A mathematical model of rat ascending Henle limb. I. Cotransporter function

Abstract

Kinetic models of Na⁺-K⁺-2Cl⁻ costransporter (NKCC2) and K⁺-Cl⁻ cotransporter (KCC4), two of the key cotransporters of the Henle limb, are fashioned with inclusion of terms representing binding and transport of NH₄⁺. The models are simplified using assumptions of equilibrium ion binding, binding symmetry, and identity of Cl⁻ binding sites. Model parameters are selected to be consistent with flux data from expression of these transporters in oocytes, specifically inwardly directed coupled transport of rubidium. In the analysis of these models, it is found that despite the simplifying assumptions to reduce the number of model parameters, neither model is uniquely determined by the data. For NKCC or KCC there are two- or three-parameter families of “optimal” solutions. As a consequence, one may specify several carrier translocation rates and/or ion affinities before fitting the remaining coefficients to the data, with no loss of fidelity in simulating the experiments. Model calculations suggest that with respect to NKCC2 near its operating point, the curve of ion flux as a function of cell Cl⁻ is steep, and with respect to KCC4, its curve of ion flux as a function of peritubular K⁺ is also steep. The implication is that the kinetics are suitable for these two transporters in series to act as a sensor for peritubular K⁺, to modulate AHL Na⁺ reabsorption, with cytosolic Cl⁻ as the intermediate variable. The models also reveal the potential for luminal NH₄⁺ to be a potent catalyst for NKCC2 Na⁺ reabsorption, provided suitable exit mechanisms for NH₄⁺ (from cell-to-lumen) are operative. It is found that KCC4 is likely to augment the secretory NH₄⁺ flux, with peritubular NH₄⁺ uptake driven by the cell-to-blood K⁺ gradient.

the principal function of the ascending Henle limb (AHL) is NaCl reabsorption, and the components of this process have been well defined for some time (12). On the luminal membrane, uptake of one Na⁺ and two Cl⁻ is mediated by the NKCC2 transporter. In this process, K⁺ is catalytic in the sense that it participates in the reabsorptive flux, and is recycled back to the lumen via a channel. At the peritubular membrane, Na⁺ reabsorption is effected by the Na⁺-K⁺-ATPase, which again is facilitated by K⁺ recycling. Peritubular K⁺ exit can proceed as a channel current, but the conductance is insufficient to account for the magnitude of the flux, and K⁺-Cl⁻ cotransport must mediate a substantial component. Similarly, estimates of peritubular membrane Cl⁻ conductance identify the necessity for electroneutral K⁺-Cl⁻ cotransport for a proper bookkeeping of Cl⁻ reabsorption. Thus it has been recognized that the series configuration of NKCC and KCC in AHL confers a natural role for cytosolic Cl⁻ concentration as a mediator of peritubular-to-luminal membrane cross talk (9). One complication of the configuration of AHL transporters is the linkage to acid-base metabolism. All of these transporters, NKCC2, the Na⁺-K⁺-ATPase, K⁺ channels, and the peritubular KCC can transport NH₄⁺ in place of K⁺ (23). In addition, the luminal membrane Na⁺/H⁺ (or Na⁺/NH₄⁺) exchanger NHE3 provides another link between acid-base metabolism and Na⁺ reabsorption.

These considerations of AHL as a series membrane system quickly become complex and can be facilitated by mathematical models of the key transporters. Considerable information from a variety of preparations has accrued over decades on the kinetics of NKCC, and to a lesser extent on KCC function (8, 19). The ability to express specific transporters in oocytes has permitted more recent investigations to examine the function of what are likely to be the renal isoforms of these transporters. Despite this wealth of data, there have been relatively few full kinetic models of NKCC (and none of KCC) that provide a complete set of binding and translocation parameters, which could be used to construct a model AHL. Benjamin and Johnson (3) were first to provide a full set of kinetic coefficients for NKCC from several sources. More recently, Marcano et al. (17) have used NKCC transport data from AHL isoforms to formulate a more kidney-specific model transporter, and they focused their attention on numerical methods that would search for an optimal parameter set. In the present work, the data from Plata et al. (20) are reconsidered for an NKCC model, and prior data from the same group (18) are used to generate a KCC model. In the analysis of these models, it will be shown that neither model is uniquely determined by the data, so that for NKCC or KCC there are two- or three-parameter families of “optimal” solutions. Representative models of these transporters will be extended with terms for NH₄⁺, to illustrate possibilities for interplay of K⁺ and NH₄⁺ fluxes, which have not received consideration in prior work.

NKCC COTRANSPORT

Figure 1 is a scheme for the NKCC2 cotransporter, with an empty carrier denoted by X′ or X″ according to its availability on the external (luminal) or internal (cytosolic) face of the cell membrane. Ion binding is assumed to be ordered, with first-on-first-off symmetry, as has been the assumption for the two prior NKCC models (3, 17). What is added with the present scheme is the possibility for NH₄⁺ binding in lieu of K⁺, so that a fully loaded carrier may transport either NaKCl₂ or NaNH₄Cl₂. As in previous models of this and other cotransporters, ion binding is assumed rapid relative to translocation, so that equilibrium concentrations are assumed for each of the bound transporter species. Where it has been examined quantitatively, this approximation has been validated (5). Another simplifying assumption is equality of external and internal ion binding affinities, but this is made in view of the fact that the experimental data are not sufficient to identify asymmetries. Of note, when experimental data were available to try to distinguish external and internal binding kinetics for the Cl⁻/HCO₃⁻ exchanger, AE1, the data were compatible with symmetric affinities (25). Finally, it will also be assumed here that the two Cl⁻ binding sites have equal affinity. When this assumption was relaxed by Marcano et al. (17), it gave little improvement to the fit of experimental data.

Fig. 1.

Reaction scheme for the Na⁺-K⁺-2Cl⁻ costransporter (NKCC2), with empty carrier denoted by X′ or X″ according to its availability on the external (luminal) or internal (cytosolic) face of the cell membrane. Ion binding is assumed to be ordered, with first-on-first-off symmetry. NH₄⁺ may bind in lieu of K⁺, so that fully loaded carrier may transport either NaKCl₂ or NaNH₄Cl₂. Ion binding is rapid relative to translocation, so that equilibrium concentrations are assumed for each of the bound transporter species.

Denote by x, x_n, x_nc, x_nkc, and x_nkcc the concentrations of X, X-Na, X-NaCl, X-NaKCl, and X-NaKCl₂ on each membrane face, with the appropriate (′) or (″) designation. Similarly, x_nmc and x_nmcc represent the concentrations of X-NaNH₄Cl and X-NaNH₄Cl₂. Transported ion concentrations are n′, c′, k′, and m′ (n″, c″, k″, and m″) for luminal (and cytosolic) Na⁺, Cl⁻, K⁺, and NH₄⁺. Set K_n, K_c, K_k, and K_m as the equilibrium concentrations for Na⁺, Cl⁻, K⁺, and NH₄⁺ binding to the carrier [and with the symmetry assumption, there is no (′) or (″) designation]. Then one can define the normalized concentrations

$$\begin{array}{cccc}\alpha \prime =\frac{\mathrm{n}\prime }{K_{\mathrm{n}}}\hfill & \beta \prime =\frac{\mathrm{k}\prime }{K_{\mathrm{k}}}\hfill & \gamma \prime =\frac{\mathrm{c}\prime }{K_{\mathrm{c}}}\hfill & \mu \prime =\frac{\mathrm{m}\prime }{K_{\mathrm{m}}}\hfill \\ \alpha ″=\frac{\mathrm{n}″}{K_{\mathrm{n}}}\hfill & \beta ″=\frac{\mathrm{k}″}{K_{\mathrm{k}}}\hfill & \gamma ″=\frac{\mathrm{c}″}{K_{\mathrm{c}}}\hfill & \mu ″=\frac{\mathrm{m}″}{K_{\mathrm{m}}}\hfill \end{array}$$ (1)

With these definitions, the equilibrium assumption yields the concentrations of the bound carrier species:

$$\begin{array}{cccc}{x}_{\mathrm{n}}^{\prime }=\mathrm{\alpha }\prime x\prime & x_{\mathrm{\text{nc}}}^{\prime }=\mathrm{\alpha }\prime \mathrm{\gamma }\prime x\prime & x_{\mathrm{\text{nkc}}}^{\prime }=\mathrm{\alpha }\prime \mathrm{\beta }\prime \mathrm{\gamma }\prime x\prime & x_{\mathrm{\text{nkcc}}}^{\prime }=\mathrm{\alpha }\prime \mathrm{\beta }\prime {(\mathrm{\gamma }\prime )}^{2}x\prime \\ & x_{\mathrm{\text{nmc}}}^{\prime }=\mathrm{\alpha }\prime \mathrm{\mu }\prime \mathrm{\gamma }\prime x\prime & x_{\mathrm{\text{nmcc}}}^{\prime }=\alpha \prime \mu \prime {(\gamma \prime )}^{2}x\prime & \\ x_{\mathrm{c}}^{″}=\gamma ″x″& x_{\mathrm{\text{kc}}}^{″}=\beta ″\gamma ″x″& x_{\mathrm{\text{kcc}}}^{″}=\beta ″{(\gamma ″)}^{2}x″\hfill & x_{\mathrm{\text{nkcc}}}^{″}=\alpha ″\beta ″{(\gamma ″)}^{2}x″\hfill \\ & x_{\mathrm{\text{mc}}}^{″}=\mu ″\gamma ″x″& x_{\mathrm{\text{mcc}}}^{″}=\mu ″{(\gamma ″)}^{2}x″& x_{\mathrm{\text{nnmc}}}^{″}=\alpha ″\mu ″{(\gamma ″)}^{2}x″\end{array}$$ (2)

The coefficients α, β, γ, and μ, depend only on the known solute concentrations and affinities, and the binding equilibrium assumption allows the species of bound carrier to be expressed in terms of the free carrier on either side of the membrane. Thus the conservation of total carrier, x_T, is represented

$$[1+\alpha \prime +\alpha \prime \gamma \prime +\alpha \prime \beta \prime \gamma \prime +\alpha \prime \beta \prime {(\gamma \prime )}^2+\alpha \prime \mu \prime \gamma \prime +\alpha \prime \mu \prime {(\gamma \prime )}^2]\cdot x\prime +[1+\gamma ″+\beta ″\gamma ″+\beta ″{(\gamma ″)}^2+\alpha ″\beta ″{(\gamma ″)}^2+\mu ″\gamma ″+\mu ″{(\gamma ″)}^2+\alpha ″\mu ″{(\gamma ″)}^2]\cdot x″=x_{\mathrm{T}}$$ (3)

or more compactly,

$$\sigma \prime \cdot x\prime +\sigma ″\cdot x″=x_{\mathrm{T}}$$ (3a)

$$\begin{array}{c}{\sigma }^{\prime }=1+\alpha \prime +\alpha \prime \gamma \prime +\alpha \prime \beta \prime \gamma \prime +\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2+\alpha \prime \mu \prime \gamma \prime \\ +\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2\\ \sigma ″=1+\gamma ″+\beta ″\gamma ″+\beta ″{\left(\gamma ″\right)}^2+\alpha ″\beta ″{\left(\gamma ″\right)}^2+\mu ″\gamma ″\\ +\mu ″{\left(\gamma ″\right)}^2+\alpha ″\mu ″{\left(\gamma ″\right)}^2\end{array}$$ (3b)

In the scheme in Fig. 1, only empty or fully loaded carrier can traverse the membrane. Translocation of empty carrier is represented by rate coefficients, $P_0^{\prime }$ and $P_0^{″}$, for influx and efflux, and for loaded carrier by $P_{\mathrm{\text{nkcc}}}^{{}^{\prime }}$ and $P_{\mathrm{\text{nkcc}}}^{″}$ for X-NaKCl₂, and $P_{\mathrm{\text{nmcc}}}^{\prime }$ and $P_{\mathrm{\text{nmcc}}}^{″}$ for X-NaNH₄Cl₂. By virtue of model symmetry (for equilibrium binding concentrations),

$$P_0^{\prime }\cdot P_{\mathrm{\text{nkcc}}}^{″}=P_0^{″}\cdot P_{\mathrm{\text{nkcc}}}^{\prime },\mathrm{\text{and}}{P}_0^{\prime }\cdot P_{\mathrm{\text{nmcc}}}^{″}=P_0^{″}\cdot P_{\mathrm{\text{nmcc}}}^{\prime }$$ (4)

The second model equation states that at stationary state, there is no net flux of carrier (unloaded plus loaded):

$$\begin{array}{c}[P_0^{\prime }+P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {(\gamma \prime )}^2+P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {(\gamma \prime )}^2]\cdot x\prime \\ -[P_0^{″}+P_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{(\gamma ″)}^2+P_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{(\gamma ″)}^2]\cdot x″=0\end{array}$$ (5)

or with simplified notation,

$$\rho \prime \cdot x\prime -\rho ″\cdot x″=0$$ (5a)

$$\begin{array}{c}\rho \prime =P_0^{\prime }+P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2+P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2\\ \rho ″=P_0^{″}+P_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2+P_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{\left(\gamma ″\right)}^2\end{array}$$ (5b)

When these Eqs. 3 and 5 are solved for x′ and x″, one obtains

$$\begin{array}{c}x\prime =\frac{x_{\mathrm{T}}}{\sum }\rho ″=\frac{x_{\mathrm{T}}}{\sum }[P_0^{″}+P_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{(\gamma ″)}^2+P_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{(\gamma ″)}^2]\\ x″=\frac{x_{\mathrm{T}}}{\sum }\rho \prime =\frac{x_{\mathrm{T}}}{\sum }[P_0^{\prime }+P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {(\gamma \prime )}^2+P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {(\gamma \prime )}^2]\end{array}$$ (6)

in which

$$\sum =\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime$$ (6a)

Thus the unidirectional Na⁺ influx, $J_{\mathrm{\text{Na}}}^{\prime }$, and efflux, $J_{\mathrm{\text{Na}}}^{″}$, are

$$\begin{array}{c}{J}_{\mathrm{N}\mathrm{a}}^{\prime }=\left[P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2+P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2\right]\cdot x\prime \\ J_{\mathrm{N}\mathrm{a}}^{″}=\left[P_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2+P_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{\left(\gamma ″\right)}^2\right]\cdot x″\end{array}$$ (7)

in which the first and second terms correspond to fluxes with K⁺ and NH₄⁺, respectively. Substituting for x′ and x″, and taking the difference, the net Na⁺ flux into the cell is J_Na = $J_{\mathrm{\text{Na}}}^{\prime }$ − $J_{\mathrm{\text{Na}}}^{″}$,

$$\begin{array}{c}{J}_{\mathrm{N}\mathrm{a}}=\frac{x_{\mathrm{T}}}{\sum }\cdot \{\left[P_0^{″}{P}_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2-P_0^{\prime }{P}_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2\right]\\ +\left[P_0^{″}{P}_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2-P_0^{\prime }{P}_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{\left(\gamma ″\right)}^2\right[\}\end{array}$$ (8)

In Eq. 8 there are two terms for net Na⁺ flux, one dependent upon K⁺ and one on NH₄⁺, but these terms cannot be identified as the net fluxes of K⁺ and NH₄⁺ (except when the other ion is absent from the system). If this analysis is repeated to find the net K⁺ flux across NKCC, the expression is

$$\begin{array}{c}{J}_{\mathrm{K}}=\frac{x_{\mathrm{T}}}{\sum }\cdot \{\left[P_0^{″}{P}_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2-P_0^{\prime }{P}_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2\right]\\ +[P_{\mathrm{\text{nmcc}}}^{″}\alpha ″\mu ″{\left(\gamma ″\right)}^2{P}_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\\ -P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2{P}_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2]\}\end{array}$$ (9)

The last two terms in this expression represent the capacity of the scheme in Fig. 1 to have K⁺ uptake driven by an NH₄⁺ gradient (cell to lumen), or just facilitated by the presence of NH₄⁺ in the system, allowing the transporter to shuttle more rapidly.

Much of the experimental data relating to NKCC function has been obtained as rubidium flux in ammonia-free systems. In this situation, the notation simplifies

$$\begin{array}{c}\sigma \prime =1+\alpha \prime +\alpha \prime \gamma \prime +\alpha \prime \beta \prime \gamma \prime +\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\\ \sigma ″=1+\gamma ″+\beta ″\gamma ″+\beta ″{\left(\gamma ″\right)}^2+\alpha ″\beta ″{\left(\gamma ″\right)}^2\end{array}$$ (10a)

$$\begin{array}{c}\rho \prime ={\text{P}}_0^{\prime }+P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\\ \rho ″={\text{P}}_0^{″}+P_{\mathrm{\text{nkcc}}}^{″}\alpha ″\beta ″{\left(\gamma ″\right)}^2\end{array}$$ (10b)

and the unidirectional uptake of K⁺ and Na⁺ are identical (Eq. 7)

$$J_{\mathrm{K}}^{\prime }=P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {(\gamma \prime )}^{2}x\prime =\frac{x_{\mathrm{T}}{P}_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {(\gamma \prime )}^2\cdot \rho ″}{\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime }$$ (11)

For large luminal ion concentrations, the maximal unidirectional flux, $J_{\mathrm{\text{K}},\mathrm{\text{max}}}^{\prime }$, is

$$J_{\mathrm{K},\mathrm{\text{max}}}^{\prime }=\frac{x_{\mathrm{T}}{P}_{\mathrm{\text{nkcc}}}^{\prime }\cdot \rho ″}{\rho ″+\sigma ″\cdot P_{\mathrm{\text{nkcc}}}^{\prime }}$$ (12)

so that the relative (or normalized) unidirectional flux is

$$\begin{array}{c}\frac{J_{\mathrm{K}}^{\prime }}{J_{\mathrm{K},\max }^{\prime }}=\frac{\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\cdot \left[\rho ″+\sigma ″+P_{\mathrm{\text{nkcc}}}^{\prime }\right]}{\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime }\\ =\frac{\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\cdot \left[\rho ″+\sigma ″+P_{\mathrm{\text{nkcc}}}^{\prime }\right]}{\left[1+\alpha \prime +\alpha \prime \gamma \prime +\alpha \prime \beta \prime \gamma \prime +\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2\right]\cdot \rho ″+\sigma ″\cdot \left[P_0^{\prime }+P_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta {\left(\gamma \prime \right)}^2\right]}\\ =\left[\rho ″+\sigma ″+P_{\mathrm{\text{nkcc}}}^{\prime }\right]\cdot {\left[\frac{\left[\rho ″+\sigma ″+P_0^{\prime }\right]}{\alpha \prime \beta \prime {\left(\gamma \prime \right)}^2}+\frac{\rho ″}{\beta \prime {\left(\gamma \prime \right)}^2}+\frac{\rho ″}{\beta \prime \gamma \prime }+\frac{\rho ″}{\gamma \prime }+\left(\rho ″+\sigma ″+P_{\mathrm{\text{nkcc}}}^{\prime }\right)\right]}^{-1}\end{array}$$ (13)

The importance of Eq. 13 is that expansion of the denominator allows grouping into five terms: a constant term, and factors of γ′, β′γ′, β′(γ′)², and α′β′(γ′)². This implies that in any experimental investigation in which only the external concentrations, α′, β′, and γ′, are varied, even perfect measurement of unidirectional fluxes can yield at most four independent model parameters. For the model under consideration here, even with the assumptions of internal-external symmetry, and identical Cl⁻ binding sites, (setting aside transporter number, x_T) there are six model parameters, namely three affinities and three translocation rates. Specifically, even with the symmetry assumptions, variation of external concentrations alone cannot fully determine the model parameters.¹

The study of Plata et al. (20) was a systematic exploration of transport kinetics of the three renal isoforms of NKCC2 expressed in oocytes. In those experiments, oocytes underwent a period of incubation in a medium that should have depleted some internal electrolytes, although cytosolic concentrations were not measured. The basic experiment was individual variation of external Na⁺, K⁺, or Cl⁻ (with impermeant ion replacement), and determination of Rb⁺ uptake (in ammonia-free medium). These data were used by Marcano et al. (17) to determine coefficients for kinetic models of the B, A, and F isoforms of NKCC. In the present work, the data of Plata et al. (20) for each isoform were first tabulated as J_K/J_K,max (with J_K,max the highest measured flux for the ion under consideration).² Internal solute concentrations were assumed Na⁺ = 20, K⁺ = 80, and Cl⁻ = 10 mM, and Eq. 13 was used to define a (difference squared) error function for all of the data points for an isoform. A Levenberg-Marquardt search was done to minimize this error, to yield an optimal set of kinetic parameters. From the considerations above, it is impossible to solve for all kinetic coefficients, so the program of calculations assumed a translocation rate for fully loaded carrier ( $P_{\mathrm{\text{nkcc}}}^{\prime }$) and then determined optimal values for the remaining coefficients (K_Na, K_K, K_Cl, and $P_0^{\prime }$, $P_0^{″}$ and $P_{\mathrm{\text{nkcc}}}^{″}$), constrained by the energy conservation Eq. 4. Table 1 shows the results of this program over several orders of magnitude range in the choice of $P_{\mathrm{\text{nkcc}}}^{\prime }$: for the F-isoform, this translocation rate was varied from 10³ to 10⁸ s⁻¹. For each isoform, there are several sets of coefficients in Table 1, and the error function of each of these sets is identical, as would be expected with a single set of composite parameters in Eq. 13.³

Table 1.

Optimal NKCC2 parameter sets for selected choices of forward translocation rates of loaded carrier

Pnkcc′	Pnkcc′/Pnkcc″	P0″/Pnkcc′	Kn	Kc	Kk
B-isoform
1.00 × 103	0.688	37.7	0.354	8.16 × 10−5	4.46
1.00 × 104	1.03	25.2	0.275	8.16 × 10−5	5.58
1.00 × 105	2.17	12.1	0.169	8.16 × 10−5	8.66
1.00 × 106	0.913	32.3	0.282	8.16 × 10−5	4.87
1.00 × 107	11.2	3.31	0.0457	8.16 × 10−5	21.7
A-isoform
1.00 × 103	0.687	37.8	0.371	8.83 × 10−5	6.65
1.00 × 104	3.44	7.54	0.119	8.83 × 10−5	18.7
1.00 × 105	17.2	1.50	0.0311	8.83 × 10−5	69.1
F-isoform
1.00 × 103	0.951	38.0	0.329	7.14 × 10−3	9.14 ×10−3
1.00 × 104	9.11	3.93	0.0589	0.0131	9.15 ×10−3
1.00 × 105	77.0	0.404	0.0186	0.0231	9.15 ×10−3
1.00 × 106	735.0	0.0406	0.0161	0.0250	9.16 ×10−3
1.00 × 107	735.0	4.06 × 10−3	0.0161	0.0249	9.16 ×10−3
1.00 × 108	73500	4.06 × 10−4	0.0161	0.0249	9.16 ×10−3

NKCC2, Na⁺-K⁺-2Cl⁻ cotransporter; Pnkcc′, forward translocation rate of loaded carrier (s⁻¹); Pnkcc″, backward translocation rate of loaded carrier (s⁻¹); P0″, backward translocation rate of empty carrier (s⁻¹); K_n, K_c, and K_k, transporter binding affinity (M) for Na⁺, Cl⁻, and K⁺, respectively.

In medullary AHL, peritubular K⁺ concentration has been identified as a critical modulator of NKCC activity, via changes in cellular composition (22). This directs attention to F-isoform kinetics, and Fig. 2 displays the solution of the optimization problem for this isoform over a range of values of $P_{\mathrm{\text{nkcc}}}^{\prime }$ (plotted as a function of the log₁₀ $P_{\mathrm{\text{nkcc}}}^{\prime }$), from 10³ to 10⁶ s⁻¹. In the top panes, the logarithms of ion binding affinities (in mM) are shown. The bottom left pane shows the ratio of entry to exit of loaded carrier ( $P_{\mathrm{\text{nkcc}}}^{\prime }$/ $P_{\mathrm{\text{nkcc}}}^{″}$), and in the middle is the ratio of entry of free carrier to loaded carrier ( $P_0^{\prime }$/ $P_{\mathrm{\text{nkcc}}}^{\prime }$). On the bottom right is the residual error for the model prediction and data points from Plata et al. (20): labels Na, K, and Cl denote the driving ion in the influx experiments; “total” is the sum of the residuals. Over the five orders of magnitude for $P_{\mathrm{\text{nkcc}}}^{\prime }$ shown in Table 1 (with 3 orders of magnitude shown in this figure), the total residual error is essentially constant, varying from 0.215 to 0.216; for the individual driving ions, however, residual errors vary from 0.091 to 0.087, from 0.069 to 0.073, and from 0.054 to 0.056 for Na⁺, K⁺, and Cl⁻, respectively. Overall, this figure suggests that for the F-isoform, one is forced to a K⁺ binding affinity of 9 mM, but could have virtually any ratio of translocation rates. Unfortunately, this conclusion regarding K⁺ biding is specific to the F-isoform, and does not extend to the other isoforms. It is also difficult to identify a criterion by which to select a preferred set of model coefficients, although it should be safe to assume that the loaded translocation rate will be somewhere between the turnover rate for an ion channel (10⁶ s⁻¹) and the Na-K-ATPase (10² s⁻¹). Thus for each isoform, a loaded translocation rate of 10⁴ s⁻¹ has been assumed, and optimal kinetic coefficients corresponding to this choice are summarized in Table 2a.

Fig. 2.

Solution of the optimized parameters for the NKCC F-isoform, solved against the data of Plata et al. (20) over a range of values of Pnkcc′ (s⁻¹), plotted as a function of the log₁₀ Pnkcc′. In the top panes, the logarithms of ion binding affinities (in mM) are shown. The bottom left pane shows the ratio of entry to exit of loaded carrier (Pnkcc′/Pnkcc″), and in the middle is the ratio of entry of free carrier to loaded carrier (P0′/Pnkcc′). On the bottom right is residual error for the model prediction and data points: labels Na, K, and Cl denote the driving ion in influx experiments; “total” is their sum.

Table 2a.

Selected NKCC2 parameter sets

Isoform	B	A	F
Kn	0.2750	0.1188	0.05893
Kc	0.8157 × 10−4	0.8834 × 10−4	0.01312
Kk	5.577	18.71	0.009149
Km	5.577	18.71	0.009149
Pnkcc′	10,000	10,000	10,000
Pnkcc″	9,695	2,904	1,098
P0″	251,700	75,350	39,280
P0′	259,600	259,400	357,800
Pnmcc′	2,000	2,000	2,000
Pnmcc″	1,939	580.8	219.6

P0′, forward translocation rate of empty carrier (s⁻¹); Pnmcc′, Pnmcc″, forward, backward translocation rate of Na⁺-NH₄⁺-2Cl⁻-loaded carrier (s⁻¹); K_n, K_c, K_k, K_m, transporter binding affinity (M) for Na⁺, Cl⁻, K⁺, and NH₄⁺.

Using the parameters of Table 2a, Fig. 3 displays simulations of the nine uptake experiments of Plata et al. (20). For each column of panes, the abscissas are the concentration of the variable external ion. The ordinates are unidirectional influxes relative to the influx at 100, 20, and 100 mM for Na⁺, K⁺, and Cl⁻. Continuous curves are model-derived, and the data points are from Plata et al. (20). The top, middle, and bottom rows correspond to the B-, A-, and F-isoforms. Of note, the internal solute concentrations used in determining each of these parameter sets was 20, 80, and 10 mM, for Na⁺, K⁺, and Cl⁻. Table 2b displays calculations of optimal NKCC parameters when these internal ion concentrations are perturbed by 50% above and below baseline; what is shown are the parameter values relative to their baseline values. Notably, translocation rates for the NKCC isoforms are insensitive to internal ion concentrations; ion affinities (Na⁺ and K⁺ for the A- and B-isoforms, or Na⁺ and Cl⁻ for the F-isoform) adjust to fit the assumed internal concentrations. For each isoform, the aggregate error between model prediction and data is identical, whether computed using baseline parameters or using those derived from perturbed internal concentrations.

Fig. 3.

Simultaneous fit for the affinities and translocation rates corresponding to Pnkcc′ = 10⁴ s⁻¹. Using the parameters of Table 2a, the 9 uptake experiments of Plata et al. (20) have been simulated. For all experiments, internal ion concentrations are 20, 80, and 10 mM, and the baseline external concentrations are 96, 10, and 96 mM for Na⁺, K⁺, and Cl⁻. For each column of panes, abcissas are the concentration of the variable external ion. The ordinates are unidirectional influxes relative to influx at 100, 20, and 100 mM for Na⁺, K⁺, and Cl⁻. Continuous curves are model-derived, and the data points are from Plata et al. (20). The top, middle, and bottom rows correspond to the B-, A-, and F-isoforms.

Table 2b.

Optimal NKCC2 parameter sets: sensitivity to internal ion concentrations

Internal concentrations, mM
Na+	10	30	20	20	20	20
K+	80	80	40	120	80	80
Cl−	10	10	10	10	5	15
B-isoform
Kn	0.889	1.100	1.139	0.936	1.668	0.835
Kc	1.000	1.000	1.000	1.000	1.000	1.000
Kk	1.113	0.919	0.891	1.062	0.639	1.179
Pnkcc′	1.000	1.000	1.000	1.000	1.000	1.000
Pnkcc″	1.000	1.000	1.000	1.000	1.000	1.000
P0″	1.000	1.000	1.000	1.000	1.000	1.000
P0′	1.000	1.000	1.000	1.000	1.000	1.000
A-isoform
Kn	0.916	1.077	1.051	0.970	1.721	0.813
Kc	1.000	1.000	1.000	1.000	1.000	1.000
Kk	1.087	0.932	0.955	1.030	0.602	1.219
Pnkcc′	1.000	1.000	1.000	1.000	1.000	1.000
Pnkcc″	1.000	1.000	1.000	1.000	0.999	1.000
P0″	1.000	1.000	1.000	1.000	1.000	1.000
P0′	1.000	1.000	1.000	1.000	1.001	1.000
F-isoform
Kn	1.037	0.995	1.509	0.795	1.906	0.706
Kc	0.983	0.999	0.846	1.104	0.773	1.164
Kk	1.000	1.000	1.000	1.000	1.000	1.000
Pnkcc′	1.000	1.000	1.000	1.000	1.000	1.000
Pnkcc″	1.001	1.000	0.990	1.012	0.986	1.019
P0″	1.002	1.000	0.999	1.003	0.998	1.003
P0′	1.001	1.000	1.010	0.991	1.012	0.984

Values are relative to Table 2a, with internal Na⁺, K⁺, and Cl⁻ =20, 80, and 10 mM.

Ultimately, this NKKC model will serve in the context of an AHL cell model, in which Na⁺ entry is modulated by cytosolic Cl⁻ concentration, and the sensitivity of transport to trans-Cl⁻ is illustrated in Fig. 4. For these calculations, luminal conditions are fixed with Na⁺, K⁺, and Cl⁻ at 140, 5, and 105 mM, and cytosolic Na⁺ and K⁺ are set to 25 and 140 mM; cytosolic Cl⁻ is varied from 10 to 60 mM and appears as the abscissa. For each of the parameter sets for the three isoforms (Table 2a), Na⁺ transport, J_Na, is plotted relative to its value when cytosolic Cl⁻ is 10 mM. In each case, J_Na = 0 when cytosolic Cl⁻ is 47 mM, and the shapes of these curves are not substantially different. The values indicated in each pane (and indicated by “+”) are the relative Na⁺ fluxes when cytosolic Cl⁻ is 20 or 40 mM.

Fig. 4.

Impact of cytosolic Cl on NKCC fluxes. Luminal conditions are fixed with Na⁺, K⁺, and Cl⁻ at 140, 5, and 105 mM, and cytosolic Na⁺ and K⁺ are set to 25 and 140 mM; cytosolic Cl⁻ is varied from 10 to 60 mM and appears as the abcissa. For each of the parameter sets for the 3 isoforms (Table 2a), Na⁺ transport, J_Na, is plotted relative to its value when cytosolic Cl⁻ is 10 mM. In each case, J_Na = 0 when cytosolic Cl⁻ is 47 mM; values indicated in each pane (and indicated by “+”) are the relative Na⁺ fluxes when cytosolic Cl⁻ is 20 or 40 mM.

With respect to parameter selection relating to NH₄⁺ transport by NKCC2, the data base is less substantial. Kinne et al. (15) demonstrated that NH₄⁺ competitively inhibits NKCC2 K⁺ flux, and that NH₄⁺ itself sustains NKCC2 Na⁺ flux (i.e., that NH₄⁺ is transported in lieu of K⁺). In a model that assumes (inside-outside) binding symmetry, there are only two NH₄⁺ parameters that are freely chosen: K_NH4 and $P_{\mathrm{\text{nmcc}}}^{\prime }$ (with $P_{\mathrm{\text{nmcc}}}^{″}$ determined by Eq. 4). Kinne et al. prepared membrane vesicles from the medullary thick AHL of the rabbit kidney and assayed the impact of external NH₄⁺ on 0-trans-rubidium uptake. The analysis of those experiments begins with Eq. 9 for K⁺ transport, with application of the 0-trans condition:

$$J_{\mathrm{K}}=\frac{x_{\mathrm{T}}}{\sum }\cdot P_0^{″}{P}_{\mathrm{\text{nkcc}}}^{\prime }\alpha \prime \beta \prime {(\gamma \prime )}^2$$ (14a)

where

$${\displaystyle \sum =\sigma \prime \cdot P_0^{″}}+\rho \prime$$ (14b)

With reference to Eqs. 3 and 5, denote by the subscript, K, the composite variables in the absence of ammonia:

$$\begin{array}{c}\sigma \prime =\sigma {\prime }_K+\alpha \prime \mu \prime \gamma \prime +\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2\\ \rho \prime ={\rho }_K^{\prime }+P_{\mathrm{\text{nmcc}}}^{\prime }\alpha \prime \mu \prime {\left(\gamma \prime \right)}^2\end{array}$$ (15)

so that Eq. 14b may be rewritten,

$$\sum =\sum {}_{\mathrm{\text{K}}}+\left[P_0^{″}\left(1+\frac{1}{\gamma \prime }\right)+P_{\mathrm{\text{nmcc}}}^{\prime }\right]\alpha \prime \mu \prime {(\gamma \prime )}^2$$ (16)

At the K_l of ammonium, K⁺ flux is half that measured in the absence of ammonium:

$$2=\frac{J_{\mathrm{K},\max }}{J_{\mathrm{K}}}=\frac{\sum }{\sum {}_{\mathrm{\text{K}}}}=1+\frac{\sum -\sum {}_{\mathrm{\text{K}}}}{\sum {}_{\mathrm{\text{K}}}}$$ (17)

which yields the relation

$$\left[(1+\frac{1}{{\gamma }^{\prime }})+\frac{P_{\mathrm{\text{nmcc}}}^{\prime }}{P_0^{″}}\right]\mu \prime =\frac{\sum {}_{\mathrm{\text{K}}}}{P_0^{″}\alpha \prime {(\gamma \prime )}^2}$$ (18)

In Eq. 18, the expression on the right contains only non-ammonia terms and can be calculated from the values already selected in Table 2a. On the left, the term in brackets will be close to 1.0, provided that substantial Cl⁻ is present, and that translocation of the NH₄⁺-loaded carrier is no faster than translocation of the K⁺-loaded carrier. In particular, Eq. 18 provides relatively small constraint on the translocation rate, $P_{\mathrm{\text{nmcc}}}^{\prime }$, but depends principally on μ′, the ratio of the observed K_l for ammonium and its affinity for the carrier, K_NH4. Parenthetically, Eq. 18 is also obtained as the condition for half-maximal transport of ammonium, in the presence of non-zero luminal potassium.

Figure 5 is a simulation of the experiments of Kinne et al. (15), in which external bath concentrations of Na⁺, K⁺, and Cl⁻ are either 79.5, 0.5, and 80.0 mM or 79.0, 1.0, and 80 mM. Internal bath concentrations are set to zero; external NH₄⁺ is varied from 0.0 to 10.0 mM (as addition of NH₄Cl); and the reciprocal of K⁺ entry is plotted. Under these conditions, Kinne et al. reported half-maximal inhibitory ammonium concentrations of 3.0 and 5.8 mM. In the left pane, model parameters are those of the F-isoform (Table 2a), and NH₄⁺ parameters are identical to those for K⁺; when external K⁺ is 0.5 or 1.0 mM, fluxes are inhibited 50% at 3.2 or 4.0 mM ammonium. With reference to Eq. 18, the NKCC parameters of Table 2a yield a right-hand side of 0.47 or 0.55, respectively, for external K⁺ of 0.5 or 1.0 mM. Under these conditions, 1/γ′ = 0.16 and $P_{\mathrm{\text{nmcc}}}^{\prime }$/ $P_0^{″}$ = 0.26, so that μ′ is 0.33 or 0.39, and the estimated K_l for ammonium is 3.0 or 3.5 mM. The discrepancy between the numerical calculations of Fig. 5 with Eq. 18 may reflect the equation's assumption of fixed luminal Cl⁻; in the simulation there is variation of luminal NH₄Cl. Looking ahead to a model of the AHL, it will not be suitable to have identical translocation rates for K⁺ and NH₄⁺. Luminal concentrations of K⁺ and NH₄⁺ are roughly comparable, while cytosolic NH₄⁺ concentration is likely to be less than that for K⁺. Furthermore, replenishment of luminal NH₄⁺ via Na⁺/NH₄⁺ exchange by NHE3 is likely to be slower than replenishment of luminal K⁺ via ROMK. [Rates of NHE3-mediated NaHCO₃ transport by rat AHL are typically 10–20% of those reported for overall Na⁺ reabsorption, e.g., Good et al. (11).] In the right panel of Fig. 5, the translocation rate of the carrier loaded with NaNH₄Cl₂ is 20% of that for NaKCl₂. With this parameter set, 50% inhibition of K⁺ flux occurs at 3.8 or 4.7 mM NH₄⁺, corresponding to 0.5 or 1.0 mM lumen K⁺. With these parameters, the corresponding estimates of Eq. 18 are 3.5 or 4.1 mM.

Fig. 5.

Inhibition of 0-trans K flux by cis NH₄⁺ (F-Isoform). External bath has baseline concentrations of Na⁺, K⁺, and Cl⁻ of either 79.5, 0.5, and 80.0 mM or 79.0, 1.0, and 80 mM; internal bath concentrations are set to zero. External NH₄⁺ is varied from 0.0 to 10.0 mM (as addition of NH₄Cl); and the reciprocal of K⁺ entry is plotted. (Fluxes have dimension mol·s⁻¹·mol transporter⁻¹.) In the left pane, NH₄⁺ parameters are identical to those for K⁺, and fluxes are inhibited 50% at 3.2 or 4.0 mM, corresponding to external K⁺ of 0.5 or 1.0 mM. In the right pane, translocation of carrier loaded with NaNH₄Cl₂ is 20% of that for NaKCl₂; 50% inhibition of K⁺ flux occurs at 3.8 or 4.7 mM NH₄⁺, corresponding to 0.5 or 1.0 mM lumen K⁺.

The presence of NH₄⁺ transport by NKCC2 adds complexity to its function within AHL. In the calculations of Fig. 6, the parameters of the three isoforms have been extended to include NH₄⁺, with the assumptions that K_NH4 = K_K and $P_{\mathrm{\text{nmcc}}}^{\prime }$ = 0.2 × $P_{\mathrm{\text{nkcc}}}^{\prime }$. Luminal concentrations of Na⁺, K⁺, and Cl⁻ have been set at 140, 5, and 105 mM, and cytosolic concentrations at 25, 140, and 35 mM, which provide a positive inward driving force for NaKCl₂. In these calculations, ambient (luminal and cytosolic) NH₄ is increased from 0 to 10 mM, and net fluxes of Na⁺, K⁺, and NH₄⁺ via NKCC2 are plotted. For each isoform, the qualitative impact of NH₄⁺ is the same, namely, a small increase in overall Na⁺ transport, with a considerable decrease in K⁺ absorption in favor of NH₄⁺ uptake. For the F-isoform, increasing NH₄⁺ concentration shifts the NaKCl₂ flux to NaNH₄Cl₂. For both B- and A-isoforms, NaKCl₂ flux becomes secretory and large, and balanced by NaNH₄Cl₂ absorption; in essence, Na⁺ and Cl⁻ are catalytic for NH₄⁺ and K⁺ exchange via NKCC2. The kinetics underlying this shift in fluxes is displayed in Fig. 7, in which variables from the model F-isoform are displayed. In the top left are the values of $x_{\mathrm{\text{nkcc}}}^{\prime }$ and $x_{\mathrm{\text{nkcc}}}^{″}$, and on the right ${\text{x}}_{\mathrm{\text{nmcc}}}^{\prime }$ and $x_{\mathrm{\text{nmcc}}}^{″}$, namely, the concentrations of X- NaKCl₂ and X-NaNH₄Cl₂ on the external and internal faces of the transporter. With the increase in ambient NH₄⁺, concentrations of both X- ${\mathrm{\text{NaKCl}}}_2^{\prime }$ and X- ${\text{NaKCl}}_2^{″}$ decrease, but because the external K⁺ concentration is so much lower than that in the cell, the fractional decrease in external bound carrier is much greater. (Specifically, $x_{\mathrm{\text{nkcc}}}^{\prime }$ decreases from 3.0 to 2.1% of total carrier.) At the point that the ratio of internal to external carrier concentration increases above the ratio of the translocation rates of bound carrier, the net flux of NaKCl₂ shifts from uptake to exit. With respect to NH₄⁺, the increase in ambient concentration has a greater effect on the amount of bound carrier oriented to the lumen, so that net influx increases progressively. It should be noted that for all of these calculations, there is no change in the chemical potential of either NaKCl₂ or NaNH₄Cl₂ across the carrier, so that the flux events are outside the scope of any thermodynamic model of the NKCC2 transporter.⁴

Fig. 6.

Impact of ambient NH₄⁺ on NKCC fluxes. Lumen Na⁺ = 140, K⁺ = 5, Cl⁻ = 105 mM, and cytosolic Na⁺ = 25, K⁺ = 140, Cl⁻ = 35 mM. Translocation rate for NaNH₄Cl₂ is 20% that for NaKCl₂, and K_NH4 = K_K. Ambient (luminal and cytosolic) NH₄ is increased from 0 to 10 mM, and net fluxes of Na⁺, K⁺, and NH₄⁺ via NKCC2 are plotted (mol·s⁻¹·mol transporter⁻¹).

Fig. 7.

Impact of ambient NH₄⁺ on fully-loaded carrier states (F-isoform). For the calculations of Fig. 6, the top panes are the abundance of fully loaded carrier for NaKCl₂ (left) and NaNH₄Cl₂ (right), on the external (X′) and internal (X″) faces of the transporter. With the increase in ambient NH₄⁺, concentrations of both X-NaKCl2′ and X-NaKCl2″ decrease, but because the external K⁺ concentration is much lower than that in the cell, the fractional decrease in external bound carrier is much greater. At the point that the ratio of internal to external carrier concentration increases above the ratio of translocation rates of bound carrier, net flux of NKCl2 shifts from uptake to exit. For either K⁺ or NH₄⁺, influx occurs when P′x′ > P″x″.

KCC COTRANSPORT

Figure 8 is a reaction scheme for KCC function, which allows for the possibility that NH₄ can be transported in lieu of K⁺. The ordered binding and unbinding parallels that for NKCC, although it is acknowledged that this is speculative. The analysis of this scheme follows that for NKCC. The notation continues the convention that inward fluxes are positive, so that for a KCC of interest on the peritubular membrane of AHL, x′ and x″ represent concentrations of free carrier facing blood and cytosol, respectively. The assumptions of equilibrium ion binding and binding symmetry are retained, so that Eqs. 1 and 2 apply for calculating the densities of bound carrier. Thus, following Eq. 3, the equations for carrier conservation are

$$\sigma \prime \cdot x\prime +\sigma ″\cdot x″=x_{\mathrm{T}}$$ (19a)

$$\begin{array}{l}\sigma \prime =1+\beta \prime +\beta \prime \gamma \prime +\mu \prime +\mu \prime \gamma \prime \\ \sigma ″=1+\gamma ″+\beta ″\gamma ″+\mu ″\gamma ″\end{array}$$ (19b)

Fig. 8.

Reaction scheme for the KCC cotransporter, with empty carrier denoted by X′ or X″ according to its availability on the external (luminal) or internal (cytosolic) face of the cell membrane. The ordered binding and unbinding parallels that for NKCC, and ion binding is rapid relative to translocation, so that equilibrium concentrations are assumed for each of the bound transporter species.

Of note, if the binding is random, the form of Eq. 19b changes to

$$\sigma =1+\beta +\gamma +\beta \gamma +\mu +\mu \gamma$$ (19c)

but the analysis is basically unaltered. Following Eq. 5, the equations for the steady-state balance of carrier flux

$$\rho \prime \cdot x\prime -\rho ″\cdot x″=0$$ (20a)

$$\begin{array}{l}\rho \prime =P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime +P_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime \\ \rho ″=P_0^{″}+P_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″+P_{\mathrm{\text{mcc}}}^{″}\mu ″\gamma ″\end{array}$$ (20b)

Equations 19 and 20 for the unbound carrier concentrations have the solution

$$\begin{array}{c}x\prime =\frac{x_{\mathrm{T}}}{\sum }\rho ″=\frac{x_{\mathrm{T}}}{\sum }[P_0^{″}+P_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″+P_{\mathrm{\text{mcc}}}^{″}\mu ″\gamma ″]\\ x″=\frac{x_{\mathrm{T}}}{\sum }\rho \prime =\frac{x_{\mathrm{T}}}{\sum }[P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime +P_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime ]\end{array}$$ (21)

in which

$$\sum =\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime$$ (21a)

Net fluxes for K⁺ and NH₄⁺ across the cotransporter are thus

$$\begin{array}{c}{J}_{\mathrm{K}}=\frac{x_{\mathrm{T}}}{\sum }\cdot \{\left[P_0^{″}{P}_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime -P_0^{\prime }{P}_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″\right]\\ +\left[P_{\mathrm{\text{mcc}}}^{″}\mu ″\gamma ″P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime =P_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime P_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″\right]\}\end{array}$$ (22a)

$$\begin{array}{c}{J}_{\mathrm{N}{\mathrm{H}}_4}=\frac{x_{\mathrm{T}}}{\sum }\cdot \{\left[P_0^{″}{P}_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime -P_0^{\prime }{P}_{\mathrm{\text{mcc}}}^{″}\mu ″\gamma ″\right]\\ +\left[P_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime P_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″-P_{\mathrm{\text{mcc}}}^{″}\mu ″\gamma ″P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime \right]\}\end{array}$$ (22b)

Given the concentrations of cytosolic solutes and of peritubular Cl⁻, Eq. 22 can be solved for the peritubular K⁺ (β′) and NH₄⁺ (μ′) concentrations which bring cotransport flux to zero, namely:

$$\frac{\beta \prime \gamma \prime }{\beta ″\gamma ″}=\frac{P_0^{\prime }+P_{\mathrm{\text{mcc}}}^{\prime }\mu \prime \gamma \prime }{P_0^{\prime }+P_{\mathrm{\text{mcc}}}^{\prime }\mu ″\gamma ″}$$ (23a)

$$\frac{\mu \prime \gamma \prime }{\mu ″\gamma ″}=\frac{P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime }{P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta ″\gamma ″}$$ (23b)

in which the translocation rate condition (Eq. 4) has been used in the simplification. In the case that NH₄⁺ is absent, Eq. 23a displays the intuitive condition that the equilibrium condition is that of the chemical potential of the salt, KCl. The presence of the counterion shifts this equilibrium according to its direction of transport. Thus, if the direction of K⁺-Cl⁻ flux is out of the cell (β″γ″ > β′γ′), then the right-hand side of Eq. 23b will be <1, and the equilibrium μ′γ′ for transport will be smaller than its chemical equilibrium. In physiological terms, K⁺-Cl⁻ efflux from the cell can drive NH₄⁺-Cl⁻ uptake even when the peritubular concentrations are reduced. Conversely, for a cell in which NH₄⁺ - Cl⁻ is entering (μ″γ″ < μ′γ′), the right-hand side of Eq. 23a is >1, so that the equilibrium β′γ′ for transport is greater than its chemical equilibrium. Thus NH₄⁺-Cl⁻ uptake can enhance K⁺-Cl⁻ exit.

In an ammonia-free system, the model equations simplify to

$$\begin{array}{c}\sigma \prime =1+\beta \prime +\beta \prime \gamma \prime \\ \sigma ″=1+\gamma ″+\beta ″\gamma ″\end{array}$$ (24a)

$$\begin{array}{c}\rho \prime =P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime \\ \rho \prime =P_0^{″}+P_{\mathrm{\text{kcc}}}^{″}\beta ″\gamma ″\end{array}$$ (24b)

and the unidirectional uptake of K⁺ is

$$J_{\mathrm{K}}^{\prime }=P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime x\prime =\frac{x_{\mathrm{T}}{P}_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime \cdot \rho ″}{\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime }$$ (25)

For large external ion concentrations, the maximal unidirectional flux, J′_K,max, is

$$J_{\mathrm{K},\mathrm{\text{max}}}^{\prime }=\frac{x_{\mathrm{T}}{P}_{\text{kcc}}^{\prime }\cdot \rho ″}{\rho ″+\sigma ″\cdot P_{\mathrm{\text{kcc}}}^{\prime }}$$ (26)

so that the relative (or normalized) unidirectional flux is

$$\begin{array}{c}\frac{J_{\mathrm{K}}^{\prime }}{J_{\mathrm{K},\mathrm{\text{max}}}^{\prime }}=\frac{\beta \prime \gamma \prime \cdot [\rho ″+\sigma ″\cdot P_{\mathrm{\text{kcc}}}^{\prime }]}{\sigma \prime \cdot \rho ″+\sigma ″\cdot \rho \prime }\\ =\frac{\beta \prime \gamma \prime \cdot [\rho ″+\sigma ″\cdot P_{\mathrm{\text{kcc}}}^{\prime }]}{[1+\beta \prime +\beta \prime \gamma \prime ]\cdot \rho ″+\sigma ″\cdot [P_0^{\prime }+P_{\mathrm{\text{kcc}}}^{\prime }\beta \prime \gamma \prime ]}\hfill \\ =[\rho ″+\sigma ″\cdot P_{\mathrm{\text{kcc}}}^{\prime }]\cdot {\left[\frac{\rho ″+\sigma ″\cdot P_0^{\prime }}{\beta \prime \gamma \prime }+\frac{\rho ″}{\gamma \prime }+(\rho ″+\sigma ″\cdot P_{\mathrm{\text{kcc}}}^{\prime })\right]}^{-1}\hfill \end{array}$$ (27)

Equation 27 has the same significance as Eq. 13 for the NKCC transporter, namely, that expansion of the denominator allows grouping into three terms: a constant term, and factors of γ′ and β′γ′. This implies that in any experimental investigation in which only the external concentrations, β′, and γ′, are varied, even perfect measurement of unidirectional fluxes can yield at most two independent model parameters. For the KCC model under consideration (without NH₄⁺), with the assumption of internal-external affinity symmetry, there are two affinities and three independent translocation rates. Thus variation of external solutes can provide only two of the five required coefficients.

The data of Mercado et al. (18) can inform selection of the KCC kinetic parameters. As was done with NKCC, these investigators expressed either KCC4 or KCC1 transporters in oocytes, and rubidium entry was measured as either external K⁺ or Cl⁻ was varied. From the considerations above, in fitting this model to their data, two of the five model parameters were assumed, namely, the external-to-internal translocation rates for both loaded and empty carrier were taken to be identical to those for the NKCC F-isoform. With that assumption, the remaining three coefficients (K⁺ and Cl⁻ affinities and internal-to-external translocation rate) were fit to the data of Mercado et al. (18) using a Levenberg-Marquardt search. (As with the NKCC model, it is acknowledged that this fit is arbitrary, in the sense that there remains one additional degree of freedom in selecting the elementary parameters.) The coefficients appear in Table 3a, and notably, even with one free translocation rate, only the ion affinities are substantially different from values for NKCC, specifically higher affinity for K⁺ binding and lower for Cl⁻. With these parameters, simulations of the experiments appear in Fig. 9, calculated with the assumption that (as was the case for the NKCC experiments), internal K⁺ and Cl⁻ concentrations are 80 and 10 mM. The panes on the left correspond to experiments with KCC4, and on the right to KCC1. The top panes show the effect of varying external K⁺ from 0 to 50 mM, while external Cl⁻ is held at 50 mM; the bottom panes vary external Cl⁻ over the same range, while K⁺ is 50 mM. The solid curves are model calculations and the points marked (x) are experimental observations. For each pane, the fluxes are normalized to 1.0 when external K⁺ and Cl⁻ are both 50 mM. It is apparent that the derived model coefficients are compatible with the data, but they are by no means a definitive parameter set. Table 3b displays a sensitivity analysis similar to that for NKCC, in which internal ion concentrations are perturbed by 50% above and below baseline, and what is shown are parameter values relative to baseline. In contrast to NKCC, both translocation rates and affinities adjust to fit the assumed internal concentrations. For each isoform, the aggregate error between model prediction and data is identical, whether computed using baseline parameters or those from perturbed internal concentrations.

Table 3a.

Selected KCC parameter sets

	KCC4	KCC1
Kk	0.00145	0.00145
Kc	0.02108	0.04465
Km	0.00145	0.00145
Pkcc′	10,000	10,000
Pkcc″	1,098	1,009
P0′	39,280	39,280
P0″	357,700	389,300
Pmcc′	2,000	2,000
Pmcc″	219.6	201.8

K_c, K_k, K_m, transporter binding affinity (M) for Cl⁻, K⁺, and NH₄⁺.

Fig. 9.

Ion entry across KCC as a function of external concentration: fit for both affinities and translocation rates. Panes on the left correspond to experiments with KCC4, and on the right to KCC1. Top panes show the effect of varying external K⁺ from 0 to 50 mM, while external Cl⁻ is held at 50 mM; the bottom panes vary external Cl⁻ over the same range, while K⁺ is 50 mM. The solid curves are model calculations, and the points marked (x) are experimental observations of Mercado et al. (18). For each pane, fluxes are normalized to 1.0 when external K⁺ and Cl⁻ are both 5 mM.

Table 3b.

Optimal KCC parameter sets: sensitivity to internal ion concentrations

Internal concentrations, mM
K+	40	120	80	80
Cl−	10	10	5	15
KCC4
Kk	1.003	0.999	1.003	0.999
Kc	0.836	1.090	0.830	1.093
Pkcc′	1.000	1.000	1.000	1.000
Pkcc″	0.801	1.107	0.794	1.110
P0′	1.006	0.996	1.006	0.996
P0″	1.255	0.900	1.267	0.898
KCC1
Kk	1.003	0.999	1.003	0.999
Kc	0.815	1.116	0.810	1.119
Pkcc′	1.000	1.000	1.000	1.000
Pkcc″	0.751	1.155	0.744	1.158
P0′	1.006	0.996	1.006	0.996
P0″	1.339	0.863	1.352	0.860

Values are relative to Table 3a, with internal K⁺ and Cl⁻ = 80 and 10 mM.

With respect to ammonia transport, Amlal et al. (1) provided indirect evidence for NH₄Cl transport by KCC within rat medullary AHL, and Bergeron et al. (4) demonstrated that when mouse KCC4 was expressed in oocytes, it could facilitate NH₄Cl entry. In the experiments of Bergeron et al., the kinetic behavior of NH₄⁺ and Rb⁺ appeared comparable, although it is not clear that one can extract an NH₄⁺ affinity or translocation rate for NH₄⁺-loaded carrier from the data. In view of this uncertainty, ammonia coefficients for KCC were taken to be similar to those for ammonia transport by NKCC, namely, equal affinity of NH₄⁺ and K⁺ and translocation of the NH₄⁺-loaded carrier at 20% the rate of the K⁺-loaded carrier (Table 3a). With those parameter choices, Fig. 10 shows the impact on KCC fluxes of increasing ambient NH₄⁺ over the range of 0 to 10 mM. This calculation uses the KCC4 parameters, with cytosolic K⁺ and Cl⁻ concentrations 140 and 35 mM, and external concentrations 5 and 105 mM. The top panel shows a relatively small decrease in K⁺ exit (denoted as a positive flux) with the increase in NH₄⁺, while the bottom pane displays NH₄⁺-Cl⁻ uptake (denoted as a negative flux) driven by the K⁺-Cl⁻ gradient. This is secondary active transport of NH₄⁺, driven by the asymmetry of carrier binding (K⁺ vs. NH₄⁺) at internal and external surfaces.

Fig. 10.

Impact of ambient NH₄⁺ on KCC4 fluxes. The calculation uses the KCC4 parameters, with cytosolic K⁺ and Cl⁻ concentrations 140 and 35 mM, and external concentrations 5 and 105 mM. (Fluxes have dimension mol·s⁻¹·mol transporter⁻¹.)

In Fig. 11, the strength of transport of K⁺-Cl⁻ and NH₄⁺-Cl⁻ is examined in KCC4. In the calculations of the left panel, ammonia is absent; cytosolic K⁺ and Cl⁻ are 140 and 35 mM, and peritubular Cl⁻ is 105 mM. Peritubular K⁺ is varied from 0 to 50 mM, and is shown on the abscissa. The ordinate is KCC flux (with cellular exit taken as positive), and the curve passes through J_K = 0, when peritubular K⁺ is 46.7 mM, in accord with Eq. 23a. The notable aspect of this curve is that it is steep over the physiological range of peritubular K⁺ values. This would be a necessary feature for KCC4 to act as the sensor of medullary interstitial K⁺ concentrations. The right panel addresses the issue of the direction of NH₄⁺ transport across KCC4. In these calculations, cytosolic K⁺ and Cl⁻ concentrations are 140 and 35 mM, and the blood concentrations for K⁺, Cl⁻ and NH₄⁺ are 5, 105, and 5 mM. What is plotted are the KCC fluxes of K⁺ and NH₄⁺ as cytosolic NH₄⁺ is varied from 0 to 60 mM. The flux intercept (J_NH4 = 0) is indicated when cytosolic NH₄⁺ = 55.6 mM. Equation 23b estimates this intercept, and with regard to Table 3a, the products β′γ′ and β″γ″ are 17.2 and 161, respectively, and the ratio $P_0^{\prime }$/ $P_{\text{kcc}}^{\mathrm{\prime }}$ is 35.8. Thus the right-hand side of Eq. 23b has the value 0.269, so that the predicted equilibrium ratio of NH₄⁺ concentrations is 11.1, as shown. The important observation is that by virtue of the K⁺-Cl⁻ gradient, for any value of cytosolic NH₄⁺ <55.6 mM, KCC will be a pathway for NH₄⁺ entry into the cell, albeit a minor flux that is only ∼10% of the K⁺ exit.

Fig. 11.

Strength of transport of KCl and NH₄Cl are examined in KCC4. In the calculations of the left pane, ammonia is absent; cytosolic K⁺ and Cl⁻ are 140 and 35 mM; and peritubular Cl⁻ is 105 mM. Peritubular K⁺ is varied from 0 to 50 mM and is shown on the abscissa. The ordinate is KCC flux (mmol·s⁻¹·mmol transporter⁻¹), and the curve passes through J_K = 0, when peritubular K⁺ is 46.7 mM, an equilibrium ratio of 3. In the right pane, cytosolic K⁺ and Cl⁻ concentrations are 140 and 35 mM, and the blood concentrations for K⁺, Cl⁻ and NH₄⁺ are 5, 105, and 5 mM. KCC fluxes of K⁺ and NH₄⁺ are plotted as cytosolic NH₄⁺ is varied from 0 to 60 mM. The flux intercept (J_NH4 = 0) is indicated when cytosolic NH₄⁺ = 55.6 mM, so that the predicted equilibrium ratio of NH₄⁺ concentrations is 11.1.

DISCUSSION

In this work, standard kinetic schemes have been used to formulate mathematical models for two of the cotransporters of the AHL of the rat kidney. The key feature of the NKCC binding diagram is sequential binding of Na⁺-Cl⁻-K⁺(NH₄⁺)-Cl⁻, with first-on-first-off kinetics. The rationalization for this scheme derives from studies of NKCC in erythrocytes, but has not been studied in the kidney isoforms. In terms of parameter reduction, the most important mathematical assumption of the analysis is that ion biding is rapid relative to translocation. This simplification has been used in the formulation of other kidney cotransporters, NHE3 (24), AE1 (25), and NCC (26), although its inaccuracy may not always be trivial. The importance of this assumption is that it allows a model which would otherwise be 15 equations to be cast as two linear equations, which can be analyzed; the payoff from this analysis is a view of the parametric dependence of experimental results. For the F-isoform, the inaccuracy appears to be trivial (^∼2%), while for B- and A-isoforms, the inaccuracy may be more substantial (^∼25%). Considering the gross uncertainties which have been identified here in formulating kinetic models for NKCC and KCC, the equilibrium binding assumption has been retained. The second simplifying assumption is that of binding symmetry, namely, that external and internal ion affinities are equal, and this also appeared in the model of Benjamin and Johnson (3). This assumption was examined by Marcano et al. (17), and with respect to the fit to data, dropping the assumption gave virtually no improvement. Where the data exist (e.g., for AE1), the symmetry assumption was legitimate (25); however, its suitability for NKCC remains to be examined. The last of the simplifying assumptions is that the affinities for the two Cl⁻ binding sites are equal; its rationalization is lack of evidence to the contrary, and that dropping this assumption fails to improve the fit to data (17).

By virtue of all of these simplifications, the calculation of NaKCl₂ transport by the NKCC in Fig. 1 requires assignment of six kinetic parameters, three affinities and three rate coefficients. Benjamin and Johnson (3) computed several sets of coefficients for different NKCC transporters using several data sources, including two from the kidney (Madin-Darby canine kidney cells, LLC-PK₁ cells). They did not examine the question of uniqueness of their sets of optimal coefficients for each transporter. Marcano et al. (17) used the most comprehensive data available for NKCC from mouse AHL (20) to determine coefficients for the three isoforms. They compared performance of their NKCC models with those of Benjamin and Johnson (3), and the differences did not appear substantial. Marcano et al. (17) used a careful search procedure to examine the space of parameter coefficients to achieve an optimal set. Nevertheless, despite the optimization, their calculations revealed a number of equally suitable parameter sets. What the analysis of the present work adds, is an explicit equation (Eq. 13) for the tracer fluxes measured by Plata et al. (20), and the demonstration that there are only four composite parameters that can be determined from such data. Indeed, it is shown (Table 1, Fig. 2) that the freedom to chose a translocation rate for a loaded carrier can vary over several orders of magnitude with no impairment of the fit of the model to the data of Plata et al. (20). A significant contribution of this study is to have made explicit the assumption of this translocation rate, and thus acknowledge the uncertainty of even these basic NKCC coefficients.

Although no model was required to establish the role of NKCC in AHL Na⁺ reabsorption, a model assumes importance in examining the impact of cytosolic Cl⁻ in modulating this transport. Qualitatively, it has been understood for some time that the series configuration of NKCC and KCC in AHL renders cytosolic Cl⁻ a natural mediator of luminal-peritubular membrane cross talk (9). In prior NKCC models, the authors had focused attention on simulating the uptake experiments that had generated their data base; specifically, all of their calculations varied luminal ion composition, while none examined the impact of cytosolic Cl⁻. The calculations of the present model (Fig. 4) indicate that in the range of cytosolic Cl⁻ concentrations, from low to equilibrium, there is steep flux dependence. It is understood that variations in cytosolic Cl⁻ may also modulate NKCC via nonthermodynamic mechanisms, such as Cl⁻-dependent kinase activity (reviewed in Ref. 13). Nevertheless, studies which documented changes in NKCC phosphorylation as a consequence of cell Cl⁻ necessarily employed large variations in Cl⁻ concentration. The extent to which phosphorylation is important during perturbations of cell Cl⁻ within a physiological range remains to be defined. Ultimately, a more comprehensive model will be required to sort out the relative contribution of ion gradients and transporter activation in regulating transport. It must be acknowledged again that in formulating the present model, there are no input data from any experiments in which cell Cl⁻ has been varied.

This is the first NKCC model to include NH₄⁺ as a transported solute. Indeed, other than NH₄⁺ vs. H⁺ transport by NHE3 (24), it appears to be the first carrier model to accommodate competitive transport. The experimental underpinnings of this aspect of the model are that NH₄⁺ can drive NKCC Na⁺ flux, and that NH₄⁺ and K⁺ are competitive in this process (15). Kinne et al. (15) had noted an overall NH₄⁺ transport affinity about one-sixth that of K⁺, but the finding here was that comparable ion binding constants (but reduced translocation rates of NH₄⁺-loaded carrier) could yield something close to the observed transport behavior. The importance of NKCC NH₄⁺ transport derives from the possibility that NH₄⁺ may be “catalytic” for Na⁺ reabsorption. Namely, the configuration of NKCC in parallel with a pathway for luminal membrane NH₄⁺ exit (secretion) can mediate net Na⁺-2Cl⁻ reabsorption, just as it occurs with K⁺ in parallel with ROMK channel activity. The parallel path for NH₄⁺ exit is uncertain; there are several candidates, all with reservations. There is NHE3 within the luminal membrane, which can mediate both Na⁺/H⁺ and Na⁺/NH₄⁺ exchange. However, the rate of AHL proton secretion (as assessed by HCO₃ reabsorption) has never been found to be >10–20% of overall Na⁺ reabsorption (10, 11). Thus it is uncertain that the rate of NHE3 Na⁺/NH₄⁺ exchange would be of sufficient magnitude to supply NH₄⁺ for NKCC uptake. There could be diffusive NH₃ exit from the AHL cell in parallel with luminal proton secretion, to generate new luminal NH₄⁺. However, luminal membrane permeability to NH₃ is relatively low (21), and ultimately NH₄⁺ formation would be limited by NHE3 proton extrusion. Luminal membrane ROMK is permeable to NH₄⁺ (6), but both the relative permeability and the electrochemical gradient for NH₄⁺ secretion are likely to be substantially less than that for K⁺. Although it is possible that the luminal membrane might contain a dedicated NH₄⁺ pore, no member of the Rh family of ammonia transporters has been identified in AHL (23). Of note, luminal fluid concentrations of NH₄⁺ and K⁺ are comparable (14), and cytosolic K⁺ concentration is high (2). There does not seem to have been a determination of AHL cytosolic NH₄⁺, but unless it is surprisingly high, the driving force for NaNH₄Cl₂ reabsorption likely exceeds that for NaKCl₂. The relative fluxes of these two species through NKCC2, in normal conditions and in hypokalemia, remain an important question.

The mathematical representation of the AHL K⁺-Cl⁻ cotransporter is less certain than that for the NKCC. One salient feature of this transporter is that the overall affinity for K⁺-Cl⁻ is lower than for ion binding to NKCC. The most intensely studied KCC is that of the erythrocyte (16). For this transporter, an asymmetric kinetic scheme has been proposed, in which the binding is ordered at the external face (Cl⁻ followed by K⁺) and random on the internal face (7). The molecular identity of the erythrocyte KCC is not known with certainty, but Gamba (8) has summarized arguments that it is likely to be KCC1. Mercado et al. (18) have argued that the AHL transporter is likely to be KCC4. With only ion entry data to work with, a very tentative model of KCC4 has been fabricated, which is a truncated form of the NKCC scheme. Even for this approximate scheme, available data allow for determination of only two of the five required model coefficients. Nevertheless, the KCC that has been fashioned reproduces experimental uptake measurements, and provides a steep sensitivity of ion flux to peritubular K⁺ concentration. When this KCC is allowed to transport NH₄⁺, it is seen to mediate NH₄⁺ uptake to cytosolic values well above thermodynamic equilibrium for the NH₄⁺-Cl⁻ ion pair. This is due to transport driven by the K⁺-Cl⁻ gradient, and the model analysis shows that the greater the translocation rate for loaded carrier (relative to unloaded carrier), the greater the effect of the K⁺-Cl⁻ gradient to increase the equilibrium NH₄⁺ concentration in the cytosol. For the model parameters taken here, the predicted limiting internal-to-external NH₄⁺ concentration ratio is several-fold greater than the equilibrium ratio for the NH₄⁺-Cl⁻ ion pair (11.1 vs. 3.0 in Fig. 11).

In sum, kinetic models of NKCC2 and KCC4, two of the key cotransporters of the Henle limb, have been fashioned consistent with flux data from expression studies in oocytes. The effort to schematize these transporters and assign parameters has highlighted gaps in our database. Specifically, available data are all outside-to-inside fluxes as a function of outside conditions. The analysis has shown that with such data, even a symmetry assumption cannot rescue the effort to fully determine the model coefficients: for the NKCC2 model with six parameters, oocyte fluxes depend on four composite coefficients; for the KCC4 model with five parameters, the fluxes yield two composite coefficients. Of necessity, some parameters must be supplied with a priori guesses, and attempts to fit a full set of coefficients using sophisticated numerical techniques cannot yield more certain results. With respect to ammonia transport, the kinetic data base for NKCC2 is thin, and for KCC4, virtually nonexistent. What the present models do suggest is that with respect to NKCC2 near its operating point, the curve of ion flux as a function of cell Cl⁻ is steep, and with respect to KCC4, its curve of ion flux as a function of peritubular K⁺ is also steep. In short, the kinetics are suitable for these two transporters in series to act as a sensor for peritubular K⁺, to modulate AHL Na⁺ reabsorption. The data are not sufficient for the model to address the issue of whether, or under what circumstances, additional biochemical signals augment this cross talk. The models also reveal the potential for luminal NH₄⁺ to be a potent catalyst for NKCC2 Na⁺ reabsorption, provided suitable exit mechanisms for NH₄⁺ (from cell-to-lumen) are operative. It is found that KCC4 is likely to augment the secretory NH₄⁺ flux, with peritubular NH₄⁺ uptake driven by the cell-to-blood K⁺ gradient.

GRANTS

This investigation was supported by Public Health Service Grant R01-DK-29857 from the National Institute of Arthritis, Diabetes, and Digestive and Kidney Disease.

DISCLOSURES

No conflicts of interest are declared by the author.