INVITED REVIEWMathematical models of renal fluid and electrolyte transport:ackn - 《American Journal of Physiology》美国生理学杂志干细胞之家



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发表于 2009-4-21 13:36 |只看该作者 |倒序浏览 |打印

作者：Alan M.Weinstein作者单位：Department of Physiology and Biophysics, Weill MedicalCollege of Cornell University, New York, New York 10021
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   Mathematical models of renaltubular function, with detail at the cellular level, have beendeveloped for most nephron segments, and these have generally beensuccessful at capturing the overall bookkeeping of solute and watertransport. Nevertheless, considerable uncertainty remains aboutimportant transport events along the nephron. The examples presentedinclude the role of proximal tubule tight junctions in water transportand in regulation of Na transport, the mechanism by whichaxial flow in proximal tubule modulates solute reabsorption, the effectof formate on proximal Cl transport, the assessment ofpotassium transport along collecting duct segments inaccessible tomicropuncture, the assignment of pathways for peritubularCl exit in outer medullary collecting duct, and theinteraction of carbonic anhydrase-sensitive and -insensitive pathwaysfor base exit from inner medullary collecting duct. Some of these uncertainties have had intense experimental interest well before theywere cast as modeling problems. Indeed, many of the renal tubularmodels have been developed based on data acquired over two or threedecades. Nevertheless, some uncertainties have been delineated as theresult of model exploration and represent communications from themodelers back to the experimental community that certain issues shouldnot be considered closed. With respect to model refinement,incorporating more biophysical detail about individual transporterswill certainly enhance model reliability, but ultimate confidence intubular models will still be contingent on experimental development ofcritical information at the tubular level. * z8 o9 C. F: P0 L  d1 x
      【关键词】 proximal tubule distal tubule collecting duct sodium potassium chloride acid/base
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IN THE PRESENTATION OF A PHYSIOLOGICAL model, declaration of success typically comes with the demonstration offaithful predictions of function in simulations of a number ofexperiments. Arguably, one may endorse a standard of model presentationin which the model builder shows not only what works but where themodel fails, or where it makes novel predictions that have yet to betested. In this regard, "model failure" may vary from small tolarge, where "large" implies an important observation that justcannot be captured by a reasonable model with realistic parameters. In some sense, this situation is the most interesting, because its resolution may provide new insights. In renal physiology, models ofglomerular filtration have probably enjoyed the closest working relationship with experimental data, initially representinghemodynamics and subsequently examining issues of glomerularpermselectivity. The successes and limitations of these models haverecently been reviewed in this series ( 20 ). Perhaps theoldest and most intense renal modeling effort has been to represent thetubules and vasculature of the kidney medulla in antidiuresis and theformation of a concentrated urine. The serious difficulties encounteredwith representations of inner medullary function have been welldocumented ( 56, 101 ) and will not be taken up here. Withrespect to tubular models, the greatest attention has been given to theproximal tubule, initially with regard to forces and routes of watertransport and subsequently focused on ion transport through thetranscellular pathway. More recently, segments of the distal nephronhave been modeled, and these simulations have been used to extrapolatein vitro transport observations to tubules in vivo. It is the aim ofthis review to survey models of renal tubular transport, specifically with regard to uncertainties encountered by the model builders. Thepoints of interest will be model limitations and the authors' responses, either to question experimental data or to propose testablemechanisms that might render the models satisfactory.. m( H- L1 b: z# K; [$ ?8 V
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PROXIMAL TUBULE: WATER TRANSPORT
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Figure 1 is a schematic of theproximal tubule epithelium, in which the lateral intercellular space(LIS) is separated from luminal and peritubular solutions by the tightjunction (TJ) and basement membrane (BM). All of the early models ofproximal tubule function had focused on water transport, specificallyto try to understand the forces responsible for isotonic waterreabsorption and the pathways for transepithelial water flow ( 45, 79, 80, 91, 114 ). The last review of these issues here( 116 ) was principally concerned with the paracellularpathway. Models of the proximal tubule had agreed that most of thetransepithelial water flowed through the LIS and out across the BM.Points of disagreement among the models included 1 ) themagnitude of the flow that arrived in the LIS via the TJ and via atranscellular route across luminal and lateral cell membranes in seriesand 2 ) the magnitude of the solute permeability of theoutlet BM (i.e., whether there was any significant resistance to soluteflux across this barrier). A finite solute permeability of the BM wouldresult in "middle compartment" behavior by the LIS. This refers toactive solute transport into the LIS across the lateral cell membrane,creating a region of local hypertonicity, which acts to pull water from cell to LIS, and ultimately from lumen to cell. The net result is thepossibility of transepithelial reabsorptive water flux in the absenceof a transepithelial osmotic gradient (coupled water transport).Furthermore, a finite solute permeability for the LIS BM would resultin solute polarization effects, namely, an overall epithelial waterpermeability less than that of the cell membranes in series and anoverall epithelial solute reflection coefficient less than that of theTJ and cell in parallel ("pseudo-solvent drag"). A previous modelhad argued that large flows of isotonic water reabsorption could existdespite relatively low proximal tubule water permeability if there weresubstantial solute polarization within the LIS ( 114 ). Inthe review ( 116 ), it was estimated that coupled watertransport was anywhere from 55-81% of isotonic waterreabsorption, depending on which of the experimental measurements ofoverall epithelial water permeability one believed. Furthermore, it wasestimated that proximal tubule could transport water against an adverseosmotic gradient (hypertonic lumen) of 8-23mosmol/kgH 2 O. These predictions were subsequently testedexperimentally by Green et al. ( 37 ) using in vivomicroperfusion of rat proximal tubules and peritubular capillaries. Itwas found that coupled water transport was ~75% of isotonictransport and that the adverse osmotic gradient required to null volumereabsorption was between 13.2 and 29.4 mosmol/kgH 2 O,depending on the concentration of peritubular protein.

Fig. 1. Transport pathways across luminal and peritubular cellmembranes of rat proximal convoluted tubule cell. All of theperitubular transport pathways shown on the basal surface also line thelateral cell membrane and communicate with the lateral intercellularspace (LIS). There is a permeable tight junction. Adapted from Ref. 118.  L, X" A) [! q" ~( V: \. h& w
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The fraction of reabsorptive water flow that actually traverses theproximal tubule TJ is unknown. Arguments in favor of transjunctional water flow have included substantial solvent drag of ionic species ( 31, 44, 80 ); the appearance of streaming potentials with the application of an impermeant osmotic agent ( 30, 104 );and ionic permeabilities roughly in proportion to their mobility in free solution ( 49 ). More direct evidence for TJ water fluxin rabbit tubules came from Whittembury and associates ( 15, 34, 127 ), whose estimate of the water permeability of theperitubular cell membrane indicated a transcellular water permeabilityless than the overall epithelial tubular water permeability. Inthe rat, evidence for TJ water flow was the observation of convective entrainment of sucrose, despite relatively small diffusional flux ( 126 ). An important discrepant observation was made bySchnermann et al. ( 85 ), who found that mice, geneticallydefective for the proximal tubule cell membrane water channelaquaporin-1, had a reduction in proximal tubule epithelial waterpermeability of nearly 80% compared with control mice. These data aredifficult to rationalize with significant TJ water permeability.However, before one can fully assess the implications of thisobservation, it must be acknowledged that there are no measurements ofproximal tubule solute reflection coefficients in any strain of mouse, so as yet we have no idea whether there is significant solute-water interaction that must be modeled in these tubules. Perhaps the mostdirect measure of TJ water flow was the work of Kovbasnjuk et al.( 50 ), who were able to visualize standing concentration gradients of a fluorescent indicator trapped within the lateral interspaces of confluent Madin-Darby canine kidney cells. A sweeping away of marker from the neighborhood of the TJ would have indicated transjunctional convection, but none was observed in this tight epithelium.

Consistent with early observations, the first proximal tubule modelsall included substantial TJ convective solute flux ( 45, 79, 80, 91, 114 ). Nevertheless, Rector and Berry ( 8, 74 )resisted ascribing substantial water flux to the TJ based onpore-theoretic calculations that indicated that the junctions were notlarge enough to allow anything but a small fraction of transepithelialwater flow. Preisig and Berry ( 72 ) measured the permeationof sucrose and mannitol across the rat proximal tubule. Applying theRenkin equations to their data, they computed the dimensions of the"sucrose pore" and indicated that it could be responsible for atmost 2% of the tubular water permeability. These arguments provoked aquantitative examination of whether apparent convective epithelialsolute flux could derive from solute polarization within the LIS( 115 ). Specifically, could one construct a model ofproximal tubule with just the right solute polarization to yieldrealistic reflection coefficients? In the reconsideration of rat datafrom Frömter et al. ( 31 ), all of the acceptable interspace models required substantial TJ convective Cl flux. An important contribution to this discussion came with thesuggestion of Fraser and Baines ( 28 ) that the TJ might be represented as a fiber matrix, rather than as a collection of pores.The critical feature of the fiber matrix equations is that for a givensolute permeability, the water permeability can be substantiallygreater than that predicted from the Renkin equations. This formulationwas compatible with the permeabilities of rat proximal tubule, althoughit was a phenomenologic equation and not based on the fine structure ofjunctional strands. Most recently, Guo et al. ( 40 )returned to this problem and examined representations of the TJ as atwo-pore structure. It was found that an abundant small pore(consistent with interstices between claudin-2 molecules) could be usedto represent small-solute permeability, and an infrequent large pore(such as 18 × 100-nm breaks in the TJ strands) could be used torepresent sucrose permeability. This large pore could also beresponsible for a substantial fraction of proximal tubule water flow,although it is not at all clear how such a pore could give rise to thedifferences in reflection coefficients that have been observed betweenNa and Cl, or between Cl andHCO 3 −. Ultimately, a true model of hindered transport,which incorporates electrical effects, will be required to represent TJfluid and electrolyte fluxes.6 @. Z  b+ a( `- L# ?2 I
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PROXIMAL TUBULE: REGULATION OF N A REABSORPTION! Y% Y1 k& X9 R
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Once the TJ had been established as a route for solute flux, anhypothesis was advanced that the junction might be a key locus for theregulation of proximal Na reabsorption. Lewy and Windhager( 57 ) demonstrated a correlation between single-nephronfiltration fraction and proximal tubule Na reabsorption( 57 ). Because a lower filtration fraction reduces proteinoncotic pressure within peritubular capillaries, they surmised thatthis would lead to reduced capillary uptake of fluid from the renalinterstitium and LIS, and, hence, elevated interspace pressure. Inturn, this would produce a backflux of Na alreadytransported into the interspace, that is, backflux across the TJ intothe lumen. Before this proposal, it was known that proximal tubuleNa reabsorption was depressed during extracellular volumeexpansion ( 22 ). In the intact dog, the ability to reversethis natriuresis with infusion of hyperoncotic albumin indicated thatperitubular oncotic pressure could influence sodium reabsorption, andEarley et al. ( 24, 63 ) had proposed that renalinterstitial pressure might be an intermediate variable. Micropunctureexperiments in the rat reproduced the findings in the dog, namely, thatdepression of proximal sodium reabsorption that occurs with salineinfusion could be reversed by perfusion of the efferent arteriole with a solution whose protein is at the control concentration ( 10, 90 ). Microperfusion of both proximal tubules and peritubular capillaries in the rat showed that peritubular oncotic force sharply increased isotonic Na reabsorption, well beyond a simpleosmotic effect on water flux, thus suggesting a qualitative change inthe epithelium ( 38 ). The precise mechanism by which LISpressures modulate TJ sodium flux is uncertain. One possibility is thatwith increased interstitial pressure there is junctional widening andback-diffusion of sodium from interspace to lumen. Evidence fromseveral sources has documented increased junctional permeability withvolume expansion, both in Necturus ( 9 ) and inthe rat ( 88 ). A second possibility is that backflux ofsodium across the TJ occurs by convective flow. The TJs of leakyepithelia are sensitive to hydrostatic pressures applied from thecontraluminal side, and volume expansion was found to decrease theproximal tubule NaCl reflection coefficient ( 7 ). In thisregard, convective backflux across the tight junction of rat proximaltubule has been invoked by Ramsey et al. ( 73 ) to explaintheir observation that the luminal appearance of lanthanum depositedwithin the renal interstitium is enhanced during saline volume expansion.
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To examine the backflux hypothesis quantitatively, a mathematical modelof rat proximal nephron was developed, comprising tubular epithelium,glomerulus, peritubular capillary, and interstitium ( 117 ).In this model, the TJ was compliant in the sense that both junctionalsalt and water permeability increased and the salt reflectioncoefficient decreased in response to small pressure differences fromlateral interspace to tubular lumen. Although these complianceproperties were empirical, they provided a model in which a decrease inperitubular protein concentration (which increased interspacehydrostatic pressure) could open the TJ and produce a secretory saltflux. This backflux was a combination of both diffusive and convectiveterms and did not specifically require either component to dominate. Inthis model of the TJ, once the interspace pressure fell below that ofthe lumen, the junction closed and junctional properties were fixed.The consequence of junctional closure is that beyond a certain value ofperitubular protein, one may expect little influence of peritubularStarling forces on volume reabsorption. Figure 2 displays the results of calculationswhen afferent arteriolar tone was varied, changing both glomerularplasma flow and filtration fraction. What is shown in the log-log plotare the changes in proximal reabsorption (APR) as a function offiltration fraction (FF; Fig. 2 A ) or as a function ofglomerular filtration rate (GFR; Fig. 2 B ). Within the region in which the TJ are "open" and influenced by pressure, the relative changes in APR and FF are identical. However, when related to changesin GFR, the fractional change in APR is only 41% (Fig. 2 B ).This derives from the fact that (within the model glomerulus) increasesin renal plasma flow produce GFR increases even in the absence ofchanges in glomerular capillary pressure, i.e., without any change infiltration fraction. This model prediction was at odds with the nearlyperfect glomerulotubular balance that has been observed in the ratkidney ( 86 ).1 r3 L$ a- R8 B5 b" T7 t
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Fig. 2. Assessment of glomerulotubular balance during afferentarteriolar constriction. The model is composed of a glomerulus,peritubular capillary, interstitial compartment, and proximal tubulewith compliant tight junction and is solved for the case of a singlenonelectrolyte salt plus plasma proteins. A : a log-log plotof predicted absolute proximal reabsorption (APR) as a functionof filtration fraction (FF). B : a log-log plot of APR as afunction of glomerular filtration rate (GFR). In the range oflower GFR, the peritubular capillary protein concentrations are lower,so that renal interstitial pressures are higher, and tight junctionsare open and modulated by pressure changes. The dotted lines are linearregressions and indicate the sensitivity of APR to either FF or GFR.Adapted from Ref. 117.
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A prerequisite for the precise glomerulotubular balance that has beenobserved is that luminal fluid flow modulates epithelial Na reabsorption. This has been termed"perfusion-absorption balance" ( 128 ) and has beendemonstrated in rat microperfusion studies ( 4, 41, 68, 78 ). One of the best illustrations of this phenomenon are themicropuncture data of Chan et al. ( 17 ), in which athree-fold increase in luminal perfusion rate (with trivial changes inluminal HCO 3 − concentration) produced a doubling ofthe rate of HCO 3 − reabsorption. It must beacknowledged that examination of rabbit tubules in vitro did not showflow-dependent reabsorption ( 13 ), but whether this is dueto species difference or to the preparation of tubules for perfusion invitro is not known. Although rat proximal tubules have been perfused invitro ( 32 ), flow-dependent reabsorption has not beenexamined. In this regard, flow-dependent Na andHCO 3 − transport has been reported recently in mousetubules both in vivo and in vitro ( 23 ). The underlyingmechanism for flow-dependent changes in reabsorption has not beenestablished. At one point, the proximal tubule brush border had beenconsidered a possible unstirred layer. However, model calculationsindicated that there was unlikely to be any appreciable convectivestirring within this pile ( 5 ). More to the point, thediffusion barrier between the bulk luminal fluid and the cell membranewas not predicted to hinder Na /H exchange( 52 ). Two studies raised the possibility that increases inaxial flow velocity recruit new transporters into the luminal membrane.Preisig ( 70 ) examined recovery of cellular pH from anacute acid load in vivo (ammonium pulse). With increases in luminalflow rate, the pH recovery mediated by Na /H exchange was enhanced. Maddox et al. ( 59 ) subjected ratsto acute changes in vascular volume to obtain hydropenic, euvolemic, and volume-expanded groups, with respective grouping according todecreased, normal, and increased GFR. When brush-border membrane vesicles were prepared from each of these groups andNa /H kinetic parameters were assessed, it wasfound that the V max determinations stratified inparallel with GFR.0 c* C; V8 u7 K- @

Ultimately, perfusion-absorption balance must derive from an afferentsensor of fluid flow rate in series with a cascade of effector stepsthat activate luminal transporters or insert new membrane transporters.Model calculations ( 39 ) have indicated that the proximaltubule microvilli are physically suitable to function as such a sensor.A striking feature of proximal tubule epithelium is the observationthat the microvilli are remarkably uniform in height and form a highlyorganized hexagonal array ( 64 ). Although the main functionthat has been attributed to the brush border has been luminal membranearea amplification, such regular organization is not necessary toaccommodate more transporters. However, the model in Guo et al.( 39 ) shows that such regularity in height and spacing ishighly advantageous if the microvilli are to function as a flow sensor,because the bending deformation of the microvilli would be both smalland uniform. The critical component of this system may well be theactin cytoskeleton, which is abundant within and beneath the brushborder ( 64 ). The model in Guo et al. ( 39 )describes how the actin filament bundle that is the central core of themicrovillus deforms under hydrodynamic loading. The proposed role forthe microvilli is that they can not only sense fluid drag forces butare also capable of greatly amplifying these stresses as the forces aretransferred to the intracellular cytoskeleton. This is due to thehydrodynamic torque exerted on the terminal web, where the actinfilament bundle within the microvillus attaches at its roots to themain cell body. To serve the hypothesized function, the microvillishould be relatively stiff structures that are able to transmit,without significant bending, the torque due to the hydrodynamic dragacting on the microvilli tips ( 113 ). In this scheme ofsignal transduction, specific interaction between the proximal tubulecytoskeleton and the apical cell membraneNa /H exchanger is a critical feature. In thisregard, Lamprecht et al. ( 55 ) have shown that theNa /H exchanger in brush-border microvilli islinked via ezrin, a kinase anchoring protein, to the actincytoskeleton. Finally, implication of the cytoskeleton in theflow-dependent modulation of luminal Na entry invites animmediate means for coordinating peritubular solute exit in response tochanging throughput. In sum, model failure to represent a fundamentalaspect of glomerulotubular balance has been the impetus to formulatetestable hypotheses for regulation of proximal tubule Na transport.! [1 O: \; @2 W0 I# Z/ u8 y, W
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PROXIMAL TUBULE: CL REABSORPTION

Proximal tubule Cl reabsorption proceeds via bothparacellular and transcellular pathways ( 3 ). Evidence fora transcellular component to Cl flux includesobservations that in the absence of transepithelial electrochemicaldriving force, there is still substantial Cl reabsorption( 1 ) and that a substantial component of Cl reabsorption can be blocked by specific inhibitors of membrane transporters ( 84, 110 ). The principal candidates forluminal membrane Cl uptake are all Cl /baseexchangers, in which the base may be HCO 3 −, hydroxyl,formate, or oxalate (Fig. 1 ). A major advance in this inquiry came withthe finding by Karniski and Aronson ( 46 ) that formatecould catalyze Cl uptake into proximal tubulebrush-border membrane vesicles, and, in the presence of avesicle-to-medium formate gradient, vesicle Cl concentration could be driven above that of the ambient medium. Inrabbit proximal tubules perfused in vitro with a high-Cl,low-HCO 3 − solution, addition of luminal formate (0.5 mM) increased volume reabsorption by 60% ( 84 ) and wasassociated with a small increase in cell volume ( 82 ). In rat tubules perfused in vivo with the same high-Cl solution, luminal formate increased volume reabsorption by 45% ( 110 ). This increase in volume and Cl reabsorption could be blocked, not only by an inhibitor of the Cl /HCO 2 − transporter but also by aninhibitor of the luminal membrane Na /H exchanger ( 109 ). Underscoring the importance of theNa /H exchanger in this observation, theeffect of formate to enhance proximal Cl reabsorption(present in normal mice) was absent in Na /H exchanger 3 (NHE3)-deficient mice studied with in vivo microperfusion ( 111 ). For formate-enhanced Cl transport tobe significant, submillimolar concentrations of formate (and micromolarconcentrations of formic acid) must mediate reabsorption of asubstantial portion of the filtered Cl load. The schemethat has emerged is one in which cellular formate exchanges for luminalCl, the formate is protonated to formic acid within thetubular lumen, and formic acid recycles back into the cell. At minimum, the luminal membrane Na /H exchanger is aproton source, but it may be more tightly coupled to the formate fluxpathways ( 3 ).

Luminal membrane Cl / HCO 2 − exchange hasbeen problematic for the modeling of proximal tubule. In modelsimulations with a luminal membrane Cl /HCO 2 − exchanger, the addition of formate produced cellswelling, and increased cytosolic Cl concentration,( 118 ). Unfortunately, variation of the density of theCl / HCO 2 − exchanger had virtually noeffect on overall NaCl reabsorption along the tubular segment, whereas in similar simulations, the density of theNa /H exchanger had a powerful impact on NaClreabsorption; i.e., the rate of Na reabsorption wasclearly rate limiting (Fig. 3 ). In thismodel, although transcellular Cl flux was substantial, itwas only about one-half the estimate of paracellular Cl flux. When the transcellular pathway was diminished, the forces favoring paracellular flux were augmented. In these model calculations, the luminal membrane permeability to formic acid was about one-third that of a lipid bilayer to CO 2. Unfortunately, the onlymeasurement of the formic acid permeability of the proximal tubule cellmembrane is ~5% of the value selected for the model parameter( 71 ), and using the measured value, the recycling schemefails. This prompted a modeling investigation into the possibility thatthe microvillous configuration of the proximal tubule brush bordercould provide a diffusion barrier, so that the experimental assessmentof membrane formic acid permeability might have been artifactually low.The result of these calculations was that the brush border onlydepressed formic acid permeability measurement by ~10%; even if theformic acid diffusion coefficient within the brush border wereone-tenth that in free solution, the permeability assessment would only be off by 25% ( 51 ). Additional calculations, whichincluded the diffusion of CO 2 and the finite rate ofcatalysis of carbonic anhydrase, confirmed the original estimates( 52 ). It is not clear how this proximal tubule modelshould be modified to represent formate's enhancement of proximalCl reabsorption. Certainly if formate were to directlymodulate the ion translocation rate through NHE3 (or NHE3 density), the overall transport effect could be achieved. However, this still leavesunresolved the problem of how enough formic acid could be recycled backinto the cell despite a low membrane permeability.
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Fig. 3. Proximal reabsorption of Na  ,Cl, and HCO 3 − predicted by anelectrolyte model of proximal tubule (Fig. 1 ). Perfusate and bath areidentical bicarbonate-Ringer solutions, and perfusion rate is 30 nl/mininto a 5-mm tubule segment. In the top panel, electrolytereabsorption is computed over a range of values for the activity of theluminal NH 4 + /H exchanger. The arrowindicates the reference value for this parameter. In the bottom panel, the independent variable is the activity ofluminal Cl /HCO 2 exchange. Adapted from Ref. 118.

With respect to peritubular exit of Cl, the earlyelectrophysiological data indicated only a small role for a conductivepathway ( 6, 12, 14 ). Welling and O'Neil( 124 ) found that in rabbit proximal straight tubule theconductive Cl pathway in peritubular membrane couldaccount for ~6% of the total membrane conductance. However, aftercell swelling induced by a 150 mosmol/kgH 2 O hypotonicosmotic shock, Cl conductance increased to 20% of thetotal peritubular membrane conductance. In rabbit proximal convolutedtubule, cell volume was unaffected by changes in peritubularCl concentration. However, prior hypotonic cell swellingrendered the cell volume sensitive to peritubular Cl concentration, and this effect was eliminated by application of aCl channel blocker ( 83 ).Electrophysiological study of rabbit convoluted tubule during osmoticshock estimated the fractional Cl conductance to increasefrom 3 to a maximum of 16%, with relaxation to 8%. During these sameexperiments, the fractional conductance of theNa -3HCO 3 − pathway declined from 41 to 16%, with little change in the absolute conductance through this pathway ( 125 ). The data from this type of experiment givesome guidance for estimating the reabsorptive flux of Cl through peritubular channels in rat experiments. If one assumes thatthe peritubular membrane conductance under control conditions is 10 mS/cm 2 ( 29 ), then the conductance of theNa -3 HCO 3 − pathway is ~4mS/cm 2, and in the relaxation phase after cell swelling,the steady-state Cl conductance increased to 2 mS/cm 2. For a peritubular membrane electrical potential of 75 mV, and cytosolic and peritubular Cl concentrationsof 18 and 118 mM ( 16 ), respectively, the cytosolic Cl potential is ~25 mV. Multiplication by thesteady-state Cl conductance of the swollen cell, 2 mS/cm 2, yields a Cl current of 50 µA/cm 2, or 0.5 nmol · s 1 · cm 2, or 25 pmol · min 1 · mm 1 for atubule with a 25-µm diameter. If in the conditions of these experiments, cytosolic Cl had been doubled to 36 mM, thenthe Cl potential might have been as high as 45 mV, andthe conductive flux 45 pmol · min 1 · mm 1. Theseestimates must be considered in light of the control volume fluxes of2.5 nl · min 1 · mm 1 ( 110 ), which for isotonic transport corresponds toCl reabsorption of 350 pmol · min 1 · mm 1. Inthe experiments of Wang et al. ( 110 ), the application of DPC (which had no effect in control conditions) was able to abolish theformate-induced 45% increase in volume reabsorption. This correspondsto a reduction in Cl flux by 160 pmol · min 1 · mm 1,several-fold higher than the conductive maximum. In the mathematical model, the formate-induced increases in luminal Cl entryexited largely via the potentNa -2HCO 3 − /Cl exchangerwithin the peritubular membrane ( 118 ).0 v1 a3 O9 q4 [+ O

DISTAL NEPHRON: K SECRETION% Y/ x" I5 r! j1 B* L6 Y$ f* q
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Micropuncture study of K handling by the rat kidneyhas identified the accessible portion of the distal convoluted tubule (DCT) as the principal site for K secretion ( 61, 62 ). Further along the nephron, there is little change inK flow, at least from a comparison of K delivery to the collecting duct (CD) with its appearance in the finalurine. In rats on a low-Na diet and treated withmineralocorticoid, maneuvers designed to enhance renal K excretion, determinations of late distal fluid-to-plasma K concentration yielded a ratio of 3.8 ( 62 ). In amicroperfusion study, the limiting K concentration of ratDCT has been estimated to be 13-15 mM by extrapolating betweensecretory (10 mM) and reabsorptive (25 mM) luminal K concentrations ( 35 ). Perhaps the most direct determination of the limiting K concentration was the "stationaryK concentration" observed in a split drop, 30.6 mM undercontrol conditions ( 2 ). These values are about an order ofmagnitude lower than urinary K concentrations ofantidiuretic rats on a control diet ( 61 ) and reflect thefact that K secretion in the distal tubule precedes finalwater abstraction from CD fluid. The idea of the CD as aK -passive conduit to the final urine stands in contrast toobservations of cortical CD (CCD) function in vitro, revealing that inrabbit ( 66, 87, 97, 95 ) and rat tubules ( 81, 102 ), the CCD is a site of Na reabsorption andK secretion, with transport enhanced by aldosterone andantidiuretic hormone. Thus one issue is understanding the observed DCTK concentrations in terms of the measured fluxes andpermeabilities of this segment, and a second issue isrationalizing the CCD K fluxes in vitro with thenegligible K secretion in vivo.

The only mathematical model of rat DCT that has been developed was doneby Chang and Fujita ( 18, 19 ) and includes K fluxes and acid-base transport in this segment. The late DCT principalcell, which is responsible for the K secretion of thissegment, is depicted in Fig. 4. Chang andFujita applied a novel method to obtain model parameters by devising apenalty function that examined the results of simulations of severaldifferent experiments, and notably none of the reference experimentsincluded measurements of the limiting DCT K concentration.What their results show is that K secretion occurspredominantly in the first half of the late DCT and then goes to zeroby the end of this segment, when luminal K concentrationreaches a limiting value of 15 mM ( 18 ). This representedan approximately sixfold increase from the entering K concentration and reflects an approximate doubling of axialK flow in conjunction with reabsorption of two-thirds ofthe delivered water. Their model prediction for the limitingconcentration was derived from data obtained under control conditionsand thus appears realistic. A model of rat CCD by Weinstein( 122 ) was developed with parameters designed to yieldfluxes and permeabilities characteristic of tubules exposed to bothaldosterone and antidiuretic hormone stimulation. With theseparameters, the limiting K concentration for CCD was 23 mM. When simulations were run for a 2-mm tubule in which entering fluidwas hypotonic with 12 mM K  , there was rapid waterreabsorption to isotonicity, a prompt doubling of luminalK  , and virtually no change in the axial flow ofK  . When entering K was 24 mM, the luminalconcentration was driven above 40 mM, and 23% of deliveredK was reabsorbed. This model was subsequently extended toa model of the whole CD, by appending outer medullary (OMCD) and inner medullary (IMCD) segments ( 123 ). Under antidiureticconditions, the model predicted that ~80% of the K delivered to the CD would be reabsorbed; most of this flux occurred within the OMCD and was paracellular (Fig. 5 ). In these CD segments, the modelK permeabilities had been guided by experimentalmeasurements of rat tubules perfused in vitro: NH 4 + permeability in OMCD ( 25 ) and K permeabilityin IMCD ( 76 ). The most immediate rationalization of thepredicted CD K reabsorption with the micropuncture data isthat the tubules in vivo either have a greater K secretorycapacity or a lower K permeability, or both.
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Fig. 4. Transport pathways across luminal and peritubularmembranes from the principal cell of the late distal convoluted tubuleof the rat. Adapted from Ref. 19.  J  H, A$ \- n5 N) W
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Fig. 5. Electrolyte transport along the model collecting duct (CD) underantidiuretic conditions. Left : luminal potential difference(PD; mV) and the concentrations of Na and K  . Right : volume flow and axial solute flows within all CD of asingle kidney. The abcissa is distance along the CD, with x = 0 the initial cortical point, and cortical (CCD),outer medullary (OMCD), and inner medullary (IMCD) CD accounting for 2, 2, and 5 mm of CD length. Peritubular conditions along the OMCD includea doubling of interstitial NaCl and KCl; along the IMCD, there is nochange in interstitial NaCl, but urea increases from 20 to 500 mM andKCl increases from 10 to 20 mM. Fluid reabsorption within the CCD,OMCD, and IMCD is 32, 35, and 22% of entering flow. Luminal deliveryof NaCl and KCl is ~5 and 50% of that filtered, and overall abouttwo-thirds of the delivered Na and 80% of deliveredK is reabsorbed. K reabsorption occurs withinthe OMCD and IMCD, due in part to transcellular uptake by luminalH-K-ATPase but most importantly due to the concentration of luminalK by water abstraction and diffusive backflux across thetight junctions and within IMCD, luminal cation channels. Adapted fromRef. 123.

Plasma HCO 3 − concentration has a profound effect onrenal K handling, with metabolic alkalosis increasingexcretion ( 27, 75 ) and acidosis decreasing excretion( 103 ). Distal micropuncture by Malnic et al.( 60 ) localized much of this effect to the accessible DCT.Microperfusion of this segment by Stanton and Giebisch( 93 ) established that the effect of HCO 3 − concentration to modulate DCT K secretion derived fromperitubular events and not the luminal concentration. These workersfound a 65% increase in K secretion with alkalosis and a50% decrease with acidosis, whereas the effects on Na transport were substantially less. Examination of rabbit CCD in vitroconfirmed that reduction in peritubular HCO 3 − decreased K secretion, with little effect onNa reabsorption, but with an increase in apical membraneresistance ( 100 ). With respect to underlying mechanisms,pH dependence of cation channels is a prime consideration, and in thisregard, Palmer and Frindt ( 67 ) patch clamped rabbit CCDand demonstrated that alkalosis opened and acidosis closed theNa channel of the luminal membrane. Wang et al.( 112 ) observed a comparable pH effect on thesmall-conductance luminal K channel. For these channels,there was an ~60% reduction in open probability as cytosolic pH wasreduced from 7.4 to 7.0. Strieter et al. ( 98, 99 ) used amathematical model of rabbit CCD to try to assess the relevance of thechannel kinetics to the pH effect on overall tubular K transport, and a summary of their observations is displayed in Table 1. Each section of the table shows theion fluxes and epithelial PD for three values of the peritubularHCO 3 − concentration,C S (HCO 3 − ). When all permeabilities arefixed or when luminal membrane K permeability( P K mp ) is the only pH-dependent permeability, peritubular HCO 3 − engenders only trivialchanges in Na and K fluxes. When luminalNa permeability ( P Na mp ) is pHdependent, alone or in combination with K permeability,the modulation of K flux is substantial, but this comeswith an increase in Na reabsorption that appears to be toohigh to be compatible with observation. Strieter et al.( 99 ) considered the possibility that, as in gallbladder( 130 ), tight junctional Cl permeability( P Cl me ), might decrease with alkaline pH.This would act to hyperpolarize the epithelium, increasing K secretion while decreasing Na reabsorption,so that when all three permeabilities are pH dependent, there is adoubling of K secretion in going from acidosis toalkalosis, with virtually no change in Na flux. Whetherthere is such pH dependence of CCD TJ permeabilities remains to beexamined., V+ |# c9 e8 k8 E2 f& z! D1 Z

Table 1. Tubular fluxes and luminal potential difference during acidosis,control, and alkalosis

CD: H SECRETION
4 M! Q1 d8 T$ j5 q) V0 T4 \
Net acid excretion by the kidney is determined within the CD as aconsequence of luminal proton secretion and buffer availability. Ourinformation about this derives from studies of CCD and OMCD segments invitro (albeit with significant uncertainty over extrapolation toconditions in vivo), and to a limited number of micropuncture andmicrocatheterization studies and far fewer perfusions of rat IMCD.Recently, component models of rat CD segments have been concatenated toyield a simulation of the complete structure, from cortex to papilla( 123 ). In this model, the OMCD emerges as the mostimportant site for CD acidification, and the -intercalated cell ofthis segment is shown in Fig. 6. In therat, when luminal HCO 3 − is 24 mM, luminal membraneproton secretion is apportioned 5:2 between the H-K-ATPase andH -ATPase transporters ( 33 ). Althoughcytosolic carbonic anhydrase (CA) is present ( 58 ), thereis no membrane-bound luminal CA ( 11, 26 ). Luminal protonsecretion is balanced by peritubular base exit, which for OMCD isalmost exclusively HCO 3 −, and it is believed that exitis almost exclusively via AE1, the peritubularCl /HCO 3 − exchanger. The first evidence for this was the finding that rabbit OMCD proton secretion is eliminated by removal of peritubular Cl or by applicationof a stilbene inhibitor of AE1 ( 96 ). Confirming theimportance of Cl /HCO 3 − exchange areexperiments in which OMCD cell pH was monitored and in which removal ofambient Cl reduced peritubular HCO 3 − permeability by 90% ( 42 ). Cl that entersvia AE1 can exit the cell via peritubular Cl channels.Although the absolute conductance of the membrane is not known, it hasbeen established that the major conductive pathway is Cl selective ( 47, 65 ). In the model OMCD, assignment of all Cl exits to a peritubular conductive pathway did yield anestimate for the absolute conductance of this pathway (~22mS/cm 2 ) but also provided a dilemma ( 121 ). Themodel attempted to accommodate the fact that Cl channelstypically have a HCO 3 − permeability ( 54, 69 ) and used a conservative estimate of 1:8 for theHCO 3 − -to-Cl permeability ratio. Thisprovided a peritubular exit pathway for HCO 3 − thatcarried about one-half of the generated HCO 3 −. Thus inthis model, even when peritubular AE1 activity was reduced to near zero, model proton secretion decreased by only one-third. Anumber of explanations could be invoked to rationalize this important discrepancy, but one attractive hypothesis is that a substantial portion of peritubular Cl exits occurs via electroneutralK-Cl cotransport and not all via Cl channels. Koeppen( 48 ) had suggested the existence of this pathway torationalize the slow hyperpolarization of SITS-inhibited OMCD cells interms of loss of cell Cl via an electroneutral pathway.One additional appeal of this hypothesis is that it may also provide apossible mechanism for blunting cell volume perturbations that arepredicted to accompany any changes in flux through the luminalH-K-ATPase ( 121 ).
$ |4 T. X! A1 b) s
Fig. 6. Transport pathways across luminal and peritubularmembranes from an intercalated cell of the rat OMCD. Adapted from Ref. 121.7 J. {4 a: S1 o7 Z1 ?

The IMCD cell, like that of the proximal tubule or thickascending limb of Henle, must coordinate both Na reabsorption and H secretion within a single cell type. Aschematic is shown in Fig. 7 ( 119 ). The Na fluxes are variable but can beperhaps nearly as large as those of the proximal tubule ( 21, 89 ) and occur without generating a significant transepithelialPD ( 36, 43 ). In view of the reports of thiazide inhibitionof IMCD Na reabsorption ( 77, 129 ), a luminalNa-Cl cotransporter appears to be the dominant pathway. To accommodatethis large transcellular Cl flux, along with theobservation that the peritubular membrane conductance is predominantlyK ( 92 ), a peritubular K-Cl cotransporter hasbeen included in the model cell. Luminal membrane H-K-ATPase has beenidentified as the major proton transporter ( 108 ). As inrat OMCD, cytosolic CA is present ( 58 ) but not within theluminal membrane ( 11, 106 ). Peritubular base exit mayoccur as HCO 3 −, via aCl /HCO 3 − exchanger and is thussusceptible to inhibition with CA inhibition ( 53, 94 ).What has been noteworthy about IMCD is a second mechanism forperitubular base exit, which has been identified by Wall( 105 ), and involves ammonia recycling. In this scheme,peritubular NH 4 + enters on the Na-K-ATPase incompetition for K ( 107 ), elevates cytosolicammonia, and thus promotes diffusive exit of NH 3.Predictions from this mechanism are that base exit and thus luminalacid secretion 1 ) should have a CA-insensitive componentthat would vary directly with peritubular NH 4 + concentration; 2 ) should vary inversely with peritubularK concentration; and 3 ) should vary directlywith the rate of IMCD Na reabsorption (i.e.,Na flux through the Na-K-ATPase). In the model IMCD, thisscheme was represented with competition of K andNH 4 + on the Na-K-ATPase, and the first prediction wasrealized ( 120 ). Inhibition of proton secretion byperitubular K was stronger than expected, becauseincreasing K also produced an increase in cellCl (via K-Cl cotransport) and thus also inhibited theCA-sensitive component of base exit. The predicted effect ofNa reabsorption on proton secretion turned out to be false(at least in the model) and the results of that calculation are shownin Fig. 8 ( 120 ). In thissimulation, the abcissa for all panels is luminal NaCl concentration,from 2 to 110 mM. Figure 8 A indicates cytosolic conditions,and Fig. 8 C shows the rate of luminal H secretion via the H-K-ATPase. Figure 8 B, left,displays the peritubular membrane fluxes of NH 4 + viathe Na-K-ATPase and K channels( G NH 4 ), and in Fig. 8 B, right, are peritubular HCO 3 − fluxes( G HCO 3 ) viaCl /HCO 3 − exchange and Cl channels. Over the range of luminal NaCl concentrations, the peritubular pump rate for Na varied from 0.9 to 6.1 nmol · s 1 · cm 2. In thesecalculations, however, there was virtually no change in luminalmembrane H secretion (Fig. 8 C ) or in cell pH(Fig. 8 A, left ). Reference to Fig. 8 B, left and right shows that the changes inNa transport led to opposing effects onNH 4 + cycling and HCO 3 − exit. Asluminal entry of NaCl decreased, there was a decrease in cytosolicCl (Fig. 8 A, right ) and thusincreased peritubular HCO 3 − exit via theCl /HCO 3 − exchanger (Fig. 8 B, right ). Thus this model predicted that with these two baseexit mechanisms operating in parallel, acid secretion would be stableover a wide range of IMCD Na transport. At present,dissected and perfused IMCD do not transport Na well, ifat all, and in situ peritubular conditions are not easily assessed, sothat the prospects for subjecting these model predictions to testingappear distant.
8 P. g4 o5 X8 V0 ~9 P
Fig. 7. Transport pathways across luminal and peritubularmembranes from rat IMCD. Adapted from Ref. 119.5 A2 m- r9 t1 M
1 F4 n. w- H7 A# k
Fig. 8. Impact of luminal Na on predicted acidsecretion by a model of rat IMCD epithelium (Fig. 7 ). Peritubularconditions are representative of the outer-inner medullary junction.The abcissa for each panel is luminal NaCl concentration, which variedfrom 2 to 110 mM. A : cytosolic conditions. B :peritubular (Peritub.) membrane fluxes of NH 4 + via theNa-K-ATPase and K channels( G NH 4; left ) andperitubular HCO 3 − fluxes viaCl /HCO 3 − exchange and Cl channels. C : rate of luminal proton secretion via theH-K-ATPase. Adapted from Ref. 120.3 m2 P* L# \$ F$ d" U, F, J
/ q3 w3 b  f% M$ i. Q6 P) y
CONCLUSION, M# ]# n, b0 j! A& T
9 S6 D. h2 Z; U9 _8 ?: m2 |5 s
It should be clear that there remains considerable uncertainty asto our ability to model renal tubular function. The issues presentedhere can be formulated as specific questions: How much proximal tubulewater flux traverses the TJ? What is the role of proximal TJs in theregulation of Na reabsorption? Which luminal cell membranetransporters of proximal tubule have flow-dependent changes in density,and what is the signaling pathway? What are the important transcellularpathways for proximal tubule Cl flux, and whichtransporters are modulated by formate? Do CD K fluxes andtubular permeabilities measured in vitro correspond to those in vivo?What are the Cl exit pathways in OMCD, and in particular,is a K-Cl cotransporter important? Does variation in H-K-ATPaseactivity in OMCD produce derangements in cell volume, or is therecoordinate activation of other pathways? Does the IMCD shift base exitbetween CA-dependent and CA-independent mechanisms to maintain stableproton secretion in the face of variable Na fluxes? Thislist of questions hardly exhausts the uncertainties that have come tothe fore in the published model investigations. Some of the issues havehad intense experimental interest well before they were cast asmodeling problems, but some truly did arise out of model exploration.What should also be clear is that some of these questions arequantitatively very important; i.e., the essence of some phenomena havenot been captured, and this is not just an effort to fine-tune a nearlycompleted picture. What must also be acknowledged is that, for all itsobvious value to understanding the kidney, structural information alonewill not suffice to answer many of our most important questions, and there is no escape from the conclusion that additional functional dataare required. It may be legitimate to question whether existing experimental technology is up to the task, or whether new techniques are necessary. However, at the level of detail considered here, theavailable numerical methods and computing power are certainly up to thetask of simulating renal tubular transport. With respect to modelrefinement, incorporating more biophysical detail about individualtransporters will no doubt enhance model reliability, but ultimateconfidence in these tubular models will still be contingent on criticalexperimental information to be developed at the tubular level." q2 F, X: k" }/ q4 j# [5 m

ACKNOWLEDGEMENTS

This work was supported by Public Health Service Grant1-RO1-DK-29857 from the National Institutes of Health.
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