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

Effects of electrical polarization on the opening rate constant of a voltage-gated ion channel

Ion channel gating kinetics can be described using Kramers' diffusion theory of reaction rates between several closed and open states, where transition rates between states depend exponentially on the membrane potential V. It has been suggested that transition rates have a more complex dependence on V at voltage extremes, but this has never been quantified. We measured the rate constant of the last closed to open transition in a voltage-gated ion channel and show that it does not depend exponentially on membrane potential at values of V greater than ≈150 mV. To explain this behavior, we estimate the effects of electrical polarization of the water contained in crevices within the channel protein, using an electrostatic model of the approximate three-dimensional geometry and the nonlinear effects of charges on the polarization of water.

Physics of ion channels

We review the basic physics involved in transport of ions across membrane channels in cells. Electrochemical forces that control the diffusion of ions are discussed both from microscopic and macroscopic perspectives. A case is made for use of Brownian dynamics as the minimal phenomenological model that provides a bridge between experiments and more fundamental theoretical approaches. Application of Brownian and molecular dynamics methods to channels with known molecular structures is discussed.

Quantum dynamics in continuum for proton transport--generalized correlation

As a key process of many biological reactions such as biological energy transduction or human sensory systems, proton transport has attracted much research attention in biological, biophysical, and mathematical fields. A quantum dynamics in continuum framework has been proposed to study proton permeation through membrane proteins in our earlier work and the present work focuses on the generalized correlation of protons with their environment. Being complementary to electrostatic potentials, generalized correlations consist of proton-proton, proton-ion, proton-protein, and proton-water interactions. In our approach, protons are treated as quantum particles while other components of generalized correlations are described classically and in different levels of approximations upon simulation feasibility and difficulty. Specifically, the membrane protein is modeled as a group of discrete atoms, while ion densities are approximated by Boltzmann distributions, and water molecules are represented as a dielectric continuum. These proton-environment interactions are formulated as convolutions between number densities of species and their corresponding interaction kernels, in which parameters are obtained from experimental data. In the present formulation, generalized correlations are important components in the total Hamiltonian of protons, and thus is seamlessly embedded in the multiscale/multiphysics total variational model of the system. It takes care of non-electrostatic interactions, including the finite size effect, the geometry confinement induced channel barriers, dehydration and hydrogen bond effects, etc. The variational principle or the Euler-Lagrange equation is utilized to minimize the total energy functional, which includes the total Hamiltonian of protons, and obtain a new version of generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation and generalized Kohn-Sham equation. A set of numerical algorithms, such as the matched interface and boundary method, the Dirichlet to Neumann mapping, Gummel iteration, and Krylov space techniques, is employed to improve the accuracy, efficiency, and robustness of model simulations. Finally, comparisons between the present model predictions and experimental data of current-voltage curves, as well as current-concentration curves of the Gramicidin A channel, verify our new model.

Ionic permeation free energy in gramicidin: a semimicroscopic perspective

Ion permeation through the gramicidin channel is studied using a model that circumvents two major difficulties inherent to standard simulational methods. It exploits the timescale separation between electronic and structural contributions to dielectric stabilization, accounting for the influence of electronic polarization by embedding the channel in a dielectric milieu that describes this polarization in a mean sense. The explicit mobile moieties are the ion, multipolar waters, and the carbonyls and amides of the peptide backbone. The model treats the influence of aromatic residues and the membrane dipole potential. A new electrical geometry is introduced that treats long-range electrostatics exactly and avoids problems related to periodic boundary conditions. It permits the translocating ion to make a seamless transition from nearby electrolyte to the channel interior. Other degrees of freedom (more distant bulk electrolyte and nonpolar lipid) are treated as dielectric continua. Reasonable permeation free energy profiles are obtained for potassium, rubidium, and cesium; binding wells are shallow and the central barrier is small. Estimated cationic single-channel conductances are smaller than experiment, but only by factors between 2 (rubidium) and 50 (potassium). When applied to chloride the internal barrier is large, with a corresponding miniscule single-channel conductance. The estimated relative single-channel conductances of gramicidin A, B, and C agree well with experiment.

Dynamics and energetics of water permeation through the aquaporin channel

Structural properties of water inside bovine aquaporin-1 are investigated by molecular simulation. The calculations, which are based on the recently determined X-ray structure at 2.2 A resolution (Sui et al., Nature 2001;414:872-878), are carried out on one monomeric subunit immersed in a water-n-octane-water bilayer. Molecular dynamics (MD) simulations suggest that His182, a fully conserved residue in the channel pore, is protonated in the delta position. Furthermore, they reveal a highly ordered water structure in the channel, induced by the electrostatic properties of the protein. Multiple-steering MD simulations are used to calculate the free-energy of water diffusion. To the best of our knowledge, this represents the first free-energy calculation based on the new, high-resolution structure of the pore. The calculated barrier is 2.5 kcal/mol, and it is associated to water permeation through the Asn-Pro-Ala (NPA) region of the pore, where water molecules are only hydrogen-bonded with themselves. These findings are fully consistent with those based on the previous MD studies on the human protein (de Groot and Grubmüller, Science 2001;294:2353-2357).

The role of the dielectric barrier in narrow biological channels: a novel composite approach to modeling single-channel currents

A composite continuum theory for calculating ion current through a protein channel of known structure is proposed, which incorporates information about the channel dynamics. The approach is utilized to predict current through the Gramicidin A ion channel, a narrow pore in which the applicability of conventional continuum theories is questionable. The proposed approach utilizes a modified version of Poisson-Nernst-Planck (PNP) theory, termed Potential-of-Mean-Force-Poisson-Nernst-Planck theory (PMFPNP), to compute ion currents. As in standard PNP, ion permeation is modeled as a continuum drift-diffusion process in a self-consistent electrostatic potential. In PMFPNP, however, information about the dynamic relaxation of the protein and the surrounding medium is incorporated into the model of ion permeation by including the free energy of inserting a single ion into the channel, i.e., the potential of mean force along the permeation pathway. In this way the dynamic flexibility of the channel environment is approximately accounted for. The PMF profile of the ion along the Gramicidin A channel is obtained by combining an equilibrium molecular dynamics (MD) simulation that samples dynamic protein configurations when an ion resides at a particular location in the channel with a continuum electrostatics calculation of the free energy. The diffusion coefficient of a potassium ion within the channel is also calculated using the MD trajectory. Therefore, except for a reasonable choice of dielectric constants, no direct fitting parameters enter into this model. The results of our study reveal that the channel response to the permeating ion produces significant electrostatic stabilization of the ion inside the channel. The dielectric self-energy of the ion remains essentially unchanged in the course of the MD simulation, indicating that no substantial changes in the protein geometry occur as the ion passes through it. Also, the model accounts for the experimentally observed saturation of ion current with increase of the electrolyte concentration, in contrast to the predictions of standard PNP theory.

Modeling permeation energetics in the KcsA potassium channel

The thermodynamics of cation permeation through the KcsA K(+) channel selectivity filter is studied from the perspective of a physically transparent semimicroscopic model using Monte Carlo free energy integration. The computational approach chosen permits dissection of the separate contributions to ionic stabilization arising from different parts of the channel (selectivity filter carbonyls, single-file water, cavity water, reaction field of bulk water, inner helices, ionizable residues). All features play important roles; their relative significance varies with the ion's position in the filter. The cavity appears to act as an electrostatic buffer, shielding filter ions from structural changes in the inner pore. The model exhibits K(+) vs. Na(+) selectivity, and roughly isoenergetic profiles for K(+) and Rb(+), and discriminates against Cs(+), all in agreement with experimental data. It also indicates that Ba(2+) and Na(+) compete effectively with permeant ions at a site near the boundary between the filter and the cavity, in the vicinity of the barium blocker site.

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.

Continuum and atomistic modeling of ion partitioning into a peptide nanotube

Continuum and atomistic descriptions of the partitioning of ions into a self-assembled (D,L)-octapeptide nanotube, cyclo[-(L-Ala-D-Ala)(4)-], are presented. Perturbation free energy calculations, including Ewald electrostatics, are used to estimate the electrostatic component of the excess free energy of charging Li(+), Na(+), Rb(+), and Cl(minus sign) ions inside the nanotube. The radial density and orientational distribution of water around the ion is calculated for the ion at two different positions inside the tube; it is seen that the calculated distributions are sensitive to the location of the ions. Two different continuum electrostatic models are formulated to describe the ion solvation inside the nanotube. When enhanced orientational structuring of water dipoles is evidenced, explicitly including the first solvation shell as part of the low dielectric nanotube environment provides good agreement with molecular dynamics simulations. When water orientational structuring is as in the reference bulk solvent, we find that treating the first shell water explicitly or as a high dielectric continuum leads to similar results. These results are discussed, and their importance for continuum electrostatic modeling of ion channels are highlighted.

Cystic fibrosis transmembrane conductance regulator. Physical basis for lyotropic anion selectivity patterns

The cystic fibrosis transmembrane conductance regulator (CFTR) Cl channel exhibits lyotropic anion selectivity. Anions that are more readily dehydrated than Cl exhibit permeability ratios (P(S)/P(Cl)) greater than unity and also bind more tightly in the channel. We compared the selectivity of CFTR to that of a synthetic anion-selective membrane [poly(vinyl chloride)-tridodecylmethylammonium chloride; PVC-TDMAC] for which the nature of the physical process that governs the anion-selective response is more readily apparent. The permeability and binding selectivity patterns of CFTR differed only by a multiplicative constant from that of the PVC-TDMAC membrane; and a continuum electrostatic model suggested that both patterns could be understood in terms of the differences in the relative stabilization of anions by water and the polarizable interior of the channel or synthetic membrane. The calculated energies of anion-channel interaction, derived from measurements of either permeability or binding, varied as a linear function of inverse ionic radius (1/r), as expected from a Born-type model of ion charging in a medium characterized by an effective dielectric constant of 19. The model predicts that large anions, like SCN, although they experience weaker interactions (relative to Cl) with water and also with the channel, are more permeant than Cl because anion-water energy is a steeper function of 1/r than is the anion-channel energy. These large anions also bind more tightly for the same reason: the reduced energy of hydration allows the net transfer energy (the well depth) to be more negative. This simple selectivity mechanism that governs permeability and binding acts to optimize the function of CFTR as a Cl filter. Anions that are smaller (more difficult to dehydrate) than Cl are energetically retarded from entering the channel, while the larger (more readily dehydrated) anions are retarded in their passage by "sticking" within the channel.

(In)validity of the constant field and constant currents assumptions in theories of ion transport

Constant electric fields and constant ion currents are often considered in theories of ion transport. Therefore, it is important to understand the validity of these helpful concepts. The constant field assumption requires that the charge density of permeant ions and flexible polar groups is virtually voltage independent. We present analytic relations that indicate the conditions under which the constant field approximation applies. Barrier models are frequently fitted to experimental current-voltage curves to describe ion transport. These models are based on three fundamental characteristics: a constant electric field, negligible concerted motions of ions inside the channel (an ion can enter only an empty site), and concentration-independent energy profiles. An analysis of those fundamental assumptions of barrier models shows that those approximations require large barriers because the electrostatic interaction is strong and has a long range. In the constant currents assumption, the current of each permeating ion species is considered to be constant throughout the channel; thus ion pairing is explicitly ignored. In inhomogeneous steady-state systems, the association rate constant determines the strength of ion pairing. Among permeable ions, however, the ion association rate constants are not small, according to modern diffusion-limited reaction rate theories. A mathematical formulation of a constant currents condition indicates that ion pairing very likely has an effect but does not dominate ion transport.

CFTR: mechanism of anion conduction

CFTR: Mechanism of Anion Conduction. Physiol. Rev. 79, Suppl.: S47-S75, 1999. - The purpose of this review is to collect together the results of recent investigations of anion conductance by the cystic fibrosis transmembrane conductance regulator along with some of the basic background that is a prerequisite for developing some physical picture of the conduction process. The review begins with an introduction to the concepts of permeability and conductance and the Nernst-Planck and rate theory models that are used to interpret these parameters. Some of the physical forces that impinge on anion conductance are considered in the context of permeability selectivity and anion binding to proteins. Probes of the conduction process are considered, particularly permeant anions that bind tightly within the pore and block anion flow. Finally, structure-function studies are reviewed in the context of some predictions for the origin of pore properties.

Electrostatics of a simple membrane model using Green's functions formalism

The electrostatics of a simple membrane model picturing a lipid bilayer as a low dielectric constant slab immersed in a homogeneous medium of high dielectric constant (water) can be accurately computed using the exact Green's functions obtainable for this geometry. We present an extensive discussion of the analysis and numerical aspects of the problem and apply the formalism and algorithms developed to the computation of the energy profiles of a test charge (e.g., ion) across the bilayer and a molecular model of the acetylcholine receptor channel embedded in it. The Green's function approach is a very convenient tool for the computer simulation of ionic transport across membrane channels and other membrane problems where a good and computationally efficient first-order treatment of dielectric polarization effects is crucial.

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.

Modelling packing interactions in parallel helix bundles: pentameric bundles of nicotinic receptor M2 helices

The transbilayer pore of the nicotinic acetylcholine receptor (nAChR) is formed by a pentameric bundle of M2 helices. Models of pentameric bundles of M2 helices have been generated using simulated annealing via restrained molecular dynamics. The influence of: (a) the initial C alpha template; and (b) screening of sidechain electrostatic interactions on the geometry of the resultant M2 helix bundles is explored. Parallel M2 helices, in the absence of sidechain electrostatic interactions, pack in accordance with simple ridges-in-grooves considerations. This results in a helix crossing angle of ca. +12 degrees, corresponding to a left-handed coiled coil structure for the bundle as a whole. Tilting of M2 helices away from the central pore axis at their C-termini and/or inclusion of sidechain electrostatic interactions may perturb such ridges-in-grooves packing. In the most extreme cases right-handed coiled coils are formed. An interplay between inter-helix H-bonding and helix bundle geometry is revealed. The effects of changes in electrostatic screening on the dimensions of the pore mouth are described and the significance of these changes in the context of models for the nAChR pore domain is discussed.