobilized enzymes: electrokinetic effects on reaction rates in a porous medium

The effect of the internal diffusion and electrical surface charge on the overall rate of a reaction catalyzed by an enzyme immobilized on a porous medium are examined. Effectiveness factors have been calculated which compare the global reaction rate to that existing in the absence of the internal diffusion and/or the electrical field. The surface charge, assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme, generates an electrical double layer at the pore surface. The double-layer potential is governed by the Poisson-Boltzmann equation. It is shown that the diffusion potential can be characterized by a modulus which depends upon the surface reaction rate, the charges and diffusivities of the substrate and products, the ionic strength, and the pore dimensions. The flux of a charged species in the pore occurs under the influences of the concentration gradient and the electrical potential gradient. The governing equations are solved by an iterative numerical method. The effects of pH, enzyme concentration, and substrate concentration on the rates of two different hydrolysis reactions catalyzed by immobilized papain are examined. The release of H(+) in one of the reactions causes the lowering of internal pH, and also a constancy of the internal pH when the external pH in creases beyond a certain value. The latter reaction also shows a maximum in the reaction rate with respect to enzyme concentration. The reaction not involving H(+) as a product shows a maximum in the reaction rate with respect to external pH, but a monotonic increase in the reaction rate as the enzyme concentration increases.