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.

An optimization problem, formulated using a nonlinear least-squares approach, was used to estimate parameters for kinetic models of the three isoforms of the kidney-specific Na-K-2Cl (NKCC2) cotransporter. Specifically, the optimization problem estimates the magnitude of model parameters (i.e., off-binding and translocation rate constants) by minimizing the distance between model unidirectional fluxes and published unidirectional 86 Rb ϩ uptake curves for the A, B, and F isoforms of the NKCC2 cotransporter obtained in transfected Xenopus oocytes. By using different symmetry assumptions, NKCC2 models with five, six, seven, or eight parameters were evaluated. The optimization method identified parameter sets that yielded computed unidirectional fluxes consistent with the uptake data. However, the parameter values were not unique, in that systematic exploration of the parameter space revealed alternative parameter sets that fit the data with similar accuracy. Finally, we demonstrate that the optimization method can identify parameter sets for the three transporter isoforms that differ only in ion binding affinities, a result that is consistent with a published mutagenesis analysis of the molecular and structural bases for the differences in 86 Rb ϩ uptake among the A, B, and F isoforms. These NKCC2 cotransporter models will facilitate the development of larger scale models of ion transport by thick ascending limb cells. epithelial transport; thick ascending limb; kidney MODELING THICK ASCENDING LIMB (TAL) cell ion transport requires derivation of suitable models of the Na-K-2Cl (NKCC2) cotransporter, which is the primary mechanism of ion uptake across the apical membrane. Because of the wide variation in luminal ion concentration in the TAL, the NKCC2 model must be thermodynamically consistent (e.g., display net flux reversal when the chemical potential difference of the ligands changes sign). For a given parameter set (binding and translocation rate constants) and intra-and extracellular ligand concentrations, the solution of a model of the NKCC2 cotransporter yields the distribution of the transporter among the possible states in an assumed kinetic scheme (see Fig. 1). From this distribution, unidirectional and net ion fluxes can be computed. For recent reviews of the NKCC2 cotransporter, see Refs. 9 and 15.The goal of this study was to determine whether estimates of the necessary model parameters (i.e., off-binding and translocation rate constants) could be obtained from binding kinetic curves for the A, B, and F isoforms of the NKCC2 cotransporter as measured by Plata et al. (13) in transfected Xenopus oocytes, an expression technique that has been used to characterize the kinetic properties of the NKCC2 isoforms. However, kinetic data from oocyte experiments present some potential difficulties in parameter estimation. First, the plateau regions of the binding-uptake curves are often not well defined, especially for Cl Ϫ . The reason for this is that these cells express an endogenous NKCC1 cotransporter that ...

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...

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PurposeThis study examines the tracking error (TE) of a sample of sector exchange traded funds (ETFs) using spectral techniques.Design/methodology/approachTE is examined by computing its power spectrum using the wavelet transform. The wavelet transform maps the TE time series from the time domain to the time–frequency domain. Albeit the wavelet transform is a more complicated mathematical tool compared with the Fourier transform, it also has important advantages such as that it allows to analyze non-stationary data and to detect transient behavior.FindingsResults show that changes in the TE of a sample of sector ETFs are captured by the wavelet transform. Moreover, the authors also find that the wavelet coherence function can be used as a measure of TE in the time–frequency domain.Originality/valueThe study shows that the wavelet coherence function can be used as a reliable measure of TE.