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R Soc Open Sci
R Soc Open Sci
RSOS
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Royal Society Open Science
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rsos231484
10.1098/rsos.231484
1008100111918144Computer Science and Artificial Intelligence
Research
How the kidney regulates magnesium: a modelling study
How the kidney regulates magnesium: a modelling study
https://orcid.org/0000-0001-9342-4176
Dutta Pritha 1 Data curation Formal analysis Investigation Methodology Software Supervision Validation Visualization Writing – original draft Writing – review and editing p7dutta@uwaterloo.ca

Hakimi Shervin 1 Writing – review and editing shervin.hakimi@uwaterloo.ca

Layton Anita T. 1 2 3 4 Conceptualization Funding acquisition Methodology Resources Supervision Writing – original draft Writing – review and editing anita.layton@uwaterloo.ca

1 Department of Applied Mathematics, University of Waterloo , Waterloo, Ontario N2L 3G1, Canada
2 Department of Biology, University of Waterloo , Waterloo, Ontario N2L 3G1, Canada
3 Cheriton School of Computer Science, University of Waterloo , Waterloo, Ontario N2L 3G1, Canada
4 School of Pharmacology, University of Waterloo , Waterloo, Ontario N2L 3G1, Canada
3 2024
20 3 2024 March 20, 2024
20 3 2024 March 20, 2024
11 3 23148429 9 2023 September 29, 2023
23 1 2024 January 23, 2024
15 2 2024 February 15, 2024
© 2024 The Authors.
2024
https://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

The kidneys are crucial for maintaining Mg2+ homeostasis. Along the proximal tubule and thick ascending limb, Mg2+ is reabsorbed paracellularly, while along the distal convoluted tubule (DCT), Mg2+ is reabsorbed transcellularly via transient receptor potential melastatin 6 (TRPM6). TRPM6 and other renal transporter expressions are regulated by sex hormones. To investigate renal Mg2 handling, we have developed sex-specific computational models of electrolyte transport along rat superficial nephron. Model simulations indicated that along the proximal tubule and thick ascending limb, Mg2+ and Na+ transport occur parallelly, but they are dissociated along the DCT. In addition, our models predicted higher paracellular Mg2+ permeability in females to attain similar cortical thick ascending limb fractional Mg2+ reabsorption in both sexes. Furthermore, DCT fractional Mg2+ reabsorption is higher in females than in males, allowing females to better fine-tune Mg2+ excretion. We validated our models by simulating the administration of three classes of diuretics. The model predicted significantly increased, marginally increased and significantly decreased Mg2+ excretions for loop, thiazide and K-sparing diuretics, respectively, aligning with experimental findings. The models can be used to conduct in silico studies on kidney adaptations to Mg2+ homeostasis alterations during conditions such as pregnancy, diabetes and chronic kidney disease.

magnesium homeostasis
; electrolyte transport
; renal transport
; sex differences
National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RGPIN-2019-03916 Canada Institutes of Health Research (CIHR) Project Grant TNC-174963 Canada 150 Research Chairs program
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pmc1. Introduction

Magnesium, the second most abundant intracellular cation, is an important cofactor for numerous biological processes, including protein synthesis, nucleic acid stability and neuromuscular excitability. However, the clinical significance of Mg2+ has only been properly acknowledged in recent years. As such, Mg2+ was once referred to as ‘the forgotten electrolyte’ [1]. Extracellular magnesium is tightly regulated, with plasma (Mg2+) maintained relatively constant between 1.6 and 2.3 mg/dl [2] under normal physiological conditions in humans. Almost all of the body’s Mg2+ (approximately 99%) is stored either in bone or within cells [2]. The normal daily Mg2+ intake of humans averages around 300 mg, about half of which is absorbed by the intestine [3].

Together with the intestine, the kidneys have an important role in maintaining Mg2+ homeostasis. About 70% of the circulating Mg2+ is non-protein bound and is thus filtered by the glomerulus, accounting for 2,400 mg in humans. Much of our understanding of Mg2+ handling in the various nephron segments has been derived from micropuncture and microperfusion studies in rodent nephrons [4,5]. More recently, genetic studies have expanded our knowledge about the protein mediators of Mg2+ transport [6]. The kidney reabsorbs 95–98% of the filtered Mg2+ primarily along three nephron segments: the proximal tubule (15–25%), cortical thick ascending limb (cTAL) (60–70%) and distal convoluted tubule (DCT) (up to ~23%) [7]. In the rat kidney, along both the proximal tubule and cTAL, Mg2+ is reabsorbed via the paracellular pathway, driven by a favourable electrochemical gradient that arises primarily from the active transport of Na+. Along the DCT, Mg2+ is reabsorbed transcellularly, mediated by the transient receptor potential melastatin 6 (TRPM6) that is expressed along the apical membrane, and extruded from the cell by the Na+/Mg2+ exchanger that is expressed along the basolateral membrane.

The expression and activity of TRPM6 are upregulated by oestrogens [8]. Across species, many other aspects of kidney function and structure are known to be regulated by sex hormones as well [9–11]. The kidney mass of a female rat is approximately half that of a male rat [11]. The single-nephron glomerular filtration rate (SNGFR) in female rat kidneys is lower than in male rat kidneys, even though they have similar glomerulus population [12]. Similar to urinary output, which is not substantially different between the sexes [11], Mg2+ excretion has also been observed to be only ~8% higher in male rats compared with female rats [13]. Major sex differences in the abundance pattern of electrolyte transporters, channels and claudins, collectively referred to as transporters, have been reported [11,14]. Overall, in rats and mice, the female-to-male ratios for Na+ and water transporter abundance are ≤1 along the proximal tubule to medullary thick ascending limb, and ≥1 along the downstream segments, from cTAL through the collecting duct. The proximal pattern is partially attributed to shorter proximal tubules in females versus males in rodents [11].

Given these morphological, hemodynamic and transport capacity differences between the sexes, how does nephron segmental transport differ between the sexes to yield the reported Mg2+ excretion rates? Furthermore, to what extent do these differences reflect life stages that are unique to females? Specifically, while males develop to adulthood, mate and age, females may undergo serial pregnancies and lactation and then menopause—normal life cycle changes that challenge renal transporter expression, abundance and activity to maintain homeostasis. As such, the female-specific renal transport abundance pattern may give female rats the reserve transport capacity to meet the markedly altered demands of Mg2+ and other electrolytes in pregnancy and lactation. A better understanding of the above issue would shed insight into some of the different male and female responses to physiological, pathophysiological and pharmaceutical challenges. Towards this goal, we formulated the first set of computational models of renal epithelial transport and conducted simulations to predict the transport of Mg2+, as well as other electrolytes and water, along the superficial nephron of the male and female rat kidney. We formulated separate models for a male rat and a female rat and applied those models to investigate the functional implications of sex differences in kidney structure and transport.

2. Methods

We have previously published a series of epithelial cell-based computational models of transporter-mediated solute and water transport along the nephron of a rat kidney [15–18]. The focus of those studies was on the renal handling of Na+, K+, glucose and water in physiological and pathophysiological conditions [16,17], the associated oxygen consumption [15,16], and the functional impacts of sex differences in renal morphology, hemodynamics and transporter pattern [18]. Those models were recently extended to simulate Ca2+ transport along the rat kidney [19–22]. In this study, we extend the models to investigate Mg2+ transport along a superficial nephron in the kidney of a male rat and a female rat.

The superficial nephron model includes the proximal tubule, short descending limb, thick ascending limb, DCT, 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 +, HPO4 2−, H2PO4 −, H+, HCO2 −, H2CO2, urea, glucose, Ca2+ and Mg2+. The segment and cell types 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 either by moving across the apical and basolateral membranes in the transcellular pathway, mediated by specialized membrane transporters or channels, or via the paracellular pathway between neighbouring cells. The model is defined by a large system of coupled differential and algebraic equations that describe mass conservation and determine transmembrane and paracellular fluxes [23]. 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.

The details of how Mg2+ transport is modelled along the proximal tubule, cTAL and DCT are given in the following sections. A schematic diagram of the model nephron and the three epithelial cell types that mediate Mg2+ transport is given in figure 1. Male and female rat models differ in parameters describing SNGFR, tubular dimensions, membrane transporter and channel activities, and paracellular permeabilities. Model parameters that describe Mg2+ transport are given in table 1. Additional model parameters can be found in [22].

Figure 1. Model diagram of epithelial transport of Mg2+ and selected electrolytes along the superficial nephron. Mg2+ transport occurs along the proximal tubule (PCT and S3), cTAL and DCT. Only the major Na+, K+, Cl−, Ca2+ and Mg2+ transporters are shown. PCT, proximal convoluted tubule; SDL, short descending limb; mTAL, medullary thick ascending limb; CNT, connecting tubule; CCD, cortical collecting duct; OMCD, outer-medullary collecting duct; IMCD, inner-medullary collecting duct. Nephron image adapted from [15].

Table 1. Mg2+-specific parameters for all the segments along the superficial nephron. Values marked (*) are adjusted and those marked (**) are the same for both male and female models. PT, proximal tubule; TAL, thick ascending limb; DCT, distal convoluted tubule; CNT, connecting tubule; CD, collecting duct; OMCD, outer-medullary collecting duct; IMCD, inner-medullary collecting duct; NCC, Na+–Cl− cotransporter.

parameter	value	details	
	male	female		
PT		
tight junction permeability to Mg2+ at the lumen–LIS interface ( PMgPT,LI )	1.1 × 10−5cm/s**	1.1 × 10−5 cm/s [24]	single nephron microperfusion experiment on adult female Wistar rats	
reflection coefficient of tight junction to Mg2+	0.89*	0.89**	assumed to be the same as the reflection coefficient of tight junction to Ca2+	
cTAL		
tight junction permeability at the lumen–LIS interface in the absence of Mg2+ ( PMgTAL,LI* )	38 × 10−5 cm/s*	56 × 10−5 cm/s*	estimated (refer to §2.5)	
maximum half concentration of Ca2+ ( EC50,Ca )	1.25 mM [25]	1.25**	in vitro study on bovine parathyroid cells	
maximum half concentration of Mg2+ ( EC50,Mg )	2.5 mM [25]	2.5**	in vitro study on bovine parathyroid cells	
Hill function coefficient, n	4 [25]	4**	in vitro study on bovine parathyroid cells	
inhibitory coefficient of Ca2+ on tight junction permeability ( αP,Ca )	−4/7 [26]	−4/7**	in vitro microperfusion experiment on male Sprague Dawley rats	
inhibitory coefficient of Mg2+ on tight junction permeability ( αP,Mg )	−0.34*	−0.34**	estimated (refer to §2.4)	
inhibitory coefficient of Ca2+ on NKCC2 activity ( αNKCC2,Ca )	−0.4 [27]	−0.4**	in vivo study on mice (sex not specified)	
inhibitory coefficient of Mg2+ on NKCC2 activity ( αNKCC2,Mg )	−0.24*	−0.24**	estimated (refer to §2.4)	
inhibitory coefficient of Ca2+ on ROMK activity ( αROMK,Ca )	−0.8 [28,29]	−0.8**	in vitro study on rat (sex not specified)	
inhibitory coefficient of Mg2+ on ROMK activity ( αROMK,Mg )	−0.48*	−0.48**	estimated (refer to §2.4)	
DCT		
TRPM6 channel density ( NTRPM6 )	22.5 × 1041/cm2*	45 × 104 1/cm2*	estimated (refer to §2.5)	
single channel conductance of TRPM6 at pH 7.4 ( gpH7.4 )	83.6 pS [30]	83.6 pS**	in vitro study using CHOK1 cells	
luminal pH for half-maximal conductance of TRPM6 ( pH1/2 )	4.3 [30]	4.3**	in vitro study using CHOK1 cells	
maximum Mg2+ flux through Na+/Mg2+ exchanger ( JMgNaMgX,max )	8.6 × 10−9 mmol/cm2/s*	9 × 10−9 mmol/cm2/s*	estimated (refer to §2.5)	
intracellular Mg2+ half-saturation constant ( KM,MgC )	3.59 M [31]	3.59 M**	in vitro study using rat erythrocyte cells	
extracellular Mg2+ half-saturation constant ( KM,MgS )	1.3 mM [31]	1.3 mM**	in vitro study using rat erythrocyte cells	
intracellular Na+ half-saturation constant ( KM,NaC )	12.29 mM [31]	12.29 mM**	in vitro study using rat erythrocyte cells	
extracellular Na+ half-saturation constant ( KM,NaS )	87.5 mM [31]	87.5 mM**	in vitro study using rat erythrocyte cells	
excitatory coefficient of Ca2+ on NCC activity ( αNCC,Ca )	0.5 [32]	0.5**	in vivo study on C57BL/6 male mice	
excitatory coefficient of Mg2+ on NCC activity ( αNCC,Mg )	0.3*	0.3**	estimated (refer to §2.4)	
CD		
Ca2+ promoting coefficient for apical HATPase activity of type A OMCD cells ( αHATP,Ca )	2 [26]	2**	in vivo study on mice	
Mg2+ promoting coefficient for apical HATPase activity of type A OMCD cells ( αHATP,Mg )	1.2*	1.2**	estimated (refer to §2.4)	
Ca2+ inhibitory coefficient for apical water permeability of IMCD cells ( αPf,Ca )	−3/8 [26]	−3/8**	in vitro study on Sprague Dawley rat cells (sex not specified)	
Mg2+ inhibitory coefficient for apical water permeability of IMCD cells ( αPf,Mg )	−0.225*	−0.225**	estimated (refer to §2.4)	

2.1. Proximal tubule

The proximal tubule reabsorbs 15–25% of the filtered Mg2+ load through the paracellular pathway, driven by the favourable electrochemical gradient established by Na+/H+ exchanger 3 (NHE3)-mediated Na+ reabsorption [5,33,34]. The paracellular electrodiffusive flux ( JMgPT,LI ) is given by

(2.1) JMgPT,LI=PMgPT,LIζMgLI(CMgL−CMgIe−ζMgLI1−e−ζMgLI),

where the superscripts L and I denote lumen and lateral intercellular space (LIS), respectively; PMgPT,LI denotes the permeability of Mg2+ at the lumen and LIS interface; ζMgLI=ZMgFRTψL-ψI ; ZMg is the valence of Mg2+ (+2); CMgL and CMgI denote Mg2+ concentrations in the lumen and LIS, respectively; ψL and ψI denote the luminal and LIS membrane potentials, respectively; RT = 2.57 J/mmol; and F = 96.5 C/mmol represents Faraday’s constant.

Paracellular Mg2+ transport is believed to be mediated by claudin-2 and claudin-12 [35]. The permeability of Mg2+ at the LIS has been measured in adult female rats to be ~1.1 × 10−5 cm/s [24]. The permeability for the basement membrane is estimated based on the permeability to Na+, given by a Mg2+-to-Na+ free diffusivity ratio of 7.05:13.3 [36]. The reflection coefficient of Mg2+ has not been measured and is assumed to be the same as Ca2+ (0.89), as the two divalent ions have similarly limited paracellular convective transport (solvent drag). We varied the reflection coefficient of Mg2+ from 0.5 to 1.0. The resulting fractional excretion of Mg2+ varied from 2.2% to 3% for male rats and 2.4% to 3.5% for female rats. These values are within the physiological range of 2–5% of fractional excretion of Mg2+. Thus, varying the reflection coefficient of Mg2+ does not cause a major change in Mg2+ excretion in either male or female rats.

2.2. Thick ascending limb

The cTAL reabsorbs 60–70% of the filtered Mg2+ via the paracellular route driven by the electrochemical gradient established by Na+–K+–Cl− cotransporter 2 (NKCC2)-mediated Na+ transport [33,34]. Claudin-16 and claudin-19 are the main claudins that regulate Mg2+ permeability [7]. The medullary thick ascending limb has negligible Mg2+ reabsorption [37–39]. Paracellular Mg2+ transport in the cTAL JMgcTAL,LI is represented by equation (2.1) with the superscript ‘PT’ replaced by ‘cTAL’.

2.3. Distal convoluted tubule

As previously noted, the DCT reabsorbs 5–23% of the filtered Mg2+. This wide range of fractional reabsorption allows the segment to play a crucial role in fine-tuning urinary Mg2+ excretion, as nephron segments that are further downstream do not have the capacity for significant Mg2+ transport. The net Mg2+ reabsorption in the DCT is unidirectional as no secretion of Mg2+ has been reported [40]. Since the DCT has a lumen-negative transepithelial potential and high epithelial resistance, Mg2+ reabsorption in this segment is active and transcellular and is mediated by the TRPM6 channels, expressed on the apical membranes [41]. TRPM6 is inhibited by intracellular Mg2+ concentration ( CMgC ) and the effect is given by

(2.2) fMg=11+(CMgC0.51)2.

Lowering extracellular pH increases TRPM6 inward currents [30]. This effect is modelled as

(2.3) fpH=gpH7.4(2−11+7.4−pHL7.4−pH1/2),

where gpH7.4 denotes the single channel conductance of TRPM6 at pH 7.4, pHL denotes the luminal fluid pH and pH1/2 denotes the luminal fluid pH for half-maximal conductance.

The Mg2+ flux through TRPM6 also depends on the luminal Mg2+ concentration [42] and is modelled as

(2.4) fconcL=11+0.8CMgL.

Hence, the flux of Mg2+ through TRPM6 is given by

(2.5) JMgTRPM6=NTRPM6×fMg×fpH×ΔψLC−EMgLC2F×fconcL,

where NTRPM6 denotes the TRPM6 channel density, ΔψLC denotes the potential difference across the apical membrane and EMgLC denotes the Nernst potential of Mg2+:

(2.6) EMgLC=RT2F×ln⁡(CMgCCMgL).

Mg2+ efflux through the basolateral membrane is not well characterized. Mg2+ efflux likely requires an anti-porter or ATPase, since no chemical gradient exists for Mg2+ (with both intracellular and extracellular Mg2+ concentrations being in the range of 0.4–1 mM), while the potential gradient favours Mg2+ influx. A Na+/Mg2+ exchanger has been demonstrated as the mechanism for Mg2+ efflux in multiple cell types [41,43]. The molecular identity of the Na+/Mg2+ exchanger remains to be definitively identified, but the most promising candidate is the solute carrier family 41 member 1 (SLC41A1), which has been shown to facilitate Mg2+ efflux [41]. Na+/Mg2+ exchanger activity is regulated by the following [43]:

(i) extracellular Na+, fNaS=(cNaS)2((cNaS)2+(KM,NaS)2),

(ii) intracellular Na+, fNaC=KM,NaC(cNaC+KM,NaC),

(iii) extracellular Mg2+, fMgS=KM,MgS(cMgS+KM,MgS), and

(iv) intracellular Mg2+, fMgC=(cMgC)2((cMgC)2+(KM,MgC)2),

where cNaS and cMgS denote interstitial fluid Na+ and Mg2+ concentrations, respectively, and KM,NaS , KM,NaC , KM,MgS and KM,MgC denote the Michaelis–Menten constants. Hence, the Na+/Mg2+ exchanger-mediated Mg2+ flux is formulated as

(2.7) JMgNaMgX=JMgNaMgX,max×fNaS×fNaC×fMgS×fMgC,

where JMgNaMgX,max denotes the maximum Mg2+ flux through Na+/Mg2+ exchanger. The estimation of the Mg2+ transporter parameters, NTRPM6 and JMgNaMgX,max , is explained in §2.5.

2.4. Calcium-sensing receptor

The calcium-sensing receptor (CaSR) plays an important role in Ca2+ and Mg2+ homeostasis by controlling parathyroid hormone (PTH) secretion from the parathyroid gland as well as regulating Ca2+ and Mg2+ reabsorption in the kidneys. CaSR is ubiquitously expressed in the kidney along both the apical (proximal convolulted tubule, DCT and inner-medullary collecting duct) and basolateral membranes (thick ascending limb, DCT, cortical and outer-medullary collecting duct), with its highest expression being at the basolateral membrane of the cTAL [44]. 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 [31,45]. We model the effect of CaSR on a given parameter v (v may denote paracellular permeability, NKCC2 activity, renal outer-medullary potassium channel (ROMK) activity or Na+–Cl− cotransporter (NCC) activity; see below) with the following function:

(2.8) v= v∗(1+αv,Ca((CCai)n(CCai)n+(EC50,Ca)n))(1+αv,Mg((CMgi)n(CMgi)n+(EC50,Mg)n)),

where v* is the value of v in the absence of the effect of CaSR, cCai and cMgi denote the concentration of Ca2+ and Mg2+, respectively, in the luminal (i = L) or interstitial (i = S) fluid, EC50,Ca = 1.25 mM and EC50,Mg = 2.5 mM represent the half-maximal concentrations for Ca2+ and Mg2+ [25], respectively, and n is the Hill coefficient, which is set to 4 because of the steep relationship between extracellular Ca2+ and Mg2+ concentrations and PTH release [25]. The parameters αv,Ca and αv,Mg are negative if CaSR has an inhibitory effect on v and positive otherwise. Since the effect of Mg2+ on CaSR activation is ~50–66% of that of Ca2+, we set αv,Mg = 0.6 αv,Ca . We set the value for αv,Ca and αv,Mg for each of the following parameters as follows:

— Paracellular permeability in the thick ascending limb: Inhibition of CaSR increases paracellular permeability of Ca2+ by 41% along the thick ascending limb without any changes in Na+ and Cl− fluxes [46]. So, we set αP,Ca to −4/7 [26] and αP,Mg = 0.6 × (−4/7) = −0.34. The cTAL permeability to Mg2+ in the absence of the effect of CaSR ( PMgcTAL,LI* ) is taken as 38 × 10−5 cm/s in male rats and 56 × 10−5 cm/s in female rats, so that under basal conditions, PMgcTAL,LI is equal to 27.2 × 10−5 cm/s in male rats and 40 × 10−5 cm/s in female rats. These sex differences are explained in §2.5.

— NKCC2 activity in the thick ascending limb: NKCC2 phosphorylation decreased by ~40% in mice treated with calcimimetic [32]. In another study, targeted deletion of CaSR in the kidney of mice resulted in a significant increase in NKCC2 phosphorylation [27]. Hence, we set αNKCC2,Ca to −0.4 and αNKCC2,Mg = 0.6 × (−0.4) = −0.24.

— ROMK activity in the thick ascending limb: Increasing extracellular Ca2+ concentration from 1.1 to 5 mM decreased ROMK channel activity in the thick ascending limb by ~84% [28,29]. Thus, we set αROMK,Ca to −0.8 and αROMK,Mg = 0.6 × (−0.8) = −0.48.

— NCC activity in the DCT: NCC phosphorylation increased by ~50% in mice treated with calcimimetic [32]. Hence, we set αNCC,Ca to 0.5 and αNCC,Mg = 0.6 × 0.5 = 0.3.

— H + -ATPase flux in outer-medullary collecting duct type A cells: Increasing luminal Ca2+ concentration from 0.1 to 5 mM increased the rate of intracellular pH recovery in OMCD; that is, the proton flux across H+-ATPase in outer-medullary collecting duct type A cells increased. Hence, αHATP,Ca is set to 2 [26] and αHATP,Mg = 0.6 × 2 = 1.2.

— Water permeability in the inner-medullary collecting duct: Increasing luminal Ca2+ concentration from 1 to 5 mM decreased the water permeability of the apical membrane in the inner-medullary collecting duct of rat by 30% [47]. Hence, we set αPf,Ca to −3/8 [26] and αPf,Mg = 0.6 × (−3/8) = −0.225.

2.5. Sex differences in Mg2+ transport parameters

Along the proximal tubule, Mg2+ paracellular permeability is assumed to be the same in both sexes. Sex differences in the transport properties for Na+ and other electrolytes yield a higher fractional reabsorption of Na+ and water in male rats relative to females. Consequently, the luminal Mg2+ concentration rises more in males. This does not manifest in any notable differences in proximal tubule Mg2+ reabsorption due to its low Mg2+ permeability. However, Mg2+ transport rate is several times higher along the cTAL due to its higher Mg2+ paracellular permeability. The absolute reabsorption rate of Mg2+ along this segment is ~7–11 pmol/min for male and female rats (the experiment did not report sex-specific values) [40]. For the male model, we estimate the paracellular permeability of Mg2+, PMgcTAL,LI , to be 27.2 × 10−5 cm/s so that the absolute reabsorption rate of Mg2+ along this segment is ~10 pmol/min, which corresponds to ~69% fractional Mg2+ reabsorption. To maintain a similar fractional Mg2+ reabsorption along the cTAL in both sexes despite the higher driving force (luminal Mg2+ concentration) in males, we assume that the Mg2+ paracellular permeability is higher in females (40 × 10−5 cm/s). The absolute reabsorption rate of Mg2+ along the DCT lies in the range of 1–1.7 pmol/min for male and female rats (the experiment did not report sex-specific values) [40]. Hence, for the male rat model, the parameters NTRPM6 and JMgNaMgX,max are estimated so that the absolute reabsorption rate of Mg2+ along this segment is ~1 pmol/min. TRPM6 expression was found to be twofold higher in non-pregnant female mice than in male mice [48]. Hence, we assume the female NTRPM6 value to be double the male NTRPM6 value. Finally, Mg2+ excretion has been observed to be ~8% higher in male rats compared with female rats [13].

2.6. Simulating the 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 administrated 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 concentration of Mg2+ at the outer–inner medullary boundary is lowered to 0.77 mM (from a baseline value of 0.96 mM). At the papillary tip, the interstitial concentration of Mg2+ is reduced from 1.54 to 1.01 mM. For changes in the interstitial concentrations of Na+, K+, Cl−, urea and Ca2+, refer to [16,22].

2.7. Simulating the effect of thiazide diuretics

Thiazide diuretics inhibit NCC, which is expressed along the apical membrane of the DCT. 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.

2.8. Simulating the effect of K-sparing diuretics

K-sparing diuretics, such as amiloride, block Na+ uptake through epithelial Na+ channels (ENaC), expressed on the apical membrane of the late DCT as well as along the full length of the connecting tubule and collecting ducts. K-sparing diuretics are weak diuretics often used in combination with others to lower blood pressure. By inhibiting ENaC, these diuretics hyperpolarize the luminal membrane potential and increase K+, Ca2+ and Mg2+ uptake [49–52]. In our model, we simulated the effect of K-sparing diuretics by reducing ENaC activity by 70%.

3. Results

3.1. Baseline results

Figure 2 shows the segmental delivery, transport and luminal fluid concentration of Mg2+ and Na+ in male and female rats. Results for other electrolytes and water can be found in [18]. The addition of Mg2+ has negligible impacts on the renal transport of other major electrolytes and water, because luminal Mg2+ flow is orders of magnitude smaller. Filtered Mg2+ load is 25% higher in males due to their higher SNGFR [12,53]. Model simulations predict that fractional reabsorption of Mg2+ along the proximal tubule is higher in males than females (22% versus 16%), due to the more favourable electrochemical gradient generated by the higher NHE3-mediated Na+ transport rate in males [14]. Since Mg2+ reabsorption is low along the proximal tubule, Mg2+ concentration increases gradually along this segment. Micropuncture experiments in male rats showed that the tubular fluid-to-ultrafiltrate Mg2+ concentration ratio at the start of the loop of Henle was 3.24 ± 0.12 [4]. Our male model predicted this ratio to be 3.17, which is in line with the experimental observations. Single nephron microperfusion experiments with female rats showed that the luminal Mg2+ concentration in the late proximal tubule was 0.70 ± 0.16 mM [54]. Our female rat model predicted the luminal Mg2+ concentrations in the late proximal tubule to be 0.95 mM. The majority of the overall Mg2+ transport occurs downstream along the cTAL (69% of the filtered load in both males and females), where the lumen-positive membrane potential drives Mg2+ reabsorption via the paracellular pathway. Micropuncture studies in male rats found the tubular fluid-to-ultrafiltrate Mg2+ concentration ratio in the early distal tubule to be 0.80 ± 0.06 [4]. Our male model predicted this ratio to be 0.67. Microperfusion experiments in female rats showed that the luminal Mg2+ concentration in the early distal tubule was 0.37 ± 0.27 mM [54] and our female rat model predicted this value to be 0.32 mM. The final nephron segment that transports Mg2+ is the DCT, where the higher TRPM6 expression in females yields a higher Mg2+ reabsorption rate (6.6% and 12% of the filtered Mg2+ in males and females, respectively). Urinary Mg2+ excretion is predicted to be ~8% higher in males compared with females, consistent with experimental data, partially validating the baseline models [13]. However, given the 25% lower filtered load in females, fractional Mg2+ excretion is higher in females (3.3%) than males (2.8%) [13].

Figure 2. Baseline model predictions. (a, b) Delivery of Mg2+ and Na+ to key nephron segments in male and female rats, given per kidney. (c, d) Mg2+ and Na+ transepithelial transport along key nephron segments in male and female rats, given per kidney. (e, f) Luminal Mg2+ and Na+ concentrations along key nephron segments in male and female rats. PT, proximal tubule; SDL, short descending limb; mTAL/cTAL, medullary/cortical thick ascending limb; DCT, distal convoluted tubule; CNT, connecting tubule; CD, collecting duct.

3.2. Effect of varying proximal tubule and thick ascending limb Mg2+ transport

We first vary proximal tubule Mg2+ paracellular permeability by ±50%. This generates almost 20% changes in proximal tubular Mg2+ transport, which are then offset by compensatory transport along the cTAL, driven by the higher luminal Mg2+ concentration, resulting in essentially no change in DCT transport and only small (<6%) changes in urinary Mg2+ excretion rates for both male and female rats (figure 3). The effects on other electrolytes and water are minimal.

Figure 3. Changes in Mg2+ excretion by varying Mg2+ specific parameters. Male and female rat Mg2+ excretions are normalized by their respective baseline values. ‘PT perm (±50%)’, proximal tubule permeability to Mg2+ increased or decreased by 50%; ‘TAL perm (±50%)’, thick ascending limb permeability to Mg2+ increased or decreased by 50%; ‘TRPM6+NaMgX (–70%)’ and ‘TRPM6+NaMgX (–100%)’, transient receptor potential melastatin 6 and Na+/Mg2+ exchanger activity co-inhibited by 70% and 100%, respectively.

When Mg2+ paracellular permeability is increased by 50% along the cTAL, the models predict small relative increases in segmental transport (2% in males and 3% in females), which are partially compensated by opposing changes (decreases) in DCT transport. Because only 2–5% of the filtered Mg2+ is typically excreted in urine, Mg2+ excretion decreases by ~50% in both sexes (figure 3). Decreasing the Mg2+ paracellular permeability along the cTAL by 50% has a larger effect, with a ~10% decrease in segmental transport in males and females. With partial compensation by the DCT, Mg2+ urinary excretion increases by 220% in males and 196% in females (figure 3). Further model validation can be done by comparing these predictions with knockout measurements. Mutations in claudin-16 and claudin-19, which determine the paracellular permeability of Mg2+ in the thick ascending limb, are associated with increased Mg2+ excretion [55,56]. Since this segment is responsible for the majority of Mg2+ reabsorption and since the downstream distal segment has limited Mg2+ reabsorption capacity, a 50% decrease in permeability is predicted to significantly increase Mg2+ excretion, as is observed from experimental data of claudin-16 and claudin-19 knockout rodents [55,56]. The effect on urinary excretion is smaller in females due to their higher distal transport capacity. But even in this case, the effects on other electrolytes are minimal.

3.3. Effect of TRPM6 and Na+/Mg2+ exchanger co-inhibition

The DCT reabsorbs Mg2+ through TRPM6 on the apical membrane and Na+/Mg2+ exchanger on the basolateral membrane. We conduct simulations where we inhibit TRPM6 and Na+/Mg2+ exchanger simultaneously by 70% and then completely. Results for these two cases suggest that distal Mg2+ uptake scales linearly with inhibition. Without compensatory transport in any downstream segments, inhibition of DCT transport has marked effects on Mg2+ excretion. With 70% co-inhibition, Mg2+ excretion increases by 160% and 250% in male and female rats, respectively; with full inhibition, excretion rates increase by 240% and 380% in males and females, respectively. This is in line with the observation that mutations in TRPM6 lead to high Mg2+ wasting and hypomagnesemia [57,58]. In contrast to the previous case where thick ascending limb Mg2+ permeability is varied, here the effect on urinary excretion is larger in females because their DCT is responsible for transporting a larger fraction of the filtered Mg2+. Again, there is little impact on other electrolytes.

3.4. Effects of sex differences in transporter patterns

The model represents the sexual dimorphism in expression patterns not only for Mg2+ but also for other electrolytes. Specifically, female rats have lower proximal tubule NHE3 activity and higher NKCC2, NCC and distal segmental Na+–K+–ATPase activities (these transporters are coupled to Mg2+ transport) [14]. To what extent does each of these individual factors, and the sex differences in Mg2+ transporter properties, contribute to the sex differences in Mg2+ excretion?

To answer that question, we conduct simulations to analyse the extent to which each sex difference in transporter activity contributes to the differences in Mg2+ excretion. To that end, we vary each sex-specific parameter individually in the male rat nephron model and compute the fractional change in Mg2+ excretion. For example, the ‘PT NHE3’ case consists of male rat parameters, except for proximal tubule NHE3 activity, which is set to the female value. Simulation results are summarized in figure 4.

Figure 4. Effect of individual sex-specific model parameters. Changes in Mg2+ excretion relative to baseline male values when individual parameters in the male rat model are set to female rat values. PT NHE3, proximal tubule Na+–H+ exchanger; NKCC2, Na+–K+–2Cl− cotransporter; TAL NaKATPase, thick ascending limb Na+–K+–ATPase; cTAL perm, cortical thick ascending limb permeability to Mg2+; TRPM6, transient receptor potential melastatin 6; NCC, Na+–Cl− cotransporter; DCT NaKATPase, distal convoluted tubule Na+–K+–ATPase.

When NHE3 activity in the male model is lowered by 17% to match the female value, the favourable electrochemical gradient that drives Mg2+ is attenuated, reducing proximal tubule fractional Mg2+ reabsorption from the baseline by 22% to 20%. Compensatory transport by downstream segments is limited, unlike the case when proximal tubule Mg2+ permeability was reduced (see above). This discrepancy can be attributed to the effect of NHE3 inhibition on water transport. Here, the reduction in NHE3 activity lowers the reabsorption not only of Mg2+ but also of Na+, K+, Cl− and water. Consequently, the luminal Mg2+ concentration does not differ substantially from the base case (unlike when proximal tubule Mg2+ permeability was reduced), resulting in limited compensatory transport along the thick ascending limb and DCT, and a marked 64% increase in Mg2+ excretion.

Along the cTAL, NKCC2 activity, Na+–K+–ATPase activity and paracellular Mg2+ permeability are higher in females, by 100%, 100% and 47%, respectively, compared with males [14]. When each of these parameters are set to the female value, cTAL fractional Mg2+ reabsorption increases, resulting in a decrease in Mg2+ excretion (by 46%, 44% and 51%, respectively). Similarly, increasing TRPM6 channel density by 100% to match the female value increases Mg2+ reabsorption along the DCT and decreases Mg2+ excretion by 96% to almost nil. Also, along the DCT, NCC and Na+–K+–ATPase activities are higher in females [14]. Setting each of these parameters to the female value decreases Mg2+ excretion, albeit the reductions are lower than the other changes (refer to figure 4). The higher (almost doubled) TRPM6 channel density in females has the largest impact on Mg2+ excretion because it directly affects Mg2+ transport and also because there is no downstream segment to compensate for Mg2+ transport. Among transporters that do not immediately mediate Mg2+ excretion, the lower proximal tubule NHE3 activity in females has the largest impact on Mg2+ excretion, even though less Mg2+ is reabsorbed along the proximal tubule than along the cTAL, indicating a stronger coupling between Na+ and Mg2+ transport along the proximal tubule.

3.5. Effect of diuretics

To validate the model, we simulate the administration of three classes of diuretics (loop diuretics, thiazide diuretics and K-sparing diuretics) and compute model predictions with observations in rodents.

3.5.1. Loop diuretics

Administration of furosemide to male rats increased Mg2+ excretion to 240% of the control excretion value [59]. The predicted Mg2+ transport along the cTAL and DCT and urinary Mg2+ excretion following NKCC2 inhibition in male and female rats are shown in figure 5. NKCC2 inhibition decreased fractional Mg2+ reabsorption along male and female cTALs to 57% and 52%, respectively, from the baseline fractional reabsorption of 69% (figure 5). Females have a higher reduction in Mg2+ reabsorption because they have higher NKCC2 activity. In male rats, this 12% decrease in Mg2+ reabsorption along the cTAL should ideally increase Mg2+ excretion to ~430% of the control value. However, according to the experimental study [59], Mg2+ excretion increases to 240% of the control value following furosemide treatment. This indicates that there must be a compensatory increase in Mg2+ reabsorption along the DCT. Indeed, TRPM6 mRNA expression was found to be increased by 30% in male mice undergoing furosemide treatment [60]. Our male model simulations indicated that TRPM6 activity must be increased by 68% to account for the experimental increase in Mg2+ excretion. Due to lack of data for female rodents, we assumed the same percentage increase (68%) in TRPM6 for female rats. With a 68% increase in TRPM6 activity, the male and female model simulations predicted fractional Mg2+ reabsorptions along the DCT to increase to 12% and 20%, respectively (figure 5). Finally, the predicted Mg2+ excretions are 241% and 270% of the baseline male and female excretion values, respectively (figure 5). Furosemide has been found to cause stronger diuretic, natriuretic and kaliuretic responses in female rats [61]. Since Mg2+ transport is strongly coupled to Na+ uptake through NKCC2, it can be inferred that the stronger natriuretic response in female rats on furosemide treatment would lead to a higher increase in Mg2+ excretion compared with male rats.

Figure 5. (a–d) Delivery and transport of Mg2+ along key nephron segments in male and female rats under normal condition and 70% inhibition of NKCC2, NCC and ENaC. The values are given per kidney. Notations are analogous to figure 2.

3.5.2. Thiazide diuretics

Administration of bendrofluazide, a thiazide diuretic, did not cause any significant change in Mg2+ excretion in male rats [62]. In agreement with the experimental data, the predicted male and female fractional Mg2+ excretions after NCC inhibition increased to 3.1% and 3.7%, respectively, from the baseline values of 2.8% and 3.3% (figure 5). In addition, we also performed simulations with 50% and 90% NCC inhibitions. The predicted Mg2+ excretion did not change significantly at 50%, and 90% NCC inhibitions compared with that at 70% NCC inhibition in both male and female rats. This further validates that Mg2+ and Na+ transport along the DCT is dissociated, as we had observed from our simulation results shown in figure 4, where setting the male NCC activity with the female value did not lower Mg2+ excretion significantly (refer to §3.4).

3.5.3. K-sparing diuretics

K-sparing diuretics, amiloride and triamterene, reduced Mg2+ excretion in male rats by ~40% [62]. By inhibiting ENaC, these diuretics hyperpolarize the luminal membrane potential and increase K+, Ca2+ and Mg2+ uptake [49–52]. Our model simulations predicted Mg2+ reabsorption along the DCT to increase by 8.2% and 8.6% in male and female rats, respectively (figure 5). Females have a higher percentage increase in Mg2+ reabsorption because they have higher ENaC activity. These increased reabsorptions decreased male and female Mg2+ excretions by 31% and 46%, respectively. In addition, we also performed simulations with 50% and 90% ENaC inhibitions. At 50% and 90% ENaC inhibitions, the predicted Mg2+ excretions decreased by 26% and 37%, respectively, compared with baseline Mg2+ excretion in male rats. In female rats, the corresponding values were 40% and 54% decreases, respectively. Thus, varying the ENaC inhibition percentage has a significant impact on urinary Mg2+ excretion.

4. Discussion

Magnesium balance is achieved by the coordinated actions of the kidneys and the intestine. The average daily Mg2+ intake for an adult human is around 250–300 mg, of which approximately 50% is absorbed by the intestine [6]. Within the physiological range of Mg2+ intake, urinary Mg2+ excretion scales approximately linearly with intestinal absorption. Indeed, the kidney’s essential role in maintaining extracellular Mg2+ concentration can be seen in its rapid response to changes in serum Mg2+: urine Mg2+ has been reported to acutely increase as serum Mg2+ increases during a 30-min infusion [63].

Although no sex differences have been reported in serum Mg2+ in children [64] or adults [65], Mg2+ excretion is typically higher in males [66]. This difference may be attributable, in part, to sex differences in kidney structure and function. Renal transporter patterns are poorly characterized in humans; thus, we seek to understand the underlying molecular mechanisms by considering rodents. Veiras et al. [14] reported markedly different transport capacities in different tubular nephron segments of male and female rat kidneys. In the proximal tubule, female rats exhibit greater NHE3 phosphorylation and redistribution to the base of the microvilli, where activity is lower, compared with male rats, as well as lower abundance of Na+–Pi cotransporter 2 (NaPi2), aquaporin 1 (AQP1) and claudin-2 [67]. As a result, the proximal tubule in the female rat reabsorbs a substantially smaller fraction of filtered Na+ compared with the male rat [14]. Model simulations indicate that, because Mg2+ transport is coupled to Na+ transport, the female rat proximal tubule also reabsorbs a smaller fraction of filtered Mg2+ compared with the male (see figure 2a–d ).

For many major solutes—for example, Na+, Cl− and Ca2+—most of their reabsorption occurs along the proximal tubule, where their luminal concentrations remain close to plasma. In contrast, the proximal tubule is responsible for only 15–25% of the filtered Mg2+ load, and because proportionally less Mg2+ is reabsorbed relative to water, its concentration rises significantly along the proximal tubule (figure 2e ). For Mg2+, most of the reabsorption occurs along the cTAL, and interestingly, none along the medullary thick ascending limb. The activities of NKCC2, K+–Cl− cotransporter (KCC) and Na+–K+–ATPase are higher in female rats. Taken in isolation, this might suggest a higher female capacity to drive Mg2+ reabsorption. However, the luminal Mg2+ concentration at the cTAL entrance is higher in males compared with females (figure 2e ), and the male segment has a larger transport area. With these competing factors, our models predict that to attain similar fractional Mg2+ reabsorption along the cTAL in males and females, paracellular Mg2+ permeability should be significantly higher in females (the model assumes approximately 50% higher).

Downstream of the macula densa, female rats exhibit higher abundance and phosphorylation of NCC, ENaC and claudin-7 [14]. Given the twofold-higher expression of TRPM6 on the apical membrane of the DCT in female mice, the model assumes that TRPM6 channel density is higher in female rats (by twofold). With these sex-specific Mg2+ transporter patterns, together with the higher filtered Mg2+ load and larger transport areas in males, the models predict fractional Mg2+ excretion consistent with reported values, with urinary excretion ~8% higher in male rats compared with females [13].

Having the cTAL and DCT, instead of the proximal tubule (as in the case of Na+ and Cl−), handle most of the Mg2+ transport may give the kidney a better ability to regulate the Mg2+ balance. The ability to fine-tune renal Mg2+ transport may be particularly crucial because the serum Mg2+ level is orders of magnitude lower than Na+ or Cl−. Renal Mg2+ transport is under hormonal control. PTH increases Mg2+ reabsorption in both the cTAL and DCT [68]. PTH increases paracellular Mg2+ permeability in the cTAL, likely via direct hormonal control of the function and/or expression of tight-junction proteins. PTH also stimulates Mg2+ reabsorption in the DCT [69,70], but the molecular mechanisms are unknown. In particular, PTH does not affect TRPM6 gene expression in the kidney [8]. Other hormones, including calcitonin, vasopressin, glucagon, and β-adrenergic agonists, also enhance Mg2+ reabsorption in the cTAL and DCT [71].

Compared with males, the female rats transport a larger fraction of the filtered Mg2+ along the DCT (figure 2c ). As a result, females are more sensitive than males to inhibition of TRPM6 (figure 3). This result is consistent with the observation that hypomagnesemia is more prevalent among female patients with diabetes than males [72–74]. Insulin upregulates TRPM6 activity; thus, patients with lower insulin receptor activity are more susceptible to hypomagnesemia [75]. The higher sensitivity of females to inhibition of TRPM6 may partially explain the higher prevalence of hypomagnesemia in female patients with impaired insulin sensitivity and, as such, attenuated TRPM6 activation.

The DCT is the segment responsible for fine-tuning Na+ transport and potentially Mg2+ as well. Another notable feature of the DCT Mg2+ transport is that, unlike the proximal tubule and thick ascending limb, it is decoupled from Na+ transport. This may allow females to better adapt their kidney function to conditions under which the electrolyte balance is altered. Two such examples are pregnancy and lactation. During pregnancy, the female body undergoes major adaptations to support the solute and volume demands of the developing fetus and placenta. In the non-pregnant state, almost all Na+ and K+ intake is excreted. In contrast, net Na+ retention begins from early pregnancy [76,77] and drives the large plasma volume expansion that is seen in a healthy pregnancy and is often sustained during lactation. Similarly, there is net K+ retention during late pregnancy. The requirements of Mg2+ in pregnancy are less well understood. The serum Mg2+ concentration has been shown to decrease in pregnancy [78], but this may be due in part to hemodilution. In lactation, human breast milk provides approximately 42  mg/day in 750  ml [78]. In a modelling study, we conducted simulations to demonstrate that the higher distal Na+ transporters in a female rat may allow its kidneys to better adapt to the increased electrolyte and fluid demands in pregnancy [79]. Having a proportionally larger distal Mg2+ transport capacity may present females with a similar advantage in the altered Mg2+ requirements in pregnancy and lactation. A notable challenge is to simultaneously meet the different demands of a number of vital electrolytes. Due to their coupled transport, regulating Na+ transporters would affect K+ and Mg2+ reabsorption as well. How can the kidney selectively retain Na+ but not K+ or Mg2+? For Mg2+ this is made possible, in part, by having a nephron segment where Mg2+ transport is decoupled from Na+, and by having the ability to adjust Mg2+ transport in multiple segments (e.g. PTH regulates Mg2+ transport along the cTAL, where Na+ and Mg2+ transport changes together, and the DCT, where changes occur in opposite directions).

To illustrate how renal Mg2+ transport is regulated under perturbed conditions, consider Mg2+ restriction. When dietary Mg2+ decreases, urinary Mg2+ excretion rapidly decreases. However, plasma [Mg2+] remains unchanged initially, indicating an upregulation in renal tubular reabsorption of Mg2+ [80]. Studies have reported increased Mg2+ reabsorption across the epithelium of the cTAL (together with increased absorption of Ca2+ but not Na+ or Cl−) [80,81]. Also observed was an increase in both the transcript and protein levels of claudin-16 [82], which is involved in Mg2+ reabsorption. These findings support the idea of an adaptive increase in the paracellular pathway for Mg2+ transport along the cTAL. Moreover, dietary Mg2+ restriction is associated with an increase in Mg2+ reabsorption in the DCT [80,83], where both the gene expression and protein expression of TRPM6 increase during Mg2+ restriction [8], facilitating the reduction in urinary Mg2+ excretion.

In addition to pregnancy, lactation and dietary restriction, Mg2+ homeostasis is also altered in diseases such as diabetes [84] and chronic kidney disease [85]. To conduct in silico studies of how the kidney adapts in terms of Mg2+ transport in these physiological and pathophysiological conditions, the present model can be incorporated into computational models of kidney function for a pregnant rat [79], a diabetic rat [86] and a nephrectomized rat [17]. For better translational value, human kidney models [87] can be expanded to include Mg2+. However, to construct accurate computational models, more data are needed that describe the adaptation of Mg2+ transporter in these physiological and pathophysiological conditions.

Computational models of renal tubular function developed in the past two decades [15–22] have provided an accurate accounting of solute and water transport and yielded insights into transport pathways, driving forces and coupling mechanisms. Despite these achievements, the limitations of these models primarily include some that stem from the paucity of experimental data and others that are inherent to the model structure (e.g. not considering spatial inhomogeneity within a compartment or intracellular signalling pathways) [88,89]. As can be seen in table 1, many of the model parameters that characterize Mg2+ transport have not been measured, especially in the female kidney. To build more accurate sex-specific computational models, we require more experimental studies on the differences in segmental Mg2+ reabsorption between male and female rodents. Notwithstanding the uncertainties in some of the model parameters, the present models yield predictions that are consistent with measurements in wild-type and genetically mutated rats (see §3). In terms of model structure, the present models simulate electrolyte and water transport along a superficial nephron, which constitutes only two-thirds of the nephron population in a rat kidney. The remainder of the nephron population is made up of juxtamedullary nephrons whose loops of Henle extend to various depths of the inner medulla. There are differences in the SNGFR, transport area and transporter activities between the two types of nephrons. This study focuses on a superficial nephron model to gain a clearer understanding of segmental Mg2+ transport. In future studies, we will develop a kidney model that incorporates both types of nephrons [15] to obtain more accurate predictions of urinary Mg2+ excretion rates. We will also investigate the adaptations/dysregulation of renal Mg2+ handling in diseases such as Bartter syndrome and Gitelman syndrome, which cause mutations in Na+ transporters that are involved in the regulation of Mg2+ transport. In addition, some parameters in our model (such as the ratio of TRPM6 channel density between male and female rats) have been taken from experimental studies conducted on mice because corresponding studies on rats were not available. There are several differences between rat kidney and mouse kidney in terms of SNGFR, tubular dimensions and activities of some transporter proteins [90]. Thus, using some mouse parameters for our rat model might affect the accuracy of our predictions to some extent. In the future, if these parameters are available from experimental studies on rats, we will substitute the mouse parameters with them in our model.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

Data and relevant code for this research work are stored in GitHub: https://github.com/Pritha17/Nephron-Mg_Ca_transport and have been archived within the Zenodo repository: [91].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

P.D.: data curation, formal analysis, investigation, methodology, software, supervision, validation, visualization, writing—original draft, writing—review and editing; S.H.: writing—review and editing; A.T.L.: conceptualization, funding acquisition, methodology, resources, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work was supported by the Canada 150 Research Chair program, National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (RGPIN-2019-03916) and Canada Institutes of Health Research (CIHR) Project Grant (TNC-174963) to A.T.L.
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References

1. Elin RJ . 1994 Magnesium: The fifth but forgotten electrolyte. Am. J. Clin. Pathol. 102 , 616–622. (10.1093/ajcp/102.5.616)7942627
2. Taal MW , Chertow GM , Marsden PA , Skorecki K , Alan S , Brenner BM . 2011 Brenner and Rector’s The Kidney E-Book. Elsevier Health Sciences.
3. Institute of Medicine (US) Standing Committee on the scientific evaluation of dietary reference intakes . 1997 Dietary reference intakes for calcium, phosphorus, magnesium, vitamin D, and fluoride. Washington, DC: National Academies Press (US). See http://www.ncbi.nlm.nih.gov/books/NBK109825/
4. Brunette MG , Vigneault N , Carriere S . 1974 Micropuncture study of magnesium transport along the nephron in the young rat. Am. J. Physiol. 227 , 891–896. (10.1152/ajplegacy.1974.227.4.891)4429138
5. Le Grimellec C , Giocondi MC , Philippe P . 1975 Micropuncture study along the proximal convoluted tubule. Electrolyte reabsorption in first convolutions. Pflugers Arch. 354 , 133–150. (10.1007/BF00579944)1167675
6. Curry JN , Yu ASL . 2018 Magnesium Handling in the Kidney. Adv. Chronic Kidney Dis. 25 , 236–243. (10.1053/j.ackd.2018.01.003)29793662
7. de Baaij JHF , Hoenderop JGJ , Bindels RJM . 2015 Magnesium in man: implications for health and disease. Physiol. Rev. 95 , 1–46. (10.1152/physrev.00012.2014)25540137
8. Groenestege WMT , Hoenderop JG , van den Heuvel L , Knoers N , Bindels R . 2006 The epithelial Mg2+ channel transient receptor potential melastatin 6 is regulated by dietary Mg2+ content and estrogens. J. Am. Soc. Nephrol. 17 , 1035–1043. (10.1681/ASN.2005070700)16524949
9. Hatano R , Onoe K , Obara M , Matsubara M , Kanai Y , Muto S , Asano S . 2012 Sex hormones induce a gender-related difference in renal expression of a novel prostaglandin transporter, OAT-PG, influencing basal PGE2 concentration. Am. J. Physiol. Renal Physiol. 302 , F342–9. (10.1152/ajprenal.00366.2011)22031854
10. Hilliard LM , Nematbakhsh M , Kett MM , Teichman E , Sampson AK , Widdop RE , Evans RG , Denton KM . 2011 Gender differences in pressure-natriuresis and renal autoregulation: role of the angiotensin type 2 receptor. Hypertension J. 57 , 275–282. (10.1161/HYPERTENSIONAHA.110.166827)
11. Sabolić I , Asif AR , Budach WE , Wanke C , Bahn A , Burckhardt G . 2007 Gender differences in kidney function. Pflugers Arch. 455 , 397–429. (10.1007/s00424-007-0308-1)17638010
12. Munger K , Baylis C . 1988 Sex differences in renal hemodynamics in rats. Am. J. Physiol. 254 , F223–F231. (10.1152/ajprenal.1988.254.2.F223)3344806
13. Grikiniene J , StakisaitiS D , Dzidziapetriene J , Tschaika M . 2005 Gender-related magnesium urinary excretion in rats: influence of furosemide. Acta Poloniae Pharmaceutica ñ. 62 , 307–311.
14. Veiras LC et al . 2017 Sexual dimorphic pattern of renal transporters and electrolyte homeostasis. J. Am. Soc. Nephrol. 28 , 3504–3517. (10.1681/ASN.2017030295)28774999
15. Layton AT , Vallon V , Edwards A . 2016 A computational model for simulating solute transport and oxygen consumption along the nephrons. Am. J. Physiol. Renal Physiol. 311 , F1378–F1390. (10.1152/ajprenal.00293.2016)27707705
16. Layton AT , Laghmani K , Vallon V , Edwards A . 2016 Solute transport and oxygen consumption along the nephrons: effects of Na+ transport inhibitors. Am. J. Physiol. Renal Physiol. 311 , F1217–F1229. (10.1152/ajprenal.00294.2016)27707706
17. Layton AT , Vallon V . 2018 SGLT2 inhibition in a kidney with reduced nephron number: modeling and analysis of solute transport and metabolism. Am. J. Physiol. Renal Physiol. 314 , F969–F984. (10.1152/ajprenal.00551.2017)29361669
18. Hu R , McDonough AA , Layton AT . 2019 Functional implications of the sex differences in transporter abundance along the rat nephron: modeling and analysis. Am. J. Physiol. Renal Physiol. 317 , F1462–F1474. (10.1152/ajprenal.00352.2019)31566436
19. Tournus M , Seguin N , Perthame B , Thomas SR , Edwards A . 2013 A model of calcium transport along the rat nephron. Am. J. Physiol. Renal Physiol. 305 , F979–94. (10.1152/ajprenal.00696.2012)23761679
20. Edwards A . 2015 Regulation of calcium reabsorption along the rat nephron: a modeling study. Am. J. Physiol. Renal Physiol. 308 , F553–66. (10.1152/ajprenal.00577.2014)25520011
21. Edwards A , Bonny O . 2018 A model of calcium transport and regulation in the proximal tubule. Am. J. Physiol. Renal Physiol. 315 , F942–F953. (10.1152/ajprenal.00129.2018)29846115
22. Hakimi S , Dutta P , Layton AT . 2023 Coupling of renal sodium and calcium transport: a modeling analysis of transporter inhibition and sex differences. Am. J. Physiol. Renal Physiol. 325 , F536–F551. (10.1152/ajprenal.00145.2023)37615047
23. Layton AT . 2022 A complete set of equations for a computational model of electrolyte and water transport along the Nephrons in a mammalian kidney. bioRxiv. (10.1101/2022.09.23.509286)
24. Murayama Y , Morel F , Le Grimellec C . 1972 Phosphate, calcium and magnesium transfers in proximal tubules and loops of Henle, as measured by single nephron microperfusion experiments in the rat. Pflugers Arch. 333 , 1–16. (10.1007/BF00586037)5064074
25. Brown EM , Chen CJ . 1989 Calcium, magnesium and the control of PTH secretion. Bone Miner. 5 , 249–257. (10.1016/0169-6009(89)90003-2)2655774
26. Edwards A . 2015 Regulation of calcium reabsorption along the rat nephron: a modeling study. Am. J. Physiol. Renal. Physiol. 308 , F553–F566. (10.1152/ajprenal.00577.2014)25520011
27. Toka HR et al . 2012 Deficiency of the calcium-sensing receptor in the kidney causes parathyroid hormone–independent hypocalciuria. J. Am. Soc. Nephrol. 23 , 1879–1890. (10.1681/ASN.2012030323)22997254
28. Wang WH , Lu M , Hebert SC . 1996 Cytochrome P-450 metabolites mediate extracellular Ca(2+)-induced inhibition of apical K+ channels in the TAL. Am. J. Physiol. 271 , C103–C111. (10.1152/ajpcell.1996.271.1.C103)8760035
29. Wang W , Lu M , Balazy M , Hebert SC . 1997 Phospholipase A2 is involved in mediating the effect of extracellular Ca2+ on apical K+ channels in rat TAL. Am. J. Physiol. 273 , F421–F429. (10.1152/ajprenal.1997.273.3.F421)9321915
30. Li M , Jiang J , Yue L . 2006 Functional characterization of homo- and heteromeric channel kinases TRPM6 and TRPM7. J. Gen. Physiol. 127 , 525–537. (10.1085/jgp.200609502)16636202
31. Bräuner-Osborne H , Jensen AA , Sheppard PO , O’Hara P , Krogsgaard-Larsen P . 1999 The agonist-binding domain of the calcium-sensing receptor is located at the amino-terminal domain. J. Biol. Chem. 274 , 18382–18386. (10.1074/jbc.274.26.18382)10373443
32. Bazúa-Valenti S et al . 2018 The calcium-sensing receptor increases activity of the renal NCC through the WNK4-SPAK pathway. JASN. 29 , 1838–1848. (10.1681/ASN.2017111155)29848507
33. Lee HY et al . 2009 Simulation and prediction of the adaptive immune response to influenza A virus infection. J. Virol. 83 , 7151–7165. (10.1128/JVI.00098-09)19439465
34. Alexander RT , Dimke H . 2017 Effect of diuretics on renal tubular transport of calcium and magnesium. Am. J. Physiol. Renal Physiol. 312 , F998–F1015. (10.1152/ajprenal.00032.2017)28274923
35. Alexander RT , Dimke H . 2022 Molecular mechanisms underlying paracellular calcium and magnesium reabsorption in the proximal tubule and thick ascending limb. Ann. N. Y. Acad. Sci. 1518 , 69–83. (10.1111/nyas.14909)36200584
36. Yuan-Hui L , Gregory S . 1974 Diffusion of ions in sea water and in deep-sea sediments. Geochim. Cosmochim. Acta 38 , 703–714. (10.1016/0016-7037(74)90145-8)
37. Wittner M , di Stefano A , Wangemann P , Nitschke R , Greger R , Bailly C , Amiel C , Roinel N , de Rouffignac C . 1988 Differential effects of ADH on sodium, chloride, potassium, calcium and magnesium transport in cortical and medullary thick ascending limbs of mouse nephron. Pflugers Arch. 412 , 516–523. (10.1007/BF00582541)3194173
38. Quamme GA . 1989 Control of magnesium transport in the thick ascending limb. Am. J. Physiol. 256 , F197–210. (10.1152/ajprenal.1989.256.2.F197)2644845
39. Di Stefano A , Wittner M , Nitschke R , Braitsch R , Greger R , Bailly C , Amiel C , Roinel N , de Rouffignac C . 1990 Effects of parathyroid hormone and calcitonin on Na+, Cl-, K+, Mg2+ and Ca2+ transport in cortical and medullary thick ascending limbs of mouse kidney. Pflugers Arch. 417 , 161–167. (10.1007/BF00370694)2084613
40. Quamme GA , Dirks JH . 1980 Intraluminal and contraluminal magnesium on magnesium and calcium transfer in the rat nephron. Am. J. Physiol. 238 , F187–98. (10.1152/ajprenal.1980.238.3.F187)7369360
41. Franken GAC , Adella A , Bindels RJM , de Baaij JHF . 2021 Mechanisms coupling sodium and magnesium reabsorption in the distal convoluted tubule of the kidney. Acta Physiol. 231 , e13528. (10.1111/apha.13528)
42. Verschuren EHJ , Hoenderop JGJ , Peters DJM , Arjona FJ , Bindels RJM . 2019 Tubular flow activates magnesium transport in the distal convoluted tubule. FASEB J. 33 , 5034–5044. (10.1096/fj.201802094R)30596515
43. Günther T , Vormann J , Höllriegl V . 1990 Characterization of Na(+)-dependent Mg2+ efflux from Mg2(+)-loaded rat erythrocytes. Biochim. Biophys. Acta 1023 , 455–461. (10.1016/0005-2736(90)90139-f)2139797
44. Riccardi D , Valenti G . 2016 Localization and function of the renal calcium-sensing receptor. Nat. Rev. Nephrol. 12 , 414–425. (10.1038/nrneph.2016.59)27157444
45. Ruat M , Snowman AM , Hester LD , Snyder SH . 1996 Cloned and expressed rat Ca2+-sensing receptor: differential cooperative responses to calcium and magnesium (∗). J. Biol. Chem. 271 , 5972–5975. (10.1074/jbc.271.11.5972)8626377
46. Loupy A et al . 2012 PTH-independent regulation of blood calcium concentration by the calcium-sensing receptor. J. Clin. Invest. 122 , 3355–3367. (10.1172/JCI57407)22886306
47. Sands JM , Naruse M , Baum M , Jo I , Hebert SC , Brown EM , Harris HW . 1997 Apical extracellular calcium/polyvalent cation-sensing receptor regulates vasopressin-elicited water permeability in rat kidney inner medullary collecting duct. J. Clin. Invest. 99 , 1399–1405. (10.1172/JCI119299)9077550
48. Cuffe JSM , Steane S , Moritz KM , Paravicini TM . 2015 Differential mRNA expression and glucocorticoid-mediated regulation of TRPM6 and TRPM7 in the heart and kidney throughout murine pregnancy and development. PLoS One 10 , e0117978. (10.1371/journal.pone.0117978)25692682
49. Costanzo LS . 1984 Comparison of calcium and sodium transport in early and late rat distal tubules: effect of amiloride. Am. J. Physiol. 246 , F937–F945. (10.1152/ajprenal.1984.246.6.F937)6742137
50. Devane J , Ryan MP . 1981 The effects of amiloride and triamterene on urinary magnesium excretion in conscious saline-loaded rats. Br. J. Pharmacol. 72 , 285–289. (10.1111/j.1476-5381.1981.tb09127.x)7214099
51. Devane J , Ryan MP . 1983 Dose-dependent reduction in renal magnesium clearance by amiloride during frusemide-induced diuresis in rats. Br. J. Pharmacol. 80 , 421–428. (10.1111/j.1476-5381.1983.tb10711.x)6640198
52. Friedman PA , Gesek FA . 1995 Stimulation of calcium transport by amiloride in mouse distal convoluted tubule cells. Kidney Int. 48 , 1427–1434. (10.1038/ki.1995.432)8544399
53. Remuzzi A , Puntorieri S , Mazzoleni A , Remuzzi G . 1988 Sex related differences in glomerular ultrafiltration and proteinuria in Munich-Wistar rats. Kidney Int. 34 , 481–486. (10.1038/ki.1988.206)3199667
54. Murayama Y , Morel F , Le Grimellec C . 1972 Phosphate, calcium and magnesium transfers in proximal tubules and loops of henle, as measured by single nephron microperfusion experiments in the rat. Pflugers Arch. 333 , 1–16. (10.1007/BF00586037)5064074
55. Hou J , Shan Q , Wang T , Gomes AS , Yan Q , Paul DL , Bleich M , Goodenough DA . 2007 Transgenic RNAi depletion of claudin-16 and the renal handling of magnesium. J. Biol. Chem. 282 , 17114–17122. (10.1074/jbc.M700632200)17442678
56. Hou J , Goodenough DA . 2010 Claudin-16 and claudin-19 function in the thick ascending limb. Curr. Opin. Nephrol. Hypertens. 19 , 483–488. (10.1097/MNH.0b013e32833b7125)20616717
57. Schlingmann KP et al . 2002 Hypomagnesemia with secondary hypocalcemia is caused by mutations in TRPM6, a new member of the TRPM gene family. Nat. Genet. 31 , 166–170. (10.1038/ng889)12032568
58. Walder RY et al . 2002 Mutation of TRPM6 causes familial hypomagnesemia with secondary hypocalcemia. Nat. Genet. 31 , 171–174. (10.1038/ng901)12032570
59. Devane J , Ryan M . 1981 Diuretics and magnesium excretion. Magnesium Bull. 3 , 122–123.
60. van Angelen AA , van der Kemp AW , Hoenderop JG , Bindels RJ . 2012 Increased expression of renal TRPM6 compensates for Mg(2+) wasting during furosemide treatment. Clin. Kidney J. 5 , 535–544. (10.1093/ckj/sfs140)26069797
61. Brandoni A , Villar SR , Torres AM . 2004 Gender-related differences in the pharmacodynamics of furosemide in rats. Pharmacology 70 , 107–112. (10.1159/000074675)14685014
62. Ryan MP , Devane J , Ryan MF , Counihan TB . 1984 Effects of diuretics on the renal handling of magnesium. Drugs 28 , 167–181. (10.2165/00003495-198400281-00017)6389077
63. Blanchard A et al . 2001 Paracellin-1 is critical for magnesium and calcium reabsorption in the human thick ascending limb of Henle. Kidney Int. 59 , 2206–2215. (10.1046/j.1523-1755.2001.00736.x)11380823
64. Jagarinec N , Flegar-Mestrić Z , Surina B , Vrhovski-Hebrang D , Preden-Kereković V . 1998 Pediatric reference intervals for 34 biochemical analytes in urban school children and adolescents. Clin. Chem. Lab. Med. 36 , 327–337. (10.1515/CCLM.1998.055)9676391
65. Bohnen N , Degenaar CP , Jolles J . 1992 Influence of age and sex on 19 blood variables in healthy subjects. Z. Gerontol. 25 , 339–345.1441715
66. Jankûnas R , Driþienë Þ , Stakiðaitis D , Kulieðienë I . 2001 Gender-dependent magnesium urinary excretion in healthy adolescents and adults. Acta Med. Litu. 8 , 167–172.
67. Brasen JC , Burford JL , McDonough AA , Holstein-Rathlou NH , Peti-Peterdi J . 2014 Local pH domains regulate NHE3-mediated Na. Am. J. Physiol. Renal. Physiol. 307 , F1249–F1262. (10.1152/ajprenal.00174.2014)25298526
68. Bailly C , Roinel N , Amiel C . 1984 PTH-like glucagon stimulation of Ca and Mg reabsorption in Henle’s loop of the rat. Am. J. Physiol. 246 , F205–F212. (10.1152/ajprenal.1984.246.2.F205)6696122
69. Bailly C , Roinel N , Amiel C . 1985 Stimulation by glucagon and PTH of Ca and Mg reabsorption in the superficial distal tubule of the rat kidney. Pflugers Arch. 403 , 28–34. (10.1007/BF00583277)3982956
70. Harris CA , Burnatowska MA , Seely JF , Sutton RA , Quamme GA , Dirks JH . 1979 Effects of parathyroid hormone on electrolyte transport in the hamster nephron. Am. J. Physiol. 236 , F342–8. (10.1152/ajprenal.1979.236.4.F342)434207
71. Quamme GA . 1997 Renal magnesium handling: new insights in understanding old problems. Kidney Int. 52 , 1180–1195. (10.1038/ki.1997.443)9350641
72. Bertinato J , Wu Xiao C , Ratnayake WMN , Fernandez L , Lavergne C , Wood C , Swist E . 2015 Lower serum magnesium concentration is associated with diabetes, insulin resistance, and obesity in South Asian and white Canadian women but not men. Food Nutr. Res. 59 , 25974. (10.3402/fnr.v59.25974)25947295
73. Pham PCT , Pham PMT , Pham PAT , Pham SV , Pham HV , Miller JM , Yanagawa N , Pham PTT . 2005 Lower serum magnesium levels are associated with more rapid decline of renal function in patients with diabetes mellitus type 2. Clin. Nephrol. 63 , 429–436. (10.5414/CNP63429)15960144
74. Hamarshih M , Hamshari S , Nazzal Z , Snobar F , Mletat R , Abu-Mazen O , Maraqa B . 2022 Hypomagnesemia and poor glycemic control among type 2 diabetic patients: a cross-sectional study. Indian J. Endocr. Metab. 26 , 575. (10.4103/ijem.ijem_213_22)
75. Nair AV et al . 2012 Loss of insulin-induced activation of TRPM6 magnesium channels results in impaired glucose tolerance during pregnancy. Proc. Natl. Acad. Sci. USA. 109 , 11324–11329. (10.1073/pnas.1113811109)22733750
76. Beers K , Patel N . 2020 Kidney Physiology in Pregnancy. Adv. Chronic Kidney Dis. 27 , 449–454. (10.1053/j.ackd.2020.07.006)33328060
77. Churchi-l SE , Bengele HH , Alexander EA . 1980 Sodium balance during pregnancy in the rat. Am. J. Physiol. 239 , R143–R148. (10.1152/ajpregu.1980.239.1.R143)7396030
78. Dalton LM , Ní Fhloinn DM , Gaydadzhieva GT , Mazurkiewicz OM , Leeson H , Wright CP . 2016 Magnesium in pregnancy. Nutr. Rev. 74 , 549–557. (10.1093/nutrit/nuw018)27445320
79. Stadt MM , Layton AT . 2022 Adaptive changes in single-nephron GFR, tubular morphology, and transport in a pregnant rat nephron: modeling and analysis. Am. J. Physiol. Renal. Physiol. 322 , F121–F137. (10.1152/ajprenal.00264.2021)34894726
80. Shafik IM , Quamme GA . 1989 Early adaptation of renal magnesium reabsorption in response to magnesium restriction. Am. J. Physiol. 257 , F974–F977. (10.1152/ajprenal.1989.257.6.F974)2603964
81. Wittner M , Jounier S , Deschênes G , de Rouffignac C , Di Stefano A . 2000 Cellular adaptation of the mouse cortical thick ascending limb of Henle’s loop (CTAL) to dietary magnesium restriction: enhanced transepithelial Mg2+ and Ca2+ transport. Pflugers Arch. 439 , 765–771. (10.1007/s004249900197)10784351
82. Efrati E , Hirsch A , Kladnitsky O , Rozenfeld J , Kaplan M , Zinder O , Zelikovic I . 2010 Transcriptional regulation of the claudin-16 gene by Mg2+ availability. Cell Physiol. Biochem. 25 , 705–714. (10.1159/000315090)20511716
83. Quamme GA , Carney SL , Wong NLM , Dirks JH . 1980 Effect of parathyroid hormone on renal calcium and magnesium reabsorption in magnesium deficient rats. Pflugers Arch. 386 , 59–65. (10.1007/BF00584188)7191964
84. Lee CT , Lien YHH , Lai LW , Chen JB , Lin CR , Chen HC . 2006 Increased renal calcium and magnesium transporter abundance in streptozotocin-induced diabetes mellitus. Kidney Int. 69 , 1786–1791. (10.1038/sj.ki.5000344)16557223
85. Felsenfeld AJ , Levine BS , Rodriguez M . 2015 Pathophysiology of calcium, phosphorus, and magnesium dysregulation in chronic kidney disease. Semin. Dial. 28 , 564–577. (10.1111/sdi.12411)26303319
86. Layton AT , Vallon V , Edwards A . 2016 Predicted consequences of diabetes and SGLT inhibition on transport and oxygen consumption along a rat nephron. Am. J. Physiol. Renal Physiol. 310 , F1269–83. (10.1152/ajprenal.00543.2015)26764207
87. Hu R , McDonough AA , Layton AT . 2021 Sex differences in solute and water handling in the human kidney: Modeling and functional implications. iScience 24 , 102667. (10.1016/j.isci.2021.102667)34169242
88. Layton AT . 2011 A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture. Am. J. Physiol. Renal Physiol 300 , F372–F384. (10.1152/ajprenal.00204.2010)21068088
89. Chen J , Edwards A , Layton AT . 2009 A mathematical model of O2 transport in the rat outer medulla. II. Impact of outer medullary architecture. Am. J. Physiol. Renal. Physiol. 297 , F537–F548. (10.1152/ajprenal.90497.2008)19403645
90. Stadt MM , Layton AT . 2022 Sex and species differences in epithelial transport in rat and mouse kidneys: modeling and analysis. Front. Physiol. 13 , 991705. (10.3389/fphys.2022.991705)36246142
91. Zenodo . Data from: Pritha17/Nephron-Mg_Ca_Transport: regulation of magnesium by kidney. Zenodo repository. See https://zenodo.org/records/10658961
