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.     

Constant fields and constant gradients in open ionic channels.

Ions enter cells through pores in proteins that are holes in dielectrics. The energy of interaction between ion and charge induced on the dielectric is many kT, and so the dielectric properties of channel and pore are important. We describe ionic movement by (three-dimensional) Nemst-Planck equations (including flux and net charge). Potential is described by Poisson's equation in the pore and Laplace's equation in the channel wall, allowing induced but not permanent charge. Asymptotic expansions are constructed exploiting the long narrow shape of the pore and the relatively high dielectric constant of the pore's contents. The resulting one-dimensional equations can be integrated numerically; they can be analyzed when channels are short or long (compared with the Debye length). Traditional constant field equations are derived if the induced charge is small, e.g., if the channel is short or if the total concentration gradient is zero. A constant gradient of concentration is derived if the channel is long. Plots directly comparable to experiments are given of current vs voltage, reversal potential vs. concentration, and slope conductance vs. concentration. This dielectric theory can easily be tested: its parameters can be determined by traditional constant field measurements. The dielectric theory then predicts current-voltage relations quite different from constant field, usually more linear, when gradients of total concentration are imposed. Numerical analysis shows that the interaction of ion and channel can be described by a mean potential if, but only if, the induced charge is negligible, that is to say, the electric field is spatially constant.     

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 channel's 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 channel's 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.     

The Excitable Membrane: A Physiochemical Model

The model of the excitable membrane assumes common channels for Na⁺ and K⁺; the two ion species interact within the pores through their electrostatic forces. The electric field varies across the membrane and with time, as a result of ionic redistribution. Ionic flow is primarily controlled by energy barriers at the two interfaces and by Ca⁺⁺ adsorption at the external interface. When the membrane is polarized, the high electric field at the external interface acting on the membrane fixed charge keeps the effective channel diameter small, so that only dihydrated ions can cross the interface. The higher energy required to partially dehydrate Na⁺ accounts for its lower permeability when polarized. Depolarized, the channel entrance can expand, permitting quadrihydrated ions to pass; the large initial Na⁺ flow is the result of the large concentration ratio across the interface. The effect at the internal interface is symmetric; Na⁺ crosses with greater difficulty when the membrane is depolarized. Na⁺ inactivation occurs when the ion distribution within the membrane has assumed its new steady-state value. Calculations based on parameters consistent with physicochemical data agree generally with a wide range of experiments. The model does not obey the two fundamental Hodgkin-Huxley (HH) postulates (independence principle, ion flow proportional to thermodynamic potential). In several instances the model predicts experimental results which are not predicted by the HH equations.     

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.     

Brownian dynamics study of flux ratios in sodium channels

Measurements of unidirectional fluxes in ion channels provide one of the experimental methods for studying the steps involved in ion permeation in biological pores. Conventionally, the number of ions in the pore is inferred by fitting the ratio of inward and outward currents to an exponential function with an adjustable parameter known as the flux ratio exponent. Here we investigate the relationship between the number of ions in the pore and the flux ratio exponent in a model sodium channel under a range of conditions. Brownian dynamics simulations enable us to count the precise number of ions in the channel and at the same time measure the currents flowing across the pore in both directions. We show here that the values of the flux ratio exponent n′ ranges between 1 and 3 and is highly dependent on the ionic concentrations in which measurements are made. This is a consequence of the fact that both inward and outward currents are susceptible to saturation with increasing concentration. These results indicate that measurements of the flux ratio exponent cannot be directly related to the number of ions in the pore and that interpretation of such experimental measurements requires careful consideration of the conditions in which the study is made.     

Ansätze zur quantitativen Analyse der Nachrichtenaufnahme und -verarbeitung in biologischen Rezeptoren

The flux equilibrium theory, used for interpretating active and passive ion transport, can explain the generation of receptor potentials. In a model, driving forces and velocity coefficients are represented by the parameters of electric circuits. From these membrane models ionic fluxes can be calculated quantitatively on the basis of transport equations. These equations are derived from the theory of irreversible thermodynamic processes. Receptor models allow a simulation and prediction of the bioelectric potentials which were recorded by other authors in neuro-physiological experiments under various stimulus conditions. The information capacity of a single receptor channel is determined by the ionic flux and the stimulus parameters. In combination with the network of neuron models, receptor models can be used in a perception. The problems of on-off-activation and lateral inhibition were investigated with such a network.     

The independence principle. A reconsideration

The electrodiffusion model presented in the previous paper, which specifically excludes ion-ion interactions, is analyzed for the ratio of one-way fluxes (flux ratio) as a function of the ionic driving force across the membrane. Significant deviations from the behavior expected on the basis of the Ussing relation are found. These are sufficient to explain the “nonindependent” ion movement noted in some biological flux ratio data. One-way fluxes are dependent on the ionic concentration on both sides of the membrane. The coupling of these fluxes to ionic concentrations comes from the dependence of ionic mobility and the diffusion coefficient on the equilibrium potential. It is concluded that nonindependent behavior in experimental data is not sufficient to implicate ion-ion interaction as the source of the discrepancy.     

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.     

The concept of channel permeability

The one-way fluxes of a certain species of ion through a passive channel in a fixed state are assumed to be well-behaved functions of seven variables only, all of which are macroscopically observable: the cis and trans concentrations of that ion, the charge of the ion, the voltage difference between the cis and trans solutions, the thermodynamic temperature, the ion concentration at half maximal conductance of the channel, and the channel permeability. It is then proved that, if these fluxes are independent and Nernstian, the absolute permeability is uniquely and explicitly defined in terms of the channel current near zero voltage; no assumptions about the detailed workings of the permeation process are necessary. It is also proved that, if the one-way fluxes of the species under consideration are merely unaffected by the other species present, then the channel's absolute permeability can still be uniquely and explicitly defined if a weak constraint on the flux ratio is assumed; again, no assumptions on the inner workings of the channel are necessary.     

Dynamic Dielectrophoresis Model of Multi-Phase Ionic Fluids

Ionic-based dielectrophoretic microchips have attracted significant attention due to their wide-ranging applications in electro kinetic and biological experiments. In this work, a numerical method is used to simulate the dynamic behaviors of ionic droplets in a microchannel under the effect of dielectrophoresis. When a discrete liquid dielectric is encompassed within a continuous fluid dielectric placed in an electric field, an electric force is produced due to the dielectrophoresis effect. If either or both of the fluids are ionic liquids, the magnitude and even the direction of the force will be changed because the net ionic charge induced by an electric field can affect the polarization degree of the dielectrics. However, using a dielectrophoresis model, assuming ideal dielectrics, results in significant errors. To avoid the inaccuracy caused by the model, this work incorporates the electrode kinetic equation and defines a relationship between the polarization charge and the net ionic charge. According to the simulation conditions presented herein, the electric force obtained in this work has an error exceeding 70% of the actual value if the false effect of net ionic charge is not accounted for, which would result in significant issues in the design and optimization of experimental parameters. Therefore, there is a clear motivation for developing a model adapted to ionic liquids to provide precise control for the dielectrophoresis of multi-phase ionic liquids.     

Intrinsic electric fields and proton diffusion in immobilized protein membranes. Effects of electrolytes and buffers.

We present a theory for proton diffusion through an immobilized protein membrane perfused with an electrolyte and a buffer. Using a Nernst-Planck equation for each species and assuming local charge neutrality, we obtain two coupled nonlinear diffusion equations with new diffusion coefficients dependent on the concentration of all species, the diffusion constants or mobilities of the buffers and salts, the pH-derivative of the titration curves of the mobile buffer and the immobilized protein, and the derivative with respect to ionic strength of the protein titration curve. Transient time scales are locally pH-dependent because of protonation-deprotonation reactions with the fixed protein and are ionic strength-dependent because salts provide charge carriers to shield internal electric fields. Intrinsic electric fields arise proportional to the gradient of an "effective" charge concentration. The field may reverse locally if buffer concentrations are large (greater to or equal to 0.1 M) and if the diffusivity of the electrolyte species is sufficiently small. The "ideal" electrolyte case (where each species has the same diffusivity) reduces to a simple form. We apply these theoretical considerations to membranes composed of papain and bovine serum albumin (BSA) and show that intrinsic electric fields greatly enhance the mobility of protons when the ionic strength of the salts is smaller than 0.1 M. These results are consistent with experiments where pH changes are observed to depend strongly on buffer, salt, and proton concentrations in baths adjacent to the membranes.     

Time resolved secretion of chloride from a monolayer of mucin-secreting epithelial cells

Short-circuit current (Isc) measurement is used to quantify transepithelial ion flux. This technique provides a direct measure of net charge transport across a cell monolayer. Isc however, lacks chemical selectivity. Chemically resolved ion fluxes may be much greater than Isc, and differ in different biological processes. This work describes a novel experimental approach and deconvolution method to obtain temporally resolved ion fluxes at epithelial cell monolayers. HT29-Cl.16E cells, a sub clone of the human colonic cancer cell line HT29 was used as a model cell line to validate this approach in the context of epithelial transport studies. This cell line is known to secrete chloride in response to purinergic stimulation. Changes in chloride concentration after stimulation with 1 mM ATP plus 50 nM phorbol-myristate acetate (PMA) are recorded with a chloride ion-selective electrode (ISE) at a short distance (∼50 μm) from the monolayer. The recorded concentrations are transformed to corresponding chloride flux across the monolayer using a deconvolution algorithm for extracellular mass transport based on minimization of the shape error function (Nair and Gratzl in Anal Chem 77:2875–2888, 2005). Simultaneous voltage clamp yields the associated net electrical charge flux (Isc). The dynamics of Cl⁻ flux did correlate with that of the electrical flux, but was found to be greater in amplitude. This suggests that Cl⁻ may not be the only ion secreted. The method of simultaneously assessing ionic and electrical fluxes with a temporal resolution of seconds provides unique information about the dynamics of solute fluxes across the apical membrane. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

(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.     

Ionic selectivity, saturation, and block in sodium channels. A four- barrier model

Ionic fluxes in Na channels of myelinated axons show ionic competition, block, and deviations from simple flux independence. These phenomena are particularly evident when external Na+ ions are replaced by other permeant or impermeant ions. The observed currents require new flux equations not based on the concepts of free diffusion. A specific permeability model for the Na channel is developed from Eyring rate theory applied to a chain of saturable binding sites. There are four energy barriers in the pore and only one ion is allowed inside at a time. Deviations from independence arise from saturation. The model shows that ionic permeability ratios measured from zero-current potentials can differ from those measured from relative current amplitudes or conductances. The model can be fitted to experiments with various external sodium substitutes by varying only two parameters: For each ion the height of the major energy barrier (the selectivity filter) determines the biionic zero-current potential and the depth of the energy well (binding site) just external to that barrier then determines the current amplitudes. Voltage clamp measurements with myelinated nerve fibers are given showing numerous examples of deviations from independence in ionic fluxes. Strong blocks of ionic currents by guanidinium compounds and Tl+ ions are fitted by binding within the channel with apparent dissociation constants in the range 50-122 mM. A small block with high Na+ concentrations can be fitted by Na+ ion binding with a dissociation constant of 368 mM. The barrier model is given a molecular interpretation that includes stepwise dehydration of the permeating ion as it interacts with an ionized carboxylic acid.     

The Double Fixed Charge Membrane: Solution-Membrane Ion Partition Effects and Membrane Potentials

An analysis is made of the effect of solution-membrane partition of ions on the electrostatic potential and ion concentration profiles in fixed charge membranes. It is shown that the inclusion of partition effects gives rise to large solution-membrane “Donnan” potentials even when the concentration of fixed charges is of the same order as the concentration of the external solution. This effect renders the system and the simplified analysis of the double fixed charge membrane (FCM) previously given more applicable to biological membranes. An analysis is also given of the voltage dependence of the fluxes of individual ion species in the double FCM when it separates different ionic solutions and an expression is deduced for the membrane resting potential. Although the latter is similar in form to the Goldman-Hodgkin-Katz (GHK) equation the corresponding value of the permeability ratio P_C1/P_K is under certain specified conditions both concentration and potential dependent.     

Synchronization modulation of Na/K pump molecules can hyperpolarize the membrane resting potential in intact fibers

Previously, we have theoretically studied the possibility of electrical rhythmic entrainment of carrier-mediated ion transporters, and experimentally realized synchronization and acceleration of the Na/K pumping rate in the cell membrane of skeletal muscle fibers by a specially designed synchronization modulation electric field. In these studies we either used cut fibers under a voltage clamp or intact fibers, but in the presence of ion channels blockers. A question remained as to whether the field-induced activation observed in the pump molecules could effectively increase the intracellular ionic concentration and the membrane potential at physiological conditions. In this paper, we studied the effects of the field on intact fibers without any channel blockers. We monitored the field-induced changes in the ionic concentration gradient across the cell membrane and the membrane potential non-invasively by using a fluorescent probe and confocal microscopic imaging techniques. The results clearly show that the entrainment of the pump molecules by the synchronization modulation electric field can effectively increase the ionic concentration gradient, and hence, hyperpolarize the membrane potential.     

The Electrical Conductance of Semipermeable Membranes: I. A Formal Analysis

A kinetic analysis of membrane conductance under conditions of stationary flow is presented. The semipermeable membrane is idealized as a homogeneous laminar phase separating ionic solutions on either side. It is assumed, without consideration of the mechanisms involved, that some ion species permeate the membrane while others do not. The flux of a given species is taken to be linearly related to the gradient of its concentration and to the electric field. The resulting flow equations, when combined with Poisson's equation, permit the formulation of the conductance problem in terms of a set of non-linear differential equations. They describe the spatial variation of the electric displacement and contain the ion current densities as parameters. Their integration, subject to appropriate boundary conditions, fixes the values of these parameters and of the corresponding transmembrane potential. The solution of the conductance problem cannot, however, be carried through in analytic form. The numerical analysis of a number of special cases will be presented in subsequent publications.     

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.     

Ion permeation and selectivity of OmpF porin: a theoretical study based on molecular dynamics,Brownian dynamics,and continuum electrodiffusion theory

Three different theoretical approaches are used and compared to refine our understanding of ion permeation through the channel formed by OmpF porin from Escherichia coli. Those approaches are all-atom molecular dynamics (MD) in which ions, solvent, and lipids are represented explicitly, Brownian dynamics (BD) in which ions are represented explicitly, while solvent and lipids are represented as featureless dielectrics, and Poisson-Nernst-Planck (PNP) electrodiffusion theory in which both solvent and local ion concentrations are represented as a continuum. First, the ability of the different theoretical approaches in reproducing the equilibrium average ion density distribution in OmpF porin bathed by a 1M KCl symmetric salt solution is examined. Under those conditions the PNP theory is equivalent to the non-linear Poisson-Boltzmann (PB) theory. Analysis shows that all the three approaches are able to capture the important electrostatic interactions between ions and the charge distribution of the channel that govern ion permeation and selectivity in OmpF. The K(+) and Cl(-) density distributions obtained from the three approaches are very consistent with one another, which suggests that a treatment on the basis of a rigid protein and continuum dielectric solvent is valid in the case of OmpF. Interestingly, both BD and continuum electrostatics reproduce the distinct left-handed twisted ion pathways for K(+) and Cl(-) extending over the length of the pore which were observed previously in MD. Equilibrium BD simulations in the grand canonical ensemble indicate that the channel is very attractive for cations, particularly at low salt concentration. On an average there is 1.55 K(+) inside the pore in 10mM KCl. Remarkably, there is still 0.17 K(+) on average inside the pore even at a concentration as low as 1microM KCl. Secondly, non-equilibrium ion flow through OmpF is calculated using BD and PNP and compared with experimental data. The channel conductance in 0.2M and 1M KCl calculated using BD is in excellent accord with the experimental data. The calculations reproduce the experimentally well-known conductance-concentration relation and also reveal an asymmetry in the channel conductance (a larger conductance is observed under a positive transmembrane potential). Calculations of the channel conductance for three mutants (R168A, R132A, and K16A) in 1M KCl suggest that the asymmetry in the channel conductance arises mostly from the permanent charge distribution of the channel rather than the shape of the pore itself. Lastly, the calculated reversal potential in a tenfold salt gradient (0.1:1M KCl) is 27.4(+/-1.3)mV (BD) and 22.1(+/-0.6)mV (PNP), in excellent accord with the experimental value of 24.3mV. Although most of the results from PNP are qualitatively reasonable, the calculated channel conductance is about 50% higher than that calculated from BD probably because of a lack of some dynamical ion-ion correlations.