ABSTRACT
Amino acid/K+ symport (cotransport) across a model epithelium, the lepidopteran midgut, is energized by an electrogenic H+ V-ATPase (H+ pump) in parallel with an electrophoretic K+/H+ antiporter (exchanger). Attempts to analyze this process using well-known equilibrium thermodynamic equations (Nernst, Gibbs), diffusion equations (Nernst, Planck, Einstein, Goldman, Hodgkin, Katz) and equations based on Ohm’s law (Hodgkin, Huxley) have all encountered major difficulties. Although they are useful for analyzing nerve/muscle action potentials, these state equations assume that brief perturbations in membrane conductance, gm, and membrane voltage, Vm, occur so rapidly that no other parameters are significantly disturbed. However, transport studies often extend for minutes, even for hours. Perturbation of one parameter in complex transport systems invariably results in a state change as all of the other elements adjust to the prolonged stress.
The development of a comprehensive mathematical treatment for transport systems that contain pumps and porters (transporters) has been hampered by the empirical nature of the concept of membrane permeability and conductance. The empirical definition of permeability was developed before pumps and porters were known. Thus, ‘permeability’ is a gross parameter that, in practice if not in theory, could describe all transport pathways including pumps, porters and channels. To surmount these difficulties, we have applied ionic circuit analysis to vesicular systems containing insect midgut transport proteins. In this analysis, pumps, porters and channels, as well as ionic concentration gradients and membrane capacitance, are components of ionic circuits that function to transform metabolic energy (e.g. from ATP hydrolysis) into useful metabolic work (e.g. amino acid uptake). Computer-generated time courses reproduce key aspects of the coupling of the proton-motive force generated by an H+ V-ATPase to K+/2H+ antiport and amino acid/K+ symport in the lepidopteran midgut.
INTRODUCTION
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, Vm, whereas signalling membranes, energized by ion gradients, pass tiny fluxes with large changes in Vm. 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 Vm. 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). Vm 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 Vm 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 Vm (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 Vm 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 Vm 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.
Hodgkin–Huxley strategy in retrospect
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
Thermodynamic analysis, based on Willard Gibbs chemical potential μi=δG/δni|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.
Diffusion theory
Equivalent circuit analysis
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 Vm 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 Em and Vm are often used to represent the membrane voltage. Ek and Vk have both been used to represent ionic equilibrium potentials. In the equations that follow, ΔΨ represents theoretically calculated membrane voltages and Vm empirically measured membrane voltages. The symbol E with a subscript represents equilibrium potentials for specific ions.
Classical thermodynamics of lepidopteran midgut ion homeostasis
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).
IONIC CIRCUIT ANALYSIS OF PUMPS, PORTERS AND CHANNELS
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 and compartmental capacitance
Membrane capacitance, Cm, is the rate of accumulation of charge, Q, as Vm develops, Cm=δQ/δVm. 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, Cv, 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; Cv=δQ/δEk. The δEk term can be calculated from the Nernst equation (equation 3); in this case, Ek=(RT/zkF)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.
Sequential versus systematic analyses
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.
Circuit diagram of a hypothetical vesicular system
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) (PP), a K+/H+ antiporter (PA) and an amino acid/K+ symporter (PS); each component containing a finite internal conductance (gP, gA and gS) 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 (gH and gK) and for the symported amino acids (gAA). A fourth leakage conductance (gG) deals with gegenions that are not specifically pumped or ported, but move through channels. Just as the membrane capacitance (Cm) 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 (CH, CK, CG and CAA) 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 (EATP).
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).
Algorithm for the analysis of the hypothetical vesicular system
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 Ek 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 (Ikl) is calculated using the equation Ikl=gkl(ΔΨ-Ek), where ΔΨ and Ek are the membrane voltage and the specific ionic equilibrium potential, respectively, and gkl is the conductance of the pathway. The current through the pump pathway is calculated using the equation IkP=gkP(ΔΨ–Ek–EATP/n), where IkP 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.
CIRCUIT ANALYSIS OF DIFFERENT EXPERIMENTAL CONDITIONS
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.
H+ V-ATPase with gegenion (acidification)
Fig. 2 shows the development of ΔΨ and the equilibrium potentials for protons (EH) and gegenions (EG) across a vesicular membrane containing an H+ V-ATPase (H+ pump) (gP=10×7 S cm−2, n=2 protons per ATP) and a negative gegenion (G−) pathway (gG=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 EH and positive EG, and the membrane voltage ΔΨ. The negative EH 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 EG approaches ΔΨ as the experiment proceeds. This pattern arises because CG is effectively in parallel with Cm but CG charges through a lower conductance (series combination of gP and gG) than Cm (gP alone). Since the parallel combination of CG and Cm is in series with CH, at equilibrium ΔΨ=EG and ΔΨ+EH=VATP/n.
H+ V-ATPase and K+/H+ antiporter in parallel (alkalization)
Fig. 3 shows the development of ΔΨ, EH and EK across a vesicular membrane containing an H+ V-ATPase (gP=10×7 S cm−2, n=2 protons per ATP) and a K+/H+ antiporter (gA=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 EK 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 Cm (ΔΨ=EATP/n). The large ΔΨ which results drives protons through the antiporter and out of the vesicle, leaving the vesicle contents alkaline.
Potassium-driven amino acid cotransport (AA/K+ symport)
Fig. 4 shows the change in intravesicular K+ and anionic amino acid (AA−) activities in a system with an AA/K+ symporter (gS=10×9 S cm−2, mS=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.
H+ V-ATPase, K+/H+ antiporter, AA/K+ symporter and gegenion (amino acid uptake)
Fig. 5 shows the development of ΔΨ, EH, EK, EG and EAA across a vesicular membrane containing an H+ V-ATPase (gP=10×7 S cm−2, n=2H+ per ATP), a K+/H+ antiporter (gA=10×7 S cm−2, mA=-2K+ per H+), an AA/K+ symporter (gS=10×7 S cm−2, n=1AA per K+) and a gegenion (G−) leakage (gG=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 Vm is much higher.
H+ V-ATPase, K+/H+ antiporter and gegenion, with a switch
Fig. 6 shows a simulation which mimics the results of an actual experiment performed by Wieczorek et al. (1989). ΔΨ, EH, EK and EG are plotted for a vesicular system containing an H+ V-ATPase (gP=10×5 S cm−2, n=2H+ per ATP) and a gegenion (G−) leakage (gG=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 (gA=10×30 S cm−2). No K+ movement is observed, but the vesicle acidifies owing to the action of the V-ATPase. ΔΨ and EG 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 (gA=10×5 S cm−2, mA=-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 EG.
CONCLUSION
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.
ACKNOWLEDGEMENTS
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.