The concept of electrical circuit analysis is extended to include components found in membrane ionic transport systems. As in classical electrical equivalent circuits, resistors and capacitors are used to represent ion channels and the membrane capacitances, respectively; batteries represent energy sources driven by chemical reactions. In the extensions proposed, energy stored in various ionic concentrations is treated as charges on compartmental capacitors; symporters and antiporters are treated as energy-coupling devices analogous to transformers in alternating current electrical circuits. Pumps are shown to be special cases of porters in which the input circuit derives its energy from a chemical reaction. Using these components, circuit diagrams are drawn for several examples of membrane ion transport systems. By applying appropriate circuit analysis techniques, these diagrams facilitate the quantitative description of the energy distributions throughout the system.
Models of membrane-associated ionic events have long provided a conceptual framework for organizing and communicating data obtained from real systems. Such models usually take the form of equivalent electrical circuits which can be subjected to standard electrical circuit analysis techniques to describe quantitatively the voltage and current relationships among the components.
Probably the most familiar of the equivalent electrical circuits is the model of the squid giant axon used by Hodgkin and Huxley (1952) to describe the ionic events associated with the generation of nerve action potentials. Finkelstein and Mauro (1963) extended the use of equivalent electrical circuits to model membrane systems and developed the concept of the pure electrical equivalent of the membrane. The models proposed by Hodgkin and Huxley and by Finkelstein and Mauro considered only the movement of ions down their electrochemical gradients. Several years elapsed before electrogenic ion pumps were included in the equivalent electrical circuits (e.g. Kishimoto et al. 1981).
The analysis of equivalent electrical circuit models has proved useful in studying transmembrane ionic movements associated with transient events, like sensory receptor potentials (Martin and Mote, 1980) and action potentials, which do not significantly affect the transmembrane ionic gradients. By contrast, in the complex world of porters, pumps and channels, the application of circuit analysis techniques has been scant. This paper proposes an extension of electrical circuit analysis technique to model ionic systems typical of biological plasma membranes in single cells and epithelia. The fundamental techniques of circuit analysis include Ohm’s law, Kirchhoffs laws, the Thevenin and Norton theorems, as well as mesh and nodal network analysis (e.g. Edminister, 1965). These techniques are as applicable to ionic systems as they are to electrical systems. However, ionic circuits are characterized by a multiplicity of current carriers, by energy coupling between carriers, and by energy storage being distributed between ionic activity gradients and the membrane capacitance. For these complex circuits the definitions of some components have been refined. The components and their assembly into ionic circuits is discussed.
Included in all equivalent electrical circuits are resistors, capacitors and batteries. Each of these electrical components represents a corresponding physical component in the membrane systems. Resistors represent the various ionic channels found in the membrane. Capacitors represent the plasma membrane with its insulating lipid core separating two conductive aqueous solutions. Batteries represent ionic gradients having an electromotive force (emf) equal to the corresponding Nemst equilibrium potential. However, symporters, antiporters and pumps do not have a typical component designation in their own right, but are usually represented as constant current or constant voltage generators in most electrical equivalent circuits (Läuger, 1991).
In the present analysis, resistors and capacitors are used to represent ionic pathways and membrane capacitances in the usual way. Batteries, however, are restricted to instances where chemical bonds are broken to generate a fixed emf or, more appropriately, an ion-motive force (imf), related to the free energy of the reaction. The ion activity gradients, porters and pumps are treated in a novel way using the concept of compartmental capacitance.
The storage of energy in activity gradients presents a conceptual problem. The usual method of representing them as batteries is particularly awkward. Ideal batteries have a constant output potential determined by the free energy of the chemical reaction involved. As batteries discharge, the open-circuit potential of the battery remains constant whereas the closed-circuit output voltage decays. This condition can be schematized by including an internal resistance in series with an idealized battery. As the battery discharges, the amount of substrate decreases and the overall rate of the reaction decreases, raising the effective internal resistance. The discharge of an activity gradient is not like the discharge of a battery. When a gradient discharges, withdrawal of energy actually decreases the magnitude of the free energy involved, a condition more like the discharging of a capacitor than that of a battery.
A membrane always separates two compartments. If each compartment has an ionic compartmental capacitance for each ionic species involved then a total ionic compartmental capacitance for a two-compartment system can be determined (again, for simplicity assume no change in compartmental volume or ionic activity coefficient).
In cases where just two compartments are separated by a membrane (e.g. vesicles in cytoplasm or vesicles in a bath) it is probably more convenient to lump both ionic compartmental capacitances into a single combined capacitance. Since, in most cases, the vesicular or cellular volume will be much smaller than the cytoplasmic or bath volume, respectively, the vesicular or cellular ionic compartmental capacitances will be very close to the combined ionic capacitance. When considering epithelia, which are, by definition, polycompartmental systems (at least lumen, intracellular space and blood), it is necessary to consider the ionic capacitances of each compartment individually since each compartment can be ionically coupled to more than one other compartment by ionic transport mechanisms.
Porters and pumps
Symporters and antiporters are commonly thought of as current-coupling devices. They are often defined in terms of their coupling ratios of 1:1, 2:1, 3:2, etc. But they can also be thought of as energy-coupling devices. Energy delivered to the device by an ionic current will be dissipated by the device. The dissipated energy can take two forms: energy lost as heat and energy used to do work. In the ideal porter, no energy would be lost as heat and all the energy delivered by an ionic current would be used to do work. When the work done is to generate an imf for a secondary ionic species, the relationship between the primary and secondary ionic species is analogous to the relationship between the primary and secondary coils in an a.c. transformer. The current, voltage and impedance relationships for porters will be described using this transformer analogy.
Fig. 1A shows the circuit symbol for porters used in this paper. It was derived from the typical symbol used to represent coupling devices in membrane diagrams (Fig. 1B); its origins can be traced back at least as far as Hodgkin and Keynes (1955). Input and output circuits are arbitrarily assigned in the ideal coupler, i.e. the ideal porter is considered reversible. The coupling coefficient (m) is considered positive if the input and output currents through the device are in opposite directions.
This analysis yields the following relationships. Anion/anion or cation/cation antiporters would have positive coupling coefficients. Anion/anion or cation/cation symporters would have negative coupling coefficients. Anion/cation antiporters would have negative coupling coefficients whereas anion/cation symporters would have positive coupling coefficients. This definition of the coupling ratio simplifies mesh analysis of the ionic circuits and is consistent with the definition of positive current as the passive movement (i.e. downhill movement driven by ionic electrochemical gradients) of positive ions into cells. By implication, the active movement (i.e. uphill movement driven directly or indirectly by metabolic processes) of positive ions out of cells is also positive current.
For a more realistic depiction of porters, it is necessary that frictional losses of the porter mechanism and also possible ‘slip’ of ions through the mechanism without accompanying coupling be considered. Frictional losses can be modeled by adding a resistance (Rs) in series with either the input or output sides of the porter. Slippage can be modeled by adding a resistance in parallel (Rp) with either the input or the output side of the porter. Exactly where these resistances are placed depends on what is known about the particular porter being modeled and the degree to which a faithful representation of the mechanism is desired. The impedance coupling relationship (equation 13) ensures that appropriate values placed on either side will have an equivalent value on the opposite side (see Fig. 2).
Pumps are special cases of porters. The input side of a pump is driven by the free energy, expressed in volts, of the chemical reaction (usually ATP hydrolysis) driving the pump, i.e. the pump potential (Vp). The pump coupling coefficient (mk) is the ratio of the number of energy substrate molecules split to the number of ions pumped. Fig. 3A depicts an ideal pump whereas Fig. 3B depicts a more realistic pump with frictional and slippage resistances added. All power, current, voltage and impedance relationships hold for pumps as they do for porters.
Sample circuits (corresponding to the models in Harvey, 1992)
Example 1: a proton pump in a vesicular membrane
An extremely simple system consisting of a pump for a single ionic species in a membrane separating two aqueous compartments (Fig. 4A) is considered first. Based on this hypothetical situation the circuit shown in Fig. 4B can be drawn. In the circuit, the ionic (in this case, proton) pump is depicted by the Vp/porter combination. For the purposes of these analyses the frictional and slippage resistances are considered to be negligible and are ignored. In the steady state, in this case equilibrium, the pump will generate a proton-motive force (pmf) which is the product of the pump potential (Vp) and the proton porter coupling coefficient (mp):
Example 2: a proton pump in a vesicular membrane with gegenion leakage
To acidify a compartment significantly the value of the effective membrane capacitance must be close to, or greater than, the proton compartmental capacitance. This condition can be achieved by adding a channel for a non-pumped ionic species (i.e. a gegenion), which moves under the influence of the membrane potential and its own activity gradient only. A gegenion channel is included in the system shown in Fig. 5A,B. In this circuit the EM is the sum of two voltages: the voltage drop across the gegenion resistance (Rk) and that of the total gegenion compartmental capacitance Ctk. In the steady state no current will flow through Rk, effectively putting Ctk and CM in parallel. The equilibrium relationship of the membrane potential to the other components of the system is described by equation 20:
In this example, acidification is made possible because the larger capacitors dominate the system in the steady state. Since both the protons and the gegenions occupy the same compartments, it is reasonable to assume that the compartmental capacitances for the two ions will be similar. In this situation, significant charging of the proton compartmental capacitance could be effected by the proton pump. It makes no difference whether the gegenion is an inwardly moving anion like chloride or an outwardly moving cation like sodium. Either can increase the effective steady-state membrane capacitance of the system and cause development of a pH gradient with the inside of the vesicle being acid with respect to the bath.
Example 3: a vesicular membrane with a proton pump, a porter and gegenion leakage
A proton pump can be used to make a compartment alkaline if its energy is used to drive an ionic cotransporter and the effective membrane capacitance is made correspondly large by passive gegenion movement. Diagrams for this circuit can be seen in Fig. 6A,B. In these diagrams the cotransporter transfers energy in the proton circuit to the potassium circuit. Consistent with the direction of energy transfer, the coupling coefficient is designated mHK, and the cotransporter is designated an H+/K+ antiporter. The potassium-motive force (Kmf) is related to the pmf by:
Example 4: the goblet cell in the midgut epithelium of the larval lepidopteran
This example describes a polycompartmental system; it provides a basis for a discussion of the salient points that must be considered whenever complex ion circuits are to be analyzed.
Fig. 7A describes the principal ionic pathways found in the goblet cell of the larval midgut of Manduca sexta (Harvey, 1992). The system represented has four compartments: the midgut lumen, the goblet cavity, the goblet cell intracellular space and the blood. A proton pump and proton/potassium antiporter are located in the membrane lining the goblet cavity. Bicarbonate, or perhaps more likely carbonate, is generated by metabolic activity and passes preferentially across the apical side of the epithelium. The apical membrane consists of the membrane lining the goblet cavity, continuing through the valve region, and across the lumen/intracellular interface.
Fig. 7B is a schematic diagram of the goblet cell system showing the component parts of the pathways shown in Fig. 7A. Each compartment has ionic compartmental capacitances associated with it: CH, CK and Ck. These are single compartment ionic capacitances rather then the combined compartmental capacitances used in the previous circuits of this paper. Along with the polycompartmental considerations mentioned previously, the single compartmental capacitances are used here to isolate each ionic pathway from the others, d.c. connections between the different ionic pathways do exist and will be the topic of a later publication.
MemBrane capacitances are found associated with each portion of membranes where they separate different compartments; Cv is a composite capacitance for the valve assembly. Although it is probably very small and not representative of the complexity of this structure (see Moffett and Koch, 1992), it does allow the goblet cavity/lumen interface to be drawn like all the others. Membrane potentials are measured at the locations indicated by the test point circles and, hence, are only capacitatively coupled to the voltage drops associated with the ionic currents.
As drawn here, the goblet cell has a ‘power take-off in the potassium ionic circuit. Potassium currents traveling longitudinally in the lumen and the blood can drive amino acid uptake in neighboring columnar cells (see Fig. 8). These extra goblet cell pathways would add to the load on the porter and hence to the load on the pump.
During, build-up to the steady state, membrane capacitances and ionic compartmental capacitances (all non-pumped ionic compartmental capacitances are lumped with the membrane capacitance in Fig. 8 to simplify the diagram) are charged in accordance with the principles mentioned earlier. If the system is closed and reaches a steady state, the load on the pump is a function of the load presented by the parallel combination of proton leakage currents and proton currents through the potassium antiporters. The load contributed by the antiporter is a function of the net resistance of the potassium circuit. Also, the size of the various membrane capacitances and ionic compartmental capacitances will not affect the steady-state load on the pump in any way, except as compartmental pH, transmembrane potential or the like can activate or inhibit membrane channels in the pumped or ported circuits.
However, no real biological system is ever closed. For the living insect larvae, the materials in the gut lumen and the materials in the blood are constantly in flux. In the gut and the interstitial fluids, fresh material is constantly being added and old material is being removed, giving these compartments a very large effective capacity and preventing them from becoming fully charged. Under these conditions, the steady state is like a sustained transition state. Currents would constantly be flowing in non-pumped and nonported ionic pathways and, to the extent that these currents were a function of the pump/porter system, the magnitudes of their pathway conductances and capacitances would affect the loading on the pump.
Viewing the energy distributions in membranes as functions of the voltage, cunent, capacitance and resistance parameters of the pumps, porters, channels, membranes and compartments of the system presents an opportunity to write and use computer programs like those successfully used in electrical and electronic engineering. These programs allow investigators and students to study how different circuit configurations can quantitatively affect the operation of a system. Circuit analysis programs able to handle the ionic systems of unit and epithelial membranes must include provisions for d.c. transformers of pumps and porters and provisions for handling the ionic compartmental capacities. They must also be able to handle the inherent nonlinearity of the compartmental capacitances and other components and the nonlinearities on circuit components affected by protein conformational changes in response to membrane potential, pH or ionic strength.
Another possible line of study with ionic circuit analysis is to consider the possible role of impedance matching between various ionic circuits on fitness of the organism to its environment and behavior. Maximum power delivered to a load is achieved when the impedance of the load is equal to the internal impedance of the power supply circuit. Whether the loads put on pumping circuits are relatively steady, as may be the case with the constantly eating larva of the tobacco hornworm Manduca sexta, or periodic, as is the case for the gypsy moth Lymantria dispar, which eats in spurts (M. G. Wolfersberger, personal communication), the matching of the energy demand to the energy supply would seem to be important, d.c. circuit analysis techniques described here could easily handle conditions in which the loads are constant. Varying loads in capacitative circuits introduce phase differences between the currents in the different ionic circuits that must be considered. In circuit analysis terminology such varying loads are called reactive loads. With periods of hours, days or perhaps weeks and months, matching the reactive loads to the energy distribution system would require circuits with very large capacitors such as the ionic compartmental capacitances described here. It would be interesting to study the degree to which evolutionary processes have selected for impedance matching as an energy optimizing strategy.
Determination of change in equilibrium potential with a movement of ions from one compartment to another
The series combination of ionic compartmental capacitances
Portetr coupling considerations
Minimal value of coupling coefficient required to alkalize a compartment
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-22444 to William R. Harvey. I wish to thank my colleagues at Temple University, Dr William Harvey, Dr Michael Wolfersberger and Dr Ranganath Parthasarathy for keeping me abreast of the current trends in transport physiology and the role of V-ATPases and, along with Dr Allan Koch of Washington State University, for reviewing this work and making many helpful suggestions. Finally, I thank Mr Daniel J. Harvey for his help in preparing the figures for publication.