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

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.

Glucose transporters in the small intestine in health and disease

Absorption of monosaccharides is mainly mediated by Na+-D-glucose cotransporter SGLT1 and the facititative transporters GLUT2 and GLUT5. SGLT1 and GLUT2 are relevant for absorption of D-glucose and D-galactose while GLUT5 is relevant for D-fructose absorption. SGLT1 and GLUT5 are constantly localized in the brush border membrane (BBM) of enterocytes, whereas GLUT2 is localized in the basolateral membrane (BLM) or the BBM plus BLM at low and high luminal D-glucose concentrations, respectively. At high luminal D-glucose, the abundance SGLT1 in the BBM is increased. Hence, D-glucose absorption at low luminal glucose is mediated via SGLT1 in the BBM and GLUT2 in the BLM whereas high-capacity D-glucose absorption at high luminal glucose is mediated by SGLT1 plus GLUT2 in the BBM and GLUT2 in the BLM. The review describes functions and regulations of SGLT1, GLUT2, and GLUT5 in the small intestine including diurnal variations and carbohydrate-dependent regulations. Also, the roles of SGLT1 and GLUT2 for secretion of enterohormones are discussed. Furthermore, diseases are described that are caused by malfunctions of small intestinal monosaccharide transporters, such as glucose-galactose malabsorption, Fanconi syndrome, and fructose intolerance. Moreover, it is reported how diabetes, small intestinal inflammation, parental nutrition, bariatric surgery, and metformin treatment affect expression of monosaccharide transporters in the small intestine. Finally, food components that decrease D-glucose absorption and drugs in development that inhibit or downregulate SGLT1 in the small intestine are compiled. Models for regulations and combined functions of glucose transporters, and for interplay between D-fructose transport and metabolism, are discussed.

Evolution of our understanding of cell volume regulation by the pump-leak mechanism

Kay and Blaustein recount the history of the pump-leak mechanism, which stabilizes cell volume by pumping sodium ions out of cells.

All animal cells are surrounded by a flexible plasma membrane that is permeable to water and to small ions. Cells thus face a fundamental problem: the considerable tension that their membranes would experience if the osmotic influx of water, driven by the presence of impermeant intracellular ions, was left unopposed. The pivotal study that described the cell’s remedy for this impending osmotic catastrophe—the “pump-leak mechanism” (PLM)—was published in the Journal of General Physiology by Tosteson and Hoffman in 1960. Their work revealed how the sodium pump stabilizes cell volume by eliminating the osmotic gradient. Here we describe the mechanistic basis of the PLM, trace the history of its discovery, and place it into the context of our current understanding.

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:

Physiological relevance of epithelial geometry: New insights into the standing gradient model and the role of LI cadherin

We introduce a mathematical model of an absorbing leaky epithelium to reconsider the problem formulated by Diamond and Bossert in 1967: whether "… some distinctive physiological properties of epithelia might arise as geometrical consequences of epithelial ultrastructure". A standing gradient model of the intercellular cleft (IC) is presented that includes tight junctions (TJ) and ion channels uniformly distributed along the whole cleft. This nonlinear system has an intrinsic homogeneous concentration and the spatial scale necessary to establish it along the cleft. These parameters have not been elucidated so far. We further provide non-perturbative analytical approximations for a broad range of parameters. We found that narrowing of the IC increases ion concentration dramatically and can therefore prevent outflow through tight junctions (TJs) and the lateral membrane, as long as extremely high luminal osmolarities are not reached. Our model predicts that the system is to some extent self-regulating and thereby prevents fluxes into the lumen. Recent experimental evidence has shown that liver-intestine (LI) cadherin can control the up/down flux in intestines via regulation of the cleft width. This finding is in full agreement with predictions of our model. We suggest that LI-cadherin may increase water transport through epithelia via sequential narrowing of the cleft, starting from the highest concentration area at the beginning of the cleft and triggering a propagating squeezing motion.

Aquaporins in the lung

The lung is the interface between air and blood where the exchange of oxygen and carbon dioxide occurs. The surface liquid that is directly exposed to the gaseous compartment covers both conducting airways and respiratory zone and forms the air-liquid interface. The barrier that separates this lining fluid of the airways and alveoli from the extracellular compartment is the pulmonary epithelium. The volume of the lining fluid must be kept in a range that guarantees an appropriate gas exchange and other functions, such as mucociliary clearance. It is generally accepted that this is maintained by balancing resorptive and secretory fluid transport across the pulmonary epithelium. Whereas osmosis is considered as the exclusive principle of fluid transport in the airways, filtration may contribute to alveolar fluid accumulation under pathologic conditions. Aquaporins (AQP) facilitate water flux across cell membranes, and as such, they provide a transcellular route for water transport across epithelia. However, their contribution to near-isosmolar fluid conditions in the lung still remains elusive. Herein, we discuss the role of AQPs in the lung with regard to fluid homeostasis across the respiratory epithelium.

Epithelial transport in The Journal of General Physiology

JGP hosts key papers that shaped the epithelial transport field.

Epithelia define the boundaries of the body and often transfer solutes and water from outside to inside (absorption) or from inside to outside (secretion). Those processes involve dual plasma membranes with different transport components that interact with each other. Understanding those functions has entailed breaking down the problem to analyze properties of individual membranes (apical vs. basolateral) and individual transport proteins. It also requires understanding of how those components interact and how they are regulated. This article outlines the modern history of this research as reflected by publications in The Journal of General Physiology.

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.

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.

Fluid transport and ion fluxes in mammalian kidney proximal tubule: a model analysis of isotonic transport

AIM

By mathematical modelling, we analyse conditions for near-isotonic and isotonic transport by mammalian kidney proximal tubule.

METHODS

The model comprises compliant lateral intercellular space (lis) and cells, and infinitely large luminal and peritubular compartments with diffusible species: Na+, K+, Cl- and an intracellular non-diffusible anion. Unknown model variables are solute concentrations, electrical potentials, volumes and hydrostatic pressures in cell and lis, and transepithelial potential. We used data mainly from rat proximal tubule to model epithelial cells and interspace with luminal and peritubular baths of identical composition.

RESULTS

The model of the tubular epithelium with physiological water permeability and paracellular electrical resistance generates solute coupled water uptake with an approx. 3% hypertonic absorbate. This function remains unperturbed following 'blocking' of apical water channels and in 'aquaporin-null' simulation. Reduced rate of volume reabsorption in AQP(-/-) mice would also require decreased apical sodium permeability. Paracellular convection accounts for approx. 36% of the net Na+ absorption, and the model epithelium accomplishes uphill water transport similar to rat proximal tubule. Na+ recirculation is required for truly isotonic transport. The tonicity of the absorbate and the recirculation flux depend critically on ion permeabilities of interspace basement membrane.

CONCLUSION

Our model based on solute-solvent coupling in lateral space simulates major physiological features of proximal tubule, including significantly lower water permeability of the AQP1-null preparation, and a ratio of net sodium uptake and oxygen consumption exceeding that predicted from stoichiometry of the Na+/K+-pump. Physical properties of interspace basement membrane are critical for obtaining near-isotonic and truly isotonic transport.

A two-barrier compartment model for volume flow across amphibian skin

The amphibian skin has long been used as a model tissue for the study of ion transport and osmotic water movement across tight epithelia. To understand the mechanism of water uptake across amphibian skin, we model the skin as a well-stirred compartment bounded by an apical barrier and a tissue barrier. The compartment represents the lateral intercellular space between cells in the stratum granulosum. The apical barrier represents the stratum corneum, the principal/mitochondria-rich cells, and the junctional area between cells. This barrier is hypothesized to have the ability to actively transport solutes through Na+-K+-ATPase. The actively transported solute flux is assumed to satisfy the Michaelis-Menten relationship. The tissue barrier represents a composite barrier comprising the stratum spinosum, the stratum germinativum, the basal lamina, and the dermis. Our model shows that 1) the predicted rehydration rates from apical bathing solutions are in good agreement with the experiment results in Hillyard and Larsen (J Comp Physiol 171: 283-292, 2001); 2) under their experimental conditions, there is a substantial volume flux coupled to the active solute flux and this coupled volume flux is nearly constant when the osmolality of the apical bathing solution is >100 mosmol/kgH2O; 3) the molar ratio of the actively transported solute flux to the coupled water flux is about 1:160, which is the same as that reported in Nielsen (J Membr Biol 159: 61-69, 1997).

Hans H. Ussing--scientific work: contemporary significance and perspectives

As a zoologist, Hans H. Ussing began his scientific career by studying the marine plankton fauna in East Greenland. This brought him in contact with August Krogh at the time George de Hevesy, Niels Bohr and Krogh planned the application of artificial radioactive isotopes for studying the dynamic state of the living organism. Following his studies of protein turnover of body tissues with deuterium-labeled amino acids, Ussing initiated a new era of studies of transport across epithelial membranes. Theoretical difficulties in the interpretation of tracer fluxes resulted in novel concepts such as exchange diffusion, unidirectional fluxes, flux-ratio equation, and solvent drag. Combining methods of biophysics with radioactive isotope technology, Ussing introduced and defined the phrases 'short-circuit current', 'active transport pathway' and 'shunt pathway', and with frog skin as experimental model, he unambiguously proved active transport of sodium ions. Conceived in his electric circuit analogue of frog skin, Ussing associated transepithelial ion fluxes with the hitherto puzzling 'bioelectric potentials'. The two-membrane hypothesis of frog skin initiated the study of epithelial transport at the cellular level and raised new questions about cellular mechanisms of actions of hormones and drugs. His theoretical treatment of osmotic water fluxes versus fluxes of deuterium labeled water resulted in the discovery of epithelial water channels. His discovery of paracellular transport in frog skin bridged studies of high and low resistance epithelia and generalized the description of epithelial transport. He devoted the last decade of his scientific life to solute-coupled water transport. He introduced the sodium recirculation theory of isotonic transport, and in an experimental study, he obtained the evidence for recirculation of sodium ions in toad small intestine. In penetrating analyses of essential aspects of epithelial membrane transport, Ussing provided insights of general applicability and powerful analytical methods for the study of intestine, kidney, respiratory epithelia, and exocrine glands-of equal importance to biology and medicine.

Paracellular fluid transport by epithelia

The evidence that a major fraction of water crosses the paracellular route during isotonic fluid transfer is reviewed together with a description of the theory and experimental results derived from extracellular probe studies. Four transporting epithelia which have been studied using the method are gallbladder, intestine, Malpighian tubule, and salivary gland. It is concluded that paracellular probe flows are not due to simple convection generated by osmotic flow through the junctions but are generated by active fluid transport within the junction: a mechano-osmotic process. The geometry of the pathway involved would indicate that some salt accompanies the paracellular fluid, representing a hypo-osmotic flow. Transport of salt by the cell route, which may be accompanied by some water, represents a hypertonic flow. The problem then becomes one of balancing the two to produce an isotonic transportate. We suggest, using recent data from knockout mice, that some aquaporins are functioning in different epithelial tissues as osmo-comparators within a feedback loop that regulates the paracellular fluid flow rate. This results in an overall quasi-isotonic transport by the epithelium. The model is applied to forward-facing systems such as proximal tubule and backward-facing systems such as exocrine glands.

Analysis of the sodium recirculation theory of solute-coupled water transport in small intestine

Our previous mathematical model of solute-coupled water transport through the intestinal epithelium is extended for dealing with electrolytes rather than electroneutral solutes. A 3Na+-2K+ pump in the lateral membranes provides the energy-requiring step for driving transjunctional and translateral flows of water across the epithelium with recirculation of the diffusible ions maintained by a 1Na+-1K+-2Cl- cotransporter in the plasma membrane facing the serosal compartment. With intracellular non-diffusible anions and compliant plasma membranes, the model describes the dependence on membrane permeabilities and pump constants of fluxes of water and electrolytes, volumes and ion concentrations of cell and lateral intercellular space (lis), and membrane potentials and conductances. Simulating physiological bioelectrical features together with cellular and paracellular fluxes of the sodium ion, computations predict that the concentration differences between lis and bathing solutions are small for all three ions. Nevertheless, the diffusion fluxes of the ions out of lis significantly exceed their mass transports. It is concluded that isotonic transport requires recirculation of all three ions. The computed sodium recirculation flux that is required for isotonic transport corresponds to that estimated in experiments on toad small intestine. This result is shown to be robust and independent of whether the apical entrance mechanism for the sodium ion is a channel, a SGLT1 transporter driving inward uphill water flux, or an electroneutral Na+-K+-2Cl- cotransporter.