Adaptive changes in GFR, tubular morphology, and transport in subtotal nephrectomized kidneys: modeling and analysis

Abstract

Removal of renal mass stimulates anatomical and functional adaptations in the surviving nephrons, including elevations in single-nephron glomerular filtration rate (SNGFR) and tubular hypertrophy. A goal of this study is to assess the extent to which the concomitant increases in filtered load and tubular transport capacity preserve homeostasis of water and salt. To accomplish that goal, we developed computational models to simulate solute transport and metabolism along nephron populations in a uninephrectomized (UNX) rat and a 5/6-nephrectomized (5/6-NX) rat. Model simulations indicate that nephrectomy-induced SNGFR increase and tubular hypertrophy go a long way to normalize excretion, but alone are insufficient to fully maintain salt balance. We then identified increases in the protein density of Na⁺-K⁺-ATPase, Na⁺-K⁺-2Cl⁻ cotransporter, Na⁺-Cl⁻ cotransporter, and epithelial Na⁺ channel, such that the UNX and 5/6-NX models predict urine flow and urinary Na⁺ and K⁺ excretions that are similar to sham levels. The models predict that, in the UNX and 5/6-NX kidneys, fractional water and salt reabsorption is similar to sham along the initial nephron segments (i.e., from the proximal tubule to the distal convoluted tubule), with a need to further reduce Na⁺ reabsorption and increase K⁺ secretion primarily along the connecting tubules and collecting ducts to achieve balance. Additionally, the models predict that, given the substantially elevated filtered and thus transport load among each of the surviving nephrons, oxygen consumption per nephron segment in a UNX or 5/6-NX kidney increases substantially. But due to the reduced nephron population, whole animal renal oxygen consumption is lower. The efficiency of tubular Na⁺ transport in the UNX and 5/6-NX kidneys is predicted to be similar to sham.

when renal mass is reduced, either by disease or by surgery, the remaining nephrons can adapt with appropriately elevated urinary sodium (Na⁺) and potassium (K⁺) excretion. Such compensatory response involves adaptive changes in renal blood flow and glomerular filtration rate (GFR), in tubular growth, and in transepithelial transport (8). With these adaptive changes, the surviving nephrons can preserve homeostasis of water and salt until total renal mass and GFR become critically low.

Surgical reduction of renal mass results in increases in renal blood flow and GFR in the remaining nephrons. This compensatory response in renal hemodynamics has been demonstrated in dogs (22), rats (14), and humans (21). The renal blood flow increase correlates with the amount of renal tissue that is surgically removed. In studies involving progressive surgical ablation of renal tissue in the rat, mean glomerular blood flow rose twofold above control in the remaining renal tissue of uninephrectomized (UNX) animals and fourfold after 70% nephrectomy (16). GFR changes in proportion to renal blood flow.

After loss of renal mass, the rise in filtered load in the surviving nephrons is accompanied by an increase in transport capacity. That is accomplished, in part, by tubular hypertrophy. Studies in rats have indicated that the extent of compensatory growth correlates with the amount of renal tissue removed (15) and thus also correlates with the nephrectomy-induced increase in single-nephron GFR (SNGFR). Given the concomitant increases in filtered load and tubular transport capacity, a reasonable question is: To what extent do these two compensatory responses preserve homeostasis of water and salt? And what are potential advantages or disadvantages of these combined changes? In other words, why is not a simple increase in GFR or a reduction in tubular transport sufficient to restore excretion?

To answer these questions, we applied a recently published computational model of nephron function (19). The model simulates solute transport and metabolism along different nephron populations of a (healthy) rat kidney. To simulate solute transport in a nephrectomized kidney, we reduced nephron number and represented the compensatory responses. We considered two common subtotal nephrectomy models: the kidney in a UNX rat and in a 5/6-nephrectomized (5/6-NX) rat. By means of model simulations, we assessed the extent to which the compensatory GFR increase and tubular hypertrophy can achieve water and salt balance. And, if not, what additional changes in epithelial transporter density are required to achieve that balance?

MODELING METHODOLOGY

Model simulations were performed using our published model of epithelial transport along rat nephrons (19). The model represents flow-dependent transport along the proximal tubule, connecting tubule, and cortical collecting duct. Model parameters were adjusted from baseline to simulate a kidney in a UNX rat and one in a 5/6-NX rat (see below). Kim et al. (17) have demonstrated that, for 3 mo after 5/6-NX, the remaining kidney undergoes continued changes. Our 5/6-NX kidney model is intended to represent 4 wk after the surgery.

Changes in GFR and tubular dimensions.

In UNX, SNGFR was increased 50 and 25% in the superficial and juxtamedullary nephrons, respectively, based on measurements in Ref. 16. Four weeks after 5/6-NX, GFR was reported to be 48% of the corresponding sham rat kidney (17). We assumed in 5/6-NX that SNGFR was increased by 110 and 50% in the superficial and juxtamedullary nephrons, respectively. This results in the same SNGFR in both types of nephrons, consistent with measurements reported in Ref. 16.

Hayslett et al. (9) reported that, in a UNX kidney, the length of the proximal convoluted tubule and distal convoluted tubule increased by 35 and 17%, respectively. We assumed those fractional increases for the proximal tubule and downstream segments. Kaufman et al. (16) reported that, 4 wk after surgery, average tubular mass in a 75% nephrectomized kidney was ~80% higher than the corresponding UNX kidney. We infer from this measurement that 4 wk after 75% nephrectomy, tubular length is 21% more than in UNX, or 63 and 42% (1.35 × 1.21 = 1.63, 1.17 × 1.21 = 1.42) above sham for the proximal convoluted tubule and distal convoluted, respectively, using data from Ref. 9. We assumed for 5/6-NX similar increases of 68 and 44%, respectively, above sham in the length of the proximal tubules and downstream segments.

Hayslett et al. (9) reported that, in a UNX kidney, the luminal diameter of the proximal convoluted tubule and distal convoluted tubule increased by 17 and 12%, respectively. We assumed those increases for the proximal tubule and downstream segments. Based on measurements reported by Kaufman et al. (16) (see above), we assumed for 5/6-NX 48 and 32% increases, respectively, above baseline in the luminal diameter of the proximal tubules and downstream segment. Apical and basolateral membrane amplification due to microvilli is assumed unchanged in the nephrectomized kidneys.

The above changes in SNGFR and tubular dimensions are summarized in Table 1. We first determined fluid and electrolyte excretion based only on the reported GFR and tubular size changes, assuming proportional changes in transporter activities. Then we varied selected transporter densities to establish excretion rates close to sham. Results were then compared with experimental findings.

Table 1.

SNGFR and tubular dimension changes for UNX and 5/6-NX models

	SNGFR (SF)		SNGFR (JM)				PT Length
	Change, %	Value, nl/min	Change, %	Value, nl/min	PT Diameter Change, %	Post-PT Diameter Change, %	Change, %	Value, mm	Post-PT Length Change, %
Sham		30		43				1.1
UNX	+50	45	+25	54	+35	+17	+17	1.3	+12
5/6-NX	+110	63	+50	65	+68	+44	+48	1.6	+32

SF, superficial; JM, juxtamedullary; PT, proximal tubule.

Tubular pressure.

Tubular fluid flow is described by pressure-driven Poiseuille flow. Along a noncoalescing tubule i, the hydrostatic pressure in the lumen, P_i, is related to volume flow Q_i (per tubule) and luminal radius r_i by

$$\frac{d{\mathrm{P}}_i}{dx}=-\frac{8\eta {\text{Q}}_i}{\pi r_i^4}$$ (1)

where η is the luminal fluid viscosity (taken as 6.4 × 10^–6 mmHg⋅s^–1). As the connecting tubules and inner medullary collecting ducts coalesce, fluid flow Q_i in a given tubule increases, even as water is reabsorbed.

Flow-dependent tubular transport.

Proximal tubule reabsorption varies proportionally to SNGFR (23). To model flow-dependent transepithelial transport, we follow the approach of Weinstein et al. (27). Specifically, the proximal tubule is assumed to be compliant, with luminal radius given by

$$r_{\text{PT}}=r_{\text{PT}}^0\left[1+{\mu }_{\text{PT}}\left({\text{P}}_{\text{PT}}-{\text{P}}_{\text{PT}}^0\right)\right]$$ (2)

where the reference radius $r_{\text{PT}}^0$ is taken as 11.2 μm in sham, and increased by 17 and 48% in UNX and 5/6-NX, respectively. The reference pressure ${\text{P}}_{\text{PT}}^0$ is taken as 9 mmHg, and μ_PT, which characterizes tubular compliance, is set to 0.030 in sham, and to 0.012 in UNX and 5/6-NX.

To account for the modulation of transporter density by luminal flow, we determine the microvillus torque as

$${\tau }_{\text{PT}}=\frac{8\mu {\text{Q}}_{\text{PT}}{l}_{\text{PT},\text{mv}}}{r_{\text{PT}}^2}\left(1+\frac{l_{\text{PT},\text{mv}}+{\delta }_{\text{PT},\text{mv}}}{r_{\text{PT}}}+\frac{l_{\text{PT},\text{mv}}^2}{2r_{\text{PT}}^2}\right)$$ (3)

where l_PT,mv = 2.5 μm is the microvillous length, and δ_PT,mv = 0.15 μm denotes the height above the microvillous tip where drag is considered (5). The density of apical and basolateral transporters in proximal tubule cells is scaled by:

$$1+s\left(\frac{{\tau }_{\mathrm{P}\mathrm{T}}}{{\tau }_{\mathrm{P}\mathrm{T}}^0}-1\right)$$ (4)

where the reference torque ${\tau }_{\text{PT}}^0$ is evaluated at the reference flow, which is set to the inflow of the proximal tubule. The scaling factor s is taken to be 1.30 in sham and 1.25 in UNX and 5/6-NX along the S1–S2 segment; it is reduced by one-half along the S3 segment.

Additional changes in model parameters.

In UNX, proximal tubule ammoniagenesis rate is assumed unchanged from sham level (24); in 5/6-NX, proximal tubule ammoniagenesis rate per unit length is increased fourfold to approximate the ammonia excretion rate reported by Buerkert et al. (2).

In UNX and 5/6-NX, interstitial concentrations are the same as in sham (see Table 2 in Ref. 19), except for urea and pH. In 5/6-NX, plasma urea concentration is assumed to increase fourfold to 32 mM (17). Within the outer medulla, interstitial urea concentration increases linearly from 32 mM to sham value (60 mM) at the outer-inner medullary boundary. Within the inner medulla, the interstitial urea concentration profile is assumed unchanged. In sham, plasma pH is 7.323, and interstitial fluid pH decreases linearly to 7.0 at the papillary tip. To simulate metabolic acidosis, plasma and interstitial fluid pH is uniformly reduced by 0.1 in UNX and by 0.15 in 5/6-NX.

Table 2.

Transport parameter changes for UNX and 5/6-NX models

	Na+-K+-ATPase	NKCC2	NCC	ENaC
UNX, case 1	no change	+5	+10	+30
UNX, case 2	+3	+5	no change	+30
5/6-NX, case 1	no change	+20	+20	+60
5/6-NX, case 2	+6	+8	no change	+50

Values are in percent.

RESULTS

Adjusting GFR and tubular dimensions only.

We conducted model simulations for sham, UNX, and 5/6-NX groups to assess the impacts of the changes in GFR and tubular dimensions (as described above). In UNX, SNGFR is taken to be 45 and 54 nl/min in the superficial and juxtamedullary nephrons, respectively; in 5/6-NX, SNGFR is taken to be ~64 nl/min in all nephrons. For comparison, in sham, SNGFR is taken to be 30 and 45 nl/min in the superficial and juxtamedullary nephrons, respectively. Transporter expression densities (i.e., the numbers of transporter per epithelial cell surface area) are assumed to remain at sham levels, with the exception of the torque effects. With these assumptions, the increase in the total amount of transport proteins can be computed from the product of tubular length and tubular circumference. In UNX, the total amount of transport proteins increases by 58 and 31%, respectively, in the proximal tubule and downstream tubular segments. In 5/6-NX, the total amount of transport proteins increases by 149 and 90%, respectively, in the proximal tubule and downstream tubular segments. In these simulations, plasma and interstitial fluid pH are lowered (see above), and plasma urea concentration is increased fourfold in 5/6-NX (which reflects the ~3/4 reduction in GFR).

We conducted model simulations in which only GFR and tubular dimensions are adjusted to assess the extent to which urinary excretion rate can be maintained when transporter density is maintained at sham level (sham). Key results are summarized in Table 3, rows labeled “Case 0.” Furthermore, the predicted Na⁺, K⁺, and Cl⁻ excretion rates are shown in Fig. 1 for UNX and Fig. 2 for 5/6-NX; compare bars labeled “Sham” and “UNX, Sham transport” or “5/6-NX, Sham transport.”

Table 3.

Predicted urinary excretion rates

Urine	Flow, μl/min	Rel. Sham	Na+, μmol/min	Rel. Sham	K+, μmol/min	Rel. Sham	Cl−, μmol/min	Rel. Sham	Urea, μmol/min	Rel. Sham	TA, μmol/min	Rel. Sham	Net Acid, μmol/min	Rel. Sham
Sham	30.3		5.0		2.1		3.7		7.4		2.1		3.1
UNX
Case 0	31.5	1.0	5.9	1.2	1.9	0.89	5.4	1.4	6.8	0.92	1.7	0.82	2.6	0.81
Case 1	29.6	0.98	5.0	1.0	2.1	1.0	4.8	1.3	6.6	0.89	1.7	0.83	2.6	0.82
Case 2	29.3	0.97	4.9	1.0	2.1	0.99	4.7	1.3	6.6	0.88	1.7	0.83	2.6	0.82
5/6-NX
Case 0	27.9	0.92	5.1	1.0	1.4	0.66	5.6	1.5	6.2	0.83	0.71	0.34	1.2	0.39
Case 1	31.6	1.0	5.2	1.1	2.2	1.0	6.4	1.7	6.9	0.92	0.75	0.36	1.3	0.43
Case 2	31.6	1.0	5.3	1.1	2.1	1.0	6.5	1.7	6.8	0.92	0.74	0.36	1.3	0.42

Rel. Sham, relative to sham. Case 0, GFR and tubular dimensions only were adjusted. Cases 1 and 2, changes in transporter expression were accounted for as well. Values are given per kidney. TA, titratable acid.

Fig. 1.

Effects of varying individual transporter expression in UNX on urinary excretion rates, given per animal. Results were obtained for sham and three UNX cases: one case with only GFR and tubular dimensions adjusted, and two cases with transport expression varied as well. A: urine flow. B: Na⁺ excretion. C: K⁺ excretion. D: Cl⁻ excretion. Ammonium transport (${\mathrm{Q}}_{\mathrm{N}{\mathrm{H}}_{\mathrm{4}}}$): a = 2.0, b = 5.0; Na⁺-K⁺-ATPase: a = 1.02, b = 1.05; NKCC2 and NCC: a = 1.1, b = 1.2; ENaC: a = 1.2, b = 1.5.

Fig. 2.

Effects of varying individual transporter expression in 5/6-NX on urinary excretion rates, given per animal. Results were obtained for sham and three 5/6-NX cases: one case with only GFR and tubular dimensions adjusted, and two cases with transport expression varied as well. A: urine flow. B: Na⁺ excretion. C: K⁺ excretion. D: Cl⁻ excretion. ${\mathrm{Q}}_{\mathrm{N}{\mathrm{H}}_{\mathrm{4}}}$: a = 2.0, b = 5.0; Na⁺-K⁺-ATPase: a = 1.02, b = 1.05; NKCC2 and NCC: a = 1.1, b = 1.2; ENaC: a = 1.2, b = 1.5.

In the absence of changes in filtration or transport (i.e., no change in GFR or tubular dimensions), urinary excretion would decrease at the same rate as nephron population, i.e., by 50% in UNX. By increasing both GFR and tubular dimensions, the UNX model predicts a urine flow unchanged from sham. Concomitantly, urinary excretion of Na⁺ and Cl⁻ increases by 14 and 35%, respectively, whereas urinary excretion of K⁺ is 15% below sham level.

In 5/6-NX, changes in GFR and tubular dimensions alone would decrease urine flow by 10%. Somewhat surprisingly, Na⁺ homeostasis is attained with Na⁺ excretion approximately the same as sham. The excretion of Cl⁻ is 44% above sham, whereas that of K⁺ is 35% below. It is noteworthy that, in the absence of changes in filtration or transport, urinary excretion would decrease by 83% in 5/6-NX (same as nephron population reduction). Thus, while changes in GFR and tubular dimensions alone do not maintain water, Na⁺, or K⁺ balance, they result in substantial progress toward that goal.

Taken together, the adaptations in GFR and tubular dimensions following nephron removal appear to generally enhance the renal excretion of NaCl at the expense of that of K⁺. Transport protein expression would need to adjust accordingly to attain salt balance.

Adjusting transporter expression individually.

Changes in transporter expression in nephrectomized kidneys have been reported (3, 17, 18). In the next set of simulations, we separately varied proximal tubule ammoniagenesis rate, and the protein density (amount per epithelial surface) of Na⁺-K⁺-ATPase, Na⁺-K⁺-2Cl⁻ cotransporter (NKCC2), Na⁺-Cl⁻ cotransporter (NCC), and epithelial Na⁺ channel (ENaC), and we assessed the effects on tubular transport and urine excretion. We focused on these transporters because changes in their protein expression levels have been reported in remnant kidneys (17); furthermore, together these transporters form a small set of transport parameters, the appropriate adjustments of which yield urine flow and urinary salt excretion rates in UNX and 5/6-NX models that are similar to sham levels. In these simulations, GFR and tubular dimensions were adjusted as described above. Results are summarized in Figs. 1 and 2.

We conducted UNX and 5/6-NX simulations in which proximal tubule ammoniagenesis rate per unit length was increased two- and fivefold. A higher ammoniagenesis rate reduces proximal reabsorption and generally increases the excretion of fluid, Na⁺, K⁺, and Cl⁻. When proximal tubule ammoniagenesis rate was increased fivefold, urine flow was increased by 64% in UNX and by 38% in 5/6-NX, whereas urinary excretion of Na⁺, K⁺, and Cl⁻ increased by 73, 71, and 100%, respectively, in UNX, and by 45, 35, and 49%, respectively, in 5/6-NX. Ammoniagenesis is accompanied by the production of ${\mathrm{HCO}}_3^{-}$ (26), which escapes via the Na⁺-${\mathrm{HCO}}_3^{-}$ cotransporter on the basolateral membrane of the proximal tubule. The higher intracellular [${\mathrm{NH}}_4^{+}$] (where brackets denote concentration) increases transport across the Na⁺/H⁺ exchanger NHE3, with some of the H⁺ replaced by ${\mathrm{NH}}_4^{+}$. The higher NHE3 flux increases transcellular Na⁺ reabsorption. The higher intracellular [${\mathrm{NH}}_4^{+}$] also increases the secretion of ${\mathrm{NH}}_4^{+}$ into the lumen, which, taken in isolation, should raise luminal fluid osmolality and reduce water reabsorption and lower the concentrations of other solutes. (The model assumes that the interstitial fluid composition is known a priori.) As a result, the paracellular diffusion of Na⁺, K⁺, and Cl⁻ decreases. The competing effects of increased transcellular Na⁺ transport (denoted T_Na) and decreased paracellular T_Na result in lower overall T_Na along the proximal tubule. Lower Na⁺ reabsorption is accompanied by reduced water reabsorption. The elevated NHE3-mediated Na⁺ reabsorption increases intracellular [Na⁺], and thus Na⁺-K⁺-ATPase activity and basolateral K⁺ influx across the pump, thereby reducing K⁺ reabsorption along the proximal tubule.

Increasing the expression of Na⁺-K⁺-ATPase, NKCC2, or NCC has the similar effect of decreasing urine flow as well as the excretion rates of Na⁺, K⁺, and Cl⁻. Increasing Na⁺-K⁺-ATPase expression in all segments by 5% raises the reabsorption of Na⁺, K⁺, and Cl⁻ along the nephron, resulting in a slight decrease in urine flow, as well as in Na⁺, K⁺, and Cl⁻ excretion rates (<5%). Increasing NKCC2 expression raises the reabsorption of Na⁺, K⁺, and Cl⁻ along the thick ascending limb, therefore decreasing their excretion rates, whereas increasing NCC expression raises the reabsorption of Na⁺ and Cl⁻ along the distal convoluted tubule, and eventually decreases their excretion rates, as well as that of K⁺, due to the lower delivery of Na⁺ downstream in exchange for K⁺.

Increasing ENaC expression augments the exchange of Na⁺ and K⁺ along the connecting tubule and collecting duct, thereby increasing K⁺ excretion, while reducing Na⁺, Cl⁻, and fluid excretion. In particular, a 50% increase in ENaC expression raises urinary K⁺ excretion rate by 25 and 26% in UNX and 5/6-NX, respectively.

Simulating UNX and 5/6-NX kidney function.

We assume that our model UNX and 5/6-NX kidneys belong to rats that are otherwise healthy, despite the nephrectomy. Moreover, we assume that the surgery has no impact on the animals’ fluid and salt intake. As a result, at steady state, one expects the sham, UNX, and 5/6-NX models to predict similar urine flow and salt excretion rates. Model simulations above indicate that nephrectomy-induced SNGFR increase and tubular hypertrophy go a long way to normalize excretion, but alone are insufficient to fully maintain salt balance. Thus, in a series of simulations, we identified several sets of changes in key transport parameters with which the UNX and 5/6-NX models predict urine flow and urinary Na⁺ and K⁺ excretions that are similar to sham levels.

Four selected parameter sets (two for each of the UNX and 5/6-NX models) are shown in Table 2. These transport parameters were applied to the model, together with changes in GFR, tubular dimensions, ammoniagenesis rate, plasma urea concentration, and pH (see above). Each of these parameter sets was chosen so that it yields urinary Na⁺ and K⁺ excretion rates that are close to sham levels. Table 3 shows urine flow and excretion rates of Na⁺, K⁺, Cl⁻, urea, titratable acid (TA), and net acid, for each of these cases. We computed the ratios of UNX and 5/6-NX excretion rates relative to sham. By design, the UNX and 5/6-NX models attain water, Na⁺, and K⁺ balance by maintaining excretion rates similar to sham. Cl⁻ excretion rate is predicted to be 27 and 73% higher in UNX and 5/6-NX, respectively, whereas urea excretion is 12 and 8% lower. Similarly, both TA and net acid excretion rates are significantly lower in UNX and 5/6-NX, a result that is consistent with the observations of Buerkert et al. (2), who reported that, in 2/3-NX, TA and net acid excretion rates are 86 and 65%, respectively, of sham. The decrease in urine [${\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{4}}^{\mathrm{-}}$] is balanced, to some extent, by the increase in [Cl⁻].

Below we provide a more in-depth analysis of the simulation results for case 1. In this case, the expression density of NKCC2, NCC, and ENaC is upregulated, but that of NHE3 remains unchanged, consistent with the protein expression findings of Kim et al. (17). Results for case 2, in which the expression of Na⁺-K⁺-ATPase is upregulated (14), are qualitatively similar. Figure 3 shows delivery of key solutes (Na⁺, K⁺, Cl⁻, TA, and urea, obtained for case 1) and fluid to the inlets of individual nephron segments. Values are given per animal, separately for superficial and juxtamedullary nephrons (denoted with colored and white bars, respectively). That is, superficial and juxtamedullary delivery values are scaled by 2/3 and 1/3, respectively, according to nephron populations. Generally, filtered loads are lower in the nephrectomized models due to the reduction in overall GFR. Exceptions are TA, because of the assumption of lower pH values in UNX and 5/6-NX, and urea, because of the elevated plasma urea concentration in 5/6-NX.

Fig. 3.

Comparison of solute delivery [Na⁺ (A), K⁺ (B), Cl⁻ (C), TA (E), and urea (F)] and fluid delivery (D) obtained for sham, UNX (case 1), and 5/6-NX (case 1), given per animal. Colored bars denote superficial nephron values and are shown above white bars, which denote juxtamedullary nephron values. Height of colored bars denotes whole kidney values. PT, proximal tubule; DL, descending limb; mTAL, medullary thick ascending limb; DCT, distal convoluted tubule; CNT, connecting tubule; CCD, cortical collecting duct.

Fluid pressure along the superficial nephron is shown in Fig. 4. Pressure profiles along the juxtamedullary nephrons are qualitatively similar. SNGFR and fluid flow per tubule are higher in UNX and 5/6-NX; however, tubular diameter is also increased in UNX and 5/6-NX. These competing factors yield similar pressure decreases along the initial nephron segments. Along the collecting duct, the NX-to-sham fluid flow ratios become sufficiently large that pressure drops in UNX and 5/6-NX are 143 and 284% of sham.

Fig. 4.

Profiles of tubular fluid pressure along a superficial nephron, obtained for sham, UNX, and 5/6-NX. Tubular segments are labeled for sham and 5/6-NX. SDL, short descending limb; TAL, thick ascending limb; CD, collecting duct.

We then compared fractional salt and fluid deliveries among the three models. To that end, we added up the delivery values in Fig. 3, divided them by the filtered values, and computed whole kidney fractional delivery of Na⁺, K⁺, Cl⁻, and fluid to individual nephron segments (see Fig. 5). All three models predict that the proximal tubule reabsorbs ~2/3 to 3/4 of the filtered loads. Also, >90% of the filtered K⁺ is reabsorbed upstream of the distal convoluted tubule, with K⁺ balance attained via secretion by the downstream segments. Generally, these results show a progressively more significant increase in fractional delivery of solutes and fluid to downstream segments in the nephrectomized kidney models, especially in 5/6-NX. This is not unexpected, since these models are designed to have the same urinary excretion rates as shams, despite reduced GFR. Fractional excretion of Na⁺ was determined to be 1.4, 2.0, and 4.7%; K⁺, 17, 25, and 57%; Cl⁻, 1.3, 2.3, and 6.7%; and fluid, 1.2, 1.7, and 4.1% for sham, UNX, and 5/6-NX, respectively.

Fig. 5.

Comparison of fractional solute delivery [Na⁺ (A), K⁺ (B), Cl⁻ (C)] and fluid delivery (D) obtained for sham, UNX, and 5/6-NX, given per animal. Fractional delivery of solutes and fluid to downstream segments is significantly higher in 5/6-NX, especially for K⁺. Notations are as defined in Fig. 3 legend.

We further analyzed simulation results to address the question: Which segments are primarily responsible for fluid and salt balance in the UNX and 5/6-NX models? At the inlet of each nephron segment, we compared the predicted whole animal solute and fluid deliveries between the UNX and 5/6-NX models and sham. Fractional deviations from sham were computed for Na⁺, K⁺, Cl⁻, and fluid and are shown in Fig. 6. A value of zero indicates that whole animal segmental delivery in UNX or 5/6-NX matches that of sham; a negative value indicates that whole animal segmental delivery is lower than sham. With this notation, a negative “urine” value implies a lower urinary excretion than sham and thus a positive fluid or solute balance, i.e., retention. Because of the lower overall GFR in UNX and, even more substantially, in 5/6-NX, the filtered fluid and solute loads of the two nephrectomized kidney models are substantially below sham levels. Those negative deviations are maintained and left essentially uncorrected along the proximal tubules and the descending limbs. This indicates that fractional reabsorption along these segments is similar to sham. Downstream from the thick ascending limbs, Na⁺ and Cl⁻ delivery values gradually converge to sham levels. This implies that, considering the animal as a whole, these downstream segments reabsorb less NaCl on a fractional basis compared with sham. However, because only a fraction of the nephron population remains in the UNX and 5/6-NX kidneys, each individual nephron segment actually reabsorbs more NaCl than their sham counterpart, owing to the larger transport area, increased delivery, and, in some cases, increased transporter density. For K⁺, the collecting duct makes the largest contribution to achieving homeostasis, especially in 5/6-NX (Fig. 6B). As can be seen also in Fig. 3B, the sham collecting duct system reabsorbs much more K⁺ (65% of K⁺ entering the cortical collecting duct) than the 5/6-NX collecting duct system (only 30% of entering K⁺). Recall, however, that there are only 1/6 as many collecting ducts represented in the 5/6-NX model as in sham. Thus each of the 5/6-NX collecting duct actually reabsorbs ~40% more K⁺ than sham. Water balance is achieved, in large part, by reduced reabsorption (in a whole kidney sense) along the connecting tubules and collecting ducts.

Fig. 6.

Deviations of solute [Na⁺ (A), K⁺ (B), Cl⁻ (C)] and volume (D) along the UNX and 5/6-NX nephron from sham levels. A value of “0” (dotted line) indicates that solute delivery to a given segment, given per animal, matches that in sham. Notations are as defined in Fig. 3 legend.

Next we assessed the extent to which O₂ consumption (denoted ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$) is affected by the changes in solute transport in UNX and 5/6-NX, with transporter adaptation (case 1) incorporated. As in a previous study (19), we assume that ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ consists of an active component and a basal component. Active ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ (or ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$) provides the energy needed to actively reabsorb Na⁺. We assume that ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$ is proportional to (specifically, 1/15 of) the rate of T_Na across Na⁺-K⁺-ATPase pumps (19).

We computed 1) transcellular active Na⁺ transport (denoted ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$), which incurs ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$; 2) total Na⁺ transport (denoted ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$), i.e., the sum of transcellular and paracellular Na⁺ transport; 3) total ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ (denoted ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$), given by the sum of basal and active ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$; and 4) transport efficiency, given by the ratio ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$/${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$.

Results for segmental T_Na and ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ are shown in Figs. 7 and 8. We analyzed these results separately for superficial and juxtamedullary nephrons (see Fig. 7), and we also determined overall nephron function and metabolism (see Fig. 8). As done in Fig. 3, values in Fig. 7 are scaled by the respective nephron populations. In the UNX and 5/6-NX kidneys, GFR and thus filtered Na⁺ load are reduced by 29 and 68%, respectively. Thus total Na⁺ reabsorption decreases (Fig. 8B). However, the nephron population reduction (50 and 83%, respectively) far exceeds the reduction in GFR. That results in a higher workload per nephron, yielding higher ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$, ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, and ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ along all nephron segments (Figs. 7, A–C, and 8A).

Fig. 7.

Na⁺ transport (T_Na) and O₂ consumption (${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$) in sham, UNX, and 5/6-NX kidney, given per nephron. A: segmental ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}$, taken positive for reabsorption. B: segmental ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$. C: segmental ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$. D: segmental ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$-to-${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ ratios. Colored bars denote superficial nephron values and are shown above white bars, which denote juxtamedullary nephron values.

Fig. 8.

Nephron function obtained for sham, UNX, and 5/6-NX, given per nephron (A) and per animal (B). C: tubular transport efficiency given by the ${\mathrm{T}}_{\mathrm{N}\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$-to-${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ ratio. White segments, outer medullary values; bottom colored segments, cortical values; blue segments, inner medulla (values not shown). Whole nephron (A) and whole animal (B) values are indicated above the bars.

While the transport efficiency of the proximal tubule is similar between sham and UNX, it is significantly reduced in 5/6-NX, due to the elevated transcellular T_Na induced by the higher ammoniagenesis rate (see above). Downstream from the proximal tubule, transport efficiency does not differ significantly among the three groups, except for the collecting duct. Along the collecting duct, the higher luminal [Na⁺] in UNX and 5/6-NX facilitates paracellular Na⁺ reabsorption (albeit by only a small amount) and increases transport efficiency (Fig. 7D).

The net effect of these competing changes in segmental transport efficiency is a minimal change in whole animal renal transport efficiency (Fig. 8C): a slight increase in UNX and a slight decrease in 5/6-NX. Despite the small increase in transport efficiency along some of the downstream segments, the fact that per-nephron ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ increases everywhere (Fig. 7C) suggests that, without compensatory increase in per-nephron O₂ supply, the UNX and 5/6-NX kidneys may have a higher risk of hypoxia.

The above analysis focuses on case 1. Simulation results obtained using parameters in case 2 are qualitatively similar. Despite the differences in transporter expression, overall T_Na and ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ results corresponding to the case 2 parameter set differ minimally from case 1. That similarity may be attributed to the fact that these parameter adjustments were chosen to “fine-tune” urinary Na⁺ and K⁺ excretion rates. As previously noted, Na⁺ excretion is a small fraction of its filtered load. As can be seen in Table 3, even without changes in transporter expression, UNX and 5/6-NX urinary excretions do not differ substantially (albeit significantly) from sham. Thus the different transport expressions in the three cases yield only minimal differences in overall T_Na and ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$.

DISCUSSION

The kidney has the remarkable ability to maintain the homeostasis of water and electrolytes, by adjusting the filtration rate and tubular reabsorptive and secretory processes. After ablation of renal mass, assuming no change in dietary intake, the surviving nephrons must maintain approximately the same urine flow and solute excretion. Indeed, Na⁺ and K⁺ balance has been demonstrated in subtotal nephrectomy models (8, 9). Given the reduction in nephron population, Na⁺ and K⁺ excretion rate per nephron must increase (inversely proportional to the amount of renal tissue remaining). That can be achieved by reducing the reabsorptive capacity of the surviving nephrons, or by increasing their filtered load, or by combining an even larger increase in their filtered load with an increase in the reabsorptive capacity. We consider the implications of each of these responses.

What would it take to maintain salt balance by reducing the reabsorptive capacity of the surviving nephrons?

We will determine the extent to which tubular reabsorptive capacity must be altered to achieve homeostasis. The excretion of Na⁺ constitutes a very small fraction of its filtered load (1.4% in controls). In contrast, substantially larger fractions of filtered K⁺ (17%) and urea (37%) are excreted. If SNGFR were not increased, then, following UNX, 2 × 17 = 34% and 2 × 37 = 74% of the filtered K⁺ and urea, respectively, would have to be excreted to maintain solute balance. Even though that may be within the physiological capacity, the remaining kidney would likely have trouble handling additional K⁺ or urea loads. The situation becomes even more challenging in 5/6-NX. In the absence of SNGFR increases, the remaining nephrons would have to excrete the same amount of K⁺ as its filtered amount (6 × 17 = 104%). Taking into account the elevated plasma urea concentration, the surviving nephrons would have to excrete more than one-half of the filtered urea (6 × 37 × 1/4 = 56%). Thus the kidney’s capacity to respond to a K⁺ or urea load would be limited. These considerations suggest the necessity of compensatory increases in renal blood flow and SNGFR following reduction in renal mass.

What are the implications of increased SNGFR in remnant kidneys?

Increases in SNGFR have been demonstrated following the surgical removal of one kidney. A marked increase of ~30–70% above sham values in GFR for one kidney has been observed in experimental animals and humans (4, 14, 21). This implies that the remaining kidney handles 65–85% of the filtered loads previously handled by both of the intact kidneys. After the ablation of 80% of renal mass in the rat, mean nephron GFR has been reported to increase by 136% (16). In that case, the remnant kidney filters ~30% of the amount previously filtered by the two intact kidneys. Maintaining filtered loads as close to control levels as possible means that the fractional solute excretion rates are not substantially increased. This may help preserve the kidney’s ability to handle acute solute loads, especially for K⁺ and urea.

After nephrectomy, renal blood flow appears to redistribute among different nephron populations, with the fractional increase in SNGFR in superficial nephrons about twice as high as that in juxtamedullary nephrons (15). Consequently, in a 5/6-NX kidney, SNGFR may be approximately the same in all nephrons, whereas, in controls, the SNGFR of juxtamedullary nephrons is 50% higher than that of superficial nephrons. The reason for the heterogeneous increase in SNGFR in nephrectomized kidneys is unclear. One possible explanation may be to limit hyperfiltration in juxtamedullary nephrons, while maintaining sufficiently high overall GFR. Indeed, it has been suggested that single-nephron hyperfiltration may have maladaptive consequences by damaging the remnant glomeruli (10), which may contribute to the progression to end-stage renal failure. A stronger increase in GFR in juxtamedullary nephrons may also further impair urine concentration by overloading the descending loops, given that the number of nephrons to establish the corticomedullary concentration gradient is reduced.

Maintaining a sufficiently high GFR in a nephrectomized kidney means that the reabsorptive capacity of the surviving nephrons must increase. Within 1–2 days after UNX, the weight of the contralateral kidney has been observed to increase in the rat (13) and mouse (20). The extent of compensatory renal growth in the rat has been shown to correlate closely with the amount of renal tissue that is surgically removed. In a study involving progressive ablation of renal mass in rats, Kaufman et al. (15) reported that the weight of the remaining renal tissue increased by 81% in UNX rats and by 168% in rats with surgical ablation of 70% of renal mass. That compensatory growth involves both hyperplasia and hypertrophy. It is noteworthy that kidney hypertrophy has been observed as a response to increases in protein intake, which is also associated with nephron hyperfiltration (1, 6).

The key question of this study is: With concomitant increases in filtered load and tubular transport capacity, to what extent can the UNX and 5/6-NX kidneys achieve water and salt balance, without adjusting the density of transporter expression? In UNX, we simulated 50 and 25% increases, respectively, in SNGFR of superficial and juxtamedullary nephrons, which correspond to a 42% increase in overall filtration. Those increases are accompanied by the 58 and 31% increases in the transport areas of proximal tubule and downstream tubular segments. The model predicts that, taken together, these changes in GFR and tubular growth yield urine excretion that is surprisingly close to control values: urine flow is almost the same as control, whereas Na⁺, K⁺, and urea excretion rates all deviate by <15% from control values [see Table 3, row “UNX (Case 0)”]. Perfect homeostasis is not achieved, but, given the drastic change in simulated kidney mass, the fact that the model predicts urinary excretion so close to control values, having taken into account only adaptations in GFR and tubular growth, is indeed quite remarkable. In the 5/6-NX simulation, the even more drastic reduction in renal tissue is accompanied by more substantial increases in GFR and tubular transport capacity. Again, the model predicts that these changes yield urinary excretion that deviates from, but remains reasonably close to, control levels [Table 3, row “5/6-NX (Case 0)”]. These results evince the exceptional, efficient, and well-coordinated adaptivity of glomerular filtration and tubular growth.

The above results also indicate that, to attain full homeostasis of water and electrolytes, transport protein expression density needs to adjust accordingly. Capasso et al. (3) reported that, in rats 15 days after 4/6-NX surgery, NHE3 protein expression increases in the remaining thick ascending limbs. Kim et al. (17) analyzed Na⁺ and K⁺ transporters in remnant rat kidneys 4, 8, and 12 wk after 5/6-NX surgery and found changes in NKCC2, NCC, ENaC, and ROMK protein expression levels at different time points. Kwon et al. (18a) reported reduced whole-kidney aquaporin expression (AQP1, AQP2, and AQP3) in kidneys of rats having undergone 5/6-NX surgery. The experimental protocols differ somewhat from the nephrectomized kidneys simulated in the present study (e.g., amount of renal mass removed or the time between surgery and experiment). Also, these studies report transporter mRNA or protein expression levels, which correlate with, but may not translate directly to, transporter density and activity. Given these considerations, we adopted the “reverse engineering” approach: instead of choosing model transport parameters based on these studies, we chose those parameters so that urinary fluid, Na⁺, and K⁺ excretion rates in UNX and 5/6-NX match the corresponding control values. These parameter sets are shown in Table 2.

Kim et al. (17) observed that, 4 wk after 5/6-NX, the protein abundance of NHE3 did not change, whereas the abundance of NKCC2, NCC, and ENaC in the remnant rat kidney increased substantially (to 14.3-, 6.7-, and 13.0-fold of control, respectively, computed per nephron). Consistent with those findings, our model simulations suggest that increasing the density of NKCC2, NCC, and ENaC may allow the body to reach salt balance (case 1). The predicted increases in transporter density (20, 20, and 60% for NKCC2, NCC, and ENaC, respectively) are given in Table 2, row “5/6-NX, case 1”. Taking into account tubular hypertrophy and reduction in nephron population, these values correspond to 2.3-, 2.3-, and 3.0-fold increases in NKCC2, NCC, and ENaC transporter expression per nephron, respectively. These increases are lower than the reported increases in protein levels by Kim et al. (17). That discrepancy may, in part, be attributed to 1) the quantitative uncertainty (12) in the immunoblotting results of Kim et al., and 2) the difficulties in converting these data into transport expression per nephron, which requires assumptions that may be invalid (e.g., the actual reduction in nephron population following the 5/6-NX procedure in Ref. 17), and in converting these data into transport activity (e.g., which protein fraction is in the cell membrane and active).

In a micropuncture study, Hayslett et al. (9) reported that fractional fluid reabsorption in the proximal convoluted tubule and distal convoluted tubule was similar in control and UNX rats. Our simulation results agree, indicating that water homeostasis in UNX and 5/6-NX is achieved primarily through appropriate transport adjustments in the connecting tubule and collecting duct (Fig. 6D). Similarly, and according to case 1, balance of Na⁺ and K⁺ in UNX and 5/6-NX can be attributed primarily to the fine-tuning of transepithelial transport in distal nephron segments (Fig. 6, A and B). The model predicts enhanced Cl⁻ excretion in UNX and 5/6-NX. The Cl⁻/${\mathrm{HCO}}_3^{-}$ exchanger pendrin in intercalated cells has been implicated in Cl⁻ reabsorption in physiological and pathophysiological states (e.g., hypertension) (7, 11). Thus pendrin would be a potential candidate to normalize chloride excretion, if upregulated in response to reduced nephron number.

Given that the filtered load among each of the surviving nephrons is significantly elevated in a nephrectomized kidney, we computed the resulting changes in ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$. Model simulations suggest that ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ per nephron segment increases substantially: compared with sham, ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ increases by 36 and 108% along the proximal tubule of the UNX and 5/6-NX models, respectively, and by 33 and 88% along the thick ascending limb. However, because nephron populations decrease even more, whole animal renal ${\mathrm{Q}}_{{\mathrm{O}}_{\mathrm{2}}}$ is reduced, to ~2/3 and 1/3 of sham levels in the UNX and 5/6-NX models, respectively. Those decreases are distributed uniformly between the cortical and medullary segments.

Given that ${\mathit{Q}}_{{\mathit{O}}_{\mathit{2}}}$ decreases overall but increases per nephron, is a nephrectomized kidney at higher risk for hypoxia?

To answer that question, we must consider O₂ supply. After nephrectomy, blood flow to surviving nephrons is found to correlate inversely with the amount of functioning tissues. In UNX rats, renal blood flow per surviving nephron was found to double in the contralateral kidney, and to increase fourfold in a kidney with 80% of renal tissue removed (16). Thus total renal blood flow increases more than whole animal renal O₂ requirement in UNX and 5/6-NX. However, the nephron segments most vulnerable to hypoxic injury are likely the S3 segment and the medullary thick ascending limb, both of which are located in the renal medulla. Thus it is more pertinent to consider changes in medullary blood flow. Given that SNGFR increases more in superficial nephrons compared with juxtamedullary nephrons, it is possible that a larger fraction of the renal blood flow is directed to the cortex rather than the medulla. A better characterization of medullary blood flow in remnant kidneys would provide the necessary information for determining the associated risk for hypoxia.

GRANTS

This research was supported by the Department of Veterans Affairs (to V. Vallon) and by the National Institutes of Health National Institute of Diabetes and Digestive and Kidney Diseases Grants R01-DK-56248 (to V. Vallon), R01-DK-106102 (A. T. Layton and V. Vallon), and the University of Alabama at Birmingham/University of California San Diego O’Brien Center for Acute Kidney Injury Research NIH-P30-DK-079337 (to V. Vallon).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

A.T.L. and V.V. conceived and designed research; A.T.L. performed experiments; A.T.L. analyzed data; A.T.L., A.E., and V.V. interpreted results of experiments; A.T.L. prepared figures; A.T.L. drafted manuscript; A.T.L., A.E., and V.V. edited and revised manuscript; A.T.L., A.E., and V.V. approved final version of manuscript.