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

Abstract

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 O₂ 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⁺], [NH₄⁺], 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.

thick ascending limb (TAL) cells are thought to transport Na⁺ efficiently owing to passive paracellular Na⁺ absorption. In the most basic model of TAL cell transport [4, 17], apical Na⁺ uptake is mediated by the electroneutral Na⁺-K⁺/NH₄⁺-2Cl⁻ cotransporter (NKCC2). Back diffusion of K⁺ through apical K⁺ channels produces a lumen-positive transepithelial potential, which drives passive Na⁺ reabsorption through the cation-permeable tight junctions. Ideally in this transport scheme, for each Na⁺ transported through the cell, which requires energy utilization, a second Na⁺ ion is transported passively via the paracellular pathway. TAL transport efficiency is of particular importance in the outer medulla (OM) where O₂ availability is limited by low blood flow (14).

The driving force for paracellular Na⁺ reabsorption depends on two competing factors. The first is the positive transepithelial potential that develops subsequent to the diffusion of K⁺ from the cytosol into the lumen. The second is the Na⁺ concentration difference between the luminal and serosal compartments. Because the TAL tubular fluid is diluted by Na⁺ transport, the transepithelial chemical potential is expected to oppose the electrical potential, but the electrical potential also increases as luminal Na⁺ concentration falls. The net effect of this is that transport efficiency may be reduced. Previously, Fernandes and Ferreira (10) implemented a cortical TAL (cTAL) cell model based on the experimental work of Greger and Schlatter (16). More recently, Weinstein (43, 44) proposed a model for TAL epithelium as well as tubule models for medullary TAL (mTAL) and cTAL tubules and used those models to study how NH₄⁺ affects solute transport in the TAL. The effects of NH₄⁺ are likely to be important due to the significant concentrations of NH₄⁺ in the luminal and serosal compartments and to the complicated and intertwined transport pathways that couple NH₄⁺ and other solutes, in particular Na⁺ and K⁺.

In this investigation, we used a mathematical model of a TAL cell to investigate the effect of luminal Na⁺ concentration on TAL transport efficiency. Akin to the modeling studies mentioned above, we implemented a detailed mathematical model of a TAL cell, in which most carrier-based transport pathways are represented. In contrast to previous modeling efforts, our model includes a cell volume regulation (CVR) mechanism that controls solute transport, and therefore short-term cell volume changes, by regulating the activities of the NKCC2 and K⁺/NH₄⁺-Cl⁻ cotransporter (KCC4) membrane transporters. This mechanism is based on observations by Komlosi et al. (25) and Kahle et al. (21). This study focuses on transport efficiency and its relation to CVR and tubular fluid dilution. Our study shows that tubular fluid dilution, a primary function of the thick ascending limb, will reduce Na⁺ transport efficiency, with the largest reduction in the cortical region of the TAL. Hence, the concept that Na⁺ transport in the TAL is highly efficient needs reconsideration.

MATHEMATICAL MODEL AND METHODS

Our TAL cell model represents three compartments: luminal, serosal, and cytosolic, which are coupled by the transport processes. The concentrations in luminal and the serosal compartments are assumed to be fixed (time independent). The model consists of a system of ordinary differential equations (ODEs) that represent the time evolution of cell volume and cytosolic concentrations. 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. A schematic diagram of the model TAL cell is shown in Fig. 1.

Fig. 1.

A schematic diagram of the thick ascending limb (TAL) cell model.

Conservation laws.

The luminal (apical) membrane of the TAL cell is assumed to be water impermeable. The cytosol is assumed to be dilute and well stirred. With these assumptions, equations for solute and water conservation are given by Latta et al. (27).

$$\frac{d}{dt}{C}_i^c(t)=\left(C_i^c(t)A^b{J}_w^b\left(C^c(t),C^b\right)-\left(A^a{J}_i^a\left(C^c(t),C^a\right)+A^b{J}_i^b\left(C^c(t),C^b\right)\right)\right)V^{-1}(t),$$ (1)

$$\frac{d}{dt}V(t)=-A^b{J}_w^b\left(C^c(t),C^b\right),$$ (2)

where t is time; the superscripts a, b, and p denote apical, basolateral, or paracellular, respectively; the subscripts i and w denote either the ith solute and water, respectively; C_i^a is the luminal concentration of solute i; C_i^c(t) is the cytosolic concentration of solute i; C_i^b is the serosal (basolateral) concentration of solute i, which is assumed to be fixed; V is cell volume; and A^a and A^b are the apical and basolateral membrane areas, respectively. The transmural solute fluxes are denoted by J_m^k (. , .) (k = a, b, p and m = i, w), where the boldface symbol C^k denotes an array of concentrations. Using this notation in the flux functions emphasizes the potential dependence of solute flux on the intra/extracellular concentrations of all solutes (see below).

Fluxes and electroneutrality constraints.

The solute fluxes have two components: the electrodiffusive part, given by the Goldman-Hodgkin-Katz constant-field flux equation, and the carrier mediated contribution (denoted as J_i^T). For a TAL cell, J^T includes the NKCC2, KCC4, Na⁺-K⁺-ATPase (pump), Na⁺-H⁺/NH₄⁺ exchanger (NHE3), and HCO₃⁻-Cl⁻ exchanger (BCE) transport. Thus, the apical and basolateral solute fluxes are given by

$$J_i^k\left(C^c(t),C^k\right)=\frac{z_i{P}_i^k{u}^k\left(C_i^c(t)\exp (u^k{z}_i)-C_i^k\right)}{\exp (u^k{z}_i)-1}+J_i^T(C^c(t),C^k),$$ (3)

where a negative flux indicates transport into the cell. In the above expression u^k = FE^k/RT, where E^k (k = a, b) is the apical or basolateral membrane potential; z_i is the valence of the ith solute; P_i^k (k = a, b) is the apical or basolateral, permeability of the ith solute; and F/RT is the ratio of Faraday's constant to the product of the gas constant and the absolute temperature. Note that the expression for paracellular flux is similar to equation 3, but with J^T = 0, the permeability set to be the paracellular permeability (P_i^p), the membrane potential set to be the transepithelial potential (E^p), and the concentrations are the luminal and serosal concentrations. Also, note that a straight-forward circuit analysis shows that E^p = E^b − E^a.

The carrier-mediated fluxes are computed from steady-state expressions of kinetic models that describe the transporter in question. The kinetic model and its steady-state expression for NHE3 is from the study of Weinstein (41); BCE is from the modeling study by Chang and Fujita (7); the model for the Na⁺-K⁺ pump is a modification of the model of Luo and Rudy (29) that allows NH₄⁺ transport (see appendix a); and the models for NKCC2 and KCC4 are discussed below. It is noteworthy that not all carriers are expressed in both cell membranes (see Fig. 1). The apical (or luminal) side has NKCC2, NHE3, and BCE, whereas the basolateral (or serosal) side has KCC4, the Na⁺-K⁺ pump, NHE3, and BCE. The NKCC2 transporter has three different isoforms: A, B, and F isoforms. The isoforms have different Cl⁻ binding affinity (B > A > F) and are expressed at different regions along the TAL: the F isoform in inner stripe of OM, the A isoform in outer stripe of OM and cortex, and the B isoform in distal cortical (cTAL) and macula densa (13, 36, 37).

Water flux arises from transmembrane osmotic pressure:

$$J_w^b\left(C^c(t),C^b\right)=P_w^b{\displaystyle \sum _{i=1}^n{\sigma }_i^b(C_i^b-C_i^c(t)),}$$ (4)

where P_w^b is the water permeability and σ_i^b is the reflection coefficient for solute i in the basolateral membrane.

Electroneutrality in the cytosol and extracellular compartments (lumen and serosa) is given by

$$I^a+I^b=F{\displaystyle \sum _{i=1}^n\left[A^a{J}_i^a(C^c,C^a)+A^b{J}_i^b(C^c,C^b)\right]z_i=0,}$$ (5)

$$I^a+I^p=F{A}^a{\displaystyle \sum _{i=1}^n\left[J_i^a(C^c,C^a)+J_i^P(C^a,C^b)\right]z_i=0.}$$ (6)

Equations 5 and 6 must be solved simultaneously for the membrane potentials E^a and E^b at each time that the solution of the system of differential equations is computed.

NKCC2 and KCC4.

The models for NKCC2 and KCC4 are based on the models published by Benjamin and Johnson (1) and Marcano et al. (32), which in turn are based on the kinetic model proposed by Lytle and McManus (30) and Lytle et al. (31). In the construction of the models, first-order binding kinetics and conservation of cotransporters is assumed, and parameter constraints are imposed to satisfy the laws of thermodynamics. The models are illustrated in Figs. 2 and 3. In each figure there are two cycles: the inner cycle represents the binding sequence of the KCC (Fig. 3) and NKCC (Fig. 2); the outer cycle represents the binding sequence when NH₄⁺ substitutes for K⁺.

Fig. 2.

Kinetic model for the Na⁺-K⁺/NH₄⁺-2Cl⁻ cotransporter (NKCC2) cotransporter. See Table 2 for definitions.

Fig. 3.

Kinetic model for the K⁺/NH₄⁺-Cl⁻ cotransporter (KCC4) cotransporter. See Table 1 for definitions.

In these models we assume symmetrical binding, in that, for each ion, the off-binding rate constant in the external side of the cell is equal to the value in the intracellular side as has been assumed by others (1, 42). Further, for the NKCC model, the two Cl⁻ binding sites have the same off-binding rate, an assumption made in earlier models (1, 42). We assume first-on first-off binding order (glide symmetry) as in Refs. 1, 31, 32, 42 and as suggested by experimental results by Gagnon et al. (11, 12). In a previous model of the NKCC, Marcano et al. (32) evaluated four models with different symmetry assumptions, which included different off-binding rates for the Cl⁻ sites and different off-binding rates for K⁺ in the luminal side of the cell and in the cytosolic side; comparison of the fits of the four resultant models to experimental data showed only small differences.

The differential equations that represent the models for the KCC4 and NKCC2 transporters are extensions of the model equations in Marcano et al. (32) in that they include the NH₄⁺ cycle. The differential equations were solved at steady state, and therefore, the model reduces to a system of linear equations from which the solute fluxes can be computed. The steady-state models of KCC4 and NKCC2 have many parameters that include binding, release, and translocation rate constants. Thereby, to estimate the unknown parameters, a nonlinear least-squares problem (curve-fitting problem) was solved as in Marcono et al. (32). The data used in the curve fitting were the fluxes reported by Bergeron et al. (2) and Mercado et al. (35) for KCC1 and KCC4, and the NKCC2A, NKCC2B, and NKCC2F fluxes reported by Bergeron et al. (2) and Plata et al. (37). A detailed explanation of the curve-fitting of the KCC4 and NKCC2 models can be found in Ref. 33. Tables 1 and 2 show the KCC4 and NKCC2 parameters used in the present study, and Figs. 4 and 5 show the actual curve fits.

Table 1.

Parameters for KCC model

Symbol	KCC1	KCC4
KCl, l/mol	46.2	998
KK, l/mol	11.3	4.75
KNH4, l/mol	6.49	0.3798
kff, s−1	1,000	2,220
kbf, s−1	2,830	1,001.3
Kfe, s−1	1,017	4,870
Kbe, s−1	360	10,810
kffN, s−1	2,190	66,030
kbfN, s−1	6,190	29,700

Data used for the fitting are in Ref. 35. KCC, K⁺/NH₄⁺-Cl⁻ cotransporter.

Table 2.

Parameters for the NKCC model

Symbol	NKCCA	NKCCB	NKCCF
KNa, l/mol	430	728	138
KCl, l/mol	137	1,000	751
KK, l/mol	0.709	0.135	0.1
KNH4, l/mol	2.69	1.75	1.33
kff, s−1	100,000	100,000	100,000
kbf, s−1	1,000	1,050	1,000
k\fe, s−1	1,200.7	7,630	4,870
kbe, s−1	12,0070	727,000	487,000
kffN, s−1	1,407	3,730	4,180
kbfN, s−1	141	39.1	41.8

Data used for the fitting are in Refs. 2, 37. NKCC, Na⁺-K⁺/NH₄⁺-2Cl⁻ cotransporter.

Fig. 4.

Rubidium influx as function of external concentration for the KCC1 and KCC4 cotransporters. A: K⁺ concentration. B: Cl⁻ concentration. C: NH₄⁺ concentration. Data from Refs. 2 and 35.

Fig. 5.

Rubidium influx as function of external concentration for the NKCC2 isoforms A, B, and F. A: Na⁺ concentration. B: Cl⁻ concentration. C: K⁺ concentration. D: NH₄⁺ concentration. Data from Refs. 2 and 37.

Weinstein (42) formulated simplified models for NKCC and KCC with ammonium transport by assuming rapid equilibrium. This assumption allowed the systems of linear equations to be reduced to two linear equations which could be solved analytically to compute the unidirectional fluxes. Weinstein used data reported in Refs. 35 and 37 to obtain rate constants by fitting the NKCC and KCC models to the kinetic curves. Then, physiological arguments were used to choose the NH₄⁺ binding rate constant and the outer-cycle translocation rate for both models. Weinstein reported that his approach yielded fitting errors of ∼25% for the NKCC2A and NKCC2B.

To compare the results obtained using our approach (which does not assume rapid equilibrium) with the ones obtained using Weinstein's approach (42), we computed unidirectional fluxes for the optimal parameters using the model in this work and the model used in Marcano et al. (32) for each cotransporter. Tables 3 and 4 report half-maximum concentration values (K_m) for the isoforms of the KCC and the NKCC2, respectively. Each table shows the K_m values for each modeling approach and the values reported in the experiments where the data were obtained. We observe that most of the values from both modeling approaches are close to one another with some exceptions. The differences may be related to the fitting errors of both modeling approaches.

Table 3.

Half-maximum concentration for each ion for the KCC models

Ion Species/Estimation Method	KCC1	KCC4
Cl−
Present study	31.3	17.0
Equilibrium modela	31.3	17.0
Published valueb	17.2 ± 8.3	16.1 ± 4.2
K+
Present study	26.9	17.0
Equilibrium modela	26.9	16.9
Published valueb	25.5 ± 3.2	17.5 ± 2.7
NH4+
Present study	22.8	15.3
Equilibrium modela	22.8	30.0
Published valuec	22.9 ± 13.5	13.5 ± 5.5

Concentrations are in mM.

^aWeinstein (42);

^bMercado et al. (35);

^cBergeron et al. (2).

Table 4.

Half-maximum concentration (mM) for each ion for the NKCC models

Ion Species/Estimation Method	NKCC2A	NKCC2B	NKCC2F
Na+
Present study	3.06	2.88	14.4
Equilibrium modela	2.66	1.66	24.5
Published valueb	5.0 ± 3.9	3.0 ± 0.6	20.6 ± 7.2
Cl−
Present study	19.3	15.1	36.7
Equilibrium modela	20.1	18.7	55.0
Published valueb	22.2 ± 4.8	11.6 ± 0.7	29.2 ± 2.1
K+
Present study	0.875	0.866	2.16
Equilibrium modela	0.953	1.30	4.30
Published valueb,d	0.96 ± 0.16 (0.30)	0.76 ± 0.07	1.54 ± 0.16
NH4+
Present study	1.54	0.982	2.41
Equilibrium modela	1.55	0.928	2.35
Published valuec,d	1.7 ± 0.5 (1.90)	NAe	NAe

^aWeinstein (42);

^bPlata et al. (37);

^cBergeron et al. (2);

^din parentheses Kinne et al. (23);

^eNA, no data available.

pH homeostasis.

The acid-base solutes H⁺, NH₄⁺, NH₃, H₂CO₃, HCO₃⁻, H₂PO₄⁻, and HPO₄²⁻ constitute three buffer systems, and the concentrations of those solutes depend on transmural fluxes and on their ionization reactions. We denote the three buffer systems and the total amount of acid as

$$\begin{gathered} B_1=\left[{\text{NH}}_4^{+}\right]+\left[{\text{NH}}_3\right],B_2=\left[{\text{H}}_2{\text{CO}}_3\right]+\left[{\text{HCO}}_3^{-}\right], \\ {B}_3=\left[{\text{H}}_2{\text{PO}}_4^{-}\right]+\left[{\text{HPO}}_4^{2-}\right], \\ {H}_{tot}=\left[{\text{H}}^{+}\right]+\left[{\text{NH}}_4^{+}\right]+\left[{\text{H}}_2{\text{CO}}_3\right]+\left[{\text{H}}_2{\text{PO}}_4^{-}\right]\mathrm{.} \end{gathered}$$ (7)

If the buffers are in equilibrium and the isohydric principle is assumed, one obtains

$$[{\text{H}}^{+}]+{\displaystyle \sum _{k=1}^3\frac{B_k[{\text{H}}^{+}]}{[{\text{H}}^{+}]+K_k}-H_{tot}=0,}$$ (8)

where K_k is the corresponding equilibrium constant. This equation was solved for [H⁺], and from Eq. 7 the buffer constituents were recovered. To account for the hydration/dehydration of CO₂, which is slow compared with the ionization of carbonic acid, a chemical “flux” or source was added to the conservation law involving H₂CO₃ (i.e., B₂). That chemical source is given by

$$J_{{\text{H}}_2{\text{CO}}_3}^{\text{chem}}=(k_d{C^c}_{{\text{H}}_2{\text{CO}}_3}-k_h{C^2}_{{\text{CO}}_2})V_{\text{cell}},$$ (9)

where k_d and k_h are the hydration and dehydration constants for CO₂. This approach allows us to compute the time evolution of the total buffers and total acid (B₁, B₂, B₃, and H_tot) in Eq. 1 and use the specific values of the buffer constituents in the cytosol.

CVR.

TAL cells swell because of an increase in luminal solute uptake or a decrease in serosal bath osmolarity with respect to the cytosol. Cells respond to swelling with a regulatory volume decrease. Such a response can be long term, which involves the release of uncharged impermeant solutes, or short term by regulation of solute transport. The long-term response was simulated by initially setting the concentration of impermeant solutes sufficiently high to match the osmolarity in the serosal bath, particularly in the inner stripe regions of the TAL where serosal osmolarity reaches values of ∼500 mOsm. The short-term response was modeled by defining functions that map cell volume to the total activity (or transporter density) of the NKCC and KCC transporters (see Fig. 6). These functions are based on observations by Kahle et al. (20). Details can be found in appendix b.

Fig. 6.

A: cell volume regulation (CVR) functions. B: NKCC and KCC maximum and minimum enzyme density (E_T) for those CVR functions. OM, outer medulla.

Model parameters.

Table 5 lists the physical dimensions of the TAL and its chemical properties. Table 6 shows the values for apical, basolateral, and paracellular permeabilities chosen for the electrodiffusion of each solute across the TAL epithelium. The parameters for the membrane-embedded carriers and the specification of carrier vs. membrane mapping are given in Table 7. In Table 6 and Fig. 6B, permeabilities and max/min NKCC/KCC activities (min/max E_T in Fig. 6A) were chosen such that transepithelial electrical and chemical gradients computed by the model are close to experimental values (15). Table 8 shows the chemical composition of the serosal bath at different locations along the length of the TAL: OM, cortico-medullary junction and cortex and distal TAL. Notice the high osmolarity (∼500 mOsm) in the deepest part of the TAL (OM). The values for this set of concentrations, in particular Na⁺, are consistent with measurements summarized by Layton (28). Baseline values for luminal concentrations are shown in Table 9.

Table 5.

Basic physicochemical parameters

	Description	Value	Reference
L	TAL length, cm	0.6	(24)
R	TAL radius, μm	10	(24)
Vlum	TAL tubule volume, cm3/cm	2.21 × 10−4	calculateda
Aa	Apical membrane area per unit length, cm2/cm	0.0152	(26)
Ab	Basolateral membrane area per unit length, cm2/cm	0.0944	(26)
pK1	pK for NH4+/NH3 buffer	9.15	(19)
pK2	pK for H2CO3/HCO3−buffer	3.57	(19)
pK3	pK for H2PO42−/HPO4− buffer	6.80	(19)
kh	CO2 hydration constant, s−1	1450	(43)
kd	CO2 dehydration constant, s−1	4.96 × 105	(43)
pCO2	Partial pressure of CO2, mmHg	55.0	(9)

^aWe assumed a cylindrical geometry for the tubule: πr²Δx. This is luminal volume, so walls are not included. Δx is a fraction of the tubule length.

Table 6.

Apical, basolateral, and paracellular permeabilities

	OM Cell			Cortico-Medullary Cell			cTAL Cell			Distal Cell
	Pa	Pb	Pp	Pa	Pb	Pp	Pa	Pb	Pp
Na+	0	0.005	0.304	0	0.005	0.470	0	0.005	0.522	0	0.005	0.596
K+	15.3	4.80	0.339	6.24	4.80	1.83	4.54	4.80	2.29	2.14	4.80	2.96
Cl−	0	1.20	0.0142	0	1.2	0.0219	0	1.20	0.0243	0	1.20	0.0278
X−	0	0	0	0	0	0	0	0	0	0	0	0
Y	0	0	0	0	0	0	0	0	0	0	0	0
Z+	0	0	0	0	0	0	0	0	0	0	0	0
H+	400	400	0.609	400	400	0.939	400	400	1.04	400	400	1.19
NH4+	0.0327	0.08	0.06	1.13	0.08	0.06	1.8	0.08	0.06	2.74	0.08	0.06
NH3	300	200	0.6	300	200	0.6	300	200	0.6	300	200	0.6
H2CO3	0	0	0	0	0	0	0	0	0	0	0	0
HCO3−	0	0.02	0.0001	0	0.02	0.0001	0	0.02	0.0001	0	0.02	0.0001
H2PO4−	0	0	0	0	0	0	0	0	0	0	0	0
HPO42−	0	0	0	0	0	0	0	0	0	0	0	0
H2O	0	40	0	0	40	0	0	40	0	0	40	0

All permeabilities are in units of 10⁻⁴cm/s, with the exception of H₂O which is in units of 10⁻⁴ mM⁻¹·cm/s. OM, outer medulla, cTAL, cortical TAL; P^a, P^b, P^p, apical, basolateral, and paracellular permeabilities.

Table 7.

Carrier-mediated transport parameters

Carrier	Apical	Basolateral	Reference
NKCC2 isoform A	*		†
NKCC2 isoform B	*		†
NKCC2 isoform F	*		†
KCC4		*	†
NHE3	*	*	(41)
BCE		*	(7)
Na-K pump		*	(29)

Isoform F is expressed by inner stripe OM cells; isoform A is expressed by outer-stripe OM cells and cortical cells; isoform B is expressed by distal cortical cells and macula densa cells. BCE, HCO₃⁻-Cl⁻ exchanger. * Carrier is embedded in the cell membrane. †Value is given in the technical report: “Equations for Models of NH₄⁺ Transport by the Renal Na-K-2Cl and K-Cl Cotransporters.” Ref. 33.

Table 8.

Serosal concentrations

	OM Cell	Cortico-Medullary Cell	cTAL cell	Distal Cell
Na+	281	145	145	145
K+	7.87	3.50	3.50	3.50
Cl−	261	105	105	105
X−	1.24	16.8	16.8	16.8
Y	8.09	1	1	1
Z+	0.769	1.06	1.41	1.90
pH	7.31	7.30	7.30	7.30
NH4+	3.91	1.86	1.52	1.03
NH3	0.0561	0.0279	0.0279	0.0279
H2CO3	0.00459	0.00466	0.00466	0.00466
HCO3−	25	25	25	25
H2PO4−	0.918	0.625	0.625	0.625
HPO42−	2.94	1.98	1.98	1.98

Concentrations are in mM.

Table 9.

Baseline luminal concentrations used in the efficiency simulations and computed cytosolic concentrations

	OM Cell		Cortico-Medullary Cell		cTAL Cell		Distal Cell
	Lum	Cyt	Lum	Cyt	Lum	Cyt	Lum	Cyt
Na+	250	11	93.8	5.5	59.2	5	38.0	3.55
K+	7.59	136	1.92	138	1.49	139	1.54	141
Cl−	233	20	233	4.5	45.8	4	29.5	3.49
X−	27.7		27.7		27.7		27.7
Y	0.3		0.3		0.3		0.3
Z+	33.7		33.7		33.7		33.7
pH	7.29	7.25	7.17	7.27	7.06	7.26	6.90	7.23
NH4+	4.05	8.25	2.66	4.45	2.20	4.1	1.91	3.54
NH3	0.0558		0.0279		0.0179		0.0106
H2CO3	0.0048		0.0048		0.0048		0.0048
HCO3−	25.3		19.3		14.9		10.2
H2PO4−	1.22		1.49		1.77		2.23
HPO42−	3.78		3.51		3.23		2.77

Concentrations are in mM. When the simulation protocol requires a change in luminal cation, [Cl⁻] is also changed such that the luminal (Lum) bath remains electroneutral. Cytosolic (Cyt) concentrations are the concentrations obtained when solving the model for the efficiency results as function of changes in luminal Na⁺ (baseline case).

Initial conditions.

The TAL cell model requires the specification of initial conditions for cytosolic solute concentrations and cell volume. To obtain these values, we conducted a long-time simulation using a different set of physiologically plausible initial conditions and then used the resulting steady-state model solutions as the initial conditions given in Table 10.

Table 10.

Values for initial conditions

	Cytosol
[Na+]	7.00
[K+]	136
[Cl−]	4.40
[X−]	113
[Y]	[6.00, 306]a
[Z+]	0.908
pH	7.28
[NH4+]	5.01
[NH3]	0.0682
[H2CO3]	0.00483
[HCO3−]	25.0
[H2PO4−]	0.988
[HPO42−]	3.01
Vcellb	8.20 × 10−8

Concentrations are in mM, and volumes are in cm³.

^aRange from cTAL to medullary TAL;

^bcomputed by assuming the TAL radius, including the walls, is 2r (see Table 5). Then, V_cell = π(2r)² Δx − V_lum, where Δx is cell height.

Numerical solution.

Model Eqs. 1 and 2 were solved using the second-order Backward Differentiation Formula (BDF) that is part of the MATLAB suite of ODE solvers. The electroneutrality constraints given in Eqs. 5 and 6 were solved with a generalized secant with Broyden update method. Equation 8, the pH homeostasis equation, can be written as a fourth-degree polynomial in [H⁺], and its roots were computed by a Dekker-Brent method. The numerical methods were implemented in MATLAB. Computations were performed in a Linux box with 8 GB of memory and two Intel quad-core Xeon X5473 3.0-GHz CPUs.

RESULTS

Cell volume regulation.

A major task of TAL cells is to dilute the tubular fluid by solute transport to the interstitium. Because the chemical composition and osmolarity of the luminal and interstitial fluid vary substantially along the cortico-medullary axis (see Table 8), TAL cells face challenges in maintaining cell volume. Experimental observations by Komlosi et al. (25) indicate that rabbit cTAL cells are capable of a regulatory volume decrease of up to ∼50% in response to cell swelling induced by increased apical Na⁺ and Cl⁻ uptake.

To study our cTAL model response to variations in Na⁺ and Cl⁻ uptake, we conducted a set of simulations in which step perturbations similar to those in Komlosi et al. (25) were applied to luminal Na⁺ and Cl⁻ concentrations. In these simulations, cTAL luminal Na⁺ and Cl⁻concentrations were each initialized to be 1 mM and then increased to 20 mM, reduced to 1 mM, increased to 40 mM, reduced to 1 mM, increased to 60 mM, and then reduced to 1 mM. Each step perturbation lasted 120 s. Concentrations of the impermeants X⁻ and Z⁺ were adjusted to maintain electroneutrality. Cell volume changes were computed in time for the base case (not shown) and for three alternative cases. In the base case, the CVR mechanism was assumed to be instantaneous; i.e., E_T^NKCC = g_NKCC and E_T^KCC = g_KCC, where E_T^NKCC and E_T^KCC are the total transporter density for NKCC2 and KCC4 respectively; and where g_NKCC and g_KCC are CVR functions for NKCC2 and KCC4 respectively (see Eq. 13 and 14 in appendix b). In alternative cases 1 and 2, we introduced a delay by modifying the dependence of the total activity of NKCC2 and KCC4 transporters on cell volume. Specifically, the evolution of E_T^NKCC and E_T^KCC were given by

$$\frac{d}{dt}{E}_T^{NKCC}=\frac{g_{NKCC}(V)-E_T^{NKCC}}{{\tau }_{NKCC}}$$ (10)

and

$$\frac{d}{dt}{E}_T^{KCC}=\frac{g_{KCC}(V)-E_T^{KCC}}{{\tau }_{KCC}}$$ (11)

where τ_NKCC and τ_KCC are the delays for the NKCC2 and KCC4 CVR function, respectively. The delays were taken to be 10 s for case 1 and 30 s for case 2. In case 3, we assumed the absence of a CVR mechanism, and the NKCC2 and KCC4 activities were set to yield a short-circuit current close to the values measured by Greger et al. (18) (discussed below). Specifically, we set E_T^NKCC = 5.46 × 10⁻⁷ μmol/cm² and E_T^KCC = 2.50 × 10⁻⁶ μmol/cm².

For each of the above cases, we computed normalized cell volume, given the ratio of transient cell volume to the steady-state cell volume (V_ss). Results are shown in Fig. 7. The two cell models with the CVR mechanism (i.e., case 1 and case 2) are able to diminish swelling by ∼50% relative to case 3. Those CVR responses, which are consistent with observations reported in Komlosi et al. (25), provide the cTAL cell with adequate protection against swelling. The effect of delaying the CVR response is evident in Fig. 7, where the cell volume variation corresponding to case 2 (with 30-s delay) lags those of case 1 (with 10-s delay). The overshoot in cell volume corresponds to the regulatory volume decrease produced by CVR, while the undershoot corresponds to the cell shrinkage caused by solute extrusion in the course of the regulatory volume decrease. Those dynamic features are reasonably consistent with data by Komlosi et al. (25).

Fig. 7.

Time course of normalized cell volume (V = V_ss, where V_ss is the steady state volume) after luminal NaCl perturbations, based on experiments by Komlosi (25) is shown. Simulations were conducted with a cortical TAL (cTAL) cell. Solid line: case 1, 10-s CVR delay. Dotted line: case 2, 30-s CVR delay. Dashed line: case 3, no CVR, but with NKCC and KCC activities set to yield a short-circuit current close to values measured by Greger (see Fig. 8).

Electrophysiological properties.

One of the challenging aspects of mathematical models for the TAL is the relative scarcity of electrophysiological data, which is important for deriving cell membrane parameters (permeabilities and transporter activity) that are linked to solute transport across the epithelium. Nonetheless, some key electrophysiological data are available for TAL cells (15). To verify that the cTAL cell model (Eqs. 1-9), with the appropriate serosal concentrations (Table 8, column “cTAL cell”), yields predictions consistent with a cTAL cell in vitro, we calculated electrophysiological properties for comparison with published data. In a simulation, we advanced the cell model in time to steady state, then injected a 100 μA/cm² current for 10 s, and computed the membrane potential (apical and basolateral) deflections and the corresponding resistances. Table 11 shows the experimental and computed short-circuit current (I_sc), that is, the current when there is no transepithelial electrochemical gradient; the table also shows apical, basolateral, and transepithelial membrane resistances (R_a, R_b, and R_t), and the voltage divider ratio (R_a/R_b). Table 11 suggests that our cell model with the chosen set of parameters has electrophysiological properties consistent with a cTAL cell. However, this result exhibits a marked dependency on apical and basolateral membrane area (results not shown).

Table 11.

Electrophysiological properties of a model cTAL cell

	Isc, μA/cm2	Ra, Ωcm2	Rb, Ωcm2	Rt, Ωcm2	Ra/Rb
Computed	260.8	69.4	22.0	23.3	3.16
Experimental	285 ± 33	30–87	20–47	10–35	2.8 ± 0.5

Experimental short-circuit current (I_sc) and apical and basolateral membrane resistances (R_a/R_b) are from Ref. 18; transepithelial membrane resistances (R_t) are from Ref. 15.

Greger et al. (18) measured short-circuit current (I_sc) through cTAL epithelium as a function of parallel, isotonic increases in luminal and serosal Cl⁻ concentration. Figure 8 shows short-circuit current simulations conducted according to protocol of Greger et al. (18); Fig. 8A shows simulation results conducted with a cell model “extracted” from differing locations along the length of the TAL, whereas Fig. 8B displays results from a cTAL cell with varying CVR regimes. We considered two cases: case 1 represents and case 2 neglects CVR but with NKCC and KCC activities that yield the optimal coefficient of determination (R²) value, i.e., a value close to one.

Fig. 8.

A: Na⁺ short-circuit current (I_sc) for cells with CVR at differing locations along the TAL. Dashed line is for a TAL cell near the inner-outer medullary boundary (the TAL inlet), solid line is for a cell at the cortico-medullary junction, dotted line is a cell at the late cortex, and diamonds are the data of Greger et al. (18). B: Na⁺ I_sc for a cTAL cell under a different CVR cases: case 1 is with CVR (solid line) and case 2 is no CVR with NKCC and KCC activities that yield the optimal R² value (dashed line). Diamonds are the data of Greger et al. (18).

We observe in Fig. 8A that the degree to which the model predictions fit data by Greger et al. (18) depends on the cell's location along the tubule. Specifically, after calculating the values for the coefficient of determination (R²) to assess the fit's quality for each simulation, we found that for an OM cell R² can rise as high as 0.9534 and for a cortical cell R² lowers to 0.8769. It is also noteworthy that, since data in Greger et al. (18) was obtained from cTAL cells, the departure from the data of the cell model at the chosen locations, as determined by the R² value, is modest (only an 8% relative difference between the best and worst cases mentioned above). Nevertheless, for a cell near the outer-inner medullary boundary, the departure from data increases markedly, because certain parameters from the cell model, e.g., permeabilities or transporter activity, are not uniform for all cells (see Table 6 and Fig. 6B). Also, cells have to adapt to the environment in which they developed; hence short-circuit current and other electrophysiological properties will show variations among cells along the TAL segment.

The results in Fig. 8B suggest that CVR has an impact in the goodness of fit and in the transport properties of the cell as embodied by the short-circuit current measure. This was expected because NKCC and KCC are major players in transepithelial Na⁺ transport, and their activities are determined by cell volume through the CVR function depicted in Fig. 6A. Figure 8B also shows that setting reasonable bounds for NKCC and KCC transporter density and letting the cell regulate its volume (case 1 in Fig. 8) yields a fit to data akin to the case in which we did not activate CVR but set the NKCC and KCC enzyme transporter to yield optimal R². From the modeler's perspective, the CVR mechanism is a biologically driven and autonomous way to set the NKCC and KCC transporter density parameters. Together, these results show that our TAL cell model predicts behaviors that are generally consistent with key measurements of Na⁺ and Cl⁻ transport in this segment (like membrane potential, membrane resistances, etc.).

Electrophysiological properties and CVR.

We computed normalized cell volumes (V = V_set, where V_set is the set point of the CVR mechanism) obtained from the short-circuit current simulations for three different locations along the TAL (results shown in Fig. 9A). At each location, cell volume is well regulated, in that there is only a modest increase in volume and luminal and serosal Cl⁻ concentration increases, although the mTAL cell has a slightly greater slope than the more distal cells (see Fig. 9A). Further, the model predicts a higher steady-state cell volume in the mTAL cells relative to the cTAL cells, and cells in both locations differ from the nominal value of V_set specified when the initial conditions were computed. This is a consequence of the cytosolic Cl⁻ concentration in mTAL cells vs. the cTAL cells (see Table 9). Figure 9, B and C, shows cytosolic Na⁺ and Cl⁻ concentrations for short-circuit current simulations in the model mTAL cells.

Fig. 9.

Normalized cell volume (V = V_set where V_set is the set volume of the CVR mechanism). A: cell volume at differing locations along the TAL. Dashed line is for a TAL cell close to inner-outer medullary boundary (the TAL inlet), solid line is for a cell at the cortico-medullary junction, and dotted line is a cell at the late cortex (note the overlap between the solid and dotted lines). B and C: cytosolic concentrations as function of isotonic increases in luminal and serosal concentrations. This is for a cell close to the inner-outer medullary junction and under the cases of CVR (solid curve), and no CVR with NKCC and KCC set to yield optimal R² value (dashed curve).

Ammonium transport.

In contrast to other organs, NH₄⁺ is present in the kidneys in nonnegligible amounts. NH₄⁺ is produced by the proximal tubular cells from the buffering of a free proton by NHE3. Furthermore, as shown by Watts and Good (40), the expression of NKCC2 in mTAL cells gives ammonium another entry pathway into the cytosol (NH₄⁺ also permeates K⁺ channels), and that yields a cytosolic acidification when the cell is submitted to an NH₄⁺pulse perturbation. Following the protocol of Watts and Good (40), we simulated luminal NH₄⁺ pulse perturbations and recorded the time course of the changes in cytosolic pH. The results of those simulations, shown in Fig. 10, A1–D1, are similar to the pH time course reported by Watts and Good (40), except for the gradual pH recovery after the ammonium challenge, which may reflect the effects of cell volume increase on NHE3 activity. The NKCC2 transporter is the major uptake pathway for ammonium, although it is not the only one since ammonium can also permeate K⁺ channels. Figure 10A1 shows the predicted cytosolic acidification that follows an ammonium perturbation on the luminal side. Figure 10B1 also shows a pH decrease but of lower magnitude since NKCC2 activity was diminished to simulate the presence of the diuretic furosemide. Figure 10C1 shows a drop similar to that in Figure 10B1, but in this case it is the NH₄⁺ and K⁺ permeabilities that are reduced to simulate channel blockage due to the presence of Ba²⁺. Figure 10D1 shows a substantially smaller cytosolic acidification than in the other cases; this is due to inhibition of both NKCC2 and the electrodiffusive pathway for NH₄⁺ entry into the cell. It is noteworthy that the pH decrease shown in Fig. 10 can be diminished or augmented by upregulation or downregulation of NHE3 activity. In particular, we have observed that increasing NHE3 activity on the basolateral side substantially reduces the pH decrease under all of the above-mentioned simulation protocols (results not shown). The observation can be attributed to the extrusion of NH₄⁺ and H⁺ by NHE3, which dampens the pH decrease and absorbs Na⁺, resulting in an increase in cytosolic Na⁺ and a lower concentration gradient that drives uptake by NKCC2 on the apical side.

Fig. 10.

Simulations based on experiments by Watts and Good (40). In A–D, luminal [NH₄⁺] was increased from 1.6 mM to 20 mM for 8 min. A1–D1: simulation results. A2–D2: experimental results of Watts. A1 and A2: control. B1 and B2: cell was washed in furosemide to inhibit NKCC. C1 and C2: solution in the luminal side contained Ba²⁺ to inhibit the apical K⁺/NH₄⁺ channels. D1 and D2: luminal side solutions have barium and furosemide. Bars indicate the content of the chemical solution in which the cell was washed.

Ammonium cycling.

Ammonium cycling across the apical membrane is a prominent phenomenon observed in simulations that mimic the experiments of Watts and Good (40). This phenomenon, also observed in a modeling study by Weinstein and Krahn (44), consists of NH₄⁺ uptake into the cell by NKCC2 and the apical channels, and extrusion via NHE3 on the apical side. Since NH₄⁺ and K⁺ compete for a binding site on NKCC2 (44, 33), NH₄⁺ loading implies a reduction in K⁺ loading through the NKCC2 transporter. Figure 11 shows the NKCC2, apical NHE3, and apical ammonium electrodiffusion fluxes that correspond to the NH₄⁺ pulse simulation in Fig. 10A1. The NH₄⁺ cycling is clearly seen in the flux plots (Fig. 11). NH₄⁺ enters the cell via NKCC2 and the apical NH₄⁺ channel and exits through the apical NHE3 (positive flux is efflux). The ammonium cycling phenomenon is one of the features of TAL cells that enhance Na⁺ transport, because it is a mechanism that prevents K⁺ depletion in the lumen. If luminal K⁺ depletion were to occur, then NaCl transport would be impaired because the NKCC transporter requires binding by either K⁺ or NH₄⁺.

Fig. 11.

NH₄⁺ fluxes for simulations, based on experiments by Watts and Good (40). In A–C, luminal [NH₄⁺ ] was increased from 1.6 to 20 mM for 8 min, and a negative flux is an influx. A: NKCC2 flux. B: apical electrodiffusive flux. C: apical NHE3 flux.

Transport efficiency.

TAL cells must transport significant amounts of solute to dilute the tubular fluid. That transport is driven by transcellular Na⁺ reabsorption, which involves energy consumption (via Na⁺- K⁺-ATPase). Given the limited O₂ availability in the OM, transport efficiency becomes important. The efficiency of TAL cell transport has been attributed to paracellular electrodiffusive transport through cation-selective junctions, which depends on the transepithelial chemical gradient and on the establishment of a lumen positive transepithelial potential (4, 17). Thus solute transport is expected to reduce or reverse the transepithelial chemical gradient while increasing the electrical driving force for paracellular transport, thereby reducing efficiency. In a number of simulations, we computed transport efficiency of the TAL cell model under differing conditions.

We quantified Na⁺ transport efficiency as the ratio of transepithelial to transcellular Na⁺ flux, i.e.,

$$ϵ=\frac{J_{{\text{Na}}^{+}}^a+J_{{\text{Na}}^{+}}^p}{J_{{\text{Na}}^{+}}^a}=\frac{\text{net transepithelial Na flux}}{\text{net transcellular Na flux}}.$$ (12)

The transport efficiency measure ϵ ranges from 0 to 2, where 0 represents complete backleak (serosa to lumen) through the paracellular pathway, whereas 2 means that for each Na⁺ that goes transcellularly one is transported paracellularly from lumen to serosa. It is important to emphasize two points. First, although our definition of Na⁺ transport efficiency has the virtue of simplicity, it does not include the transport of other solutes by the TAL. Second, although we do not represent O₂ nor ATP in our model, ϵ can be translated to ATP consumption by assuming two things: the first is that all transcellular Na⁺ transport ultimately involves Na-K-ATPase; the second is that the stoichiometry of the pump yields 3 moles of Na⁺ translocated per mole of ATP hydrolyzed. This gives a conversion factor that maps the unitless ϵ to transport efficiency in units of moles of Na⁺ transported per mole of ATP used.

We first examined the effect of luminal Na⁺ concentration on TAL efficiency. We varied luminal Na⁺ concentration and computed transepithelial potential difference (E^p), Na⁺ fluxes, and transport efficiency for cells at various locations along the TAL. Note that we also varied Cl⁻ such that the luminal bath remained electroneutral. Moreover, for all efficiency simulations that require a change in the luminal concentration of a cation, we proceeded in a similar fashion and changed [Cl⁻] to preserve electroneutrality in the lumen.

TAL efficiency as a function of luminal Na⁺ is shown in Fig. 12, A1–D1. The results indicate that the transepithelial membrane potential (E^p) decreases as luminal [Na⁺] increases. It is notable that E^p becomes negative only at luminal concentrations that are likely too high for the given types of cells. This suggests that, under physiological conditions, an ensemble of TAL cell models could reproduce important features of the TAL like that of a positive E^p. Fig. 12, A2–D2, exhibit net apical, paracellular, and transepithelial Na⁺ fluxes. Here, a positive paracellular flux is from lumen to serosa, and positive apical flux is from the lumen to the cytosol (influx). Then, for sufficiently high luminal Na⁺, both fluxes favor Na⁺ reabsorption (removal from lumen). Efficiency results, shown in Fig. 12, A3–D3, suggest that under the given conditions the cell model would not attain an ϵ value close to 2 unless the luminal [NaCl] exceeds 350 mM. Further, ϵ reaches a maximum in the medullary or cortico-medullary region, and ϵ decreases in the early cortical and distal TAL segments. These changes in efficiency as a function of cell location along the TAL can be attributed to the changes in paracellular uptake (Fig. 12, A2–D2), which are in turn a consequence of the changes in transepithelial chemical gradient at different locations along the TAL. Transport efficiency as a function of changes in luminal Na⁺ for cell model without CVR mechanism was also computed and it is shown in Fig. 13. The overall behavior is similar to the CVR case albeit the efficiency measure is lower for the no CVR case, particularly, at the cortical and distal TAL.

Fig. 12.

Membrane potential, fluxes, and efficiency as a function of luminal [Na⁺] computed with the cell model. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is the solid line, paracellular flux is the dotted-dashed line, and transepithelial flux is the dotted line with positive flux meaning reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of Na⁺ changes in the luminal side of cells at differing locations along the TAL segment. Dots mark the baseline case.

Fig. 13.

Membrane potential, fluxes, and efficiency as function of luminal [Na⁺] computed with the cell model with no CVR mechanism enabled. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is the solid line, paracellular flux is the dotted-dashed line, and transepithelial flux is the dotted line with positive flux meaning reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of Na⁺ changes in the luminal side of cells at differing locations along the TAL segment. Dots mark the baseline case.

In the next set of simulations, we assessed the effect of NH₄⁺ on TAL efficiency. We varied luminal NH₄⁺ for cells at various locations along the TAL and then explored the impact of such changes on membrane potential, Na⁺ fluxes and TAL efficiency. Model results are exhibited in Fig. 14. The model predicts that the transepithelial membrane potential becomes more positive as luminal NH₄⁺ concentrations diminish. Also, the paracellular flux is absorptive (positive) for low NH₄⁺ concentrations for a cell in the cortico-medullary region. It is noteworthy that regardless of cell location the efficiency decreases as luminal NH₄⁺ increases. This is caused by the increased transcellular Na⁺ uptake through the apical NHE3 that is a component of NH₄⁺ cycling. Hence, the efficiencies are lower than the cases in where we changed luminal NaCl (cf. efficiencies in Figs. 12 and 14). The trend of the efficiency to attain a maximum for a cell in the corticomedullary region is akin to the case where we varied luminal Na⁺.

Fig. 14.

Membrane potential, fluxes, and efficiency as function of luminal [NH₄⁺ ] computed with the cell model. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is the solid line, paracellular is the dotted-dashed line, and transepithelial is dotted line with positive flux indicating reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of NH₄⁺ changes in the luminal side of cells at various locations along the TAL segment. Dots indicate the baseline case.

We then conducted simulations in which we varied luminal K⁺ concentration. The predicted transepithelial membrane potential and efficiency, shown in Fig. 15, are qualitatively different from the previous cases. For an OM cell with low luminal K⁺, the model predicts a substantial positive transepithelial potential that drives paracellular reabsorption (see Fig. 15, A1–A3) to a level that is almost equal to the transcellular uptake, and thus the efficiency approaches two. The efficiency drops markedly as luminal K⁺ increases because the K⁺ backleak diminishes (from −6 × 10⁻³ nmol/s/cm² to 3× 10⁻³ nmol/s/cm²), which yields a lower transepithelial membrane potential. Also, in contrast to the previous cases, here the efficiency monotonically decreases as the cell location is changed from the medullary to the cortical regions. Nonetheless, the decrease in efficiency is substantial in the cortical regions, similar to the previous cases, owing to the establishment of a chemical gradient that favors paracellular backleak.

Fig. 15.

Membrane potential, fluxes, and efficiency as function of luminal [K⁺] computed with the cell model. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is the solid line, paracellular is the dotted-dashed line, and transepithelial is dotted line with positive flux indicating reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of K⁺ changes in the luminal side of cells at differing locations along the TAL segment. Dots represent the baseline case.

Given the results above, which suggest that the transepithelial electrochemical gradient is a key factor for the efficiency along the TAL, we then sought to explore how efficiency changes as a function of paracellular Na⁺ permeability. Figure 16 shows simulation results obtained for a cell at different locations along the TAL. Closing the junctions minimizes Na⁺ backleak but also prevents paracellular Na⁺ transport. These two competing effects result in an efficiency that is ∼1 when the junctions are closed, and that efficiency decreases as the junctions are opened.

Fig. 16.

Membrane potential, fluxes, and efficiency as function of paracellular Na⁺ permeability computed with the cell model. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is solid line, paracellular is dotted-dashed line, and transepithelial is dotted line with positive flux indicating reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of changes in paracellular Na⁺ permeability (P_Na^p) at differing locations along the TAL segment. Dots mark the baseline case.

We also assessed the effects of apical K⁺ permeability on transepithelial membrane potential, Na⁺ fluxes, and transport efficiency. Model results, shown in Fig. 17, indicate that increasing the apical K⁺ permeability increases the paracellular Na⁺ reabsorption, consistent with the general belief that apical electrodiffusive K⁺ backleak establishes a lumen-positive transepithelial membrane potential, which promotes electrodiffusive paracellular Na⁺ uptake. Nonetheless, overall efficiency remains well below 2, owing to the substantial inward-directed Na⁺ concentration gradient.

Fig. 17.

Membrane potential, fluxes, and efficiency as function of apical K⁺ permeability computed with the cell model. A1–D1: transepithelial membrane potential (E^p). A2–D2: fluxes (J_Na), where apical flux is solid line, paracellular is dotted-dashed line, and transepithelial is dotted line with positive flux indicating reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of changes in apical K⁺ permeability (P_K^a) at different locations along the TAL segment. The dots mark the baseline case.

The effect of changes in paracellular Cl⁻ permeability on membrane potentials, fluxes, and transport efficiency can be seen in Fig. 18. It shows that substantial increases in Cl⁻ permeability have an adverse effect in transport efficiency. This stems from the decrease in transepithelial membrane potential. In all the above simulations, we observed the low Na⁺ concentrations (3–6 mM) and high K⁺ concentrations (135–142 mM) that are typical of cytosolic concentrations, as well as pH values that are well within reasonable bounds (7.2–7.3). Also, changes in cytosolic concentrations were predicted to be small, relative to the changes in luminal concentrations or membrane permeabilities, which illustrate the model TAL cell's ability to maintain cytosolic homeostasis.

Fig. 18.

Membrane potential, fluxes, and efficiency as function of paracellular Cl− permeability computed with the cell model. A1–D1: transepithelial membrane potential (Ep). A2–D2: fluxes (JNa), where apical flux is solid line, paracellular is dotted-dashed line, and transepithelial is dotted line with positive flux indicating reabsorption. A3–D3: efficiency index (ϵ). All rows are given as a function of changes in paracellular Cl− permeability (PCl−p) at differing locations along the TAL segment. The dots mark the baseline case.

DISCUSSION

To study efficiency of Na⁺ transport in the TAL, we developed a detailed mathematical model of a TAL cell, which includes a CVR mechanism that couples changes in cell volume with transcellular Na⁺ transport. Transport efficiency, defined here as the ratio of transepithelial to transcellular transport, is believed to be high owing to electrodiffusive paracellular transport secondary to the establishment of a positive transepithelial potential caused by K⁺ cycling across the apical membrane of TAL cells (4, 17). Here we explore these concepts and discuss in greater detail how our results compare with the modeling and experimental work of other investigators.

Comparison with other modeling studies of the TAL.

Fernandes and Ferreira (10) and Weinstein and Krahn (44) independently developed mathematical models of TAL cells. cTAL cell model of Fernandes and Ferreira (10) is based on experimental work by Greger and Schlatter (16). Their model, as all subsequent modeling efforts, is a set of conservation relations that includes electrodiffusive and carrier-mediated transport by NKCC, KCC, and Na⁺-K⁺-ATPase. Even though their model does not include kinetic representations of the major cotransporters (NKCC and KCC), it reproduces major electrophysiological and functional properties of cTAL cells.

Weinstein and Krahn (44) presented a much more detailed TAL cell model that includes detailed kinetic models of NKCC2, KCC4, NHE3, and BCE, as well as all the intricate transcellular transport pathways that involve NH₄⁺. Inasmuch as our mathematical model of a TAL cell is similar in many ways to Weinstein and Krahn's model, we proceed with an enumeration of the key differences and how they influence model predictions.

First, we implemented a CVR mechanism that controls solute transport and therefore short-term cell volume changes by regulating the activities of the NKCC2 and KCC4 membrane transporters. In Weinstein's model, cell volume in the medulla was adjusted only by specifying the amounts of cytosolic impermeants. In our work that technique was used to simulate long-term CVR (osmolyte accumulation) in cells in the OM. The major consequence of the inclusion of cell volume regulation by altering the activities of NKCC2 and KCC4 transporters is that our model predicts somewhat lower Na⁺ fluxes and cytosolic Cl⁻ concentrations, and a lesser degree of cell swelling when luminal Na⁺ increases.

A second difference concerns the choice of micropuncture data used in the calibration of the models. Our goal was to adjust parameters to ensure that cTAL cells were able to reproduce the TAL outflow composition (Na⁺ ∼25 mM) as measured by Vallon et al. (39) in rats with superficial glomeruli where it is possible to collect tubular fluid in the earliest part of the distal tubule. In contrast, Weinstein calibrated his model to reproduce micropuncture measurements in the first superficial segment of the distal tubule, where Na⁺ and Cl⁻ concentrations are higher (∼50 mM) owing to secretion of these ions in the earliest segment of the distal tubule (6, 34). The consequence of this is that the permeabilities used by Weinstein are somewhat different than ours, but the relationships between them (especially for the major solutes: Na⁺, K⁺, Cl⁻, and NH₄⁺) are similar in both models. Further, since we employ lower luminal Na⁺ concentrations in the cTAL than Weinstein and Krahn (44), the larger transepithelial Na⁺ gradients in our model results in lower Na⁺ transport efficiency in this segment.

Third, although the acid-base handling is similar in both works, Weinstein tracks the time evolution of CO₂ whereas we treat it as a parameter and assume a fixed concentration and immediate gas equilibration. Further, they include a basolateral Na⁺-HCO₃⁻ cotransporter, which we do not.

Despite these differences between Weinstein's model and ours, the steady-state values of the concentration of major solutes are close, and important phenomena, e.g., K⁺ and NH₄⁺ cycling, are captured in both models. Specifically, if one compares Weinstein's steady-state cytosolic concentrations for the symmetrical Ringer's type solution with our results for the baseline case of the efficiency simulations for an OM cell, which have luminal and serosal solutions close to one another but not identical, we find that our cytosolic Na⁺ and K⁺ concentrations are slightly lower than those predicted by Weinstein's model and that our cytosolic Cl⁻ is about half of that reported by Weinstein. This is a consequence of lower NKCC2 fluxes in our model that arise from the workings of our CVR scheme. The same reasoning explains the lower cytosolic Cl⁻ concentrations for cells at non-OM locations along the TAL.

Finally, the goals of the two studies are different, in that we address transport efficiency along the TAL, while Weinstein addresses the impact of ammonium on the different transport pathways of TAL cell. Thus the two models are, to a certain extent, complementary.

Comparison with experimental studies.

The impact of luminal Na⁺ concentration on TAL transport efficiency has been studied in dogs by Kiil and Sejersted (22). The authors sought to determine whether the TAL reabsorbs NaCl mostly via the transcellular pathway or through the paracellular pathway driven by K⁺ cycling across the apical membrane of TAL cells. Kiil and Sejersted measured Na⁺ reabsorption, OM heat production, and r = ΔNa/ΔO₂. The r values were used to estimate energy metabolism. They also estimated the magnitude of the Na⁺ transepithelial concentration difference at which paracellular transport would reverse, assuming a 10-mV transepithelial potential. Based on this and their data, they concluded that the thermodynamic conditions for substantial paracellular Na⁺ reabsorption are not fulfilled. Our modeling results extend the analysis of Kiil and Sejersted and lead to a similar conclusion, namely that paracellular Na⁺ reabsorption, a process that would raise our Na⁺ transport efficiency measure, makes only a limited contribution to TAL tubular fluid dilution in the OM. In the cTAL segment, net paracelluar Na⁺ transport reverses and leads to Na⁺ cycling across the TAL wall.

CVR responses elicited by altering luminal NaCl concentration have been observed by Komlosi et al. (25) and are clearly reproduced by our cell model: see Fig. 7. The changes in cell volume of our TAL OM cell model are smaller compared with the predictions of Weinstein's model (44). This is expected, inasmuch our model includes a CVR mechanism that can tightly restrict changes in cell volume by reciprocally changing NKCC2 and KCC4 transporter densities. To asses the importance of CVR responses in controlling Na⁺ transport across the TAL epithelium, we used the TAL cell model to simulate short-circuit current experiments (18). The results not only stress the importance of NKCC2 as the main NaCl pathway but also emphasize that a side-effect of CVR is the regulation of Na⁺ reabsorption across the epithelium (see Fig. 8B).

The presence of significant levels of ammonium in the kidney and its role in Na⁺ transport in the TAL motivated the simulation of NH₄⁺ pulse experiments [40]. In these simulations, the cell model was able to reproduce the experimentally observed decrease in cytosolic pH when a luminal NH₄⁺ perturbation is introduced. We observed the phenomenon of NH₄⁺ cycling, that is, NH₄⁺ uptake through NKCC2 and apical K⁺ channels, which caused the cytosolic pH to decrease and which resulted in apical extrusion of NH₄⁺ via NHE3. NH₄⁺ cycling is also seen when NH₄⁺ enters via NKCC2 and NH₃ diffuses back into the lumen while H⁺ is extruded by the apical NHE3. A comparison with similar simulations by Weinstein (42) reveals that our model predicts a smaller decline in pH. The difference stems from our setting of a higher activity for NHE3 in the apical membrane. As we mentioned above, the pH drop in our model can be increased by decreasing the activity of NHE3 in the basolateral membrane.

Because of the differences in experimental conditions between the short-circuit and the ammonium perturbation experiments, both sets of simulations illustrate the difficulty of fitting the cell model to differing sets of experimental data. Indeed, simply embedding the fitted-to-data cell model into a TAL segment model will likely not yield a model with reasonable physiologic behaviors. The variability of the environment, both luminal and serosal, along the TAL makes it difficult to find a parameter set for the cell model that will yield reasonable behavior at all possible locations.

Na⁺ transport efficiency.

The notion of highly efficient Na⁺ TAL transport as a consequence electrodiffusive paracellular uptake secondary to K⁺ cycling across the apical membrane requires careful reconsideration for several reasons. First, because the function of the TAL segment is to generate dilute tubular fluid, the transepithelial [Na⁺] gradient that is created substantially reduces transport efficiency. Indeed, Na⁺ transport efficiency decreases markedly in the cTAL as the luminal Na⁺ concentration decreases. As the tubular fluid Na⁺ concentration approaches limiting static head values, where Na⁺ backleak balances transcellular Na⁺ transport, the transport efficiency approaches zero.

Second, the competition of NH₄⁺ and K⁺ for binding on the NKCC2, as well as the NH₄⁺ cycling across the apical membrane, adds a layer of complexity. On the one hand, competition for transport by NKCC2 clearly prevents the lumen from being K⁺ depleted. However, on the other hand, NH₄⁺ cycling will increase cellular Na⁺ uptake via the NHE3 transporter which in turns decrease the efficiency (see Fig. 14).

Third, although lower luminal K⁺ concentration yields a more positive transepithelial membrane potential, only cells deep in the OM exhibit a transepithelial chemical gradient small enough to permit paracellular Na⁺ reabsorption at rates approaching the rate of transcellular Na⁺ reabsorption. However, still, as Fig. 15 shows, the transport efficiency drops markedly as luminal K⁺ concentration increases.

Increasing paracellular Na⁺ permeability is not predicted to markedly improve Na⁺ transport efficiency regardless of cell location along the TAL. When junctional Na⁺ permeability is decreased, transport efficiency approaches unity, and the tighter the junction (close to zero permeability) the closer to an efficiency measure of one, as expected from the definition of efficiency (Eq. 12). Further, opening the junctions for a cortical or a distal cell will drive the efficiency toward zero because of increased backleak. In contrast, increasing the apical K⁺ permeability increases transport efficiency, albeit to values far from two.

In summary, because the TAL normally establishes a significant transepithelial Na⁺ gradient, our model predicts that transport efficiency will be well below 2 along most of the outer medullary and cTAL and even lower in the distal portion of the TAL, where the transepithelial gradient is greatest.

Limitations and extensions of the TAL cell model.

The above conclusions inevitably lead us toward exploring the Na⁺ transport efficiency with a cell-based mathematical model of the TAL segment. In such model the walls of the TAL would be represented by mathematical models of TAL cells (like the ones used here and in Refs. 43, 44). Such a model would certainly allow us to address the issue of efficiency along the TAL (as we did here), but it would also permit us to determine whether the TAL outflow is properly diluted, the paramount function of the TAL segment.

The model does not consider divalent cations such as Ca²⁺ and Mg²⁺. Although the lumen positive transepithelial potential drives substantial paracellular divalent cation reabsorption (5), it is unlikely that this would have a significant impact on Na⁺ transport efficiency. The reason is that transepithelial transport of Na⁺, Cl⁻, and apical K⁺ cycling are the main factors in the establishment of the lumen positive membrane potential (3, 15). In our model, as in real TAL cells, the transepithelial membrane potential depends not only on paracellular permeabilities but also on transport across the apical and basolateral membranes.

Inasmuch as this model assumes a well-stirred cytosol, any spatial association between the membrane carriers or between the membrane carriers and the solutes they load in the cytosolic compartment cannot be captured in this model. To our knowledge it has not been shown experimentally that any such spatial association exists. However, if there are, in particular for NKCC2 and K⁺ channels, then the K⁺ and NH₄⁺ cycling across the apical membrane might be enhanced, which would lead to a stronger electrical gradient that would drive more Na⁺ paracellularly, thus leading to more efficient Na⁺ transport. In terms of the model presented in this work such spatial association could be implemented by defining a subcompartment within the cytosol.

Additionally, given the involvement of NHE3 in Na⁺ transport across the epithelium, a natural extension of our model would be to include representation of the regulation of the NHE3 transport by cell volume (8). Another extension for this work stems from the model we used (29) to represent the Na⁺-K⁺ pump. This model does not depend explicitly on ATP, but we could include a submodel of ATP production and usage within the cell and link this to both the phosphorylation of the NKCC2 and KCC4 transporters and to Na⁺-K⁺-ATPase activity. Although this would add more layers of complexity, it might result in an even tighter regulation of TAL cell transport and provide the basis of a model of the outer medulla that explicitly represents vascular O₂ delivery to the TAL cells. Further, a model that includes key aspects of cellular metabolism would make it possible to more completely analyze the overall efficiency of TAL cells. Here, we have only focused on Na⁺ transport efficiency using a simple efficiency index. Although other solutes are transported across the TAL secondary to the reabsorption of Na⁺, it is unlikely that inclusion of all transported solutes would substantially alter the conclusions we have reached because the amount of Na⁺ and Cl⁻ reabsorption by the TAL dwarfs the amounts of other transported solutes.

APPENDIX A: CARRIER MEDIATED TRANSPORT

Na⁺-K⁺-ATPase.

The formulation for the Na⁺-K⁺-ATPase that we used in this work was that of Luo and Rudy (29), which includes dependence on the basolateral membrane potential. Similar to Weinstein (44), we also added a pathway for NH₄⁺ through the pump as summarized below:

1) Use the Na⁺-K⁺ pump (29) to compute the total current through the pump I.

2) Compute the Na⁺ and the combined K⁺ and NH₄⁺ fluxes using the stoichiometry 3:2; hence, J_Na⁺ = 3I/F and J_K⁺ + J_Na⁺ = −2I/F, is the combined ammonium and potassium flux.

3) Use the ratio of equilibrium constants r = K_K⁺/K_NH4⁺ = 0.2 (from Ref. 44) and set r = J_K⁺/J_NH4⁺. Therefore the K⁺ and NH₄⁺ fluxes are given by,

$$J_{{\text{K}}^{+}}=\frac{J_{{\text{K}}^{+}-{\text{NH}}_4^{+}}}{(1+r)}\text{ and }{J}_{{\text{NH}}_{\text{4}}^{\text{+}}}=J_{{\text{K}}^{+}-{\text{NH}}_4^{+}}-J_{{\text{K}}^{+}}.$$

APPENDIX B: CVR FUNCTIONS

The CVR functions map cell volume (V) to the total activity (or enzyme or transporter density) of the NKCC and KCC transporters (E_T^NKCC and E_T^KCC). The functions are based on observations by Kahle (20) and are depicted in Fig. 6 of the present study, and they are denoted here as g_NKCC and g_KCC. The equations are:

$$g_{NKCC}(V)=\{\begin{array}{cc}{E}_{\max }^{NKCC},\hfill & \text{if }V\le V_{set}-{\delta }_1^{NKCC}\hfill \\ m_{NKCC}V+b_{NKCC},\hfill & V_{set}-{\delta }_1^{NKCC}<V<V_{set}+{\delta }_2^{NKCC}\hfill \\ E_{\min }^{KCC},\hfill & V_{set}+{\delta }_2^{NKCC}\le V\hfill \end{array}$$ (13)

$$g_{KCC}(V)=\{\begin{array}{cc}{E}_{\min }^{KCC},\hfill & \text{if}V\le V_{set}-{\delta }_1^{KCC}\hfill \\ m_{KCC}V+b_{NKCC},\hfill & V_{set}-{\delta }_1^{KCC}<V<V_{set}+{\delta }_2^{KCC},\hfill \\ E_{\max }^{KCC},\hfill & V_{set}+{\delta }_2^{KCC}\le V\hfill \end{array}$$ (14)

where

$$\begin{gathered} m_{NKCC}=\frac{E_{\min }^{NKCC}-E_{\max }^{NKCC}}{{\delta }_2^{NKCC}+{\delta }_1^{NKCC}}, \\ {b}_{NKCC}=E_{\min }^{NKCC}-m_{NKCC}(V_{\text{set}}+{\delta }_2^{NKCC}), \end{gathered}$$

$$\begin{gathered} m_{KCC}=\frac{E_{\max }^{KCC}-E_{\min }^{KCC}}{{\delta }_1^{KCC}+{\delta }_2^{KCC}},\text{and} \\ {b}_{KCC}=E_{\min }^{KCC}-m_{KCC}(V_{\text{set}}+{\delta }_1^{KCC}). \end{gathered}$$

and the upper and lower bounds on NKCC and KCC activity are E_max^NKCC, E_min^NKCC, E_max^KCC, and E_min^KCC, respectively. These values are given in Fig. 6B; V_set is the volume set point, whose value is the steady-state cell volume; and δ₁ and δ₂ determine the slope of the transition between the minimum and maximum transporter activities.

GRANTS

This research was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-42091 and DK-89066 (to H. E. Layton and A. T. Layton, respectively), National Science Foundation Grant DMS-0340654 (to A.T. Layton), and National Institute of General Medical Sciences Grant SC1-GM-084744 (to M. Marcano).

DISCLOSURES

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