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

Abstract

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.

Fig. 1.

Thick ascending limb (TAL) segment model.

Fig. 2.

Na⁺-K⁺/NH₄⁺-2Cl⁻ cotransporter (NKCC) and KCC maximum, minimum, and steady-state enzyme density for cell volume regulation function defined in Ref. .

Fig. 3.

Steady-state luminal concentration for major solutes and steady-state membrane potential. A–D: interstitial concentration profile. F: membrane potentials (MP) where E^a, E^b, and E^p are the apical, basolateral, and transepithelial membrane potentials, respectively.

Fig. 4.

A–F: steady-state cytosolic concentration for major solutes and steady-state cell volume. In F, the solid line denotes cell volume regulation (CVR), and dashed line is for no CVR enabled with NKCC2 and KCC4 density set to its maximum allowed value.

Fig. 5.

A–F: steady-state luminal concentration for major solutes. Solid curve denotes an inflow of 6 nl/min, dotted an inflow of 8 nl/min, dotted-dashed an inflow of 4 nl/min, and dashed line is the fixed serosal concentration profile. OM, outer medulla.

Fig. 6.

Steady-state outflow for major solutes. A: Na⁺. B: Cl⁻. C: K⁺. Circles denote CVR, squares denote no CVR enabled with NKCC2 and KCC4 density set to its maximum allowed value (see Fig. 2), and triangles denote no CVR enabled with NKCC2 and KCC4 density set to its minimum allowed value.

Fig. 7.

Efficiency index is defined in Eq. 10. A: 4 nl/min tubular flow. B: 6 nl/min tubular flow. C: 8 nl/min tubular flow. Solid line denotes CVR, dashed line is for no CVR enabled with NKCC and KCC activity set to its maximum allowed value, and dotted line is for no CVR enabled with NKCC and KCC activity set to its minimum allowed value.

Fig. 8.

In A, the dashed line is the serosal concentration profile. In A-C, solid line is for the case with NKCC2 and normal NH₄⁺ levels, dotted line is for NKCC2 without the NH₄⁺ cycle (wing) and normal NH₄⁺ levels, and dotted-dashed line is for NKCC2 and low levels of NH₄⁺. C: corresponding apical and paracellular fluxes with a positive flux being a reabsorptive flux.

Fig. 9.

A: dashed line is the serosal concentration profile, and the rest are the luminal concentration profiles for different boundary conditions for NaCl where the solid line is the baseline case. B: corresponding efficiencies. C: corresponding apical and paracellular fluxes with a positive flux being a reabsorptive flux.

Fig. 10.

A: dotted line is the serosal profile, solid line is the case with 4 nl/min inflow, dotted line is low-gain TGF, and dot-dashed and asterisked lines correspond to high-gain TGF, with oscillating macula densa Na⁺ concentration at minimum and maximum, respectively. C: apical and paracellular fluxes are shown (positive flux is reabsorption).