George Eisenman

Ionic selectivity revisited: The role of kinetic and equilibrium processes in ion permeation through channels

The intent of this paper is to bring up to date an earlier theory of equilibrium selectivity [23, 24], with emphasis on making it applicable to the permeation of membrane channels and carriers, which involve kinetic considerations [8, 56]. We will review critically and unify a number of conceptual advances that have occurred since 1961, particularly the recognition by Hille [56] of how the same energetic principles that describe equilibrium selectivity of binding sites also apply to the peaks of the energy barriers (the so-called selectivity filters) involved in the kinetics of permeation. In the process, the precision of the term selectivity will gradually be increased. We will start with the intuitively simple comparison of effects of one ion versus another. Then we will proceed through more theoretically based classical concepts such as permeability ratios, conductance ratios, and ratios of binding affinities, all of which can be interrelated quite directly in sufficiently simple channels. It will become clear that these concepts lose their crispness and usefulness in multi-barrier channels as a consequence of asymmetry and/or possible multiple occupancy. Our formulation is influenced by a number of useful concepts from classical rate theory [40, 48, 126, 128], where the channel is viewed as having a static energy profile for a given state of ion occupancy; but we will also consider the consequences of viewing the channel as a dynamic structure with a fluctuating energy profile [83, 851. Besides presenting a topical review of selectivity, we have found it necessary to consider two subjects which are so new that they have received little attention as yet in a selectivity context. These are the effects in multi-barrier channels of asymmetry and multiple occupancy. This is because a number of classical concepts, and even ways of thinking about selectivity, are implicitly conditioned by conclusions that have been reached from considerations heretofore largely restricted to channels occupiable by no more than one ion at a time, or to channels with only one significantly rate determining barrier 1, or to channels which are symmetrical. To exemplify the effects of channel asymmetry, we examine in a simple two-barrier one-site model the consequences of asymmetry for reversal potential selectivity. To exemplify the effects of multiple rate-determining barriers, we also examine some implications of the energy profile which can be inferred for the gramicidin channel. In the process of presenting this material, we describe certain relationships between selectivity as seen in different phenomena such as conductance, reversal potential, and binding at least in systems which are simple enough for these classical concepts to retain their utility. This should lead to an understanding of why different measures of selectivity sometimes lead to different apparent selectivities, even in very simple channels. We also will define some guidelines for the applicability of these classical concepts and give some suggestions as to how to define selectivity when they fail, as fail they must, in those biological channels which are asymmetric and/or occupied by more than one ion at a time.

Freezing and Melting of Lipid Bilayers and the Mode of Action of Nonactin, Valinomycin, and Gramicidin

An abrupt loss of effectiveness of the presumed carriers, nonactin and valinomycin, in mediating ion conductance occurred at the same temperature as the membrane fluidity, judged visually, was lost. By contrast, the effects of the presumed channel-former, gramicidin, were the same on solid and liquid membranes. Taken together, these findings imply that freezing the membrane primarily reduces the mobility of these antibiotics with little effect on their solubility.

The effects of the macrotetralide actin antibiotics on the electrical properties of phospholipid bilayer membranes.

SummaryThis paper, the last in a series of three, characterizes the electrical properties of phospholipid bilayer membranes exposed to aqueous solutions containing nonactin, monactin, dinactin, and trinactin and Li+, Na+, K+, Rb+, Cs+, and NH4+ ions. Not only are both the membrane resistance at zero current and the membrane potential at zero current found to depend on the aqueous concentrations of antibiotic and ions in the manner expected from the theory of the first paper, but also these measurements are demonstrated to be related to each other in the manner required by this theory for “neutral carriers”. To verify that these antibiotics indeed are free to move as carriers of cations, cholesterol was added to the lipid to increase the “viscosity” of the interior of the membrane. Cholesterol decreased by several orders of magnitude the ability of the macrotetralide antibiotics to lower the membrane resistance; nevertheless, the permeability ratios and conductance ratios remained exactly the same as in cholesterolfree membranes. These findings are expected for the “carrier” mechanism postulated in the first paper and serve to verify it. Lastly, the observed effects of nonactin, monactin, dinactin, and trinactin on bilayers are compared with those predicted in the preceding paper from the salt-extraction equilibrium constants measured there; and a close agreement is found. These results show that the theory of the first paper satisfactorily predicts the effects of the macrotetralide actin antibiotics on the electrical properties of phospholipid bilayer membranes, using only the thermodynamic constants measured in the second paper. It therefore seems reasonable to conclude that these antibiotics produce their characteristic effects on membranes by solubilizing cations therein as mobile positively charged complexes.

The Effects of the Macrotetralide Actin Antibiotics on the Equilibrium Extraction of Alkali Metal Salts into Organic Solvents

SummaryIn order to clarify the mechanism by which neutral molecules such as the macrotetralide actin antibiotics make phospholipid bilayer membranes selectively permeable to cations, we have studied, both theoretically and experimentally, the extraction by these antibiotics of cations from aqueous solutions into organic solvents. The experiments involve merely shaking an organic solvent phase containing the antibiotic with aqueous solutions containing various cationic salts of a lipid-soluble colored anion. The intensity of color of the organic phase is then measured spectrophotometrically to indicate how much salt has been extracted. From such measurements of the equilibrium extraction of picrate and dinitrophenolate salts of Li, Na, K, Rb, Cs, and NH4 into n-hexane, dichloromethane, and hexane-dichloromethane mixtures, we have verified that the chemical reactions are as simple as previously postulated, at least for nonactin, monactin, dinactin, and trinactin. The equilibrium constant for the extraction of each cation by a given macrotetralide actin antibiotic was also found to be measurable with sufficient precision for meaningful differences among the members of this series of antibiotics to be detected. It is noteworthy that the ratios of selectivities among the various cations were discovered to be characteristic of a given antibiotic and to be completely independent of the solvent used. This finding and others reported here indicate that the size and shape of the complex formed between the macrotetralide and a given cation is the same, regardless of the species of cation bound. For such “isosteric” complexes, notable simplifications of the theory become possible which enable us to predict not only the electrical properties of a membrane made of the same solvent and having the thinness of the phospholipid bilayer but also, and more importantly, the electrical properties of the phospholipid bilayer membrane itself. These predictions will be compared with experimental data for phospholipid bilayer membranes in the accompanying paper.

The Steady-State Properties of an Ion Exchange Membrane with Mobile Sites

A study of the properties of the steady states of a system composed of two solutions separated by an ion exchange membrane having mobile sites is presented. It is assumed that the membrane is impermeable to coions; the solutions contain no more than two species of counterions, both of the same valence; and no flow of bulk solution occurs. Assuming that all ions are completely dissociated, behave ideally, and have constant mobilities throughout the membrane, explicit expressions are derived for the steady states of the electric current, individual fluxes, and concentration profiles as functions of the compositions of the solutions and of the difference of electric potential between them. The derived expressions are compared with those for an ion exchange membrane having fixed sites; and it is found that the expressions of certain quantities, such as the difference of electric potential between the two solutions for zero current or the ratio of the fluxes of the counterions as functions of the external parameters of the system, are the same for both types of membranes. On the other hand, differences in the behavior of the two types of membranes are found from other expressions-for example, the current-voltage relationship. In the mobile site ion exchanger the current asymptotically approaches finite limiting values for high positive and negative voltages while in the fixed site ion exchanger it is the conductance which approaches finite limiting values.

Ionic probes of membrane structures.

In this paper we have examined the possibility of identifying those membrane structural variables (polar head groups and the nature of hydrocarbon tails) that modulate membrane ionic permeability. Altering the bilayer lipid composition produces variations in physical parameters (surface potential, partition coefficient, and mobility) governing the conductance mediated by neutral carriers of anions and cations. Specifically, the effects of the charged polar head groups are shown to be understandable in terms of the surface potential they produce through the formation of a diffuse double layer, whereas the effects of the viscosity may be demonstrated by “freezing” the membrane. The effects of membrane composition on membrane conductance are illustrated by a third, less well understood, example of how cholesterol alters bilayer conductances. The results indicate the possibility of using positive and negative permeant species as probes of membrane structures.

Electrical potentials and ionic fluxes in ion exchangers: I. “n type” non-ideal systems with zero current

In this paper a derivation, based on the Nernst-Planck flux equations extended for non-ideal behavior, is given for the electrical potentials and cationic fluxes occurring across a one-phase ion exchange system separating two aqueous solutions. It is assumed that the activities of the cations in the exchanger are proportional to thenth power of their concentrations, a relationship of broad empirical validity. This paper deals only with the case of the quasi-stationary state in which no net electrical current flows. The profles of the cationic concentrations and of the diffusion potential in the exchanger are found to be, in general, exponential and linear respectively. An equation is derived for the stationary-state potential of an ion exchange phase interposed between two aqueous solutions containing mixtures of two cations. The potential selectivity constant of this equation is found to equal the product of the ion exchange equilibrium constant and thenth power of the mobility ratio.

Calculation of ion currents from energy profiles and energy profiles from ion currents in multibarrier, multisite, multioccupancy channel model.

Publisher Summary This chapter describes the theory of multibarrier channels, illustrating the example of how to find the parameters of a model channel using our programs. Barrier models describe the ion transport through a channel as a series of ion movements from one binding site of the channel to another binding site. The channel can have several binding sites and more than one site can be occupied at any given time. The chapter describes AJUSTE, which is a nonlinear curve-fitting procedure based on the Gauss-Newton method. AJUSTE includes ways of fixing or adjusting any of the parameters of the model and allows constraints to be defined on the allowable values of the parameters. It gives information on the errors of the parameter estimations and parameter correlations. The program is simple, fast, and has provisions to escape from local minima.

Theory for carrier-mediated zero-current conductance of bilayers extended to allow for nonequilibrium of interfacial reactions, spatially dependent mobilities and barrier shape

SummaryA generalized form of the electrodiffusion equation, allowing for any shape of symmetrical energy barrier and any spatial dependence of the diffusion coefficient, is used to deduce theoretically the carrier-mediated conductance for thin (e.g., bilayer) membranes in the limit of low applied current. Both the Nernst-Planck and the Eyring single-barrier treatments are special cases of this more general approach, which allows for the effect of non-uniform properties of the lipid and non-uniform profiles of the forces acting within the membrane interior. Two independent mechanisms for ions to cross the membrane-solution interfaces are considered; namely, (1) the reaction at the interface between ions from solution and carriers from the membrane, and (2) the partition across the interfaces of complexes already formed in the solution. The rates of these reactions are taken into account using the rate equations of chemical kinetics; and the Poisson-Boltzmann equation is integrated in the aqueous solutions to evaluate the effect of charged polar head groups of the lipid. The analysis leads to an expression for the conductance, which, in the approximation of constant field, is an explicit function of such experimentally variable parameters as the concentrations and types of permeant ions and carriers in the aqueous phases, the total ionic strength and the nature of the polar head groups of the lipid. The functional relationship observable in an unknown membrane can, in principle, enable one to deduce such information as the mechanism of ion permeation across the interfaces, the magnitude of the surface charge, and the degree of ion-carrier complexation in the aqueous solutions.

Gramicidin as an example of a single-filing ionic channel.

The cationic channel formed by the uncharged polypeptide backbone of the gramicidin A dimer is a simple and well characterizable prototype for the less easily studied types of ion-selective channels of the cell membrane. Excitable membranes are known to contain several types of ionic channels, each of which has come to be recogmzed’ to be a macromolecular pore capable of passing millions of ions per second and yet narrow enough to exhibit some selectivity as to the charge and size of the ions allowed to pass as well as to show signs of “singlefiling*’.’-3 A number of important conceptions about the general properties of channels have been confirmed by experimental findings in gramicidin A and by the theory that is developing to account for For example, recent findings on gramicidin channels indicate that such channels are able to contain several interacting permeant ions simultaneo~sly‘*’~.~ and are also SQ narrow that ions and water molecules are not permitted to pass each other?t.28*29 This supports the current view as to the likelihood of such behavior in the Na+ and K+ channels of The findings, in roughly chronological sequence, for excitable membranes and for gramicidin, are: single-filing flux coupling,20*2t concentrationdependent permeability ratio^.^-^^ block in mixture of cations’. ’% and coupling between ion-flux and water flu^.^*^ The conductanceconcentration behavior for the gramicidin channel in single salts also indicates the existence of at least two classes of sites with different binding constants for the group la cations: t % Here we will provide further evidence for multiple occupancy from the changes in shape that occur in the current-voltage relationshp of single gramicidin channels when the ionic concentration is changed. Finally, although the highly selective H+ permeation of the gramicidin channel has long been k n ~ w n : * ~ * ~ new, and surprising, data on the H+ permeation properties of the gramicidin channel, which we will present here, indicate that this channel may provide a much more %*