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

Renal oxygenation: From data to insight

Computational models have made a major contribution to the field of physiology. As the complexity of our understanding of biological systems expands, the need for computational methods only increases. But collaboration between experimental physiologists and computational modellers (ie theoretical physiologists) is not easy. One of the major challenges is to break down the barriers created by differences in vocabulary and approach between the two disciplines. In this review, we have two major aims. Firstly, we wish to contribute to the effort to break down these barriers and so encourage more interdisciplinary collaboration. So, we begin with a "primer" on the ways in which computational models can help us understand physiology and pathophysiology. Second, we aim to provide an update of recent efforts in one specific area of physiology, renal oxygenation. This work is shedding new light on the causes and consequences of renal hypoxia. But as importantly, computational modelling is providing direction for experimental physiologists working in the field of renal oxygenation by: (a) generating new hypotheses that can be tested in experimental studies, (b) allowing experiments that are technically unfeasible to be simulated in silico, or variables that cannot be measured experimentally to be estimated, and (c) providing a means by which the quality of experimental data can be assessed. Critically, based on our experience, we strongly believe that experimental and theoretical physiology should not be seen as separate exercises. Rather, they should be integrated to permit an iterative process between modelling and experimentation.

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.

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.

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.