Stochastic theory of ion movement in channels with single-ion occupancy. Application to sodium permeation of gramicidin channels.

The electrodiffusion equations were solved for the one-ion channel both by the analytical method due to Levitt and also by Brownian dynamic simulations. For both types of calculations equilibration of ion distribution between the bath and the ends of the channel was assumed. Potential profiles were found that give good fits to published data on Na+ permeation of gramicidin channels. The data were best fit by profiles that have no relative energy maximum at the mouth of the channel. This finding suggests that alignment of waters or channel charged groups inside the channel in response to an ion's approach may provide an energetically favorable situation for entry sufficient to overcome the energy required for removing bulk waters of hydration. An alternative possibility is that the barrier to ion entry is situated outside the region restricted to single-ion occupancy. Replacement of valine with more polar amino acids at the No. 1 location was found to correspond to a deepening of the potential minima near the channel mouths, an increase in height of the central barrier to ion translocation across the channel, and possibly a reduction in the mobility of the ion-water complex in the channel. The Levitt theory was extended to calculate passage times for ions to cross the channel and the blocking effects of ions that entered the channel but didn't cross. These quantities were also calculated by the Brownian dynamics method.     

Stochastic models for ion channels: introduction and bibliography.

This paper provides an introduction to and overview of the use of stochastic models and statistical analysis in the study of ion channels in cell membranes. An extensive bibliography is included.     

Tests of continuum theories as models of ion channels. II. Poisson-Nernst-Planck theory versus brownian dynamics

We test the validity of the mean-field approximation in Poisson-Nernst-Planck theory by contrasting its predictions with those of Brownian dynamics simulations in schematic cylindrical channels and in a realistic potassium channel. Equivalence of the two theories in bulk situations is demonstrated in a control study. In simple cylindrical channels, considerable differences are found between the two theories with regard to the concentration profiles in the channel and its conductance properties. These differences are at a maximum in narrow channels with a radius smaller than the Debye length and diminish with increasing radius. Convergence occurs when the channel radius is over 2 Debye lengths. These tests unequivocally demonstrate that the mean-field approximation in the Poisson-Nernst-Planck theory breaks down in narrow ion channels that have radii smaller than the Debye length.     

Statistical mechanical equilibrium theory of selective ion channels.

A rigorous statistical mechanical formulation of the equilibrium properties of selective ion channels is developed, incorporating the influence of the membrane potential, multiple occupancy, and saturation effects. The theory provides a framework for discussing familiar quantities and concepts in the context of detailed microscopic models. Statistical mechanical expressions for the free energy profile along the channel axis, the cross-sectional area of the pore, and probability of occupancy are given and discussed. In particular, the influence of the membrane voltage, the significance of the electric distance, and traditional assumptions concerning the linearity of the membrane electric field along the channel axis are examined. Important findings are: 1) the equilibrium probabilities of occupancy of multiply occupied channels have the familiar algebraic form of saturation properties which is obtained from kinetic models with discrete states of denumerable ion occupancy (although this does not prove the existence of specific binding sites; 2) the total free energy profile of an ion along the channel axis can be separated into an intrinsic ion-pore free energy potential of mean force, independent of the transmembrane potential, and other contributions that arise from the interfacial polarization; 3) the transmembrane potential calculated numerically for a detailed atomic configuration of the gramicidin A channel embedded in a bilayer membrane with explicit lipid molecules is shown to be closely linear over a distance of 25 A along the channel axis. Therefore, the present analysis provides some support for the constant membrane potential field approximation, a concept that has played a central role in the interpretation of flux data based on traditional models of ion permeation. It is hoped that this formulation will provide a sound physical basis for developing nonequilibrium theories of ion transport in selective biological channels.     

Electrostatic modeling of ion pores. II. Effects attributable to the membrane dipole potential.

This paper presents calculations of the shielded dipole potential in the interior of a pore piercing a lipid membrane that is at a potential V0 with respect to the aqueous solution. Except in the case of long narrow pores, there is substantial shielding of the membrane dipole potential. The associated dipole field never extends a significant distance into the aqueous region. The fact that the single-channel conductance of gramicidin B is only twice as large in glyceryl monooleate membranes as in phosphatidyl choline (PC) membranes, even though PC is approximately 120 mV more positive with respect to water, is interpreted in terms of the potential energy profile calculated for a gramicidin-like channel. It is demonstrated that the membrane dipole potential can significantly affect channel conductance only if the pore is narrow and if the peak in the potential energy profile occurs in the pore interior.     

Quantitation of hydrogen ion and potential gradients in gastric plasma membrane vesicles.

The ATP-dependent uptake of H+ by hog gastric parietal cell vesicles was quantitated by using the pH indicator dyes bromcresol green and malachite green, the weak bases, aminopyrine and 9-aminoacridine, and the pH electrode. A K+-dependent H+ uptake was found, with a significant difference between the quantity of H+ disappearing from the medium (deltaHo) and the quantity appearing inside the vesicle (deltaHi). 9-Aminoacridine gave a lower value for the deltaHi than any of the other probes. Probes of potential such as diethyloxadicarbocyanine or oxonol dyes showed that only secondary diffusion potentials occurred during H+ uptake and that the cationic dyes in the presence of protonophores could also be used to quantitate H+ uptake. The potential in the presence of protonophore indicated a deltaHi greater than that found with the other probes. Binding sites for acridine orange were generated either by ATP or an artificial pH gradient and corresponded to the deltaHi indicated by aminopyrine. SCN- (30mM) only partially inhibited the H+ gradient, and this, coupled with the failure to detect the physiological deltapH of 6.6, indicated that these vesicles may be an incomplete model of gastric acid secretion.     

Reconstitution of vacuolar ion channels into planar lipid bilayers.

Vacuolar ion channels were characterized after reconstitution into planar lipid bilayers. (1) Channel activity was observed after incorporation of tonoplast-enriched microsomal membranes, purified tonoplast membranes or of solubilized tonoplast proteins. (2) Channels of varying single-channel conductances were detected after reconstitution. In symmetrical 100 mmol l-1 KCl, conductances between 1 and 110 pS were frequently measured; the largest number of independent reconstitution events was seen for single-channel conductances of 16-25 pS (28 experiments), 30-42 pS (26), 49-56 pS (15) and 64-81 pS (15). Channel current usually increased linearly with voltage. (3) In asymmetrical solutions, cation-, non-selective and, for the first time for the tonoplast, anion-selective channels were detected. Ca(2+)-dependent regulation of channel opening was not observed in our reconstitution system. (4) Permeability was also observed for Cl-, NO3-, SO4(2-) and phosphate. (5) After fractionation of tonoplast proteins by size exclusion chromatography, ion channel activity was recovered in specific fractions. (6) Some of these fractions catalyzed sulfate transport after reconstitution into liposomes. The results suggest that different channels are active at the tonoplast membrane at a larger number than has been concluded from previous work.     

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.     

Excitation of skinned muscle fibers by imposed ion gradients. IV. Effects of stretch and perchlorate ion

Depolarizing ion gradients stimulate 45Ca release in skeletal muscle fibers skinned by microdissection. Several lines of indirect evidence suggest that sealed transverse (T) tubules rather than sarcoplasmic reticulum (SR) are the locus of such stimulatory depolarization. Two implications of this hypothesis were tested. (a) A requirement for signal transmission was evaluated from the stimulation of 45Ca efflux in fibers that had been highly stretched, an intervention that can impair the electrical stimulation of intact fibers. Length was increased over approximately 95-115 s, after loading with 45Ca and rinsing at normal length; prestimulus 45Ca loss due to stretch itself was very small. In the first study, stimulation of 45Ca release by KCl replacement of K propionate was inhibited completely in fibers stretched to twice slack length, compared with fibers at 1.05-1.1 times slack length. Identical protocols did not alter 45Ca release stimulated by caffeine or Mg2+ reduction, implying that SR Ca release per se was fully functional and inhibition was selective for a preceding step in ionic stimulation. In a second study, stimulation by choline Cl replacement of K methanesulfonate, at constant [K+] [Cl-] product, was inhibited strongly; total 45Ca release decreased 69%, and stimulation above control loss decreased 78%, in segments stretched to twice the length at which sarcomere spacing had been 2.2 micron, compared with paired controls from the same fibers kept at 2.3 micron. (b) Perchlorate potentiation of T tubule activation was evaluated in fibers stimulated at constant [K+] [Cl-] at normal length (2.3 micron); this anion shifts the voltage dependence of intramembrane charge movement and contractile activation in intact fibers. Perchlorate (8 mM) potentiated both submaximal stimulation of Ca2+-dependent 45Ca release by partial choline Cl replacement of K methanesulfonate and the small Ca2+-insensitive 45Ca efflux component stimulated by nearly full replacement in the presence of 5 mM EGTA. These results provide independent support for the hypothesis that the T tubules are the locus of stimulation by depolarizing ion gradients, with junctional transmission of this signal causing SR 45Ca release.     

Comparison of Nernst-Planck and reaction rate models for multiply occupied channels.

The Nernst-Planck continuum equation for a channel that can be occupied by at most two ions is solved for two different physical cases. The first case is for the assumption that the water and ion cannot get around each other anywhere in the channel, so that if there are two ions in the channel the distance between them is fixed by the number of water molecules between them. The second case is for the assumption that there are regions at he ends of the channel where the ions and water can get around each other. For these two cases, the validity of the simple two-site reaction-rate approximation when there is a continuously varying central energy barrier was evaluated by comparing it with the exact Nernst-Planck solution. For the first continuum case, the kinetics for the continuum and reaction-rate models are nearly identical. For the second case, the agreement depends on the strength of the ion-ion interaction energy. For a low interaction energy (large channel diameter) a high ion concentrations, there is a large difference in the flux as a function of voltage for the two models-with the continuum flux becoming more than four times larger at 250 mV. Simple analytical expressions are derived for the two-ion continuum channel for the case where the ends are in equilibrium with the bulk solution and for the case where ion mobility becomes zero when there are two ions in the channel. The implications of these results for biological channels are discussed.     

Ion flow in the bath and flux interactions between channels.

We present an exact solution to the linearized Nernst-Planck-Poisson equation for spherically symmetric current flow. This solution differs from Levitt's solution (Levitt, D. G. 1992. Biophys. J., Eq. A5) by its dependence on an additional parameter, which is equal to the net ion flux for monovalent ion-selective channels. For ion-selective channels, this solution may provide better boundary conditions to modelling the flow in the channel pore itself, although only at low salt concentrations. We use the solution to estimate the effects of flux interaction between closely packed channels.     

Tracing the roots of ion channels.

Two sets of recent findings draw our attention to questions concerning the origin of ion channels. First, there is sequence similarity among five classes of channels: voltage-gated channels, a putative Ca(2+)-activated K+ channel, cyclic nucleotide-gated cation channels, a putative Ca2+ channel for phosphoinositide-mediated Ca2+ entry, and a plant K+ channel/transporter. Like voltage-gated K+ channels, the most recently identified members of the superfamily share the basic design of one set of six potential membrane-spanning segments plus the H5 sequence; as such, they may resemble more closely the ancestral channel, which is likely to predate the separation of the animal and plant kingdoms. Second, several members of the ABC superfamily function as ion channels, even though they were previously known as transporters or enzymes. Did some ancestral enzymes subsequently acquire channel/transporter function? Or could it be the other way around? Aside from evolutionary considerations, enzymes and ion channels can no longer be treated as separate and nonoverlapping groups of proteins. When one molecule exhibits both functions, there are interesting mechanistic questions: How might the enzyme activity such as ATP hydrolysis be coupled to activation/regulation of the intrinsic channel activity? How might interactions between the permeant ions and the channel pore in turn regulate the enzymatic function of the same molecule? It seems possible that the latter is an extension of the observed coupling between permeant ions and the gating machinery of an ion channel (Swenson and Armstrong, 1981). Finally, the potential cross-regulation between channel activity and enzyme activity within the same molecule offers many intriguing possibilities for the integration of different cellular functions.     

Acetylcholinesterase in singly and multiply innervated muscles of normal and dystrophic chickens. II. Effects of denervation.

Neural regulation of mature normal fast twitch muscle of the chicken suppresses high activity, extrajunctional localization, and isozyme forms of acetylcholinesterase (AChE) characteristic of embryonic, denervated and dystrophic muscle. Normal adult slow tonic muscle ofthe chicken retains intermediate levels of activity and embryonic isozyme forms but not extrajunctional activity; it is not affected by muscular dystrophy. The hypothesis that neural regulation of the AChE system is lacking in slow tonic muscle and thus not affected by dystrophy was tested by denervating the fast twitch posterior latissimus dorsi and slow tonic anterior latissimus dorsi muscles of normal and dystrophic chickens. Extrajunctional AChE activity and embryonic isozyme forms increased, then declined, in both muscles. The results suggest that ocntrol of AChE is qualitatively similar in slow tonic and fast twitch muscle of the chicken.     

Surface potential of phosphatidylserine monolayers II. Divalent and monovalent ion binding

Ion binding constants for phosphatidylserine membranes have been derived from the variation of the surface potential of phosphatidylserine monolayers with divalent cation concentrations in the presence of various monovalent salts in the aqueous subphase. The observed surface potential data for the monolayers, analyzed by use of the Gouy-Chapman diffuse potential theory, together with a simple binding reaction formula, yield, for Ca²⁺, Mg²⁺, Na⁺ and (Me)₄N⁺ binding constant values of 30 M^?1, 10 M^?1, 0.6 M^?1 and 0.05 M^?1, respectively. The effect of pH on surface potential of phosphatidylserine monolayers was found to be dependent upon ionic species other than H⁺ in the subphase solution. The distinction between apparent and intrinsic dissociation constants of H⁺ for biomolecules was made in terms of ion binding due to other ions at the same site as for H⁺ in biomolecules.     

Receptor classes and the transmitter-gated ion channels.

Transmitter-gated channels, which can be selective for cations or for anions, form an important class among the membrane receptors responsible for signal transduction. Thirteen principal types of these channels can now be recognized and most of these are available for analysis in recombinant form. It is instructive to contrast their characteristic structural features with those of the two other primary classes of the signal-transducing receptors of membranes.     

Genesis of the monophasic action potential: role of interstitial resistance and boundary gradients

The extracellular potential at the site of a mechanical deformation has been shown to resemble the underlying transmembrane action potential, providing a minimally invasive way to access membrane dynamics. The biophysical factors underlying the genesis of this signal, however, are still poorly understood. With the use of data from a recent experimental study in a murine heart, a three-dimensional anisotropic bidomain model of the mouse ventricular free wall was developed to study the currents and potentials resulting from the application of a point mechanical load on cardiac tissue. The applied pressure is assumed to open nonspecific pressure-sensitive channels depolarizing the membrane, leading to monophasic currents at the electrode edge that give rise to the monophasic action potential (MAP). The results show that the magnitude and the time course of the MAP are reproduced only for certain combinations of local or global intracellular and interstitial resistances that form a resting tissue length constant that, if applied over the entire domain, is smaller than that required to match the wave speed. The results suggest that the application of pressure not only causes local depolarization but also changes local tissue properties, both of which appear to play a critical role in the genesis of the MAP.     

Distribution of electric potential and ion transport in the hypocotyl of Vigna sesquipedalis II. Axial potential difference

Vital staining with pH indicator dyes made it possible to identifythe xylem with the specific channel A, reported previously,through which hydrogen ions flow generating a resting potentialdifference along the germ axis as their diffusion potential.The distribution of K+ concentration within this channel showedno similarity to electric potential distribution, in contrastto the distribution of H+. The axial P.D. between both ends of a segment cut from a hypocotylresponded reversibly to the change in O₂ tension of the surroundinggas phase. After air had been quickly replaced by N₂, a lagperiod appeared before the sudden potential drop took place.The lag period () was largely dependent on temperature. Apparentactivation energy of the process characterized by 1/ was 18Kcal/mole between 14–30?C, approximately equal to thatof the O₂-uptake within the same temperature range. The relationbetween O₂ concentration and the maximum rate of recovery ofP.D. from anoxia was of the Michaelis-Menten type; the apparentKm was calculated as 2.1 ? 10^–5M O₂ being of the sameorder as that of cytochrome oxidase in higher plants. The O₂-uptakerate "per unit of hypocotyl length" showed a distinct maximumin the elongating region where the axial distribution of bothelectric potential and pH within channel A had their minimums. (Received July 21, 1972; )