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.

along the thick ascending limb (TAL) of a short loop of Henle in a rat kidney, active transport of Na⁺ reduces luminal NaCl concentration from ∼250 to 300 mM at the loop bend to ∼25 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⁺ transport be efficient. TAL cells are believed to attain that efficiency by means of apical Na⁺ uptake that is mediated by the electroneutral Na⁺-K⁺/NH₄⁺-2Cl⁻ cotransporter (NKCC2; Refs. 1, 11). Back diffusion of K⁺ through apical K⁺ channels produces a lumen-positive transepithelial potential that drives passive Na⁺ 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⁺ transport were solely transcellular.

However, the above scheme is complicated by several factors. As tubular Na⁺ concentration ([Na⁺]) decreases along the TAL, the transepithelial Na⁺ chemical potential reduces, or perhaps reverses, the driving force for paracellular Na⁺ transport. Also, the distribution of Na⁺ transport along the TAL likely influences transport efficiency, especially if luminal [Na⁺] approaches low static-head levels. Similarly, apical uptake of NH₄⁺ via NKCC2 (in competition with K⁺) and its extrusion via Na⁺-H⁺ exchanger (NHE3) will affect Na⁺ transport efficiency along the TAL by preventing K⁺ depletion in the lumen and by making NHE3 another transcellular pathway for Na⁺ reabsorption. Additionally, the rate of tubular fluid flow, which is regulated by tubuloglomerular feedback (TGF), affects transport efficiency by altering the luminal Na⁺ concentration profile along the TAL. Thus a thorough understanding of TAL Na⁺ transport efficiency can only be gained by taking all of these factors into account.

In a companion study (31), we developed a mathematical model of a TAL cell. In the present study, we extended that model to represent a TAL segment, and we used the resulting model to study the effects of tubular fluid dilution on Na⁺ transport efficiency. The results suggest that a quasi-uniform workload distribution along the TAL is established by processes at both the cellular and tubular levels. At the cellular level, TAL cell volume regulation (CVR) distributes transport workload by reciprocally regulating NKCC2 and KCC4 activities in the cells that comprise the TAL (15, 17). This has the effect of balancing, in each cell, apical uptake with basolateral transport capacity and leads to a more uniform distribution of Na⁺ transport (i.e., workload) along the corticomedullary axis. At the tubular level, the TGF system acts to maintain reasonable efficiency of TAL Na⁺ transport by balancing glomerular filtration with tubular reabsorptive capacity such that the TAL effluent Na⁺ concentration is maintained well above the static head.

MATHEMATICAL MODEL

We developed a cell-based model of the TAL of a short loop of Henle of the rat kidney. The TAL segment model consists of an ensemble of N cells that make up the walls of the TAL. A schematic diagram of the model configuration is shown in Fig. 1. The model equations describe the time evolution of cell volume and the cytosolic and luminal concentrations along the TAL. The solutes represented are Na⁺, K⁺, Cl⁻, H⁺, NH₄⁺, NH₃, H₂CO₃, HCO₃⁻, H₂PO₄⁻, HPO₄²⁻, X⁻, Y, and Z⁺, where X⁻, Y, and Z⁺ represent charged and uncharged impermeants. The equations that represent solute conservation in the cytosolic compartment as well as the expressions for solute flux, water flux, and electroneutrality constraints were described in the companion study (31), as are also the mechanisms by which the model maintains pH homeostasis and regulates cell volume.

Fig. 1.

Thick ascending limb (TAL) segment model.

Within the TAL lumen, water and solute conservation equations are given by

$$\frac{\partial }{\partial x}{F}_w(x,t)=0,$$ (1)

$$\frac{\partial }{\partial t}{C}_i^a(x,t)+\frac{F_w(t)}{A}\frac{\partial }{\partial x}{C}_i^a(x,t)=\frac{2\pi r}{A}\left[J_i^a(C^c,C^a)+J_i^p(C^a,C^b)\right],$$ (2)

where x is the axial coordinate along the TAL segment, such that x = 0 at the loop bend and x = L at the end of the cTAL; t is time; the superscripts a, b, p, c indicate apical, basolateral, paracellular, or cytosolic, respectively; the subscripts i, w denote either the ith solute or water; C_i^a (x, t) is the luminal concentration of solute i; C_i^c(t) is the concentration of solute i in a cell at a given position along the TAL; r is the TAL luminal radius; A is the luminal cross-sectional area; and C_i^b is the serosal concentration of solute i, which is assumed to be time independent. F_w(x, t) is the water flow entering the TAL segment. Equation 1 implies that, because the TAL is water impermeable, F_w is a function of time only, so we need not represent the spatial dependency and thus we write F_w(x, t) = F_w(t). The transmural solute fluxes are denoted by J_i^m(·, ·), (m = a, b, p), with each flux a function of its own intra/extracellular concentration as well as the intra/extracellular concentration of other solutes (C^m, m = a, b, c). Finally, the hydration/dehydration of CO₂ in the lumen was represented by the inclusion of a chemical source flux (see eq 9 in the companion paper (31)) into the lumenal volume.

Boundary and initial conditions.

To complete the specification of the model equations, we specify the solute concentrations of the tubular fluid at the loop bend (i.e., at x = 0). Those concentration values are listed in Table 1.

Table 1.

Values for initial and boundary conditions

	Initial
	Lumen*	Cytosol	Boundary
[Na+]	252	7.00	252
[K+]	8.00	136	8.00
[Cl−]	236	4.40	236
[X−]	27.7	112.5	27.7
[Y]	0.300	[6.00, 306]†	0.300
[Z+]	33.7	0.908	33.7
pH	7.27	7.28	7.27
[NH4+]	3.97	5.01	3.97
[NH3]	0.0519	0.0682	0.051875
[H2CO3]	0.00504	0.00483	0.00504
[HCO3−]	25.0	25.0	25.0
[H2PO4−]	1.28	0.988	1.28
[HPO42−]	3.73	3.01	3.73
Vcell	N/A	8.20 × 10−8	N/A

Concentrations indicated by brackets and are in mM, and volume is in cm³. V_cell is defined in mathematical model.

^*Uniform along the length of the thick ascending limb (TAL).

^†Range from cortical TAL to medullary TAL.

Another boundary condition is the loop-bend water flow rate F_w(t), which determines the water flow rate into and along the TAL segment. We considered two approaches in specifying F_w(t). In the first case, we made the simple assumption that TAL inflow is constant, i.e.,

$$\begin{array}{cc}{F}_w(t)=\alpha Q,& \text{constant}\end{array}$$ (3)

where Q is the steady-state single nephron glomerular filtration rate (SNGFR) and α is the fraction of SNGFR reaching the TAL. In the second case, we coupled the TAL model with the TGF system, so that TAL inflow depends on macula densa [Cl⁻] with a time delay (22):

$$F_w(t)=\alpha \left(Q_{op}+0.5\Delta Q\tanh \left(0.5k(C_{op}-C_{\text{MD}}(t))\right)\right)$$ (4)

$$C_{\text{MD}}(t)={\displaystyle {\int }_{-\infty }^t\Psi (t-s-0.5{\tau }_d)C_{{\text{Cl}}^{-}}^a(L,s-\tau )ds,}$$ (5)

$$\Psi (u)=\{\begin{array}{cc}\left[1+\cos \left(\frac{2\pi u}{{\tau }_d}\right)\right]{\tau }_d^{-1},\hfill & \text{if}\left|u\right|\le 0.5{\tau }_d,\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$$ (6)

Equation 4 is the TGF response, where k is the sensitivity of the TGF response, C_op is the steady-state [Cl⁻]_MD, Q_op is steady-state SNGFR, and ΔQ is the TGF-mediated range of SNGFR. Equation 5 represents the delay between changes in luminal Cl⁻ at the end of the TAL (x = L) and effective Cl⁻ concentration at the macula densa. The delay consists of a pure delay τ that represents the elapsed time from the stimulus until muscle tension in the afferent arteriole changes, and τ_d is the total time over which the effect (change in afferent arteriole diameter) is distributed. Equation 6 characterizes the distributed delay.

To compute the time evolution of the model variables, we must also specify their values at initial time, and Table 1 shows the assumed initial conditions unless stated otherwise. Cell volume is computed by assuming the TAL radius, including the walls, is 2r. Thereby, V_cell = π(2r)²Δx − V_lum, where V_lum denotes the luminal volume, given by V_lum = πr²Δx and Δx = L/N.

Numerical solution.

The numerical solution of the model equations for the cell model is described in the companion study (31). The luminal flow and solute conservation Eqs. 1 and 2 were solved using a parallelized, simple-splitting, min-mod flux limiter (27, 38) with a Δx = L/N. The implementation of the numerical methods were done in MATLAB^®. The computing resource used was a Linux box with 8 GB of memory and two Intel^® quad-core Xeon^® X5473 3 GHz CPUs.

Model parameters.

The physical dimensions and chemical properties of the TAL cell can be found in the companion study (31). The physical dimensions of the TAL are shown in Table 2. Baseline TAL inflow is taken to be 6 nl/min, based on α = 0.2 and Q = 30 nl/min (5, 30). Table 3 shows the chemical composition of the serosal bath or the interstitium along the TAL. Note the high osmolarity (∼500 mOsm) near the outer-inner medullary boundary (loop bend) and that halfway along the TAL segment is the cortico-medullary junction. The values for this set of concentrations are consistent with values summarized in Ref. 21.

Table 2.

Basic physical parameters

	Description	Value	Reference
N	Number of cells	70	N/A
L	TAL length, cm	0.6	(16)
R	TAL radius, μm	10	(16)
Vlum	TAL luminal volume, cm3	1.885 × 10−6	Calculated*
α	Fraction of SNGFR reaching TAL	0.2	(30)
Q	Steady-state SNGFR, nl/min	30	(5)

SNGFR, single nephron glomerular filtration rate; N/A, not applicable.

^*Luminal volume V_lum = πr²Δx, where Δx = L/N.

Table 3.

Serosal concentrations

	OM	Cortex
Na+	285.6→145.0	145.0
K+	8.000→3.500	3.500
Cl−	265.7→105.0	105.0
X−	0.7626→16.85	16.85
Y	8.300	1.000
Z+	0.7625→1.000	1→1.921
pH	7.306→7.300	7.300
NH4+	4.000→2.000	2.000→1.000
NH3	0.0569→0.0279	0.0279
H2CO3	4.591 × 10−3→4.658 × 10−3	4.658×10−3
HCO3−	25.00	25.00
H2PO4−	0.9271→0.6250	0.6250
HPO42−	3.000→2.000	2.000

All concentrations are in mM. Arrow (→) denotes a linear increase (or decrease) along the TAL. OM, outer medulla.

Table 4 shows apical, basolateral, and paracellular solute permeabilities along the TAL. The parameters for the membrane-embedded carriers and the carrier representation for each membrane are described in the companion study (31). In Fig. 2 and Table 4, max/min NKCC/KCC transporter densities and permeabilities were chosen such that outflow and luminal concentration profiles, and transepithelial electrical and chemical gradients computed by the model approximate experimental values reported by Greger (10) and by Vallon et al. (40). Finally, to illustrate the effects of the degree of tubular dilution on Na⁺ transport efficiency and TGF gain, two different No-CVR cases were studied: one with the NKCC/KCC densities at the maximum limit, and a second with the densities set to the minimum limit.

Table 4.

Apical, basolateral, and paracellular permeabilities

	Pa	Pb	Pp
Na+	0	0.05	[0.3, 0.6]
K+	[16, 11]0.2L, [11, 2]	4.8	[0.3, 3]
Cl−	0	1.2	[0.014, 0.028]
X−	0	0	0
Y	0	0	0
Z+	0	0	0
H+	40	40	[0.6, 1.2]
NH4+	[0.014, 0.75]0.2L, [0.75, 2.8]	0.08	0.06
NH3	30	20	0.6
H2CO3	0	0	0
HCO3−	0	0.02	0.001
H2PO4−	0	0	0
HPO42−	0	0	0
H2O	0	40	0

Solute permeabilities are in units of ×10⁻⁴ cm/s; H₂O permeability is in ×10⁻⁴ mM⁻¹ cm/s. Brackets give the range of a linear decrease (or increase) along the TAL. The brackets' subscript denotes the distance from the inlet of the TAL for which the range applies.

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. 31.

RESULTS

TAL function.

Steady-state model solutions were computed by solving model equations (Eqs. 1 and 2), and cell model equations in Ref. 31, using parameters in Tables 2, 3, and 4 and constant TAL inflow (Eq. 3). Luminal and cytosolic concentrations are shown in Figs. 3 and 4, respectively. A generally descending gradient of all major luminal concentrations is established along the TAL length, with the exception of H₂PO₄⁻ (not shown), which increases because of buffering of secreted H⁺ by HPO₄²⁻. Likewise, in spite of the transepithelial NaCl concentration gradient favoring paracellular backleak, Na⁺ is diluted from 250 to ∼28 mM (see below). The pH decreases along the segment and tubular fluid becomes slightly acidic (∼6.9) when it reaches the end of the TAL owing to the presence of NH₄⁺. Also noteworthy are the localized changes in concavity, which we call “jumps,” in the K⁺ and NH₄⁺ profiles at certain locations along the TAL (see Fig. 3, B and D). The reasons for the “jumps” are twofold: first, there is a sharp change in slope of the serosal concentration profile at the cortico-medullary junction (x = 0.3 cm) for most of the tracked solutes, in particular Na⁺, K⁺, Cl⁻, and NH₄⁺ (see Table 3). Second, there is an abrupt change from NKCC2 isoform-A to isoform-B in the distal cTAL (x = 0.5 cm). Since different NKCC2 isoforms have different Cl⁻ binding affinities (B > A > F) (9, 33) and translocation rates, when isoform-A is switched to isoform-B, a discontinuous change in model parameters associated with NKCC2 is introduced.

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.

Figure 3F shows that the apical and basolateral membrane potentials decrease from approximately −75 mV at the loop bend to approximately −100 and −120 mV, respectively, at the late cortex. Hence, the transepithelial membrane potential (E_p = E_b − E_a) is always positive and increases up to a value of 20 mV at the late cortex. This is important because a positive transepithelial potential favors paracellular Na⁺ reabsorption and partially counteracts the chemical gradient for Na⁺ that favors backleak through the tight junctions. Figure 4 shows the steady-state cytosolic concentrations and normalized cell volume. We observe the low-Na⁺/high-K⁺ regime typical of all eukaryotic cells and a pH that is within reasonable bounds. A change in monotonicity, i.e., the slope of concentration profile changes sign, is observed in the pH profile. This is attributed to the change in the slope of the interstitial pH at the cortico-medullary junction (x = 0.3 cm and see Table 3). The model also predicts that cell volume is well controlled except for the distal part of the cTAL, where the luminal fluid becomes very dilute (osmolarity of ∼60 mOsm, and see Fig. 3, A and C). Without CVR and high NKCC/KCC densities, cell volume progressively decreases along the TAL segment, owing to the very low luminal Na⁺ and Cl⁻ concentrations, which blunts NKCC Na⁺ uptake.

Influence of CVR on TAL function.

We assessed the impact of CVR on transport efficiency and workload distribution along the TAL by computing steady-state model solutions for different tubular fluid flow rates (F_w = 4, 6, and 8 nl/min). Simulations were conducted with and without CVR. When CVR was neglected, the NKCC and KCC densities were either set to a prescribed maximum value (case no-CVR) or to their minimum (case no-CVR-min). Alternative strategies (described in Ref. 31) were tried, with results similar to those shown below, although in some cases the changes were either more subtle or led to failure to adequately dilute the tubular fluid.

Figure 5 shows the concentration profiles for Na⁺, Cl⁻, and K⁺ at three flows; the dashed lines are the corresponding fixed interstitium concentration profiles. At baseline flow (6 nl/min), the TAL model with CVR predicts concentration profiles for Na⁺ and Cl⁻ that indicate that reabsorption is distributed such that there is substantial dilution in the cortex, while at low flow the majority of the transport occurs in the medulla and luminal concentrations approach static head such that net transport is almost zero. At high flows, the profile's slope and profiles themselves for Na⁺ and Cl⁻ rise all along the TAL. Most of the K⁺ is reabsorbed in the medulla, and the same holds for NH₄⁺ (not shown). Without CVR and NKCC and KCC at maximum activity, net transport takes place mostly within the medulla and the concentration profiles are relatively flat in the cortex, except at the highest flow. A flat [Cl⁻] profile near the macula densa would render TGF ineffective. This can be shown by computing the steady-state gain (G_ss) and the compensation of the TGF system for the cases of CVR and no CVR. The calculation is based on the rationale presented in (23, 26). The gain is calculated as

$$G_{ss}=\left(\frac{\partial }{\partial C_{\text{MD}}}{F}_w\right)\left[{\frac{\partial }{\partial F_w}{C}_{C{l}^{-}}^a(x)|}_{x=L}\right]$$ (7)

$$\approx \left(\frac{-\alpha k\mathrm{\Delta }Q}{4}\right)\left[\frac{{{C^a}_{Cl}-(L)}_{|F_w^{op}+\mathrm{\Delta }{F}_w}-{{C^a}_{C{l}^{-}}(L)}_{|F_w^{op}-\mathrm{\Delta }{F}_w}}{2\mathrm{\Delta }{F}_w}\right]$$ (8)

where the concentrations are the steady-state values, F_w^op is the flow at the TGF operating point, and ΔF_w = 0.6 = 0.1F^op nl/min is a small perturbation in tubular inflow fluid from baseline value (observe that ∂F_w/∂C_MD is computed by taking the derivative of Eq. 4 and evaluating at C_MD = C_op). The compensation is defined as

$$\text{compensation}=\frac{\left|G_{ss}\right|}{1+\left|G_{ss}\right|}\times 100\%.$$ (9)

The no CVR case yields a G_ss = (−0.14625)(13 − 12.55)/1.2 = −0.055 and the compensation is equal to 5.2%, whereas for the CVR case G_ss = (−0.14625)(35.79 − 22.79)/1.2 = −1.584 and the compensation is equal to 61.3%. These results illustrate that CVR acts to distribute work among all of the individual TAL cells and helps establish an axial Cl⁻ gradient consistent with measured TGF gain values (12, 29, 39).

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.

The graphs in Fig. 6 further illustrate the influence of CVR on TAL effluent ion composition. With CVR, the flow dependence of Na⁺, Cl⁻, and K⁺ is comparable to measurements by Vallon et al. (40). Without CVR and high transporter density, the model predicts that the slopes of the outflow curves are reduced because the slopes of the axial concentration profiles in the cTAL (see Fig. 5) are small in magnitude. As a reference, Fig. 6 also illustrates the effect in TAL outflow concentration when CVR was disabled but NKCC2 activity was set to a low minimal value. In that case, the TAL effluent [Na⁺] is flow dependent, but the concentration range is much higher than measured. Hence, CVR responses may in part explain the observed dependency of effluent ion composition on tubular fluid inflow. It is noteworthy that, in essence, the CVR responses that protect individual cells from volume perturbations also organizes the Na⁺ transport in the ensemble of cells along the TAL.

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.

CVR and transport efficiency.

We studied the transport efficiency of the TAL using the following measure:

$$\epsilon =\frac{J_{{\text{Na}}^{+}}^a+J_{{\text{Na}}^{+}}^p}{J_{{\text{Na}}^{+}}^a}=\frac{\text{net transepithelial Na flux}}{\text{net transcellular Na flux}}.$$ (10)

The index ε ranges from 0 to 2 where ε = 0 means complete backleak (serosa to lumen) through the paracellular pathway, whereas ε = 2 means that for each Na⁺ that goes transcellularly, one is transported paracellularly from lumen to serosa.¹ An alternative measure of Na⁺ transport efficiency over the whole TAL segment can be obtained by integrating ε over the length of the TAL, i.e.,

$$\overline{ϵ}=\frac{1}{L}{\displaystyle {\int }_0^L\epsilon \text{d}x}$$ (11)

Table 5 shows the values of ϵ̄ for cells with and without CVR and for different tubular inflow. The reported values show that increasing the tubular inflow yields greater transport efficiency and an overall greater transport efficiency for the cases with CVR. Higher tubular inflows yield greater ϵ̄ because Na⁺ concentration profile along the TAL is higher, and thus the transepithelial concentration gradient that favors backleak is smaller compared with the low tubular inflow case.

Table 5.

Integrated Na⁺ transport efficiency

	ϵ̄	ϵ̄OM	ϵ̄cortex
CVR, 4 nl/min	0.676	1.032	0.327
No CVR, 4 nl/min	0.510	0.883	0.141
CVR, 6 nl/min	0.895	1.074	0.726
No CVR, 6 nl/min	0.691	1.083	0.304
CVR, 8 nl/min	0.983	1.083	0.892
No CVR, 8 nl/min	0.844	1.151	0.546

The ϵ̄_MO and ϵ̄_cortex are for the average efficiency along the OM region and cortex respectively. CVR, cell volume regulation.

Figure 7 illustrates how the transport efficiency varies along the TAL. After reaching its peak in the outer medulla, the efficiency falls as paracellular transport reverses. For the case of no CVR and NKCC2 and KCC4 transporter set to a maximal value (no-CVR case and dashed line), the lower efficiency in the cortex is a consequence of the greater dilution of the tubular fluid. In contrast, in the case of no CVR and transporter density set to minimum (no-CVR-min case and dotted line), the efficiency is higher because of the high luminal [Na⁺] (hence a lower transepithelial concentration gradient). Also, the jumps in the curves in the cortex results from the transition from isoform-A to isoform-B of the NKCC2 transporter.

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.

Four things are noteworthy in Fig. 7: first, the efficiency does not monotonically decrease from the outer medullary region to the late cortex; instead, it reaches a maximum before crossing the cortico-medullary junction. The location of the maximum is determined by the location of maximum paracellular flux, which depends on the transepithelial concentration gradient, membrane potential differences, and tubular flow (inasmuch as the flow changes the transepithelial concentration difference along the TAL segment).

Second, the transport efficiency with no-CVR case is higher than the CVR case along part of the outer medulla. This is caused by an increase in paracellular transport, relative to the CVR case, driven by the rise in transepithelial potential difference that is a consequence of the increased NKCC2 uptake, which has been set to its maximum, and passive apical K⁺ and NH₄⁺ backleak. Nevertheless, the increased NKCC2 uptake also yields a much greater tubular fluid dilution close to the TAL inlet that establishes a transepithelial chemical gradient that favors backleak. As a result, efficiency decreases toward the cortex.

Third, the efficiency does not rise to the maximum permitted value of 2. This means that either the TAL Na⁺ transport system is not as efficient as suspected or that a second system that regulates TAL workload by controlling TAL inflow, i.e., TGF, needs to be accounted for. The possible synergism between CVR and TGF is explored below.

Fourth, merely permitting the individual TAL cells to autonomously regulate their volume results in a distribution of transport workload that increases ε and yields concentration profiles consistent with experimental data.

Using the TAL cell model (31), we showed that the presence of NH₄⁺ in the TAL lumen and the expression of the NKCC2 in the cell apical membrane gave rise to NH₄⁺ cycling. To determine the impact of NH₄⁺ cycling on efficiency and on the dilution capacity of the TAL, we computed the steady-state concentration profiles and the efficiency index (ε) for three cases with and without NH₄⁺ cycling impaired. In Fig. 8, the solid lines indicate the baseline case in terms of NH₄⁺levels and NKCC2 NH₄⁺ transport; the dotted lines represent the results obtained when the NKCC2 model without the ammonium wing is used (28), hence hampering NH₄⁺cycling. The dash-dotted lines indicate the case in which [NH₄⁺] in all compartments was reduced to low levels (10⁻⁶ mM).

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.

Results in Fig. 8A indicate that, with low NH₄⁺ in the lumen, cytosol, and serosa, the diluting ability of the TAL is greatly diminished, with an effluent [Na⁺] of 137 mM. This stems from overall lower fluxes through NKCC and apical NHE3 (Na⁺ uptake pathways), lower flux via Na⁺-K⁺/NH₄⁺ pump (Na⁺ extrusion pathway), lower transepithelial membrane potential, and higher luminal K⁺. Eliminating NKCC as an NH₄⁺uptake pathway also reduces the dilution but not as dramatically as in the previous case (outflow just above 50 mM). Specifically, in this case ammonium permeates through the apical and basolateral K⁺ channels, enters via the basolateral Na⁺-K⁺/NH₄⁺ pump, and is extruded via apical and basolateral NHE3. Figure 8B shows that the reward for poor diluting ability is higher efficiency, which just reflects the fact that high luminal Na⁺ concentrations reduce paracellular backleak. There, although we observe the same trend for efficiency that was pointed out above (Fig. 7), the cases in which NH₄⁺ cycling was impaired have a higher efficiency in the outer medulla. This result was unexpected at first, but its origin is revealed in Fig. 7C. Figure 8C shows Na⁺ fluxes along the TAL segment with positive flux indicating net reabsorption (lumen to serosa). From Fig. 8C and from the definition of the efficiency measure (Eq. 10), the higher efficiency for the diminished NH₄⁺cycling cases is explained by a reduction of the net apical flux. In other words, when the NH₄⁺ cycling is reduced, the transcellular and paracellular fluxes are closer to one another. This increases the efficiency but at the expense of reducing transcellular transport rather than increasing paracellular transport, and this ultimately leads to poor tubular fluid dilution. The possibility that this presents is that it may not be possible to have a highly efficient TAL, in terms of paracellular vs. transcellular transport, without sacrificing diluting ability.

Motivated by recent findings (20, 41) that suggest that the osmolarity of tubular fluid in descending thin limbs may be lower than the medullary interstitium, we explored the impact on Na⁺ transport efficiency of the transepithelial osmolarity gradient in the outer medulla. We did a series of simulations for which the luminal NaCl concentration at the TAL inlet was changed (the inlet of the TAL is at the inner-outer medullary boundary). These changes were perturbations around the baseline values in Fig. 3. The results are in Fig. 9. It is clear from Fig. 9 that increasing the NaCl concentration at the TAL inlet raises the efficiency because of the reduction in transepithelial chemical gradient and that if the tubular NaCl is high compared with serosa, then paracellular transport will favor Na⁺ reabsorption. However, we still observe the nonmonotonic trend in efficiency where the maximum is close to the cortico-medullary junction and the global minimum is at the end of the segment in the cortical region.

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.

TAL-TGF system and transport efficiency.

Figures 5 and 7 show that CVR may result in a distribution of Na⁺ transport that establishes an axial luminal Cl⁻ gradient consistent with the known flow dependency of TAL outflow [Na⁺] (40) and to account for the gain of the TGF system (12, 29, 39). In addition, Fig. 7 shows that CVR responses raise transport efficiency in part of the outer medulla and cortical regions by limiting the degree of tubular dilution. Another system that monitors tubular fluid dilution and can alter the luminal axial Na⁺ gradient by adjusting TAL flow is the TGF system. This motivated us to add the TGF loop Eqs. 4 and 6 to the TAL model Eqs. 1 and 2, to analyze the impact of this intranephron reflex on the distribution of transport and the efficiency along the length of the TAL. Table 6 shows the parameters for the TGF system.

Table 6.

Tubuloglomerular feedback parameters

	Description	Value	Reference
k	Sensitivity of TGF response, mM−1	0.1625	N/A
Cop	Steady-state [Cl−]MD, mM	28.4	N/A
Qop	Steady-state SNGFR, nl/min	30.0	(5)
ΔQ	TGF-mediated range of SNGFR, nl/min	18.0	(22, 35)
τ	Pure delay interval at JGA, s	2.00	(3, 25)
τd	Distributed delay interval at JGA, s	3.00	(3, 25)

TGF, tubuloglomerular feedback; JGA, juxta-glomerular apparatus.

The TGF system regulates TAL flow by sensing TAL outflow [Cl⁻]. Figure 10 shows the effect TGF on ε. Starting from the 4 nl/min inflow steady-state profile which exhibits low transport efficiency (see Fig. 5), we closed the TFG loop and let the TAL-TGF system reach a steady-state. We examined two TGF modes: a low-gain mode (sensitivity of TGF response, k = 0.12; see Eq. 5), and a high-gain mode (k = 0.1625) that triggered limit-cycle oscillations in TAL flow [as observed in experiments by Holstein-Rathlou and Marsh (13)]. Model predictions summarized in Fig. 10 indicate that TGF, in both low- and high-gain cases, increases ε by readjusting the tubular inflow such that very low luminal Na⁺ concentrations near static head are avoided. As a result the dynamic gain of the TGF system, which depends in part on the slope of the axial Cl⁻ gradient at the end of the TAL, also increases.

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).

The increase in efficiency is explained, in part, by Fig. 10C, which shows greater transcellular reabsorption and slightly smaller backleak in the cortical regions. To understand those trends we consider the transit time, that is, the amount of time that a fluid particle takes to travel the length of the TAL. A higher flow rate implies a shorter transit time, and as seen on Fig. 5, a more even workload distribution. This (as shown in Figs. 5 and 6) simultaneously raises the concentration profiles along the TAL, which reduces paracellular backleak and yields a less diluted outflow. Together, the CVR responses and the regulation of TAL flow by TGF result in a quasi-uniform distribution of NaCl transport and an axial [Cl⁻] gradient sufficiently steep to yield a TGF system gain consistent with experimental data.

DISCUSSION

A major function of the TAL is to reabsorb NaCl to build and sustain the interstitial osmolarity gradient that drives passive water reabsorption from the collecting duct and generates a concentrated urine. As much as 90% of the NaCl flow entering the thick limb is reabsorbed along the medullary and cortical TAL, with a part of that reabsorption driven metabolically, via Na⁺-K⁺-ATPase, against a substantial electrochemical gradient. Thus, relative to other tubular segments, the TAL has a high metabolic need. However, the availability of oxygen is limited in the rat kidney, with renal O₂ tension ranging from 40 to 50 mmHg in the cortex to ∼20 mmHg in the outer medulla. With the oxygen supply to the renal medulla barely exceeding its oxygen consumption (7), the kidney is particularly vulnerable when hypoperfused, because the associated hypoxia can lead to localized tissue injury. The availability of oxygen to mTALs is further limited by their distance from the oxygen-supplying descending vasa recta. Anatomic findings (18, 19) have revealed a highly structured organization of tubules and vessels in the rat outer medulla, in which the vasa recta form tightly packed vascular bundles that appear to dominate the histotopography of the outer medulla, especially in the inner stripe, and in which collecting ducts and mTALs are found distant from the vascular bundles. As a result, under experimental conditions resulting in reduced medullary O₂ tension, mTALs are the most injured segments (7, 8). Taken together, the high metabolic demands of the mTAL and the low oxygen tension make the efficiency of NaCl transport an important issue.

Our TAL model predicts that TAL transport efficiency is affected by the distribution of Na⁺ transport and that cell volume regulatory responses influences workload distribution along the TAL. Our representation of CVR is based on recent experimental findings in nonepithelial cells and nonrenal secretory epithelial cells that indicate that reciprocal regulation of NKCC1 (a KCl loading pathway) and the potassium chloride cotransporter (KCC) (a KCl exit pathway) stabilizes cell volume and intracellular chloride. These responses are mediated by a chloride-sensitive signaling pathway that involves WNK and Ste20-type serine/threonine kinases (14, 15). This mechanism also appears to mediate CVR in mTAL cells, as WNK-3 has been shown to enhance phosphorylation of the renal NKCC2 isoforms (34). Regardless of the mechanism, in TAL cells where the KCl loading and exit pathways are on different membranes, this CVR mechanism also balances apical uptake with KCl extrusion capacity. This permits TAL cells to adapt to variations in local availability of metabolic substrates and luminal ion concentrations. This would not only protect the cells from ischemia, but our model also predicts that it distributes the reabsorptive workload along the length of the TAL, an effect that is needed to generate an axial tubular fluid gradient in Na⁺ concentration. There is indirect evidence that an axial gradient is present. The well-known flow dependency of TAL effluent [Na⁺] (see Fig. 3 in Ref. 22) requires an axial TAL concentration gradient, and the gain of the TGF system is strongly influenced by the magnitude of that gradient (23).

TAL transport efficiency.

The standard textbook model (1, 11) proposes that the TAL efficiently reabsorbs Na⁺ from the luminal fluid because of paracellular electrodiffusion of Na⁺ driven by a transepithelial membrane potential established by K⁺ cycling across the apical membrane of TAL cells. However, our model predicts that this elegant scheme is complicated by several factors including CVR, NH₄⁺ transport and the transepithelial chemical gradient established by the tubular fluid dilution. The complexity of solute transport along the TAL and the inability of the K⁺ cycling scheme to raise efficiency were observed using our TAL cell model (31). That model predicted that efficiency drops conspicuously as the luminal concentration decreases (Fig. 12 of Ref. 31) and that not even a substantially positive transepithelial potential can blunt the backleak in the distal TAL where the tubular fluid is very dilute with respect to the interstitium. The TAL model presented here yielded the same result. Nevertheless, the following observations are noteworthy.

First, when cells regulate their volume and when the TGF system is enabled, transport efficiency, albeit low (i.e., far from 2), is still higher than the non-CVR or non-TGF cases, in particular in the cortical region (see Figs. 7 and 8). Second, CVR and the TGF system work synergistically to increase transport efficiency by distributing the transport workload and by ensuring that TAL flow is sufficiently high to keep luminal [Na⁺] well above static head. Third, “perfect” cycling of K⁺ might raise the transport efficiency substantially. By “perfect” we mean that all K⁺ reabsorbed by NKCC2 is extruded by apical K⁺ channels. If so, a stronger transepithelial electrical potential would develop and enhance paracellular reabsorption. Close spatial association or direct interaction between the NKCC2 transporter and the K⁺ channels within the apical membrane could provide a means to enhance K⁺ cycling back into the lumen. In the model, this possibility could be examined by creating a cytosolic subcompartment that delays mixing of K⁺ transported in the cytosol, but experimental verification would be essential. Fourth, when NH₄⁺ cycling is abolished, higher transport efficiency is obtained but at the expense of poor dilution of tubular fluid by our TAL model (Fig. 8). Thus the role of NH₄⁺ and of the associated pH control mechanism particularly the NHE3 transporter must be taken into account. Apical membrane ammonium cycling, while preventing luminal K⁺ depletion, also increases NHE3 Na⁺ uptake, which reduces transport efficiency. These findings, along with the volume sensitivity of NHE3 (6), suggest a complicated but interesting interaction between the regulation of pH, cell volume, and transport efficiency. Finally, our model predicts that CVR may play an important role in the functional organization of the TAL. By defending its volume, each cell selects NKCC2 and KCC transporter activities that produce physiologically reasonable behavior for the ensemble of cells. This greatly facilitated the development of the present TAL model in that explicit parameter tuning in each cell was not necessary.

CVR and TGF gain.

Our model simulations using baseline parameters suggest that in the absence of CVR and with NKCC2 activity defaulting to a preset maximum (see Fig. 2), NaCl transport in the outer medulla may be so vigorous that transcellular NaCl reabsorption equals paracellular electrodiffusive NaCl backleak in cortical TAL near the macula densa (see Fig. 5). This rather extreme case was chosen because it illustrates the effect of the TAL luminal [Cl⁻] profile near the macula densa reaching “static head” (4), a condition of essentially constant concentration along the TAL near the macula densa. In this situation, variations in TAL luminal fluid inflow would have no impact on [Cl⁻] at the macula densa, no TGF signal would be generated, and transport efficiency would be zero. If the TGF loop were closed and functional, TAL inflow would increase to better match the NaCl load entering the TAL with its transport capacity. The result is an increase in transport efficiency and the establishment of a more linear axial Na⁺ gradient. On the other hand, if CVR is disabled but NKCC2 density defaults to a preset minimum, flow dependence of TAL outflow concentration is observed but at the expense of tubular dilution. In this case, with an active TGF loop, TAL inflow would be decreased to better match the load of Na⁺ with TAL transport capacity. The result is a decrease in transport efficiency but more appropriate dilution of the tubular fluid. In contrast, with CVR active, the model predicted flow dependence of [Na⁺], [Cl⁻], and [K⁺] that is comparable to experimental measurements by Vallon et al. (40). The controlled variable of the TGF system is the [Cl⁻] of the TAL effluent (36), and the gain of the TGF system is determined by the axial Cl⁻ gradient. CVR, by distributing the transport workload, and the TGF system, by balancing TAL Na⁺ load with transport capacity, act together to keep TGF gain high. This also elevates cortical TAL transport efficiency.

Comparison with previous models.

A detailed comparison with a TAL cell model developed by Weinstein and Krahn (43) is presented in (31). Here we compare our model with the in vivo TAL model by Weinstein (42). First, the luminal diameter used in Weinstein's model is the same as ours but the segment length is shorter (4 mm vis-a-vis 6 mm), with the outer medulla and the cortex taken to be 2 mm each, whereas ours are 3 mm each. In part, the differences in length explain the differences in outflow concentrations of major solutes between the models. Weinstein's model (42) predicted higher Na⁺, K⁺, and Cl⁻ outflow concentrations compared with ours: 69.0 vs. 36.6 mM for Na⁺; 2.05 vs. 1.9 mM for K⁺; and 57.4 vs. 28.4 mM for Cl⁻. However, if we compare the tubular concentrations at 4 mm from TAL inlet (which is the TAL outlet in Weinstein's model but not in ours) then we have similar outflow concentrations: 69.0 vs. 69.0 mM for Na⁺; 1.60 vs. 2.05 mM for K⁺; and 54.7 vs. 57.4 mM for Cl⁻. Additionally, the two models predict luminal concentration profiles that are similarly monotonic (cf. Fig. 3 with Fig. 8 in Ref. 42). It is noteworthy that, in terms of monotonicity, only the ammonium profile near the TAL inlet differs significantly from the ammonium profile in Fig. 8 of (42). The divergence arises from the different isoform distributions and binding affinities of the NKCC2 models used in the two modeling studies. Nevertheless, both modeling studies (Weinstein and ours) agree with experimental findings about NKCC2 isoform expression along the TAL (32, 33, 44), and the differences in the concentration profiles, even for NH₄⁺ (see above), are relatively minor. Finally, as discussed in the companion study, the transport parameter values of the cells that comprise the TAL models differ in magnitude between the Weinstein's models and ours but have the similar relationships between the various solutes. Nevertheless, both models predict behaviors that are physiological reasonable given the limited functional data available for the TAL segment. This suggests that the functional characteristics of the TAL are robust with respect to modest variation in transport parameter values.

Limitations and further extensions.

Our model assumes that the serosal concentrations are known at each level along the corticomedullary axis. A natural extension would be to embed the current TAL model into a model of the outer medulla. Such a model would include collecting ducts, a descending limb, capillary blood flow, and an interstitium whose composition is determined by the balance of solute and water reabsorption from all these structures. Also, a minimal model of cell metabolism and ATP production and utilization would permit a more thorough examination of the efficiency of the TAL on a metabolic basis, as opposed to the simple measure we have employed here. That model would permit a more direct study of the interaction between blood flow, oxygen availability, and TAL function including transport efficiency. Additionally, the model in its current form can be used to determine the relation between the dynamic features of the TGF-TAL system (e.g., the establishment of limit cycle oscillations and nonlinear filtering properties of the TAL; Ref. 24) including the dynamic features of the individual TAL cells.

Finally, as discussed in the companion paper (31), additional modeling studies are needed to understand why rat cortical TAL dilutes the tubular fluid to concentrations where transport efficiency is low. It cannot be related to the urine concentrating mechanism, as the reabsorbed Na⁺ is taken up by the cortical peritubular capillaries. Further, in the earliest segment of the distal tubule, there is enough secretion of Na⁺ and Cl⁻ to substantially increase their tubular fluid concentrations (2). Hence, the impact of the strong dilution of the tubular fluid in the cortical TAL on free water clearance would be small. One intriguing hypothesis is that the high transepithelial potential difference that develops with strong tubular fluid dilution enhances the passive paracellular reabsorption of divalent cations like Ca²⁺ and Mg²⁺.

GRANTS

This research was supported by the National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-42091 and DK-89066 (to H. E. Layton and A.T. Layton, respectively) and by the National Science Foundation Grant DMS-0701412 (to A. T. Layton).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).