The use of linear nonequilibrium thermodynamics in the study of renal physiology.

The use of linear nonequilibrium in the study of renal physiology

thermodynamics

ESSIG, ALVIN, AND S. ROY CAPLAN. The use of linear nonequilibrium thermodynamics in the study of renal physiology. Am. J. Physiol. 236(3): F21bF219, 1979 or Am. J. Physiol.: Renal Fluid Electrolyte Physiol. S(3): F21LF219, 1979.-Classical formulations for the analysis of membrane transport processes, which ignored possible interactions between flows of diverse permeant species, often led to inconsistencies in the evaluation of permeability coefficients. For water flow induced by an osmotic pressure difference this difficulty was resolved by Staverman’s introduction of the reflection coefficient 0, a parameter which incorporates the interaction between solute and solvent in the course of their passage through a membrane. A comprehensive nonequilibrium thermodynamic (NET) formalism suitable for many biological systems was provided by Kedem and Katchalsky. For an n-flow system each flow is in general dependent on n forces; the assumption of Onsager reciprocity, however,‘reduces the number of independent phenomenological coefficients. Although NET is widely applied in the study of renal physiology, fundamental theoretical and practical problems remain. Basic considerations are the need to control or evaluate the influence of all coupled flows and to establish conditions fostering linear dependencies of flows on forces. When this is done a transport system may be characterized in terms of intrinsic membrane parameters, facilitating the systematic study of the effects of drugs, hormones, and various experimental perturbations. transport;

permeability;

experimental

physiology in terms of mechanism, it has long been appreciated that it is desirable also to analyze transport processes systematically in terms of formalisms. Ideally such formalisms should define intrinsic membrane parameters, permitting the self-consistent correlation of diverse data and the prediction of function in a wide variety of conditions. Understandably, early attempts to describe permeation processes in this manner were simple. In analogy to Ohm’s law for electric current, Fourier’s law for heat flow, and Fick’s law for diffusion in solution, it was considered that both solute flow and solvent flow across a membrane could be described appropriately by simple proportionality relationships. Accordingly, the rate of solute flow per unit area of membrane, which we may call J,, was considered proportional to the concentration difference AC,

perturbations;

DESPITE THE ULTIMATE GOAL~OFunderstanding

J, = P&,

(1)

with the proportionality constant P, being the permeability coefficient. Similarly, with a hydrostatic or osmotic pressure difference resulting in water flow it was considered that 0363.6127/79/oooO-0000901.25

renal physiology

J = P,(Ap - ANT) W

or introducing

(2 a )

the van’t Hoff relationship 3 = P,(Ap - RTAc) W

(2b)

where AC here represents the total concentration of all solute species, whether permeant or not. Unfortunately, with the progressive accumulation of experimental data it became clear that these relationships failed to provide a self-consistent description of solute and water transport, and that the permeability coefficients evaluated in ‘a given membrane differed markedly, according to the experimental protocol employed. The inadequacy of these formulations may be easily appreciated by considering the ambiguities associated with the use of equations 2a and 2b to evaluate the osmotic pressure of a solution containing only a single solute. Classically, one means of doing this was to determine the hydrostatic pressure head necessary to prevent the flow of pure water across a semipermeable membrane separating it from the test solution. When the membrane is impermeable to the solute, this technique, although slow, offers no significant problem: in time an equilibrium is established in which J, = 0, so that A?Tis given by the

Copyright 0 1979 the American Physiological Society

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A. ESSIG AND S. R. CAPLAN

F212

hydrostatic pressure difference at equilibrium. If, however, in an attempt to speed the process a more permeable membrane is employed, permitting permeation by the solute, the situation becomes less satisfactory. Now, with both solute and solvent flowing across the membrane, equilibrium is reached only when mixing is complete. If, in an attempt to eliminate the influence of solute permeation, measurements of Ap are extrapolated to time zero, when no solute has yet crossed the membrane, inconsistent results are obtained, depending on the nature of the solute and membrane employed. Coupling of Flows; Introduction Nonequilibrium Thermodynamics

of

This difficulty was resolved by Staverman (52, 53), who employed the formalism of linear nonequilibrium thermodynamics (NET) for the analysis of interacting flows. On this basis Staverman pointed out that in the presence of permeant solutes the classical procedure for the determination of An was in principle unsatisfactory, and that it was necessary to consider a new parameter, which he called the reflection coefficient. To quote, “If the leakage of a membrane to a molecular species i is described by a reflection coefficient ai, ranging from zero for completely permeable membranes to unity for completely impermeable membranes, the membrane behaves from the start as if not ni molecules were present in the solution, but only Gini” (52). Accordingly, 0 is the ratio of the apparent to the real osmotic pressure. At this point it is only necessary to add that deviation of a from 1 is a manifestation of the coupling of solute and water flows. That is, whatever factors influence solute flow will influence solvent flow, and vice versa; therefore, CFwill depend on the interactions between water, the specific solute, and the specific membrane under study. Coupling may be direct, as in frictional interaction between solute and water (28,50). On the other hand, in complex membranes it may be indirect, reflecting devious routes of solute and water movement. The concept of the reflection coefficient has, of course, been usefully applied to many problems of biological salt and water transport. As an early and important example, we may consider near-isosmotic fluid reabsorption. This common finding and the occasional description of transport of water against a concentration gradient led certain workers to invoke active water transport. Durbin (11) and Curran and Macintosh (8), however, suggested mechanisms by which such observations could be the consequence of passive water movement coupled to active salt transport, and Curran and Macintosh constructed an artificial model with which they demonstrated the feasibility of such a mechanism. Their model consisted of three compartments in series, A and B being separated by “tight” dialysis tubing, and B and C by a “loose” sintered glass disk; compartments A and C were open, while B was closed. With this arrangement it was possible to demonstrate net flow of solution from A to C when the solute concentration in A was less than in B, despite the fact that the solute concentration in A was greater than in C. The explanation for this phenomenon lies, of course,, in the different values of CFfor the two “membranes.” With a higher value of 0 for the dialysis

tubing, the effective” osmotic pressure difference (-ohv) between A and B is large, resulting in water flow into B, with creationof a hydrostatic pressure difference AP. With a low value of u for the glass disk, -OAT between B and C is small, so that the orientation of the net force W - UAT) is such as to cause solution to flow from B to C. Clearly, if 0 were the same for both membranes there could be no net flow from A to C, since, in this case, whatever the relative hydraulic conductivity of the two membranes, the orientation of the solute concentration gradients would result in water flow from both A and C into B. Although precise details remain to be defined, increasing knowledge of structure and function supports strongly the general validity of such a model for isosmotic water reabsorption in several organs. Such evidence speaks to the fundamental importance of the concept of the reflection coefficient, but in itself the evaluation of 0 does not, of course, constitute a suitable basis for the comprehensive analysis of transport processes. However, a selfconsistent framework was provided by Kedem and Katchalsky (23, 27.31), who in a classic series of papers applied linear NET to biological systems, following the approach of Staverman. The spirit of their treatment may be appreciated by considering the simple case of two steady-state flows, that of a nonelectrolyte solute and that of water. Various impermeant solutes may also be present. According to the fundamental concept of NET it is not assumed a priori that the flows are independent, but rather it is considered that they may be coupled. Therefore, in general, it is considered that the flow of solute Js will depend not only on the chemical potential difference of solute Ab, but also on Apw, whereas Jw will depend not only on Apw, but also on Aps. It is then postulated that in the steady state (when flows and forces are near constant) there will be an experimentally useful range in which the flows will be linear functions of the forces. Such a formulation offers the promise of consistency and predictability, but seemingly at the cost of great complexity, since the two permeability coefficients of equations 1 and 2 have now been replaced by four “phenomenological” coefficients. Here, however, there is recourse to the powerful reciprocal relation of Onsager, according to which, for a system sufficiently near equilibrium, the cross-coefficients will be equal, i.e., the dependence of J, on Apw [Lw = (aJ,/~Ap,)~] will be equal to the dependence of 3, on b8 [L8 = (M,/~ApJ~]. Incorporating this relationship, the description of two coupled flows involves only three independent coefficients. Although this approach is useful for analytic purposes, in practice one does not usually measure Ap, or APT, nor is water flow discriminated from volume flow. Therefore, the above relationships. were manipulated algebraically to provide equations suitable for practical calculations on experimental data. Representing the rate of volume flow by J,, the osmotic pressure difference of impermeant solutes by ATi, and the “mean” solute concentration by c,,l gives ’ The mean concentration cBis defined as the logarithmic mean, AC,/ Aln ce.For small Ace, csapproximates the arithmetic mean concentration of the two bathing solutions.


EDITORIAL

F213

REVIEW

J V = Lp(Ap 0 Awi 0 ~RTAcJ

at the molecular level. Like NET, the Nernst-Planck equation constitutes a phenomenological, macroscopic and approach. However, the Nernst-Planck equation and the J 8 = oRTAc, + (1 - a)cBJ, (4 Goldman-Hodgkin-Katz equation derived from it are limited in applicability, omitting as they do a consideration to NET, the possibility of interaction of These relationships, in contrast to eqz&ions 1 and 2, fundamental incorporate the coupling between solute and water flow. flows. It is true, of course, that many biological systems (Remembering that for dilute solutions J, = t, J,, where show strikingly nonlinear behavior, or respond too rapVW is the molar volume of water, it is seen that the two idly to permit their ready analysis in the steady state; sets of equations become practically equivalent only for such systems may possibly justify analysis by means of (42). It is also. true, however, the special case where 0 = 1.) Therefore, equations 3 and network thermodynamics that various transport processes are characterized bY 4 provide a basis for a systematic, self-consistent treatlong mlterm steady states with simple linear behavior. As ment. Assuming. the validity of the Onsager reciprocal will be discussed, there are means by which the investirelationship, a small series of practical measurements gator can foster linearity, thereby permitting analysis bY will characterize a system in terms of three fundameans of a simple formalism. Although on the basis of u, and 0, specific for a given mental parameters, Lp, analysis of a model certain workers have solute-membrane system, and, it is hoped, constant over theoretical claimed failure of the Onsager reciprocal . relations (3), a useful range of operating conditions. these arguments have been disputed (49). We are aware of no instance in which reciprocity fails to apply in the Present Status of NET linear range of concern to us here. Although the original It cannot be denied that renal physiology has been basis for assumption of the general validity of the reciimportantly influenced by the concepts discussed above. procity relations was Onsager’s statistical proof, valid The use of equations 3 and 4 and more complex relationonly very near equilibrium, reciprocal relations have now ships incorporating multiple nonelectrolyte and/or elec- been shown experimentally to apply to a great variety of trolyte flows is commonplace. Few physiologists will now- coupled processes well rem ,oved from equilibrium; their adays attempt to evaluate solute permeability without specific validity for membrane transport is supported by controlling volume flow, or water permeability without their equivalence to the well-established Saxen relationaccounting for solute flow. Nevertheless, it is also true ships of classical electrokinetics (41). Finally, although that in some ‘hinds there are substantial doubts as to the NET cannot, of course, discover mechanisms, the evalutility of NET. Many feel that the formalism is too uation of intrinsic membrane parameters permits tests of abstract and difficult. Why is it not possible. to start with the validity of models. Newton’s laws of motion or the Nernst-Planck equation, In sum, although it cannot be assumed that all biologfamiliar and well-understood fundamental relationships ical systems can be studied in the steady state ‘or that of accepted validity? Others question the range of applithey will always be linear, we believe linear NET to be a cability of linear relationships. Granting linearity in sim- discipline of broad utility, providing important benefits ple homogeneous membranes, what about the complex for the study of physiology, and, in particular, renal heterogeneous structures of biology? Even Katchalsky physiology. However, it is not a discipline that can be and his colleagues (42) have written, “It has become applied by rule of thumb. Care must be exercised in the almost a cliche to acknowledge that living entities are choice of systems and experimental conditions and in the extremely complicated, heterogeneous, non-linear sys- interpretation of data. These considerations may be tems.” Doubts are also expressed as to the general validmade explicit by examination of various problems to ity of the Onsager reciprocal relationships. If these fail to which NET has been applied, and the methods of study apply in complex biological systems how can one deal employed. with the multiplicity of necessary phenomenological coefficients? Finally, even granting the validity of the for- Analysis of Osmotic Water Flow malism, how can thermodynamics tell us anything about One useful example is osmotic water flow, which has mechanism? Given such doubts, many wonder whether it is worth the time and effort necessary to learn to use been studied in many systems. We shall consider first NET. Rather than forcing observations to conform to a Diamond’s (10) studies in the rabbit gallbladder, in which osmotic gradients were produced by use of the impertheoretical construct, would it not be more meaningful meant nonelectrolyte sucrose. Under the conditions emand satisfying to interpret them in terms of plausible ployed, the influence of active solute transport on water models? We take a different point of view. Admittedly the transport may be neglected (62). With no quantitatively significant coupled flows, and Ap’= 0, equation 3 becomes physical and statistical underpinnings of NET are subtle, simply but, as with many techniques, its successful experimental application is not dependent on full understanding at the J = -L,An (5) most fundamental level. The phenomenological coeffiAt first sight, an attempt to use equatio n 5 to evaluate a cients have been interpreted in terms of frictional interactions obeying Newton’s third law (23, 28, 50), but well-defined Ltp might seem futile in this system, since, to quote Diamond, “As in many other tissues, water flow present-day understanding of solutions and membranes was found to vary with gradient in a markedly non-linear does not permit a satisfactory theory of permeation mechanisms based on the application of Newton’s laws fashion. There was no consistent relation between the (3)

V


F214 water permeability and either the direction or the rate of water flow.” Such observations raise forcefully the question of the utility of a linear thermodynamic formalism to characterize even relatively simple processes in complex biological systems. Thermodynamics cannot, of course, predict whether the response of a flow to perturbation of a force will be linear. As pointed out by Prigogine (44) in speaking of linear laws, “It is clear that the existence of such laws is an extra-thermodynamic hypothesis. . . .” But it is also true, as emphasized repeatedly by Sauer (49), that for perturbation of forces in the neighborhood of equilibrium there will be some range in which the response of flows will be linear. In general, however, the range of linearity (as well as the values of the phenomenological coefficients) will vary with the choice of conditions (“reference state”) (49). For example, it is possible to perturb an osmotic gradient by varying the concentration of a given solute in either one or both bathing solutions; if both are varied, it is possible to maintain either the arithmetic or logarithmic mean concentration constant. The behavior of the system will, of course, reflect the choice made as well as the initial state of the system. To put this in terms of mechanism, flows depend not only on forces, but also on such characteristics as membrane geometry and concentration and electrical potential profiles, i.e., characteristics not controlled in the average experiment. Under these circumstances it is to be expected that whether or not a membrane can be characterized by a unique L, in a given situation will depend on whether or not rate-determining features of membrane structure happen to be altered by the means chosen to measure L. The range of possible linearity must, of course, be determined experimentally in every case. The pertinence of these considerations becomes clear when it is discovered that in the cited experiments ‘.‘the resistance to water flow increased linearly with osmolarity over the range 186-825 m-osm” and “the resistance to water flow was the same when the gall bladder separated any two bathing solutions with the same average osmolarity, regardless of the magnitude of the gradient” (10). That is to say, although it was impossible to measure a well-defined L, by addition of sucrose to only the mucosal or serosal bathing solution, the symmetrical perturbation of sucrose in both solutions, so as to maintain constancy of the mean solution concentration (and thus, presumably, of membrane parameters), resulted in a strikingly linear relationship between the magnitude of the sucrose gradient and the streaming potential (under conditions in which the streaming potential was considered proportional to water flow), thereby permitting the calculation of a well-defined unique-valued L, (Ref. 10, Fig. 9). It must be emphasized, however, that this value of Lp does not apply to the membrane under all circumstances, but only under those referable to the same conditions of study (49). The choice of conditions is, of course, infinite (e.g., the sucrose gradient may be altered while maintaining the arithmetic mean concentration at 100 mM or the external solution concentration at zero) but it is quite possible that some choices may be particularly convenient, resulting in stability and large ranges of linearity. However this may be, there is little point in


attempting to define the effect of an experimental variable (e.g., a drug or hormone) on Lp (or u or a) unless comparisons are carried out with the same reference state.2 Similar considerations apply to the study of Fromter et al. of osmotic water flow across the proximal tubule of rat kidney (Ref. 15, cited in Ref. 54, in which see Fig. 3). Here when streaming potentials were plotted against the raffinose concentrations in the luminal or capillary perf&ate, they gave a nonlinear but nearly symmetrical curve. Again, this finding indicates a strong dependence of L, on the mean solution concentration of the osmotic agent, and suggests the possibility that if the tubule were f”lrst exposed to the Same concentration of raEnose at each surface, and the concentrations were then perturbed symmetrically, J, might be a linear function of Ar, permitting evaluation of a unique-valued Lp characterizing the state of the system under the conditions of its study. It is to be expected that measures taken to minimize changes of the membranes in the course of their study will be likely to facilitate measurements not only of a unique-valued L,, but also of a, 0, and other intrinsic membrane parameters. Analysis

of Flows in Heterogeneous

Membranes

Additional complexities will, of course, be anticipated when dealing with electrolyte flow across heterogeneous membranes. Sometimes these may prove intractable, but this is a matter to be determined for each case individually. As previously, the approach will be to identify the significant coupled flows and to analyze the relationships between flows and forces by means of formalisms and techniques most conveniently applicable to the situation at hand. For example, in the study of electrolyte flows it is often adequate and convenient to evaluate total electric current rather than all component cation and anion flows; ease of analysis may then dictate the measurement of the electrochemical potential difference (Afii) of one of the ions by the use of reversible electrodes rather than measurement of the electrical potential difference (A#) (22, 29). As always, the emphasis will be on a search for conditions leading to simple linear relationships, facilitating the definition and evaluation of intrinsic membrane parameters. The difficulties and the approach may be appreciated by comparing the flow of a nonelectrolyte and that of a uni-univalent salt in the absence of significant volume flow. We consider first a system composed of parallel membrane elements (30). For this case the flow of non2 A later study of rabbit gallbladder by Wright et al. (62) confirmed the fmding.of nonlinear osmosis, but differed in also demonstrating asymmetry of water flow, attributable largely to changes in the dimensions of lateral intercellular spaces, and a limited range of proportionality between streaming potentials and flows, attributable to structural changes and solute polarization. These differences do not alter our point, namely, that in every case in which it is desired to evaluate an intrinsic membrane parameter it is important to carry out experiments in such a way as to minimize changes in membrane structure and function. In the study of Wright et al. the addition of only 50 mM sucrose to either the mucosal or serosal bath appeared to have little influence. on the membrane, since in this range the values of LP measured from water flow in the two directions differed insignificantlv.


EDITORIAL

F215

REVIEW

electrolyte may be described simply. Since total flow is the sum of the flows through alI the elements, whereas the force acting across each element is the same, it is seen that the overall phenomenological coefficient is the sum of the individual elemental values, weighted appropriately for element area. For the electrolyte the same is true, provided that the discrete elements are of similar permselectivity. For both cases J tends to be insensitive to mean concentration, and is likely to be a linear function of AC. Very different behavior is observed if the elements differ significantly in permselectivity. In this case, in the absence of net current flow cations and anions move in the same direction through separate channels, resulting in a circulation of current and marked enhancement of salt permeability. Although it might seem that this type of system would show highly nonlinear behavior, it was found that such a mosaic membrane was, in fact, highly insensitive to mean concentration (in the regime of membrane control), with salt flow a linear function not of AC but of ApS (59,60). For nonelectrolyte flow across a composite series membrane (3l), again there is additivity, but now of resistance phenomenological coefficients. The same is true for electrolyte flow, provided that the series elements are of similar permselectivity. If we consider, however, a series array of two membranes differing in permselectivity, dramatically different behavior may be observed, with a buildup or depletion of internal salt concentration, depending on the direction of current flow, with resultant rectification. Rectification is, of course, commonly observed in biological membranes, but more commonly in excitable tissues than in epithelia. As would be expected from the analysis of behavior of synthetic membranes, the importance of the effect will be related to the magnitude and duration of current flow, and it often is possible to analyze electrolyte flow in epithelia under conditions in which current-voltage relationships are linear. Determination of Permeability Coefficients; Tracer Flows Special problems are associated with the measurement of solute permeability coefficients. As discussed above, the evaluation of an intrinsic membrane parameter requires the measurement of a flow induced by a welldefined “force,” with control of significant coupled flows. This is generally more easily accomplished in isolated epithelial sheets than in minute nephron structures, in which the identification and control of coupled flows may constitute a difficult technical problem. An influence which is often overlooked is that of active transport, i.e., the coupled flow of metabolism. When an agent alters the apparent permeability coefficient of a species which is actively transported, it cannot be assumed that the effect is only on intrinsic membrane permeability rather than on the energetics of the process as well. For evident reasons, measurements of permeability in the nephron often involve the use of tracer isotopes. This raises another problem in the possibility that the flow of tracer may be influenced not only by its electrochemical potential difference and coupled flows of other species

and metabolism, but also by the flow of the abundant form of the test species (“isotope interaction”) (26). In this case the tracer permeability coefficient o* will differ from the permeability coefficient for net flow ~3.This has long been known to be the case for water flow, since soon after the introduction of isotopes it was found that the permeability measured osmotically far exceeded that measured from tracer isotope flow in the absence of net flow. This discrepancy was explained in part by KoefoedJohnsen and Ussing (33), who pointed out that a diffusing molecule “may possess, superimposed upon its rate of diffusion, that rate at which the solvent flows.” Related effects were described also for solutes, e.g., for K+ flux in the squid axon, where again a exceeds ~3” (commonly attributed to “single-file” diffusion) (19), and for Na+ flux in the frog sartorius, in which o* exceeds ~3 (generally attributed to “exchange diffusion” via a carrier) (38). Although these discrepancies have .been attributed to specific mechanisms, the observations are purely phenomenological, and therefore consistent with a variety of mechanisms. Indeed, values of o/w* = 0.2 have been demonstrated in polyelectrolyte-impregnated collodion membranes which are incompatible with transmembrane flux of carrier (39), and even lower values have been demonstrated in mosaic heteroporous membranes (40). Despite the well-known examples mentioned above, although the difference between tracer permeability and net permeability is well appreciated for the case of water flow, coupling of ionic flows is often considered to be of little significance in the characterization of nephron electrophysiology, the reasoning being that ionic interactions are weak in dilute aqueous media. This argument overlooks the fact that the rate-limiting barrier for ion movements will often be the plasma membrane, in which the routes of flux and the extent of interaction have been only poorly characterized. Furthermore, it is worth mention that a modest degree of negative isotope interaction (O < o*) has been suggested for flux (presumably paracellular) of Na+ and Cl- in toad bladder sacs (6) and for flux of Cl- in toad bladders mounted in chambers (48). If such isotope interaction occurs in regions of the nephron, uncorrected measurements of tracer permeability will lead to inaccurate estimates of partial conductance. Although comparisons of tracer and electrical measurements suggest that tracer flux coupling as well as ion-ion interaction are unimportant in rat proximal tubules for Na’ and Cl- (16,54), there are currently too few available data to permit a definitive evaluation of the overall significance of these factors in other regions of the nephron. Applications

to Renal Physiology

Many workers have applied one or another aspect of NET to study of the physiology of the nephron. The most comprehensive treatment has been‘ made by Ullrich, Fromter, and Sauer and their colleagues (16). Their work and the related work of others has been well summarized in a recent extensive article by Ullrich (54), accompanied by a fundamental analysis of its theoretical basis by Sauer (49). We will comment briefly on aspects


F216

A. ESSIG

related to the general issues discussed above. Although we will not consider details of technique (55), it is worth mentioning features which may affect the feasibility and accuracy of evaluation of the NET parameters. Continuous microperfusion entails multiple transmembrane flows, with the possibility of significant interaction as discussed above. Stop-flow microperfusion provides the advantage that with the passage of time net solute and water fl.ow come to a halt, facilitating the evaluation of tracer permeability coefficients. The simultaneous perfusion .of blood capillaries (15,51) permits the measurement of backflux. The study of isolated kidney tubules where technically feasible may allow precise control of transtubular concentration and electrical potential differences. Measurements

of Lp

As has been discussed by Ullrich (54), measurements of L, (and (3) in different structures are not precisely comparable, owing to uncertainties in regard to the area of permeable surface. As mentioned above, physiologists are now well aware of the importance of interactions between solute and solvent flows, and, therefore, measurements of L, are generally carried out under conditions which minimize the influence of solute flow. It is disconcerting, however, to note in a survey of a large number of articles in the recent literature that only rarely is mention made of tests done to assure that hydraulic conductivity is being measured in the linear range. Examination of the means employed to induce water flow, interpreted in the light of the studies cited above in the rabbit gallbladder and the rat proximal tubule, suggests that in most instances water flow is probably measured in a nonlinear range. If so, there is no basis for the application of equation 3 or analogous equations and, even in the complete absence of ‘coupled flows, L, is undefined. Under these circumstances, putative values of L, are not comparable, even if measured in the same region of the nephron, unless by chance the different investigators happened to employ precisely the same means of perturbing water flow (e.g., the same mean concentration and gradient of sucrose). The obvious solution to this problem is for researchers to describe their experimental protocols precisely and to alter solution concentrations in a way as to obtain a linear response, thereby permitting the determination of a unique-valued L characteristic of the conditions employed. LP values determined in this manner can then usefully be compared before and after perturbation of pertinent experimental variables. When L, is evaluated appropriately, as for example in the studies of Hays and Leaf (US), it is useful to compare its value to the water tracer diffusion coefficient (with appropriate dimensional corrections). This comparison has led to insights concerning the significance of unstirred layers in the toad bladder and the mechanism of action of antidiuretic hormone (17). (In the terminology of the present article Hays’s c3is represented by o*.) Measurements

of w

In addition to the general issues discussed above, certain snecific asnects deserve consideration with respect

AND

S. R. CAPLAN

to studies of the nephron. Apparent permeability coefficients are often cited without taking account of the value of the transmembrane electrical potential difference (A$). For electrolytes, this means that the coefficient incorporates a dependence on A$ which, although small in the proximal regions of the mammalian nephron, may well be appreciable distally. Since in some cases this factor might influence interpretations as to mechanism, it would seem desirable to use the Goldman-Hodgkin-Katz equation or flux ratio equation, where applicable, to calculate corrected permeability coefficients more likely to represent intrinsic membrane parameters. (Of course, these formulations entail assumptions.) The problem of distinguishing passive from active contributions to apparent permeability coefficients measured by tracer efflux from the lumen is not always soluble with techniques at hand.3 However, where A+ is not large it may be possible to evaluate passive permeability by measuring the reverse flow (influx) from the capillary or bathing fluid into the lumen (correcting for A+). The legitimacy of this approach is supported by the observation in the toad bladder that reverse flux through the transepithelial active transport pathway becomes measurable only when the transmembrane potential A# approaches the electromotive force of Na transport (ENS) (7, 61). In regions in which A+ is large, the tracer efflux is likely to be largely by way of the active pathway. The efflux may then be used to calculate a permeability coefficient for the ‘active pathway. It must be remembered, however, that in the presence of net flow this quantity will incorporate not only kinetic factors, but also the contribution of metabolism to solute flow. This is easily seen by realizing that tracer efflux-will persist even if the tracer concentration difference AC* is set at zero, in which case the apparent permeability coefficient J*/Ac* becomes infinite. We have discussed’ the fact that the standard treatments of tracer flux data for the calculation of partial conductances are contingent on the assumption that isotope interaction is insignificant and have cited evidence that this may be the case for the mammalian proximal tubule (16, 54). We agree, however, with Ullrich’s (54) statement that “The investigation of electrolyte permeabilities of collecting tubules and distal tubules, as compared to proximal tubules, offers a series of additional problems that have not yet been solved. . . the presence of ion-ion interaction and tracer flux coupling cannot be evaluated.” This is an area clearly deserving systematic study which may be amenable to recently improved techniques. Measurements

of 0

Unlike LP and ~3, 0 is dimensionless, and, therefore, directly comparable from one structure to another, irre3 Ullrich (54) and Fromter et al. (16) have suggested that in the case of simultaneous passive and active transport, the tracer data can be corrected by application of the factor (1 - AC/C), where AC and the mean concentration 5 are measured at zero net transtubular water and solute flux. Such a calculation, applicable for a specific model, was used only to estimate the possibility of significant error (F. Sauer, personal communication).


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spective of errors in measurement of the area of permeable surface. For purposes of general understanding we here ignore precise formulations in looking at some of the ways in which 0 has been determined. Assuming then that it is possible to minimize the influence of active transport and other coupled solute flows, we set AP = 0 and make A?T = 0 for aIl substances other th .an our experimental solutes; equation 3 then becomes J, = -L&WAC,, This equation points to two methods commonly employed to evaluate CT:1) determining the volume flow induced by equal concentrations of the test solute a and a reference solute b of known 0, in which case (J~/(J~ = Jva/Jvb; or 2) determining the concentrations of species a and b necessary to induce equal rates of volume flow, in which case o&, = Acb/Ac*. (It is, of course, convenient to have ob = 1.) Both of these methods obviously depend on the assumption that the two agents have equivalent effects on L,, which may not always be the case. An ingenious method which in principle eliminates this difficulty is the simultaneous placement of a and b on opposite sides of the membrane in concentrations adequate to bring J, to zero (34). (Of course, this method depends on the sensitive detection of low rates of volume flow.) Under these circumstances o&b = c&~, irrespective of any effects on L,. It is encouraging that values of proximal tubule ON&l determined in these various ways agreed well (54). Kedem and Leaf (32) have pointed out that individual ionic reflection coefficients o+ and D- are meaningfully measured in the absence of an electrical field, whereas amIt is best measured at open circuit, and they have shown how these quantities are related. The values of Ullrich et al. (54) for ONa,acl, and ON&1 are consistent with this relationship. Comprehensive

Measurements

of NET

Parameters

Fromter et al. (16) have carried out a comprehensive analysis of ion transport in the rat proximal tubule, altering J and thereby the several solute flows Ji by addition 0: raffinose to the 1umen and/or capillary perfusion fluids. The linear region of plots of Ji against their respective electrochemical potential differences (Abi), combined with tracer flux measurements, allowed estimates of L,, single ion permeabilities, and reflection coefficients. Finite values of Api at zero flow indicated active transport for Na+ and HCOa-, but not for Cl-. In examining these data it is noteworthy that linearity in the plots of Ji against A& was seen in the same range of raffinose concentrations as found earlier by Fromter et al. (15) to result in linear responses in volume flow. This raises the possibility that the same measures suggested above to extend the range of linearity for J, might, by minimizing changes in tissue structure and composition, extend the range of linearity in electrophysiological studies, thereby enhancing the accuracy of values of derived parameters. Fromter et al. (16) have pointed out that the analysis which they made is valid only near equilibrium, and, therefore, not easily applied to regions other than the proximal tubule. There are important problems in dealing with the influence of active transport on permeability coefficients. On fundamental grounds it is clear that, in

analogy to the need to evaluate or control the influence of coupled material flows in the case of passive transport, it is necessary to take account of coupling to the “flow” of metabolism in the case of active transport. In addition, since active transport presumably is by means of a carrier, there are theoretical reasons to expect that isotope interaction (“exchange diffusion”) may be important. To determine whether this is indeed so would require a comprehensive analysis both of transport and metabolism (26). The only attempts to estimate the extent of isotope interaction in an active transport pathway involved measurements of sodium fluxes and oxygen consumption in different toad bladders. Estimates of w*/ cd*a ranged from 0.82 (13) to 0.67 (61), but obviously a precise study would require the simultaneous measurement of the rate of active Na transport, the rate of suprabasal metabolism, and tracer flux. One useful approach to determination of the conductance of the transepithelial active pathway is to attempt to differentiate it from the conductance of the parallel passive pathway (presumably paracellular) by the use of an agent which eradicates the former without significantly affecting the latter. This appears to be the case for amiloride in the toad bladder (20,35,37). Ultimately, of course, it would be desirable to have a systematic treatment of the energetic factors influencing transport. Energetics

of Active Transport

A comprehensive thermodynamic treatment of the energetics of active transport would require the concurrent measurement of all active flows and the associated metabolism in a voltage-clamped perfused isolated region of the nephron-a formidable procedure with present day technology! It seems worthwhile to ask, however, what can be learned from studies in tissues in which these requirements can be satisfied (12,21,24). Very convenient tissues for the study ‘of transepithelial active sodium transport are the isolated frog skin and toad urinary bladder. Tissues from appropriate species carry out significant active transport only of sodium, and in the absence of osmotic gradients there is no appreciable volume flow or other coupled material flow. In air, metabolism is very largely aerobic. Studies in the presence and absence of amiloride (which eliminates active transport and reduces oxygen consumption to the basal level) permit measurement of the rate of active sodium transport (JNa*) and the rate of suprabasal 02 consumption (JFb). Again postulating linearity, in analogy to the treatment of passive transport above, we write (5, 12) JNaa =

LNa&Na

+

LNa,rA

(6)

J r sb = LNa,rAbNa + LrA

(7) Here A is the “affinity” (negative Gibbs free energy) of the metabolic reaction driving transport. It is to be noted that although there is no a priori basis for expectation of an extended range of linearity, in fact experiment shows linearity of both JNa* and JFb in A# over a range of t80 mV or more when A\c, is perturbed symmetrically at 6min intervals (37, 48, 58).4 Tests for linearity in the ’ It is of great importance results in linear responses

that Ai&, be perturbed in a manner of both ~~~~ and JTb, since otherwise

which equa-


F218 affinity A are as yet not possible because the means of setting A at desired levels are lacking. Similarly, it has not yet been possible to test for validity of the Onsager relationship [but we point out reciprocity in a synthetic model of an active transport system (2) and in mitochondrial oxidative phosphorylation (45)]. Presuming the validity of equations 6 and 7, it is possible to evaluate all the phenomenological coefficients and A by noninvasive means. It is to be noted that A here represents the free energy specifically available to the sodium pump, in contrast to the value of [ATP]/ [ADP] [Pi] determined by standard techniques, which reflects a mean free energy applicable to all tissue functions. Values of A are reasonable, ranging from 20-80 kcal/molO2 in the control state (37,43,47, 58), presumably corresponding to -3-13 kcal/mol ATP, and respond in appropriate fashion to the administration of amiloride (47), ouabain, and 2-deoxyglucose (43). However, more model compounds deserve to be tested. Studies with aldosterone indicate effects on both kinetic factors (phenomenological coefficients) and energetics (the affinity) (46, 47).

Although both ~~~~ and J,“” are linearly dependent on A$, their ratio appears not to be constant, falling to low levels as the transepithelial potential approaches ENa (5, 36, 37). It appears, therefore, that there is no unique “stoichiometric” ratio Na+/OQ, but rather the tissues are “incompletely coupled” (12, 25, 36, 37).5 This consideration raises compellingly the question of the utility of the classical equivalent-circuit model for active sodium transport for the analysis of energetics. In its original form this model considers only one flow, that of electrical current (5,20,56). This being the case, the electromotive force of Na transport (ENa) cannot in principle evaluate


a purely energetic parameter. This is readily seen by Setthg ApNa = -FEN, in equation 6, SO that ~~~~ becomes zero, giving FEN,

= (ha,r/ha)A

(8)

where LN a,r and L incorporate kinetic factors (permeability and rate coefficients).6 For the completely coupled system LNa,r/LNa rr=Jrsb/JNaa9 but in the absence of such fixed stoichiometry the kinetic factor LNa,r/LNe cannot be determined without measuring the potential dependence of JNaa and JFb. (It will be noted that the need to consider the flow of metabolism in the analysis of a coupled active transport process is closely analogous to the need to consider the flow of solvent in the analysis of the diffusion of a solute. More generally, for any process, it is necessary to consider all coupled flows.) It is not our intention to claim that this as yet incompletely tested formulation provides the correct basis for analysis of the energetics of active transport in even the relatively uncomplicated cases we have examined. But for a transport system which is incompletely coupled it is the simplest conceivable basis. As such, it underlines the fundamental considerations which we have emphasized throughout this Editorial Review, that to define and evaluate intrinsic membrane kinetic (and energetic) parameters in a self-consistent manner it is necessary to study all significant coupled flows under conditions min. imizing changes in the state of the system. It is hoped that one day this will be done for the kidney.’ We are most grateful to Drs. E. Fromter, F. Sauer, and K. J. Ullrich for critical reading and advice on the manuscript. This study was supported by a grant from the National Science Foundation (PCM 76-23295) and a grant from the U.S.-Israel Binational Science Foundation (B.S.F.), Jerusalem, Israel.

REFERENCES Q., A. MUELLER, AND P. R. STEINMETZ. Transport of H+ against electrochemical potential gradients in turtle urinary bladder. Am. J. Physiol. 233: F502-F508, 1977 or Am. J. Physiol.: Renal FZuid EZectroZyte Physiol. 2: F502-F508, 1977. 2. BLUMENTHAL, R., S. R. CAPLAN, AND 0. KEDEM. The coupling of an enzymatic reaction to transmembrane flow of electric current in a synthetic “active transport” system. Biophys. J. 7: 735-757, 1967. 3. BRESLER, E. H., AND R. P. WENDT. Onsager’s reciprocal relation. An examination of its application to a simple membrane transport process. J. Phys. Chem. 73: 264-266, 1969. 4. CANESSA, M., P. LABARCA, AND A. LEAF. Reply to: Metabolic cost of sodium transport and the degree of coupling of transport and metabolism in toad urinary bladder. J. Membr. BioZ. 41: 193-194, 1. AL-AWQATI,

bladder. J. Membr. BioZ. 18: 365-378, 1974. J. S., AND M. WALGER. Sodium fluxes through the active transport pathway in toad bladder. J. Membr. BioZ. 21: 87-98,1975. 8. CURRAN, P. F., AND J. R. MACINTOSH. A model system for biological water transport. Nature 193: 347-348,1962. 9. DANISI, G., AND F. L. VIEIRA. Nonequilibrium thermodynamic analysis of the coupling between active sodium transport and oxygen consumption. J. Gen. Physiul. 64: 372.391,1974. 10. DIAMOND, J. M. Non-linear osmosis. J. Physiol. London 183: 587. CHEN,

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5. CAPLAN, S. R., AND A. ESSIG. A thermodynamic treatment of active sodium transport. In: Current Topics in Membranes and Transport, edited by F. Bronner and A. Kleinzeller. New York: Academic, 1977, p. 145-175. 6. CHEN, J. S., AND M. WALSER. Passive ion fluxes across toad

R. P. Osmotic flow of water across permeable cellulose membranes. J. Gen. Physiol. 44: 315-326,196O. 12. ESSIG, A., AND S. R. CAPLAN. Energetics of active transport processes. Biophys. J. 8: 1434.1457,1968. 13. ESSIG, A., AND M. LANG. Letter to the editor. Comments on: sodium fluxes through the active transport pathway in toad bladder. J. Membr. BioZ. 24: 401-403, 1975. 14. ESSIG, A., D. WOLFF, S. ROSENTHAL, M. A. LANG, J. G. KING, AND S. R. CAPLAN. Metabolic cost of sodium transport and the degree

tions 6 and 7 are inapplicable. When AJ/ is perturbed asymmetrically or for longer periods, nonlinearity is observed. In the toad skin linearity is observed when Ai& is altered by change of external Na concentration (9), but not by change of internal Na concentration (57). It is felt that both of these instances of nonlinearity are likely to reflect changes in tissue microstructure and/or composition, as with the effects of asymmetric perturbations of nonelectrolyte concentrations cited above. 5 This view is not accepted by Labarca et al. (35), who feel that the Na’/Oz ratio is essentially constant with variation of A+ This difference in opinion reflects different criteria for the definition of experimentally accentable steadv states (4. 14).

6 In the toad bladder 2-deoxy-D-glucose depressed JN~*without effect on EN* (20), and in the turtle urinary bladder glucose addition, or deoxygenation, depletion of metabolic substrate, or addition of %deoxyD-&COSe all affected active H+ transport, with little or no effect on the apparent protonmotive force ( 1). ’ We have omitted any reference to the dissipation function. Although its derivation is necessary for the analysis of complex systems with nonconservative flows, we feel that it is not necessary for a practical treatment such as this in which pertinent independent forces and flows are evident.

1978.


EDITORIAL

F219

REVIEW

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A., AND P. F. CURRAN. Nonequilibrium Thermodynamics in Biophysics. Cambridge, MA: Harvard Univ. Press, 1965. 23. KATCHALSKY, A., AND 0. KEDEM. Thermodynamics of flow processes in biological systems. Biophys. J. 2: 53-78, 1962. 24. KEDEM, 0. Criteria of active transport. In: Proceedings of the Symposium on Transport and Metabolism, edited by A. Kleinzeller and A. Kotyk. New York: Academic, 1961, p. 87-93. 25. KEDEM, O., AND S. R. CAPLAN. Degree of coupling and its relation to efficiency of energy conversion. Trans. Faraday Sot. 61: 18971911,1965. 26. KEDEM, O., AND A. ESSIG. Isotope flows and flux ratios in biological membranes. J. Gen. Physiol. 48: 1047.1070,1965. 27. KEDEM, O., AND A. KATCHALSKY. Thermodynamic analysis of the

permeability of biological membranes to non-electrolytes. Biochim. Biophys. Acta 27: 229-246,1958. 28. KEDEM, O., AND A. KATCHALSKY. A physical interpretation of the phenomenological coefficients of membrane permeability. J. Gen. Physiol. 45: 1430179,196l. 29. KEDEM, O., AND A. KATCHAL~KY. Permeability of composite membranes. I. Electric current, volume flow, and flow of solute through membranes. Trans. Faraday Sot. 59: 1918-1930, 1963. 30. KEDEM, O., AND A. KATCHALSKY. Permeability of composite membranes. II. Parallel elements. Trans. Faraduy Sot. 59: 1931-1940,

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The effect of aldosterone on the energetics of sodium transport in the frog skin. B&him. Biophys. Acta 318: 371.382,1973. 48. SAITO, T., P. D. LIEF, AND A. ESSIG. Conductance of active and passive pathways in the toad bladder. Am. J. Physiol. 226: 12651271,1974. 49. SAUER, F. Appendix:

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ions across the fibre membrane of the isolated frog sartorius. Acta Physiol. Stand. 16: 2320249,1948. LI, J. H., R. C. DESOUSA, AND A. ESSIG. Kinetics of tracer flows and isotope interaction in an ion exchange membrane. J. Membr. BioZ. 19: 93.104,1974. LI, J. H., AND A. ESSIG. Ion tracer flows and flux ratios in heterogeneous membranes. Biochim. Biophys. Acta 465: 421-425, 1977. MILLER, D. G. Thermodynamics of irreversible processes. The experimental verification of the Onsager reciprocal relations. Chem. Rev. 60: 15-37,196O. OSTER, G. F., A. S. PERELSON, AND A. KATCHALSKY. Network thermodynamics: dynamic modelling of biophysical systems. Q. Rev. Biophys. 6: l-134,1973. OWEN, A., S. R. CAPLAN, AND A. ESSIG. A comparison of the effects of ouabain and %deoxy-n-glucose on the thermodynamic variables of the frog skin. Biochim. Biophys. Acta 394: 438-448,1975. PRIGOGINE, I. Introduction to Thermodynamics of Irreversible Processes. Springfield, IL: Thomas, 1955. ROTTENBERG, H. The thermodynamic description of enzyme-catalyzed reactions. Bbphys. J. 13: 503.511,1973. SAITO, T., AND A. E~~IG. Effects of aldosterone on active and passive conductance and ENS in the toad bladder. J. Membr. BioZ.

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E. M., A. P. SMULDERS, AND J. M. TORMEY. The role of the lateral intercellular spaces and solute polarization effects in the passive flow of water across the rabbit gallbladder. J. Membr. Biol. 7: 198-219,1972.

Alvin Essig and S. RQY Caplan Departments of Physiology and Medicine, Boston University Boston, Massachusetts 02118; and Department of Membrane Weizmann Institute of Science, Rehovot, Israel

School of Medicine, Research,


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