Mathematical Models of the Mammalian Urine Concentrating Mechanism

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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 ϩ uptake. We used a multicell mathematical model of the TAL to estimate the efficiency of Na ϩ 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 ϩ transport (i.e., paracellular and transcellular transport) to transcellular Na ϩ transport. Model predictions suggest that 1) the transepithelial Na ϩ concentration gradient is a major determinant of transport efficiency; 2) CVR in individual cells influences the distribution of net Na ϩ transport along the TAL; 3) CVR responses in conjunction with TGF maintain luminal Na ϩ 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 ϩ transport along the TAL is quasi-uniform, the tubular fluid axial Cl Ϫ concentration gradient near the macula densa is sufficiently steep to yield a TGF gain consistent with experimental data. tubuloglomerular feedback; autoregulation; NaCl transport; cell volume regulation ALONG THE THICK ASCENDING limb (TAL) of a short loop of Henle in a rat kidney, active transport of Na ϩ reduces luminal NaCl concentration from ϳ250 to 300 mM at the loop bend to ϳ25 mM at the end of the cortical TAL (cTAL; Refs. 2, 37, 40). Given the low availability of oxygen within the renal medulla and the high metabolic demands of the medullary TAL (mTAL), it appears essential that TAL Na ϩ transport be efficient. TAL cells are believed to attain that efficiency by means of apical Na ϩ uptake that is mediated by the electroneutral NaϪ cotransporter (NKCC2; Refs. 1, 11). Back diffusion of K ϩ through apical K ϩ channels produces a lumen-positive transepithelial potential that drives passive Na ϩ reabsorption through cation-permeable tight junctions. It is thought that this reduces net ATP utilization to a level much lower than what would be required if Na ϩ transport were solely transcellular. However, the above scheme is complicated by several factors. As tubular Na ϩ concentration ([Na ϩ ]) decreases along the TAL, the transepithelial Na ϩ chemical potential reduces, or perhaps reverses, the driving force for paracellular Na ϩ transport. Also, the distribution of Na ϩ transport along the TAL likely influences transport efficiency, especially if luminal [Na ϩ ] approaches low static-head lev...

“…The values for this set of concentrations are consistent with values summarized in Ref. 21. Table 4 shows apical, basolateral, and paracellular solute permeabilities along the TAL.…”

Section: Mathematical Modelsupporting

confidence: 87%

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

Nieves‐Gonzalez

Clausen

Layton

et al. 2013

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“…Solute advection along the tubule also depends only on the spatial variable along the TAL, and both axial and radial diffusion are ignored. These assumptions, which have been standard for sufficiently long tubular segments, have been previously justified (25). Also, as in other modeling efforts by us (28) and by others (15), we have assumed that transepithelial transport rates respond to variations in tubular fluid NaCl concentration instantaneously.…”

Section: Discussionmentioning

confidence: 92%

Signal transduction in a compliant thick ascending limb

Moore

Layton

2012

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Layton AT, Moore LC, Layton HE. Signal transduction in a compliant thick ascending limb. Am J Physiol Renal Physiol 302: F1188 -F1202, 2012. First published January 18, 2012 doi:10.1152/ajprenal.00732.2010In several previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model, which represents the TAL as a rigid tube, predicts that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time interval required for an element of tubular fluid to traverse the TAL, as a function of time, is oscillatory and periodic but not sinusoidal. As a consequence, NaCl concentration in tubular fluid alongside the macula densa will be nonsinusoidal and thus contain harmonics of the original sinusoidal frequency. We hypothesized that the complexity found in power spectra based on in vivo time series of key TGF variables arises in part from those harmonics and that nonlinearities in TGF-mediated oscillations may result in increased NaCl delivery to the distal nephron. To investigate the possibility that a more realistic model of the TAL would damp the harmonics, we have conducted new studies in a model TAL that has compliant walls and thus a tubular radius that depends on transmural pressure. These studies predict that compliant TAL walls do not damp, but instead intensify, the harmonics. In addition, our results predict that mean TAL flow strongly influences the shape of the NaCl concentration waveform at the macula densa. This is a consequence of the inverse relationship between flow speed and transit time, which produces asymmetry between up-and downslopes of the oscillation, and the nonlinearity of TAL NaCl absorption at low flow rates, which broadens the trough of the oscillation relative to the peak. The dependence of waveform shape on mean TAL flow may be the source of the variable degree of distortion, relative to a sine wave, seen in experimental recordings of TGF-mediated oscillations. kidney; mathematical model; renal hemodynamics; sodium chloride transport; nonlinear system IN A SERIES OF STUDIES WE have used a mathematical model of the tubuloglomerular feedback (TGF) loop to propose plausible explanations for phenomena that have been reported in experimental studies and to predict phenomena that might be found in new experimental studies (23,24,28,29,30,31,32,34,36). Our mathematical model consists of simple components that have been individually well characterized in our publications. The thick ascending limb (TAL) is represented by a rigid tube with plug flow that carries only the chloride ion, which is believed to be the principal species sensed by the macula densa (MD; Ref. 39). The time delay in the TGF signal transmission from the tubular fluid chloride concentration alongside the MD to the action of the smooth muscle cells in the distal portion of the afferent arteriole is represented by either a discrete time delay ...

“…It may well be that further spatial discretization would be desirable in our model formulation, and it could be introduced by means of additional concentric regions. However, we believe it likely that little would be gained by concentric regions that are separated by less than a typical tubule diameter (i.e., ϳ20 m), owing to rapid diffusive equilibration, in the radial direction, for relevant length and time scales (44).…”

Section: Model Limitations and Potential Extensionsmentioning

confidence: 99%

“…This difference, frequently called the "single effect," is augmented (or "multiplied") by the countercurrent flow configuration of renal tubules and vasa recta, to generate a substantial osmolality gradient along the corticomedullary axis, a gradient that is believed to be common to all OM structures. In the inner medulla (IM), however, the means by which water is absorbed from CDs remains undetermined (20, 67).Substantial effort has been directed to constructing mathematical models of the mammalian urine concentrating mechanism (UCM) (27,44). None of these models (including the one proposed in this study) has incorporated more than one fully realized spatial dimension, viz., the dimension corresponding to the corticomedullary axis.…”

mentioning

confidence: 99%

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

Layton¹

2005

Self Cite

144

Layton, Anita T., and Harold E. Layton. A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol 289: F1346 -F1366, 2005. First published May 24, 2005 doi:10.1152/ajprenal.00346.2003.-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. kidney; countercurrent multiplication; countercurrent exchange; NaCl transport; urea transport CONCENTRATED URINE, i.e., urine having an osmolality exceeding that of blood plasma, is produced by the absorption of water, in excess of solute, from the medullary collecting ducts (CDs). In the outer medulla (OM), water absorption from CDs is driven by vigorous active transport of NaCl from the thick ascending limbs (TALs) into the interstitium; at each medullary level, this transport results in an osmolality difference between the TALs and other medullary structures. This difference, frequently called the "single effect," is augmented (or "multiplied") by the countercurrent flow configuration of renal tubules and vasa recta, to generate a substantial osmolality gradient along the corticomedullary axis, a gradient that is believed to be common to all OM structures. In the inner medulla (IM), however, the means by which water is absorbed from CDs remains undetermined (20, 67).Substantial effort has been directed to constructing mathematical models...