A kinetically defined Na+/H+ antiporter within a mathematical model of the rat proximal tubule

A modular and reusable model of epithelial transport in the proximal convoluted tubule

We review a collection of published renal epithelial transport models, from which we build a consistent and reusable mathematical model able to reproduce many observations and predictions from the literature. The flexible modular model we present here can be adapted to specific configurations of epithelial transport, and in this work we focus on transport in the proximal convoluted tubule of the renal nephron. Our mathematical model of the epithelial proximal convoluted tubule describes the cellular and subcellular mechanisms of the transporters, intracellular buffering, solute fluxes, and other processes. We provide free and open access to the Python implementation to ensure our multiscale proximal tubule model is accessible; enabling the reader to explore the model through setting their own simulations, reproducibility tests, and sensitivity analyses.

The SLC9A-C Mammalian Na+/H+ Exchanger Family: Molecules, Mechanisms, and Physiology

Na⁺/H⁺ exchangers play pivotal roles in the control of cell and tissue pH by mediating the electroneutral exchange of Na⁺ and H⁺ across cellular membranes. They belong to an ancient family of highly evolutionarily conserved proteins, and they play essential physiological roles in all phyla. In this review, we focus on the mammalian Na⁺/H⁺ exchangers (NHEs), the solute carrier (SLC) 9 family. This family of electroneutral transporters constitutes three branches: SLC9A, -B, and -C. Within these, each isoform exhibits distinct tissue expression profiles, regulation, and physiological roles. Some of these transporters are highly studied, with hundreds of original articles, and some are still only rudimentarily understood. In this review, we present and discuss the pioneering original work as well as the current state-of-the-art research on mammalian NHEs. We aim to provide the reader with a comprehensive view of core knowledge and recent insights into each family member, from gene organization over protein structure and regulation to physiological and pathophysiological roles. Particular attention is given to the integrated physiology of NHEs in the main organ systems. We provide several novel analyses and useful overviews, and we pinpoint main remaining enigmas, which we hope will inspire novel research on these highly versatile proteins.

Model annotation and discovery with the Physiome Model Repository

BACKGROUND

Mathematics and Phy sics-based simulation models have the potential to help interpret and encapsulate biological phenomena in a computable and reproducible form. Similarly, comprehensive descriptions of such models help to ensure that such models are accessible, discoverable, and reusable. To this end, researchers have developed tools and standards to encode mathematical models of biological systems enabling reproducibility and reuse, tools and guidelines to facilitate semantic description of mathematical models, and repositories in which to archive, share, and discover models. Scientists can leverage these resources to investigate specific questions and hypotheses in a more efficient manner.

RESULTS

We have comprehensively annotated a cohort of models with biological semantics. These annotated models are freely available in the Physiome Model Repository (PMR). To demonstrate the benefits of this approach, we have developed a web-based tool which enables users to discover models relevant to their work, with a particular focus on epithelial transport. Based on a semantic query, this tool will help users discover relevant models, suggesting similar or alternative models that the user may wish to explore or use.

CONCLUSION

The semantic annotation and the web tool we have developed is a new contribution enabling scientists to discover relevant models in the PMR as candidates for reuse in their own scientific endeavours. This approach demonstrates how semantic web technologies and methodologies can contribute to biomedical and clinical research. The source code and links to the web tool are available at https://github.com/dewancse/model-discovery-tool.

Computational Modeling of Glucose Uptake in the Enterocyte

Absorption of glucose across the epithelial cells of the small intestine is a key process in human nutrition and initiates signaling cascades that regulate metabolic homeostasis. Validated and predictive mathematical models of glucose transport in intestinal epithelial cells are essential for interpreting experimental data, generating hypotheses, and understanding the contributions of and interactions between transport pathways. Here we report on the development of such a model that, in contrast to existing models, incorporates mechanistic descriptions of all relevant transport proteins and is implemented in the CellML framework. The model is validated against experimental and simulation data from the literature. It is then used to elucidate the relative contributions of the sodium-glucose cotransporter (SGLT1) and the glucose transporter type 2 (GLUT2) proteins in published measurements of glucose absorption from human intestinal epithelial cell lines. The model predicts that the contribution of SGLT1 dominates at low extracellular glucose concentrations (<20 mM) and short exposure times (<60 s) while the GLUT2 contribution is more significant at high glucose concentrations and long durations. Implementation in CellML permitted a modular structure in which the model was composed by reusing existing models of the individual transporters. The final structure also permits transparent changes of the model components and parameter values in order to facilitate model reuse, extension, and customization (for example, to simplify, or add complexity to specific transporter/pathway models, or reuse the model as a component of a larger framework) and carry out parameter sensitivity studies.

Meeting the multiscale challenge: representing physiology processes over ApiNATOMY circuits using bond graphs

We introduce, and provide examples of, the application of the bond graph formalism to explicitly represent biophysical processes between and within modular biological compartments in ApiNATOMY. In particular, we focus on modelling scenarios from acid-base physiology to link distinct process modalities as bond graphs over an ApiNATOMY circuit of multiscale compartments. The embedding of bond graphs onto ApiNATOMY compartments provides a semantically and mathematically explicit basis for the coherent representation, integration and visualisation of multiscale physiology processes together with the compartmental topology of those biological structures that convey these processes.

Proximal tubule NHE3 activity is inhibited by beta-arrestin-biased angiotensin II type 1 receptor signaling

Physiological concentrations of angiotensin II (ANG II) upregulate the activity of Na+/H+ exchanger isoform 3 (NHE3) in the renal proximal tubule through activation of the ANG II type I (AT1) receptor/G protein-coupled signaling. This effect is key for maintenance of extracellular fluid volume homeostasis and blood pressure. Recent findings have shown that selective activation of the beta-arrestin-biased AT1 receptor signaling pathway induces diuresis and natriuresis independent of G protein-mediated signaling. This study tested the hypothesis that activation of this AT1 receptor/beta-arrestin signaling inhibits NHE3 activity in proximal tubule. To this end, we determined the effects of the compound TRV120023, which binds to the AT1R, blocks G-protein coupling, and stimulates beta-arrestin signaling on NHE3 function in vivo and in vitro. NHE3 activity was measured in both native proximal tubules, by stationary microperfusion, and in opossum proximal tubule (OKP) cells, by Na+-dependent intracellular pH recovery. We found that 10−7 M TRV120023 remarkably inhibited proximal tubule NHE3 activity both in vivo and in vitro. Additionally, stimulation of NHE3 by ANG II was completely suppressed by TRV120023 both in vivo as well as in vitro. Inhibition of NHE3 activity by TRV120023 was associated with a decrease in NHE3 surface expression in OKP cells and with a redistribution from the body to the base of the microvilli in the rat proximal tubule. These findings indicate that biased signaling of the beta-arrestin pathway through the AT1 receptor inhibits NHE3 activity in the proximal tubule at least in part due to changes in NHE3 subcellular localization.

Mathematical modeling of urea transport in the kidney

Mathematical modeling techniques have been useful in providing insights into biological systems, including the kidney. This article considers some of the mathematical models that concern urea transport in the kidney. Modeling simulations have been conducted to investigate, in the context of urea cycling and urine concentration, the effects of hypothetical active urea secretion into pars recta. Simulation results suggest that active urea secretion induces a "urea-selective" improvement in urine concentrating ability. Mathematical models have also been built to study the implications of the highly structured organization of tubules and vessels in the renal medulla on urea sequestration and cycling. The goal of this article is to show how physiological problems can be formulated and studied mathematically, and how such models may provide insights into renal functions.

Local pH domains regulate NHE3-mediated Na⁺ reabsorption in the renal proximal tubule

The proximal tubule Na(+)/H(+) exchanger 3 (NHE3), located in the apical dense microvilli (brush border), plays a major role in the reabsorption of NaCl and water in the renal proximal tubule. In response to a rise in blood pressure NHE3 redistributes in the plane of the plasma membrane to the base of the brush border, where NHE3 activity is reduced. This NHE3 redistribution is assumed to provoke pressure natriuresis; however, it is unclear how NHE3 redistribution per se reduces NHE3 activity. To investigate if the distribution of NHE3 in the brush border can change the reabsorption rate, we constructed a spatiotemporal mathematical model of NHE3-mediated Na(+) reabsorption across a proximal tubule cell and compared the model results with in vivo experiments in rats. The model predicts that when NHE3 is localized exclusively at the base of the brush border, it creates local pH microdomains that reduce NHE3 activity by >30%. We tested the model's prediction experimentally: the rat kidney cortex was loaded with the pH-sensitive fluorescent dye BCECF, and cells of the proximal tubule were imaged in vivo using confocal fluorescence microscopy before and after an increase of blood pressure by ∼50 mmHg. The experimental results supported the model by demonstrating that a rise of blood pressure induces the development of pH microdomains near the bottom of the brush border. These local changes in pH reduce NHE3 activity, which may explain the pressure natriuresis response to NHE3 redistribution.

Transepithelial glucose transport and Na+/K+ homeostasis in enterocytes: an integrative model

The uptake of glucose and the nutrient coupled transcellular sodium traffic across epithelial cells in the small intestine has been an ongoing topic in physiological research for over half a century. Driving the uptake of nutrients like glucose, enterocytes must have regulatory mechanisms that respond to the considerable changes in the inflow of sodium during absorption. The Na-K-ATPase membrane protein plays a major role in this regulation. We propose the hypothesis that the amount of active Na-K-ATPase in enterocytes is directly regulated by the concentration of intracellular Na(+) and that this regulation together with a regulation of basolateral K permeability by intracellular ATP gives the enterocyte the ability to maintain ionic Na(+)/K(+) homeostasis. To explore these regulatory mechanisms, we present a mathematical model of the sodium coupled uptake of glucose in epithelial enterocytes. Our model integrates knowledge about individual transporter proteins including apical SGLT1, basolateral Na-K-ATPase, and GLUT2, together with diffusion and membrane potentials. The intracellular concentrations of glucose, sodium, potassium, and chloride are modeled by nonlinear differential equations, and molecular flows are calculated based on experimental kinetic data from the literature, including substrate saturation, product inhibition, and modulation by membrane potential. Simulation results of the model without the addition of regulatory mechanisms fit well with published short-term observations, including cell depolarization and increased concentration of intracellular glucose and sodium during increased concentration of luminal glucose/sodium. Adding regulatory mechanisms for regulation of Na-K-ATPase and K permeability to the model show that our hypothesis predicts observed long-term ionic homeostasis.

Fluid dilution and efficiency of Na(+) transport in a mathematical model of a thick ascending limb cell

Thick ascending limb (TAL) cells are capable of reducing tubular fluid Na(+) concentration to as low as ~25 mM, and yet they are thought to transport Na(+) efficiently owing to passive paracellular Na(+) absorption. Transport efficiency in the TAL is of particular importance in the outer medulla where O(2) availability is limited by low blood flow. We used a mathematical model of a TAL cell to estimate the efficiency of Na(+) transport and to examine how tubular dilution and cell volume regulation influence transport efficiency. The TAL cell model represents 13 major solutes and the associated transporters and channels; model equations are based on mass conservation and electroneutrality constraints. We analyzed TAL transport in cells with conditions relevant to the inner stripe of the outer medulla, the cortico-medullary junction, and the distal cortical TAL. At each location Na(+) transport efficiency was computed as functions of changes in luminal NaCl concentration ([NaCl]), [K(+)], [NH(4)(+)], junctional Na(+) permeability, and apical K(+) permeability. Na(+) transport efficiency was calculated as the ratio of total net Na(+) transport to transcellular Na(+) transport. Transport efficiency is predicted to be highest at the cortico-medullary boundary where the transepithelial Na(+) gradient is the smallest. Transport efficiency is lowest in the cortex where luminal [NaCl] approaches static head.

A model of Na+/H+ exchanger and its central role in regulation of pH and Na+ in cardiac myocytes

A new kinetic model of the Na(+)/H(+) exchanger (NHE) was developed by fitting a variety of major experimental findings, such as ion-dependencies, forward/reverse mode, and the turnover rate. The role of NHE in ion homeostasis was examined by implementing the NHE model in a minimum cell model including intracellular pH buffer, Na(+)/K(+) pump, background H(+), and Na(+) fluxes. This minimum cell model was validated by reconstructing recovery of pH(i) from acidification, accompanying transient increase in [Na(+)](i) due to NHE activity. Based on this cell model, steady-state relationships among pH(i), [Na(+)](I), and [Ca(2+)](i) were quantitatively determined, and thereby the critical level of acidosis for cell survival was predicted. The acidification reported during partial blockade of the Na(+)/K(+) pump was not attributed to a dissipation of the Na(+) gradient across the membrane, but to an increase in indirect H(+) production. This NHE model, though not adapted to the dimeric behavioral aspects of NHE, can provide a strong clue to quantitative prediction of degree of acidification and accompanying disturbance of ion homeostasis under various pathophysiological conditions.

A mathematical model of rat ascending Henle limb. II. Epithelial function

A mathematical model of ascending Henle limb (AHL) epithelium has been fashioned using kinetic representations of Na+-K+-2Cl- cotransporter (NKCC2), KCC4, and type 3 Na+/H+ exchanger (NHE3), with transporter densities selected to yield the reabsorptive Na+ flux expected for rat tubules in vivo. Of necessity, this model predicts fluxes that are higher than those measured in vitro. The kinetics of the NKCC and KCC are such that Na+ reabsorption by the model tubule is responsive to variation in luminal NaCl concentration over the range of 30 to 130 mM, with only minor changes in cell volume. Peritubular KCC accounts for about half the reabsorptive Cl- flux, with the remainder via peritubular Cl- channels. Transcellular Na+ flux is turned off by increasing peritubular KCl, which produces increased cytosolic Cl- and thus inhibits NKCC2 transport. In the presence of physiological concentrations of ammonia, there is a large acid challenge to the cell, due primarily to NH4+ entry via NKCC2, with diffusive NH3 exit to both lumen and peritubular solutions. When NHE3 density is adjusted to compensate this acid challenge, the model predicts luminal membrane proton secretion that is greater than the HCO3(-)-reabsorptive fluxes measured in vitro. The model also predicts luminal membrane ammonia cycling, with uptake via NKCC2 or K+ channel, and secretion either as NH4+ by NHE3 or as diffusive NH3 flux in parallel with a secreted proton. If such luminal ammonia cycling occurs in vivo, it could act in concert with luminal K+ cycling to facilitate AHL Na+ reabsorption via NKCC2. With physiological ammonia, peritubular KCl also blunts NHE3 activity by inhibiting NH4+ uptake on the Na-K-ATPase, and alkalinizing the cell.

A mathematical model of rat ascending Henle limb. I. Cotransporter function

Kinetic models of Na+-K+-2Cl- costransporter (NKCC2) and K+-Cl- cotransporter (KCC4), two of the key cotransporters of the Henle limb, are fashioned with inclusion of terms representing binding and transport of NH4+. The models are simplified using assumptions of equilibrium ion binding, binding symmetry, and identity of Cl- binding sites. Model parameters are selected to be consistent with flux data from expression of these transporters in oocytes, specifically inwardly directed coupled transport of rubidium. In the analysis of these models, it is found that despite the simplifying assumptions to reduce the number of model parameters, neither model is uniquely determined by the data. For NKCC or KCC there are two- or three-parameter families of "optimal" solutions. As a consequence, one may specify several carrier translocation rates and/or ion affinities before fitting the remaining coefficients to the data, with no loss of fidelity in simulating the experiments. Model calculations suggest that with respect to NKCC2 near its operating point, the curve of ion flux as a function of cell Cl- is steep, and with respect to KCC4, its curve of ion flux as a function of peritubular K+ is also steep. The implication is that the kinetics are suitable for these two transporters in series to act as a sensor for peritubular K+, to modulate AHL Na+ reabsorption, with cytosolic Cl- as the intermediate variable. The models also reveal the potential for luminal NH4+ to be a potent catalyst for NKCC2 Na+ reabsorption, provided suitable exit mechanisms for NH4+ (from cell-to-lumen) are operative. It is found that KCC4 is likely to augment the secretory NH4+ flux, with peritubular NH4+ uptake driven by the cell-to-blood K+ gradient.

Modeling the cardiac Na(+)/H (+) exchanger based on major experimental findings

Na(+)-H(+) exchanger (NHE) is the main acid extruder in cardiac myocytes. We review the experimental findings of ion-dependency of NHE activity, and the mathematical modeling developed so far. In spite of extensive investigation, many unsolved questions still remain. We consider that the precise description of NHE activity with mathematical models elucidates the roles of NHE in maintaining ionic homeostasis, especially under pathophysiological conditions.

Modeling proximal tubule cell homeostasis: tracking changes in luminal flow

During normal kidney function, there are routinely wide swings in proximal tubule fluid flow and proportional changes in Na(+) reabsorption across tubule epithelial cells. This "glomerulotubular balance" occurs in the absence of any substantial change in cell volume, and is thus a challenge to coordinate luminal membrane solute entry with peritubular membrane solute exit. In this work, linear optimal control theory is applied to generate a configuration of regulated transporters that could achieve this result. A previously developed model of rat proximal tubule epithelium is linearized about a physiologic reference condition; the approximate linear system is recast as a dynamical system; and a Riccati equation is solved to yield the optimal linear feedback that stabilizes Na(+) flux, cell volume, and cell pH. The first observation is that optimal feedback control is largely consigned to three physiologic variables, cell volume, cell electrical potential, and lateral intercellular hydrostatic pressure. Parameter modulation by cell volume stabilizes cell volume; parameter modulation by electrical potential or interspace pressure act to stabilize Na(+) flux and cell pH. This feedback control is utilized in a tracking problem, in which reabsorptive Na(+) flux varies over a factor of two, in order to represent a substantial excursion of glomerulotubular balance. The resulting control parameters consist of two terms, an autonomous term and a feedback term, and both terms include transporters on both luminal and peritubular cell membranes. Overall, the increase in Na(+) flux is achieved with upregulation of luminal Na(+)/H(+) exchange and Na(+)-glucose cotransport, with increased peritubular Na(+)-3HCO(3)(-) and K(+)-Cl(-) cotransport, and with increased Na(+), K(+)-ATPase activity. The configuration of activated transporters emerges as a testable hypothesis of the molecular basis for glomerulotubular balance. It is suggested that the autonomous control component at each cell membrane could represent the cytoskeletal effects of luminal flow.

Flow-activated transport events along the nephron

PURPOSE OF REVIEW

It is well recognized that an increase in glomerular filtration rate leads to an increase in fluid, Na+ and HCO3- absorption in proximal tubules; however, underlying mechanisms of this modulation have not been delineated. This review provides an update of flow-activated transport events along the nephron. Transporters, flow-sensors and secondary messengers that may modulate flow are also discussed.

RECENT FINDINGS

We have demonstrated that both NHE3 and H-ATPase activities are modulated by axial flow in mouse proximal tubules in vitro. Experimental data and modeling calculations provide strong evidence that brush-border microvilli function as flow sensors in the proximal tubule. In addition, AngII receptor localization is regulated by flow in cultured proximal tubule cells, and flow induces eNOS translocation in TAL.

SUMMARY

Flow-modulated NHE3 activity is the regulatory mechanism for GTB. It is independent of neuron and systemic hormonal regulation, but requires the intact actin cytoskeleton to transmit the signal of altered axial flow sensed by brush-border microvilli. Unanswered questions include the identification of specific signaling transduction mechanisms and second messengers in response to flow. Whether the Na+/2Cl-/K+-cotransporter in TAL is flow-activated, and whether a divalent cation, Ca2+ and Mg2+ transport, can be regulated by flow is unknown.

Flow-dependent transport in a mathematical model of rat proximal tubule

The mathematical model of rat proximal tubule has been extended to include calculation of microvillous torque and to incorporate torque-dependent solute transport in a compliant tubule. The torque calculation follows that of Du Z, Yan Q, Duan Y, Weinbaum S, Weinstein AM, and Wang T (Am J Physiol 290: F289-F296, 2006). In the model calculations, torque-dependent scaling of luminal membrane transporter density [either as an ensemble or just type 3 Na(+)/H(+) exchanger (NHE3) alone] had a relatively small impact on overall Na(+) reabsorption and could produce a lethal derangement of cell volume; coordinated regulation of luminal and peritubular transporters was required to represent the overall impact of luminal flow on Na(+) reabsorption. When the magnitude of torque-dependent Na(+) reabsorption in the model agrees with that observed in mouse proximal tubules, the model tubule shows nearly perfect perfusion-absorption balance at high luminal perfusion rates, but enhanced sensitivity of reabsorption at low flow. With a slightly lower coefficient for torque-sensitive transporter insertion, perfusion-absorption balance in the model tubule is closer to observations in the rat over a broader range of inlet flows. In simulation of hyperglycemia, torque-dependent transport attenuated the diuretic effect and brought the model tubule into closer agreement with experimental observation in the rat. The model was also extended to represent finite rates of hydration and dehydration of CO(2) and H(2)CO(3). With carbonic anhydrase inhibition, torque-dependent transport blunted the diuretic effect and enhanced the shift from paracellular to transcellular NaCl reabsorption. The new features of this model tubule are an important step toward simulation of glomerulotubular balance.

Differences in pHi recovery in CO2-chemosensitive and non-chemosensitive cells: predictions from a mathematical model

In this paper, we present a mathematic model designed to identify potential mechanisms responsible for the observed differences in pHi recovery in CO(2)-chemosensitive versus non-chemosensitive cells. The model suggests that differences in pHi regulation may be dependent upon differences in the activation set-point of the internal modifier site of the Na(+)/H(+) exchanger (NHE).

A mathematical model of rat distal convoluted tubule. I. Cotransporter function in early DCT

A model of rat early distal convoluted tubule (DCT) is developed in conjunction with a kinetic representation of the thiazide-sensitive NaCl cotransporter (TSC). Realistic constraints on cell membrane electrical conductance require that most of the peritubular Cl(-) reabsorption proceeds via a KCl cotransporter,along with most of the K(+) recycled from the Na-K-ATPase. The model tubule reproduces the saturable Cl(-) reabsorption of DCT but not the micropuncture finding of linear Na(+) flux in response to load, more likely a feature of late DCT (CNT). As in proximal tubule, early DCT HCO(3)(-) reabsorption is mediated by a luminal Na(+)/H(+) exchanger (NHE), but in contrast to proximal tubule, the DCT exchanger is operating closer to equilibrium. In the model DCT, two consequences of the lesser driving force for NHE exchange are an acidic cytosol and wider swings in NHE flux with perturbations of luminal composition. Variations in luminal NaCl provide a challenge to cell volume, which can be blunted by volume dependence of the KCl cotransporter. Cell swelling can also be induced by increases in peritubular K(+) concentration. In this case, volume-dependent inhibition of TSC could provide volume homeostasis that also enhances distal Na(+) delivery, and ultimately enhances renal K(+) excretion. In the model DCT, proton secretion is blunted by peritubular HCO(3)(-), so that there is little contribution by this segment to the maintenance of metabolic alkalosis. During alkalosis, the model predicts that increasing luminal NaCl concentration enhances NHE flux, so that these calculations provide no support for a role of early DCT in recovery from Cl(-) depletion alkalosis.

A numerical model of acid-base transport in rat distal tubule

The purpose of this study is to develop a numerical model that simulates acid-base transport in rat distal tubule. We have previously reported a model that deals with transport of Na(+), K(+), Cl(-), and water in this nephron segment (Chang H and Fujita T. Am J Physiol Renal Physiol 276: F931-F951, 1999). In this study, we extend our previous model by incorporating buffer systems, new cell types, and new transport mechanisms. Specifically, the model incorporates bicarbonate, ammonium, and phosphate buffer systems; has cell types corresponding to intercalated cells; and includes the Na/H exchanger, H-ATPase, and anion exchanger. Incorporation of buffer systems has required the following modifications of model equations: new model equations are introduced to represent chemical equilibria of buffer partners [e.g., pH = pK(a) + log(10) (NH(3)/NH(4))], and the formulation of mass conservation is extended to take into account interconversion of buffer partners. Furthermore, finite rates of H(2)CO(3)-CO(2) interconversion (i.e., H(2)CO(3) &rlharr; CO(2) + H(2)O) are taken into account in modeling the bicarbonate buffer system. Owing to this treatment, the model can simulate the development of disequilibrium pH in the distal tubular fluid. For each new transporter, a state diagram has been constructed to simulate its transport kinetics. With appropriate assignment of maximal transport rates for individual transporters, the model predictions are in agreement with free-flow micropuncture experiments in terms of HCO reabsorption rate in the normal state as well as under the high bicarbonate load. Although the model cannot simulate all of the microperfusion experiments, especially those that showed a flow-dependent increase in HCO reabsorption, the model is consistent with those microperfusion experiments that showed HCO reabsorption rates similar to those in the free-flow micropuncture experiments. We conclude that it is possible to develop a numerical model of the rat distal tubule that simulates acid-base transport, as well as basic solute and water transport, on the basis of tubular geometry, physical principles, and transporter kinetics. Such a model would provide a useful means of integrating detailed kinetic properties of transporters and predicting macroscopic transport characteristics of this nephron segment under physiological and pathophysiological settings.

Modeling cell volume regulation in nonexcitable cells: the roles of the Na+ pump and of cotransport systems

The purpose of this study is to contribute to understanding the role of Na+-K+-ATPase and of ionic cotransporters in the regulation of cell volume, by employing a model that describes the rates of change of the intracellular concentrations of Na+, K+, and Cl-, of the cell volume, and of the membrane potential. In most previous models of dynamic cellular phenomena, Na+-K+-ATPase is incorporated via phenomenological formulations; the enzyme is incorporated here via an explicit kinetic scheme. Another feature of the present model is the capability to perform short-term cell volume regulation mediated by cotransporters of KCl and NaCl. The model is employed to perform numerical simulations for a "typical" nonpolarized animal cell. Basically, the results are consistent with the view that the Na+ pump mainly plays a long-term role in the maintenance of the electrochemical gradients of Na+ and K+ and that short-term cell volume regulation is achieved via passive transport, exemplified in this case by the cotransport of KCl and NaCl.