Bob Eisenberg

Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel

The synthetic channel [acetyl-(LeuSerSerLeuLeuSerLeu)3-CONH2]6 (pore diameter approximately 8 A, length approximately 30 A) is a bundle of six alpha-helices with blocked termini. This simple channel has complex properties, which are difficult to explain, even qualitatively, by traditional theories: its single-channel currents rectify in symmetrical solutions and its selectivity (defined by reversal potential) is a sensitive function of bathing solution. These complex properties can be fit quantitatively if the channel has fixed charge at its ends, forming a kind of macrodipole, bracketing a central charged region, and the shielding of the fixed charges is described by the Poisson-Nernst-Planck (PNP) equations. PNP fits current voltage relations measured in 15 solutions with an r.m.s. error of 3.6% using four adjustable parameters: the diffusion coefficients in the channels pore DK = 2.1 x 10(-6) and DCl = 2.6 x 10(-7) cm2/s; and the fixed charge at the ends of the channel of +/- 0.12e (with unequal densities 0.71 M = 0.021e/A on the N-side and -1.9 M = -0.058e/A on the C-side). The fixed charge in the central region is 0.31e (with density P2 = 0.47 M = 0.014e/A). In contrast to traditional theories, PNP computes the electric field in the open channel from all of the charges in the system, by a rapid and accurate numerical procedure. In essence, PNP is a theory of the shielding of fixed (i.e., permanent) charge of the channel by mobile charge and by the ionic atmosphere in and near the channels pore. The theory fits a wide range of data because the ionic contents and potential profile in the channel change significantly with experimental conditions, as they must, if the channel simultaneously satisfies the Poisson and Nernst-Planck equations and boundary conditions. Qualitatively speaking, the theory shows that small changes in the ionic atmosphere of the channel (i.e., shielding) make big changes in the potential profile and even bigger changes in flux, because potential is a sensitive function of charge and shielding, and flux is an exponential function of potential.

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.

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.

Proteins, channels and crowded ions

Ion channels are proteins with a hole down their middle that control a vast range of biological function in health and disease. Selectivity is an important biological function determined by the open channel, which does not change conformation on the biological time scale. The challenge is to predict the function-the current of ions of different types and concentrations through a variety of channels-from structure, given fundamental physical laws. Walls of ion channels, like active sites of enzymes, often contain several fixed charges. Those fixed charges demand counter ions nearby, and the density of those counter ions is very high, greater than 5 molar, because of the tiny volumes of the channels pore. Physical chemists can now calculate the free energy per mole of salt solutions (e.g. the activity coefficient) from infinite dilution to saturation, even in ionic melts. Such calculations of a model of the L-type calcium channel show that the large energies needed to crowd charges into the channel can account for the substantial selectivity and complex properties found experimentally. The properties of such crowded charge are likely to be an important determinant of the properties of proteins in general because channels are nearly enzymes.

Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-Nernst- Planck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a singular parameter, the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Application of the exchange lemma from geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the solution of the singular boundary value problem near each singular orbit. A set of simultaneous equations appears in the construction of singular orbits. Multiple solutions of such equations in this or similar problems might explain a variety of multiple valued phenomena seen in biological channels, for example, some forms of gating, and might be involved in other more complex behaviors, for example, some kinds of active transport.

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.

Molecular dynamics in physiological solutions: Force fields, alkali metal ions, and ionic strength

The monovalent ions Na(+) and K(+) and Cl(-) are present in any living organism. The fundamental thermodynamic properties of solutions containing such ions is given as the excess (electro-)chemical potential differences of single ions at finite ionic strength. This quantity is key for many biological processes, including ion permeation in membrane ion channels and DNA-protein interaction. It is given by a chemical contribution, related to the ion activity, and an electric contribution, related to the Galvani potential of the water/air interface. Here we investigate molecular dynamics based predictions of these quantities by using a variety of ion/water force fields commonly used in biological simulation, namely the AMBER (the newly developed), CHARMM, OPLS, Dang95 with TIP3P, and SPC/E water. Comparison with experiment is made with the corresponding values for salts, for which data are available. The calculations based on the newly developed AMBER force field with TIP3P water agrees well with experiment for both KCl and NaCl electrolytes in water solutions, as previously reported. The simulations based on the CHARMM-TIP3P and Dang95-SPC/E force fields agree well for the KCl and NaCl solutions, respectively. The other models are not as accurate. Single cations excess (electro-)chemical potential differences turn out to be similar for all the force fields considered here. In the case of KCl, the calculated electric contribution is consistent with higher level calculations. Instead, such agreement is not found with NaCl. Finally, we found that the calculated activities for single Cl(-) ions turn out to depend clearly on the type of counterion used, with all the force fields investigated. The implications of these findings for biomolecular systems are discussed.

Protein structure and ionic selectivity in calcium channels: Selectivity filter size, not shape, matters

Calcium channels have highly charged selectivity filters (4 COO(-) groups) that attract cations in to balance this charge and minimize free energy, forcing the cations (Na(+) and Ca(2+)) to compete for space in the filter. A reduced model was developed to better understand the mechanism of ion selectivity in calcium channels. The charge/space competition (CSC) mechanism implies that Ca(2+) is more efficient in balancing the charge of the filter because it provides twice the charge as Na(+) while occupying the same space. The CSC mechanism further implies that the main determinant of Ca(2+) versus Na(+) selectivity is the density of charged particles in the selectivity filter, i.e., the volume of the filter (after fixing the number of charged groups in the filter). In this paper we test this hypothesis by changing filter length and/or radius (shape) of the cylindrical selectivity filter of our reduced model. We show that varying volume and shape together has substantially stronger effects than varying shape alone with volume fixed. Our simulations show the importance of depletion zones of ions in determining channel conductance calculated with the integrated Nernst-Planck equation. We show that confining the protein side chains with soft or hard walls does not influence selectivity.