Ionic Circuit Analysis of K⁺/H⁺ Antiport and Amino Acid/k⁺ Symport Energized by a Proton-Motive Force in Manduca Sexta Larval Midgut Vesicles

We describe here a membrane biologist’s ‘theory of everything’, which applies to the coupling between pumps, porters and channels in bacterial cells, intracellular organelles, plasma membrane vesicles and even in multicellular epithelia. Transporting cells, on the one hand, and signalling cells, on the other, are at opposite ends of the flux/voltage spectrum. Epithelial cell membranes, energized by highly regulated ATPases, pass large ionic fluxes with little change in membrane voltage, V_m, whereas signalling membranes, energized by ion gradients, pass tiny fluxes with large changes in V_m. Transporting membranes of bacterial cells, eukaryotic organelles and membrane vesicles are like those of epithelial cells in that they can sustain large fluxes at constant V_m. Yet membrane transport theories for epithelia, organelles and vesicles are usually patchwork modifications of the Hodgkin–Huxley theory for signalling membranes. Terms for pumps and porters are ‘added on’ to equations that were designed to deal with transient openings and closings of channels. We propose to start afresh. We will follow the strategy of Hodgkin and Huxley, but we will replace awkward concepts by tractable ones and emerge with a comprehensive, quantitative treatment for integrated membrane transport events.

The lepidopteran midgut (Fig. 1A) is a particularly useful model for this fresh start because it is very different from well-known mammalian systems and from the squid axon model (Harvey et al. 1986). V_m is not a combination of Na⁺ and K⁺ diffusion potentials, but is a component of an H⁺ pump potential. In fact, no Na⁺/K⁺-ATPase is detectable in the basal or lateral membranes or any other membranes of the midgut epithelium; instead, an electrogenic H⁺ V-ATPase (H⁺ pump) energizes the apical membrane directly (Wieczorek et al. 1989). The [Na⁺] is low both inside the cells and in the lumen. The transapical V_m is approximately 240 mV, lumen positive. The pH is 4, outside alkaline. The [K⁺] is high in the extracellular space on the apical side (lumen) and the K⁺lumen/K⁺cell activity ratio is maintained at approximately 1, despite massive K⁺ transport from cell to lumen in vivo. Although electrical forces drive massive K⁺/2H⁺ antiport (Wieczorek et al. 1991; Lepier et al 1994) and amino acid/K⁺ symport across the apical membrane, they do not significantly change the transapical V_m (see reviews in Harvey and Nelson, 1992).

In many of these respects, the lepidopteran midgut more closely resembles plant or bacterial membranes than classical animal cell membranes. The membrane is energized by a proton-motive force, not a sodium-motive force. Indeed a V_m generated by a proton-motive force is emerging as a common intermediate in many animal cells (Nelson, 1994). Already recognized as the primary energizer of endomembranes, H⁺ V-ATPases are now recognized to be energizers of animal cell plasma membranes. The list already includes the apical membrane of all insect sensory epithelial cells and of many insect Malpighian tubule cells (Klein, 1992). It also includes frog skin mitochondria-rich cells (Ehrenfeld et al. 1989), two types of kidney cells (Brown et al. 1992), osteoclasts (Chatterjee et al. 1992) and macrophages (Grinstein et al. 1992). Membranes energized by a proton-motive force using V_m as a common intermediate may eventually turn out to be more important, even in animal cells, than membranes energized by a sodium-motive force using ion gradients as an intermediate.

The Hodgkin and Huxley (1952) equations were the ‘jewels’ in the mathematical ‘crown’ that allowed the reconstruction of squid axon action potentials from voltage-clamp data. This crown developed through the three stages described below.

Thermodynamic analysis, based on Willard Gibbs chemical potential μ_i=δG/δn_i|T,P,nj, provided a means for calculating the limits of diffusion phenomena (Snell et al. 1965). δG/δn is the partial change in molar free energy attributable to the addition of 1 mole of one molecular species (i) to a compartment when temperature (T), pressure (P) and the number (n) of molecules of all other species (j) are not allowed to change. This potential allows us to think about what happens to a single solute species and to combine that result with what happens to other species that are the components of a chemical reaction, such as ATP hydrolysis.

This approach of Hodgkin and Huxley was so successful that it led to a Nobel Prize and dominated subsequent thought concerning transport physiology. However, its very success may have been unfortunate for scientists attempting to deal with complex systems including pumps and porters as well as channels. Attempts to modify the Hodgkin–Huxley equations (developed for nerve and muscle membranes containing fast-opening and fast-closing channels) in order to obtain a mathematical treatment suitable for transporting epithelia (containing pumps, porters and chronically open channels) have met with limited success. The state equations of Nernst, Goldman and others do not consider the presence of pumps and porters. Mullens and Noda (1963) added a factor to the Goldman equation to include the effects of an electrogenic ion pump on the membrane potential. Others have modified variations of the Hodgkin–Huxley equation to solve for V_m in the presence of electrogenic pumps (see Kishimoto et al. 1981). However, the effects of porters on electrical parameters have never been adequately addressed. Gerencser and Stevens (1994) address this problem in this volume, using non-equilibrium thermodynamics (Kedem and Katchalsky, 1958) to analyze transport in animal cells. We address the problem here, using circuit analysis techniques (Blitzer, 1964; Edminister, 1965) to examine transport in lepidopteran midgut components.

As can be seen from the previous paragraphs, the symbols representing membrane voltage are different for each set of workers. Along with Hodgkin and Katz’s E and Hodgkin and Huxley’s V, the symbols E_m and V_m are often used to represent the membrane voltage. E_k and V_k have both been used to represent ionic equilibrium potentials. In the equations that follow, ΔΨ represents theoretically calculated membrane voltages and V_m empirically measured membrane voltages. The symbol E with a subscript represents equilibrium potentials for specific ions.

The lepidopteran midgut has proved to be a favorable model for epithelial ion and metabolite transport. Its large size, enormous fluxes (Harvey and Nedergaard, 1964) and relatively simple structure (Anderson and Harvey, 1966) enabled a physiological and biochemical characterization of transport activities (reviews in Harvey and Nelson, 1992) that led to the identification (Wieczorek et al. 1989), as well as the cloning and sequencing of most of the major subunits of its plasma membrane H⁺-translocating V-ATPase (Dow et al. 1992; Gill and Ross, 1990; Gräf et al. 1992, 1994a,b; Lepier et al. 1994; Novak et al. 1992; reviewed by Wieczorek, 1992).

At this point we must part company with Hodgkin and Huxley. A simple equivalent circuit, such as the one they used to derive equation 5, will not suffice to deal with changing pump, porter and channel conductances or with the resulting changes in transmembrane ionic currents, ionic equilibrium potentials and membrane voltages. Martin (1992) used circuit analysis based upon Kirchoff’s laws as well as mesh and nodal analysis (Blitzer, 1964; Edminister, 1965) to develop a new theory of transport. He then wrote an algorithm that included not just channels, but pumps and porters as well, and used the algorithm to write a computer program. This analysis departs from all previous ones by representing ionic gradients as charged compartmental capacitances instead of batteries; the use of batteries is limited to the chemical driving force behind pumps.

Membrane capacitance, C_m, is the rate of accumulation of charge, Q, as V_m develops, C_m=δQ/δV_m. Because the dielectric thickness of all biological membranes is approximately 3 nm, the membrane capacitance has a constant value of about 1 μF cm⁻². By analogy, the compartmental capacitance, C_v, is the rate of accumulation of charge in a bulk compartment, such as the interior of a cell or a vesicle, as the concentration of an ion changes; C_v=δQ/δE_k. The δE_k term can be calculated from the Nernst equation (equation 3); in this case, E_k=(RT/z_kF)ln ⁠, where is the concentration of the ionic species in the compartment and is the concentration under standard conditions, taken as 1 mmol l⁻¹, for convenience. The compartmental capacitance is clearly not a constant, but is a function of concentration, and therefore of volume; its value can range from less than 1 μF for a small vesicle to more than 1 MF for an intestinal lumen.

Just as the membrane capacitances depend upon the dielectric constant of the lipid bilayer, the compartmental capacitances depend upon a dielectric-like parameter, the activity coefficient of the ion under consideration. Unlike membrane capacitances, which are usually assumed to be linear and to remain constant under large changes in charge, compartmental capacitances are very nonlinear since they are very sensitive to the amount of charge present. As with all capacitances, compartmental capacitances combine with themselves and other capacitances to form series-parallel circuits.

Transduction of energy in a membrane system is an integrated process, in which all ionic species contribute to the load on cell metabolism as transport work is being done. A perturbation of one component in a system has the ability to affect the whole system. If a system consists of just two or three components, an intuitive or semiquantitative analysis can be undertaken. In such an approach, the action of one component sets up a condition, e.g. a concentration gradient or a transmembrane voltage, which then determines the action of a second component. If sufficient isolation exists between components, then such a sequential analysis may be adequate to treat the effects of one component or another component or two. However, if there are many components and/or if component activities are coupled, then such an intuitive and semiquantitative sequential approach rapidly becomes unwieldy if not impossible.

Ionic circuit analysis overcomes these difficulties. It provides a way to analyze quantitatively the complex interactions among the various components of complex membrane energy transduction systems in vivo or in vitro. Ionic circuit analysis is an extension of electrical circuit analysis which includes components that are unique to ionic circuits of biomembranes (Martin, 1992). Whereas equivalent electrical circuit analysis depicts systems metaphorically, ionic circuit analysis represents systems as they actually exist.

A circuit diagram of a hypothetical vesicular system, containing the major transporters found in the apical membranes of midgut cells, demonstrates the use of ionic circuit analysis (Fig. 1B). The main components are a proton-translocating V-ATPase (pump) (P_P), a K⁺/H⁺ antiporter (P_A) and an amino acid/K⁺ symporter (P_S); each component containing a finite internal conductance (g_P, g_A and g_S) that limits its efficiency. Also present are specific transmembrane leakage conductances (e.g. uniporters, channels and slips) for each of the two major transported ions (g_H and g_K) and for the symported amino acids (g_AA). A fourth leakage conductance (g_G) deals with gegenions that are not specifically pumped or ported, but move through channels. Just as the membrane capacitance (C_m) is the linear capacitance formed by the two conducting aqueous solutions (intravesicular and bath) and the dielectric property of the lipid bilayer membrane that separates them, the ionic compartmental capacitances (C_H, C_K, C_G and C_AA) are the nonlinear capacitances of the intravesicular (i) and bath (o) compartments themselves, as defined above. The only power source for the system is the driving force of ATP hydrolysis, expressed as an electrical potential (E_ATP).

In ionic circuit analysis, leakage conductances and the membrane capacitance are standard components that are used in the same way as in equivalent electrical circuits. The pump, porters and compartmental capacitances, in contrast, are unique to ionic circuits. The pumps and porters are energy-transduction devices analogous to transformers in a.c. electrical circuits. They have primary and secondary sides depending on the direction of energy flow and are step-up or step-down transformers according to their stoichiometric coupling ratios. They are current-inverting if they are antiporters or current-noninverting if they are symporters. They can match impedances from one side to the other. The other components, the ion-specific compartmental capacitances, are the electrical manifestations of the solution compartments (see above). Just as the membrane capacitance is a function of the area and thickness of the membrane, the compartmental capacitance is a function of the volume of the compartment. The compartmental capacitances shown in Fig. 1B are distinguished from the membrane capacitance by a different symbol. Unlike the two halves of the symbol for the membrane capacitance, the two halves of the symbol for compartmental capacitance are not meant to represent the aqueous compartments on either side of the membrane. A compartmental capacitance is the specific ionic capacitance of the compartment itself as explained above; hence, compartmental capacitances are shown for each ion on each side of the membrane. A comprehensive mathematical analysis of pumps and porters that uses such compartmental capacitances has been published previously (Martin, 1992).

Using the circuit diagram in Fig. 1B as a foundation, an iterative program was written. The program depicts the flow of ionic currents, the formation of ionic activity gradients and the development of transmembrane voltage as they change over time. The program uses nodal network analysis, which takes advantage of two facts: (1) that all of the ionic pathways through the membrane are essentially in parallel with each other and (2) that the membrane voltage is a common force that affects the flow of all the ionic species involved. During each iterative step, the net current flow for each ionic species is calculated and the charges on the membrane and compartmental capacitances are then adjusted accordingly.

Each iterative step of the analysis begins with the calculation of E_k across the vesicular membrane. This is done for each species and each compartment using the number of ions present, the activity coefficients of the ions, and the volumes of the vesicle and the bath compartments. Each step also begins with the calculation of the ΔΨ based upon the charge on the membrane capacitance (1 μF cm⁻²).

The net current flow for a given ionic species is the sum of the current flows through all of the pump, porter and leakage pathways available to the ion. Current flow through the leakage pathway (I_kl) is calculated using the equation I_kl=g_kl(ΔΨ-E_k), where ΔΨ and E_k are the membrane voltage and the specific ionic equilibrium potential, respectively, and g_kl is the conductance of the pathway. The current through the pump pathway is calculated using the equation I_kP=g_kP(ΔΨ–E_k–E_ATP/n), where I_kP is the current flow through the pump and n is the stoichiometric coupling ratio, i.e. the number of ions pumped per molecule of ATP hydrolysed.

Calculating the current flow through porters is a bit more complicated. It involves two different ionic species, k and j, one of which (assume it to be k) moves through the primary (′) side of the porter (left side in Fig. 1B) while the other (assume it to be j) moves through the secondary (″) side of the porter (right side in Fig. 1B). Also, one must consider the primary:secondary coupling ratio of the porter m; symporters have positive values of m whereas antiporters have negative values of m.

In order to demonstrate the usefulness of the preceding algorithm, a program written in Pascal was used to plot the time courses of events in several different simulated experiments. The program allows the investigator to control all of the variables which could potentially change the results of an experiment. In all the simulated experiments depicted here, vesicles (1 μm in diameter) are bathed by a volume (1 ml per vesicle) of medium at room temperature (25°C) and standard pressure (1 atmosphere=101.3 kPa). Fig. 1B is the general schematic diagram for all the experiments, but pathways not explicitly turned on in the description of the experiment are rendered inoperative by the assignment of a very low conductance (10⁻³⁰ S cm⁻²). Where the data were available, the values for component conductance and ionic activities were adjusted so that the computer-generated experiments were similar to actual laboratory experiments.

Fig. 2 shows the development of ΔΨ and the equilibrium potentials for protons (E_H) and gegenions (E_G) across a vesicular membrane containing an H⁺ V-ATPase (H⁺ pump) (g_P=10^×7 S cm⁻², n=2 protons per ATP) and a negative gegenion (G⁻) pathway (g_G=10^×6 S cm⁻²). The initial solutions both inside and outside the vesicle are the same (the activity of H⁺=10^×7 mol l⁻¹ with an activity coefficient of 10^×7 and activity of G⁻=50 mmol l⁻¹ with an activity coefficient of 10^×2. (Note that the dissociation of water is the primary source of protons in the examples described in this section and is the reason that the very low activity coefficient is assigned to H⁺.) The plot shows the development of three different voltages, a negative E_H and positive E_G, and the membrane voltage ΔΨ. The negative E_H is to be expected from the acidification of the vesicular contents by the inwardly directed H⁺ V-ATPase. Close examination reveals that ΔΨ develops faster initially, but that its development slows and E_G approaches ΔΨ as the experiment proceeds. This pattern arises because C_G is effectively in parallel with C_m but C_G charges through a lower conductance (series combination of g_P and g_G) than C_m (g_P alone). Since the parallel combination of C_G and C_m is in series with C_H, at equilibrium ΔΨ=E_G and ΔΨ+E_H=V_ATP/n.

Fig. 3 shows the development of ΔΨ, E_H and E_K across a vesicular membrane containing an H⁺ V-ATPase (g_P=10^×7 S cm⁻², n=2 protons per ATP) and a K⁺/H⁺ antiporter (g_A=10^×7 S cm⁻², m=-2 protons per K⁺). The initial solutions both inside and outside the vesicle are the same [activity of H⁺=10^×7 mol l⁻¹ with an activity coefficient of 10^×7 (see note above) and activity of K⁺=5 mmol l⁻¹ with an activity coefficient of 10^×2]. The negative E_K is the result of potassium accumulation inside the vesicle as inwardly pumped protons move out through the antiporter. With no gegenion leakage in this experiment, all of the pump potential is developed across C_m (ΔΨ=E_ATP/n). The large ΔΨ which results drives protons through the antiporter and out of the vesicle, leaving the vesicle contents alkaline.

Fig. 4 shows the change in intravesicular K⁺ and anionic amino acid (AA⁻) activities in a system with an AA/K⁺ symporter (g_S=10^×9 S cm⁻², m_S=1AA⁻ per K⁺) in the vesicular membrane. Initial AA⁻ activity is the same in both the intravesicular and bathing solutions (5×10^×4 mol l⁻¹ with an activity coefficient of 1) but an initial K⁺ gradient exists between the bathing medium (5×10^×2 mol l⁻¹ with an activity coefficient of 1) and the intravesicular compartment (5×10^×4 mol l⁻¹ with an activity coefficient of 1). As K⁺ moves down its electrochemical gradient, it drives AA⁻ into the vesicle through the AA/K⁺ symporter. However, as the K⁺ electrochemical gradient dissipates, the accumulation of AA⁻ reaches a peak and then slowly reverses, coming to equilibrium at an activity higher than the initial, but lower than the peak, value. Amino acid uptake experiments performed by Hennigan et al. (1993a,b) show a similar pattern of K⁺-driven amino acid accumulation over time.

Fig. 5 shows the development of ΔΨ, E_H, E_K, E_G and E_AA across a vesicular membrane containing an H⁺ V-ATPase (g_P=10^×7 S cm⁻², n=2H⁺ per ATP), a K⁺/H⁺ antiporter (g_A=10^×7 S cm⁻², m_A=-2K⁺ per H⁺), an AA/K⁺ symporter (g_S=10^×7 S cm⁻², n=1AA per K⁺) and a gegenion (G⁻) leakage (g_G=10^×6 S cm⁻²). The initial solutions both inside and outside the vesicle are the same [activity of H⁺=10^×7 mol l⁻¹ with an activity coefficient of 10^×7 (see note above) and the activities of K⁺, AA⁺ (cationic amino acid) and G⁻ at 50 mmol l⁻¹ each with activity coefficients of 10^×2]. Here, a strong AA⁺ accumulation occurs, with some initial acidification followed by a return to pH neutrality. The ΔΨ here is low compared with the in vivo condition of the midgut apical membrane (Dow and Peacock, 1989). This discrepancy can be accounted for by the presence of gegenions in the simulation which are needed to prevent chaotic behavior when the time increment is conveniently large. In contrast, the apical membranes of the lepidopteran larval midgut are not very permeable to gegenions and V_m is much higher.

Fig. 6 shows a simulation which mimics the results of an actual experiment performed by Wieczorek et al. (1989). ΔΨ, E_H, E_K and E_G are plotted for a vesicular system containing an H⁺ V-ATPase (g_P=10^×5 S cm⁻², n=2H⁺ per ATP) and a gegenion (G⁻) leakage (g_G=10^×4 S cm⁻²). The initial solutions both inside and outside the vesicle are the same [activity of H⁺=10^×7 mol l⁻¹ with an activity coefficient of 10^×7 (see note above) and the activities of K⁺ and G⁻ at 5 mmol l⁻¹ each with an activity coefficient of 10^×2 each]. In the first part of the experiment, the antiporter is turned off (g_A=10^×30 S cm⁻²). No K⁺ movement is observed, but the vesicle acidifies owing to the action of the V-ATPase. ΔΨ and E_G develop just as they did in the section above describing the circuit for an H⁺ V-ATPase with gegenion (acidification). Again, as in the preceding paragraph, without the gegenion ion leakage, ΔΨ would be much higher and the degree of acidification would be considerably less. Fifty seconds into the experiment, the antiporter is turned on (g_A=10^×5 S cm⁻², m_A=-2K⁺ per H⁺) and immediately the acidification is lost and K⁺ accumulates owing to the action of the antiporter. This shift from H⁺ to K⁺ accumulation and loss of acidification occurs with only a transient perturbation in the development of ΔΨ and E_G.

Thinking in terms of integrated pump, porter and channel circuits makes better sense than thinking in terms of sequential primary and secondary active transport mechanisms and passive transport mechanisms. The dynamic approach is required to appreciate fully the interplay of primary and secondary events in a membrane.

This research has been supported in part by Faculty Incentive Grants and Faculty Development Grants from Immaculata College to F.G.M. and by NIH Research Grant AI-30464 to W.R.H. We thank our colleagues at Temple University, Dr Michael G. Wolfersberger and Dr Ranganath Parthasarathy, for critically evaluating the work in progress. We thank them and Dr Clifford Slayman of Yale University, Dr Bruce Stevens of the University of Florida and Mr James Mooney of Immaculata College, who also provided suggestions as we prepared this manuscript. We also thank Mr Daniel J. Harvey for his assistance in preparing the figures.