Tests of the carrier model for ion transport by nonactin and trinactin.

The fluxes of K+ and NH+4 carried by nonactin and trinactin across thin lipid membranes have been measured as functions of ion activity, electric potential and time. In agreement with the predictions of a version of the carrier model in common use, the shape of the initial current-voltage relation is independent of the activity of the electrolyte, alpha-i, while the ratio of the initial conductance, G-o, to the steady-state conductance, G infinity, increases according to G-o/G infinity equals const1+const2 times alpha-i. For trinactin the data presented allow the estimation of the rate constants of the carrier process (in the limit of zero potential) in a manner which does not assume any particular variation with potential for the constants. Using empirically determined functions of potential, a complete set of values is also available for nonactin. The curve fitting which is necessary is described in the following paper. The data presently available for valinomycin are sufficient neither to test the model nor to determine a complete set of constants.     

Steady-state ion transport by nonactin and trinactin

The fluxes of K⁺ and NH₄⁺ carried by nonactin and trinactin across thin lipid membranes have been measured as functions of ion activity, electric potential and time. In agreement with the predictions of a version of the carrier model in common use, the shape of the initial current-voltage relation is independent of the activity of the electrolyte, a_i while the ratio of the initial conductance, G₀, to the steady-state conductance, G_∞, increases according to G₀/G_∞ = const₁ + const₂ × a_i. For trinactin the data presented allow the estimation of the rate constants of the carrier process (in the limit of zero potential) in a manner which does not assume any particular variation with potential for the constants. Using empirically determined functions of potential, a complete set of values is also available for nonactin. The curve fitting which is necessary is described in the following paper. The data presently available for valinomycin are sufficient neither to test the model nor to determine a complete set of constants.     

Steady-state ion transport by nonactin and trinactin.

The steady-state fluxes of Na+, K+, and NH4+ carried by nonactin and trinactin across thin lipid membranes have been measured as functions of ion activity, carrier concentration, and the applied potential. In agreement with earlier studies the conductance, G(O), is found to be proportional to the carrier concentration and, for low activities, to the ion activity. The determination of the dependence of G(O) on activity at high activities is, however, apparently obscured by changes in the concentration of carrier in the membrane. Using the values for the rate constants at zero potential which were determined in the preceding paper, it is possible to adjust the potential dependence of the constants so as to achieve a reasonable fit to the current-voltage relations. The data presented provide further evidence that a single molecule of nonactin or trinactin acts cyclicly as a carrier of univalent cations.     

The energy barriers to ion transport by nonactin across thin lipid membranes

The experimental steady-state current-voltage relations for low concentrations of a neutral carrier and an ion may be fitted theoretically either by assuming a form for the potential dependence of the rate of transfer of complex across the membrane and adjusting the proposed nature of the association-dissociation reactions or by assuming equilibrium for the association and adjusting the potential dependence of the transfer process. Different dependences for the rate of transfer correspond, at least formally, to different shapes for the potential energy barrier which the complex must cross. By comparing measurements of the current-voltage relations for non-actin with Na⁺, K⁺, and NH₄⁺, it is possible to distinguish between the effeects of the various rates. For black lipid membranes made from glycerolmonooleate+$\text{n-}\text{hexadecane}$, the potential energy barrier is high with a narrow top, but the rate of association still becomes increasingly limiting for Na⁺, K⁺ and NH₄⁺, in the order given. For bacterial phosphatidylethanolamine, with $\text{n-}\text{decane}$ the barrier is much wider and no effect of the rate of association can be detected.     

A general kinetic analysis of transport Tests of the carrier model based on predicted relations among experimental parameters

The analysis of transport kinetics has lacked both a unified treatment in which general rate equations are written entirely in terms of experimental parameters, and a convention by which these parameters may be designated in a concise yet immediately recognizable manner. Such a treatment is presented here in an easily accessible form, and a simple system of nomenclature is proposed resembling that in use in enzyme kinetics. The treatment is independent of assumptions about rate-limiting steps in transport, and applies to both active and facilitated systems, including obligatory exchange. A single substrate is characterized by twelve different parameters, only five of which are required in theory to calculate the others. If a second substrate is present on the trans side of the membrane there are six more parameters. All eighteen parameters are linked by multiple relationships which provide a complete set of rejection criteria for the generalized form of the mobile carrier. Relationships among parameters are also defined that give information on the rate-limiting steps in transport. Equations governing any individual experiment, involving only experimental parameters, are easily written out from the general expressions, for example under conditions of zero trans and infinite trans flux, equilibrium exchange, or competitive inhibition.     

Infinite-cis kinetics support the carrier model for erythrocyte glucose transport

It has been claimed that the Km for infinite-cis uptake of glucose in human erythrocytes is so low that the carrier model for transport must be rejected. We redetermined this parameter for three experimental conditions and found instead that the Km values were in good agreement with the model. For each of a variety of cis glucose concentrations, cells were preequilibrated with various concentrations of glucose, and the apparent Km was determined as the intracellular concentration reducing the initial rate of net uptake by half. The dependence of the apparent Km values on the cis glucose was as predicted by the carrier model; the infinite-cis Km was determined from both this concentration dependence and the extrapolated value at infinite cis glucose. The resulting values were 15 mM for fresh blood at 0 degrees C, 39 mM for outdated blood at 0 degrees C, and 11 mM for outdated blood at 25 degrees C. Previous measurements of the Km at room temperature yielded values of 2-3 mM. These earlier studies used a time course procedure that indicated rapid changes in rates during the initial 10 s of uptake but did not directly measure such changes. We examined the uptake of 60 mM glucose at 20 degrees C into cells containing 0 and 5 mM glucose; rapid changes in rates were not observed in the first few seconds, and the time courses were more consistent with our higher Km values. Our new values, together with other initial rate measurements in the literature, support the adequacy of the carrier model to account for the kinetics of glucose transport in human erythrocytes.     

Facilitated transport of di- and trinitrophenolate ions across lipid membranes by valinomycin and nonactin

The conductance of black lipid membranes in the presence of 2,4,6-trinitrophenol (or 2,4-dinitrophenol) is considerably enhanced, if the cation carriers valinomycin, enniatin B or nonactin are added. The effect is, however, largely independent of the cation concentration and is identical for the cations Li⁺, Na⁺ and Ba²⁺. This finding, as well as the sign and magnitude of the diffusion potential in the presence of a gradient of picrate are consistent with the assumption that the transport of picrate anions is facilitated by the above-mentioned macrocyclic compounds, but that cations are not directly involved. A model is suggested which, based on the generation of mobile defect structures by the incorporation of large molecules, allows one to explain facilitated transport without the assumption of stable chemical bonds between a carrier and its transported substrate.If K⁺ is present in the aqueous phase, the conductance is largely determined by the permeation of the cation complexes of valinomycin and nonactin. The conductance is, however, increased by adsorption of picrate anions to the membrane surface. The negative surface potential generated by the adsorption layer seems to be responsible for the saturation of the conductance at high picrate concentrations in the absence of valinomycin and nonactin.     

An analysis of the adequacy of the asymmetric carrier model for sugar transport

In 1972, Lieb, W. R.; Stein, W. D. (Biochim. Biophys. Acta 265, 187–207) in their review of sugar transport in human erythrocytes concluded that the conventional two-state carrier model was inconsistent with the experimental data available at that time. Since then, other papers have appeared which question the validity of the model. In this paper, we give a brief derivation of the equations describing the two-state carrier model, and analyze the predictions of the model in the classical experiments, i.e. zero-trans, infinite-cis, and equilibrium exchange. We show that the estimate of the half saturatiion constant of 2.8 mM for glucose at the inner face of the human red cell membrane for the infinite-cis procedure reported by Hankin, B. L., Lieb, W. R. and Stein, W. D ((1972) Biochim. Biophys. Acta 288, 114–126) is unreliable. We note that all of the other experimental findings are consistent with the asymmetric carrier model.     

A general kinetic analysis of transport. Tests of the carrier model based on predicted relations among experimental parameters.

The analysis of transport kinetics has lacked both a unified treatment in which general rate equations are written entirely in terms of experimental parameters, and a convention by which these parameters may be designated in a concise yet immediately recognizable manner. Such a treatment is presented here in an easily accessible form, and a simple system of nomenclature is proposed resembling that in use in enzyme kinetics. The treatment is independent of assumptions about rate-limiting steps in transport, and applies to both active and facilitated systems, including obligatory exchange. A single substrate is characterized by twelve different parameters, only five of which are required in theory to calculate the others. If a second substrate is present on the trans side of the membrane there are six more parameters. All eighteen parameters are linked by multiple relationships which provide a complete set of rejection criteria for the generalized form of the mobile carrier. Relationships among parameters are also defined that give information on the rate-limiting steps in transport. Equations governing any individual experiment, involving only experimental parameters, are easily written out from the general expressions, for example under conditions of zero trans and infinite trans flux, equilibrium exchange, or competitive inhibition.     

Facilitated transport of di- and trinitrophenolate ions across lipid membranes by valinomycin and nonactin.

The conductance of black lipid membranes in the presence of 2,4,6-trinitrophenol (or 2,4-dinitrophenol) is considerably enhanced, if the cation carriers valinomycin, enniatin B or nonactin are added. The effect is, however, largely independent of the cation concentration and is identical for the cations Li+, Na+ and Ba2+. This finding, as well as the sign and magnitude of the diffusion potential in the presence of a gradient of picrate are consistent with the assumption that the transport of picrate anions is facilitated by the above-mentioned macrocyclic compounds, but that cations are not directly involved. A model is suggested which, based on the generation of mobile defect structures by the incorporation of large molecules, allows one to explain facilitated transport without the assumption of stable chemical bonds between a carrier and its transported substrate. If K+ is present in the aqueous phase, the conductance is largely determined by the permeation of the cation complexes of valinomycin and nonactin. The conductance is, however, increases by adsorption of picrate anions to the membrane surface. The negative surface potential generated by the adsorption layer seems to be responsible for the saturation of the conductance at high picrate concentrations in the absence of valinomycin and nonactin.     

Properties of the stochastic energization-relaxation channel model for vectorial ion transport

A model for the primary active transport by an ion pump protein is proposed. The model, the "energization-relaxation channel model," describes an ion pump as a multiion channel that undergoes stochastic transitions between two conformational states by external energy supply. When the potential profile along ion transport pathway is asymmetrical, a net ion flux is induced by the transitions. In this model, the coupling of the conformational change and ion transport is stochastic and loose. The model qualitatively reproduces known properties of active transport such as the effect of ion concentration gradient and membrane potential on the rate of transport and the inhibition of ion transport at high ion concentration. We further examined the effect of various parameters on the ion transport properties of this model. The efficiency of the coupling was almost 100% under some conditions.     

An analysis of the adequacy of the asymmetric carrier model for sugar transport.

In 1972, Lieb, W. R. and Stein, W. D. (Biochim, Biophys. Acta 265, 187-207) in their review of sugar transport in human erythrocytes concluded that the conventional two-state carrier model was inconsistent with the experimental data available at that time. Since then, other papers have appeared which question the validity of the model. In this paper, we give a brief derivation of the equations describing the two-state carrier model, and analyze the predictions of the model in the classical experiments, i.e. zero-trans, infinite-cis, and equilibrium exchange. We show that the estimate of the half saturation constant of 2.8 mM for glucose at the inner face of the human red cell membrane for the infinite-cis procedure reported by Hankin, B.L., Liev, W.R. and Stein, W.D ((1972) Biochim. Biophys. Acta 288, 114-126) is unreliable. We note that all of the other experimental findings are consistent with the asymmetric carrier model.     

An enzymatic ion exchange model for active sodium transport

An enzymatic ion exchange model for active sodium transport is described. Kinetic equations relating net flux to time, and to concentration difference across the actively transporting membrane are derived. The second of these equations is tested, using the isolated frog skin in the "short-circuit" apparatus of Ussing. Reasonable linearity, as predicted by this equation, is observed. The passive permeability coefficient for Na⁺, is calculated as 5.3 x 10^-4 ± 5.3 x 10^-4 cm./hr. If cholinesterase is assumed to be the enzyme responsible for transport, the activity required to account for the observations reported here is 17.7 x 10^-4 mmoles/cm.²/hr.     

On the cyclotron resonance model of ion transport.

The cyclotron-resonance model, which has been suggested as an explanation of a purported enhancement of transport of ions through the membranes of cells exposed to weak, low-frequency-modulated RF fields, is shown to be inconsistent with basic physical principles. Under the conditions of the model, in which the ions are presumed to circulate under the constraint of the earth's magnetic field, the radii of gyration of the ions would approximate 50 m and, thus, are much larger than the cells. Moreover, from general considerations, the collision-damping time of such ions is expected to be less than 10(-10) s, much smaller than the times of the order of 10(-2) s, shown to be necessary if the conditions for low-frequency resonance are to be satisfied.     

Effects of variation of ion and methylation of carrier on the rate constants of macrotetralide-mediated ion transport in lipid bilayers

Summary The effects of methylation on the rate constants of carrier-mediated ion transport have been studied on monooleindecane bilayers with K⁺, Rb⁺, NH ₄ ⁺ , and TI⁺ ions, using the series of homologue carriers, nonactin, monactin, dinactin, trinactin, and tetranactin, each member of the series differing from the previous one by only one methyl group. Measurements of the amplitude and time constant of the current relaxation after a voltage jump over a large domain of voltage and permeant ion concentration, together with a computer curve-fitting procedure, have allowed us, without the help of steady-state current-voltage data, to deduce and compare the values of the various rate constants for ion transport: formation (k _Ri) and dissociation (k _Di) of the ion-carrier complex at the interface, translocation across the membrane interior of the carrier (k _s) and the complex (k _is). With the additional information from steady-state low-voltage conductance measurements, we have obtained the value of the aqueous phase-membrane and torus-membrane partition coefficient of the carrier ({ie191-1} and {ie191-2}). From nonactin to tetranactin with the NH ₄ ⁺ ion,k _is, and {ie191-3} are found to increase by factors of 5 and 3, respectively,k _Di and {ie191-4} to decrease respectively by factors 8 and 2, whilek _Ri andk _s are practically invariant. Nearly identical results are found for K⁺, Rb⁺, and Tl⁺ ions.k _Ri,k _s andk _is are quite invariant from one ion to the other except for Tl⁺ wherek _Ri is about five times larger. On the other hand,k _Di depends strongly on the ion, indicating that dissociation is the determining step of the ionic selectivity of a given carrier. The systematic variations in the values of the rate constants with increasing methylation are interpreted in terms of modifications of energy barriers induced by the carrier increasing size. Within this framework, we have been able to establish and verify a fundamental relationship between the variations ofk _is andk _Di with methylation.