Magnesium and Calcium Transport along the Male Rat Kidney: Effect of Diuretics

Abstract

Calcium (Ca²⁺) and magnesium (Mg²⁺) are essential for cellular function. The kidneys play an important role in maintaining the homeostasis of these cations. Their reabsorption along the nephron is dependent on distinct trans- and paracellular pathways and is coupled to the transport of other electrolytes. Notably, sodium (Na⁺) transport establishes an electrochemical gradient to drive Ca²⁺ and Mg²⁺ reabsorption. Consequently, alterations in renal Na⁺ handling, under pathophysiological conditions or pharmacological manipulations, can have major effects on Ca²⁺ and Mg²⁺ transport. One such condition is the administration of diuretics, which are used to treat a large range of clinical conditions, but most commonly for the management of blood pressure and fluid balance. While the pharmacological targets of diuretics typically directly mediate Na⁺ transport, they also indirectly affect renal Ca²⁺ and Mg²⁺ handling through alterations in the electrochemical gradient. To investigate renal Ca²⁺ and Mg² handling and how those processes are affected by diuretic treatment, we have developed computational models of electrolyte transport along the nephrons. Model simulations indicate that along the proximal tubule and thick ascending limb, the transport of Ca²⁺ and Mg²⁺ occurs in parallel with Na⁺, but those processes are dissociated along the distal convoluted tubule. We also simulated the effects of acute administration of loop, thiazide, and K-sparing diuretics. The model predicted significantly increased Ca²⁺ and Mg²⁺ excretions and significantly decreased Ca²⁺ and Mg²⁺ excretions on treatment with loop and K-sparing diuretics, respectively. Treatment with thiazide diuretics significantly decreased Ca²⁺ excretion, but there was no significant alteration in Mg²⁺ excretion. The present models can be used to conduct in silico studies on how the kidney adapts to alterations in Ca²⁺ and Mg²⁺ homeostasis during various physiological and pathophysiological conditions, such as pregnancy, diabetes, and chronic kidney disease.

1. Introduction

The divalent cations, Ca²⁺ and Mg²⁺, are important for various physiological processes. About 99% of the body’s Ca²⁺ is stored in bones, where it forms a calcium-phosphate compound called hydroxyapatite [1]. The remaining 1% of body calcium plays an important role in various other physiological processes, such as cell signaling, both skeletal and smooth muscle contraction, and blood clotting [1]. Mg²⁺ plays a pivotal role in energy-demanding metabolic reactions, protein synthesis, ensuring membrane integrity, facilitating nervous tissue conduction, promoting neuromuscular excitability, regulating muscle contraction, influencing hormone secretion, and participating in intermediary metabolism. Nearly 99% of the body’s Mg²⁺ is distributed within cells or stored in bone, with only a small fraction in circulation [2]. Tight regulation of the serum Ca²⁺ and Mg²⁺ concentrations is essential since too much or too little Ca²⁺ or Mg²⁺ can have dangerous, potentially fatal consequences. To maintain Ca²⁺ and Mg²⁺ balance, it is crucial to regulate the fluxes of Ca²⁺ and Mg²⁺ among the primary organs involved in their regulation, namely the intestine, bone, and kidneys.

The kidneys play an important role in maintaining Mg²⁺ and Ca²⁺ homeostasis. The majority, ~60–70% of the filtered Ca²⁺, is reabsorbed along the proximal tubule through the paracellular pathway [3]. By contrast, paracellular Mg²⁺ permeability in the proximal tubule is very low, and hence only 15–25% of the filtered Mg²⁺ is reabsorbed along this segment [3]. The majority of the filtered Mg²⁺ is reabsorbed along the cortical thick ascending limb (60–70%) paracellularly [3]. The paracellular fractional reabsorption of Ca²⁺ along the thick ascending limb is ~15–25% [3]. The distal convoluted tubule is the final segment that reabsorbs Mg²⁺; hence, it plays an important role in fine-tuning urinary Mg²⁺ excretion. Approximately 5–10% of the filtered Mg²⁺ is reabsorbed transcellularly along the distal convoluted tubule, mediated by the transient receptor potential melastatin 6/7 (TRPM6/7) heteromeric complex on the apical membrane and the Na⁺/Mg²⁺ exchanger on the basolateral membrane [3]. Approximately 5–10% of the filtered Ca²⁺ is reabsorbed transcellularly along the distal convoluted tubule and connecting tubule, mediated by the transient receptor potential vanilloid 5 (TRPV5) on the apical membrane, the Na⁺/Ca²⁺ exchanger (NCX1), and plasma membrane Ca²⁺-ATPase (PMCA) on the basolateral membrane [3]. Finally, ~2–5% of the filtered Mg²⁺ and Ca²⁺ are excreted through urine [3].

Our understanding of Ca²⁺ and Mg²⁺ handling within different segments of the nephron has been greatly advanced through micropuncture and microperfusion studies in rodent nephrons [4,5]. Furthermore, recent genetic studies have expanded our knowledge about the protein mediators of Ca²⁺ and Mg²⁺ transport [6]. Despite these advances, our understanding of the renal handling of these electrolytes remains incomplete. What fraction of the renal reabsorption goes through the transcellular versus paracellular pathway? To what extent is the renal transport in each nephron segment coupled to the transport of other electrolytes, e.g., Na⁺, K⁺, and Cl⁻? To answer these questions, we developed a detailed computational model of epithelial transport of electrolytes and water along the nephrons in a male rat kidney and conducted simulations to predict the renal transport of Ca²⁺ and Mg²⁺ as well as other electrolytes and water under different physiological conditions.

Besides electrolyte and fluid homeostasis, the kidney also plays an essential role in maintaining normal blood pressure. For the management of blood pressure and fluid balance, diuretics are commonly prescribed. Although the pharmacological targets of diuretics directly affect Na⁺ transport, they also indirectly affect renal Mg²⁺ and Ca²⁺ reabsorption through changes in the electrochemical gradient. How is renal Ca²⁺ and Mg²⁺ transport affected by the administration of diuretics? To answer this question, we simulate the effect of acute administration of three classes of diuretics—loop; thiazide; and K-sparing diuretics—on renal Mg²⁺ and Ca²⁺ transport and excretion.

2. Materials and Methods

We have previously developed epithelial cell-based computational models of transporter-mediated solute and water transport along the nephron of a rat kidney [7,8,9,10], focusing on the renal handling of Na⁺, K⁺, Ca²⁺, glucose, and water in physiological and pathophysiological conditions. The superficial nephron model includes the proximal tubule, short descending limb, thick ascending limb, distal convoluted tubule, connecting tubule, and collecting duct segments. Each nephron segment is represented as a tubule lined by a layer of epithelial cells. The model tracks the transport of the following 17 solutes: Na⁺, K⁺, Cl⁻, HCO₃⁻, H₂CO₃, CO₂, NH₃, NH₄⁺, HPO₄²⁻, H₂PO₄⁻, H⁺, HCO₂⁻, H₂CO₂, urea, glucose, Ca²⁺, and Mg²⁺. The segment and cell type determine the type and abundance of transporters found on the apical and basolateral membranes of the cell. Solutes and water may be transported across the epithelium by either moving across the apical and basolateral membranes in the transcellular pathway, mediated by specialized membrane transporters or channels, or via the paracellular pathway between neighboring cells. A schematic diagram for the model nephrons is shown in Figure 1.

The model is defined by a large system of coupled differential and algebraic equations that describe mass conservation and determine transmembrane and paracellular fluxes [11]. The model predicts luminal fluid flow, hydrostatic pressure, membrane potential, luminal and cytosolic solute concentrations, transcellular and paracellular fluxes, urine volume, and urinary excretion rates of model solutes.

Below, we summarize a model representation of Mg²⁺ transport along the proximal tubule, cortical thick ascending limb, and distal convoluted tubule. Model parameters that describe Mg²⁺ transport are given in Table 1. The analogous model description and parameter for Ca²⁺ transport can be found in Ref. [10]. Additional model parameters can be found in Ref. [12].

2.1. Mg²⁺ Transport along the Proximal Tubule

The proximal tubule reabsorbs 15–25% of the filtered Mg²⁺ load through the paracellular pathway, which is mediated by claudin-2 and -12 [13] and is driven by the favorable electrochemical gradient established by Na⁺/H⁺ exchanger 3 (NHE3)-mediated Na⁺ reabsorption [5,14,15]. Paracellular electro-diffusive Mg²⁺ flux (${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}$) is given by

$${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}={\mathit{P}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}{\mathit{\zeta }}_{\mathrm{Mg}}^{\mathrm{LI}}\left(\frac{{\mathit{C}}_{\mathrm{Mg}}^{\mathrm{L}}-{\mathit{C}}_{\mathrm{Mg}}^{\mathrm{I}}{\mathit{e}}^{-{\mathit{\zeta }}_{\mathrm{Mg}}^{\mathrm{LI}}}}{\mathbf{1}-{\mathit{e}}^{-{\mathit{\zeta }}_{\mathrm{Mg}}^{\mathrm{LI}}}}\right),$$

(1)

where the superscripts L and I denote lumen and lateral intercellular space (LIS), respectively; ${\mathit{P}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}$ denotes the permeability of Mg²⁺ at the lumen and LIS interface; ${\mathit{\zeta }}_{\mathrm{Mg}}^{\mathrm{LI}}=\raisebox{1ex}{${\mathit{Z}}_{\mathrm{Mg}}\mathit{F}$}\!\left/ \!\raisebox{-1ex}{$\mathit{R}\mathit{T}$}\right.\left({\mathit{\psi }}^{\mathrm{L}}-{\mathit{\psi }}^{\mathrm{I}}\right)$; ${\mathit{Z}}_{\mathrm{Mg}}$ is the valence of Mg²⁺ (+2); ${\mathit{C}}_{\mathrm{Mg}}^{\mathrm{L}}$ and ${\mathit{C}}_{\mathrm{Mg}}^{\mathrm{I}}$ denote Mg²⁺ concentrations in the lumen and LIS, respectively; ${\mathit{\psi }}^{\mathrm{L}}$ and ${\mathit{\psi }}^{\mathrm{I}}$ denote the luminal and LIS membrane potentials, respectively; RT = 2.57 J/mmol; and F = 96.5 C/mmol represents Faraday’s constant.

2.2. Mg²⁺ Transport along the Thick Ascending Limb

The medullary thick ascending limb has negligible Mg²⁺ reabsorption [15,16,17]. In contrast, the cortical thick ascending limb reabsorbs 60–70% of the filtered Mg²⁺ via the paracellular route. That flux is mediated by claudins 16 and 19 [18], and is driven by the electrochemical gradient established by Na⁺-K⁺-Cl⁻ cotransporter 2 (NKCC2)-mediated Na⁺ transport [9,14]. Paracellular Mg²⁺ transport in the cortical thick ascending limb ${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{cTAL},\mathrm{LI}}$ is represented by an expression analogous to Equation (1), with the superscript “PT” replaced by “cTAL”.

2.3. Mg²⁺ Transport along the Distal Convoluted Tubule

Unlike the proximal tubule and cortical thick ascending limb, where Mg²⁺ transport proceeds passively via the paracelluar route, the reabsorptive process along the distal convoluted tubule is active and transcellular and is mediated by the TRPM6 and TRPM7 (TRPM6/7) heteromeric channels, expressed on the apical membranes [19]. Mg²⁺ flux through TRPM6/7 is given by

$${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{TRPM}6/7}={\mathit{N}}_{\mathrm{TRPM}6/7}\times {\mathit{f}}_{\mathrm{Mg}}\times {\mathit{f}}_{\mathrm{pH}}\times \frac{∆{\mathit{\psi }}^{\mathrm{L}\mathrm{C}}-{\mathit{E}}_{\mathrm{Mg}}^{\mathrm{L}\mathrm{C}}}{\mathbf{2}\mathit{F}},$$

(2)

${\mathit{f}}_{\mathrm{Mg}}$ and ${\mathit{f}}_{\mathrm{pH}}$ describe the effects of intracellular Mg²⁺ concentration (${\mathit{C}}_{\mathrm{Mg}}^{\mathrm{C}}$) and extracellular pH on TRPM6/7 [20], and are given by

$${\mathit{f}}_{\mathrm{Mg}}=\frac{\mathbf{1}}{\mathbf{1}+{\left(\frac{{\mathit{C}}_{\mathrm{Mg}}^{\mathrm{C}}}{\mathbf{0.51}}\right)}^{\mathbf{2}}}.$$

(3)

$${\mathit{f}}_{\mathrm{pH}}={\mathit{g}}_{\mathrm{pH}7.4}\left(\mathbf{2}-\frac{\mathbf{1}}{\mathbf{1}+\frac{\mathbf{7.4}-{\mathit{p}\mathit{H}}^{\mathrm{L}}}{\mathbf{7.4}-{\mathit{p}\mathit{H}}_{1/2}}}\right),$$

(4)

where ${\mathit{g}}_{\mathrm{pH}7.4}$ denotes the single channel conductance of TRPM6/7 channel at pH 7.4, ${\mathit{pH}}^{\mathrm{L}}$ denotes the luminal fluid pH, and ${\mathit{pH}}_{1/2}$ denotes the luminal fluid pH for half-maximal conductance.

Mg²⁺ efflux through the basolateral membrane is assumed to be mediated by an Na⁺/Mg²⁺ exchanger [19,21], given by

$${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{NaMgX}}={\mathit{J}}_{\mathrm{Mg}}^{\mathrm{NaMgX},\max }\times {\mathit{f}}_{\mathrm{Na}\mathit{S}}\times {\mathit{f}}_{\mathrm{Na}\mathit{C}}\times {\mathit{f}}_{\mathrm{Mg}\mathit{S}}\times {\mathit{f}}_{\mathrm{Mg}\mathit{C}},$$

(5)

where ${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{NaMgX},\max }$ denotes the maximum Mg²⁺ flux through the Na⁺/Mg²⁺ exchanger, and the f terms represent regulation by extracellular Na⁺, ${\mathit{f}}_{\mathrm{Na}\mathit{S}}=\raisebox{1ex}{${\left({\mathit{c}}_{\mathrm{Na}}^{\mathrm{S}}\right)}^{\mathbf{2}}$}\!\left/ \!\raisebox{-1ex}{$\left({\left({\mathit{c}}_{\mathrm{Na}}^{\mathrm{S}}\right)}^{\mathbf{2}}+{\left({\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{S}}\right)}^{\mathbf{2}}\right)$}\right.$, intracellular Na⁺, ${\mathit{f}}_{\mathrm{Na}\mathit{C}}=\raisebox{1ex}{${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{C}}$}\!\left/ \!\raisebox{-1ex}{$\left({\mathit{c}}_{\mathrm{Na}}^{\mathrm{C}}+{\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{C}}\right)$}\right.$, extracellular Mg²⁺, ${\mathit{f}}_{\mathrm{Mg}\mathit{S}}=\raisebox{1ex}{${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{S}}$}\!\left/ \!\raisebox{-1ex}{$\left({\mathit{c}}_{\mathrm{Mg}}^{\mathrm{S}}+{\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{S}}\right)$}\right.$, and intracellular Mg²⁺, ${\mathit{f}}_{\mathrm{Mg}\mathit{C}}=\raisebox{1ex}{${\left({\mathit{c}}_{\mathrm{Mg}}^{\mathrm{C}}\right)}^{\mathbf{2}}$}\!\left/ \!\raisebox{-1ex}{$\left({\left({\mathit{c}}_{\mathrm{Mg}}^{\mathrm{C}}\right)}^{\mathbf{2}}+{\left({\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{C}}\right)}^{\mathbf{2}}\right)$}\right.$. In these expressions, ${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{S}}$, ${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{C}}$, ${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{S}}$, and ${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{C}}$ denote the Michaelis-Menten constants.

2.4. Calcium-Sensing Receptor

The calcium-sensing receptor (CaSR) regulates not only Ca²⁺ and Mg²⁺ reabsorption in the kidneys but other electrolytes and water as well by modifying transporter activities. Ca²⁺ is the primary ligand for activating CaSR. At equimolar concentrations, Mg²⁺ is 1/2 to 2/3 as potent as Ca²⁺ in activating CaSR [22,23]. We model the effect of CaSR on a given parameter $\mathit{v}$ ($\mathit{v}$ may denote paracellular permeability, NKCC2 activity, ROMK activity, or NCC activity; see below) with the following expression:

$$\mathit{v}={\mathit{v}}^{\mathit{*}}\left(\mathbf{1}+{\mathit{\alpha }}_{\mathit{v},\mathrm{Ca}}\left(\frac{{\left({\mathit{C}}_{\mathrm{Ca}}^{\mathit{i}}\right)}^{\mathbf{4}}}{{\left({\mathit{C}}_{\mathrm{Ca}}^{\mathit{i}}\right)}^{\mathbf{4}}+{\left({\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Ca}}\right)}^{\mathbf{4}}}\right)\right)\left(\mathbf{1}+{\mathit{\alpha }}_{\mathit{v},\mathrm{Mg}}\left(\frac{{\left({\mathit{C}}_{\mathrm{Mg}}^{\mathit{i}}\right)}^{\mathbf{4}}}{{\left({\mathit{C}}_{\mathrm{Mg}}^{\mathit{i}}\right)}^{\mathbf{4}}+{\left({\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Mg}}\right)}^{\mathbf{4}}}\right)\right),$$

(6)

where ${\mathit{v}}^{\mathit{*}}$ is the value of $\mathit{v}$ in the absence of the effect of CaSR, ${\mathit{c}}_{\mathrm{Ca}}^{\mathit{i}}$ and ${\mathit{c}}_{\mathrm{Mg}}^{\mathit{i}}$ denote the concentration of Ca²⁺ and Mg²⁺ in the luminal (i = L) or interstitial (i = S) fluid, and ${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Ca}}$ = 1.25 mM and ${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Mg}}$ = 2.5 mM represent the half-maximal concentrations for Ca²⁺ and Mg²⁺ [24], respectively. CaSR is ubiquitously expressed in the kidney both along the apical and basolateral membranes, with its highest expression being at the basolateral membrane of the cortical thick ascending limb [25]. Hence, we represent $\mathit{v}$ for (i) paracellular permeability, NKCC2 activity, and renal outer-medullary potassium channel (ROMK) in the thick ascending limb, (ii) Na⁺-Cl⁻ cotransporter (NCC) activity in the distal convoluted tubule, (iii) H⁺-ATPase flux in outer-medullary collecting duct type A cells, and (iv) water permeability in the inner-medullary collecting duct. The parameters ${\mathit{\alpha }}_{\mathit{v},\mathrm{Ca}}$ and ${\mathit{\alpha }}_{\mathit{v},\mathrm{Mg}}$ are negative if CaSR has an inhibitory effect on $\mathit{v}$ and positive otherwise. Since the effect of Mg²⁺ on CaSR activation is ~50–66% of that of Ca²⁺, we set ${\mathit{\alpha }}_{\mathit{v},\mathrm{Mg}}$= 0.6${\mathit{\alpha }}_{\mathit{v},\mathrm{Ca}}$. The values for ${\mathit{\alpha }}_{\mathit{v},\mathrm{Ca}}$ and ${\mathit{\alpha }}_{\mathit{v},\mathrm{Mg}}$ for each of the segments are given in Table 1.

Table 1. Mg²⁺-specific parameters for all the segments along the superficial nephron. Values marked (*) are adjusted. PT, proximal tubule; TAL, thick ascending limb; DCT, distal convoluted tubule; CD, collecting duct; OMCD, outer-medullary collecting duct; IMCD, inner-medullary collecting duct.

Parameter	Value
PT
Tight junction permeability to Mg2+ at the lumen-LIS interface (${\mathit{P}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}$)	1.1 × 10−5 cm/s [26]
Reflection coefficient of tight junction to Mg2+	0.89 (*)
cTAL
Tight junction permeability at the lumen-LIS interface in the absence of Mg2+ (${\mathit{P}}_{\mathrm{Mg}}^{\mathrm{TAL},\mathrm{LI}*}$)	38 × 10−5 cm/s (*)
Maximum half concentration of Ca2+ (${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Ca}}$)	1.25 mM [24]
Maximum half concentration of Mg2+ (${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Mg}}$)	2.5 mM [24]
Hill function coefficient, n	4 [24]
Inhibitory coefficient of Ca2+ on tight junction permeability (${\mathit{\alpha }}_{\mathrm{P},\mathrm{Ca}}$)	−4/7 [27]
Inhibitory coefficient of Mg2+ on tight junction permeability (${\mathit{\alpha }}_{\mathrm{P},\mathrm{Mg}}$)	−0.34 (*)
Inhibitory coefficient of Ca2+ on NKCC2 activity (${\mathit{\alpha }}_{\mathrm{NKCC}2,\mathrm{Ca}}$)	−0.4 [28]
Inhibitory coefficient of Mg2+ on NKCC2 activity (${\mathit{\alpha }}_{\mathrm{NKCC}2,\mathrm{Mg}}$)	−0.24 (*)
Inhibitory coefficient of Ca2+ on ROMK activity (${\mathit{\alpha }}_{\mathrm{ROMK},\mathrm{Ca}}$)	−0.8 [29,30]
Inhibitory coefficient of Mg2+ on ROMK activity (${\mathit{\alpha }}_{\mathrm{ROMK},\mathrm{Mg}}$)	−0.48 (*)
DCT
TRPM6/7 channel density (${\mathit{N}}_{\mathbf{T}\mathbf{R}\mathbf{P}\mathbf{M}\mathbf{6}/\mathbf{7}}$)	26 × 104 cm-2 (*)
Single channel conductance of TRPM6/7 at pH 7.4 (${\mathit{g}}_{\mathrm{pH}7.4}$)	56.6 pS [20]
Luminal pH for half-maximal conductance of TRPM6/7 (${\mathrm{pH}}_{\mathbf{1}/\mathbf{2}}$)	5.5 [20]
Maximum Mg2+ flux through Na+/Mg2+ exchanger (${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{NaMgX},\max }$)	8.6 × 10−9 mmol/cm2/s (*)
Intracellular Mg2+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{C}}$)	3.59 M [19,21]
Extracellular Mg2+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{S}}$)	1.3 mM [19,21]
Intracellular Na+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{C}}$)	12.29 mM [19,21]
Extracellular Na+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{S}}$)	87.5 mM [19,21]
Excitatory coefficient of Ca2+ on NCC activity (${\mathit{\alpha }}_{\mathrm{NCC},\mathrm{Ca}}$)	0.5 [31]
Excitatory coefficient of Mg2+ on NCC activity (${\mathit{\alpha }}_{\mathrm{NCC},\mathrm{Mg}}$)	0.3 (*)
CD
Ca2+ promoting coefficient for apical HATPase activity of type A OMCD cells (${\mathit{\alpha }}_{\mathrm{HATP},\mathrm{Ca}}$)	2 [27]
Mg2+ promoting coefficient for apical HATPase activity of type A OMCD cells (${\mathit{\alpha }}_{\mathrm{HATP},\mathrm{Mg}}$)	1.2 (*)
Ca2+ inhibitory coefficient for apical water permeability of IMCD cells (${\mathit{\alpha }}_{\mathrm{Pf},\mathrm{Ca}}$)	−3/8 [27]
Mg2+ inhibitory coefficient for apical water permeability of IMCD cells (${\mathit{\alpha }}_{\mathrm{Pf},\mathrm{Mg}}$)	−0.225 (*)

Table 1. Mg²⁺-specific parameters for all the segments along the superficial nephron. Values marked (*) are adjusted. PT, proximal tubule; TAL, thick ascending limb; DCT, distal convoluted tubule; CD, collecting duct; OMCD, outer-medullary collecting duct; IMCD, inner-medullary collecting duct.

Parameter	Value
PT
Tight junction permeability to Mg2+ at the lumen-LIS interface (${\mathit{P}}_{\mathrm{Mg}}^{\mathrm{PT},\mathrm{LI}}$)	1.1 × 10−5 cm/s [26]
Reflection coefficient of tight junction to Mg2+	0.89 (*)
cTAL
Tight junction permeability at the lumen-LIS interface in the absence of Mg2+ (${\mathit{P}}_{\mathrm{Mg}}^{\mathrm{TAL},\mathrm{LI}*}$)	38 × 10−5 cm/s (*)
Maximum half concentration of Ca2+ (${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Ca}}$)	1.25 mM [24]
Maximum half concentration of Mg2+ (${\mathit{E}\mathit{C}}_{\mathbf{50},\mathrm{Mg}}$)	2.5 mM [24]
Hill function coefficient, n	4 [24]
Inhibitory coefficient of Ca2+ on tight junction permeability (${\mathit{\alpha }}_{\mathrm{P},\mathrm{Ca}}$)	−4/7 [27]
Inhibitory coefficient of Mg2+ on tight junction permeability (${\mathit{\alpha }}_{\mathrm{P},\mathrm{Mg}}$)	−0.34 (*)
Inhibitory coefficient of Ca2+ on NKCC2 activity (${\mathit{\alpha }}_{\mathrm{NKCC}2,\mathrm{Ca}}$)	−0.4 [28]
Inhibitory coefficient of Mg2+ on NKCC2 activity (${\mathit{\alpha }}_{\mathrm{NKCC}2,\mathrm{Mg}}$)	−0.24 (*)
Inhibitory coefficient of Ca2+ on ROMK activity (${\mathit{\alpha }}_{\mathrm{ROMK},\mathrm{Ca}}$)	−0.8 [29,30]
Inhibitory coefficient of Mg2+ on ROMK activity (${\mathit{\alpha }}_{\mathrm{ROMK},\mathrm{Mg}}$)	−0.48 (*)
DCT
TRPM6/7 channel density (${\mathit{N}}_{\mathbf{T}\mathbf{R}\mathbf{P}\mathbf{M}\mathbf{6}/\mathbf{7}}$)	26 × 104 cm-2 (*)
Single channel conductance of TRPM6/7 at pH 7.4 (${\mathit{g}}_{\mathrm{pH}7.4}$)	56.6 pS [20]
Luminal pH for half-maximal conductance of TRPM6/7 (${\mathrm{pH}}_{\mathbf{1}/\mathbf{2}}$)	5.5 [20]
Maximum Mg2+ flux through Na+/Mg2+ exchanger (${\mathit{J}}_{\mathrm{Mg}}^{\mathrm{NaMgX},\max }$)	8.6 × 10−9 mmol/cm2/s (*)
Intracellular Mg2+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{C}}$)	3.59 M [19,21]
Extracellular Mg2+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Mg}\mathit{S}}$)	1.3 mM [19,21]
Intracellular Na+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{C}}$)	12.29 mM [19,21]
Extracellular Na+ half-saturation constant (${\mathit{K}}_{\mathit{M},\mathrm{Na}\mathit{S}}$)	87.5 mM [19,21]
Excitatory coefficient of Ca2+ on NCC activity (${\mathit{\alpha }}_{\mathrm{NCC},\mathrm{Ca}}$)	0.5 [31]
Excitatory coefficient of Mg2+ on NCC activity (${\mathit{\alpha }}_{\mathrm{NCC},\mathrm{Mg}}$)	0.3 (*)
CD
Ca2+ promoting coefficient for apical HATPase activity of type A OMCD cells (${\mathit{\alpha }}_{\mathrm{HATP},\mathrm{Ca}}$)	2 [27]
Mg2+ promoting coefficient for apical HATPase activity of type A OMCD cells (${\mathit{\alpha }}_{\mathrm{HATP},\mathrm{Mg}}$)	1.2 (*)
Ca2+ inhibitory coefficient for apical water permeability of IMCD cells (${\mathit{\alpha }}_{\mathrm{Pf},\mathrm{Ca}}$)	−3/8 [27]
Mg2+ inhibitory coefficient for apical water permeability of IMCD cells (${\mathit{\alpha }}_{\mathrm{Pf},\mathrm{Mg}}$)	−0.225 (*)

3. Results

3.1. Baseline Results

Using rat parameters, we computed the model’s luminal fluid flow, luminal fluid solute concentrations, cytosolic solute concentrations, membrane potential, and fluxes. Figure 2 shows the predicted segmental delivery, transport, and luminal fluid concentration of Mg²⁺ and Ca²⁺ along the nephron. Results for other electrolytes and water can be found in Ref. [32].

The majority of the filtered Ca²⁺ is reabsorbed along the proximal tubule, accounting for 68% of the filtered load; by contrast, Mg²⁺ reabsorption along this segment accounts for only 22% of the filtered load. Since Mg²⁺ reabsorption is low along the proximal tubule, Mg²⁺ concentration increases by ~3-fold [4]. The majority of the overall Mg²⁺ transport occurs downstream along the cortical thick ascending limb (68% of the filtered load), where the lumen-positive membrane potential drives Mg²⁺ reabsorption via the paracellular pathway. The fractional reabsorption of Ca²⁺ along the medullary and cortical thick ascending limb is 20%. The final nephron segment that transports Mg²⁺ is the distal convoluted tubule, where 6.6% of the filtered Mg²⁺ is reabsorbed. The fractional reabsorption of Ca²⁺ along the distal convoluted tubule and connecting tubule is 7.9%. Finally, fractional urinary Mg²⁺ and Ca²⁺ excretions are 3.2% and 3.9%, respectively.

3.2. Effect of Loop Diuretics

Loop diuretics inhibit NKCC2, which is expressed on the apical membrane of the thick ascending limb. We simulated the effect of acute administration of loop diuretics by inhibiting NKCC2 activity by 70%. We assumed that the NKCC2 inhibitor was administered for long enough to significantly impair the kidney’s ability to generate an axial osmolality gradient. The cortical interstitial concentrations were assumed to remain unchanged. Since the concentrating mechanism of the outer medulla is significantly impaired following complete NKCC2 inhibition, the interstitial concentrations of Mg²⁺ and Ca²⁺ at the outer-inner medullary boundary are lowered to 0.77 mM (from a baseline value of 0.96 mM) and 2.0 mM (from a baseline value of 2.5 mM), respectively. At the papillary tip, the interstitial concentrations of Mg²⁺ and Ca²⁺ are reduced to 1.0 mM (from 1.54 mM) and 2.62 mM (from 4.0 mM), respectively. For changes in the interstitial concentrations of Na⁺, K⁺, Cl⁻, and urea, refer to [9].

The predicted Mg²⁺ and Ca²⁺ transport along the thick ascending limb and distal tubules and urinary Mg²⁺ and Ca²⁺ excretions following NKCC2 inhibition in male rats are shown in Figure 3. Our NKCC2 inhibition simulations predicted the fractional Mg²⁺ reabsorption along the cortical thick ascending limb to decrease to 57% from the baseline fractional reabsorption of 69% (Figure 3). Administration of furosemide, a loop diuretic, to male mice increased TRPM6 mRNA expression by 30% [33]. Our model simulations predicted a 68% increase in TRPM6/7 channel activity to account for the 240% increase in Mg²⁺ excretion in male rats undergoing furosemide treatment [34]. Fractional Ca²⁺ reabsorption along the thick ascending limb decreased by 17% following NKCC2 inhibition (Figure 3). This resulted in urinary Ca²⁺ excretion increasing to 236% of the baseline excretion value (Figure 3).

3.3. Thiazide Diuretics

Thiazide diuretics inhibit NCC, which is expressed along the apical membrane of the distal convoluted tubule. We simulated the effect of acute administration of thiazide diuretics by inhibiting NCC activity by 70%. In the NCC inhibition simulations, baseline interstitial concentration profiles were used.

The predicted fractional Mg²⁺ excretion after NCC inhibition increased to 3.1% from the baseline value of 2.8% (Figure 3). This is in agreement with experimental data where male rats treated with bendrofluazide, a thiazide diuretic, did not show any significant change in Mg²⁺ excretion [35]. Acute administration of chlorothiazide to male mice increased TRPV5 mRNA expression by 40–80% and decreased Ca²⁺ excretion by ~60% [36]. Accordingly, we increased TRPV5 activity by 52% following NCC inhibition in our model. This decreased the predicted Ca²⁺ excretion by 63% from the baseline excretion value (Figure 3).

3.4. K-Sparing Diuretics

K-sparing diuretics, such as amiloride, block Na⁺ uptake through ENaC, expressed on the apical membrane of the late distal convoluted tubule as well as along the full length of the connecting tubule and collecting ducts. In our model, we simulated the effect of K-sparing diuretics by reducing ENaC activity by 70%.

ENaC inhibition hyperpolarizes the luminal membrane potential and increases K⁺, Ca²⁺, and Mg²⁺ uptake [37,38,39,40]. Our model simulations predicted Mg²⁺ reabsorption along the distal convoluted tubule to increase by 8.2% and Ca²⁺ reabsorption along the distal convoluted tubule and connecting tubule to increase by 29% (Figure 3). These increased reabsorptions decreased Mg²⁺ and Ca²⁺ excretions by 31% and 56%, respectively (Figure 3).

3.5. TRPM6/7 Inhibition

Kidney-specific TRPM6 [41] and TRPM7 [42] knock-out mice did not display hypomagnesemia and increased urinary Mg²⁺ excretion. This indicates that in mice, there must be other Mg²⁺ uptake mechanisms in the distal convoluted tubule. In fact, Verschuren et al. [43] reported that fluid shear stress (FSS) stimulated Mg²⁺ uptake in mDCT15 cells, and this uptake was independent of the TRPM6 and TRPM7 channels. However, the pathways or regulatory mechanisms for this FSS-sensitive Mg²⁺ uptake are unclear and hence not included in our present model.

To simulate TRPM6/7 knock-out experiments, we inhibited the TRPM6/7 channel by 100%. Our model predicted fractional Mg²⁺ excretion to increase to 8.8% (from baseline 3.2%), which is significantly above the 2–5% physiological fractional Mg²⁺ excretion. How much should the Mg²⁺ uptake through the FSS-sensitive Mg²⁺ channel be for fractional Mg²⁺ excretion to be within the physiological range when TRPM6/7 is completely inhibited? Our simulations predicted that if Mg²⁺ uptake through the FSS-sensitive channel is at least 60% of the Mg²⁺ uptake through TRPM6/7, then the fractional Mg²⁺ excretion becomes 4.3% (within the physiological range).

4. Discussion

Calcium (Ca²⁺) and magnesium (Mg²⁺) are both essential for cellular function. The homeostasis of these cations must be tightly regulated, and that balance is facilitated by intestinal absorption and renal excretion. For Na⁺, Cl⁻, K⁺, Ca²⁺, and many other major filtered solutes, most of the renal reabsorption occurs along the proximal convoluted tubule (about 1/2 to 2/3 in rats); the same is true for water. Thus, the luminal concentrations of these solutes, including Ca²⁺, remain close to plasma along the proximal convoluted tubule. The majority of proximal tubule Ca²⁺ reabsorption occurs via a passive paracellular process, driven by Na⁺ reabsorption mediated primarily by NHE3 and subsequent water reabsorption. In contrast, only 15–25% of the filtered Mg²⁺ load is reabsorbed along the proximal tubule. As a result, its concentration rises significantly along the proximal tubule. For Mg²⁺, most of the reabsorption occurs along the cortical thick ascending limb (about 60–70%), while somewhat unexpectedly, essentially none occurs along the medullary thick ascending limb. Most of the remainder of the Mg²⁺ is reabsorbed along the distal convoluted tubule.

What difference does it make for the cortical thick ascending limb and distal convoluted tubule to handle most of the Mg²⁺ transport instead of the proximal tubule, as in the case of Na⁺ and Cl⁻? Having these distal segments responsible for transporting a substantial fraction of the filtered Mg²⁺ load via the pathways that can be regulated may give the kidney a better ability to regulate Mg²⁺ balance. Recall that the plasma Mg²⁺ level is orders of magnitude lower than Na⁺ or Cl⁻. Thus, to maintain plasma [Mg²⁺] within a narrow range, the ability to fine-tune renal Mg²⁺ transport is particularly crucial. Parathyroid hormone, for instance, increases Mg²⁺ reabsorption in both the cortical thick ascending limb and distal convoluted tubule [44]. Transport of Mg²⁺ along these segments can also be regulated by hormones such as calcitonin, vasopressin, glucagon, and β-adrenergic agonists [45]. Coincidentally, some common diuretics also target these segments.

The goal of this study is to better understand the impact of diuretics on renal Ca²⁺ and Mg²⁺ transport. Diuretics are medications that reduce fluid buildup in the body and are often employed in the management of hypertension, edema, and various other conditions influenced by changes in electrolyte transport. In the context of kidney function, diuretics often focus on transport proteins or mechanisms vital for the reabsorption of Na⁺, Cl⁻, and water. Considering that renal Ca²⁺ and Mg²⁺ transports are driven primarily by the electrochemical gradients established by tubular NaCl transport processes, potential modifications in the renal handling of Ca²⁺ and Mg²⁺ by the administration of diuretics deserve scrutiny.

Loop diuretics, such as furosemide, induce notable natriuresis by targeting the thick ascending limb of the nephron; specifically, they inhibit NKCC2-dependent transport by competing for the chloride (Cl⁻) binding site [46]. Given that the inhibition of NKCC2 reduces the lumen-positive transepithelial voltage gradient across the thick ascending limb epithelium [47], it is unsurprising that the transport of Mg²⁺ and Ca²⁺ in this segment is substantially decreased [48,49]. In fact, loop diuretics like furosemide induce hypercalciuria and hypermagnesuria in both experimental animals [50] and human subjects [51]. Thiazide diuretics induce a natriuretic response by inhibiting NCC and blocking NaCl transport in the distal convoluted tubule. A hypocalciuric effect has been reported following thiazide treatment [52]. Consistent with the drug’s mode of action, among hypertensive individuals undergoing chronic thiazide treatment, there is a slight decrease in serum Mg²⁺ levels compared to those not taking diuretics [53], although the effect appears to be subtle. K-sparing diuretics function by inhibiting ENaC, a protein expressed in the late distal convoluted tubule, connecting tubule, and collecting duct of the kidney. Research has confirmed that K-sparing diuretics impact the urinary excretion of Ca²⁺ and Mg²⁺ in both human subjects and animals. Hypertensive individuals undergoing K-sparing diuretics treatment exhibit reduced urinary Ca²⁺ excretion [53] and elevated serum Mg²⁺ levels compared to those not receiving treatment [53]. The effect of these three classes of diuretics on urinary Mg²⁺ and Ca²⁺ excretions has been summarized in Figure 4.

The present study considers renal Ca²⁺ and Mg²⁺ transport under normal physiological conditions. The homeostasis of these electrolytes is altered during pregnancy, lactation, and dietary restriction, as well as in diseases such as diabetes and chronic kidney disease. To utilize the present model for in silico studies of how the kidney adapts in terms of Ca²⁺ and Mg²⁺ transport under these physiological and pathophysiological conditions, one can combine the model with computational models of kidney function for a pregnant rat [54,55], a diabetic rat [56,57,58], and a nephrectomized rat [8,59]. The resulting models may provide insights into altered renal Ca²⁺ and Mg²⁺ transport under these conditions. How do changes in renal Ca²⁺ and Mg²⁺ transport impact whole-body Ca²⁺ and Mg²⁺ homeostasis? A whole-body Ca²⁺ and Mg²⁺ balance model may help answer that question. By incorporating the present model into whole-body electrolyte balance models (e.g., [60,61,62,63]), one can obtain an integrative model to study whole-body Ca²⁺ and Mg²⁺ balance.

The present model simulates electrolyte and water transport along a superficial nephron, which constitutes only 2/3 of the total nephron population in a rat kidney. The juxtamedullary nephrons, whose loops of Henle extend into various depths of the inner medulla, comprise the remaining nephron population. These two types of nephrons differ in the single-nephron glomerular filtration rate, transport area, and transporter activities. This study focuses on a superficial nephron model to gain a clearer understanding of segmental Mg²⁺ transport. In future studies, developing a kidney model that incorporates both types of nephrons will enable more accurate predictions of urinary excretion rates.

Another limitation of this model is that it does not include cyclin and CBS domain divalent metal cation transport mediator 2 (CNNM2), expressed on the basolateral membrane, which is also potentially involved in Mg²⁺ transport along the distal convoluted tubule. CNNM2 has been shown to mediate both cellular Mg²⁺ influx and efflux. The role of CNNM2 as an Mg²⁺ transporter has been openly debated (in favor [64]; opposing view [65]). A common view is that CNNM2 is not a Mg²⁺ transporter by itself but is an important protein for regulating Mg²⁺ homeostasis [65,66]. Due to these conflicting views on the role of CNNM2 as a Mg²⁺ transporter, we did not include it in our present model. CNNM2 can be incorporated into our model in the future when more consistent experimental outcomes are available.