A model of calcium transport and regulation in the proximal tubule

The objective of this study was to examine theoretically how Ca 2 (cid:2) reabsorption in the proximal tubule (PT) is modulated by Na (cid:2) and water ﬂuxes, parathyroid hormone (PTH), Na (cid:2) -glucose cotransporter (SGLT2) inhibitors, and acetazolamide. We expanded a previously published mathematical model of water and solute transport in the rat PT (Layton AT, Vallon V, Edwards A. Am J Physiol Renal Physiol 308: F1343– F1357, 2015) that did not include Ca 2 (cid:2) . Our results indicate that Ca 2 (cid:2) reabsorption in the PT is primarily driven by the transepithelial Ca 2 (cid:2) concentration gradient that stems from water reabsorption, which is itself coupled to Na (cid:2) reabsorption. Simulated variations in permeability or transporter activity elicit opposite changes in paracellular and trans- cellular Ca 2 (cid:2) ﬂuxes, whereas a simulated decrease in ﬁltration rate lowers both ﬂuxes. The model predicts that PTH-mediated inhibition of the apical Na (cid:2) /H (cid:2) exchanger NHE3 reduces Na (cid:2) and Ca 2 (cid:2) transport to a similar extent. It also suggests that acetazolamide- and SGLT2 inhibitor-induced calciuria at least partly stems from reduced Ca 2 (cid:2) reabsorption in the PT. In addition, backleak of phosphate (PO 4 ) across tight junctions is predicted to reduce net PO 4 reab- sorption by ~20% under normal conditions. When transcellular PO 4 transport is substantially reduced by PTH, paracellular PO 4 ﬂux is reversed and contributes signiﬁcantly to PO 4 reabsorption. Furthermore, PTH is predicted to exert an indirect impact on PO 4 reabsorption via its inhibitory action on NHE3. This model thus provides greater insight into the mechanisms that modulate Ca 2 (cid:2) and PO 4 reabsorption in the PT.


INTRODUCTION
Sixty to 70% of the filtered load of Ca 2ϩ is reabsorbed in the proximal tubule (PT), mostly across the paracellular route, via passive diffusion and convection (i.e., solvent drag). There is some evidence (reviewed in Ref. 18) that a small fraction (10 -20%) of Ca 2ϩ is reabsorbed across the transcellular route, but the underlying molecular transporters remain to be elucidated. The respective contribution of diffusion and convection to Ca 2ϩ fluxes in the PT and the extent to which the transport of Ca 2ϩ is coupled to that of Na ϩ and water have yet to be fully understood (27,39). The objective of the present study was to provide a quantitative description of the forces that drive Ca 2ϩ reabsorption in the PT and to examine how Ca 2ϩ transport in this segment is modulated by filtration rates, Na ϩ , parathyroid hormone (PTH), acetazolamide, and inhibitors of Na ϩ -glucose transporters (SGLTs). We expanded a model of water and solute transport in the PT that we published previously (26) but that did not include Ca 2ϩ .
We also modified the model's handling of anionic inorganic phosphate (PO 4 ). In plasma, PO 4 is mostly present as HPO 4 2Ϫ and H 2 PO 4 Ϫ in a 4:1 ratio. About 80% of the filtered load of PO 4 is reabsorbed by the PT (27,49). PO 4 entry into the cell is mediated by three types of Na ϩ -phosphate cotransporters: NaPi-IIa (SLC34A1), NaPi-IIc (SLC34A3), and PiT-2 (SLC20A2) (10). NaPi-IIa mediates cotransport of 3 Na ϩ :1 HPO 4 2Ϫ , NaPi-IIc mediates cotransport of 2 Na ϩ :1 HPO 4 2Ϫ , and PiT-2 mediates cotransport of 2 Na ϩ :1 H 2 PO 4 Ϫ . Whereas the previous version of the PT model only considered a generic apical 1 Na ϩ :1 H 2 PO 4 Ϫ cotransporter, in the present study we accounted for these specific Na ϩ -PO 4 cotransporters. We also compared the direct and indirect ways in which PTH affects PO 4 transport in the PT.
The PT model incorporates flow-dependent transport, i.e., the observation that high flow velocity augments transepithelial fluxes by increasing transporter membrane abundance (34). Du et al. (14) demonstrated that Na ϩ and HCO 3 Ϫ reabsorption varies proportionally to the microvillous torque. Following the approach of Weinstein and colleagues (62), the abundance of transporters in apical and basolateral membranes is taken to be proportional to the torque. Flow-dependent transport plays an important role in maintaining perfusion-absorption balance. It may also act to prevent large excursions in the transepithelial fluxes of water and Na ϩ at a given perfusion rate.

MODEL DESCRIPTION
Conservation equations. The mathematical model of transport along the PT of a male rat is based on conservation equations, which are solved at steady state. The PT consists of two cortical (S1-S2) segments (with a combined length taken as 0.97 cm) and a medullary (S3) segment (0.13 cm). We assume that all segments of the PT express the same types of channels, pumps, and cotransporters, with the exception of glucose transporters (26). Membrane surface areas are reduced by a factor of 2 in the S3 segment to account for decreased membrane infolding. As described above, luminal and peritubular transporter density increases linearly with the relative microvillous torque (62); the proportionality constant is set to 1.8 in the S1-S2 segment and to 0.9 in the S3 segment in the present study, so that the predicted reabsorption of Na ϩ and K ϩ equals approximately two-thirds of the filtered load.
As shown in Fig. 1, the model represents four compartments: the lumen (L), the cell cytosol (C), the lateral intercellular space (I), and the peritubular fluid (P). Conservation of Ca 2ϩ in the lumen, cells, and intercellular space is written as follows: F V represents the luminal flow rate, S NM is the surface area per unit length at the interface between compartments N and M, and J NM is the flux across that interface. Note that at steady state the net flux of Ca 2ϩ from intracellular stores to the cytosol and the net rate of Ca 2ϩ binding to Ca 2ϩ buffers in the cytosol and the sarcoplasmic reticulum are zero (for their dynamic expressions see Ref. 12). Ca 2ϩ fluxes. The Ca 2ϩ flux from the lumen to compartment M (M ϭ C, I) is computed as follows: The first term in Eq. 3a corresponds to the convective component of the flux: J V is the volume flux (superscripts are omitted for simplicity), C Ca is a (logarithmic) mean concentration, and Ca is the reflection coefficient of the membrane to Ca 2ϩ ; Ca is set to 1 for cell membranes, 0.89 for the tight junction (TJ) (41), and 0 for the basement membrane (BM; at the interface between the lateral interspace and the peritubular fluid). The second term corresponds to the electrodiffusive component: P Ca is the membrane permeability to Ca 2ϩ , and Ca is a normalized electric potential difference, a function of the valence z Ca of Ca 2ϩ and the electric potential ⌿. R and F are the ideal gas and Faraday constants, respectively.
The permeability of the TJ to Ca 2ϩ is taken as 20 ϫ 10 Ϫ5 cm/s based on the compilation in Ref. 18. The permeability of the BM to Ca 2ϩ is computed based on the permeability to Na ϩ , given a Ca 2ϩ -to-Na ϩ free diffusivity ratio of 7.93:13.3 (30). The permeability of the apical membrane to Ca 2ϩ is set to 0.005 ϫ 10 Ϫ5 cm/s so as to yield a transcellular flux amounting to 15% of the overall flux.
The molecular mechanisms by which Ca 2ϩ is extruded from PT cells remain to be identified. Based on recent transcriptomic data (28), we neglect the potential contribution of Na ϩ /Ca 2ϩ exchangers and assume that Ca 2ϩ is pumped out of the cell by plasma membrane Ca 2ϩ -ATPase (PMCA) at rate given by The affinity of the pump to Ca 2ϩ (K m PMCA ) is taken as 75.6 nM (57) and the maximum flux (J max PMCA ) as 0.5 ϫ 10 Ϫ9 mmol·cm Ϫ2 ·s Ϫ1 . Parameter values are summarized in Table 1.
Interstitial concentration gradient. Interstitial fluid concentrations are prescribed in this model: they are equal to plasma concentrations in the cortex and increase linearly along the corticomedullary axis in the medulla. Based on our macroscopic model of Ca 2ϩ transport in different populations of nephrons, vasa recta, and the interstitium (56), which did not represent processes at the cell/molecular level, we assume that the interstitial Ca 2ϩ concentration ([Ca 2ϩ ]) increases linearly from 1.25 mM at the corticomedullary junction to 1.60 mM at Ϫ cotransporter. It is now known that PO 4 entry into PT cells is mediated by the cotransporters NaPi-IIa, NaPi-IIc, and PiT-2 (11,27); their respective stoichiometries are as follows: 3 Na ϩ :1 HPO 4 2Ϫ , 2 Na ϩ :1 HPO 4 2Ϫ , and 2 Na ϩ :1 H 2 PO 4 Ϫ . Hence, NaPi-IIa and PiT-2 are electrogenic, whereas NaPi-IIc is electroneutral. Fluxes across these cotransporters are computed using the nonequilibrium thermodynamic approach (60). Based on the immunochemistry findings of Picard et al. (42), we assume that NaPi-IIa is expressed along the full PT and that NaPi-IIc and PiT-2 are only present in the convoluted PT. The transport coefficient (a measure of density) of NaPi-IIa is set to 0.30 ϫ 10 Ϫ9 mmol 2 ·J Ϫ1 ·s Ϫ1 ·cm Ϫ2 everywhere, and those of NaPi-IIc and PiT-2 are taken as 0.25 ϫ 10 Ϫ9 and 0.10 ϫ 10 Ϫ9 mmol 2 ·J Ϫ1 ·s Ϫ1 ·cm Ϫ2 , respectively, in the S1-S2 segment and zero in the S3 segment. No other PO 4 cellular entry pathways are considered. The interstitial concentration of PO 4 is taken to increase from 2.6 mM at the corticomedullary junction to 3.9 mM at the OM-IM junction (58), and the filtered load of PO 4 is 78.0 pmol/min per nephron.

Forces driving Ca 2ϩ reabsorption.
Under baseline conditions, the model predicts that the PT reabsorbs 68 -70% of the filtered load of Na ϩ , K ϩ , and Cl Ϫ . Fractional Ca 2ϩ reabsorption is 69.3%, and the tubular fluid-to-glomerular filtrate [Ca 2ϩ ] ratio [(TF/GF) Ca ] increases from 1.0 to 1.3, in accordance with reported measurements (reviewed in Ref. 18). As depicted in Fig. 2, the molar flow rate of Ca 2ϩ in the lumen decreases in parallel with volume flow. As the rate of water reabsorption increases significantly in the medullary (S3) segment, owing to the interstitial osmolality gradient, so do [Ca 2ϩ ] in the lumen and (TF/GF) Ca (Fig. 2).
Results are given on a per-nephron basis: J Ca denotes the local or average Ca 2ϩ flux (in pmol·min Ϫ1 ·mm Ϫ1 ) and T Ca denotes the rate of Ca 2ϩ reabsorption along the entire PT (in pmol/min). The base-case T Ca is computed as 26.0 pmol/min, 85% of which is paracellular. Equivalently, the average J Ca equals 2.36 pmol·min Ϫ1 ·mm Ϫ1 (Table 2). Paracellular transport of Ca 2ϩ across the TJ, which separates the lumen from the lateral interspace, is followed by Ca 2ϩ transport across the BM, which separates the lateral interspace from the interstitium (Fig. 1). Across the TJ, the Ca 2ϩ flux is predominantly governed by electrodiffusion, rather than by convection (i.e., solvent drag). As water reabsorption raises [Ca 2ϩ ] in the lumen above that in the lateral interspace and the interstitium (Fig. 2), the [Ca 2ϩ ] gradient across the TJ drives Ca 2ϩ reabsorption (Fig. 3). Since the reflection coefficient of the TJ to Ca 2ϩ is 0.89 (41), i.e., close to 1, solvent drag across this barrier is limited (see Eq. 3a).
At steady state, the flux of Ca 2ϩ across the BM is the sum of the Ca 2ϩ flux across the TJ and the Ca 2ϩ flux from the lateral membrane of cells into the lateral interspace. As shown in Fig.  3, Ca 2ϩ transport across the BM is primarily driven by con-vection. Solvent drag is significantly enhanced, relative to the TJ, because the reflection coefficient of the BM to Ca 2ϩ (and all other electrolytes) is zero; moreover, there is significant water reabsorption from the cell into the lateral interspace, so that the water flux across the BM is larger than that across the TJ. The rapid convective transport of Ca 2ϩ into the interstitium lowers [Ca 2ϩ ] in the lateral interspace below that in the interstitium along parts of the tubule (Fig. 2), thereby eliciting backdiffusion of Ca 2ϩ across the BM (Fig. 3).
Note that electrodiffusion is governed by both the transmembrane [Ca 2ϩ ] gradient and the transmembrane voltage (⌬⌿). The electric potential in the lumen is predicted to rise from Ϫ0.22 mV at the PT inlet to a maximum of 1.31 mV about halfway through the tubule and to decrease to ϩ0.94 mV at the outlet. The electric potential in the lateral interspace remains between Ϫ0.06 and ϩ0.01 mV in the cortex and the medulla. As ⌬⌿ across the TJ changes sign and becomes lumenpositive, its contribution to the electrodiffusive flux increases. Nevertheless, our computations suggest that, overall, ⌬⌿ exerts a lower driving force than the lumen-to-lateral interspace [Ca 2ϩ ] gradient.
In the renal medulla, the axial interstitial osmolarity gradient accelerates the rate of water removal from the PT lumen, thereby augmenting convective Ca 2ϩ fluxes in the S3 segment (Fig. 3). The large increase in solvent drag across the BM is partly counteracted by enhanced Ca 2ϩ diffusion in the opposite direction, as the lateral interspace [Ca 2ϩ ] increasingly lags behind the interstitial [Ca 2ϩ ] (Fig. 2).
Impact of transcellular Ca 2ϩ permeability. The contribution of transcellular Ca 2ϩ transport in the PT and its underlying mechanisms are not understood. We chose the baseline value of the apical membrane Ca 2ϩ permeability (P Ca ap ) so that transcellular T Ca represents 15% of total T Ca . The impact of varying P Ca ap is illustrated in Fig. 4. The maximum PMCA flux was varied by the same factor as P Ca ap to maintain intracellular [Ca 2ϩ ] at ϳ100 nM. The model predicts that increasing P Ca ap and, thus, the transcellular J Ca conversely reduces paracellular J Ca , because it accelerates Ca 2ϩ removal from the lumen, thereby diminishing the lumen-to-interstitium [Ca 2ϩ ] gradient. In fact, the paracellular J Ca decreases to the extent that even though the net J Ca is higher than in the base case in the early part of the PT, it is lower in the late PT. Conversely, decreasing the transcellular J Ca augments the paracellular J Ca , as the lumen-to-interstitium [Ca 2ϩ ] gradient becomes larger. Consequently, overall Ca 2ϩ reabsorption is not much affected by variations in P Ca ap ( Table 2). Values in parentheses show decomposition of Ca 2ϩ flux (JCa) into its diffusive and convective components, respectively. P Ca ap , P Ca tj , and P Ca bm , permeability of apical cell membrane, tight junction, and basement membrane to Ca 2ϩ ; Ca tj , reflection coefficient of the tight junction to Ca 2ϩ ; [Ca 2ϩ ], Ca 2ϩ concentration; OM-IM, outer medulla-inner medulla; CA, carbonic anhydrase; NBC1, basolateral Na ϩ -HCO 3 Ϫ cotransporter; SGLT2, Na ϩ -glucose cotransporter type 2; SNGFR, single-nephron glomerular filtration rate.  Table 1. Vertical line denotes the boundary between the cortex and the medulla.  Impact of solvent drag. We examined the effects of solvent drag on J Ca by varying the reflection coefficient of the TJ to Ca 2ϩ ( Ca tj ) from 0 to 1. Decreasing Ca tj from its baseline value of 0.89 to 0, that is, enhancing the convective transport of Ca 2ϩ across the TJ (see Eq. 3a), is predicted to substantially increase the paracellular J Ca in the first half of the PT, relative to the base case, and to lower it in the second half due to the resulting decrease in the lumen-to-interstitium [Ca 2ϩ ] gradient (Fig. 5). Because of these counterbalancing effects, overall Ca 2ϩ reabsorption is only 7.6% higher (28.0 pmol·min Ϫ1 ·mm Ϫ1 ) than in the base case. Conversely, increasing Ca tj from 0.89 to 1.0 slightly reduces (by 1.1%) overall Ca 2ϩ reabsorption.
Impact of BM permeability. Permeability of the BM to Ca 2ϩ (P Ca bm ) has not been measured. As shown in Table 2, the model predicts that a 10-fold decrease in P Ca bm raises Ca 2ϩ reabsorption by 3.4%, because it reduces the backdiffusion of Ca 2ϩ across the BM. A 10-fold increase in P Ca bm has opposite, albeit smaller, effects: it lowers Ca 2ϩ reabsorption by 1.2%.
Impact of axial interstitial gradient. Another uncertainty relates to the interstitial axial [Ca 2ϩ ] gradient in the medulla. In the base case, interstitial [Ca 2ϩ ] is taken to double between the cortex and the OM-IM junction. If we assume, instead, a threefold increase, the lag between luminal [Ca 2ϩ ] and interstitial [Ca 2ϩ ] in the medulla increases (results not shown).
Hence, paracellular Ca 2ϩ reabsorption is reduced along the S3 segment, from 2.51 pmol/min in the base case to 1.46 pmol/ min, and overall Ca 2ϩ reabsorption decreases from 26.0 to 24.9 pmol/min. Conversely, a smaller medullary [Ca 2ϩ ] gradient is predicted to enhance the diffusion of Ca 2ϩ across the paracellular pathway in the S3 segment and augment its overall reabsorption ( Table 2).
Effects of glomerular filtration rate variations. To examine the coupling between the transport of Ca 2ϩ and that of water and/or Na ϩ , we first simulated a 20% increase in the SNGFR. As described above, the model accounts for flow-dependent transport (14,34,62). A high flow rate in the PT lumen acts via the microvillous torque to recruit more transporters to the cell membrane, thereby enhancing transcellular fluxes and maintaining perfusion-absorption balance. The model thus predicts that a 20% increase in the filtered load of fluid and solutes results in substantially higher absolute reabsorption rates, but the fractional reabsorption of water, Na ϩ , and Ca 2ϩ increases only by 3-4% (from 69.3% to 73.1% for Ca 2ϩ ). More specifically, as the filtered load of Ca 2ϩ is raised from 37.5 to 45.0 pmol/min per nephron, T Ca increases from 26.0 to 32.9 pmol/ min, and the length-averaged J Ca increases from 2.36 to 2.99 pmol·min Ϫ1 ·mm Ϫ1 . Ca 2ϩ transport is elevated across both pathways: transcellular J Ca increases owing to enhanced membrane abundance of Ca 2ϩ transporters, and paracellular J Ca increases because the higher rate of water reabsorption augments the lumen-to-interstitium [Ca 2ϩ ] gradient.
Conversely a 20% decrease in the SNGFR reduces the microvillous torque in the PT lumen, subsequently lowering transporter abundance at the membrane and absolute reabsorption rates. The fractional reabsorption of water, Na ϩ , and Ca 2ϩ is predicted to then decrease by 5-7% (from 69.3% to 62.7% for Ca 2ϩ ). With a Ca 2ϩ filtered load of 30.0 pmol/min per nephron, T Ca equals 18.8 pmol/min, and the length-averaged J Ca equals 1.71 pmol·min Ϫ1 ·mm Ϫ1 (Table 3). Ca 2ϩ transport is reduced both across cells, due to fewer Ca 2ϩ transporters at the membrane, and between cells, due to the lower lumen-tointerstitium [Ca 2ϩ ] gradient.
Apical Na ϩ /H ϩ exchanger type 3-mediated PTH effects. PTH acts to augment Ca 2ϩ reabsorption in the thick ascending limb and below, but in the PT it has been found to reduce T Ca , at least in dogs (2,51). This may be explained by the coupling between Na ϩ and Ca 2ϩ transport in the PT. PTH is known to inhibit apical Na ϩ /H ϩ exchanger type 3 (NHE3), Na ϩ -PO 4 cotransporters, and the basolateral Na ϩ -K ϩ -ATPase pump (27); the resulting decrease in Na ϩ and water reabsorption in the PT likely reduces Ca 2ϩ transport as well. To test this hypothesis, we examined the effects of a 30% decrease in PT  JCa, Ca 2ϩ flux; SNGFR, single-nephron glomerular filtration rate. *Basal expression of Na ϩ /H ϩ exchanger type 3, Na ϩ -PO4 cotransporters (NaPi-IIa, NaPi-IIc, and PiT-2), and Na ϩ -K ϩ -ATPase is lowered by 75%, 75%, and 25%, respectively. Na ϩ reabsorption (as observed in Ref. 2) on T Ca at constant SNGFR. Reducing Na ϩ entry into the PT cell lowers intracellular Na ϩ concentration ([Na ϩ ]), thereby augmenting basolateral Na ϩ secretion via the Na ϩ -dependent Cl Ϫ /HCO 3 Ϫ exchanger NDCBE; a decrease in Na ϩ -K ϩ -ATPase activity partly counterbalances these effects. Moreover, as transcellular Na ϩ transport decreases, less water is reabsorbed, so that luminal flow decreases less rapidly; the higher microvillous torque then recruits more transporters to the membrane, which conversely enhances transcellular transport. In the following simulations, Na ϩ reabsorption was reduced by 30%, while intracellular [Na ϩ ] was maintained at Ͼ10 mM, by lowering the basal (without torque-mediated effects) expression of NHE3, Na ϩ -PO 4 transporters, and Na ϩ -K ϩ -ATPase by 75%, 75%, and 25%, respectively. Owing to the compensatory and torque-modulated effects described above, the Na ϩ fluxes across NHE3, Na ϩ -PO 4 transporters, and Na ϩ -K ϩ -ATPase were equal to 79%, 57%, and 92%, respectively, of their base-case values.
As summarized in Table 3, the 30% decrease in transepithelial Na ϩ transport is predicted to reduce T Ca also by 30%, in accordance with experimental observations (2). As water reabsorption diminishes, luminal [Ca 2ϩ ] is lowered, and both convection and diffusion of Ca 2ϩ across TJs decrease. The 44% reduction in the paracellular J Ca is, however, partially counterbalanced by a 62% increase in the transcellular J Ca , stemming from a more negative electric potential within the cell cytosol (not shown). Overall, these results indicate that PTH indeed reduces T Ca indirectly, via its effects on Na ϩ transport.
PTH is also known to reduce glomerular filtration rate (GFR), perhaps via tubuloglomerular feedback (27). Thus, in the next set of simulations, we both lowered SNGFR by 20% and reduced the basal expression of NHE3, apical Na ϩ -PO 4 transporters, and Na ϩ -K ϩ -ATPase by 75%, 75%, and 25%, respectively. As shown in Table 3, the effects of decreasing SNGFR and Na ϩ transporter activity on Ca 2ϩ transport are almost additive; fractional Ca 2ϩ reabsorption is computed as 47.0% vs. 69.3% in the base case.
Impact of paracellular Ca 2ϩ permeability. There is some evidence that PTH may also decrease paracellular permeability in the PT (23,31). To our knowledge, the specific effects of PTH on permeability of the TJ to Ca 2ϩ (P Ca tj ) have not been investigated. Our model suggests that large reductions in P Ca tj may have a substantial effect on overall Ca 2ϩ reabsorption, in spite of counteracting effects (Fig. 6). Decreasing P Ca tj lowers paracellular T Ca , thereby raising luminal [Ca 2ϩ ] and, thus, the driving force for transcellular transport; increases in transcellular T Ca are limited, however (Table 2). Moreover, if P Ca tj is diminished by a factor of Յ5, paracellular J Ca becomes higher than in the base case toward the end of the PT, where the effects of the larger [Ca 2ϩ ] gradient (namely, the greater driving force) are sufficient to overcome the effects of the permeability reduction. Nevertheless, overall T Ca decreases by Ͼ10% if P Ca tj is reduced by a factor of Ͼ2 (Table 2). Increasing P Ca tj elicits opposite effects, namely, a reduction in luminal [Ca 2ϩ ] and, therefore, in transcellular T Ca . Furthermore, paracellular J Ca becomes lower than in the base case beyond a certain point along the PT (Fig. 6), owing to the reduced lumen-to-interstitium [Ca 2ϩ ] gradient. These compen-sating mechanisms substantially mitigate the effects of even large (e.g., 10-fold) increases in P Ca tj , as recapitulated in Table  2. Overall, these results suggest that decreasing and increasing P Ca tj induce asymmetric responses. Ca 2ϩ -sensing mechanisms. Several studies have suggested that the Ca 2ϩ -sensing-receptor (CaSR) or CaSR-like molecules may be expressed in the PT (6,22,43), but the impact of the CaSR on Ca 2ϩ transport in that segment has not been examined and is a subject of debate (32). Capasso et al. demonstrated that treating PT segments with a calcimimetic agent activates NHEmediated H ϩ extrusion (13). Together, their results suggest that Ca 2ϩ -sensing mechanisms increase NHE-mediated Na ϩ reabsorption, which in turn enhances fluid reabsorption. Thus these mechanisms and PTH appear to exert opposite effects, as also suggested by other studies (6). To examine the impact of Ca 2ϩ sensing on T Ca , we increased the basal (without torquemodulated effects) expression of NHE3 by 40%. Expression of Na ϩ -K ϩ -ATPase was raised concomitantly, albeit to a lesser extent (10%), so as to maintain intracellular [Na ϩ ] below 25 mM. The basal expression of basolateral Na ϩ -HCO 3 Ϫ cotransporters was also augmented (by 40%) to prevent large fluctuations in intracellular volume: adjustment of Na ϩ -HCO 3 Ϫ cotransport is one of the mechanisms underlying cell volume regulation in the PT (16,59,61).
Augmenting the Na ϩ flux across NHE3 raises intracellular [Na ϩ ], which then decreases Na ϩ entry via Na ϩ -PO 4 cotransporters. It also increases water reabsorption, thus reducing the luminal flow and microvillous torque, such that fewer transporters are recruited to the membrane. Specifically, upregulating Na ϩ transporter activity as described above is predicted to raise fractional Na ϩ reabsorption from 70.1% to 73.5%, which in turn elevates fractional Ca 2ϩ reabsorption from 69.3% to 72.8% (Table 2). Together, our results suggest that variations in Na ϩ reabsorption induce nearly identical changes in T Ca .
Impact of acid-base status. Acetazolamide, a commonly prescribed inhibitor of carbonic anhydrase (CA), raises urinary   (3,38). To investigate the impact of acetazolamide in the PT specifically, we reduced the rates of CO 2 -H 2 CO 3 interconversion in the PT lumen by a factor of 10 Ϫ4 (to uncatalyzed values). As previously described (62), luminal CA inhibition induces PT diuresis, because it abolishes the HCO 3 Ϫ and Cl Ϫ gradients that normally drive paracellular water reabsorption. The present model predicts that Na ϩ reabsorption is then reduced by 20%, across both transcellular and paracellular pathways, as apical H ϩ recycling via NHE3 is significantly impaired and the lumen-to-interstitium [Na ϩ ] gradient is reduced. Ca 2ϩ reabsorption is predicted to similarly decrease by 23% (Table 2); an identical 23% acetazolamide-induced decrease was observed in the dog PT (8).
Acetazolamide may also indirectly inhibit the basolateral Na ϩ -HCO 3 transporter NBC1 (50), mutations of which are associated with proximal renal tubular acidosis. Our model suggests that NBC1 inhibition reduces transcellular Na ϩ reabsorption, subsequently diminishing water reabsorption and paracellular Na ϩ fluxes, as well as Ca 2ϩ reabsorption. An isolated 90% decrease in NBC1 expression is predicted to reduce T Ca from 26.0 to 19.7 pmol/min (Table 2), a 24% decrease. Luminal pH at the PT outlet is computed as 7.34 vs. 7.28 in the base case.
Effects of SGLT2 inhibitors. Inhibitors of SGLT2 are increasingly used to treat diabetes, but they have been linked to bone loss or increased risk of fracture, possibly as a result of altered Ca 2ϩ and PO 4 metabolism (1). SLGT2 inhibition in-creases urinary Ca 2ϩ excretion in rats and mice (33,35), but whether this is due to a direct effect on proximal Ca 2ϩ reabsorption is unknown. We examined the effects of blocking SGLT2 on Ca 2ϩ reabsorption in the PT under normoglycemia. SGLT2 inhibition elicits glucose-induced osmotic diuresis, which subsequently reduces the lumen-to-interstitium [Ca 2ϩ ] gradient and, therefore, paracellular Ca 2ϩ transport. On the other hand, the higher luminal flow increases microvillous torque, thereby upregulating the membrane abundance of transport proteins and enhancing transcellular Ca 2ϩ transport. As a net result, the computed T Ca decreases by 12% (Table 2), fractional Ca 2ϩ reabsorption decreases by 8.5% (from 69.3% to 60.8%), and delivery of Ca 2ϩ to the loop of Henle increases from 11.5 to 14.7 pmol/min per nephron.
By significantly reducing proximal reabsorption, SGLT2 blockers activate tubuloglomerular feedback, which in turn reduces SNGFR. Chronic SGLT2 blockade was found to lower GFR by 15% in diabetic rats (54). When we simulated a 15% decrease in SNGFR in combination with a 100% inhibition of SGLT2, fractional Ca 2ϩ reabsorption decreased even further (to 56.6%), and the distal delivery of Ca 2ϩ remained higher than in the base case (13.8 pmol/min per nephron). Together, these simulations suggest that SGLT2 blockers may significantly affect renal handling of Ca 2ϩ (see below). PO 4 reabsorption. The baseline expression of NaPi-IIa, NaPi-IIc, and PiT-2 was chosen so that NaPi-IIa mediates 70% of PO 4 reabsorption in the PT (27), with the remainder arbitrarily divided equally between NaPi-IIc and PiT-2. In the base case, the model predicts that the PT reabsorbs 79.2% of the filtered load of PO 4 , entirely across transcellular pathways. The tubular fluid-to-glomerular filtrate PO 4 concentration ratio [(TF/GF) PO4 ] is Ͻ1.0 and is comparable to micropuncture measurements in rats and dogs (4,63). Specifically, the computed value of (TF/GF) PO4 decreases from 1.0 to 0.53 along the S1-S2 segment and increases to 0.86 in the S3 segment due to the interstitial medullary concentration gradient.
The contribution of NaPi-IIa to PO 4 reabsorption in mice is thought to be~70%, based on gene knockout studies (7,46,53). To assess the importance of compensatory mechanisms, we simulated the effects of inhibiting each type of Na ϩ -PO 4 transporter in turn. Results are summarized in Table 4. Full inhibition of NaPi-IIa is predicted to reduce net PO 4 reabsorption by 26% (to 45.7 pmol/min) and net Na ϩ reabsorption by 3%. The lower PO 4 transport rate means that (TF/GF) PO4  ); NaPi-IIa, NaPi-IIc, and PiT-2, Na ϩ -PO4 cotransporters.

F948
CALCIUM TRANSPORT IN THE PROXIMAL TUBULE remains above 1.0, backleak across the TJ into the lumen is abolished, and the paracellular pathway mediates PO 4 reabsorption instead of secretion; it is predicted to represent 16% of the total flux. The contribution of NaPi-IIc to T PO4 then exceeds that of PiT-2 (Table 4), likely because NaPi-IIc carries HPO 4 2Ϫ (as does NaPi-IIa), which is more abundant than H 2 PO 4 Ϫ , the species carried by PiT-2. When either NaPi-IIc or PiT-2 is fully inhibited, the activity of NaPi-IIa increases by~5% in compensation. The lumen-tointerstitium PO 4 concentration ([PO 4 ]) gradient still favors paracellular PO 4 secretion, as in the base case, but the PO 4 flux across the TJ is nevertheless reduced. As a result of both effects, net T PO4 is predicted to decrease only slightly, by 4% (Table 4). Net Na ϩ reabsorption diminishes by Ͻ1%.
Note that as a baseline, the model assumes that NaPi-IIc and PiT-2 are only expressed in the convoluted PT (42). When we assumed a uniform distribution of NaPi-IIc and PiT-2 along the full PT, the computed T PO4 increased from 61.8 to 62.6 pmol/min, a 1.3% difference.
Effects of PTH on PO 4 transport. PTH decreases PO 4 reabsorption in the PT in a direct manner, by reducing the membrane abundance of NaPi-IIa, NaPi-IIc, and PiT-2 via different mechanisms (10,24,37,42,47). PTH stimulates the internalization of NaPi-IIa by activating PKA and PKC (10), two kinases that are also involved in NHE3 inhibition (25,27). We surmised that PTH may also affect T PO4 indirectly, via its inhibitory action on NHE3. We examined each of these effects, first separately and then simultaneously.
Shown in Table 5 are the computed average PO 4 fluxes for different degrees of Na ϩ -PO 4 cotransporter inhibition. As expected, PO 4 reabsorption is predicted to decrease significantly with increasing inhibition; 90% inhibition of all Na ϩ -PO 4 transporters reduces T PO4 from 61.8 to 36.1 pmol/min. Conversely, NHE3 inhibition by itself is predicted to enhance T PO4 , because it lowers intracellular [Na ϩ ], thereby stimulating the activity of Na ϩ -PO 4 transporters. This means that the direct effect of PTH on PO 4 transporter abundance and its indirect effect on PO 4 transporter activity counteract each other. Consequently, when the abundance of apical PO 4 transporters is reduced by Յ75%, the predicted T PO4 is higher when NHE3 inhibition is taken into account than when it is not (Table 5). Interestingly, the model predicts that when the abundance of apical PO 4 transporters is reduced by Ն80%, the predicted T PO4 is lower when NHE3 inhibition is taken into account than when it is not (Table 5). This occurs because, in the latter case (i.e., no indirect, counterbalancing effects via NHE3), transcellular J PO4 is so small that luminal [PO 4 ] increases steeply along the PT, which strongly stimulates PO 4 reabsorption across the paracellular route. In the former case (i.e., in the presence of counterbalancing effects via NHE3), transcellular J PO4 remains higher, and paracellular PO 4 reabsorption is significantly lower ( Table 5).
As described above, PTH also reduces GFR. By itself, a 20% decrease in SNGFR is predicted to reduce fractional PO 4 reabsorption by 12% (from 79.2% to 67.1%). When combined with PTH-mediated inhibition of Na ϩ -PO 4 transporters and NHE3, fractional PO 4 reabsorption is further reduced ( Table 5).

DISCUSSION
Determinants of Ca 2ϩ transport in the PT. The main objective of this study was to elucidate the physical mechanisms underlying Ca 2ϩ reabsorption and its regulation in the PT. We expanded a previously published model of transport across the PT, which did not account for Ca 2ϩ (26). Our model suggests that the lumen-to-interstitium [Ca 2ϩ ] gradient, which results from water reabsorption, is the main force driving Ca 2ϩ transport across the PT epithelium. The transepithelial electric potential difference contributes to the Ca 2ϩ flux to a lesser extent, and only in the distal PT, where it is lumen-positive. When we set the valence of Ca 2ϩ to zero in our simulations, the computed T Ca decreased by only 5% (from 26.0 to 24.8 pmol/min). If it is assumed that the reflection coefficient of the TJ to Ca 2ϩ is equal to 0.89 (41), the convective J Ca across the TJ is predicted to be approximately seven times lower than the electrodiffusive J Ca ( Table 2). As previously suggested (17), our model indicates that Ca 2ϩ reabsorption in the PT is dominated by passive diffusion.
The nature and contribution of transcellular Ca 2ϩ fluxes in the PT are poorly understood. Apical L-type Ca 2ϩ channels were identified in cultured rabbit PT cells (65), and colocalization of transient receptor potential channel 1 and aquaporin ); NHE3, Na ϩ /H ϩ exchanger type 3; SNGFR, single-nephron glomerular filtration rate. Fractional inhibition of Na ϩ -PO4 cotransporters NaPi-IIa, NaPi-IIc, and PiT-2 is taken to be the same. *Parathyroid hormone is assumed to reduce basal expression of NHE3 and Na ϩ -K ϩ -ATPase by 75% and 25%, respectively. 1 was observed in rat kidneys (21), but the specific Ca 2ϩ molecular transporters in rat PT cells remain to be determined. Interestingly, our results suggest that enhancing transcellular Ca 2ϩ transport has a small impact on overall T Ca , because it induces a counteracting decrease in paracellular Ca 2ϩ transport (by reducing the lumen-to-interstitium [Ca 2ϩ ] gradient). Conversely, reducing transcellular Ca 2ϩ transport elicits a compensatory increase in paracellular Ca 2ϩ transport, such that the overall T Ca also does not vary much (Table 2). Hence, it may be difficult to parse the contribution of each pathway unless experiments are carefully designed.
Besides the contribution of transcellular T Ca , other uncertainties include the magnitude of the corticomedullary interstitial [Ca 2ϩ ] gradient in the medulla (which controls Ca 2ϩ reabsorption in the S3 segment), the reflection coefficient of the TJ to Ca 2ϩ , and the permeability of the BM to Ca 2ϩ . Our results suggest that the latter three parameters have only a moderate impact on T Ca ( Table 2).
Regulation of Ca 2ϩ transport. Whereas PTH augments Ca 2ϩ reabsorption in the thick ascending limb and the distal tubule, studies in dogs suggest that it paradoxically reduces T Ca in the PT (2,51). This reduction is thought to result from PTHmediated inhibition of NHE3, which diminishes transepithelial Na ϩ and Ca 2ϩ fluxes (and fractional reabsorption) by the same factor according to our simulations. PTH is also known to affect GFR. Per se, a PTH-induced decrease in GFR reduces the absolute transport rates of Na ϩ and Ca 2ϩ , but fractional reabsorption is somewhat maintained by flow-dependent (torque-mediated) transport mechanisms. Since the largest proportion (nearly two-thirds) of the filtered load of Ca 2ϩ is reabsorbed in the PT, small variations in transepithelial Ca 2ϩ transport in the PT may have a considerable impact on final calciuria, even if Ca 2ϩ reabsorption increases downstream via tubular cross talk between segments.
In addition, the model predicts that a (putative) inhibitory action of PTH on TJ permeability to Ca 2ϩ may also lower T Ca substantially (Table 2); indeed, large decreases in paracellular J Ca cannot be compensated for by comparable increases in transcellular Ca 2ϩ transport, owing to its limited capacity. The model generally predicts opposite (i.e., counteracting) changes in paracellular and transcellular Ca 2ϩ fluxes ( Table 2). One exception is when GFR is varied: in this case, Ca 2ϩ transport increases (or decreases) both across and between cells, as described above.
Increasing luminal [Ca 2ϩ ] has a demonstrable impact on water and Na ϩ fluxes in the PT (13), but its effects on J Ca have not been examined to our knowledge. If it is assumed that Ca 2ϩ -sensing mechanisms exert only indirect, NHE3-mediated effects on Ca 2ϩ transport in that segment, they are predicted to affect Na ϩ and Ca ϩ reabsorption in the same proportion. The model suggests that increases in J Na are limited in vivo, owing to flow-dependent transport and the tight coupling between water and Na ϩ transport; thus, Ca 2ϩ -sensing-induced increases in J Ca may be similarly restricted.
The mechanisms by which acetazolamide augments urinary Ca 2ϩ excretion remain to be fully characterized. The present study suggests that acetazolamide-induced inhibition of CA in the PT lumen decreases transcellular Na ϩ reabsorption, which in turn lowers paracellular Ca 2ϩ fluxes. Our results thus support the hypothesis that acetazolamide-induced cal-ciuria at least partly stems from reduced Ca 2ϩ reabsorption in the PT (3). SLGT2 inhibitors, which are increasingly used to treat diabetes (55), are associated with disturbances in bone metabolism, higher plasma PO 4 levels, and elevated urinary Ca 2ϩ excretion (1,33,35). In particular, the mechanisms underlying hypercalciuria remain to be elucidated. Our model predicts that, by itself, blocking SGLT2 in the PT reduces Ca 2ϩ reabsorption by 12% in that segment, thereby lowering fractional Ca 2ϩ reabsorption from 69.3% to 60.8%. SGLT2 inhibition is also known to decrease SNGFR via tubuloglomerular feedback (54). Even when SNGFR (i.e., the filtered load) is concomitantly reduced by 15-25%, Ca 2ϩ delivery to the loop of Henle remains 15-20% higher than in the base case, according to our simulations. This significant increase in Ca 2ϩ delivery may not be fully compensated for downstream, given the limited Ca 2ϩ transport capacity of distal segments. In other words, our results suggest that the effects of SGLT2 inhibition on Ca 2ϩ transport in the PT may contribute to SGLT2 blockerinduced hypercalciuria and bone disease.
In addition, it has been postulated that the increase in plasma [PO 4 ] induced by SGLT2 inhibition may stem from increased tubular PO 4 reabsorption (52). However, our model suggests that T PO4 may not increase if SNGFR is reduced. Per se, blocking SGLT2 is predicted to raise T PO4 (by 5.8%) via two mechanisms: not only is the activity of Na-PO 4 cotransporters stimulated by the decrease in intracellular [Na ϩ ], but their membrane abundance (and that of other transcellular transporters) is also upregulated in response to the higher luminal flow. Nevertheless, even a small (5%) SGLT2 inhibition-induced decrease in SNGFR more than counterbalances these effects and lowers T PO4 below its baseline value. If it is assumed that blocking SGLT2 lowers SNGFR by 15% (54), the computed T PO4 is 20% lower than in the absence of SGLT2 inhibitors.
Impact of claudin-2 deletion. Fractional Ca 2ϩ excretion (FE Ca ) is increased threefold, from 0.13 to 0.40% (40), in claudin-2 knockout mice, likely as a result of impaired Ca 2ϩ reabsorption in the PT (19). Claudin-2 is the main cation-and water-permeable channel in the PT (19,40,44); its Ca 2ϩ -to-Na ϩ permeability ratio has been estimated as 1:4 (64). In PT segments specifically, Na ϩ , Cl Ϫ , and water reabsorption is reduced by 20 -40% following claudin-2 gene deletion (40,45). Whether the FE Ca increase stems from altered claudin-2dependent paracellular Ca 2ϩ transport in the PT or is only an indirect consequence of impaired paracellular Na ϩ and water reabsorption remains to be ascertained. To shed light on this question, we simulated a 90% decrease in the paracellular permeability of PT TJs to Na ϩ and water, with or without a concomitant 90% decrease in P Ca tj . A 10-fold decrease in the paracellular permeability to Na ϩ and water is predicted to lower their reabsorption by 25% and 24%, respectively. By itself, this reduces T Ca from 26.0 to 18.3 pmol/min and fractional Ca 2ϩ reabsorption from 69.3% to 48.7%. If P Ca tj is reduced by 90% in tandem, the computed fractional Ca 2ϩ reabsorption decreases further, to 28.8%; in other words, the load delivered to the thick ascending limb is then 2.3 times higher than under normal conditions. Under these circumstances, it seems unlikely that Ca 2ϩ reabsorption mechanisms downstream from the PT (namely, passive reabsorption in the thick ascending limb and active transport in the F950 CALCIUM TRANSPORT IN THE PROXIMAL TUBULE distal convoluted and connecting tubules) can be sufficiently upregulated to maintain urinary Ca 2ϩ excretion at ϳ1%. This suggests that the higher FE Ca observed in claudin-2 KO mice might only be an indirect effect, the consequence of reduced Na ϩ and water reabsorption in the PT. A model of transport along the entire nephron would help fully elucidate this question. Experiments designed to block Ca 2ϩ reabsorption specifically in the thick ascending limb and/or the distal convoluted tubule of claudin-2 knockout mice would also yield a better understanding of the relative contribution of these segments in compensating for the loss of Ca 2ϩ in the PT due to the absence of claudin-2.
Determinants of PO 4 transport. The model was also expanded to account for the specific stoichiometry of each apical Na ϩ -PO 4 transporter in the PT. In mice, but not in humans, the contribution of NaPi-IIa predominates (9,27), and we assumed in the base case that NaPi-IIa mediates 70% of T PO4 vs. 15% each for NaPi-IIc and PiT-2. In addition, permeability of the TJ to HPO 4 2Ϫ and H 2 PO 4 Ϫ was taken as 4.0 ϫ 10 Ϫ5 cm/s (62). With these hypotheses, the model predicts that, under normal conditions, the passive backleak of PO 4 across the TJ reduces net PO 4 reabsorption by 21% (Table 4). However, when transcellular PO 4 transport is substantially impaired, the paracellular PO 4 flux switches direction and may contribute significantly to PO 4 reabsorption (Tables 4 and 5). Moreover, when transport via NaPi-IIa is fully blocked, NaPi-IIc flux increases significantly more than PiT-2 flux. Together, these results suggest that it may be difficult to extrapolate measurements in knockout animal models to quantify the contribution of each Na ϩ -PO 4 transporter.
PTH is known to reduce PO 4 reabsorption in the PT by lowering the membrane expression of NaPi-IIa, NaPi-IIc, and PiT-2 (27). PTH may also impact T PO4 indirectly by inhibiting NHE3. According to our simulations, inhibition of NHE3 per se stimulates the activity of Na ϩ -PO 4 transporters, suggesting that the direct and indirect effects of PTH on T PO4 counteract each other. Moreover, when transcellular PO 4 reabsorption is severely reduced, paracellular PO 4 reabsorption may increase very significantly in compensation. PTH-induced decreases in GFR may also contribute to lowering T PO4 (Table 5).
Possible model extensions. Since the molecular transporters that mediate transcellular Ca 2ϩ reabsorption in the PT remain to be identified, our model assumes that the transcellular J Ca is solely driven by the Ca 2ϩ electrochemical potential gradient: this assumption is valid if Ca 2ϩ entry into the cell occurs via a Ca 2ϩ channel, but not if it is mediated by a cotransporter. Further experimental studies are needed to clarify this. Additionally, the current model could be expanded in several ways. It does not account for the binding between Ca 2ϩ and HPO 4 2Ϫ or H 2 PO 4 Ϫ , the rate of which is pH-dependent. Nor does it include phospho-and calcitropic hormones other than PTH, such as fibroblast growth factor 23 (FGF23) and vitamin D 3 . Vitamin D 3 enhances the intestinal absorption of both Ca 2ϩ and PO 4 , and its synthesis in PT cells is activated by PTH (5) and, conversely, inhibited by FGF23 (48). However, whether vitamin D 3 directly modulates Ca 2ϩ and PO 4 fluxes in the PT remains to be ascertained (27). FGF23 is an important regulator of PO 4 metabolism; in the PT, it reduces PO 4 reabsorption by decreasing NaPi-IIa and NaPi-IIc expression (20). FGF23 requires ␣-klotho as a cofactor, which is expressed mainly in the distal tubule and, to a much lower extent, in the PT (29). Thus the actions of FGF23 in the PT may be indirect and may involve a distal-to-proximal tubular feedback mechanism that has yet to be elucidated (36). Finally, this PT model should be linked to our models of Ca 2ϩ transport in the distal nephron (12,15) to yield an integrated understanding of renal Ca 2ϩ handling.
In conclusion, we have developed the first model of Ca 2ϩ transport in the PT. Our results indicate that Ca 2ϩ reabsorption in that segment is principally driven by the lumen-to-interstitium [Ca 2ϩ ] gradient that is generated by water reabsorption. Our model also provides greater insight into the different mechanisms by which the reabsorption of Ca 2ϩ and PO 4 is regulated in that segment.