A mathematical model of the outer medullary collecting duct of the rat

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.

Intraglomerular crosstalk elaborately regulates podocyte injury and repair in diabetic patients: insights from a 3D multiscale modeling study

Podocytes are mainly involved in the regulation of glomerular filtration rate (GFR) under physiological condition. Podocyte depletion is a crucial pathological alteration in diabetic nephropathy (DN) and results in a broad spectrum of clinical syndromes such as protein urine and renal insufficiency. Recent studies indicate that depleted podocytes can be regenerated via differentiation of the parietal epithelial cells (PECs), which serve as the local progenitors of podocytes. However, the podocyte regeneration process is regulated by a complicated mechanism of cell-cell interactions and cytokine stimulations, which has been studied in a piecemeal manner rather than systematically. To address this gap, we developed a high-resolution multi-scale multi-agent mathematical model in 3D, mimicking the in situ glomerulus anatomical structure and micro-environment, to simulate the podocyte regeneration process under various cytokine perturbations in healthy and diabetic conditions. Our model showed that, treatment with pigment epithelium derived factor (PEDF) or insulin-like growth factor-1 (IGF-1) alone merely ameliorated the glomerulus injury, while co-treatment with both cytokines replenished the damaged podocyte population gradually. In addition, our model suggested that continuous administration of PEDF instead of a bolus injection sustained the regeneration process of podocytes. Part of the results has been validated by our in vivo experiments. These results indicated that amelioration of the glomerular stress by PEDF and promotion of PEC differentiation by IGF-1 are equivalently critical for podocyte regeneration. Our 3D multi-scale model represents a powerful tool for understanding the signaling regulation and guiding the design of cytokine therapies in promoting podocyte regeneration.

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.

Three-dimensional simulation of urine concentrating mechanism in a functional unit of rat outer medulla. I. Model structure and base case results

The urine formation and excretion system have long been of interest for mathematicians and physiologists to elucidate the obscurities within the process happens in renal tissue. In this study, a novel three-dimensional approach is utilized for modeling the urine concentrating mechanism in rat renal outer medulla which is essentially focused on demonstrating the significance of tubule's architecture revealed in anatomic studies and physiological literature. Since nephrons and vasculatures work interdependently through a highly structured arrangement in outer medulla which is dominated by vascular bundles, a detailed functional unit is proposed based on this specific configuration. Furthermore, due to relatively lesser influence of vasa recta on interstitial medullary osmolality and osmotic gradients as well as model structure simplicity, central core assumption is employed. The model equations are based on three spatial dimensional mass, momentum and species transport equations as well as standard expressions for solutes and water transmural transport. Our model can simulate preferential interactions between different tubules and it is shown that such interactions promote solute cycling and subsequently, enhance urine-concentrating capability. The numerical results are well consistent with tissue slice experiments and moreover, our model predicts more corticomedullary osmolality gradient in outer medulla than previous influential 1-D simulations.

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.

Chloride transport

Chloride transport along the nephron is one of the key actions of the kidney that regulates extracellular volume and blood pressure. To maintain steady state, the kidney needs to reabsorb the vast majority of the filtered load of chloride. This is accomplished by the integrated function of sequential chloride transport activities along the nephron. The detailed mechanisms of transport in each segment generate unique patterns of interactions between chloride and numerous other individual components that are transported by the kidney. Consequently, chloride transport is inextricably intertwined with that of sodium, potassium, protons, calcium, and water. These interactions not only allow for exquisitely precise regulation but also determine the particular patterns in which the system can fail in disease states.

Hormonal regulation of salt and water excretion: a mathematical model of whole kidney function and pressure natriuresis

We present a lumped-nephron model that explicitly represents the main features of the underlying physiology, incorporating the major hormonal regulatory effects on both tubular and vascular function, and that accurately simulates hormonal regulation of renal salt and water excretion. This is the first model to explicitly couple glomerulovascular and medullary dynamics, and it is much more detailed in structure than existing whole organ models and renal portions of multiorgan models. In contrast to previous medullary models, which have only considered the antidiuretic state, our model is able to regulate water and sodium excretion over a variety of experimental conditions in good agreement with data from experimental studies of the rat. Since the properties of the vasculature and epithelia are explicitly represented, they can be altered to simulate pathophysiological conditions and pharmacological interventions. The model serves as an appropriate starting point for simulations of physiological, pathophysiological, and pharmacological renal conditions and for exploring the relationship between the extrarenal environment and renal excretory function in physiological and pathophysiological contexts.

Mathematical modeling of kidney transport

In addition to metabolic waste and toxin excretion, the kidney also plays an indispensable role in regulating the balance of water, electrolytes, nitrogen, and acid-base. In this review, we describe representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, epithelial transport, and regulation of renal oxygen transport. We discuss the extent to which these modeling efforts have expanded our understanding of renal function in both health and disease.

Mechanisms of pressure-diuresis and pressure-natriuresis in Dahl salt-resistant and Dahl salt-sensitive rats

Background

Data on blood flow regulation, renal filtration, and urine output in salt-sensitive Dahl S rats fed on high-salt (hypertensive) and low-salt (prehypertensive) diets and salt-resistant Dahl R rats fed on high-salt diets were analyzed using a mathematical model of renal blood flow regulation, glomerular filtration, and solute transport in a nephron.

Results

The mechanism of pressure-diuresis and pressure-natriuresis that emerges from simulation of the integrated systems is that relatively small increases in glomerular filtration that follow from increases in renal arterial pressure cause relatively large increases in urine and sodium output. Furthermore, analysis reveals the minimal differences between the experimental cases necessary to explain the observed data. It is determined that differences in renal afferent and efferent arterial resistances are able to explain all of the qualitative differences in observed flows, filtration rates, and glomerular pressure as well as the differences in the pressure-natriuresis and pressure-diuresis relationships in the three groups. The model is able to satisfactorily explain data from all three groups without varying parameters associated with glomerular filtration or solute transport in the nephron component of the model.

Conclusions

Thus the differences between the experimental groups are explained solely in terms of difference in blood flow regulation. This finding is consistent with the hypothesis that, if a shift in the pressure-natriuresis relationship is the primary cause of elevated arterial pressure in the Dahl S rat, then alternation in how renal afferent and efferent arterial resistances are regulated represents the primary cause of chronic hypertension in the Dahl S rat.

Intravoxel incoherent motion and diffusion-tensor imaging in renal tissue under hydration and furosemide flow challenges

PURPOSE

To assess the reproducibility and the distribution of intravoxel incoherent motion (IVIM) and diffusion-tensor (DT) imaging parameters in healthy renal cortex and medulla at baseline and after hydration or furosemide challenges.

MATERIALS AND METHODS

Using an institutional review board-approved HIPAA-compliant protocol with written informed consent, IVIM and DT imaging were performed at 3 T in 10 volunteers before and after water loading or furosemide administration. IVIM (apparent diffusion coefficient [ADC], tissue diffusivity [D(t)], perfusion fraction [f(p)], pseudodiffusivity [D(p)]) and DT (mean diffusivity [MD], fractional anisotropy [FA], eigenvalues [λ(i)]) imaging parameters and urine output from serial bladder volumes were calculated. (a)Reproducibility was quantified with coefficient of variation, intraclass correlation coefficient, and Bland-Altman limits of agreement; (b) contrast and challenge response were quantified with analysis of variance; and (c) Pearson correlations were quantified with urine output.

RESULTS

Good reproducibility was found for ADC, D(t), MD, FA, and λ(i) (average coefficient of variation, 3.7% [cortex] and 5.0% [medulla]), and moderate reproducibility was found for D(p), f(p), and f(p) · D(p) (average coefficient of variation, 18.7% [cortex] and 25.9% [medulla]). Baseline cortical diffusivities significantly exceeded medullary values except D(p), for which medullary values significantly exceeded cortical values, and λ(1,) which showed no contrast. ADC, D(t), MD, and λ(i) increased significantly for both challenges. Medullary diffusivity increases were dominated by transverse diffusion (1.72 ± 0.09 [baseline] to 1.79 ± 0.10 [hydration] μm(2)/msec, P = .0059; or 1.86 ± 0.07 [furosemide] μm(2)/msec, P = .0094). Urine output correlated with cortical ADC with furosemide (r = 0.7, P = .034) and with medullary λ(1) (r = 0.83, P = .0418), λ(2) (r = 0.85, P = .0301), and MD (r = 0.82, P = .045) with hydration.

CONCLUSION

Diffusion MR metrics are sensitive to flow changes in kidney induced by diuretic challenges. The results of this study suggest that vascular flow, tubular dilation, water reabsorption, and intratubular flow all play important roles in diffusion-weighted imaging contrast.

Countercurrent multiplication may not explain the axial osmolality gradient in the outer medulla of the rat kidney

It has become widely accepted that the osmolality gradient along the corticomedullary axis of the mammalian outer medulla is generated and sustained by a process of countercurrent multiplication: active NaCl absorption from thick ascending limbs is coupled with the counterflow configuration of the descending and ascending limbs of the loops of Henle to generate an axial osmolality gradient along the outer medulla. However, aspects of anatomic structure (e.g., the physical separation of the descending limbs of short loops of Henle from contiguous ascending limbs), recent physiologic experiments (e.g., those that suggest that the thin descending limbs of short loops of Henle have a low osmotic water permeability), and mathematical modeling studies (e.g., those that predict that water-permeable descending limbs of short loops are not required for the generation of an axial osmolality gradient) suggest that countercurrent multiplication may be an incomplete, or perhaps even erroneous, explanation. We propose an alternative explanation for the axial osmolality gradient: we regard the thick limbs as NaCl sources for the surrounding interstitium, and we hypothesize that the increasing axial osmolality gradient along the outer medulla is primarily sustained by an increasing ratio, as a function of increasing medullary depth, of NaCl absorption (from thick limbs) to water absorption (from thin descending limbs of long loops of Henle and, in antidiuresis, from collecting ducts). We further hypothesize that ascending vasa recta that are external to vascular bundles will carry, toward the cortex, an absorbate that at each medullary level is hyperosmotic relative to the adjacent interstitium.

Regulation of pendrin by pH: dependence on glycosylation

Mutations in the anion exchanger pendrin are responsible for Pendred syndrome, an autosomal recessive disease characterized by deafness and goitre. Pendrin is highly expressed in kidney collecting ducts, where it acts as a chloride/bicarbonate exchanger and thereby contributes to the regulation of acid–base homoeostasis and blood pressure. The present study aimed to characterize the intrinsic properties of pendrin. Mouse pendrin was transfected in HEK (human embryonic kidney) 293 and OKP (opossum kidney proximal tubule) cells and its activity was determined by monitoring changes in the intracellular pH induced by variations of transmembrane anion gradients. Combining measurements of pendrin activity with mathematical modelling we found that its affinity for Cl−, HCO3− and OH− varies with intracellular pH, with increased activity at low intracellular pH. Maximal pendrin activity was also stimulated at low extracellular pH, suggesting the presence of both intracellular and extracellular proton regulatory sites. We identified five putative pendrin glycosylation sites, only two of which are used. Mutagenesis-induced disruption of pendrin glycosylation did not alter its cell-surface expression or polarized targeting to the apical membrane and basal activity, but fully abrogated its sensitivity to extracellular pH. The hither to unknown regulation of pendrin by external pH may constitute a key mechanism in controlling ionic exchanges across the collecting duct and inner ear.

A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture

In a companion study [Layton AT. A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol. (First published November 10, 2010). 10.1152/ajprenal.00203.2010] a region-based mathematical model was formulated for the urine concentrating mechanism in the renal medulla of the rat kidney. In the present study, we investigated model sensitivity to some of the fundamental structural assumptions. An unexpected finding is that the concentrating capability of this region-based model falls short of the capability of models that have radially homogeneous interstitial fluid at each level of only the inner medulla (IM) or of both the outer medulla and IM, but are otherwise analogous to the region-based model. Nonetheless, model results reveal the functional significance of several aspects of tubular segmentation and heterogeneity: 1) the exclusion of ascending thin limbs that reach into the deep IM from the collecting duct clusters in the upper IM promotes urea cycling within the IM; 2) the high urea permeability of the lower IM thin limb segments allows their tubular fluid urea content to equilibrate with the surrounding interstitium; 3) the aquaporin-1-null terminal descending limb segments prevent water entry and maintain the transepithelial NaCl concentration gradient; 4) a higher thick ascending limb Na(+) active transport rate in the inner stripe augments concentrating capability without a corresponding increase in energy expenditure for transport; 5) active Na(+) reabsorption from the collecting duct elevates its tubular fluid urea concentration. Model calculations predict that these aspects of tubular segmentation and heterogeneity promote effective urine concentrating functions.

Maximum urine concentrating capability in a mathematical model of the inner medulla of the rat kidney

In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-to-plasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of CD surface area) along the cortico-medullary axis.

A mathematical model of rat ascending Henle limb. III. Tubular function

K+ plays a catalytic role in AHL Na+ reabsorption via Na+-K+-2Cl- cotransporter (NKCC2), recycling across luminal K+ channels, so that luminal K+ is not depleted. Based on models of the ascending Henle limb (AHL) epithelium, it has been hypothesized that NH4+ may also catalyze luminal Na+ uptake. This hypothesis requires that luminal NH4+ not be depleted, implying replenishment via either direct secretion of NH4+, or NH3 in parallel with a proton. In the present work, epithelial models of rat medullary and cortical AHL (Weinstein AM, Krahn TA. Am J Physiol Renal Physiol 298: F000-F000, 2009) are configured as tubules and examined in simulations of function in vitro and in vivo to assess the feasibility of a catalytic role for NH4+ in Na+ reabsorption. Modulation of Na+ transport is also examined by peritubular K+ concentration and by Bartter-type transport defects in NKCC2 (type 1), in luminal membrane K+ channels (type 2), and in peritubular Cl- channels (type 3). It is found that a catalytic role for NH4+, which is significant in magnitude (relative to K+), is quantitatively realistic, in terms of uptake via NKCC2, and in terms of luminal membrane ammonia backflux. Simulation of a 90% NKCC2 defect is predicted to double distal Na+ delivery; it is also predicted to increase distal acid delivery (principally as NH4+). With doubling of medullary K+, the model predicts a 30% increase in distal Na+ delivery, but in this case there is a decrease in AHL acidification. This effect of peritubular K+ on proton secretion appears to be akin to type 3 Bartter's pathophysiology, in which there is decreased peritubular HCO3- exit, cytosolic alkalinization, and a consequent decrease in luminal proton secretion by NHE3. One consequence of overlapping and redundant roles for K+ and NH4+, is a blunted impact of luminal membrane K+ permeability on overall Na+ reabsorption, so that type 2 Bartter pathophysiology is not well captured by the model.

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.

A mathematical model of O2 transport in the rat outer medulla. I. Model formulation and baseline results

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O(2) supply to the renal medulla barely exceeds its O(2) requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O(2) distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton (Am J Physiol Renal Physiol 289: F1346-F1366, 2005) to incorporate the transport of RBCs, Hb, and O(2). We considered basal cellular O(2) consumption and O(2) consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po(2) gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O(2), at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po(2) between regions, limits O(2) reabsorption from long descending vasa recta, and helps preserve O(2) delivery to the inner medulla. Under baseline conditions, significantly more O(2) is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O(2) supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. II. Parameter sensitivity and tubular inhomogeneity

In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1346-F1366, 2005), a region-based mathematical model was formulated for the urine concentrating mechanism (UCM) in the outer medulla (OM) of the rat kidney. In the present study, we quantified the sensitivity of that model to several structural assumptions, including the degree of regionalization and the degree of inclusion of short descending limbs (SDLs) in the vascular bundles of the inner stripe (IS). Also, we quantified model sensitivity to several parameters that have not been well characterized in the experimental literature, including boundary conditions, short vasa recta distribution, and ascending vasa recta (AVR) solute permeabilities. These studies indicate that regionalization elevates the osmolality of the fluid delivered into the inner medulla via the collecting ducts; that model predictions are not significantly sensitive to boundary conditions; and that short vasa recta distribution and AVR permeabilities significantly impact concentrating capability. Moreover, we investigated, in the context of the UCM, the functional significance of several aspects of tubular segmentation and heterogeneity: SDL segments in the IS that are likely to be impermeable to water but highly permeable to urea; a prebend segment of SDLs that may be functionally like thick ascending limb (TAL); differing IS and outer stripe Na(+) active transport rates in TAL; and potential active urea secretion into the proximal straight tubules. Model calculations predict that these aspects of tubular of segmentation and heterogeneity generally enhance solute cycling or promote effective UCM function.

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results

We have developed a highly detailed mathematical model for the urine concentrating mechanism (UCM) of the rat kidney outer medulla (OM). The model simulates preferential interactions among tubules and vessels by representing four concentric regions that are centered on a vascular bundle; tubules and vessels, or fractions thereof, are assigned to anatomically appropriate regions. Model parameters, which are based on the experimental literature, include transepithelial transport properties of short descending limbs inferred from immunohistochemical localization studies. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict significantly differing interstitial NaCl and urea concentrations in adjoining regions. Active NaCl transport from thick ascending limbs (TALs), at rates inferred from the physiological literature, resulted in model osmolality profiles along the OM that are consistent with tissue slice experiments. TAL luminal NaCl concentrations at the corticomedullary boundary are consistent with tubuloglomerular feedback function. The model exhibited solute exchange, cycling, and sequestration patterns (in tubules, vessels, and regions) that are generally consistent with predictions in the physiological literature, including significant urea addition from long ascending vasa recta to inner-stripe short descending limbs. In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1367-F1381, 2005), the impact of model assumptions, medullary anatomy, and tubular segmentation on the UCM was investigated by means of extensive parameter studies.

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.

Renal vacuolar H+-ATPase

Vacuolar H(+)-ATPases are ubiquitous multisubunit complexes mediating the ATP-dependent transport of protons. In addition to their role in acidifying the lumen of various intracellular organelles, vacuolar H(+)-ATPases fulfill special tasks in the kidney. Vacuolar H(+)-ATPases are expressed in the plasma membrane in the kidney almost along the entire length of the nephron with apical and/or basolateral localization patterns. In the proximal tubule, a high number of vacuolar H(+)-ATPases are also found in endosomes, which are acidified by the pump. In addition, vacuolar H(+)-ATPases contribute to proximal tubular bicarbonate reabsorption. The importance in final urinary acidification along the collecting system is highlighted by monogenic defects in two subunits (ATP6V0A4, ATP6V1B1) of the vacuolar H(+)-ATPase in patients with distal renal tubular acidosis. The activity of vacuolar H(+)-ATPases is tightly regulated by a variety of factors such as the acid-base or electrolyte status. This regulation is at least in part mediated by various hormones and protein-protein interactions between regulatory proteins and multiple subunits of the pump.

Mathematical models of renal fluid and electrolyte transport: acknowledging our uncertainty

Mathematical models of renal tubular function, with detail at the cellular level, have been developed for most nephron segments, and these have generally been successful at capturing the overall bookkeeping of solute and water transport. Nevertheless, considerable uncertainty remains about important transport events along the nephron. The examples presented include the role of proximal tubule tight junctions in water transport and in regulation of Na(+) transport, the mechanism by which axial flow in proximal tubule modulates solute reabsorption, the effect of formate on proximal Cl(-) transport, the assessment of potassium transport along collecting duct segments inaccessible to micropuncture, the assignment of pathways for peritubular Cl(-) exit in outer medullary collecting duct, and the interaction of carbonic anhydrase-sensitive and -insensitive pathways for base exit from inner medullary collecting duct. Some of these uncertainties have had intense experimental interest well before they were cast as modeling problems. Indeed, many of the renal tubular models have been developed based on data acquired over two or three decades. Nevertheless, some uncertainties have been delineated as the result of model exploration and represent communications from the modelers back to the experimental community that certain issues should not be considered closed. With respect to model refinement, incorporating more biophysical detail about individual transporters will certainly enhance model reliability, but ultimate confidence in tubular models will still be contingent on experimental development of critical information at the tubular level.

A mathematical model of rat collecting duct. I. Flow effects on transport and urinary acidification

A mathematical model of the rat collecting duct (CD) has been developed by concatenating previously published models of cortical (Weinstein AM. Am J Physiol Renal Physiol 280: F1072-F1092, 2001); outer medullary (Weinstein AM. Am J Physiol Renal Physiol 279: F24-F45, 2000); and inner medullary segments (Weinstein AM. Am J Physiol Renal Physiol 274: F841-F855, 1998). Starting with end-distal tubular flow rate and composition, plus interstitial solute profiles, the model predicts urinary solute flows, including the buffer concentrations required to assess net acid excretion. In the model CD, the interstitial corticomedullary osmotic gradient provides the basis for the flow effect on the transport of several solutes. For substances that have an interstitial accumulation and that can have diffusive secretion (e.g., urea and NH(4)(+)), enhanced luminal flow increases excretion by decreasing luminal accumulation. For substances that are reabsorbed (e.g., K+ and HCO(3)(-)), and for which luminal accumulation can enhance reabsorption, increasing luminal flow again increases excretion by decreasing luminal solute concentration. In model calculations, flow-dependent increases in HCO(3)(-) and NH(4)(+) approximately balance, so net acid excretion is little changed by flow, albeit at a higher urinary pH. The model identifies delivery flow rate to the CD as a potent determinant of urinary pH, with high flows blunting maximal acidification. At even modestly high flows (9 nl x min-1. tubule-1, with 6% of filtered Na+ entering the CD), the model cannot achieve a urinary pH <5.5 unless the delivered HCO(3)(-) concentration is extremely low (<2 mM). Nevertheless, simulation of Na2SO4 diuresis does yield both an increase in net acid excretion and a decrease in urinary HCO(3)(-) (i.e., a decrease in pH) despite the increase in urinary flow. This model should provide a tool for examining hypotheses regarding transport defects underlying distal renal tubular acidosis.

A mathematical model of rat collecting duct. III. Paradigms for distal acidification defects

The present clinical taxonomy of distal renal tubular acidoses includes "gradient," "secretory," and "voltage" defects. These categories refer to presumed collecting duct defects in which the epithelium may be abnormally permeable and unable to sustain an ion gradient, in which luminal proton ATPases are defective, or in which electrogenic Na+ reabsorption is impaired and luminal electronegativity is reduced. Classification of affected individuals is based on urinary pH and ion concentrations during spontaneous acidosis and during SO(4)(2-) infusion, as well as urinary PCO2 during HCO(3)(-) loading. A model of rat CD has been developed that has been used to examine determinants of urinary acidification (Weinstein AM. Am J Physiol Renal Physiol 283: F1252-F1266, 2002) and the interplay of HCO(3)(-) and PO(4)(3-) loads to generate a disequlibrium pH and equilibrium PCO2. In this paper, pure forms of gradient, voltage, and secretory defects are simulated, with attention to variability in the locus of the defect in the cortical (CCD), outer medullary (OMCD), or inner medullary collecting duct (IMCD). The objective of these calculations is to discover whether the intuitive description of these defects is sustained quantitatively. The most important positive finding is that the locus of the transport defect along the CD plays a critical role in the apparent severity of the lesion, with more proximal defects being less severe and more easily correctable. In particular, model calculations suggest that for gradient or secretory defects to be clinically detectable they need to involve the OMCD and/or IMCD. Additionally, the calculations reveal a possible mechanism for CD K+ wasting, which does not involve failure of H+ - K+-ATPase but derives from a paracellular anion leak and thereby a more electronegative lumen. The most important negative finding is the lack of support for the category of renal tubular acidosis associated with a voltage defect. Although CD lesions that present with both K+ and H+ secretory defects suggest mediation by transepithelial electrical potential difference (PD), both PD changes and proton pump PD sensitivity appear too small to account for the observed abnormalities.

Substrate-dependent reversal of anion transport site orientation in the human red blood cell anion-exchange protein, AE1

The tightly coupled, one-for-one exchange of anions mediated by the human red blood cell AE1 anion-exchange protein involves a ping-pong mechanism, in which AE1 alternates between a state with the anion-binding site facing inward toward the cytoplasm (Ei) and a state with the site facing outward toward the external medium (Eo). The conformational shift (Ei <--> Eo) is only permitted when a suitable substrate such as Cl(-) or HCO(3)(-) (B(-)) is bound. With no anions bound, or with Cl(-) bound, far more AE1 molecules are in the inward-facing than the outward-facing forms (Ei Eo, ECli EClo). We have constructed a model for CI(-)-B(-) exchange based on Cl(-)-Cl(-) and B(-)-B(-) exchange data, and have used it to predict the heteroexchange flux under extremely asymmetric conditions, with either all Cl(-) inside and all B(-) outside (Cli-Bo) or vice versa (Bi-Clo). The experimental values of the ratio of the exchange rate for Bi-Clo to that for Cli-Bo are only compatible with the model if the asymmetry of bicarbonate-loaded sites (A(B) = EBo/EBi) > 10, the opposite of the asymmetry for unloaded or Cl-loaded sites. Furthermore, the Eo form has a higher affinity for HCO(3)(-) than for Cl(-), whereas the Ei form has a higher affinity for Cl(-). The fact that this "passive" system exhibits changes in substrate selectivity with site orientation ("sidedness"), a characteristic usually associated with energy-coupled "active" pumps, suggests that changes in affinity with changes in sidedness are a more general property of transport proteins than previously thought.

Return of the secretory kidney

The evolution of the kidney has had a major role in the emigration of vertebrates from the sea onto dry land. The mammalian kidney has conserved to a remarkable extent many of the molecular and functional elements of primordial apocrine kidneys that regulate fluid balance and eliminate potentially toxic endogenous and xenobiotic molecules in the urine entirely by transepithelial secretion. However, these occult secretory processes in the proximal tubules and collecting ducts of mammalian kidneys have remained underappreciated in the last half of the twentieth century as investigators focused, to a large extent, on the mechanisms of glomerular filtration and tubule sodium chloride and fluid reabsorption. On the basis of evidence reviewed in this paper, we propose that transepithelial salt and fluid secretion mechanisms enable mammalian renal tubules to finely regulate extracellular fluid volume and composition day to day and maintain urine formation during the cessation of glomerular filtration.

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.

A mathematical model of rat cortical collecting duct: determinants of the transtubular potassium gradient

In assessing disorders of potassium excretion, urine composition is used to calculate the transtubular gradient (TTKG), as an estimate of tubule fluid concentration, at a point when the fluid was last isotonic to plasma, namely, within the cortical collecting duct (CCD). A mathematical model of the CCD has been developed, consisting of principal cells and α- and β-intercalated cells, and which includes Na+, K+, Cl−, HCO[Formula: see text], CO2, H2CO3, phosphate, ammonia, and urea. Parameters have been selected to achieve fluxes and permeabilities compatible with data obtained from perfusion studies of rat CCD under the influence of both antidiuretic hormone and mineralocorticoid. Both epithelial (flat sheet) and tubule models have been configured, and model calculations have focused on the determinants of the TTKG. Using the epithelial model, luminal K+ concentrations can be computed at which K+secretion ceases (0-flux equilibrium), and this luminal concentration derives from the magnitude of principal cell peritubular uptake of K+ via the Na-K-ATPase, relative to principal cell peritubular membrane K+ permeability. When the model is configured as a tubule and examined in the context of conditions in vivo, osmotic equilibration of luminal fluid produces a doubling of the initial K+ concentration, which, depending on delivered load, may be substantially greater than the zero-flux equilibrium value. Under such circumstances, the CCD will be a site for K+ reabsorption, although the relatively low permeability ensures that this reabsorptive flux is likely to be small. Osmotic equilibration may also raise luminal NH3 concentrations well above those in cortical blood. In this situation, diffusive reabsorption of NH3 provides a mechanism for base reclamation without the metabolic cost of active proton secretion.