Single-file Pore Research Articles

Capacitive energy storage in single-file pores: Exactly solvable models and simulations.

Understanding charge storage in low-dimensional electrodes is crucial for developing novel ecologically friendly devices for capacitive energy storage and conversion and water desalination. Exactly solvable models allow in-depth analyses and essential physical insights into the charging mechanisms. So far, however, such analytical approaches have been mainly limited to lattice models. Herein, we develop a versatile, exactly solvable, one-dimensional off-lattice model for charging single-file pores. Unlike the lattice model, this model shows an excellent quantitative agreement with three-dimensional Monte Carlo simulations. With analytical calculations and simulations, we show that the differential capacitance can be bell-shaped (one peak), camel-shaped (two peaks), or have four peaks. Transformations between these capacitance shapes can be induced by changing pore ionophilicity, by changing cation-anion size asymmetry, or by adding solvent. We find that the camel-shaped capacitance, characteristic of dilute electrolytes, appears for strongly ionophilic pores with high ion densities, which we relate to charging mechanisms specific to narrow pores. We also derive a large-voltage asymptotic expression for the capacitance, showing that the capacitance decays to zero as the inverse square of the voltage, C ∼ u-2. This dependence follows from hard-core interactions and is not captured by the lattice model.

Non-Fickian single-file pore transport.

Single-file diffusion exhibits anomalously slow collective transport when particles are able to immobilize by binding and unbinding to the one-dimensional channel within which the particles diffuse. We have explored this system for short porelike channels using a symmetric exclusion process with fully stochastic dynamics. We find that for shorter channels, a non-Fickian regime emerges for slow binding kinetics. In this regime the average flux 〈Φ〉∼1/L^{3}, where L is the channel length in units of the particle size. We find that a two-state model describes this behavior well for sufficiently slow binding rates, where the binding rates determine the switching time between high-flux bursts of directed transport and low-flux leaky states. Each high-flux burst is Fickian with 〈Φ〉∼1/L. Longer systems are more often in a low-flux state, leading to the non-Fickian behavior.

A theoretical analysis on the separation of mixtures by extractive adsorption

A new approach to the separation of chemical mixtures by way of extractive adsorption is proposed. The approach relies on the use of adsorbents with single file pores, for which clusters of aligned dipoles can result in a 1D ferroelectric phase. To study this class of separations, a new theory is developed for the adsorption of polar species in single file pores based on a 1D approximation of classical density functional theory (CDFT). The theory is applied to pure, binary and ternary mixtures composed of hydrogen cyanide, formaldehyde and ethylene. It is demonstrated that adsorbing polar molecules in single file channels do undergo ferroelectric transitions, under certain conditions. These transitions result in step like isotherms with wide hysteresis loops. It is then shown that inclusion of the adsorbate solvent hydrogen cyanide, can extract formaldehyde from a hydrogen cyanide, formaldehyde, ethylene mixture.

Ising models of charge storage in multifile metallic nanopores

Ising type models of charging of conductive nanopores with ions have already been proposed and investigated for single file cylindrical or single layer slit nanopores. In such pores, the state of ions, the coulombic interactions of which are exponentially screened by their images in pore walls, was named superionic. In the present work we extend the analysis of the superionic state to nanopores that can accommodate multiple rows of ions. By grouping multiple charges in the same row into ‘supercharges’, we map the arrangement of ions in polarised electrodes on a multi-row Ising model in an external field. We investigate one-, two- and three-row cases, which we solve exactly, using a purpose-built semi-numerical transfer matrix method. For pores of different radii, which can accommodate the corresponding number of ion rows, we calculate the dependence of the electrical capacitance and stored energy density on electrode potential. As in charging the single file pores, we find that in narrower pores higher energy densities can be achieved at low applied potentials, while wider pores perform better as the voltage is increased.

Osmotic and diffusive flows in single-file pores: new approach to modeling pore occupancy states

BackgroundThe relation between osmotic permeability, Pf, diffusion permeability, Pd, and the number of water molecules, Np, in the single-file membrane pore remains an open question. Theoretical analyses, empirical studies on aquaporins and nanotubes, and molecular dynamics simulations have yet to provide a consensus view.ResultsThis paper presents a new combinatorial analysis of the different pore states formed from water molecules and the presence of a vacancy that differs from the several previous combinatorial approaches to analyzing pore states. It is the first such analysis to show that Pf / Pd = Np. It is rooted in the concept of different classes of pore occupancy states, tracer states and tracer exit states, present in the pore. This includes pores with and without a single vacancy. The concepts of knock-on collisions and concerted Brownian fluctuations provide the mechanisms underlying the behaviors of the tracer and vacancy as each moves through the pore during osmotic or diffusive flow. It develops the important role of the knock-on collision mechanism for osmotic flow. An essential feature of the model is the presence, or absence, of a single vacancy in the pore. The vacancy slows down tracer translocation through the pore. Its absence facilitates osmotic flow.ConclusionsThe full pore states and the single vacancy states together with the knock-on and Brownian mechanisms account for the relative values of Pf and Pd during osmotic and diffusive flow through the single-file pore. The new approach to combinatorial analysis differs from previous approaches and is the first to show a simple intuitive basis for the relation Pf / Pd = Np. This resolves a long persisting dichotomy.

Model of osmosis in a single-file pore.

Single-file transport of water and other small molecules through narrow pores in osmosis has drawn considerable attention in recent years due to its extensive application in biology and industry. In this work, we propose a discrete model to describe nonideal osmosis through single-file pores. Every site is assumed to be occupied by a molecule according to experiments and simulations. Hence, a dense chain can always be found, and collective hopping is the only movement method enabling the molecular chain to move. The roles of solute in osmosis are clarified in this model. Those molecules reflected at the pore entrance produce osmotic pressure, and those inside the pore contribute to the flow resistance of the molecular chain. The solute molecules that can enter the pore but cannot penetrate it may significantly reduce the osmotic flux, although they are all rejected by the pore. This conclusion can help to clarify the emerging debate about whether the reflection coefficient of the fully rejected solute can be less than 1. The design of highly efficient membrane pores may also benefit from this study.

Charging Ultrananoporous Electrodes with Size-Asymmetric Ions Assisted by Apolar Solvent

We develop a statistical theory of charging quasi single-file pores with cations and anions of different sizes as well as solvent molecules or voids. This is done by mapping the charging onto a one-dimensional Blume–Emery–Griffith model with variable coupling constants. The results are supported by three-dimensional Monte Carlo simulations in which many limitations of the theory are lifted. We explore the different ways of enhancing the energy storage which depend on the competitive adsorption of ions and solvent molecules into pores, the degree of ionophilicity and the voltage regimes accessed. We identify new solvent-related charging mechanisms and show that the solvent can play the role of an “ionophobic agent”, effectively controlling the pore ionophobicity. In addition, we demonstrate that the ion-size asymmetry can significantly enhance the energy stored in a nanopore.

Aquaporins at a glance

With a concentration of 55,000 mM, water is by far the most prevalent molecule in biological systems. For many years it was assumed that biological membranes are freely water permeable, so that the existence of a family of water channels would not have been predicted. However, biophysical studies in

Cycle flux algebra for ion and water flux through the KcsA channel single-file pore links microscopic trajectories and macroscopic observables.

In narrow pore ion channels, ions and water molecules diffuse in a single-file manner and cannot pass each other. Under such constraints, ion and water fluxes are coupled, leading to experimentally observable phenomena such as the streaming potential. Analysis of this coupled flux would provide unprecedented insights into the mechanism of permeation. In this study, ion and water permeation through the KcsA potassium channel was the focus, for which an eight-state discrete-state Markov model has been proposed based on the crystal structure, exhibiting four ion-binding sites. Random transitions on the model lead to the generation of the net flux. Here we introduced the concept of cycle flux to derive exact solutions of experimental observables from the permeation model. There are multiple cyclic paths on the model, and random transitions complete the cycles. The rate of cycle completion is called the cycle flux. The net flux is generated by a combination of cyclic paths with their own cycle flux. T.L. Hill developed a graphical method of exact solutions for the cycle flux. This method was extended to calculate one-way cycle fluxes of the KcsA channel. By assigning the stoichiometric numbers for ion and water transfer to each cycle, we established a method to calculate the water-ion coupling ratio (CR w-i) through cycle flux algebra. These calculations predicted that CR w-i would increase at low potassium concentrations. One envisions an intuitive picture of permeation as random transitions among cyclic paths, and the relative contributions of the cycle fluxes afford experimental observables.

Teaching Old Coefficients New Tricks: New Insight into the Meaning of the Osmotic and Diffusive Permeation Coefficients

Teaching Old Coefficients New Tricks: New Insight into the Meaning of the Osmotic and Diffusive Permeation Coefficients

Molecular and Functional Characterization of a Na+-K+ Transporter from the Trk Family in the Ectomycorrhizal Fungus Hebeloma cylindrosporum

Ectomycorrhizal symbiosis between fungi and woody plants strongly improves plant mineral nutrition and constitutes a major biological process in natural ecosystems. Molecular identification and functional characterization of fungal transport systems involved in nutrient uptake are crucial steps toward understanding the improvement of plant nutrition and the symbiotic relationship itself. In the present report a transporter belonging to the Trk family is identified in the model ectomycorrhizal fungus Hebeloma cylindrosporum and named HcTrk1. The Trk family is still poorly characterized, although it plays crucial roles in K(+) transport in yeasts and filamentous fungi. In Saccharomyces cerevisiae K(+) uptake is mainly dependent on the activity of Trk transporters thought to mediate H(+):K(+) symport. The ectomycorrhizal HcTrk1 transporter was functional when expressed in Xenopus oocytes, enabling the first electrophysiological characterization of a transporter from the Trk family. HcTrk1 mediates instantaneously activating inwardly rectifying currents, is permeable to both K(+) and Na(+), and displays channel-like functional properties. The whole set of data and particularly a phenomenon reminiscent of the anomalous mole fraction effect suggest that the transport does not occur according to the classical alternating access model. Permeation appears to occur through a single-file pore, where interactions between Na(+) and K(+) might result in Na(+):K(+) co-transport activity. HcTrk1 is expressed in external hyphae that explore the soil when the fungus grows in symbiotic condition. Thus, it could play a major role in both the K(+) and Na(+) nutrition of the fungus (and of the plant) in nutrient-poor soils.

Sodium Flux Ratio in Na/K Pump-Channels Opened by Palytoxin

Palytoxin binds to Na+/K+ pumps in the plasma membrane of animal cells and opens an electrodiffusive cation pathway through the pumps. We investigated properties of the palytoxin-opened channels by recording macroscopic and microscopic currents in cell bodies of neurons from the giant fiber lobe, and by simultaneously measuring net current and 22Na+ efflux in voltage-clamped, internally dialyzed giant axons of the squid Loligo pealei. The conductance of single palytoxin-bound “pump-channels” in outside-out patches was ∼7 pS in symmetrical 500 mM [Na+], comparable to findings in other cells. In these high-[Na+], K+-free solutions, with 5 mM cytoplasmic [ATP], the K 0.5 for palytoxin action was ∼70 pM. The pump-channels were ∼40–50 times less permeable to N-methyl-d-glucamine (NMG+) than to Na+. The reversal potential of palytoxin-elicited current under biionic conditions, with the same concentration of a different permeant cation on each side of the membrane, was independent of the concentration of those ions over the range 55–550 mM. In giant axons, the Ussing flux ratio exponent (n') for Na+ movements through palytoxin-bound pump-channels, over a 100–400 mM range of external [Na+] and 0 to −40 mV range of membrane potentials, averaged 1.05 ± 0.02 (n = 28). These findings are consistent with occupancy of palytoxin-bound Na+/K+ pump-channels either by a single Na+ ion or by two Na+ ions as might be anticipated from other work; idiosyncratic constraints are needed if the two Na+ ions occupy a single-file pore, but not if they occupy side-by-side binding sites, as observed in related structures, and if only one of the sites is readily accessible from both sides of the membrane.

Role of Protein Flexibility in Ion Permeation: A Case Study in Gramicidin A

Proteins have a flexible structure, and their atoms exhibit considerable fluctuations under normal operating conditions. However, apart from some enzyme reactions involving ligand binding, our understanding of the role of flexibility in protein function remains mostly incomplete. Here we investigate this question in the realm of membrane proteins that form ion channels. Specifically, we consider ion permeation in the gramicidin A channel, and study how the energetics of ion conduction changes as the channel structure is progressively changed from completely flexible to a fixed one. For each channel structure, the potential of mean force for a permeating potassium ion is determined from molecular dynamics (MD) simulations. Using the same molecular dynamics data for completely flexible gramicidin A, we also calculate the average densities and fluctuations of the peptide atoms and investigate the correlations between these fluctuations and the motion of a permeating ion. Our results show conclusively that peptide flexibility plays an important role in ion permeation in the gramicidin A channel, thus providing another reason—besides the well-known problem with the description of single file pore water—why this channel cannot be modeled using continuum electrostatics with a fixed structure. The new method developed here for studying the role of protein flexibility on its function clarifies the contributions of the fluctuations to energy and entropy, and places limits on the level of detail required in a coarse-grained model.

Coupled K+–Water Flux through the HERG Potassium Channel Measured by an Osmotic Pulse Method

The streaming potential (V stream) is a signature feature of ion channels in which permeating ions and water molecules move in a single file. V stream provides a quantitative measure of the ion and water flux (the water–ion coupling ratio), the knowledge of which is a prerequisite for elucidating the mechanisms of ion permeation. We have developed a method to measure V stream with the whole-cell patch-clamp configuration. A HEK293 cell stably expressing the HERG potassium channel was voltage clamped and exposed to hyperosmotic solutions for short periods of time (<1 s) by an ultrafast solution switching system (the osmotic pulse [quick jump-and-away] method). The reversal potentials were monitored by a series of voltage ramps before, during, and after the osmotic pulse. The shifts of the reversal potentials immediately after the osmotic jump gave V stream. In symmetrical K+ solutions (10 mM), the V streams measured at different osmolalities showed a linear relationship with a slope of −0.7 mV/ΔOsm, from which the water–ion coupling ratio (n, the ratio of the flux of water to the flux of cations; Levitt, D.G., S.R. Elias, and J.M. Hautman. 1978. Biochim. Biophys. Acta. 512:436–451) was calculated to be 1.4. In symmetrical 100 mM K+ solutions, the coupling ratio was decreased significantly (n = 0.9), indicating that the permeation process through states with increased ion occupancy became significant. We presented a diagrammatic representation linking the water–ion coupling ratio to the mode of ion permeation and suggested that the coupling ratio of one may represent the least hydrated ion flux in the single-file pore.

Effect of Na(+) flow on Cd(2+) block of tetrodotoxin-resistant Na(+) channels.

Tetrodotoxin-resistant (TTX-R) Na+ channels are 1,000-fold less sensitive to TTX than TTX-sensitive (TTX-S) Na+ channels. On the other hand, TTX-R channels are much more susceptible to external Cd2+ block than TTX-S channels. A cysteine (or serine) residue situated just next to the aspartate residue of the presumable selectivity filter “DEKA” ring of the TTX-R channel has been identified as the key ligand determining the binding affinity of both TTX and Cd2+. In this study we demonstrate that the binding affinity of Cd2+ to the TTX-R channels in neurons from dorsal root ganglia has little intrinsic voltage dependence, but is significantly influenced by the direction of Na+ current flow. In the presence of inward Na+ current, the apparent dissociation constant of Cd2+ (∼200 μM) is ∼9 times smaller than that in the presence of outward Na+ current. The Na+ flow–dependent binding affinity change of Cd2+ block is true no matter whether the direction of Na+ current is secured by asymmetrical chemical gradient (e.g., 150 mM Na+ vs. 150 mM Cs+ on different sides of the membrane, 0 mV) or by asymmetrical electrical gradient (e.g., 150 mM Na+ on both sides of the membrane, −20 mV vs. 20 mV). These findings suggest that Cd2+ is a pore blocker of TTX-R channels with its binding site located in a multiion, single-file region near the external pore mouth. Quantitative analysis of the flow dependence with the flux-coupling equation reveals that at least two Na+ ions coexist with the blocking Cd2+ ion in this pore region in the presence of 150 mM ambient Na+. Thus, the selectivity filter of the TTX-R Na+ channels in dorsal root ganglion neurons might be located in or close to a multiion single-file pore segment connected externally to a wide vestibule, a molecular feature probably shared by other voltage-gated cationic channels, such as some Ca2+ and K+ channels.

Propagation Dynamics of a Particle Phase in a Single-File Pore.

AbstractWe study propagation dynamics of a particle phase in a single-file pore connected to a reservoir of particles (bulk liquid phase). We show that the total mass M(t) of particles entering the pore up to time t grows as (Mt) = 2m(J,ρF) √D0t, where D0 is the “bare” di usion coeffcient and the prefactor m(J,ρF) is a non-trivial function of the reservoir density ρF and the amplitude J of attractive particle-particle interactions. Behavior of the dynamic density pro les is also discussed.

Kinetics and thermodynamics across single-file pores: Solute permeability and rectified osmosis

We study the effects of solute-membrane interactions on osmotic transport through pores. By extending single-file, single-species kinetic models to include entrance of solute into membrane pores, we model the statistical mechanics of competitive transport of two species across membrane pores. The results have direct applications to water transport across biomembrane pores and particle movement in zeolites, and can be extended to study ion channel transport. Reflection coefficients, the reduction of osmotic fluxes measured using different solutes, are computed in terms of the microscopic kinetic parameters. We find that a reduction in solvent flow due to solute–pore interactions can be modeled by a Langmuir adsorption isotherm. Osmosis experiments are discussed and proposed. Special cases and Onsager relations are presented in the Appendixes.

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 channel’s 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.

Ion channel selectivity through stepwise changes in binding affinity.

Voltage-gated Ca2+ channels select Ca2+ over competing, more abundant ions by means of a high affinity binding site in the pore. The maximum off rate from this site is approximately 1,000x slower than observed Ca2+ current. Various theories that explain how high Ca2+ current can pass through such a sticky pore all assume that flux occurs from a condition in which the pore's affinity for Ca2+ transiently decreases because of ion interactions. Here, we use rate theory calculations to demonstrate a different mechanism that requires no transient changes in affinity to quantitatively reproduce observed Ca2+ channel behavior. The model pore has a single high affinity Ca2+ binding site flanked by a low affinity site on either side; ions permeate in single file without repulsive interactions. The low affinity sites provide steps of potential energy that speed the exit of a Ca2+ ion off the selectivity site, just as potential energy steps accelerate other chemical reactions. The steps could be provided by weak binding in the nonselective vestibules that appear to be a general feature of ion channels, by specific protein structures in a long pore, or by stepwise rehydration of a permeating ion. The previous ion-interaction models and this stepwise permeation model demonstrate two general mechanisms, which might well work together, to simultaneously generate high flux and high selectivity in single file pores.

How Fast Do Fluids Squeeze through Microscopic Single-File Pores?

A one dimensional symmetric exclusion model is used to study pressure and osmosis driven flows through molecular-sized channels, such as biological membrane channels and zeolite pores. Analytic expressions are found for the steady-state flow as a function of pore radius, pore energetics, reservoir temperature, driving force, and internal defects. We find a flux maximum as a function of particle-pore interactions: This, and other nonlinear dependences suggest numerous diagnostic experiments for biological and zeolitic systems.