Effective pore radius of the gramicidin channel. Electrostatic energies of ions calculated by a three-dielectric model

Self-consistent analytic solution for the current and the access resistance in open ion channels

A self-consistent analytic approach is introduced for the estimation of the access resistance and the current through an open ion channel for an arbitrary number of species. For an ion current flowing radially inward from infinity to the channel mouth, the Poisson-Boltzmann-Nernst-Planck equations are solved analytically in the bulk with spherical symmetry in three dimensions, by linearization. Within the channel, the Poisson-Nernst-Planck equation is solved analytically in a one-dimensional approximation. An iterative procedure is used to match the two solutions together at the channel mouth in a self-consistent way. It is shown that the current-voltage characteristics obtained are in good quantitative agreement with experimental measurements.

Continuum electrostatics fails to describe ion permeation in the gramicidin channel

We investigate the validity of continuum electrostatics in the gramicidin A channel using a recently determined high-resolution structure. The potential and electric field acting on ions in and around the channel are computed by solving Poisson's equation. These are then used in Brownian dynamics simulations to obtain concentration profiles and the current passing through the channel. We show that regardless of the effective dielectric constant used for water in the channel or the channel protein, it is not possible to reproduce all the experimental data on gramicidin A; thus, continuum electrostatics cannot provide a valid framework for the description of ion dynamics in gramicidin channels. Using experimental data and molecular dynamics simulations as guides, we have constructed potential energy profiles that can satisfactorily describe the available physiological data. These profiles provide useful benchmarks for future potential of mean force calculations of permeating ions from molecular dynamics simulations of gramicidin A. They also offer a convenient starting point for studying structure-function relationships in modified gramicidin channels.

Cellular microtransport processes: intercellular, intracellular, and aggregate behavior

Ionic and molecular transfer among cells occurs by a variety of transport processes operative at different length scales. Cell membrane permeability and electrical conductance derive from channel proteins producing pores at the molecular (ultrastructural) scale. Intracellular mobility involves the dynamics of motion through the complex ultrastructure of the cytoplasm. These phenomena unite in the larger-scale (microscopic) process of gross intercellular transfer. When such movement occurs among sufficiently many cells, it in turn begins to reflect their average collective (macroscopic) behavior as bulk tissue. This article surveys selected aspects of intercellular and intracellular transport, with emphasis on detailed mechanistic theory, experimental probes of cellular permeability, and systematic transcendence from small to large length scales.

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.

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.

Extended dipolar chain model for ion channels: electrostriction effects and the translocational energy barrier

We reinvestigate the dipolar chain model for an ion channel. Our goal is to account for the influence that ion-induced electrostriction of channel water has on the translocational energy barriers experienced by different ions in the channel. For this purpose, we refine our former model by relaxing the positional constraint on the ion and the water dipoles and by including Lennard-Jones contributions in addition to the electrostatic interactions. The positions of the ion and the waters are established by minimization of the free energy. As before, interaction with the external medium is described via the image forces. Application to alkali cations show that the short range interactions modulate the free energy profiles leading to a selectivity sequence for translocation. We study the influence of some structural parameters on this sequence and compare our theoretical predictions with observed results for gramicidin.

Influence of a channel-forming peptide on energy barriers to ion permeation, viewed from a continuum dielectric perspective

The continuum three-dielectric model for an aqueous ion channel pore-forming peptide-membrane system is extended to account for the finite length of the channel. We focus on the electrostatic influence that a channel-forming peptide may exert on energy barriers to ion permeation. The nonlinear dielectric behavior of channel water caused by dielectric saturation in the presence of an ion is explicitly modeled by assigning channel water a mean dielectric constant much less than that of bulk water. An exact solution of the continuum problem is formulated by approximating the dielectric behavior of bulk water, assigning it a dielectric constant of infinity. The validity of this approximation is verified by comparison with a Poisson-Boltzmann description of the electrolyte. The formal equivalence of high ionic strength and high electrolyte dielectric constant is demonstrated. We estimate limits on the reduction of the electrostatic free energy caused by ionic interaction with the channel-forming peptide. We find that even assigning this region an epsilon of 100, its influence is insufficient to lower permeation free energy barriers to values consistent with observed channel conductances. We provide estimates of the effective dielectric constant of this highly polarizable region, by comparing energy barriers computed using the continuum approach with those found from a semi-microscopic analysis of a simplified model of a gramicidin-like charge distribution. Possible ways of improving both models are discussed.

Electrostatic interactions in gramicidin channels. Three-dielectric model

A model based on the solution of the electrostatic potential for a geometry of three dielectric regions associated with a gramicidin A channel (GA) is presented. The model includes a cylindrical dielectric layer to represent the peptide backbone and dipole rings to account for dipolar side chains. Image potential and dipolar contributions for different orientations and positions along the channel are analyzed. The conductance of GA and two analogues obtained by substituting the amino acid at position 1 are studied. The numerical simulation reproduces experimental results (Barrett et al. 1986, Biophys J 49, 673-686) and supports the idea that electrostatic dipole-ion interactions are of primary importance in gramicidin channel function.

Energy-minimized conformation of gramicidin-like channels. I. Infinitely long poly-(L,D)-alanine beta 6.3-helix

The energy-minimized conformation of an infinitely long poly-(L,D)-alanine in single-stranded beta 6.3-helix was calculated by the molecular mechanics method. When energy minimization was started from a wide range of initial geometries, six optimized conformations were obtained and identified as the right- and left-handed counterparts of the beta 4.5-, beta 6.3-, and beta 8.2-helices. It was found that their conformation energies increase in this order, the beta 4.5-helix having the lowest energy. The backbone dihedral angles of the energy-minimized beta 6.3-helix were: phi L = -116 degrees (or -131 degrees), psi L = 122 degrees (or 111 degrees), phi D = 131 degrees (or 116 degrees), psi D = -111 degrees (or -122 degrees), omega L = 173 degrees (or 173 degrees), and omega D = -173 degrees (or -173 degrees) for the right-handed (or left-handed) helix. This helix was composed of 6.30 residues/turn with a pitch of 4.97 A. All the alpha-carbons of L- and D-configurations appeared on one common circular helix. Interestingly, small deviations (approximately 7 degrees) of the peptide bonds from the planar structure caused a considerable lowering of the conformation energy, and, at the same time, they produced more favorable fitting of the hydrogen bonds; the carbonyl oxygens and the nearest-neighbor alpha-hydrogens also took more favorable relative positions.

Theoretical perspectives on ion-channel electrostatics: continuum and microscopic approaches

Peter Läuger introduced me (P.C.J.) to the field of ion-channel electrostatics while I was a sabbatical visitor at Konstanz in 1978–79. Läuger pointed out that the relative conductance of hydrophobic ions through phosphatidyl choline (PC) and glyceryl monooleate (GMO) membranes differed by a factor of about 100 (Hladky & Haydon, 1973), quite consistent with the difference in the water-membrane potential differences in the two systems (Pickar & Benz, 1978). However, cation conductance through gramicidin channels spanning these membranes only differs by a factor of 2–3 (Bamberg et al. 1976). Why? It is the pursuit of an answer to this question which led me into my researches in this field.