Models of coupled salt and water transport across leaky epithelia

Calcium Dynamics and Water Transport in Salivary Acinar Cells

Saliva is secreted from the acinar cells of the salivary glands, using mechanisms that are similar to other types of water-transporting epithelial cells. Using a combination of theoretical and experimental techniques, over the past 20 years we have continually developed and modified a quantitative model of saliva secretion, and how it is controlled by the dynamics of intracellular calcium. However, over approximately the past 5 years there have been significant developments both in our understanding of the underlying mechanisms and in the way these mechanisms should best be modelled. Here, we review the traditional understanding of how saliva is secreted, and describe how our work has suggested important modifications to this traditional view. We end with a brief description of the most recent data from living animals and discuss how this is now contributing to yet another iteration of model construction and experimental investigation.

Fluid and solute transport across the retinal pigment epithelium: a theoretical model

The retina is composed of two main layers-the neuroretina and the retinal pigment epithelium (RPE)-that are separated by a potential gap termed the sub-retinal space (SRS). Accumulation of fluid in the SRS may result in a retinal detachment. A key function of the RPE is to prevent fluid accumulation in the SRS by actively pumping fluid from this space to the choroid. We have developed a mathematical model of this process that incorporates the transport of seven chemical species: Na⁺, K⁺, Cl^-, HCO3-, H⁺, CO₂ and H₂CO₃. This allows us to estimate solute and water fluxes and to understand the role of the different membrane ion channels. We have performed a global sensitivity analysis using the extended Fourier amplitude sensitivity test to investigate the relative importance of parameters in generating the model outputs. The model predicts that flow across the RPE is driven by an osmotic gradient in the cleft gap between adjacent cells. Moreover, the model estimates how water flux is modified in response to inhibition of membrane ion channels and carbonic anhydrase (CA). It provides a possible explanation for how CA inhibitors, which are used clinically to prevent fluid accumulation in the SRS, may be acting.

Stationary and Nonstationary Ion and Water Flux Interactions in Kidney Proximal Tubule: Mathematical Analysis of Isosmotic Transport by a Minimalistic Model

Our mathematical model of epithelial transport (Larsen et al. Acta Physiol. 195:171–186, 2009) is extended by equations for currents and conductance of apical SGLT2. With independent variables of the physiological parameter space, the model reproduces intracellular solute concentrations, ion and water fluxes, and electrophysiology of proximal convoluted tubule. The following were shown:

Cell mechanics: a dialogue

Under the microscope, eukaryotic animal cells can adopt a variety of different shapes and sizes. These cells also move and deform, and the physical mechanisms driving these movements and shape changes are important in fundamental cell biology, tissue mechanics, as well as disease biology. This article reviews some of the basic mechanical concepts in cells, emphasizing continuum mechanics description of cytoskeletal networks and hydrodynamic flows across the cell membrane. We discuss how cells can generate movement and shape changes by controlling mass fluxes at the cell boundary. These mass fluxes can come from polymerization/depolymerization of actin cytoskeleton, as well as osmotic and hydraulic pressure-driven flow of water across the cell membrane. By combining hydraulic pressure control with force balance conditions at the cell surface, we discuss a quantitative mechanism of cell shape and volume control. The broad consequences of this model on cell mechanosensation and tissue mechanics are outlined.

Directional Fluid Transport across Organ-Blood Barriers: Physiology and Cell Biology

Directional fluid flow is an essential process for embryo development as well as for organ and organism homeostasis. Here, we review the diverse structure of various organ-blood barriers, the driving forces, transporters, and polarity mechanisms that regulate fluid transport across them, focusing on kidney-, eye-, and brain-blood barriers. We end by discussing how cross talk between barrier epithelial and endothelial cells, perivascular cells, and basement membrane signaling contribute to generate and maintain organ-blood barriers.

Fluid and ion transfer across the blood-brain and blood-cerebrospinal fluid barriers; a comparative account of mechanisms and roles

The two major interfaces separating brain and blood have different primary roles. The choroid plexuses secrete cerebrospinal fluid into the ventricles, accounting for most net fluid entry to the brain. Aquaporin, AQP1, allows water transfer across the apical surface of the choroid epithelium; another protein, perhaps GLUT1, is important on the basolateral surface. Fluid secretion is driven by apical Na+-pumps. K+ secretion occurs via net paracellular influx through relatively leaky tight junctions partially offset by transcellular efflux. The blood-brain barrier lining brain microvasculature, allows passage of O2, CO2, and glucose as required for brain cell metabolism. Because of high resistance tight junctions between microvascular endothelial cells transport of most polar solutes is greatly restricted. Because solute permeability is low, hydrostatic pressure differences cannot account for net fluid movement; however, water permeability is sufficient for fluid secretion with water following net solute transport. The endothelial cells have ion transporters that, if appropriately arranged, could support fluid secretion. Evidence favours a rate smaller than, but not much smaller than, that of the choroid plexuses. At the blood-brain barrier Na+ tracer influx into the brain substantially exceeds any possible net flux. The tracer flux may occur primarily by a paracellular route. The blood-brain barrier is the most important interface for maintaining interstitial fluid (ISF) K+ concentration within tight limits. This is most likely because Na+-pumps vary the rate at which K+ is transported out of ISF in response to small changes in K+ concentration. There is also evidence for functional regulation of K+ transporters with chronic changes in plasma concentration. The blood-brain barrier is also important in regulating HCO3- and pH in ISF: the principles of this regulation are reviewed. Whether the rate of blood-brain barrier HCO3- transport is slow or fast is discussed critically: a slow transport rate comparable to those of other ions is favoured. In metabolic acidosis and alkalosis variations in HCO3- concentration and pH are much smaller in ISF than in plasma whereas in respiratory acidosis variations in pHISF and pHplasma are similar. The key similarities and differences of the two interfaces are summarized.

Flow-Driven Cell Migration under External Electric Fields

Electric fields influence many aspects of cell physiology, including various forms of cell migration. Many cells are sensitive to electric fields, and they can migrate toward a cathode or an anode, depending on the cell type. In this Letter, we examine an actomyosin-independent mode of cell migration under electrical fields. Our theory considers a one-dimensional cell with water and ionic fluxes at the cell boundary. Water fluxes through the membrane are governed by the osmotic pressure difference across the cell membrane. Fluxes of cations and anions across the cell membrane are determined by the properties of the ion channels as well as the external electric field. Results show that without actin polymerization and myosin contraction, electric fields can also drive cell migration, even when the cell is not polarized. The direction of migration with respect to the electric field direction is influenced by the properties of ion channels, and are cell-type dependent.

Regulation of glomerulotubular balance. III. Implication of cytosolic calcium in flow-dependent proximal tubule transport

In the proximal tubule, axial flow (drag on brush-border microvilli) stimulates Na(+) and HCO3 (-) reabsorption by modulating both Na/H exchanger 3 (NHE3) and H-ATPase activity, a process critical to glomerulotubular balance. We have also demonstrated that blocking the angiotensin II receptor decreases baseline transport, but preserves the flow effect; dopamine leaves baseline fluxes intact, but abrogates the flow effect. In the current work, we provide evidence implicating cytosolic calcium in flow-dependent transport. Mouse proximal tubules were microperfused in vitro at perfusion rates of 5 and 20 nl/min, and reabsorption of fluid (Jv) and HCO3 (-) (JHCO3) were measured. We examined the effect of high luminal Ca(2+) (5 mM), 0 mM Ca(2+), the Ca(2+) chelator BAPTA-AM, the inositol 1,4,5-trisphosphate (IP3) receptor antagonist 2-aminoethoxydiphenyl borate (2-APB), and the Ca-ATPase inhibitor thapsigargin. In control tubules, increasing perfusion rate from 5 to 20 nl/min increased Jv by 62% and JHCO3 by 104%. With respect to Na(+) reabsorption, high luminal Ca(2+) decreased transport at low flow, but preserved the flow-induced increase; low luminal Ca(2+) had little impact; both BAPTA and 2-APB had no effect on baseline flux, but abrogated the flow effect; thapsigargin decreased baseline flow, leaving the flow effect intact. With respect to HCO3 (-) reabsorption, high luminal Ca(2+) decreased transport at low flow and mildly diminished the flow-induced increase; low luminal Ca(2+) had little impact; both BAPTA and 2-APB had no effect on baseline flux, but abrogated the flow effect. These data implicate IP3 receptor-mediated intracellular Ca(2+) signaling as a critical step in transduction of microvillous drag to modulate Na(+) and HCO3 (-) transport.

Multiscale modelling of saliva secretion

We review a multiscale model of saliva secretion, describing in brief how the model is constructed and what we have so far learned from it. The model begins at the level of inositol trisphosphate receptors (IPR), and proceeds through the cellular level (with a model of acinar cell calcium dynamics) to the multicellular level (with a model of the acinus), finally to a model of a saliva production unit that includes an acinus and associated duct. The model at the level of the entire salivary gland is not yet completed. Particular results from the model so far include (i) the importance of modal behaviour of IPR, (ii) the relative unimportance of Ca(2+) oscillation frequency as a controller of saliva secretion, (iii) the need for the periodic Ca(2+) waves to be as fast as possible in order to maximise water transport, (iv) the presence of functional K(+) channels in the apical membrane increases saliva secretion, (v) the relative unimportance of acinar spatial structure for isotonic water transport, (vi) the prediction that duct cells are highly depolarised, (vii) the prediction that the secondary saliva takes at least 1mm (from the acinus) to reach ionic equilibrium. We end with a brief discussion of future directions for the model, both in construction and in the study of scientific questions.

Cerebrospinal Fluid Secretion by the Choroid Plexus

The choroid plexus epithelium is a cuboidal cell monolayer, which produces the majority of the cerebrospinal fluid. The concerted action of a variety of integral membrane proteins mediates the transepithelial movement of solutes and water across the epithelium. Secretion by the choroid plexus is characterized by an extremely high rate and by the unusual cellular polarization of well-known epithelial transport proteins. This review focuses on the specific ion and water transport by the choroid plexus cells, and then attempts to integrate the action of specific transport proteins to formulate a model of cerebrospinal fluid secretion. Significant emphasis is placed on the concept of isotonic fluid transport across epithelia, as there is still surprisingly little consensus on the basic biophysics of this phenomenon. The role of the choroid plexus in the regulation of fluid and electrolyte balance in the central nervous system is discussed, and choroid plexus dysfunctions are described in a very diverse set of clinical conditions such as aging, Alzheimer's disease, brain edema, neoplasms, and hydrocephalus. Although the choroid plexus may only have an indirect influence on the pathogenesis of these conditions, the ability to modify epithelial function may be an important component of future therapies.

What do aquaporin knockout studies tell us about fluid transport in epithelia?

The investigation of near-isosmotic water transport in epithelia goes back over 100 years; however, debates over mechanism and pathway remain. Aquaporin (AQP) knockouts have been used by various research groups to test the hypothesis of an osmotic mechanism as well as to explore the paracellular versus transcellular pathway debate. Nonproportional reductions in the water permeability of a water-transporting epithelial cell (e.g., a reduction of around 80-90 %) compared to the reduction in overall water transport rate in the knockout animal (e.g., a reduction of 50-60 %) are commonly found. This nonproportionality has led to controversy over whether AQP knockout studies support or contradict the osmotic mechanism. Arguments raised for and against an interpretation supporting the osmotic mechanism typically have partially specified, implicit, or incorrect assumptions. We present a simple mathematical model of the osmotic mechanism with clear assumptions and, for models based on this mechanism, establish a baseline prediction of AQP knockout studies. We allow for deviations from isotonic/isosmotic conditions and utilize dimensional analysis to reduce the number of parameters that must be considered independently. This enables a single prediction curve to be used for multiple epithelial systems. We find that a simple, transcellular-only osmotic mechanism sufficiently predicts the results of knockout studies and find criticisms of this mechanism to be overstated. We note, however, that AQP knockout studies do not give sufficient information to definitively rule out an additional paracellular pathway.

Regulation of glomerulotubular balance: II: impact of angiotensin II on flow-dependent transport

Underlying glomerulotubular balance (GTB) is the impact of axial flow to regulate Na(+) and HCO(3)(-) transport by modulating Na(+)-H(+) exchanger 3 (NHE3) and H-ATPase activity. It is not known whether the cascade of events following a change in flow relies on local angiotensin (ANG II) generation or receptor availability. Mouse tubules were microperfused in vitro at flows of 5 and 20 nl/min, and net fluid (J(v)) and HCO(3)(-) (J(HCO3)) absorption and cell height were measured. Na(+) (J(Na)) and Cl(-) (J(Cl)) absorption and changes in microvillous torque were estimated. Raising flow increased Na(+) and HCO(3)(-) reabsorption but did not change either Cl(-) transport or cell volume. Losartan reduced absolute Na(+) and HCO(3)(-) absorption at both low and high flows but did not affect fractional flow-stimulated transport. Compared with controls, in AT(1a) knockout (KO) mouse tubules, 53% of flow-stimulated Na(+) absorption was abolished, but flow-stimulated HCO(3)(-) absorption was retained at similar levels. The remaining flow-stimulated J(HCO3) was eliminated by the H-ATPase inhibitor bafilomycin. Inhibition of the AT(2) receptor by PD123319 increased both J(Na) and J(HCO3) but did not affect flow-mediated fractional changes. NHE3 expression at the protein level was reduced in AT(1a) KO mice kidneys. We conclude that 1) although the AT(1a) receptor is necessary for flow to impact NHE3, the effect on H(+)-ATPase is independent of AT(1a); 2) the small flow-mediated changes in cell volume suggest a coordinate flow effect on both luminal and basolateral transporters; and 3) there is no evidence of flow-dependent Cl(-) transport, and thus no evidence for convective paracellular Cl(-) transport in mouse tubules.

Regulation of glomerulotubular balance. I. Impact of dopamine on flow-dependent transport

In response to volume expansion, locally generated dopamine decreases proximal tubule reabsorption by reducing both Na/H-exchanger 3 (NHE3) and Na-K-ATPase activity. We have previously demonstrated that mouse proximal tubules in vitro respond to changes in luminal flow with proportional changes in Na(+) and HCO(3)(-) reabsorption and have suggested that this observation underlies glomerulotubular balance. In the present work, we investigate the impact of dopamine on the sensitivity of reabsorptive fluxes to changes in luminal flow. Mouse proximal tubules were microperfused in vitro at low and high flow rates, and volume and HCO(3)(-) reabsorption (J(v) and J(HCO3)) were measured, while Na(+) and Cl(-) reabsorption (J(Na) and J(Cl)) were estimated. Raising luminal flow increased J(v), J(Na), and J(HCO3) but did not change J(Cl). Luminal dopamine did not change J(v), J(Na), and J(HCO3) at low flow rates but completely abolished the increments of Na(+) absorption by flow and partially inhibited the flow-stimulated HCO(3)(-) absorption. The remaining flow-stimulated HCO(3)(-) absorption was completely abolished by bafilomycin. The DA1 receptor blocker SCH23390 and the PKA inhibitor H89 blocked the effect of exogenous dopamine and produced a two to threefold increase in the sensitivity of proximal Na(+) reabsorption to luminal flow rate. Under the variety of perfusion conditions, changes in cell volume were small and did not always parallel changes in Na(+) transport. We conclude that 1) dopamine inhibits flow-stimulated NHE3 activity by activation of the DA1 receptor via a PKA-mediated mechanism; 2) dopamine has no effect on flow-stimulated H-ATPase activity; 3) there is no evidence of flow stimulation of Cl(-) reabsorption; and 4) the impact of dopamine is a coordinated modulation of both luminal and peritubular Na(+) transporters.

Efficiency of primary saliva secretion: an analysis of parameter dependence in dynamic single-cell and acinus models, with application to aquaporin knockout studies

Secretion from the salivary glands is driven by osmosis following the establishment of osmotic gradients between the lumen, the cell and the interstitium by active ion transport. We consider a dynamic model of osmotically driven primary saliva secretion and use singular perturbation approaches and scaling assumptions to reduce the model. Our analysis shows that isosmotic secretion is the most efficient secretion regime and that this holds for single isolated cells and for multiple cells assembled into an acinus. For typical parameter variations, we rule out any significant synergistic effect on total water secretion of an acinar arrangement of cells about a single shared lumen. Conditions for the attainment of isosmotic secretion are considered, and we derive an expression for how the concentration gradient between the interstitium and the lumen scales with water- and chloride-transport parameters. Aquaporin knockout studies are interpreted in the context of our analysis and further investigated using simulations of transport efficiency with different membrane water permeabilities. We conclude that recent claims that aquaporin knockout studies can be interpreted as evidence against a simple osmotic mechanism are not supported by our work. Many of the results that we obtain are independent of specific transporter details, and our analysis can be easily extended to apply to models that use other proposed ionic mechanisms of saliva secretion.

Reconciling the Krogh and Ussing interpretations of epithelial chloride transport - presenting a novel hypothesis for the physiological significance of the passive cellular chloride uptake

In 1937, August Krogh discovered a powerful active Cl(-) uptake mechanism in frog skin. After WWII, Hans Ussing continued the studies on the isolated skin and discovered the passive nature of the chloride uptake. The review concludes that the two modes of transport are associated with a minority cell type denoted as the γ-type mitochondria-rich (MR) cell, which is highly specialized for epithelial Cl(-) uptake whether the frog is in the pond of low [NaCl] or the skin is isolated and studied by Ussing chamber technique. One type of apical Cl(-) channels of the γ-MR cell is activated by binding of Cl(-) to an external binding site and by membrane depolarization. This results in a tight coupling of the uptake of Na(+) by principal cells and Cl(-) by MR cells. Another type of Cl(-) channels (probably CFTR) is involved in isotonic fluid uptake. It is suggested that the Cl(-) channels serve passive uptake of Cl(-) from the thin epidermal film of fluid produced by mucosal glands. The hypothesis is evaluated by discussing the turnover of water and ions of the epidermal surface fluid under terrestrial conditions. The apical Cl(-) channels close when the electrodiffusion force is outwardly directed as it is when the animal is in the pond. With the passive fluxes eliminated, the Cl(-) flux is governed by active transport and evidence is discussed that this is brought about by an exchange of cellular HCO(3) (-) with Cl(-) of the outside bath driven by an apical H(+) V-ATPase.

Claudin-2, a component of the tight junction, forms a paracellular water channel

Whether or not significant amounts of water pass the tight junction (TJ) of leaky epithelia is still unresolved, because it is difficult to separate transcellular water flux from TJ-controlled paracellular water flux. Using an approach without differentiating technically between the transcellular and paracellular route, we measured transepithelial water flux with and without selective molecular perturbation of the TJ to unequivocally attribute changes to the paracellular pathway. To this end, MDCK C7 cells were stably transfected with either claudin-2 or claudin-10b, two paracellular cation-channel-forming TJ proteins that are not endogenously expressed in this cell line. Claudin-2 is typical of leaky, water-transporting epithelia, such as the kidney proximal tubule, whereas claudin-10b is present in numerous epithelia, including water-impermeable segments of the loop of Henle. Neither transfection altered the expression of endogenous claudins or aquaporins. Water flux was induced by an osmotic gradient, a Na+ gradient or both. Under all conditions, water flux in claudin-2-transfected cells was elevated compared with vector controls, indicating claudin-2-mediated paracellular water permeability. Na+-driven water transport in the absence of an osmotic gradient indicates a single-file mechanism. By contrast, claudin-10b transfection did not alter water flux. We conclude that claudin-2, but not claudin-10b, forms a paracellular water channel and thus mediates paracellular water transport in leaky epithelia.

Local osmosis and isotonic transport

Osmotically driven water flow, u (cm/s), between two solutions of identical osmolarity, c(o) (300 mM: in mammals), has a theoretical isotonic maximum given by u = j/c(o), where j (moles/cm(2)/s) is the rate of salt transport. In many experimental studies, transport was found to be indistinguishable from isotonic. The purpose of this work is to investigate the conditions for u to approach isotonic. A necessary condition is that the membrane salt/water permeability ratio, epsilon, must be small: typical physiological values are epsilon = 10(-3) to 10(-5), so epsilon is generally small but this is not sufficient to guarantee near-isotonic transport. If we consider the simplest model of two series membranes, which secrete a tear or drop of sweat (i.e., there are no externally-imposed boundary conditions on the secretion), diffusion is negligible and the predicted osmolarities are: basal = c(o), intracellular approximately (1 + epsilon)c(o), secretion approximately (1 + 2epsilon)c(o), and u approximately (1 - 2epsilon)j/c(o). Note that this model is also appropriate when the transported solution is experimentally collected. Thus, in the absence of external boundary conditions, transport is experimentally indistinguishable from isotonic. However, if external boundary conditions set salt concentrations to c(o) on both sides of the epithelium, then fluid transport depends on distributed osmotic gradients in lateral spaces. If lateral spaces are too short and wide, diffusion dominates convection, reduces osmotic gradients and fluid flow is significantly less than isotonic. Moreover, because apical and basolateral membrane water fluxes are linked by the intracellular osmolarity, water flow is maximum when the total water permeability of basolateral membranes equals that of apical membranes. In the context of the renal proximal tubule, data suggest it is transporting at near optimal conditions. Nevertheless, typical physiological values suggest the newly filtered fluid is reabsorbed at a rate u approximately 0.86 j/c(o), so a hypertonic solution is being reabsorbed. The osmolarity of the filtrate c(F) (M) will therefore diminish with distance from the site of filtration (the glomerulus) until the solution being transported is isotonic with the filtrate, u = j/c(F).With this steady-state condition, the distributed model becomes approximately equivalent to two membranes in series. The osmolarities are now: c(F) approximately (1 - 2epsilon)j/c(o), intracellular approximately (1 - epsilon)c(o), lateral spaces approximately c(o), and u approximately (1 + 2epsilon)j/c(o). The change in c(F) is predicted to occur with a length constant of about 0.3 cm. Thus, membrane transport tends to adjust transmembrane osmotic gradients toward epsilonc(o), which induces water flow that is isotonic to within order epsilon. These findings provide a plausible hypothesis on how the proximal tubule or other epithelia appear to transport an isotonic solution.

The lateral intercellular space as osmotic coupling compartment in isotonic transport

Solute-coupled water transport and isotonic transport are basic functions of low- and high-resistance epithelia. These functions are studied with the epithelium bathed on the two sides with physiological saline of similar composition. Hence, at transepithelial equilibrium water enters the epithelial cells from both sides, and with the reflection coefficient of tight junction being larger than that of the interspace basement membrane, all of the water leaves the epithelium through the interspace basement membrane. The common design of transporting epithelia leads to the theory that an osmotic coupling of water absorption to ion flow is energized by lateral Na(+)/K(+) pumps. We show that the theory accounts quantitatively for steady- and time dependent states of solute-coupled fluid uptake by toad skin epithelium. Our experimental results exclude definitively three alternative theories of epithelial solute-water coupling: stoichiometric coupling at the molecular level by transport proteins like SGLT1, electro-osmosis and a 'junctional fluid transfer mechanism'. Convection-diffusion out of the lateral space constitutes the fundamental problem of isotonic transport by making the emerging fluid hypertonic relative to the fluid in the lateral intercellular space. In the Na(+) recirculation theory the 'surplus of solutes' is returned to the lateral space via the cells energized by the lateral Na(+)/K(+) pumps. We show that this theory accounts quantitatively for isotonic and hypotonic transport at transepithelial osmotic equilibrium as observed in toad skin epithelium in vitro. Our conclusions are further developed for discussing their application to solute-solvent coupling in other vertebrate epithelia such as small intestine, proximal tubule of glomerular kidney and gallbladder. Evidence is discussed that the Na(+) recirculation theory is not irreconcilable with the wide range of metabolic cost of Na(+) transport observed in fluid-transporting epithelia.

Application of the Na+ recirculation theory to ion coupled water transport in low- and high resistance osmoregulatory epithelia

The theory of Na+ recirculation for isosmotic fluid absorption follows logically from Hertz's convection-diffusion equation applied to the exit of water and solutes from the lateral intercellular space. Experimental evidence is discussed indicating Na+ recirculation based upon the following approaches: (i) An isotope tracer method in small intestine. Simultaneous measurement of water flow and ion transport in toad skin epithelium demonstrating, (ii) occasional hyposmotic absorbates, and (iii) reduced fluid absorption in the presence of serosal bumetanide. (iv) Studies of the metabolic cost of net Na+ absorption demonstrating an efficiency that is lower than the 18 Na+ per O2 consumed given by the stoichiometry of the Na+/K+-pump. Mathematical modeling predicts a significant range of observations such as isosmotic transport, hyposmotic transport, solvent drag, anomalous solvent drag, the residual hydraulic permeability in proximal tubule of AQP1(-/-) mice, the adverse relationship between hydraulic permeability and the concentration difference needed to reverse transepithelial water flow, and in a non-contradictory way the wide range of metabolic efficiencies from above to below 18 Na+/O2. Certain types of observations are poorly or not at all reproduced by the model. It is discussed that such lack of agreement between model and experiment is due to cellular regulations of ion permeabilities that are not incorporated in the modeling. Clarification of these problems requires further experimental studies.

Na+ Recirculation and Isosmotic Transport

The Na(+) recirculation theory for solute-coupled fluid absorption is an expansion of the local osmosis concept introduced by Curran and analyzed by Diamond & Bossert. Based on studies on small intestine the theory assumes that the observed recirculation of Na(+) serves regulation of the osmolarity of the absorbate. Mathematical modeling reproducing bioelectric and hydrosmotic properties of small intestine and proximal tubule, respectively, predicts a significant range of observations such as isosmotic transport, hyposmotic transport, solvent drag, anomalous solvent drag, the residual hydraulic permeability in proximal tubule of AQP1 (-/-) mice, and the inverse relationship between hydraulic permeability and the concentration difference needed to reverse transepithelial water flow. The model reproduces the volume responses of cells and lateral intercellular space (lis) following replacement of luminal NaCl by sucrose as well as the linear dependence of volume absorption on luminal NaCl concentration. Analysis of solvent drag on Na(+) in tight junctions provides explanation for the surprisingly high metabolic efficiency of Na(+) reabsorption. The model predicts and explains low metabolic efficiency in diluted external baths. Hyperosmolarity of lis is governed by the hydraulic permeability of the apical plasma membrane and tight junction with 6-7 mOsm in small intestine and < or = 1 mOsm in proximal tubule. Truly isosmotic transport demands a Na(+) recirculation of 50-70% in small intestine but might be barely measurable in proximal tubule. The model fails to reproduce a certain type of observations: The reduced volume absorption at transepithelial osmotic equilibrium in AQP1 knockout mice, and the stimulated water absorption by gallbladder in diluted external solutions. Thus, it indicates cellular regulation of apical Na(+) uptake, which is not included in the mathematical treatment.

A dual-pathway ultrastructural model for the tight junction of rat proximal tubule epithelium

A dual-pathway model is proposed for transport across the tight junction (TJ) in rat proximal tubule: large slit breaks formed by infrequent discontinuities in the TJ complex and numerous small circular pores, with spacing similar to that of claudin-2. This dual-pathway model is developed in the context of a proximal tubule model (Weinstein AM. Am J Physiol Renal Fluid Electrolyte Physiol 247: F848-F862, 1984) to provide an ultrastructural view of solute and water fluxes. Tubule model paramters (TJ reflection coefficient and water permeability), plus the measured epithelial NaCl and sucrose permeabilities, provide constraints for the dual-pathway model, which yields the small-pore radius and spacing and large slit height and area. For a small-pore spacing of 20.2 nm, comparable to the distance between adjacent particle pairs in apposing TJ strands, the small-pore radius is 0.668 nm and the large slit breaks have a height of 19.6 nm, occupying 0.04% of the total TJ length. This pore/slit geometry also satisfies the measured permeability for mannitol. The numerous small circular pores account for 91.25% of TJ NaCl permeability but only 5.0% of TJ water permeability. The infrequent large slit breaks in the TJ account for 95.0% of TJ water permeability but only 8.7% of TJ NaCl permeability. Sucrose and mannitol (4.6- and 3.6-A radius) can pass through both the large slit breaks and the small pores. For sucrose, 78.3% of the flux is via the slits and 21.7% via the pores; for mannitol, the flux is split nearly evenly between the two pathways, 50.8 and 49.2%. In this ultrastructural model, the TJ water permeability is 21.2% of the entire transepithelial water permeability and thus an order of magnitude greater than that predicted by the single-pore/slit theory (Preisig PA and Berry CA. Am J Physiol Renal Fluid Electrolyte Physiol 249: F124-F131, 1985).

General models for water transport across leaky epithelia

The group of leaky epithelia, such as proximal tubule and small intestine, have several common properties in regard to salt and water transport. The fluid transport is isotonic, the transport rate increases in dilute solutions, and water can be transported uphill. Yet, it is difficult to find common features that could form the basis for a general transport model. The direction of transepithelial water transport does not correlate with the direction of the primary active Na+ transport, or with the ultrastucture as defined by the location of apical and basolateral membranes, of the junctional complex and the lateral intercellular spaces. The presence of specific water channels, aquaporins, increases the water permeability of the epithelial cell membranes, i.e., the kidney proximal tubule. Yet other leaky epithelia, for example, the retinal pigment epithelium, have no known aquaporins. There is, however, a general correlation between the direction of transepithelial transport and the direction of transport via cotransporters of the symport type. A simple epithelial model based on water permeabilities, a hyperosmolar compartment and restricted salt diffusion, is unable to explain epithelial transport phenomena, in particular the ability for uphill water transport. The inclusion of cotransporters as molecular water pumps in these models alleviates this problem.

A mathematical model of solute coupled water transport in toad intestine incorporating recirculation of the actively transported solute

A mathematical model of an absorbing leaky epithelium is developed for analysis of solute coupled water transport. The non-charged driving solute diffuses into cells and is pumped from cells into the lateral intercellular space (lis). All membranes contain water channels with the solute passing those of tight junction and interspace basement membrane by convection-diffusion. With solute permeability of paracellular pathway large relative to paracellular water flow, the paracellular flux ratio of the solute (influx/outflux) is small (2-4) in agreement with experiments. The virtual solute concentration of fluid emerging from lis is then significantly larger than the concentration in lis. Thus, in absence of external driving forces the model generates isotonic transport provided a component of the solute flux emerging downstream lis is taken up by cells through the serosal membrane and pumped back into lis, i.e., the solute would have to be recirculated. With input variables from toad intestine (Nedergaard, S., E.H. Larsen, and H.H. Ussing, J. Membr. Biol. 168:241-251), computations predict that 60-80% of the pumped flux stems from serosal bath in agreement with the experimental estimate of the recirculation flux. Robust solutions are obtained with realistic concentrations and pressures of lis, and with the following features. Rate of fluid absorption is governed by the solute permeability of mucosal membrane. Maximum fluid flow is governed by density of pumps on lis-membranes. Energetic efficiency increases with hydraulic conductance of the pathway carrying water from mucosal solution into lis. Uphill water transport is accomplished, but with high hydraulic conductance of cell membranes strength of transport is obscured by water flow through cells. Anomalous solvent drag occurs when back flux of water through cells exceeds inward water flux between cells. Molecules moving along the paracellular pathway are driven by a translateral flow of water, i.e., the model generates pseudo-solvent drag. The associated flux-ratio equation is derived.

Molecular mechanisms for passive and active transport of water

Water crosses cell membranes by passive transport and by secondary active cotransport along with ions. While the first concept is well established, the second is new. The two modes of transport allow cellular H2O homeostasis to be viewed as a balance between H2O leaks and H2O pumps. Consequently, cells can be hyperosmolar relative to their surroundings during steady states. Under physiological conditions, cells from leaky epithelia may be hyperosmolar by roughly 5 mosm liter-1, under dilute conditions, hyperosmolarities up to 40 mosm liter-1 have been recorded. Most intracellular H2O is free to serve as solvent for small inorganic ions. The mechanism of transport across the membrane depends on how H2O interacts with the proteinaceous or lipoid pathways. Osmotic transport of H2O through specific H2O channels such as CHIP 28 is hydraulic if the pore is impermeable to the solute and diffusive if the pore is permeable. Cotransport of ions and H2O can be a result of conformational changes in proteins, which in addition to ion transport also translocate H2O bound to or occlude in the protein. A cellular model of a leaky epithelium based on H2O leaks and H2O pumps quantitatively predicts a number of so-far unexplained observations of H2O transport.

Acidic pH of the lateral intercellular spaces of MDCK cells cultured on permeable supports

The pH of the lateral intercellular space (LIS) of Madin-Darby canine kidney (MDCK) cell monolayers grown on permeable supports was investigated by microspectrofluorimetry using BCECF (2',7'-bis(carboxyethyl)-5,6-carboxyfluorescein). The permeability of the support was selectively reduced by growing Zn-Al-silicate crystals inside its pores. The diffusion of BCECF across the filter was sufficiently retarded to allow measurements of fluorescence in the LIS. The LIS pH and intracellular pH of the cells surrounding them were determined in HEPES-buffered solutions. When the perfusate pH was 7.4, the LIS pH was more acidic (7.06 +/- 0.02) and equaled the cytoplasmic pH (7.08 +/- 0.05). When perfusate was changed to pH 7.0 or 7.8, the LIS changed linearly by about half the magnitude of the perfusate pH. Intracellular pH followed LIS pH variations between perfusate pH 7.0 and 7.4 but was significantly higher when perfusate pH was 7.8. Tight junctional H+ permeability was undetectably low. The low steady-state pH in the LIS was not altered by inhibitors of acid transport or low temperature. Rapid perturbations of pH in the LIS showed that protons were not immobilized in the LIS. The acidic microenvironment within the LIS may be the result of buffering by the cell surface proteins.

Osmotic water permeability of Necturus gallbladder epithelium

An electrophysiological technique that is sensitive to small changes in cell water content and has good temporal resolution was used to determine the hydraulic permeability (Lp) of Necturus gallbladder epithelium. The epithelial cells were loaded with the impermeant cation tetramethylammonium (TMA+) by transient exposure to the pore-forming ionophore nystatin in the presence of bathing solution TMA+. Upon removal of the nystatin a small amount of TMA+ is trapped within the cell. Changes in cell water content result in changes in intracellular TMA+ activity which are measured with intracellular ion-sensitive microelectrodes. We describe a method that allows us to determine the time course for the increase or decrease in the concentration of osmotic solute at the membrane surface, which allows for continuous monitoring of the difference in osmolality across the apical membrane. We also describe a new method for the determination of transepithelial hydraulic permeability (Ltp). Apical and basolateral membrane Lp's were assessed from the initial rates of change in cell water volume in response to anisosmotic mucosal or serosal bathing solutions, respectively. The corresponding values for apical and basolateral membrane Lp's were 0.66 x 10(-3) and 0.38 x 10(-3) cm/s.osmol/kg, respectively. This method underestimates the true Lp values because the nominal osmotic differences (delta II) cannot be imposed instantaneously, and because it is not possible to measure the true initial rate of volume change. A model was developed that allows for the simultaneous determination of both apical and basal membrane Lp's from a unilateral exposure to an anisosmotic bathing solution (mucosal). The estimates of apical and basal Lp with this method were 1.16 x 10(-3) and 0.84 x 10(-3) cm/s.osmol/kg, respectively. The values of Lp for the apical and basal cell membranes are sufficiently large that only a small (less than 3 mosmol/kg) transepithelial difference in osmolality is required to drive the observed rate of spontaneous fluid absorption by the gallbladder. Furthermore, comparison of membrane and transepithelial Lp's suggests that a large fraction of the transepithelial water flow is across the cells rather than across the tight junctions.

Network thermodynamic analysis and stimulation of isotonic solute-coupled volume flow in leaky epithelia: an example of the use of network theory to provide the qualitative aspects of a complex system and its verification by stimulation

A network thermodynamic model was developed to provide insights into the nature of isotonic solute-coupled volume flow in "leaky" epithelia, where the transepithelial volume flow is assumed to be primarily through the cellular pathway. The coupled flows of solute and volume at each membrane in this four membrane model are described by the practical phenomenological equations as developed by Kedem & Katchalsky (1958). The model contains one permeable non-electrolyte solute (s) and a fixed amount of an impermeable non-electrolyte (i) inside the cell. The cell is assumed to be capable of volume regulation under the steady-state experimental conditions simulated. A solute-pump, located in the basolateral membrane, uses feedback regulation to adjust Cs in the cell in order to maintain cell volume at or near control levels in all simulations. Model behavior is, in general, very consistent with experimental observations with respect to tonicity and magnitude of volume flow over a wide range of experimental conditions. Examination of the parameter space suggests the following important features when isotonic solute-coupled volume flow moves primarily through the cellular pathway: (1) the apical membrane reflection coefficient must be less than that of the basolateral membrane; (2) the basement membrane reflection coefficient must be small; (3) the apical membrane solute permeability and reflection coefficient are the two most "sensitive" parameters and need to vary in an inverse manner in order to maintain isotonicity when both solute and volume flows increase; and (4) relationships (1) and (3) above imply the need for at least two separate solute pathways in the apical membrane, one that is shared with volume flow and one that is not.

Convective paracellular solute flux. A source of ion-ion interaction in the epithelial transport equations

An electrolyte model of an epithelium (a cell and a tight junction in parallel, both in series with a lateral interspace basement membrane) is analyzed using the formalism of nonequilibrium thermodynamics. It is shown that if the parallel structures are heteroporous (i.e., reflection coefficients for two ion species differ between the components), then a cross-term will appear in the overall transport equations of the epithelium. Formally, this cross-term represents an ion-ion interaction. With respect to the rat proximal tubule, data indicating epithelial ionic reflection coefficients less than unity, together with the assumption of no transcellular solvent drag, imply the presence of convective paracellular solute flux. This means that a model applicable to a heteroporous structure must be used to represent the tubule, and, in particular, the cross-terms for ion-ion interaction must also be evaluated in permeability determinations. A series of calculations is presented that permits the estimation of the Na-Cl interaction for rat proximal tubule from available experimental data. One consequence of tubule heteroporosity is that an electrical potential may be substantially less effective than an equivalent concentration gradient in driving reabsorptive ion fluxes.

The roles of paracellular and transcellular pathways and submucosal space in isotonic water absorption by rabbit ileum

Water movements have been studied in sheets of isolated rabbit ileum using a method which measures net volume flows across the mucosal and serosal surfaces of the tissue continuously with high resolution. At 35 degrees C, with the tissues incubated in isotonic Ringer solution containing D-glucose (25 mM) on both sides, there is a steady net inflow of fluid at the rate of 24 +/- 2 microliter cm-2 h-1 across the mucosal surface (Jm) and an outflow of 8 +/- 1 microliter cm-2 h-1 across the serosal surface (Js) (n = 16). The stable transepithelial p.d. across these tissues is 2.7 +/- 0.2 mV, serosa positive. Jm can be reversibly inhibited by anoxia. Ouabain (0.1 mM) added to the serosal solution inhibits inflow across the mucosal and serosal surfaces by 75% (n = 7) within 30 min. If phlorizin (0.1 mM) is added to the mucosal Ringer solution containing glucose (20 mM) within 30 min of the commencement of in vitro absorption, Jm is reduced from 37 +/- 3 to 28 +/- 2 microliter cm-2 h-1 (n = 3). Dilution of the mucosal Ringer solution by 50 mosmol kg-1 (with the serosal solution kept isosmolar) results in a rapid transient increase in mucosal inflow. An increase of 50 mosmol kg-1 in the mucosal Ringer solution with NaCl, sucrose or mannitol causes a transient reversal of mucosal flow, followed by a return of inflow at a reduced level. Rabbit ileum can transport water against gradients of approximately 75 mosmol kg-1 of sucrose, NaCl, or mannitol. Addition of polyethylene glycol (mol. wt. 20000; 3 mosmol kg-1) causes a sustained reversal of mucosal inflow; inflow can be restored only by removing polyethylene glycol from the mucosal Ringer solution. The tissue can absorb water against an osmotic gradient of 200 mM-glycerol. The above data have been incorporated into a new model to explain isotonic flow of fluid by this epithelium. The main features are that the hydraulic conductivity (Lp) of the mucosal boundary of the lateral intercellular space is approximately 1 X 10(-8) cm s-1 cmH2O-1. This Lp is too low to sustain isotonicity of the flow emerging from the lateral intercellular space at the observed rates. Hypertonic fluid emerging from the lateral intercellular space is diluted by transcellular water flow generated by the hypertonicity of the submucosa and back-diffusion of solute via mucosal shunt channels.(ABSTRACT TRUNCATED AT 400 WORDS)

Permeability properties of the subepithelial tissues of Necturus gallbladder

The permeability properties of the subepithelial connective tissue of Necturus gallbladder were evaluated by measurement of electrical resistance, dilution potentials and hydraulic water permeability. The gallbladder epithelial cells were removed by scraping and the underlying connective tissue placed in an Ussing chamber. The electrical resistance was 2.2 +/- 0.8 omega X cm2; the tissue was slightly cation selective relative to free solution. The subepithelial tissues restricted the rate of diffusion of small solutes to 50% of the free solution value. The hydraulic water permeability averaged 2.1 X 10(-2) cm/s per atm. We conclude that limitations of the area of subepithelium available for fluid movement are the most important factors in determining the restrictions to solute and water flow offered by the subepithelial tissues.

Nonequilibrium thermodynamic model of the rat proximal tubule epithelium

The rat proximal tubule epithelium is represented as well-stirred, compliant cellular and paracellular compartments bounded by mucosal and serosal bathing solutions. With a uniform pCO2 throughout the epithelium, the model variables include the concentrations of Na, K, Cl, HCO3, H2PO4, HPO4, and H, as well as hydrostatic pressure and electrical potential. Except for a metabolically driven Na-K exchanger at the basolateral cell membrane, all membrane transport within the epithelium is passive and is represented by the linear equations of nonequilibrium thermodynamics. In particular, this includes the cotransport of Na-Cl and Na-H2PO4 and countertransport of Na-H at the apical cell membrane. Experimental constraints on the choice of ionic conductivities are satisfied by allowing K-Cl cotransport at the basolateral membrane. The model equations include those for mass balance of the nonreacting species, as well as chemical equilibrium for the acidification reactions. Time-dependent terms are retained to permit the study of transient phenomena. In the steady state the energy dissipation is computed and verified equal to the sum of input from the Na-K exchanger plus the Gibbs free energy of mass addition to the system. The parameter dependence of coupled water transport is studied and shown to be consistent with the predictions of previous analytical models of the lateral intercellular space. Water transport in the presence of an end-proximal (HCO3-depleted) luminal solution is investigated. Here the lower permeability and higher reflection coefficient of HCO3 enhance net sodium and water transport. Due to enhanced flux across the tight junction, this process may permit proximal tubule Na transport to proceed with diminished energy dissipation.

Epithelial cell volume modulation and regulation

Epithelial cell volume is a sensitive indicator of the balance between solute entry into the cell and solute exit. Solute accumulation in the cell leads to cell swelling because the water permeability of the cell membranes is high. Similarly, solute depletion leads to cell shrinkage. The rate of volume change under a variety of experimental conditions may be utilized to study the rate and direction of solute transport by an epithelial cell. The pathways of water movement across an epithelium may also be deduced from the changes in cellular volume. A technique for the measurement of the volume of living epithelial cells is described, and a number of experiments are discussed in which cell volume determination provided significant new information about the dynamic behavior of epithelia. The mechanism of volume regulation of epithelial cells exposed to anisotonic bathing solution is discussed and shown to involve the transient stimulation of normally dormant ion exchangers in the cell membrane.

Intra- and transepithelial analytical techniques

Epithelia transport a variety of solutes and water. Study of such transport requires a determination of the driving forces responsible for transport, of the pathways through which transport occurs, and of the factors controlling such transport. Transepithelial driving forces are readily determined where the composition of the bathing media can be altered and electrical forces negated. Where substances move only through a paracellular pathway such manipulations may be adequate to define the permeability and selectivity of the pathways. For substances utilizing a cellular pathway, driving forces and permeabilities across the two dissimilar apical and basolateral cellular membranes must be determined. Where a substance can be shown to move across a membrane against its electrochemical potential gradient, the source of the energy for such movement must be assessed. This review focuses on the applicability and validity of a variety of techniques utilized for the study of epithelial transport to answer these questions. These include microelectrode techniques, chemical analyses, microprobe analysis, microscopy, and techniques for assessing the coupling of metabolism to transport.