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

Clinical and genetic analysis of pseudohypoparathyroidism complicated by hypokalemia: a case report and review of the literature

BACKGROUND

Pseudohypoparathyroidism (PHP) encompasses a highly heterogenous group of disorders, characterized by parathyroid hormone (PTH) resistance caused by mutations in the GNAS gene or other upstream targets. Here, we investigate the characteristics of a female patient diagnosed with PHP complicated with hypokalemia, and her family members.

CASE PRESENTATION AND GENE ANALYSIS

A 27-year-old female patient occasionally exhibited asymptomatic hypocalcemia and hypokalemia during her pregnancy 1 year ago. Seven months after delivery, she experienced tetany and dysphonia with diarrhea. Tetany symptoms were relieved after intravenous calcium gluconate supplementation and she was then transferred to our Hospital. Laboratory assessments of the patient revealed hypokalemia, hypocalcemia and hyperphosphatemia despite elevated PTH levels. CT scanning of the brain revealed globus pallidus calcification. Possible mutations in GNAS and hypokalemia related genes were identified using WES, exon copies of STX16 were analized by MLPA and the methylation status of GNAS in three differential methylated regions (DMRs) was analyzed by methylation-specific polymerase chain reaction, followed by confirmation with gene sequencing. The patient was clinically diagnosed with PHP-1b. Loss of methylation in the A/B region and hypermethylation in the NESP55 region were detected. No other mutations in GNAS or hypokalemia related genes and no deletions of STX16 exons were detected. A negative family history and abnormal DMRs in GNAS led to a diagnosis of sporadic PHP-1b of the patient.

CONCLUSIONS

Hypokalemia is a rare disorder associated with PHP-1b. Analysis of genetic and epigenetic mutations can aid in the diagnosis and accurate subtyping of PHP.

Cardiac and renal function interactions in heart failure with reduced ejection fraction: A mathematical modeling analysis

Congestive heart failure is characterized by suppressed cardiac output and arterial filling pressure, leading to renal retention of salt and water, contributing to further volume overload. Mathematical modeling provides a means to investigate the integrated function and dysfunction of heart and kidney in heart failure. This study updates our previously reported integrated model of cardiac and renal functions to account for the fluid exchange between the blood and interstitium across the capillary membrane, allowing the simulation of edema. A state of heart failure with reduced ejection fraction (HF-rEF) was then produced by altering cardiac parameters reflecting cardiac injury and cardiovascular disease, including heart contractility, myocyte hypertrophy, arterial stiffness, and systemic resistance. After matching baseline characteristics of the SOLVD clinical study, parameters governing rates of cardiac remodeling were calibrated to describe the progression of cardiac hemodynamic variables observed over one year in the placebo arm of the SOLVD clinical study. The model was then validated by reproducing improvements in cardiac function in the enalapril arm of SOLVD. The model was then applied to prospectively predict the response to the sodium-glucose co-transporter 2 (SGLT2) inhibitor dapagliflozin, which has been shown to reduce heart failure events in HF-rEF patients in the recent DAPAHF clinical trial by incompletely understood mechanisms. The simulations predict that dapagliflozin slows cardiac remodeling by reducing preload on the heart, and relieves congestion by clearing interstitial fluid without excessively reducing blood volume. This provides a quantitative mechanistic explanation for the observed benefits of SGLT2i in HF-rEF. The model also provides a tool for further investigation of heart failure drug therapies.

Bicarbonate permeation through anion channels: its role in health and disease

Many anion channels, frequently referred as Cl^- channels, are permeable to different anions in addition to Cl^-. As the second-most abundant anion in the human body, HCO₃^- permeation via anion channels has many important physiological roles. In addition to its classical role as an intracellular pH regulator, HCO₃^- also controls the activity and stability of dissolved proteins in bodily fluids such as saliva, pancreatic juice, intestinal fluid, and airway surface liquid. Moreover, HCO₃^- permeation through these channels affects membrane potentials that are the driving forces for transmembrane transport of solutes and water in epithelia and affect neuronal excitability in nervous tissue. Consequently, aberrant HCO₃^- transport via anion channels causes a number of human diseases in respiratory, gastrointestinal, genitourinary, and neuronal systems. Notably, recent studies have shown that the HCO₃^- permeabilities of several anion channels are not fixed and can be altered by cellular stimuli, findings which may have both physiological and pathophysiological significance. In this review, we summarize recent progress in understanding the molecular mechanisms and the physiological roles of HCO₃^- permeation through anion channels. We hope that the present discussions can stimulate further research into this very important topic, which will provide the basis for human disorders associated with aberrant HCO₃^- transport.

Claudins in Renal Physiology and Pathology

Claudins are integral proteins expressed at the tight junctions of epithelial and endothelial cells. In the mammalian kidney, every tubular segment express a specific set of claudins that give to that segment unique properties regarding permeability and selectivity of the paracellular pathway. So far, 3 claudins (10b, 16 and 19) have been causally traced to rare human syndromes: variants of CLDN10b cause HELIX syndrome and variants of CLDN16 or CLDN19 cause familial hypomagnesemia with hypercalciuria and nephrocalcinosis. The review summarizes our current knowledge on the physiology of mammalian tight junctions and paracellular ion transport, as well as on the role of the 3 above-mentioned claudins in health and disease. Claudin 14, although not having been causally linked to any rare renal disease, is also considered, because available evidence suggests that it may interact with claudin 16. Some single-nucleotide polymorphisms of CLDN14 are associated with urinary calcium excretion and/or kidney stones. For each claudin considered, the pattern of expression, the function and the human syndrome caused by pathogenic variants are described.

A mathematical model of the rat kidney. II. Antidiuresis

Kidney water conservation requires a hypertonic medullary interstitium, NaCl in the outer medulla and NaCl and urea in the inner medulla, plus a vascular configuration that protects against washout. In this work, a multisolute model of the rat kidney is revisited to examine its capacity to simulate antidiuresis. The first step was to streamline model computation by parallelizing its Jacobian calculation, thus allowing finer medullary spatial resolution and more extensive examination of model parameters. It is found that outer medullary NaCl is modestly increased when transporter density in ascending Henle limbs from juxtamedullary nephrons is scaled to match the greater juxtamedullary solute flow. However, higher NaCl transport produces greater CO₂ generation and, by virtue of countercurrent vascular flows, establishment of high medullary Pco₂. This CO₂ gradient can be mitigated by assuming that a fraction of medullary transport is powered anaerobically. Reducing vascular flows or increasing vessel permeabilities does little to further increase outer medullary solute gradients. In contrast to medullary models of others, vessels in this model have solute reflection coefficients close to zero; increasing these coefficients provides little enhancement of solute profiles but does generate high interstitial pressures, which distort tubule architecture. Increasing medullary urea delivery via entering vasa recta increases inner medullary urea, although not nearly to levels found in rats. In summary, 1) medullary Na⁺ and urea gradients are not captured by the model and 2) the countercurrent architecture that provides antidiuresis also produces exaggerated Pco₂ profiles and is an unappreciated constraint on models of medullary function.

Pulsatile flow through idealized renal tubules: Fluid-structure interaction and dynamic pathologies

Kidney tubules are lined with flow-sensing structures, yet information about the flow itself is not easily obtained. We aim to generate a multiscale biomechanical model for analyzing fluid flow and fluid-structure interactions within an elastic kidney tubule when the driving pressure is pulsatile. We developed a two-dimensional macroscopic mathematical model of a single fluid-filled tubule corresponding to a distal nephron segment and determined both flow dynamics and wall strains over a range of driving frequencies and wall compliances using finite-element analysis. The results presented here demonstrate good agreement with available analytical solutions and form a foundation for future inclusion of elastohydrodynamic coupling by neighboring tubules. Overall, we are interested in exploring the idea of dynamic pathology to better understand the progression of chronic kidney diseases such as Polycystic Kidney Disease.

Mathematical models for kidney function focusing on clinical interest

We give an overview of mathematical models of renal physiology and anatomy with the clinician in mind. Beyond the past focus on issues of local transport mechanisms along the nephron and the urine concentrating mechanism, recent models have brought insight into difficult problems such as renal ischemia (oxygen and CO₂ diffusion in the medulla) or calcium and potassium homeostasis. They have also provided revealing 3D reconstructions of the full trajectories of families of nephrons and collecting ducts through cortex and medulla. The recent appearance of sophisticated whole-kidney models representing nephrons and their associated renal vasculature promises more realistic simulation of renal pathologies and pharmacological treatments in the foreseeable future.

Functional implications of sexual dimorphism of transporter patterns along the rat proximal tubule: modeling and analysis

The goal of this study is to investigate the functional implications of the sexual dimorphism in transporter patterns along the proximal tubule. To do so, we have developed sex-specific computational models of solute and water transport in the proximal convoluted tubule of the rat kidney. The models account for the sex differences in expression levels of the apical and basolateral transporters, in single-nephron glomerular filtration rate, and in tubular dimensions. Model simulations predict that 70.6 and 38.7% of the filtered volume is reabsorbed by the proximal tubule of the male and female rat kidneys, respectively. The lower fractional volume reabsorption in females can be attributed to their smaller transport area and lower aquaporin-1 expression level. The latter also results in a larger contribution of the paracellular pathway to water transport. Correspondingly similar fractions (70.9 and 39.2%) of the filtered Na+ are reabsorbed by the male and female proximal tubule models, respectively. The lower fractional Na+ reabsorption in females is due primarily to their smaller transport area and lower Na+/H+ exchanger isoform 3 and claudin-2 expression levels. Notably, unlike most Na+ transporters, whose expression levels are lower in females, Na+-glucose cotransporter 2 (SGLT2) expression levels are 2.5-fold higher in females. Model simulations suggest that the higher SGLT2 expression in females may compensate for their lower tubular transport area to achieve a hyperglycemic tolerance similar to that of males.

A mathematical model of the rat kidney: K+-induced natriuresis

A model of the rat nephron (Weinstein. Am J Physiol Renal Physiol 308: F1098-F1118, 2015) has been extended with addition of medullary vasculature. Blood vessels contain solutes from the nephron model, plus additional species from the model of Atherton et al. (Am J Physiol Renal Fluid Electrolyte Physiol 247: F61-F72, 1984), representing hemoglobin buffering. In contrast to prior models of the urine-concentrating mechanism, reflection coefficients for DVR are near zero. Model unknowns are initial proximal tubule pressures and flows, connecting tubule pressure, and medullary interstitial pressures and concentrations. The model predicts outer medullary (OM) interstitial gradients for Na+, K+, CO2, and [Formula: see text], such that at OM-IM junction, the respective concentrations relative to plasma are 1.2, 3.0, 2.7, and 8.0; within IM, there is high urea and low [Formula: see text], with concentration ratios of 11 and 0.5 near the papillary tip. Quantitative similarities are noted between K+ and urea handling (medullary delivery and permeabilities). The model K+ gradient is physiologic, and the urea gradient is steeper due to restriction of urea permeability to distal collecting duct. Nevertheless, the predicted urea gradient is less than expected, suggesting reconsideration of proposals of an unrecognized reabsorptive urea flux. When plasma K+ is increased from 5.0 to 5.5 mM, Na+ and K+ excretion increase 2.3- and 1.3-fold, respectively. The natriuresis derives from a 3.3% decrease in proximal Na+ reabsorption and occurs despite delivery-driven increases in Na+ reabsorption in distal segments; kaliuresis derives from a 30% increase in connecting tubule Na+ delivery. Thus this model favors the importance of proximal over distal events in K+-induced diuresis.

Primary cilia are not calcium-responsive mechanosensors

Primary cilia are solitary, generally non-motile, hair-like protrusions that extend from the surface of cells between cell divisions. Their antenna-like structure leads naturally to the assumption that they sense the surrounding environment, the most common hypothesis being sensation of mechanical force through calcium-permeable ion channels within the cilium. This Ca(2+)-responsive mechanosensor hypothesis for primary cilia has been invoked to explain a large range of biological responses, from control of left-right axis determination in embryonic development to adult progression of polycystic kidney disease and some cancers. Here we report the complete lack of mechanically induced calcium increases in primary cilia, in tissues upon which this hypothesis has been based. We developed a transgenic mouse, Arl13b-mCherry-GECO1.2, expressing a ratiometric genetically encoded calcium indicator in all primary cilia. We then measured responses to flow in primary cilia of cultured kidney epithelial cells, kidney thick ascending tubules, crown cells of the embryonic node, kinocilia of inner ear hair cells, and several cell lines. Cilia-specific Ca(2+) influxes were not observed in physiological or even highly supraphysiological levels of fluid flow. We conclude that mechanosensation, if it originates in primary cilia, is not via calcium signalling.

Simulation of Cl(-) Secretion in Epithelial Tissues: New Methodology Estimating Activity of Electro-Neutral Cl(-) Transporter

Transcellular Cl⁻ secretion is, in general, mediated by two steps; (1) the entry step of Cl⁻ into the cytosolic space from the basolateral space across the basolateral membrane by Cl⁻ transporters, such as Na⁺-K⁺-2Cl⁻ cotransporter (NKCC1, an isoform of NKCC), and (2) the releasing step of Cl⁻ from the cytosolic space into the luminal (air) space across the apical membrane via Cl⁻ channels, such as cystic fibrosis transmembrane conductance regulator (CFTR) Cl⁻ channel. Transcellular Cl⁻ secretion has been characterized by using various experimental techniques. For example, measurements of short-circuit currents in the Ussing chamber and patch clamp techniques provide us information on transepithelial ion movements via transcellular pathway, transepithelial conductance, activity (open probability) of single channel, and whole cell currents. Although many investigators have tried to clarify roles of Cl⁻ channels and transporters located at the apical and basolateral membranes in transcellular Cl⁻ secretion, it is still unclear how Cl⁻ channels/transporters contribute to transcellular Cl⁻ secretion and are regulated by various stimuli such as Ca²⁺ and cAMP. In the present study, we simulate transcellular Cl⁻ secretion using mathematical models combined with electrophysiological measurements, providing information on contribution of Cl⁻ channels/transporters to transcellular Cl⁻ secretion, activity of electro-neutral ion transporters and how Cl⁻ channels/transporters are regulated.

Mathematical Model of Ammonia Handling in the Rat Renal Medulla

The kidney is one of the main organs that produces ammonia and release it into the circulation. Under normal conditions, between 30 and 50% of the ammonia produced in the kidney is excreted in the urine, the rest being absorbed into the systemic circulation via the renal vein. In acidosis and in some pathological conditions, the proportion of urinary excretion can increase to 70% of the ammonia produced in the kidney. Mechanisms regulating the balance between urinary excretion and renal vein release are not fully understood. We developed a mathematical model that reflects current thinking about renal ammonia handling in order to investigate the role of each tubular segment and identify some of the components which might control this balance. The model treats the movements of water, sodium chloride, urea, NH₃ and NH4+, and non-reabsorbable solute in an idealized renal medulla of the rat at steady state. A parameter study was performed to identify the transport parameters and microenvironmental conditions that most affect the rate of urinary ammonia excretion. Our results suggest that urinary ammonia excretion is mainly determined by those parameters that affect ammonia recycling in the loops of Henle. In particular, our results suggest a critical role for interstitial pH in the outer medulla and for luminal pH along the inner medullary collecting ducts.

Recent advances in renal hemodynamics: insights from bench experiments and computer simulations

It has been long known that the kidney plays an essential role in the control of body fluids and blood pressure and that impairment of renal function may lead to the development of diseases such as hypertension (Guyton AC, Coleman TG, Granger Annu Rev Physiol 34: 13-46, 1972). In this review, we highlight recent advances in our understanding of renal hemodynamics, obtained from experimental and theoretical studies. Some of these studies were published in response to a recent Call for Papers of this journal: Renal Hemodynamics: Integrating with the Nephron and Beyond.

Regulation of calcium reabsorption along the rat nephron: a modeling study

We expanded a mathematical model of transepithelial transport along the rat nephron to include the transport of Ca2+ and probe the impact of calcium-sensing mechanisms on Ca2+ reabsorption. The model nephron extends from the medullary thick ascending limb (mTAL) to the inner medullary collecting duct (IMCD). Our model reproduces several experimental findings, such as measurements of luminal Ca2+ concentrations in cortical tubules, and the effects of furosemide or deletion of the transient receptor potential channel vanilloid subtype 5 (TRPV5) on urinary Ca2+ excretion. In vitro microperfusion of rat TAL has demonstrated that activation of the calcium-sensing receptor CaSR lowers the TAL permeability to Ca2+, PCaTAL (Loupy A, Ramakrishnan SK, Wootla B, Chambrey R, de la Faille R, Bourgeois S, Bruneval P, Mandet C, Christensen EI, Faure H, Cheval L, Laghmani K, Collet C, Eladari D, Dodd RH, Ruat M, Houillier P. J Clin Invest 122: 3355, 2012). Our results suggest that this regulatory mechanism significantly impacts renal Ca2+ handling: when plasma Ca2+ concentration ([Ca2+]) is raised by 10%, the CaSR-mediated reduction in PCaTAL per se is predicted to enhance urinary Ca2+ excretion by ∼30%. If high [Ca2+] also induces renal outer medullary potassium (ROMK) inhibition, urinary Ca2+ excretion is further raised. In vitro, increases in luminal [Ca2+] have been shown to activate H+-ATPase pumps in the outer medullary CD and to lower the water permeability of IMCD. Our model suggests that if these responses exhibit the sigmoidal dependence on luminal [Ca2+] that is characteristic of CaSR, then the impact of elevated Ca2+ levels in the CD on urinary volume and pH remains limited. Finally, our model suggests that CaSR inhibitors could significantly reduce urinary Ca2+ excretion in hypoparathyroidism, thereby reducing the risk of calcium stone formation.

A mathematical model of rat proximal tubule and loop of Henle

Proximal tubule and loop of Henle function are coupled, with proximal transport determining loop fluid composition, and loop transport modulating glomerular filtration via tubuloglomerular feedback (TGF). To examine this interaction, we begin with published models of the superficial rat proximal convoluted tubule (PCT; including flow-dependent transport in a compliant tubule), and the rat thick ascending Henle limb (AHL). Transport parameters for this PCT are scaled down to represent the proximal straight tubule (PST), which is connected to the thick AHL via a short descending limb. Transport parameters for superficial PCT and PST are scaled up for a juxtamedullary nephron, and connected to AHL via outer and inner medullary descending limbs, and inner medullary thin AHL. Medullary interstitial solute concentrations are specified. End-AHL hydrostatic pressure is determined by distal nephron flow resistance, and the TGF signal is represented as a linear function of end-AHL cytosolic Cl concentration. These two distal conditions required iterative solution of the model. Model calculations capture inner medullary countercurrent flux of urea, and also suggest the presence of an outer medullary countercurrent flux of ammonia, with reabsorption in AHL and secretion in PST. For a realistically strong TGF signal, there is the expected homeostatic impact on distal flows, and in addition, a homeostatic effect on proximal tubule pressure. The model glycosuria threshold is compatible with rat data, and predicted glucose excretion with selective 1Na(+):1glucose cotransporter (SGLT2) inhibition comports with observations in the mouse. Model calculations suggest that enhanced proximal tubule Na(+) reabsorption during hyperglycemia is sufficient to activate TGF and contribute to diabetic hyperfiltration.

CLC channel function and dysfunction in health and disease

CLC channels and transporters are expressed in most tissues and fulfill diverse functions. There are four human CLC channels, ClC-1, ClC-2, ClC-Ka, and ClC-Kb, and five CLC transporters, ClC-3 through −7. Some of the CLC channels additionally associate with accessory subunits. Whereas barttin is mandatory for the functional expression of ClC-K, GlialCam is a facultative subunit of ClC-2 which modifies gating and thus increases the functional variability within the CLC family. Isoform-specific ion conduction and gating properties optimize distinct CLC channels for their cellular tasks. ClC-1 preferentially conducts at negative voltages, and the resulting inward rectification provides a large resting chloride conductance without interference with the muscle action potential. Exclusive opening at voltages negative to the chloride reversal potential allows for ClC-2 to regulate intracellular chloride concentrations. ClC-Ka and ClC-Kb are equally suited for inward and outward currents to support transcellular chloride fluxes. Every human CLC channel gene has been linked to a genetic disease, and studying these mutations has provided much information about the physiological roles and the molecular basis of CLC channel function. Mutations in the gene encoding ClC-1 cause myotonia congenita, a disease characterized by sarcolemmal hyperexcitability and muscle stiffness. Loss-of-function of ClC-Kb/barttin channels impairs NaCl resorption in the limb of Henle and causes hyponatriaemia, hypovolemia and hypotension in patients suffering from Bartter syndrome. Mutations in CLCN2 were found in patients with CNS disorders but the functional role of this isoform is still not understood. Recent links between ClC-1 and epilepsy and ClC-Ka and heart failure suggested novel cellular functions of these proteins. This review aims to survey the knowledge about physiological and pathophysiological functions of human CLC channels in the light of recent discoveries from biophysical, physiological, and genetic studies.

Effects of NKCC2 isoform regulation on NaCl transport in thick ascending limb and macula densa: a modeling study

This study aims to understand the extent to which modulation of the Na(+)-K(+)-2Cl(-) cotransporter NKCC2 differential splicing affects NaCl delivery to the macula densa. NaCl absorption by the thick ascending limb and macula densa cells is mediated by apical NKCC2. A recent study has indicated that differential splicing of NKCC2 is modulated by dietary salt (Schieβl IM, Rosenauer A, Kattler V, Minuth WW, Oppermann M, Castrop H. Am J Physiol Renal Physiol 305: F1139-F1148, 2013). Given the markedly different ion affinities of its splice variants, modulation of NKCC2 differential splicing is believed to impact NaCl reabsorption. To assess the validity of that hypothesis, we have developed a mathematical model of macula densa cell transport and incorporated that cell model into a previously applied model of the thick ascending limb (Weinstein AM, Krahn TA. Am J Physiol Renal Physiol 298: F525-F542, 2010). The macula densa model predicts a 27.4- and 13.1-mV depolarization of the basolateral membrane [as a surrogate for activation of tubuloglomerular feedback (TGF)] when luminal NaCl concentration is increased from 25 to 145 mM or luminal K(+) concentration is increased from 1.5 to 3.5 mM, respectively, consistent with experimental measurements. Simulations indicate that with luminal solute concentrations consistent with in vivo conditions near the macula densa, NKCC2 operates near its equilibrium state. Results also suggest that modulation of NKCC2 differential splicing by low salt, which induces a shift from NKCC2-A to NKCC2-B primarily in the cortical thick ascending limb and macula densa cells, significantly enhances salt reabsorption in the thick limb and reduces Na(+) and Cl(-) delivery to the macula densa by 3.7 and 12.5%, respectively. Simulation results also predict that the NKCC2 isoform shift hyperpolarizes the macula densa basolateral cell membrane, which, taken in isolation, may inhibit the release of the TGF signal. However, excessive early distal salt delivery and renal salt loss during a low-salt diet may be prevented by an asymmetric TGF response, which may be more sensitive to flow increases.

Ammonia transport in the kidney by Rhesus glycoproteins

Renal ammonia metabolism is a fundamental element of acid-base homeostasis, comprising a major component of both basal and physiologically altered renal net acid excretion. Over the past several years, a fundamental change in our understanding of the mechanisms of renal epithelial cell ammonia transport has occurred, replacing the previous model which was based upon diffusion equilibrium for NH3 and trapping of NH4(+) with a new model in which specific and regulated transport of both NH3 and NH4(+) across renal epithelial cell membranes via specific membrane proteins is required for normal ammonia metabolism. A major advance has been the recognition that members of a recently recognized transporter family, the Rhesus glycoprotein family, mediate critical roles in renal and extrarenal ammonia transport. The erythroid-specific Rhesus glycoprotein, Rh A Glycoprotein (Rhag), was the first Rhesus glycoprotein recognized as an ammonia-specific transporter. Subsequently, the nonerythroid Rh glycoproteins, Rh B Glycoprotein (Rhbg) and Rh C Glycoprotein (Rhcg), were cloned and identified as ammonia transporters. They are expressed in specific cell populations and membrane domains in distal renal epithelial cells, where they facilitate ammonia secretion. In this review, we discuss the distribution of Rhbg and Rhcg in the kidney, the regulation of their expression and activity in physiological disturbances, the effects of genetic deletion on renal ammonia metabolism, and the molecular mechanisms of Rh glycoprotein-mediated ammonia transport.

Functional consequences of NKCC2 splice isoforms: insights from a Xenopus oocyte model

The Na(+)-K(+)-2Cl(-) cotransporter NKCC2 is exclusively expressed in the renal thick ascending limb (TAL), where it exists as three main splice isoforms, NKCC2B, NKCC2A, and NKCC2F, with the latter two predominating. NKCC2A is expressed in both medullary and cortical TAL, but NKCC2F localizes to the medullary TAL. The biochemical characteristics of the isoforms have been extensively studied by ion uptake studies in Xenopus oocytes, but the functional consequences of alternative splicing remain unclear. We developed a charge-difference model of an NKCC2-transfected oocyte. The model closely recapitulated existing data from ion-uptake experiments. This allowed the reconciliation of different apparent Km values reported by various groups, which have hitherto either been attributed to species differences or remained unexplained. Instead, simulations showed that apparent Na(+) and Cl(-) dependencies are influenced by the ambient K(+) or Rb(+) bath concentrations, which differed between experimental protocols. At steady state, under bath conditions similar to the outer medulla, NKCC2F mediated greater Na(+) reabsorption than NKCC2A. Furthermore, Na(+) reabsorption by the NKCC2F-transfected oocyte was more energy efficient, as quantified by J NKCC/J Pump. Both the increased Na(+) reabsorption and the increased efficiency were eroded as osmolarity decreased toward levels observed in the cortical TAL. This supports the hypothesis that the NKCC2F is a medullary specialization of NKCC2 and demonstrates the utility of modeling in analyzing the functional implications of ion uptake data at physiologically relevant steady 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.

Transport efficiency and workload distribution in a mathematical model of the thick ascending limb

The thick ascending limb (TAL) is a major NaCl reabsorbing site in the nephron. Efficient reabsorption along that segment is thought to be a consequence of the establishment of a strong transepithelial potential that drives paracellular Na(+) uptake. We used a multicell mathematical model of the TAL to estimate the efficiency of Na(+) transport along the TAL and to examine factors that determine transport efficiency, given the condition that TAL outflow must be adequately dilute. The TAL model consists of a series of epithelial cell models that represent all major solutes and transport pathways. Model equations describe luminal flows, based on mass conservation and electroneutrality constraints. Empirical descriptions of cell volume regulation (CVR) and pH control were implemented, together with the tubuloglomerular feedback (TGF) system. Transport efficiency was calculated as the ratio of total net Na(+) transport (i.e., paracellular and transcellular transport) to transcellular Na(+) transport. Model predictions suggest that 1) the transepithelial Na(+) concentration gradient is a major determinant of transport efficiency; 2) CVR in individual cells influences the distribution of net Na(+) transport along the TAL; 3) CVR responses in conjunction with TGF maintain luminal Na(+) concentration well above static head levels in the cortical TAL, thereby preventing large decreases in transport efficiency; and 4) under the condition that the distribution of Na(+) transport along the TAL is quasi-uniform, the tubular fluid axial Cl(-) concentration gradient near the macula densa is sufficiently steep to yield a TGF gain consistent with experimental data.

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.

Potassium excretion during antinatriuresis: perspective from a distal nephron model

Renal excretion of Na(+) and K(+) must be regulated independently within the distal nephron, but is complicated by the fact that changing excretion of one solute requires adjustments in the transport of both. It is long known that hypovolemia increases Na(+) reabsorption while impairing K(+) excretion, even when distal Na(+) delivery is little changed. Renewed interest in this micropuncture observation came with identification of the molecular defects underlying familial hyperkalemic hypertension (FHH), which also increases distal Na(+) reabsorption and impairs K(+) excretion. In this work, a mathematical model of the distal nephron (Weinstein AM. Am J Physiol Renal Physiol 295: F1353-F1364, 2008), including the distal convoluted tubule (DCT), connecting segment (CNT), and collecting duct (CD), is used to examine renal K(+) excretion during antinatriuresis. Within the model, Na(+) avidity is represented as the modulation of DCT NaCl reabsorption, and the K(+) secretion signal is an aldosterone-like effect on principal cells of the CNT and CD. The first model prediction is that changes in DCT NaCl reabsorption are not mediated by NaCl cotransporter density alone, but require additional adjustments of both peritubular Na-K-ATPase and KCl cotransport. A second observation is that the CNT response to increased DCT Na(+) reabsorption should not only stabilize CD K(+) delivery but also compensate for the compromise of K(+) excretion downstream, as low Na(+) delivery increases CD K(+) reabsorption. Such anticipatory regulation is seen with the aldosterone response of hypovolemia, while the FHH phenotype manifests enhanced DCT NaCl transport but a blunted aldosterone effect. The model emphasizes the need for two distinct signals to the distal nephron, regulating Na(+) excretion and K(+) excretion, in contrast to a single switch apportioning NaCl reabsorption and Na(+)-for-K(+) exchange.

A tool for multi-scale modelling of the renal nephron

We present the development of a tool, which provides users with the ability to visualize and interact with a comprehensive description of a multi-scale model of the renal nephron. A one-dimensional anatomical model of the nephron has been created and is used for visualization and modelling of tubule transport in various nephron anatomical segments. Mathematical models of nephron segments are embedded in the one-dimensional model. At the cellular level, these segment models use models encoded in CellML to describe cellular and subcellular transport kinetics. A web-based presentation environment has been developed that allows the user to visualize and navigate through the multi-scale nephron model, including simulation results, at the different spatial scales encompassed by the model description. The Zinc extension to Firefox is used to provide an interactive three-dimensional view of the tubule model and the native Firefox rendering of scalable vector graphics is used to present schematic diagrams for cellular and subcellular scale models. The model viewer is embedded in a web page that dynamically presents content based on user input. For example, when viewing the whole nephron model, the user might be presented with information on the various embedded segment models as they select them in the three-dimensional model view. Alternatively, the user chooses to focus the model viewer on a cellular model located in a particular nephron segment in order to view the various membrane transport proteins. Selecting a specific protein may then present the user with a description of the mathematical model governing the behaviour of that protein-including the mathematical model itself and various simulation experiments used to validate the model against the literature.

Effects of pH and medullary blood flow on oxygen transport and sodium reabsorption in the rat outer medulla

We used a mathematical model of O(2) transport and the urine concentrating mechanism of the outer medulla of the rat kidney to study the effects of blood pH and medullary blood flow on O(2) availability and Na(+) reabsorption. The model predicts that in vivo paracellular Na(+) fluxes across medullary thick ascending limbs (mTALs) are small relative to transcellular Na(+) fluxes and that paracellular fluxes favor Na(+) reabsorption from the lumen along most of the mTAL segments. In addition, model results suggest that blood pH has a significant impact on O(2) transport and Na(+) reabsorption owing to the Bohr effect, according to which a lower pH reduces the binding affinity of hemoglobin for O(2). Thus our model predicts that the presumed greater acidity of blood in the interbundle regions, where mTALs are located, relative to that in the vascular bundles, facilitates the delivery of O(2) to support the high metabolic requirements of the mTALs and raises the concentrating capability of the outer medulla. Model results also suggest that increases in vascular and tubular flow rates result in disproportional, smaller increases in active O(2) consumption and mTAL active Na(+) transport, despite the higher delivery of O(2) and Na(+). That is, at a sufficiently high medullary O(2) supply, O(2) demand in the outer medulla does not adjust precisely to changes in O(2) delivery.

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.