Wolfgang Nonner

Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux

Ion transport between two baths of fixed ionic concentrations and applied electrostatic (ES) potential is analysed using a one-dimensional drift-diffusion (Poisson–Nernst–Planck, PNP) transport system designed to model biological ion channels. The ions are described as charged, hard spheres with excess chemical potentials computed from equilibrium density functional theory (DFT). The method of Rosenfeld (Rosenfeld Y 1993 J. Chem. Phys. 98 8126) is generalized to calculate the ES excess chemical potential in channels. A numerical algorithm for solving the set of integral–differential PNP/DFT equations is described and used to calculate flux through a calcium-selective ion channel.

Anomalous Mole Fraction Effect, Electrostatics, and Binding in Ionic Channels

Ionic channels bathed in mixed solutions of two permeant electrolytes often conduct less current than channels bathed in pure solutions of either. For many years, this anomalous mole fraction effect (AMFE) has been thought to occur only in single-file pores containing two or more ions at a time. Most thinking about channels incorporates this view. We show here that the AMFE arises naturally, as an electrostatic consequence of localized ion specific binding, if the average current through a channel is described by a theory (Poisson-Nernst-Planck, PNP) that computes the average electric field from the average concentration of charges in and near the channel. The theory contains only those ion-ion interactions mediated by the mean field, and it does not enforce single filing. The AMFE is predicted by PNP over a wide range of mean concentrations of ions in the channel; for example, it is predicted when (on the average) less, or much less, than one ion is found in the channels pore. In this treatment, the AMFE arises, in large measure, from a depletion layer produced near a region of ion-specific binding. The small excess concentration of ions in the binding region repels all nearby ions of like charge, thereby creating a depletion layer. The overall conductance of the channel arises in effect from resistors in series, one from the binding region, one from the depletion zone, and one from the unbinding region. The highest value resistor (which occurs in the depletion zone) limits the overall series conductance. Here the AMFE is not the result of single filing or multiple occupancy, and so previous views of permeation need to be revised: the presence of an AMFE does not imply that ions permeate single file through a multiply occupied pore.

The effect of protein dielectric coefficient on the ionic selectivity of a calcium channel.

Calcium-selective ion channels are known to have carboxylate-rich selectivity filters, a common motif that is primarily responsible for their high Ca(2+) affinity. Different Ca(2+) affinities ranging from micromolar (the L-type Ca channel) to millimolar (the ryanodine receptor channel) are closely related to the different physiological functions of these channels. To understand the physical mechanism for this range of affinities given similar amino acids in their selectivity filters, we use grand canonical Monte Carlo simulations to assess the binding of monovalent and divalent ions in the selectivity filter of a model Ca channel. We use a reduced model where the electolyte is modeled by hard-sphere ions embedded in a continuum dielectric solvent, while the interior of protein surrounding the channel is allowed to have a dielectric coefficient different from that of the electrolyte. The induced charges that appear on the protein/lumen interface are calculated by the induced charge computation method [Boda et al., Phys. Rev. E 69, 046702 (2004)]. It is shown that decreasing the dielectric coefficient of the protein attracts more cations into the pore because the proteins carboxyl groups induce negative charges on the dielectric boundary. As the density of the hard-sphere ions increases in the filter, Ca(2+) is absorbed into the filter with higher probability than Na(+) because Ca(2+) provides twice the charge to neutralize the negative charge of the pore (both structural carboxylate oxygens and induced charges) than Na(+) while occupying about the same space (the charge/space competition mechanism). As a result, Ca(2+) affinity is improved an order of magnitude by decreasing the protein dielectric coefficient from 80 to 5. Our results indicate that adjusting the dielectric properties of the protein surrounding the permeation pathway is a possible way for evolution to regulate the Ca(2+) affinity of the common four-carboxylate motif.

ATTEMPTS TO DEFINE FUNCTIONAL DOMAINS OF GAP JUNCTION PROTEINS WITH SYNTHETIC PEPTIDES

To map the binding sites involved in channel formation, synthetic peptides representing sequences of connexin 32 were tested for their ability to inhibit cell-cell channel formation. Both large peptides representing most of the two presumed extracellular loops of connexin32 and shorter peptides representing subsets of these larger peptides were found to inhibit cell-cell channel formation. The properties of the peptide inhibition suggested that the binding site is complex, involving several segments of both extracellular loops. One of the peptides (a 12-mer) did not inhibit but instead was found to form channels in membranes. Both in oocyte membranes and in bilayers, the channels formed by the peptide were asymmetrically voltage dependent. Their unit conductances ranged from 20 to 160 pS. These data are discussed in the form of a model in which the connexin sequence represented by the peptide is part of a beta structure providing the lining of the channel pore.

Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion

A physical model of selective “ion binding” in the L-type calcium channel is constructed, and consequences of the model are compared with experimental data. This reduced model treats only ions and the carboxylate oxygens of the EEEE locus explicitly and restricts interactions to hard-core repulsion and ion–ion and ion–dielectric electrostatic forces. The structural atoms provide a flexible environment for passing cations, thus resulting in a self-organized induced-fit model of the selectivity filter. Experimental conditions involving binary mixtures of alkali and/or alkaline earth metal ions are computed using equilibrium Monte Carlo simulations in the grand canonical ensemble. The model pore rejects alkali metal ions in the presence of biological concentrations of Ca2+ and predicts the blockade of alkali metal ion currents by micromolar Ca2+. Conductance patterns observed in varied mixtures containing Na+ and Li+, or Ba2+ and Ca2+, are predicted. Ca2+ is substantially more potent in blocking Na+ current than Ba2+. In apparent contrast to experiments using buffered Ca2+ solutions, the predicted potency of Ca2+ in blocking alkali metal ion currents depends on the species and concentration of the alkali metal ion, as is expected if these ions compete with Ca2+ for the pore. These experiments depend on the problematic estimation of Ca2+ activity in solutions buffered for Ca2+ and pH in a varying background of bulk salt. Simulations of Ca2+ distribution with the model pore bathed in solutions containing a varied amount of Li+ reveal a “barrier and well” pattern. The entry/exit barrier for Ca2+ is strongly modulated by the Li+ concentration of the bath, suggesting a physical explanation for observed kinetic phenomena. Our simulations show that the selectivity of L-type calcium channels can arise from an interplay of electrostatic and hard-core repulsion forces among ions and a few crucial channel atoms. The reduced system selects for the cation that delivers the largest charge in the smallest ion volume.

Bubbles, Gating, and Anesthetics in Ion Channels

We suggest that bubbles are the bistable hydrophobic gates responsible for the on-off transitions of single channel currents. In this view, many types of channels gate by the same physical mechanism-dewetting by capillary evaporation-but different types of channels use different sensors to modulate hydrophobic properties of the channel wall and thereby trigger and control bubbles and gating. Spontaneous emptying of channels has been seen in many simulations. Because of the physics involved, such phase transitions are inherently sensitive, unstable threshold phenomena that are difficult to simulate reproducibly and thus convincingly. We present a thermodynamic analysis of a bubble gate using morphometric density functional theory of classical (not quantum) mechanics. Thermodynamic analysis of phase transitions is generally more reproducible and less sensitive to details than simulations. Anesthetic actions of inert gases-and their interactions with hydrostatic pressure (e.g., nitrogen narcosis)-can be easily understood by actions on bubbles. A general theory of gas anesthesia may involve bubbles in channels. Only experiments can show whether, or when, or which channels actually use bubbles as hydrophobic gates: direct observation of bubbles in channels is needed. Existing experiments show thin gas layers on hydrophobic surfaces in water and suggest that bubbles nearly exist in bulk water.

Volume Exclusion in Calcium Selective Channels

Another research group has proposed an interesting model for calcium channel selectivity. However, on the basis of their reported results we find it impossible to assess the merits of their model because their results and claims concerning selectivity are based on an extrapolation over four orders of magnitude to low Ca(2+) concentration. Their results and claims have been presented in several articles and reviews in several journals and, thus, need attention. In this article, we first establish that we obtain results on electrostatics and channel occupancies similar to the high-concentration simulations they present. We then perform grand canonical ensemble simulations that enable us to study micromolar Ca(2+) concentrations. We find that their model channel is only weakly Ca(2+) selective. A crucial problem with their model appears to be the placement of the negatively charged glutamate structural elements in fixed positions inside the protein rather than as flexible units inside the filter.

Monte Carlo simulations of ion selectivity in a biological Na channel: Charge–space competition

Na channels that produce the action potentials of nerve and muscle include a selectivity filter formed by both positively and negatively charged amino acid residues in a molecular pore. Here we present Monte Carlo simulations of equilibrium ion absorption in such a system. Ions are treated as charged hard spheres in a uniform dielectric. Tethered carboxylate and amino groups known to line the selectivity filter of the Na channel are represented as charged hard spheres and restricted to the filter region of the channel. Consistent with experiments, we find (1) that absorption of Ca2+ into the filter exceeds absorption of Na+ only when the concentration of Ca2+ is some tenfold larger than physiological; (2) the model channel absorbs smaller alkali metal ions preferentially compared to larger ones. The alkali metal selectivity involves volume exclusion of larger ions from the center of the filter region.

Electrodiffusion in ionic channels of biological membranes

Abstract An important class of biological molecules - proteins called ionic channels conduct ions (like Na+, K+, Ca++ and Cl−) through a narrow tunnel of fixed charge. Ionic channels are the main pathway by which substances move into cells and son are of great biological and medical importance: asubstantial fraction of all drugs used by physicians act on channels. Ionic channels can be modified by the powerful techniques of of molecular biology. Charged groups can be engineered (i.e. replaced one at a time) and the location of every atom can be determined. channels can be studied in the tradition of electrochemistry. If coupled to the Poisson equation, the drift diffusion equations (i.e. Nernst-Planck equations) form an adequate model of the current through 6 different channel proteins with quite different characteristics in 10 solutions over ±150 mV. In this theory the channel is represented as a distribution of fixed charge, and the ion as a mobile charge with a diffusion coefficient. The theory predicts the electric field (i.e. potential profile) and resulting current produced by the fixed charge and other charges in the system. In this theory, the shape of the electric field is found to be a sensitive function of ionic conditions and the potential difference across the channel, in contrast to traditional theories that assume potential profiles (or rate constants) independent of experimental conditions. Traditional theories fail to fit data, probably for because they assume the shape of the electric field. The Poisson-Nernst-Planck (PNP) theory is nearly idemntical to the drift diffusion equations used to analyze the flow of quasi-particles in semiconductors, implying that - given appropriate geometry and profiles of fixed charge - ionic channels can perform many of the useful functions of transistors, acting as resistors, voltage amplifiers, current amplifiers, or logic elements. Channels form a useful system for electrochemistry since they are biologically and clinically important, they follow the simple rules of electrodiffusion, and they promise to be of considerable use in technology.

Ca2+ transport properties and determinants of anomalous mole fraction effects of single voltage-gated Ca2+ channels in hair cells from bullfrog saccule

We studied the permeation properties of two distinct single voltage‐gated Ca2+ channels in bullfrog saccular hair cells to assess the roles of the channels as physiological Ca2+ transporters and multi‐ion pores. By varying the permeant ions (Ba2+, Ca2+) and concentrations (2–70 mm), we estimated the affinity constant (KD) of the two channels as follows (mm): L‐type channel, KD,Ba= 7.4 ± 1.0, KD,Ca= 7.1 ± 2.2 (n= 7); non‐L‐type channel, KD,Ba= 5.3 ± 3.2, KD,Ca= 2.0 ± 1.0 (n= 8). Using ionic concentrations close to physiological conditions (2 mm Ca2+ and 1.0 mm Mg2+), the conductance of the L‐type channel was ∼2 pS. We determined the mechanisms by which ions traverse the pore of these single Ca2+ channels, using mixtures of Ba2+ and Ca2+ at total concentrations above (70 mm) or close to (5 mm) the KD of the channels. We found evidence for an anomalous mole fraction effect (AMFE) only when the total divalent ion concentration was 5 mm, consistent with a multi‐ion pore. We show that AMFE arises from the boundaries between the pore and bulk solution in the atria of the channel, which is derived from the presence of depletion zones that become apparent at low divalent cation concentrations. The present findings provide an explanation as to why previous whole‐cell Ca2+ currents that were recorded in quasi‐physiological Ca2+ concentrations (∼2–5 mm) showed clear AMFE, whereas single Ca2+ channel currents that were recorded routinely at high Ca2+ concentrations (20–110 mm) did not.