Predicted effect of circadian clock modulation of NHE3 of a proximal tubule cell on sodium transport

Abstract

Major renal functions such as renal blood flow, glomerular filtration rate, and urinary excretion are known to exhibit circadian oscillations. However, the underlying mechanisms that govern these variations have yet to be fully elucidated. To better understand the impact of the circadian clock on renal solute and water transport, we have developed a computational model of the renal circadian clock and coupled that model to an epithelial transport model of the proximal convoluted cell of the rat kidney. The activity of the Na⁺-H⁺ exchanger 3 (NHE3) is assumed to be regulated by changes in transcription of the NHE3 mRNA due to regulation by circadian clock proteins. The model predicts the rhythmic oscillations in NHE3 activity, which gives rise to significant daily fluctuations in Na⁺ and water transport of the proximal tubule cell. Additionally, the model predicts that 1) mutation in period 2 (Per2) or cryptochrome 1 (Cry1) preserves the circadian rhythm and modestly raises Na⁺ reabsorption; 2) mutation in Bmal1 or CLOCK eliminates the circadian rhythm and modestly lowers Na⁺ reabsorption; 3) mutation in Rev-Erb or ROR-related orphan receptor (Ror) has minimal impact on the circadian oscillations. The model represents the first step in building a tool set aimed at increasing our understanding of how the molecular clock affects renal ion transport and renal function, which likely has important implications for kidney disease.

INTRODUCTION

Circadian clocks are cell-autonomous time-keeping mechanisms that organize physiological functions in a 24-h periodicity. In mammals, the circadian system comprises a hierarchy of oscillators that function at the cellular, tissue, and systems levels. The central circadian clock, which is located in the suprachiasmatic nucleus (SCN) of the hypothalamus, acts as a central pacemaker for the organism that contributes to rhythms in behavior, body temperature, and hormones (23). The circadian clock mechanism is present in most peripheral organs, and cells and these peripheral clocks can oscillate independently of the SCN (48). Whereas light is the dominant zeitgeber, or synchronization cue, for the SCN, food intake appears to be the most potent zeitgeber for peripheral clocks including the liver and kidney (36). Exactly how each peripheral clock receives signals from and responds to the SCN is currently an area of active investigation.

In both animals and humans, plasma Na⁺ concentration and renal Na⁺ excretion have been observed to exhibit significant diurnal fluctuations (8, 27). While the molecular mechanisms that drive these diurnal variations have yet to be completely elucidated, a number of genes that play key roles in these processes have been identified. Here, we focus on the rhythmic regulation of the Na⁺/H⁺ exchanger (NHE3). NHE3 is expressed on the apical membrane of the proximal tubule cell and catalyzes the electroneutral exchange of one extracellular Na⁺ for one intracellular H⁺ (44). The expression of NHE3 has been shown to exhibit diurnal variations at the level of mRNA, protein, and localization and is regulated directly by CLOCK:BMAL1 heterodimers of the peripheral circadian clock of the kidney (33).

In the rat, the proximal tubule is responsible for reabsorbing approximately two-thirds of the filtered Na⁺ and volume, with a large fraction of that reabsorption mediated by NHE3. Proximal tubule specific knockout of NHE3 resulted in substantial reduction in bicarbonate and volume reabsorption in mutant mice, although no significant change in systolic blood pressure was observed (21). Thus, as the expression of NHE3 fluctuates diurnally, the transport of water and sodium by the proximal tubule is expected to fluctuate correspondingly, likely resulting in variations in water and salt excretion. To assess the extent to which proximal tubule cellular transport is modulated by the clock gene circadian rhythm, we have developed the very first computational model of the renal circadian clock and connected the clock model to a published model of the proximal convoluted tubule epithelial cell model of the rat. The resulting model was applied to simulate changes in cellular transport in wild type and mutants.

MODEL FORMULATION

To simulate the modulation of NHE3 activity by the circadian clock, we first developed a mathematical model of the core clock in the rat kidney. The clock model comprises a number of transcription factors that regulate gene expression: period (Per), cryptochromes (Cry), Rev-Erb and RAR-related orphan receptor retinoic acid receptor-related orphan receptor (Ror), brain and muscle ARNT-Like 1 (Bmal1), and circadian locomotor output cycles kaput (CLOCK). A schematic diagram of the clock network is shown in Fig. 1. The model represents the direct regulation of NHE3 expression level by Bmal1 and CLOCK via a transcriptional mechanism (33). Equations for the clock model are given in the appendix. Model parameters were obtained by fitting predicted profiles for mRNA expression levels of Per, Cry, Rev-Erb, Ror, Bmal1, and NHE3 with their corresponding experimentally measured profiles reported in Ref 49 and database (http://circadb.hogeneschlab.org), obtained for the dark-dark cycles.

Fig. 1.

Schematic network of our mathematical model, which describes the interactions of core clock components with Na+/H+ exchanger (NHE3) gene. mRNAs are denoted by slanted boxes, proteins by squares, and protein complexes by ovals. Dashed arrows represent the transactivation and dotted arrows represent inhibition.

The predicted NHE3 activity was then incorporated into a detailed mathematical model of the proximal convoluted tubule cell of a rat kidney (15, 16, 18). The model was formulated for the time-dependent state (7) and represents 15 major solutes (Na⁺, K⁺, Cl⁻, ${\text{HCO}}_3^{-}$, H₂CO₃, CO₂, ${\text{HPO}}_4^{2-}$, ${\text{H}}_2{\text{PO}}_{\text{4}}^{-}$, urea, NH₃, ${\text{NH}}_4^{+}$, H⁺, ${\text{HCO}}_2^{-}$, H₂CO₂, and glucose). The proximal tubule cell is represented as a well-stirred, compliant cellular compartment surrounded by luminal and peritubular (bath) solutions, the compositions of which are assumed known a priori (Table 1), as well as a lateral, paracellular space. Transporters and channels are represented on the apical and basolateral membranes, some of which are shown in Fig. 2. Model equations are based on mass conservation and electroneutrality constraints. Model equations and parameters are described in the appendix. With this configuration, the proximal tubule cell model predicts solute concentrations of the cellular and lateral innerspace compartments, membrane potential, and solute and water fluxes across the apical membrane, basolateral membrane, and paracellular pathway, as functions of time.

Table 1.

Luminal and bath concentrations

Solute	Concentration (mM)
Na+	144.0
K+	4.9
Cl−	117.0
${\text{HCO}}_3^{-}$	25.0
${\text{H}}_2{\text{CO}}_3^{-}$	4.41 × 10−3
CO2	1.50
${\text{HPO}}_4^{2-}$ + ${\text{H}}_2{\text{PO}}_{\text{4}}^{-}$	3.9
urea	5.0
NH3 + ${\text{NH}}_4^{+}$ †	1.0
${\text{HCO}}_2^{-}$ + H2CO2	1.0
Glucose	5.0
Protein	2.0
pH	7.323

Fig. 2.

Schematic diagram of the proximal convoluted tubule cell model. The model accounts for 15 major solutes (see text). The diagram displays only the major Na⁺, K⁺, and Cl⁻ transporters.

MODEL RESULTS

Using the baseline model parameters (see appendix), the circadian clock model predicts that the expression levels of all clock components exhibit rhythmic oscillations (i.e., limit-cycle oscillations) with a period of 24 h. Time-profiles of core clock components and the expression level of NHE3 are shown in Fig. 3, together with the experimental data (49) and database http://circadb.hogeneschlab.org. Model parameters were chosen to ensure good agreement with data. Oscillations in NHE3 activity gave rise to in-phase oscillations in NHE3-mediated Na⁺ fluxes and consequently cellular [Na⁺] (Figs. 4, A and B, and 5A). Similarly, cellular pH exhibited in-phase oscillation with NHE3 expression (Fig. 5B). Changes in NHE3-mediated Na⁺ flux are proportionally smaller than changes in NHE3 expression: amplitude of the NHE3 expression and Na⁺ flux oscillation is 39% and 15% of the corresponding mean, respectively. That difference can be attributed to the changes in intracellular [Na⁺], a rise in which, taken in isolation, would attenuate the NHE3-mediated Na⁺ flux.

Fig. 3.

Comparison of predicted [solid lines, period (Per; A), cryptochrome (Cry; B), Rev-Erb (C), Ror (D), Bmal1 (E), and NHE3 (F)] with experimental (dashed lines with dots, Per2, Cry1, Rev-Erbα, Rorc, Bmal1, and NHE3 mRNA expression levels obtained in constant darkness. Gray shading and white regions correspond to activity and rest cycles, respectively.

Fig. 4.

Normalized NHE3 mRNA (with respect to the average of the baseline; A), NHE3-mediated Na⁺ flux (B), Na⁺-dependent glucose cotransporter 2 (SGLT2)-mediated Na⁺ flux (C), and Na⁺-phosphate cotransporter (NaPi)-mediated Na⁺ flux (D), obtained under baseline conditions. Gray shading and white regions correspond to activity and rest cycles, respectively.

Fig. 5.

Intracellular Na⁺ concentration (A), intracellular pH value (B), and intracellular osmolarity (C), obtained under baseline conditions. Gray shading and white regions correspond to te activity and rest cycles, respectively.

Oscillations in cellular [Na⁺] also induced oscillations in Na⁺ fluxes that are mediated by other apical Na⁺ transporters (Na⁺-glucose cotransporter SGLT2, and the Na⁺-phosphate cotransporter NaPi2); these oscillations are out-of-phase with those of the NHE3 activity and cellular [Na⁺] (Fig. 4, C and D). Together, the changes in NHE3-, SGLT2-, and NaPi2-mediated Na⁺ fluxes gave rise to oscillation in transcellular Na⁺ flux, with an amplitude that is 10% of its mean (Fig. 6A). As cellular [Na⁺] and other solute concentrations fluctuated, cellular fluid osmolarity fluctuated also (Fig. 5C). As a result, transcellular water flux exhibited oscillations that were in-phase with NHE3 expression (Fig. 6C).

Fig. 6.

Transcellular Na⁺ flux (A), paracellular Na⁺ flux (B), transcellular water flux (C), paracellular water flux (D), total Na⁺ flux (E), and total water flux (F) at baseline. Gray shading and white regions correspond to activity and rest cycles, respectively.

Furthermore, the model predicted rhythmic oscillations in intracellular concentrations of all solutes. In particular, intracellular [Cl⁻] and [${\text{HCO}}_3^{-}$] exhibit oscillations that are in phase with NHE3, whereas [K⁺] exhibits out-of-phase oscillations with NHE3 (results not shown). These oscillations in turn induced rhythmic changes in all transmembrane fluxes.

Rhythmic oscillations in basolateral Na⁺/K⁺-ATPase-mediated [Na⁺] (result not shown) yielded fluctuations in fluid osmolarity in the lateral innerspace. These oscillations, which are in phase with the NHE3 oscillations, generated in-phase oscillations in paracellular water flux and subsequently in paracellular Na⁺ flux (Fig. 6, B and D). Taken together, oscillations in NHE3 expression yielded in-phase oscillations in total (transcellular and paracellular) Na⁺ and water reabsorptive fluxes (Fig. 6, E and F), with amplitudes that are 11% and 8% of their respective means.

Below, we describe model simulations that predict the impact of eliminating individual clock genes on the circadian rhythm and on solute and water transport of the proximal tubule cell.

Per2 mutation.

In mammals, there are three known Per isoforms: Per1, Per2, and Per3. In the kidney and liver, the abundance of Per3 is low relative to Per1 and Per2. The present clock model ignores Per3, and, because the specific actions of Per1 and Per2 have yet to be sufficiently well characterized, for simplicity the model does not distinguish between Per1 and Per2. Thus, to simulate the effect of Per2 mutation on proximal tubular Na⁺ and water transport, we reduced the maximal transcription rate of Per (${\text{V}}_{\max }^{\text{per}}$ in Eq. A1) by 50%. The model predicted that, following Per2 mutation, the rhythmic oscillations in clock gene expression persisted. PER protein decreased by 30% (in mean relative to baseline), whereas CRY protein is increased by 2% (in mean). Recall that these two proteins combine in a reversible reaction to form the PER-CRY complex (see Fig. 1). Their competing changes resulted in a decrease in PER-CRY complex abundance (27% in mean), in an increase in NHE3 expression (9% in mean), and, consequently, in elevated proximal tubular Na⁺ fluxes (transcellular and paracellular; see Figs. 7 and 8). The higher transcellular Na⁺ transport raised intracellular [Na⁺] and augmented transcellular water reabsorption. Changes in transcellular fluxes of other solutes (e.g., K⁺, Cl⁻, etc.) were minimal, whereas mean ${\text{HCO}}_3^{-}$ flux increased significantly, by 7%. The circadian period was predicted to be shortened by an hour (i.e., the period became 23 h), due to the attenuated inhibition from PER-CRY complex.

Fig. 7.

Normalized NHE3 mRNA (with respect to the average of the baseline; A), NHE3-mediated Na⁺ flux (B), transcellular Na⁺ flux (C), and transcellular water flux (D). Results were obtained for baseline conditions (black solid line), Per2 mutation (black dotted line), Cry1 mutation (black dashed line), CLOCK mutation (gray solid line), and Bmal1 mutation (gray solid line). Profiles corresponding to CLOCK mutation and Bmal1 mutation are indistinguishable. Gray shading and white regions correspond to activity and rest cycles, respectively.

Fig. 8.

Paracellular Na⁺ flux (A), paracellular water flux (B), total Na⁺ flux (C), and total water flux (D). Results were obtained for baseline conditions (black solid line), Per2 mutation (black dotted line Cry1 mutation (black dashed line), CLOCK mutation (gray solid line), and Bmal1 mutation (gray solid line). Profiles corresponding to CLOCK mutation and Bmal1 mutation are indistinguishable. Gray shading and white regions correspond to activity and rest cycles, respectively.

Cry1 mutation.

The clock model does not distinguish between the two Cry isoforms Cry1 and Cry2. Due to their similar abundance in the kidney and liver (25, 26, 52), we simulated a selective mutation of Cry1 by reducing the associate rate-maximal transcription rate of Cry (${\text{V}}_{\max }^{\text{cry}}$ in Eq. A2) by 50%. The model predicted that, when Cry1 was eliminated, the rhythmic oscillations in clock gene expression would persist. Specifically, Cry1 mutation increased PER protein abundance by 55% (in mean relative to baseline) and decreased CRY protein abundance by 60% (in mean). Taken together, these competing changes yielded a 36% reduction in PER-CRY complex. The resulting attenuated inhibition (from PER-CRY complex) sped up the system. We thus observed a reduction in the circadian period of 30 min. The attenuated inhibition also elevated mean NHE3 expression (Fig. 7A) by 13% relative to baseline.

As shown in Figs. 7 and 8, transcellular and paracellular Na⁺ and water fluxes of a proximal tubule cell in a Cry1 mutant oscillated in phase with the NHE3 expression (black dashed line). Mean NHE3-mediated Na⁺ flux (Fig. 7B, black dashed line) increased by 5% relative to baseline. That increase is substantially smaller than the increase in NHE3 expression level (13%) due to the higher cellular Na⁺ concentration, which limited the driving force of Na⁺/H⁺ antiporter. The mean value of total (transcellular plus paracellular) Na⁺ and water fluxes are increased by 4% and 3%, respectively.

In another simulation, we considered the Cry1/Cry2 double mutant by reducing the maximal transcription rate of Cry (${\text{V}}_{\max }^{\text{cry}}$) to zero. The model predicted that the concentration of PER-CRY protein complex decreased to zero. The disappearance of the inhibitor of the circadian system broke the feedback loop; consequently, core clock and its target genes ceased to oscillate (results not shown). Without the inhibiting effect of PER-CRY, the NHE3 expression level increased by 157% above baseline, resulting in a 24% increase in total Na⁺ flux, a 9% increase in water flux, a 16% decrease in total K⁺ flux, an 11% decrease in total Cl⁻ flux, and a 141% increase in ${\text{HCO}}_3^{-}$ flux.

Rev-Erbα mutation.

The current model does not distinguish between Rev-Erbα and Rev-Erbβ. Assuming similar abundance of these two isoforms, we reduced the maximal transcription rate of Rev-Erb (${\text{V}}_{\text{max}}^{\text{rev}}$ in Eq. A3) by 50% to simulate the selective mutation of Rev-Erbα. The model predicted that the circadian rhythms of clock component would persist. Specifically, Rev-Erbα mutation decreased REV-ERB protein by 48% (in mean), thus leading to an increase in BMAL1 protein by 6% (in mean). That was followed by an increase in CLOCK-BMAL1 protein complex (by 6% in mean). The resulting stronger activation sped up the circadian system, modestly lowering its period by 30 min. Another consequence of the enhanced activation was the elevated mean NHE3 expression (by 5%) above baseline (results not shown).

In the proximal tubule cell of a Rev-Erbα mutant, transcellular and paracellular ${\text{Na}}^{+}$ and water fluxes oscillated in phase with NHE3 activity. The elevations in the mean NHE3-mediated Na⁺ flux, transcellular and paracellular Na⁺ flux, and water fluxes were minimal relative to the corresponding baseline fluxes. Changes in other solute (K⁺, Cl⁻, and ${\text{HCO}}_3^{-}$) fluxes were similarly small.

In another simulation, we considered a Rev-Erbα/Rev-Erbβ double mutant by reducing the maximal transcription rate of Rev-Erb (${\text{V}}_{\text{max}}^{\text{rev}}$) to zero. The model predicted that core clock and its target genes ceased to oscillate (results not shown), following disappearance of the inhibitor of the Bmal1, which broke the feedback loop.

Rorc mutation.

The model does not distinguish among the three Ror isoforms Rora, Rorb, and Rorc. To simulate the selective mutation of Rorc, we reduced the maximal transcription rate of Ror (${\text{V}}_{\max }^{\text{ror}}$ in Eq. A4) by 30, 50, and 75%. The model predicted that, in all cases following Rorc mutation, the rhythmic oscillation in clock gene expression persisted. Rorc mutation (30, 50, and 75%) yielded a decrease in mean ROR protein by 7, 13, and 28%, respectively; this led to minor reductions in BMAL1 protein and CLOCK-BMAL1 complex (by 3, 5, and 12%, respectively). Subsequently, REV-ERB protein mean abundance decreased by 2, 5, and 16%, respectively. Because REV-ERB protein inhibits the formation of Bmal1, the lower REV-ERB protein level attenuated the reduction in Bmal1. Attenuated activation of target genes resulting from a decrease in CLOCK-BMAL1 complex slowed down the circadian system. We thus observed an increase in the period of oscillation by 0.9, 1.7, and 2.8 h, respectively.

With a 75% reduction in ${\text{V}}_{\max }^{\text{ror}}$, the decrease in the activator CLOCK-BMAL1 abundance lowered NHE3 expression by 7% (in mean, relative to baseline). Following Ror mutation, the proximal tubule cell Na⁺ and water fluxes oscillated in phase with NHE3 expression. The mean NHE3-mediated Na⁺ and ${\text{HCO}}_3^{-}$ fluxes decreased by 3 and 6%, respectively. The reductions in the mean total Na⁺ and water fluxes were minimal. Changes in other solute fluxes (e.g., K⁺, Cl⁻, etc.) were similarly small.

Bmal1 mutation.

Next, we simulated Bmal1 mutation by setting the maximal transcription rate of Bmal1 (${\text{V}}_{\max }^{\text{bmal}}$ in Eq. A5) to zero. The model predicted that the concentration of Bmal1 mRNA would decrease to zero, as would the concentrations of BMAL1 protein and the activator CLOCK-BMAL complex. The feedback loop of the gene clock network was thus broken, and the concentrations of other target genes ceased to oscillate. Similarly, NHE3 expression level no longer oscillated but instead settled at a level below the mean (by 22%) baseline level (Fig. 7A). That reduction can be attributed to the disappearance of the transcriptional activator CLOCK-BMAL1 complex (i.e., attenuated activation). The lower NHE3 level decreased NHE3-mediated and overall transcellular Na⁺ flux by 10% and 6%, respectively; decreased transcellular water flux by 5% (Fig. 7). The mean total ${\text{HCO}}_3^{-}$ flux decreased by 19% relative to the baseline. The changes on total K⁺ and Cl⁻ fluxes were minimal. Model predictions are consistent with experimental observations showing that immediate and complete loss of circadian rhythmicity are seen in SCN and liver of BMAL1 mutant mice in constant darkness (3).

CLOCK mutation.

Elimination of the CLOCK protein caused the concentration of CLOCK-BMAL complex to decrease almost to zero. As in Bmal1 mutation, the rhythmic oscillations disappeared, and the model predicted a NHE3 expression level that was essentially the same as the Bmal1 mutation case (22% below baseline, Fig. 7). Model predictions are consistent with experimental reports that mice carrying CLOCK mutation exhibit abnormalities of circadian behavior in SCN, including loss of rhythmicity (43).

DISCUSSION

The principle goal of this study was to predict the extent to which Na⁺ and water transport by a proximal convoluted tubule cell of the rat kidney is modulated by the circadian rhythm. To accomplish that goal, we have developed the first computational model of the renal circadian clock. Most published modeling studies of proximal tubule epithelial transport focus on steady-state results (15, 16, 18, 45), an exception being Ref 7, where cellular response to abrupt changes in luminal or bath composition was simulated. None of the published modeling studies have considered sustained oscillations in the expression level of NHE3 or other transporters. Given that the regulation of NHE3 by clock gene has been well established (33), the impact of circadian rhythms on cellular transport of salt and water, in wild-type and various clock gene mutants, is worthy of investigation.

To simulate the regulation of NHE3 activity by the circadian clock, we combined the new computational model of the core clock with a model of epithelial transport of the proximal convoluted tubule cell of the rat kidney. The clock model simulates the interactions among transcription factors Per, Cry, Rev-Erb, Ror, Bmal1, and CLOCK; it also represents the direct regulation of NHE3 expression level by Bmal1 and CLOCK (see Fig. 1). A set of model parameters was identified that yielded mRNA profiles that are consistent with the experimental data (49). With these parameters, the model predicts circadian rhythms in clock gene expression levels, which drive NHE3 activity to fluctuate with an amplitude that is 39% of its mean baseline value (see Fig. 3).

The predicted NHE3 activity is incorporated into a computational model of proximal tubule epithelial transport to predict intracellular solute concentration, membrane potential, luminal solute and water fluxes, basolateral solute and water fluxes, and paracellular solute and water fluxes (see Fig. 2). The model predicts that fluctuations in NHE3 activity result in fluctuations in NHE3-mediated Na⁺ flux and consequently transcellular and paracellular Na⁺ flux and water flux (Figs. 4–6). However, the amplitudes of the flux fluctuations are significantly smaller than that of the NHE3 activity due to fluctuations in intracellular [Na⁺] that are out of phase with NHE3 activity and thus have the opposite effect on Na⁺ transport.

Comparison with experimental observations.

Model simulations predicted that mutation of Bmal1 or CLOCK would sufficiently eliminate the activator CLOCK-BMAL complex to break the feedback loop, leading to the disappearance of the circadian rhythms. These predictions are consistent with experimental observations in BMAL1 and CLOCK mutant mice (3, 43).

Model simulation of Per2 mutation predicted an increase in NHE3 expression level, consistent with findings in Ref 33, and a 1-h reduction in circadian period (Fig. 7A). In comparison, findings in the SCN of mPer2 mutant mice in constant darkness revealed clock gene mRNA rhythms that exhibit an even shorter (by 1.5 h) circadian period followed by a loss of circadian rhythmicity (1, 51). The differences in rhythmicity may have resulted from organ-dependent clock genes.

Simulation results of Cry1 mutation are qualitatively similar to those of Per2 mutation: a significant rise in NHE3 expression level, together with a 0.5-h reduction in circadian period (Fig. 7A). These results are consistent with findings in the liver nuclei of Rev-Erbα-deficient mice, where clock gene rhythms have been reported to exhibit a period reduction of 0.5 h (28).

Model limitations and potential extensions.

The mammalian Per gene exhibits three isoforms: Per1, Per2, and Per3. Due to the lack of data regarding Per3 action in the kidney, the present clock model ignores Per3. Furthermore, for simplicity, the model does not distinguish between Per1 and Per2. Recent experimental studies have shed light into the distinct functions of Per1 and Per2 (31, 33). Findings in Ref 31. suggest that Per1 potentially activates renal sodium transport through a Cry2-CLOCK/Bmal1-dependent mechanism, in which Per1 transcriptionally represses Cry2, preventing its repressions of CLOCK/Bmal1. In contrast, Per2 has been shown to inhibit Na⁺ reabsorption (33). In future modeling studies of the renal circadian clock, separate differential equations can be used to represent and reveal the distinct roles of Per1 and Per2 in renal Na⁺ handling.

Again for simplicity, the present model does not distinguish between the two Cry isoforms Cry1 and Cry2. It has been observed in the brain that Cry1 and Cry2 mutations have opposite effects on the circadian period (~1-h increase/decrease, respectively) (42), although the underlying mechanism has not been elucidated. Furthermore, Cry2 is suppressed by Per1, but Cry1 is not (31). Thus, a more detailed computational model that separates Cry1 and Cry2 would better capture their different roles in the renal circadian system.

Compensation is common in knockout animals; that is, when one clock gene is deleted, other isoforms may upregulate. For example, Cry2 mRNA and protein expression are increased in the liver and kidney of Per1 KO mice (31). On the other hand, kidney-specific KO of Bmal1 did not appear to result in increased expression of CLOCK mRNA (25). Because the issue of compensation is not fully understood in the kidney, we did not consider compensation in our knockout simulations. However, we consider this a benefit of computational modeling: in silico studies can simulate “clean” knockouts and clearly demonstrate the effects of deleting a given gene component without the influence of compensation.

An additional limitation of the present model is that it does not consider the role of the circadian clock in other cellular functions that may affect NHE3 activity. For example, paracellular transport pathways may be subject to circadian regulation. Indeed, claudin-1 mRNA levels oscillate in the kidney (CircaDB database), and this finding has been replicated by independent studies (25). Related to adherens and tight junctions, E-cadherin and claudin-4, respectively, exhibit rhythmic changes in mRNA and protein levels in the rat kidney (47). NHE3 activity is also likely to be affected by changes in dietary sodium which has recently been shown to affect clock gene expression in the rat kidney (38). Based on the model presented here, future studies will be designed to add additional important elements, such as clock-regulation of other Na⁺-coupled transporters, e.g., SGLT1 (26). Such investigations should increase our overall understanding of how the circadian clock in the kidney contributes to the regulation of fluid and electrolyte homeostasis, acid-base balance, and the role of general proximal tubule dysfunction in case of core-clock disruption.

The present model represents a stand-alone proximal convoluted cell, with luminal and peritubular fluid composition assumed to be known a priori. Clearly, variations in epithelial transport of a given cell would affect luminal fluid composition and thus the solute and water transport downstream. To investigate the effect of circadian rhythm on overall proximal tubule Na⁺ transport, we may extend the present epithelial model to simulate a proximal tubule, by connecting a series of cells following a standard approach (15, 16, 18). The resulting proximal tubule model can be used to answer questions such as: 1) to what extent does the circadian rhythm affect solute and water transport along the proximal tubule? 2) how does segmental transport change when a key clock gene is eliminated? and 3) is there any significant impact on glomerulotubular balance?

The impact of the renal circadian clock reaches far beyond the proximal tubule. In addition to NHE3 (33), clock genes have also be shown to regulate SGLT1 of the proximal straight tubule cell (26), Na⁺-K⁺-Cl⁻ cotransporter isoform 2 (NKCC2) and estrogen-related receptor-β (ERRβ) of the thick ascending limb cell (12, 26), Na⁺-Cl⁻ cotransporter (NCC), and with-no-lysine kinase (WNK) of the distal convoluted tubule cell (32, 40), α-subunit of epithelial sodium channel (αENaC) (10, 52), and key regulators of sodium transport of the renal collecting duct cell (39, 52). The circadian rhythm of these transporters can be simulated in a computational model of the nephron (14, 16, 18) using the approach we have used for NHE3. The resulting model can be used to predict the interactions of the clock-controlled transporters and their effects on renal tubular transport and urinary excretion.

Clinical perspectives.

It is becoming increasingly clear that circadian rhythms are directly relevant to human health. For instance, Montaigne et al. (24) recently demonstrated that afternoon surgery was protective against perioperative myocardial injury compared with morning surgery in patients undergoing aortic valve replacement. The hazard ratio was 0.50 for afternoon surgery (n = 298 per group, P < 0.01). This effect was linked to the transcriptional actions of the molecular clock. The dramatic results of this first-of-its-kind clinical study have important implications for human health and are likely to extend to the pathophysiology of other organ systems such as the kidney. Indeed, circadian blood pressure disorders are common in chronic kidney disease (11). The model presented here represents an important step in extending our understanding of the kidney clock to the level of cellular transport function. This model provides a critical hypothesis-generating tool that is necessary to increase our understanding of how the molecular clock affects renal ion transport and renal function, which is likely to have important implications for kidney disease.

GRANTS

This research was supported in part by the National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases, via Grant DK-106102 and by the National Science Foundation via Grant DMS-1263995 to A. T. Layton.

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

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

APPENDIX: CIRCADIAN CLOCK MODEL EQUATIONS

The circadian system is a network of interlocked transcriptional-translational feedback loops that drives the circadian oscillations of core clock components with a cycle length of 24 h (30). In the primary negative feedback loop, CLOCK and BMAL1 heterodimerize to initiate the transcription of target clock genes, including Per (with isoforms Per1, Per2, Per3) and Cry (with isoforms Cry1 and Cry2), by binding the E-box elements in the promoter region (3, 5, 50). PERs and CRYs then heterodimerize to inhibit their own transcription by acting on CLOCK-BMAL1 protein complex (6, 13, 35, 37). In the secondary feedback loop, activators of CLOCK and BMAL1 dimerize to initiate the transcription of ROR genes Rev-Erbα and Rorc (28, 34, 41), which compete to bind to ROR response elements (ROREs) present in Bmal1 promoter. REV-ERBs (with isoforms REV-ERBα, REV-ERBβ) and RORs (with isoforms RORa, RORb, and RORc) are shown to repress and activate Bmal1 transcription, respectively (4, 9). In addition, Rev-Erb also inhibits Cry transcription (22) to ensure the robust oscillations (2, 29). NHE3 is a transporter that is directly regulated by the circadian system (33). NHE3 is an output gene transactivated by the heterodimerizer CLOCK-BMAL1, whose activation is repressed by the primary negative feedback loop (33). The result of this molecular regulation is that NHE3 protein expression and localization change over the course of a 24-h period. Thus, the molecular clock directly regulates NHE3 expression at the level of mRNA, protein, and localization (33). Consistent with this regulation, it is interesting to note that the well-established regulator of NHE3 trafficking, Nherf1, also exhibits circadian expression at the level of mRNA (CircaDB, gene name Slc9a3r1). Indeed, Nherf1 mRNA expression showed a similar time-dependent change in expression in the kidney in an independent study by Firsov et al. (25).

For simplicity, we represent the two period homologs (Per1 and Per2) as a single Per gene and ignore Per3, and we represent the two cryptochromes (Cry1, Cry2) as a single Cry gene. Similarly, the two isoforms Rev-Erbα and Rev-Erbβ and three isoforms Rora, Rorb and Rorc are represented by single variables Rev-Erb and Ror, respectively. Expression of the Clock protein is constitutive. Nuclear entry of clock proteins like PER is regulated by phosphorylation and is assumed to be rapid relative to the 24-h period of the clock (20); therefore, posttranslational modification are not included in the model.

Our mathematical model, inspired by Ref 46, describes the time variations of mRNA and corresponding protein concentrations of clock genes Per, Cry, Rev-Erb, Ror, and Bmal1 as well as mRNA concentration for the output clock gene NHE3. The rates of change of the mRNA and protein concentrations are determined by Eqs. A1–A13. Parameters involved in the model are listed in Tables A1–A7. All parameters are unknown and were estimated from the mouse circadian kidney database (http://circadb.hogeneschlab.org), which was developed using data obtained in constant darkness. It is noteworthy that the present model is based on the rat, whereas the parameters were estimated using mouse data. Despite known species differences, key core clock gene expressions of mouse and rat kidneys exhibit many similarities (38, 52). Below we adopt the notation where names with a mix of upper and lower case letters (e.g., Per) denote mRNA, and names in all caps (e.g., PER) denote protein.

$$\frac{d\left[\text{Per}\right]}{dt}=-\text{dm\_per}\cdot \left[\text{Per}\right]+\frac{{\text{V}}_{\text{max}}^{\text{per}}\cdot \left(1+\text{fold\_per}\cdot {\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_per\_cb}}\right)}^{\text{hill\_per\_cb}}\right)}{\begin{array}{c}1+{\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_per\_cb}}\right)}^{\text{hill\_per\_cb}}\cdot \left(1+{\left(\frac{\left[\text{PER-CRY}\right]}{\text{Ki\_per\_pc}}\right)}^{\text{hill\_per\_pc}}\right)\end{array}},$$ (A1)

$$\frac{d\left[\text{Cry}\right]}{dt}=-\text{dm\_cry}\cdot \left[\text{Cry}\right]+\frac{{\text{V}}_{\text{max}}^{\text{cry}}\cdot \left(1+\text{fold\_cry}\cdot {\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_cry\_cb}}\right)}^{\text{hill\_cry\_cb}}\right)}{1+{\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_cry\_cb}}\right)}^{\text{hill\_cry\_cb}}\cdot \left(1+{\left(\frac{\left[\text{PER-CRY}\right]}{\text{Ki\_cry\_pc}}\right)}^{\text{hill\_cry\_pc}}\right)}\cdot \frac{1}{\left(1+{\left(\frac{\text{REV-ERB}}{\text{Ki\_cry\_rev}}\right)}^{\text{hill\_cry\_rev}}\right)},$$ (A2)

$$\frac{d\left[\text{REV-ERB}\right]}{dt}=-\text{dm\_rev}\cdot \left[\text{Rev-Erb}\right]+\frac{{\text{V}}_{\text{max}}^{\text{rev}}\cdot \left(1+\text{fold\_rev}\cdot {\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_rev\_cb}}\right)}^{\text{hill\_rev\_cb}}\right)}{\begin{array}{c}1+{\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_rev\_cb}}\right)}^{\text{hill\_rev\_cb}}\cdot \left(1+{\left(\frac{\left[\text{PER-CRY}\right]}{\text{Ki\_rev\_pc}}\right)}^{\text{hill\_rev\_pc}}\right)\end{array}},$$ (A3)

$$\frac{d\left[\text{Ror}\right]}{dt}=-\text{dm\_ror}\cdot \left[\text{Ror}\right]+\frac{{\text{V}}_{\text{max}}^{\text{ror}}\cdot \left(1+\text{fold\_ror}\cdot {\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_ror\_cb}}\right)}^{\text{hill\_ror\_cb}}\right)}{\begin{array}{c}1+{\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_ror\_cb}}\right)}^{\text{hill\_ror\_cb}}\cdot \left(1+{\left(\frac{\left[\text{PER-CRY}\right]}{\text{Ki\_ror\_pc}}\right)}^{\text{hill\_ror\_pc}}\right)\end{array}},$$ (A4)

$$\frac{d\left[\text{Bmal1}\right]}{dt}=-\text{dm\_bmal}\cdot \left[\text{Bmal1}\right]+\frac{{\text{V}}_{\text{max}}^{\text{bmal}}\cdot \left(1+\text{fold\_bmal}\cdot {\left(\frac{\left[\text{ROR}\right]}{\text{Ka\_bmal\_ror}}\right)}^{\text{hill\_bmal\_ror}}\right)}{\begin{array}{c}1+{\left(\frac{\left[\text{REV-ERB}\right]}{\text{Ki\_bmal\_rev}}\right)}^{\text{hill\_bmal\_rev}}+{\left(\frac{\left[\text{ROR}\right]}{\text{Ka\_bmal\_ror}}\right)}^{\text{hill\_bmal\_ror}}\end{array}},$$ (A5)

$$\frac{d\left[\text{NHE3}\right]}{dt}=-\text{dm\_NHE3}\cdot \left[\text{NHE3}\right]+\frac{{\text{V}}_{\text{max}}^{\text{NHE3}}\cdot \left(1+\text{fold\_NHE3}\cdot {\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_NHE3\_cb}}\right)}^{\text{hill\_NHE3\_cb}}\right)}{\begin{array}{c}1+{\left(\frac{\left[\text{CLOCK-BMAL1}\right]}{\text{Ka\_NHE3\_cb}}\right)}^{\text{hill\_NHE3\_cb}}\cdot \left(1+{\left(\frac{\left[\text{PER-CRY}\right]}{\text{Ki\_NHE3\_pc}}\right)}^{\text{hill\_NHE3\_pc}}\right)\end{array}},$$ (A6)

$$\frac{d\left[\text{PER}\right]}{dt}=-\text{dp\_per}\cdot \left[\text{PER}\right]+\text{kp\_per}\left[\text{Per}\right]-\left(\text{kass\_pc}\cdot \left[\text{CRY}\right]\cdot \left[\text{PER}\right]-\text{kdiss\_pc}\cdot \left[\text{PER-CRY}\right]\right),$$ (A7)

$$\frac{d\left[\text{CRY}\right]}{dt}=-\text{dp\_cry}\cdot \left[\text{CRY}\right]+\text{kp\_cry}\left[\text{Cry}\right]-\left(\text{kass\_pc}\cdot \left[\text{CRY}\right]\cdot \left[\text{PER}\right]-\text{kdiss\_pc}\cdot \left[\text{PER-CRY}\right]\right),$$ (A8)

$$\frac{d\left[\text{REV-ERB}\right]}{dt}=-\text{dp\_rev}\cdot \left[\text{REV-ERB}\right]+\text{kp\_rev}\cdot \left[\text{Rev-Erb}\right],$$ (A9)

$$\frac{d\left[\text{ROR}\right]}{dt}=-\text{dp\_ror}\cdot \left[\text{ROR}\right]+\text{kp\_ror}\cdot \left[\text{Ror}\right],$$ (A10)

$$\frac{d\left[\text{BMAL1}\right]}{dt}=-\text{dp\_bmal}\cdot \left[\text{BMAL1}\right]+\text{kp\_bmal}\cdot \left[\text{Bmal1}\right]-\text{kass\_cb}\cdot \left[\text{BMAL1}\right]+\text{kdiss\_cb}\cdot \left[\text{CLOCK-BMAL1}\right],$$ (A11)

$$\frac{d\left[\text{PER-CRY}\right]}{dt}=\left(\text{kass\_pc}\cdot \left[\text{CRY}\right]\cdot \left[\text{PER}\right]-\text{kdiss\_pc}\cdot \left[\text{PER-CRY}\right]\right)-\text{d\_pc}\cdot \left[\text{PER-CRY}\right],$$ (A12)

$$\frac{d\left[\text{CLOCK-BMAL1}\right]}{dt}=\left(\text{kass\_cb}\cdot \left[\text{BMAL1}\right]-\text{kdiss\_cb}\cdot \left[\text{CLOCK-BMAL1}\right]\right)-\text{d\_cb}\cdot \left[\text{CLOCK-BMAL1}\right],$$ (A13)

In Eqs. A1–A6, the rates of change of [Per], [Cry], [Rev-Erb], [Ror], [Bmal1], and [NHE3] are given by the mRNA degradation and transcription. The transcription process of Per, Cry, Rev-Erb, and Ror are activated by CLOCK-BMAL1 protein complex binding at the promoter region of target genes, and the CLOCK-BMAL1-dependent transcription is inhibited by the PER-CRY protein complex. Equation A2 also considers the repression of Cry by Rev-Erb. The transcription rate is formulated based on thermodynamic equilibrium and the fractions of time spent by the gene in its different states (free, bound by CLOCK-BMAL1, bound by CLOCK-BMAL1 and PER-CRY). Equations A7–A13 show that the rates of change of [PER], [CRY], [REV-ERB], [ROR], [BMAL1], [CLOCK-BMAL1], and [PER-CRY] are mediated by protein degradation, mRNA translation, and the protein-protein interactions. Model parameters are given in Tables A1–A7.

Table A1.

mRNA and protein degradation rate constant (in h^–1)

Parameter	Value	Description
dm_per	0.171401	Per mRNA degradation rate constant
dm_cry	0.353319	Cry mRNA degradation rate constant
dm_rev	0.507872	Rev-Erb mRNA degradation rate constant
dm_ror	0.168399	Ror mRNA degradation rate constant
dm_bmal	5.26234	Bmal1 mRNA degradation rate constant
dm_NHE3	0.524274	NHE3 mRNA degradation rate constant
dp_per	0.406045	PER protein degradation rate constant
dp_cry	1.81463	CRY protein degradation rate constant
dp_rev	0.249662	REV-ERB protein degradation rate constant
dp_ror	0.188237	ROR protein degradation rate constant
dp_bmal	0.229965	BMAL protein degradation rate constant
d_cb	0.164691	CLOCK-BMAL protein complex degradation rate constant
d_pc	0.167848	PER-CRY protein complex degradation rate constant

Table A2.

Maximal transcription rates (in nmol⋅l⁻¹⋅h^–1)

Parameter	Value	Description
${\text{V}}_{\max }^{\text{per}}$	0.608252	Per maximal transcription rate
${\text{V}}_{\max }^{\text{cry}}$	0.506614	Cry maximal transcription rate
${\text{V}}_{\max }^{\text{rev}}$	0.067494	Rev-Erb maximal transcription rate
${\text{V}}_{\max }^{\text{ror}}$	20.5441	Ror maximal transcription rate
${\text{V}}_{\max }^{\text{bmal}}$	0.602884	Bmal1 maximal transcription rate
${\text{V}}_{\max }^{\text{NHE3}}$	3.145644	NHE3 maximal transcription rate

Table A3.

Activation ratios (dimensionless)

Parameter	Value	Description
fold_per	2.60968	Activation ratio of Per by CLOCK-BMAL
fold_cry	9.3413	Activation ratio of Cry by CLOCK-BMAL
fold_rev	120.772	Activation ratio of Rev-Erb by CLOCK-BMAL
fold_ror	2.61103	Activation ratio of Ror by CLOCK-BMAL
fold_bmal	29.7513	Activation ratio of Bmal1 by ROR
fold_NHE3	49.0476	Activation ratio of NHE3 by CLOCK-BMAL

Table A4.

Regulation thresholds (in nmol/l)

Parameter	Value	Description
Ka_per_cb	2.20956	Regulation threshold of Per by CLOCK-BMAL
Ki_per_pc	0.13775	Regulation threshold of Per by PER-CRY
Ka_cry_cb	1.70642	Regulation threshold of Cry by CLOCK-BMAL
Ki_cry_pc	0.00286886	Regulation threshold of Cry by PER-CRY
Ki_cry_rev	0.338714	Regulation threshold of Cry by REV-ERB
Ka_rev_cb	0.253161	Regulation threshold of Rev-Erb by CLOCK-BMAL
Ki_rev_pc	233.541	Regulation threshold of Rev-Erb by PER-CRY
Ka_ror_cb	0.622771	Regulation threshold of ROR by CLOCK-BMAL
Ki_ror_pc	0.0657504	Regulation threshold of ROR by PER-CRY
Ka_bmal_ror	0.116865	Regulation threshold of bmal by ROR
Ki_bmal_rev	0.000255891	Regulation threshold of bmal by REV-ERB
Ka_NHE3_cb	7.71202	Regulation threshold of NHE3 by CLOCK-BMAL
Ki_NHE3_pc	0.0698202	Regulation threshold of NHE3 by PER-CRY

Table A5.

Hill coefficients (dimensionless)

Parameter	Value	Description
hill_per_cb	21.633	Hill coefficient regulation of Per by CLOCK-BMAL
hill_per_pc	22.0113	Hill coefficient regulation of Per by PER-CRY
hill_cry_cb	5.11758	Hill coefficient regulation of Cry by CLOCK-BMAL
hill_cry_pc	1.77601	Hill coefficient regulation of Cry by PER-CRY
hill_cry_rev	52.6298	Hill coefficient regulation of Cry by Rev-Erb
hill_rev_cb	5.37045	Hill coefficient regulation of Rev-Erb by CLOCK-BMAL
hill_rev_pc	3.96668	Hill coefficient regulation of Rev-Erb by PER-CRY
hill_ror_cb	7.16985	Hill coefficient regulation of ROR by CLOCK-BMAL
hill_ror_pc	3.68283	Hill coefficient regulation of ROR by PER-CRY
hill_bmal_ror	3.07146	Hil coefficient regulation of bmal by ROR
hill_bmal_rev	1.46683	Hill coefficient regulation of bmal by REV-ERB
hill_NHE3_cb	1.16369	Hill coefficient regulation of NHE3 by CLOCK-BMAL
hill_NHE3_pc	1.0005	Hill coefficient regulation of NHE3 by PER-CRY

Table A6.

Translation rates (in molecules per hour per mRNA)

Parameter	Value	Description
kp_per	2.20832	Per translation rate
kp_cry	1.82108	Cry translation rate
kp_rev	0.000696149	Rev-Erb translation rate
kp_ror	0.013315	Ror translation rate
kp_bmal	1.0651	Bmal1 translation rate

Table A7.

Complexation kinetic rates

Parameter	Value	Unit	Description
kass_cb	0.0063849	nmol−1⋅l⋅h–1	CLOCK-BMAL association rate
kass_pc	0.13296611	nmol−1⋅l⋅h–1	PER-CRY association rate
kdiss_cb	0.00073878	h–1	CLOCK-BMAL dissociation rate
kdiss_pc	0.195142	h–1	PER-CRY dissociation rate