Flux, coupling, and selectivity in ionic channels of one conformation

Structure and Computational Electrophysiology of Ac-LS3, a Synthetic Ion Channel

Computer simulations are reported on Ac-LS3, a synthetic ion channel, containing 21 residues with a Leu-Ser-Ser-Leu-Leu-Ser-Leu heptad repeat, which forms ions channels upon application of voltage. A hexameric, coiled-coil bundle initially positioned perpendicular to the membrane settled into a stable, tilted structure after 1.5 μs, most likely to improve contacts between the non-polar exterior of the channel and the hydrophobic core of the membrane. Once tilted, the bundle remained in this state during subsequent simulations of nearly 10 μs at voltages ranging from 200 to -100 mV. In contrast, attempts to identify a stable pentameric structure failed, thus supporting the hypothesis that the channel is a hexamer. Results at 100 mV were used to reconstruct the free energy profiles for K⁺ and Cl^- in the channel. This was done by way of several methods in which results of molecular dynamics (MD) simulations were combined with the electrodiffusion model. Two of them developed recently do not require knowledge of the diffusivity. Instead, they utilize one-sided density profiles and committor probabilities. The consistency between different methods is very good, supporting the utility of the newly developed methods for reconstructing free energies of ions in channels. The flux of K⁺, which accounts for most of the current through the channel, calculated directly from MD matches well the total measured current. However, the current of Cl^- is somewhat overestimated, possibly due to a slightly unbalanced force field involving chloride. The current-voltage dependence was also reconstructed by way of a recently developed, efficient method that requires simulations only at a single voltage, yielding good agreement with the experiment. Taken together, the results demonstrate that computational electrophysiology has become a reliable tool for studying how channels mediate ion transport through membranes.

Electrophysiological Properties from Computations at a Single Voltage: Testing Theory with Stochastic Simulations

We use stochastic simulations to investigate the performance of two recently developed methods for calculating the free energy profiles of ion channels and their electrophysiological properties, such as current–voltage dependence and reversal potential, from molecular dynamics simulations at a single applied voltage. These methods require neither knowledge of the diffusivity nor simulations at multiple voltages, which greatly reduces the computational effort required to probe the electrophysiological properties of ion channels. They can be used to determine the free energy profiles from either forward or backward one-sided properties of ions in the channel, such as ion fluxes, density profiles, committor probabilities, or from their two-sided combination. By generating large sets of stochastic trajectories, which are individually designed to mimic the molecular dynamics crossing statistics of models of channels of trichotoxin, p7 from hepatitis C and a bacterial homolog of the pentameric ligand-gated ion channel, GLIC, we find that the free energy profiles obtained from stochastic simulations corresponding to molecular dynamics simulations of even a modest length are burdened with statistical errors of only 0.3 kcal/mol. Even with many crossing events, applying two-sided formulas substantially reduces statistical errors compared to one-sided formulas. With a properly chosen reference voltage, the current–voltage curves can be reproduced with good accuracy from simulations at a single voltage in a range extending for over 200 mV. If possible, the reference voltages should be chosen not simply to drive a large current in one direction, but to observe crossing events in both directions.

Conic shapes have higher sensitivity than cylindrical ones in nanopore DNA sequencing

Nanopores have emerged as helpful research tools for single molecule detection. Through continuum modeling, we investigated the effects of membrane thickness, nanopore size, and pore shape on current signal characteristics of DNA. The simulation results showed that, when reducing the pore diameter, the amplitudes of current signals of DNA increase. Moreover, we found that, compared to cylindrically shaped nanopores, conical-shaped nanopores produce greater signal amplitudes from biomolecules translocation. Finally, we demonstrated that continuum model simulations for the discrimination of DNA and RNA yield current characteristics approximately consistent with experimental measurements and that A-T and G-C base pairs can be distinguished using thin conical solid-state nanopores. Our study not only suggests that computational approaches in this work can be used to guide the designs of nanopore for single molecule detection, but it also provides several possible ways to improve the current amplitudes of nanopores for better resolution.

Probing the oxidative etching induced dissolution of palladium nanocrystals in solution by liquid cell transmission electron microscopy

A microscopic study of dissolution process of nanocrystals, an opposite while functioning cooperatively with growth in many cases, is an essential issue in variety aspects of research on nanocrystals. In this work, an in situ study of the dynamic dissolution process of palladium nanocrystals by liquid cell transmission electron microscope (TEM) is presented. The effective critical size (R_critical) for monodispersed nanocrystals is determined to be about 5nm in the experimental condition of this article. When the size of nanocrystal is above R_critical, the dissolution rate (dr/dt) is nearly a constant. For the nanocrystal sizing below R_critical, the dissolution rate (dr/dt) increases with the decrease of the nanocrystal radius r, indicating that high equilibrium solubility must be taken into account in the dissolution rate of small nanocrystals in solution. It is found that the aggregation kinetics and confinement effect between adjacent nanocrystals have effects on the dissolution rate during the reaction, and it has been analyzed in details and discussed in terms of the underlying physics involved. Lastly, the effects of electron beam-water interaction and the iron (III) agents on the oxidative etching are also compared.

Cation Selectivity in Biological Cation Channels Using Experimental Structural Information and Statistical Mechanical Simulation

Cation selective channels constitute the gate for ion currents through the cell membrane. Here we present an improved statistical mechanical model based on atomistic structural information, cation hydration state and without tuned parameters that reproduces the selectivity of biological Na+ and Ca2+ ion channels. The importance of the inclusion of step-wise cation hydration in these results confirms the essential role partial dehydration plays in the bacterial Na+ channels. The model, proven reliable against experimental data, could be straightforwardly used for designing Na+ and Ca2+ selective nanopores.

Modeling and Simulation of Thermo-Fluid-Electrochemical Ion Flow in Biological Channels

In this articlewe address the study of ion charge transport in the biological channels separating the intra and extracellular regions of a cell. The focus of the investigation is devoted to including thermal driving forces in the well-known velocity-extended Poisson-Nernst-Planck (vPNP) electrodiffusion model. Two extensions of the vPNP system are proposed: the velocity-extended Thermo-Hydrodynamic model (vTHD) and the velocity-extended Electro-Thermal model (vET). Both formulations are based on the principles of conservation of mass, momentum and energy, and collapse into the vPNP model under thermodynamical equilibrium conditions. Upon introducing a suitable one-dimensional geometrical representation of the channel,we discuss appropriate boundary conditions that depend only on effectively accessible measurable quantities. Then, we describe the novel models, the solution map used to iteratively solve them, and the mixed-hybrid flux-conservative stabilized finite element scheme used to discretize the linearized equations. Finally,we successfully apply our computational algorithms to the simulation of two different realistic biological channels: 1) the Gramicidin-A channel considered in [12]; and 2) the bipolar nanofluidic diode considered in [45].

Auxin-driven patterning with unidirectional fluxes

The plant hormone auxin plays an essential role in the patterning of plant structures. Biological hypotheses supported by computational models suggest that auxin may fulfil this role by regulating its own transport, but the plausibility of previously proposed models has been questioned. We applied the notion of unidirectional fluxes and the formalism of Petri nets to show that the key modes of auxin-driven patterning-the formation of convergence points and the formation of canals-can be implemented by biochemically plausible networks, with the fluxes measured by dedicated tally molecules or by efflux and influx carriers themselves. Common elements of these networks include a positive feedback of auxin efflux on the allocation of membrane-bound auxin efflux carriers (PIN proteins), and a modulation of this allocation by auxin in the extracellular space. Auxin concentration in the extracellular space is the only information exchanged by the cells. Canalization patterns are produced when auxin efflux and influx act antagonistically: an increase in auxin influx or concentration in the extracellular space decreases the abundance of efflux carriers in the adjacent segment of the membrane. In contrast, convergence points emerge in networks in which auxin efflux and influx act synergistically. A change in a single reaction rate may result in a dynamic switch between these modes, suggesting plausible molecular implementations of coordinated patterning of organ initials and vascular strands predicted by the dual polarization theory.

Hemichannel composition and electrical synaptic transmission: molecular diversity and its implications for electrical rectification

Unapposed hemichannels (HCs) formed by hexamers of gap junction proteins are now known to be involved in various cellular processes under both physiological and pathological conditions. On the other hand, less is known regarding how differences in the molecular composition of HCs impact electrical synaptic transmission between neurons when they form intercellular heterotypic gap junctions (GJs). Here we review data indicating that molecular differences between apposed HCs at electrical synapses are generally associated with rectification of electrical transmission. Furthermore, this association has been observed at both innexin and connexin (Cx) based electrical synapses. We discuss the possible molecular mechanisms underlying electrical rectification, as well as the potential contribution of intracellular soluble factors to this phenomenon. We conclude that asymmetries in molecular composition and sensitivity to cellular factors of each contributing hemichannel can profoundly influence the transmission of electrical signals, endowing electrical synapses with more complex functional properties.

Critical assessment of OmpF channel selectivity: merging information from different experimental protocols

The ion selectivity of a channel can be quantified in several ways by using different experimental protocols. A wide, mesoscopic channel, the OmpF porin of the outer membrane of E. coli, serves as a case study for comparing and analysing several measures of the channel cation-anion permeability in chlorides of alkali metals (LiCl, NaCl, KCl, CsCl). We show how different insights can be gained and integrated to rationalize the global image of channel selectivity. To this end, reversal potential, channel conductance and bi-ionic potential (two different salts with a common anion on each side of the channel but with the same concentration) experiments are discussed in light of an electrodiffusion model based on the Poisson-Nernst-Planck formalism. Measurements and calculations based on the atomic crystal structure of the channel show that each protocol displays a particular balance between the different sources of selectivity.

Theory for polymer analysis using nanopore-based single-molecule mass spectrometry

Nanometer-scale pores have demonstrated potential for the electrical detection, quantification, and characterization of molecules for biomedical applications and the chemical analysis of polymers. Despite extensive research in the nanopore sensing field, there is a paucity of theoretical models that incorporate the interactions between chemicals (i.e., solute, solvent, analyte, and nanopore). Here, we develop a model that simultaneously describes both the current blockade depth and residence times caused by individual poly(ethylene glycol) (PEG) molecules in a single alpha-hemolysin ion channel. Modeling polymer-cation binding leads to a description of two significant effects: a reduction in the mobile cation concentration inside the pore and an increase in the affinity between the polymer and the pore. The model was used to estimate the free energy of formation for K(+)-PEG inside the nanopore (approximately -49.7 meV) and the free energy of PEG partitioning into the nanopore ( approximately 0.76 meV per ethylene glycol monomer). The results suggest that rational, physical models for the analysis of analyte-nanopore interactions will develop the full potential of nanopore-based sensing for chemical and biological applications.

Nanofluidic technology for biomolecule applications: a critical review

In this review, we present nanofluidic phenomena, particularly as they relate to applications involving analysis of biomolecules within nanofabricated devices. The relevant length scales and physical phenomena that govern biomolecule transport and manipulation within nanofabricated nanofluidic devices are reviewed, the advantages of nanofabricated devices are presented, and relevant applications are cited. Characteristic length scales include the Debye length, the Van der Waals radius, the action distance of hydrogen bonding, the slip length, and macromolecular dimensions. On the basis of the characteristic lengths and related nanofluidic phenomena, a nanofluidic toolbox will be assembled. Nanofluidic phenomena that affect biomolecule behavior within such devices can include ion depletion and enrichment, modified velocity and mobility, permselectivity, steric hindrance, entropy, adsorption, and hydrodynamic interaction. The complex interactions and coupled physics of such phenomena allow for many applications, including biomolecule separation, concentration, reaction/hybridization, sequencing (in the case of DNA) and detection. Examples of devices for such applications will be presented, followed by a discussion of near-term challenges and future thoughts for the field.

Single-file diffusion of confined water inside SWNTs: an NMR study

We report a nuclear magnetic resonance (NMR) study of confined water inside approximately 1.4 nm diameter single-walled carbon nanotubes (SWNTs). We show that the confined water does not freeze even up to 223 K. A pulse field gradient (PFG) NMR method is used to determine the mean squared displacement (MSD) of the water molecules inside the nanotubes at temperatures below 273 K, where the bulk water outside the nanotubes freezes and hence does not contribute to the proton NMR signal. We show that the mean squared displacement varies as the square root of time, predicted for single-file diffusion in a one-dimensional channel. We propose a qualitative understanding of our results based on available molecular dynamics simulations.

Self-consistent analytic solution for the current and the access resistance in open ion channels

A self-consistent analytic approach is introduced for the estimation of the access resistance and the current through an open ion channel for an arbitrary number of species. For an ion current flowing radially inward from infinity to the channel mouth, the Poisson-Boltzmann-Nernst-Planck equations are solved analytically in the bulk with spherical symmetry in three dimensions, by linearization. Within the channel, the Poisson-Nernst-Planck equation is solved analytically in a one-dimensional approximation. An iterative procedure is used to match the two solutions together at the channel mouth in a self-consistent way. It is shown that the current-voltage characteristics obtained are in good quantitative agreement with experimental measurements.

Geometry-induced asymmetric diffusion

Past work has shown that ions can pass through a membrane more readily in one direction than the other. We demonstrate here in a model and an experiment that for a mixture of small and large particles such asymmetric diffusion can arise solely from an asymmetry in the geometry of the pores of the membrane. Our deterministic simulation considers a two-dimensional gas of elastic disks of two sizes diffusing through a membrane, and our laboratory experiment examines the diffusion of glass beads of two sizes through a metal membrane. In both experiment and simulation, the membrane is permeable only to the smaller particles, and the asymmetric pores lead to an asymmetry in the diffusion rates of these particles. The presence of even a small percentage of large particles can clog a membrane, preventing passage of the small particles in one direction while permitting free flow of the small particles in the other direction. The purely geometric kinetic constraints may play a role in common biological contexts such as membrane ion channels.

Poisson–Nernst–Planck Theory Approach to the Calculation of Current Through Biological Ion Channels

The Poisson-Nernst-Planck (PNP) theory of electro-diffusion is reviewed. Techniques for numerical solution of the three-dimensional PNP equations are summarized, and several illustrative applications to ion transport through protein channels are presented. Strengths and weaknesses of the theory are discussed, as well as attempts to improve it via increasingly realistic evaluation of the force acting on each ion due to the protein/membrane environment.

Brownian simulations and unidirectional flux in diffusion

The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics. Existing continuum descriptions of ionic permeation fail to capture the rich phenomenology of the permeation process, so it is therefore necessary to resort to particle simulations. Brownian dynamics (BD) simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time 1/gamma of the Langevin equation. We find that the probability of Brownian trajectories that cross an interface in one direction in unit time Deltat equals that of the probability of the corresponding Langevin trajectories if gammaDeltat=2 . That is, we find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to sqrt[Deltat ] to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.

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.

Conductance and permeability of the residual state of connexin43 gap junction channels

We used cell lines expressing wild-type connexin43 and connexin43 fused with the enhanced green fluorescent protein (Cx43-EGFP) to examine conductance and perm-selectivity of the residual state of Cx43 homotypic and Cx43/Cx43-EGFP heterotypic gap junction channels. Each hemichannel in Cx43 cell-cell channel possesses two gates: a fast gate that closes channels to the residual state and a slow gate that fully closes channels; the transjunctional voltage (V(j)) closes the fast gate in the hemichannel that is on the relatively negative side. Here, we demonstrate macroscopically and at the single-channel level that the I-V relationship of the residual state rectifies, exhibiting higher conductance at higher V(j)s that are negative on the side of gated hemichannel. The degree of rectification increases when Cl(-) is replaced by Asp(-) and decreases when K(+) is replaced by TEA(+). These data are consistent with an increased anionic selectivity of the residual state. The V(j)-gated channel is not permeable to monovalent positively and negatively charged dyes, which are readily permeable through the fully open channel. These data indicate that a narrowing of the channel pore accompanies gating to the residual state. We suggest that the fast gate operates through a conformational change that introduces positive charge at the cytoplasmic vestibule of the gated hemichannel, thereby producing current rectification, increased anionic selectivity, and a narrowing of channel pore that is largely responsible for reducing channel conductance and restricting dye transfer. Consequently, the fast V(j)-sensitive gating mechanism can serve as a selectivity filter, which allows electrical coupling but limits metabolic communication.

Cellular microtransport processes: intercellular, intracellular, and aggregate behavior

Ionic and molecular transfer among cells occurs by a variety of transport processes operative at different length scales. Cell membrane permeability and electrical conductance derive from channel proteins producing pores at the molecular (ultrastructural) scale. Intracellular mobility involves the dynamics of motion through the complex ultrastructure of the cytoplasm. These phenomena unite in the larger-scale (microscopic) process of gross intercellular transfer. When such movement occurs among sufficiently many cells, it in turn begins to reflect their average collective (macroscopic) behavior as bulk tissue. This article surveys selected aspects of intercellular and intracellular transport, with emphasis on detailed mechanistic theory, experimental probes of cellular permeability, and systematic transcendence from small to large length scales.

Different ionic selectivities for connexins 26 and 32 produce rectifying gap junction channels

The functional diversity of gap junction intercellular channels arising from the large number of connexin isoforms is significantly increased by heterotypic interactions between members of this family. This is particularly evident in the rectifying behavior of Cx26/Cx32 heterotypic channels (. Proc. Natl. Acad. Sci. USA. 88:8410-8414). The channel properties responsible for producing the rectifying current observed for Cx26/Cx32 heterotypic gap junction channels were determined in transfected mouse neuroblastoma 2A (N2A) cells. Transfectants revealed maximum unitary conductances (gamma(j)) of 135 pS for Cx26 and 53 pS for Cx32 homotypic channels in 120 mM KCl. Anionic substitution of glutamate for Cl indicated that Cx26 channels favored cations by 2.6:1, whereas Cx32 channels were relatively nonselective with respect to charge. In Cx26/Cx32 heterotypic cell pairs, the macroscopic fast rectification of the current-voltage relationship was fully explained at the single-channel level by a rectifying gamma(j) that increased by a factor of 2.9 as the transjunctional voltage (V(j)) changed from -100 to +100 mV with the Cx26 cell as the positive pole. A model of electrodiffusion of ions through the gap junction pore based on Nernst-Planck equations for ion concentrations and the Poisson equation for the electrical potential within the junction is developed. Selectivity characteristics are ascribed to each hemichannel based on either pore features (treated as uniform along the length of the hemichannel) or entrance effects unique to each connexin. Both analytical GHK approximations and full numerical solutions predict rectifying characteristics for Cx32/Cx26 heterotypic channels, although not to the full extent seen empirically. The model predicts that asymmetries in the conductance/permeability properties of the hemichannels (also cast as Donnan potentials) will produce either an accumulation or a depletion of ions within the channel, depending on voltage polarity, that will result in rectification.

Mechanism of verapamil block of a neuronal delayed rectifier K channel: active form of the blocker and location of its binding domain

1. The mechanism of verapamil block of the delayed rectifier K currents (I K(DR)) in chick dorsal root ganglion (DRG) neurons was investigated using the whole-cell patch clamp configuration. In particular we focused on the location of the blocking site, and the active form (neutral or charged) of verapamil using the permanently charged verapamil analogue D890. 2. Block by D890 displayed similar characteristics to that of verapamil, indicating the same state-dependent nature of block. In contrast with verapamil, D890 was effective only when applied internally, and its block was voltage dependent (136 mV/e-fold change of the on rate). Given that verapamil block is insensitive to voltage (Trequattrini et al., 1998), these observations indicate that verapamil reaches its binding site in the uncharged form, and accesses the binding domain from the cytoplasm. 3. In external K and saturating verapamil we recorded tail currents that did not decay monotonically but showed an initial increase (hook). As these currents can only be observed if verapamil unblock is significantly voltage dependent, it has been suggested (DeCoursey, 1995) that neutral drug is protonated upon binding. We tested this hypothesis by assessing the voltage dependence of the unblock rate from the hooked tail currents for verapamil and D890. 4. The voltage dependence of the off rate of D890, but not of verapamil, was well described by adopting the classical Woodhull (1973) model for ionic blockage of Na channels. The voltage dependence of verapamil off rate was consistent with a kinetic scheme where the bound drug can be protonated with rapid equilibrium, and both charged and neutral verapamil can unbind from the site, but with distinct kinetics and voltage dependencies.

Selectivity and permeation in calcium release channel of cardiac muscle: alkali metal ions

Current was measured from single open channels of the calcium release channel (CRC) of cardiac sarcoplasmic reticulum (over the range +/-180 mV) in pure and mixed solutions (e.g., biionic conditions) of the alkali metal ions Li+, K+, Na+, Rb+, Cs+, ranging in concentration from 25 mM to 2 M. The current-voltage (I-V) relations were analyzed by an extension of the Poisson-Nernst-Planck (PNP) formulation of electrodiffusion, which includes local chemical interaction described by an offset in chemical potential, which likely reflects the difference in dehydration/solvation/rehydration energies in the entry/exit steps of permeation. The theory fits all of the data with few adjustable parameters: the diffusion coefficient of each ion species, the average effective charge distribution on the wall of the pore, and an offset in chemical potential for lithium and sodium ions. In particular, the theory explains the discrepancy between "selectivities" defined by conductance sequence and "selectivities" determined by the permeability ratios (i.e., reversal potentials) in biionic conditions. The extended PNP formulation seems to offer a successful combined treatment of selectivity and permeation. Conductance selectivity in this channel arises mostly from friction: different species of ions have different diffusion coefficients in the channel. Permeability selectivity of an ion is determined by its electrochemical potential gradient and local chemical interaction with the channel. Neither selectivity (in CRC) seems to involve different electrostatic interaction of different ions with the channel protein, even though the ions have widely varying diameters.

Structure-function study on a de novo synthetic hydrophobic ion channel

Ion conduction properties of a de novo synthesized channel, formed from cyclic octa-peptides consisting of four alternate L-alanine (Ala) and N'-acylated 3-aminobenzoic acid (Aba) moieties, were studied in bilayer membranes. The single-channel conductance was 9 pS in symmetrical 500 mM KCl. The channel favored permeation of cations over anions with a permeability ratio (PCl-/PK+) of 0.15. The selectivity sequence among monovalent cations based on permeability ratio (PX+/PK+) fell into an order: NH4+(1.4) > Cs+(1. 1) >/= K+(1.0) > Na+(0.4) >> Li+(0). The conductance-activity relationship of the channel in K+ solutions followed simple Michaelis-Menten kinetics with a half-maximal saturating activity of 8 mM and a maximal conductance of 9 pS. The permeability ratio PNa+/PK+ remained constant ( approximately 0.40) under biionic concentrations from 10 to 500 mM. These results suggests that the channel is a one-ion channel. The pore diameter probed by a set of organic cations was approximately 6 A. The single-channel current was blocked by Ca2+ in a dose-dependent manner that followed a single-site titration curve with a voltage-dependent dissociation constant of 0.6 mM at 100 mV. The electric distance of the binding site for Ca2+ was 0.07 from both entrances of the channel, indicating the presence of two symmetrical binding sites in each vicinity of the channel entrance. Correlations between conduction properties and structural aspects of the channel are discussed in terms of a three-barrier and two-binding-site (3B2S) model of Eyring rate theory. All available structural information supported an idea that the channel was formed from a tail-to-tail associated dimer of the molecule, the pore of which was lined with hydrophobic acyl chains. This is the first report to have made a systematic analysis of ion permeation through a hydrophobic pore.

(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.

A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel

A lattice relaxation algorithm is developed to solve the Poisson-Nernst-Planck (PNP) equations for ion transport through arbitrary three-dimensional volumes. Calculations of systems characterized by simple parallel plate and cylindrical pore geometries are presented in order to calibrate the accuracy of the method. A study of ion transport through gramicidin A dimer is carried out within this PNP framework. Good agreement with experimental measurements is obtained. Strengths and weaknesses of the PNP approach are discussed.

Electrostatics and the ion selectivity of ligand-gated channels

The nicotinic acetylcholine receptor (nAChR) is a cation-selective ion channel that opens in response to acetylcholine binding. The related glycine receptor (GlyR) is anion selective. The pore-lining domain of each protein may be modeled as a bundle of five parallel M2 helices. Models of the pore-lining domains of homopentameric nAChR and GlyR have been used in continuum electrostatics calculations to probe the origins of ion selectivity. Calculated pKA values suggest that "rings" of acidic or basic side chains at the mouths of the nAChR or GlyR M2 helix bundles, respectively, may not be fully ionized. In particular, for the nAChR the ring of glutamate side chains at the extracellular mouth of the pore is predicted to be largely protonated at neutral pH, whereas those glutamate side chains in the intracellular and intermediate rings (at the opposite mouth of the pore) are predicted to be fully ionized. Inclusion of the other domains of each protein represented as an irregular cylindrical tube in which the M2 bundles are embedded suggests that both the M2 helices and the extramembrane domains play significant roles in determining ion selectivity.

Nonlinear and asymmetric open channel characteristics of an ion-selective porin in planar membranes

The open channel characteristics of the bacterial porin Omp32 from Comamonas acidovorans were investigated by means of conductance measurements in planar lipid bilayers of the Montal-Mueller type. Particularly at low salt conditions (< or = 30 mM KCl) Omp32 exhibited some unusual asymmetric and nonlinear functional properties. Current-voltage relationship measurements showed that conductance depends on the orientation of porin molecules and is a nonlinear function of the applied membrane potential. Conductance also depends on the salt concentration in a manner not common to porins and the salt concentration modulates the nonlinearity of conductance-voltage relationships. Omp32 is strongly anion-selective. The nonlinear and asymmetric conductance of the open channel is a new observation in porins.

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.

Permeation through the calcium release channel of cardiac muscle

Current voltage (I-V) relations were measured from the calcium release channel (CRC) of the sarcoplasmic reticulum of cardiac muscle in 12 KCl solutions, symmetrical and asymmetrical, from 25 mM to 2 M. I-V curves are nearly linear, in the voltage range +/- 150 mV approximately 12kT/e, even in asymmetrical solutions, e.g., 2 M // 100 mM. It is awkward to describe straight lines as sums of exponentials in a wide range of solutions and potentials, and so traditional barrier models have difficulty fitting this data. Diffusion theories with constant fields predict curvilinear I-V relations, and so they are also unsatisfactory. The Poisson and Nernst-Planck equations (PNP) form a diffusion theory with variable fields. They fit the data by using adjustable parameters for the diffusion constant of each ion and for the effective density of fixed (i.e., permanent) charge P(x) along the channel's "filter" (7-A diameter, 10 A long). If P(x) is described by just one parameter, independent of x (i.e., P(x) = P0 = -4.2 M), the fits are satisfactory (RMS error/RMS current = 6.4/67), and the estimates of diffusion coefficients are reasonable D(K) = 1.3 x 10(-6) cm2/s, D(Cl) = 3.9 x 10(-6) cm2/s. The CRC seems to have a small selectivity filter with a very high density of permanent charge. This may be a design principle of channels specialized for large flux. The Appendix derives barrier models, and their prefactor, from diffusion theories (with variable fields) and argues that barrier models are poor descriptions of CRCs in particular and open channels in general.

Ion-water and water-water interactions in a gramicidinlike channel: effects due to group polarizability and backbone flexibility

In the gramicidin channel, ionic transport and water transport occur simultaneously. Gramicidin's transport properties are influenced by ionic interactions with both the polypeptide and the channel waters. We present results of molecular dynamics studies on a series of alkali metal ions interacting with a water-filled gramicidinlike channel (a configurationally constrained polyglycine analog) at the dimer junction, in mid-monomer, and near the channel entrance. We investigate details of both short and long range ion-water and water-water correlation; these are notably dependent on the explicit consideration of polarizability and the degree of backbone flexibility. The nature of ion-water and water-water correlations changes as ionic size decreases and these changes may be augmented or attenuated by manipulation of the two parameters under study. Incorporating polarizability generally shortens ion-water distances and enhances ion-induced electrostriction (decreased water-water separations), while simultaneously reducing the long range orientational correlation of the single filing waters within the channel. Increasing flexibility predictably results in a broadening of the distribution of water-water and ion-water separations and contributes to the loss of long range orientational correlations. Both effects are ion specific; Cs+ and Na+ interact with the channel in distinctly different ways, while K+ represents an intermediate case more closely resembling Cs+. Our results demonstrate that incorporation of polarizability in the potential function has significant effects on the properties of channel water and, consequently, on the ionic transport process. While ion-water and water-water distances are decreased due to this feature, thereby fostering longer ranged correlations within the channel, enhanced interactions between water molecules and peptide groups tend to mitigate this effect. Possible implications for the multiple occupancy states of gramicidin and long range information transfer via a single file water chain are considered.

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.

NMR Studies of Single-File Diffusion in Unidimensional Channel Zeolites

Single-file diffusion is the restricted propagation of particles that cannot pass each other. The occurrence of this phenomenon should be reflected by a change in the time dependence of the mean particle displacement in comparison with ordinary diffusion. Although this process is considered to be the rate-controlling mechanism in a large variety of processes, so far no direct evidence of this phenomenon has been provided. Diffusion measurements made with pulsed field gradient nuclear magnetic resonance (NMR) in unidimensional pore systems (zeolites AlPO4-5 and Theta-1) reflect the expected time dependence of single-file diffusion.

A multi-substrate single-file model for ion-coupled transporters

Ion-coupled transporters are simulated by a model that differs from contemporary alternating-access schemes. Beginning with concepts derived from multi-ion pores, the model assumes that substrates (both inorganic ions and small organic molecules) hop a) between the solutions and binding sites and b) between binding sites within a single-file pore. No two substrates can simultaneously occupy the same site. Rate constants for hopping can be increased both a) when substrates in two sites attract each other into a vacant site between them and b) when substrates in adjacent sites repel each other. Hopping rate constants for charged substrates are also modified by the membrane field. For a three-site model, simulated annealing yields parameters to fit steady-state measurements of flux coupling, transport-associated currents, and charge movements for the GABA transporter GAT1. The model then accounts for some GAT1 kinetic data as well. The model also yields parameters that describe the available data for the rat 5-HT transporter and for the rabbit Na(+)-glucose transporter. The simulations show that coupled fluxes and other aspects of ion transport can be explained by a model that includes local substrate-substrate interactions but no explicit global conformational changes.

Hydrodynamic model of temperature change in open ionic channels

Most theories of open ionic channels ignore heat generated by current flow, but that heat is known to be significant when analogous currents flow in semiconductors, so a generalization of the Poisson-Nernst-Planck theory of channels, called the hydrodynamic model, is needed. The hydrodynamic theory is a combination of the Poisson and Euler field equations of electrostatics and fluid dynamics, conservation laws that describe diffusive and convective flow of mass, heat, and charge (i.e., current), and their coupling. That is to say, it is a kinetic theory of solute and solvent flow, allowing heat and current flow as well, taking into account density changes, temperature changes, and electrical potential gradients. We integrate the equations with an essentially nonoscillatory shock-capturing numerical scheme previously shown to be stable and accurate. Our calculations show that 1) a significant amount of electrical energy is exchanged with the permeating ions; 2) the local temperature of the ions rises some tens of degrees, and this temperature rise significantly alters for ionic flux in a channel 25 A long, such as gramicidin-A; and 3) a critical parameter, called the saturation velocity, determines whether ionic motion is overdamped (Poisson-Nernst-Planck theory), is an intermediate regime (called the adiabatic approximation in semiconductor theory), or is altogether unrestricted (requiring the full hydrodynamic model). It seems that significant temperature changes are likely to accompany current flow in the open ionic channel.

Charge selectivity of the designed uncharged peptide ion channel Ac-(LSSLLSL)3-CONH2

Charge selectivity in ion channel proteins is not fully understood. We have studied charge selectivity in a simple model system without charged groups, in which an amphiphilic helical peptide, Ac-(Leu-Ser-Ser-Leu-Leu-Ser-Leu)3-CONH2, forms ion channels across an uncharged phospholipid membrane. We find these channels to conduct both K+ and Cl-, with a permeability ratio (based on reversal potentials) that depends on the direction of the KCl concentration gradient across the membrane. The channel shows high selectivity for K+ when [KCl] is lowered on the side of the membrane that is held at a positive potential (the putative C-terminal side), but only modest K+ selectivity when [KCl] is lowered on the opposite side (the putative N-terminal side). Neither a simple Nernst-Planck electrodiffusion model including screening of the helix dipole potential, nor a multi-ion, state transition model allowing simultaneous cation and anion occupancy of the channel can satisfactorily fit the current-voltage curves over the full range of experimental conditions. However, the C-side/N-side dilution asymmetry in reversal potentials can be simulated with either type of model.

Sodium in gramicidin: an example of a permion

The reaction path and free energy profile of Na+ were computed in the interior of the channel protein gramicidin, with the program MOIL. Gramicidin was represented in atomic detail, but surrounding water and lipid molecules were not included. Thus, only short range interactions were investigated. The permeation path of the ion was an irregular spiral, far from a straight line. Permeation cannot be described by motions of a single Na+ ion. The minimal energy path includes significant motion of water and channel atoms as well as motion of the permeating ion. We think of permeation as motion of a permion, a quasi-particle that includes the many body character of the permeation process, comparable with quasi-particles like holes, phonons, and electrons of solid-state physics. Na+ is accompanied by a plug of water molecules, and motions of water, Na+, and the atoms of gramicidin are highly correlated. The permion moves like a linear polymer made of waters and ion linked and moving coherently along a zigzag line, following the reptation mechanism of polymer transport. The effective mass, free energy, and memory kernel (of the integral describing time-dependent friction) of short range interactions were calculated. The effective mass of the permion (properly normalized) is much less than Na+. Friction varies substantially along the path. The free energy profile has two deep minima and several maxima. In certain regions, the dominant motions along the reaction path are those of the channel protein, not the permeating ion: there, ion waits while the other atoms move. At these waiting sites, the permion's motion along the reaction path is a displacement of the atoms of gramicidin that prepare the way for the Na+ ion.

Transport properties of single-file pores with two conformational states

Complex facilitative membrane transporters of specific ligands may operate via inner channels subject to conformational transitions. To describe some properties of these systems, we introduce here a kinetic model of coupled transport of two species, L and w, through a two-conformational pore. The basic assumptions of the model are: a) single-file of, at most, n molecules inside the channel; b) each pore state is open to one of the compartments only; c) there is at most only one vacancy per pore; d) inside the channel, a molecule of L occupies the same positions as a molecule of w; and e) there is at most only one molecule of L per pore. We develop a general representation of the kinetic diagram of the model that is formally similar to the one used to describe one-vacancy transport through a one-conformational single-file pore. In many cases of biological importance, L could be a hydrophilic (ionic or nonionic) ligand and w could be water. The model also finds application to describe solute (w) transport under saturation conditions. In this latter case, L would be another solute, or a tracer of w. We derive steady-state expressions for the fluxes of L and w, and for the permeability coefficients. The main results obtained from the analysis of the model are the following. 1) Under the condition of equilibrium of w, the expression derived for the flux of L is formally indistinguishable from the one obtainable from a standard four-state model of ligand transport mediated by a two-conformational transporter. 2) When L is a tracer of w, we can derive an expression for the ratio between the main isotope and tracer permeability coefficients (Pw/Pd). We find that the near-equilibrium permeability ratio satisfies (n - 1) < or = (Pw/Pd)eq < or = n, a result previously derived for the one-conformational, single-file pore for the case that n > or = 2. 3) The kinetic model studied here represents a generalization of the carrier concept. In fact, for the case that n = 1 (corresponding to the classical single-occupancy carrier), the near-equilibrium permeability ratio satisfies 0 < or = (Pw/Pd)eq < or = 1, which is characteristic of a carrier performing exchange-diffusion.