Modeling calcium and magnesium balance: Regulation by calciotropic hormones and adaptations under varying dietary intake

Summary

Magnesium (Mg²⁺) is crucial for several cellular and physiological processes and is tightly regulated due to health risks associated with imbalances. Mg²⁺, calcium (Ca²⁺), parathyroid hormone, and vitamin D₃ are tightly coupled, ensuring proper bone metabolism and intestinal and renal absorption of Mg²⁺ and Ca²⁺. While several Ca²⁺ homeostasis models exist, no computational model has been developed to study Mg²⁺ homeostasis. We developed a computational model of Mg²⁺ homeostasis in male rats, integrating it with an existing Ca²⁺ homeostasis model, to understand the interconnected physiological processes regulating their homeostasis. We then analyzed adaptations in these interconnected processes under (1) dietary Mg²⁺ deficiency, (2) low/high dietary Ca²⁺ with Mg²⁺ deficiency, and (3) vitamin D₃ deficiency. Model simulations predicted severe hypomagnesemia and mild hypocalcemia with significant dietary Mg²⁺ deficiency. Low dietary Ca²⁺ improved, while high dietary Ca²⁺ worsened Mg²⁺ deficiency. Finally, vitamin D₃ deficiency caused severe hypocalcemia, with minimal impact on Mg²⁺ homeostasis.

Graphical abstract

Highlights

• •We have developed a computational model of Mg²⁺ and Ca²⁺ homeostasis in a male rat
• •Severe dietary Mg²⁺ deficiency caused severe hypomagnesemia and mild hypocalcemia
• •Dietary Ca²⁺ deficiency in the presence of Mg²⁺ deficiency improved plasma Mg²⁺ level
• •Vitamin D3 deficiency significantly impacted Ca²⁺ homeostasis but not Mg²⁺ homeostasis

Physiology; Bioinformatics; Computational bioinformatics

Introduction

Magnesium (Mg²⁺) is the fourth most abundant cation in the body. Mg²⁺ is the cofactor for several enzymes and hence plays an important role in most major cellular processes such as energy metabolism, DNA transcription, and protein synthesis. In addition, Mg²⁺ is required for muscle contraction, neuromuscular stability, and bone formation. Thus, any disturbance in Mg²⁺ homeostasis can disrupt several essential cellular and physiological processes.

Extracellular Mg²⁺ is tightly regulated, with plasma [Mg²⁺] maintained relatively constant between 1.6 and 2.3 mg/dL¹ under normal physiological conditions in humans. Mg²⁺ homeostasis is maintained by three organs: the intestine, responsible for Mg²⁺ uptake from diet; bones, responsible for Mg²⁺ storage; and the kidneys, responsible for Mg²⁺ excretion. Almost all of the body’s Mg²⁺ (approximately 99%) is either stored in bone or within cells and less than 1% is present in the blood.¹ The normal daily Mg²⁺ intake of humans averages around 300 mg, about half of which is absorbed by the intestine.² About 70% of the circulating Mg²⁺ is non-protein-bound and is thus filtered by the glomerulus, accounting for 2,400 mg in humans.

A complex interconnection exists between parathyroid hormone (PTH), calcitriol (1,25(OH)₂D₃), Mg²⁺, and calcium (Ca²⁺). Mg²⁺ is an important regulator of PTH secretion and vitamin D3 metabolism. Although Mg²⁺ has only ∼60% of the effect of Ca²⁺ on PTH secretion, it is still an important regulator of PTH secretion. In fact, acute elevations and reductions in plasma [Mg²⁺] inhibit PTH secretion irrespective of plasma Ca²⁺ levels. Mg²⁺ also plays an important role in the activation of vitamin D₃ which occurs in two steps: (1) in the liver, cholecalciferol is hydroxylated to 25(OH)D, and (2) in the kidneys, 25(OH)D is converted to 1,25(OH)₂D₃. The activity of the enzymes involved in both these processes is dependent on Mg²⁺. Thus, dysregulation of Mg²⁺ homeostasis can significantly impact PTH secretion and vitamin D₃ metabolism which in turn can disrupt bone and Ca²⁺ homeostasis.

Maintaining Mg²⁺ and Ca²⁺ balance relies on the highly coupled regulation of various processes, including intestinal absorption, renal filtration and reabsorption, and bone remodeling. Given the multitude of interconnected physiological processes involved in the homeostasis of these two divalent cations, mathematical modeling proves valuable in comprehending the system’s complexities. In this study, we developed the first Mg²⁺ homeostasis model for male rats and integrated it with a previously developed calcium homeostasis model for male rats.^3,4 Figure 1 provides a schematic representation of the fundamental fluxes and hormones involved in Mg²⁺ and Ca²⁺ homeostasis. We then used this model to understand how these different interconnected processes adapt in the presence of different disorders (dietary Mg²⁺ deficiency, low/high dietary Ca²⁺ in the presence of Mg²⁺ deficiency, and vitamin D₃ deficiency) to maintain Mg²⁺ and Ca²⁺ homeostasis.

Figure 1.

Schematics of the Mg²⁺ homeostasis model

The model consists of five compartments: plasma, intestine, kidney, parathyroid gland, and bone. Arrows with triangular arrowheads indicate activation, while those with circular arrowheads indicate inhibition. All arrows are color coded. Green arrow, Ca²⁺; red arrow, Mg²⁺, blue arrow, parathyroid hormone (PTH); mauve arrow, 1,25(OH)₂D₃.

Results

Baseline results

The predicted baseline steady state concentrations of PTH, 1,25(OH)₂D₃, Mg²⁺, and Ca²⁺, and the steady state Mg²⁺ and Ca²⁺ fluxes are given in Table 1. The predicted baseline ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and ${\left[{Ca}^{2+}\right]}_p$ fall within the physiological ranges reported in the literature.

Table 1.

Baseline steady state concentrations and fluxes

	Steady-state value	Range	Source
${\left[PTH\right]}_p$, pM	6.28	1.5–13	Table 4 of ref.3 (mathematical model of Ca2+ homeostasis in rats),5,6 (experimental studies on Sprague-Dawley rats)
${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, pM	154	80–250	Table 4 of ref.3,7,8 (experimental studies on Sherman rats and Sprague-Dawley rats)
${\left[{Mg}^{2+}\right]}_p$, mM	0.65	0.45–0.85	9,10,11 (experimental studies on Sprague-Dawley rats and Wistar rats)
${\left[{Ca}^{2+}\right]}_p$, mM	1.25	1.1–1.3	Table 4 of ref.3,5,12 (experimental studies on Sprague-Dawley rats and Wistar Hannover rats)
Intestinal Ca2+ absorption, μmol/min	0.59	0.55–1.22	Table 4 of ref.3
Intestinal Mg2+ absorption, μmol/min	0.032	0.027–0.05	13 (experimental study on Wistar rats)
Urinary Ca2+ excretion, μmol/min	0.045	0.015–0.054	Table 4 of ref.3
Urinary Mg2+ excretion, μmol/min	0.024	0.01–0.038	13 (experimental study on Wistar rats)
Bone Ca2+ accretion, μmol/min	1.08	–	–
Bone Mg2+ accretion, μmol/min	0.02	–	–
Bone Ca2+ resorption, μmol/min	0.53	–	–
Bone Mg2+ resorption, μmol/min	0.014	–	–
Ca2+ flux from fast bone pool to plasma, μmol/min	0.45	–	–
Mg2+ flux from fast bone pool to plasma, μmol/min	0.16	–	–
Ca2+ flux from plasma to fast bone pool, μmol/min	1.53	–	–
Mg2+ flux from plasma to fast bone pool, μmol/min	0.18	–	–

Sensitivity analysis

Local sensitivity analysis

We performed a local sensitivity analysis by varying each model parameter listed in Table 2 by ±5% and computing the corresponding steady state. The resulting percent changes in ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and ${\left[{Ca}^{2+}\right]}_p$ are shown in Figure 4.

Table 2.

Description and values of model parameters

Parameter	Symbol	Value	Reference
Plasma volume	$V_p$	10 mL	Granjon et al. 3
PTH
Maximal rate constant of PTH secretion from the parathyroid gland	${\beta }_{exo}^{PTHg}$	1.034 min−1	estimated (refer to section: STAR Methods:Method details:Parathyroid gland and parathyroid hormone and Figure 2)
Factor controlling the maximal PTH secretion at a given plasma Ca2+ concentration	${\gamma }_{Ca}$	0.15 mM	estimated (refer to section: STAR Methods:Method details:Parathyroid gland and parathyroid hormone and Figure 2)
Factor controlling the maximal PTH secretion at a given plasma Ca2+ concentration	${\gamma }_p$	2	estimated (refer to section: STAR Methods:Method details:Parathyroid gland and parathyroid hormone and Figure 2)
Factor controlling the slope of the PTH secretion curve	$C_m$	3.8 mM	estimated (refer to section: STAR Methods:Method details:Parathyroid gland and parathyroid hormone and Figure 2)
IC50 value of Mg2+ for inhibition of PTH secretion	$K_{low-Mg}$	0.25 mM	Quitterer et al. 14
Basal rate of PTH production in the parathyroid gland	$k_{prod}^{PTHg}$	2.53 pmol/min	estimated
Inhibition of PTH synthesis by 1,25(OH)2D3	${\gamma }_{prod}^{1,25{\left(OH\right)}_2{D}_3}$	0.003 p.m.−1	Stadt et al.4
Rate of degradation of PTH in parathyroid gland	$k_{\mathit{deg}}^{PTHg}$	0.035 min−1	Granjon et al. 3
Degradation rate constant of plasma PTH	$k_{\mathit{deg}}^{PTHp}$	0.081 min−1	estimated
Vitamin D3
Inactive vitamin D3	${\left[25\left(OH\right)D\right]}_p$	25 nM	Granjon et al. 3
Minimum production rate constant of 1,25(OH)2D3	$k_{conv}^{\mathit{min}}$	4.4 x 10−6 min−1	Granjon et al. 3
Maximal increase in 1,25(OH)2D3 production rate	${\delta }_{conv}^{\mathit{max}}$	6 x 10−5 min−1	Granjon et al. 3
Stimulation of 1,25(OH)2D3 production by PTH	$K_{conv}^{PTH}$	3 p.m.	Granjon et al. 3
PTH sensitivity coefficient	$n_{conv}$	6	Granjon et al. 3
Inhibition of 1,25(OH)2D3 production by Ca2+	${\gamma }_{conv}^{Ca}$	0.3 mM−1	Granjon et al. 3
Inhibition of 1,25(OH)2D3 production by itself	${\gamma }_{conv}^{1,25{\left(OH\right)}_2{D}_3}$	1.8 x 10−2 p.m.−1	Granjon et al. 3
Factor controlling the maximal increase in 1,25(OH)2D3 production by Mg2+	${\delta }_{Mg-act}$	6.6	estimated (refer to section: STAR Methods:Method details:Plasma 1,25(OH)2D3 and Figure 3)
Michaelis-Menten constant	$K_{Mg-act}^1$	1.1 mM	estimated (refer to section: STAR Methods:Method details:Plasma 1,25(OH)2D3 and Figure 3)
Michaelis-Menten constant	$K_{Mg-act}^2$	1.2 mM	estimated (refer to section: STAR Methods:Method details:Plasma 1,25(OH)2D3 and Figure 3)
Inhibition of 1,25(OH)2D3 degradation by PTH	${\gamma }_{inact}^{PTH}$	0.52 p.m.−1	Granjon et al. 3
Michaelis-Menten constant	$K_{D3}$	1.7 mM	estimated
Degradation rate constant of 1,25(OH)2D3	$k_{\mathit{deg}}^{1,25{\left(OH\right)}_2{D}_3}$	0.008 min−1	estimated
Kidneys
Minimal fractional reabsorption of Mg2+ in proximal tubule	${\lambda }_{Mg-PT}^0$	0.185	Quamme et al.15
Stimulation of Mg2+ reabsorption in proximal tubule by PTH	${\delta }_{Mg-PT}^{\mathit{max}}$	0.015	estimated
Sensitivity of Mg2+ reabsorption in proximal tubule to PTH	${PTH}_{ref}$	12 p.m.	Granjon et al. 3
Hill coefficient	$n_{PT}$	5	Granjon et al. 3
Minimal fractional reabsorption of Mg2+ in thick ascending limb	${\lambda }_{Mg-TAL}^0$	0.66	Quamme et al.15
Stimulation of Mg2+ reabsorption in thick ascending limb by CaSR	${\delta }_{Mg-CaSR}^{\mathit{max}}$	0.028	estimated
Stimulation of Mg2+ reabsorption in thick ascending limb by PTH	${\delta }_{Mg-PTH}^{\mathit{max}}$	0.012	estimated
Sensitivity of Mg2+ reabsorption in thick ascending limb to Ca2+	${Ca}_{ref}$	1.25 mM	Brown et al. 16
Sensitivity of Mg2+ reabsorption in thick ascending limb to Mg2+	${Mg}_{ref}$	2.5 mM	Brown et al. 16
Hill coefficient	$n_{TAL}$	4	Granjon et al. 3
Sensitivity of Mg2+ reabsorption in thick ascending limb to PTH	$K_{TAL}^{PTH}$	4 p.m.	Stadt et al.4
Minimal fractional reabsorption of Mg2+ in distal convoluted tubule	${\lambda }_{Mg-DCT}^0$	0.08	Quamme et al.15
Stimulation of Mg2+ reabsorption in distal convoluted tubule by PTH and 1,25(OH)2D3	${\delta }_{Mg-DCT}^{\mathit{max}}$	0.02	estimated
Sensitivity of Mg2+ reabsorption in distal convoluted tubule to PTH	$K_{DCT}^{PTH}$	7.25 p.m.	Stadt et al. 4
Sensitivity of Mg2+ reabsorption in distal convoluted tubule to 1,25(OH)2D3	$K_{DCT}^{1,25{\left(OH\right)}_2{D}_3}$	160 p.m.	Stadt et al. 4
Glomerular filtration rate (GFR)	${\mathrm{\Phi }}_{GFR}$	1.25 mL/min	Sadick et al. 17
Intestine
Dietary Mg2+ intake	$I_{Mg}$	0.04 μmol/min	Coudray et al. 13
Maximal rate of active absorption of Mg2+	$V_{active}$	0.764 μmol/min	Hardwick et al.18
Stimulation of active Mg2+ absorption by dietary Mg2+ intake	$K_{active}$	0.17 μmol/min	Hardwick et al.18
Stimulation of Mg2+ absorption by 1,25(OH)2D3	$K_{abs}^{1,25{\left(OH\right)}_2{D}_3}$	100 p.m.	Granjon et al. 3
Bones
Rate of Mg2+ uptake from plasma by fast bone pool	$k_{p-f}^{Mg}$	0.074 min−1	estimated
Rate of Mg2+ release from fast bone pool to plasma	$k_{f-p}^{Mg}$	0.2 x 10−3 min−1	estimated
Rate of accretion into the slow bone pool	${\gamma }_{ac}^{Mg}$	3.98 x 10−4 min−1	estimated
Minimal resorption rate	${\tau }_{res}^{\mathit{min}}$	0.142 x 10−3 mmol/min	Granjon et al. 3
Maximal resorption rate	${\delta }_{res}^{\mathit{max}}$	0.7 x 10−3 mmol/min	Granjon et al. 3
Stimulation of bone resorption by PTH	$K_{res}^{PTHp}$	2.45 p.m.	Granjon et al. 3
Stimulation of bone resorption by 1,25(OH)2D3	$K_{res}^{1,25{\left(OH\right)}_2{D}_3}$	160 p.m.	Granjon et al. 3
Plasma
Fraction of magnesium bound to proteins	${\kappa }_{b-Mg}$	0.3	Jahnen-Dechent et al. 19

Figure 4.

Local sensitivity analysis

Local sensitivity analysis conducted by (A) increasing individual parameters by 5% and (B) decreasing individual parameters by 5%. The resulting percent change in model steady state concentrations from baseline is presented here. White indicates the resulting change was less than 1%.

A 5% change in minimal thick ascending limb fractional reabsorption ${\lambda }_{Mg-TAL}^0$ causes a significant change in ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and ${\left[{Ca}^{2+}\right]}_p$. Let us analyze the results when ${\lambda }_{Mg-TAL}^0$ is increased by 5% (Figure 4A). As intestinal Mg²⁺ absorption is mostly dependent on dietary Mg²⁺ intake and only a small fraction (12%) is regulated by 1,25(OH)₂D₃ (Equation 19), the kidneys play a major role in determining and maintaining ${\left[{Mg}^{2+}\right]}_p$. The thick ascending limb is the major Mg²⁺ reabsorption segment in the kidney; hence, a 5% increase in ${\lambda }_{Mg-TAL}^0$ increases ${\left[{Mg}^{2+}\right]}_p$ by 6.7% to 0.69 mM from the baseline value of 0.65 mM (Figure 4). The higher ${\left[{Mg}^{2+}\right]}_p$ increases the synthesis of 1,25(OH)₂D₃, which in turn increases the intestinal absorption of Ca²⁺ as 45% of it is regulated by 1,25(OH)₂D₃.³ Thus, ${\left[{Ca}^{2+}\right]}_p$ increases to 1.28 mM from the baseline value of 1.25 mM. ${\left[PTH\right]}_p$ decreases by 5.2% because (1) the increased ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ inhibits PTH synthesis in the parathyroid gland and (2) the increased ${\left[{Mg}^{2+}\right]}_p$ and ${\left[{Ca}^{2+}\right]}_p$ inhibits PTH secretion.

It is interesting to note that the fractional reabsorption of Ca²⁺ along the proximal tubule, which is the major segment of renal Ca²⁺ reabsorption (represented by the parameter ${\lambda }_{Ca-PT}^0$), does not significantly alter ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and ${\left[{Ca}^{2+}\right]}_p$ (Figure 4). This somewhat unintuitive result is mainly due to the negative feedback loop between ${\left[{Ca}^{2+}\right]}_p$ and 1,25(OH)₂D₃ synthesis and PTH secretion. The increased renal Ca²⁺ reabsorption increases ${\left[{Ca}^{2+}\right]}_p$. This in turn inhibits PTH secretion. The increased ${\left[{Ca}^{2+}\right]}_p$ and decreased ${\left[PTH\right]}_p$ inhibit 1,25(OH)₂D₃ synthesis, which in turn inhibits intestinal Ca²⁺ absorption. Thus, the negative feedback loop between ${\left[{Ca}^{2+}\right]}_p$ and ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ attenuates the effect on ${\left[{Ca}^{2+}\right]}_p$ when renal Ca²⁺ reabsorption is varied. By contrast, the feedback loop between ${\left[{Mg}^{2+}\right]}_p$ and ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ is reinforcing which contributes to significantly alter ${\left[{Mg}^{2+}\right]}_p$ in the face of increased/decreased renal Mg²⁺ reabsorption.

Global sensitivity analysis

We conducted global sensitivity analysis by applying the variance-based Sobol method.²⁰ This method decomposes the output variance into contributions from individual parameters and their interactions, thus identifying parameters that have the most significant effect on the output. Sobol indices are of different orders that reflect the number of parameters interacting with each other. Therefore, the 1^st-order Sobol indices measure the effect of individual parameters, 2nd-order Sobol indices measure the effect of the interaction between two parameters and so on. The total Sobol index measures the influence of a parameter, including interactions with other parameters.

We performed the Sobol sensitivity analysis with 10,000 samples. Each parameter listed in Table 2 was varied in the range of ±20%. Figure 5 shows the Sobol indices of parameters that have significant influence on ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and ${\left[{Ca}^{2+}\right]}_p$. The figure shows the 1^st-order Sobol indices (blue bars) and the other order Sobol indices indicating interactions (red bars). The other order index was calculated as total Sobol index ˗ 1^st-order Sobol index. The figure shows only those parameters which had total Sobol indices greater than 0.05. For all the parameters, we observe that the 1^st-order Sobol indices predominate over the Sobol indices due to interactions. Thus, these parameters individually make significant contributions to the model outputs. Plasma volume (V_p) has the highest influence on ${\left[PTH\right]}_p$, minimal fractional reabsorption of Mg²⁺ in thick ascending limb (${\lambda }_{Mg-TAL}^0$) has the most impact on ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ and ${\left[{Mg}^{2+}\right]}_p$, and rate of Ca²⁺ uptake from plasma by fast bone pool ($k_{pf}^{Ca}$) has the highest impact on ${\left[{Ca}^{2+}\right]}_p$.

Figure 5.

Global sensitivity analysis

Sobol indices of parameters that have significant impact on (A) ${\left[PTH\right]}_p$, (B) ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, (C) ${\left[{Mg}^{2+}\right]}_p$, and (D) ${\left[{Ca}^{2+}\right]}_p$. Parameters were varied in the range of ±20% and Sobol indices were calculated on steady state concentrations.

Parameters with total Sobol indices greater than 0.05 are shown. Blue bars indicate the 1^st-order Sobol indices and red bars indicate the other order indices representing interaction with other parameters. The other order indices were calculated as total Sobol indices ˗ 1^st-order Sobol indices.

Effect of deficiency of dietary Mg²⁺

A large portion of the population in all continents consumes less than two-thirds of the recommended dietary allowances for Mg²⁺.²¹ In the United States, the standard diet contains only about 50% of the recommended daily Mg²⁺.²² Insufficient dietary Mg²⁺ intake may lower plasma Mg²⁺ level which can have severe consequences. For instance, vitamin D metabolizing enzymes, 1α-hydroxylase and 24-hydroxylase, are Mg²⁺-dependent and hence Mg²⁺ deficiency can significantly lower plasma 1,25(OH)₂D₃ levels.^23,24 In addition, Mg²⁺ deficiency also impairs PTH response. To evaluate its effect on Ca²⁺ and Mg²⁺ homeostasis, we conducted simulations in which dietary Mg²⁺ intake ($I_{Mg}$) was reduced by 50%, 75%, and 90% for 6 months according to the experiments conducted by Rude et al.^9,25,26 The predicted fractional changes in plasma concentrations of PTH, 1,25(OH)₂D₃, Mg²⁺, and Ca²⁺ and Mg²⁺ and Ca²⁺ fluxes after 6 months of restricted $I_{Mg}$ are shown in Figure 6.

Figure 6.

Dietary Mg²⁺ deficiency

Predicted fractional changes in plasma concentrations and fluxes at 50%, 75%, and 90% dietary Mg²⁺ intake ($I_{Mg}$) restrictions from baseline (denoted by the gray line at 0) for 6 months (A) Predicted (bars) and experimental (black diamonds) fractional changes in plasma concentrations of PTH, 1,25(OH)₂D₃, Ca²⁺, Mg²⁺, bone Ca²⁺ content, and bone Mg²⁺ content.

(B) Predicted fractional changes in Mg²⁺ and Ca²⁺ fluxes into plasma.

(C) Predicted fractional changes in Mg²⁺ and Ca²⁺ fluxes out of plasma.

Figure 6A shows the predicted and experimental^9,25,26 fractional changes in ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, ${\left[{Ca}^{2+}\right]}_p$, bone Ca²⁺ content, and bone Mg²⁺ content at different $I_{Mg}$ restrictions. Bone Ca²⁺ and Mg²⁺ contents were measured from bone ash in the experimental studies.^9,25,26 The predicted fractional changes in ${\left[PTH\right]}_p$, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, ${\left[{Mg}^{2+}\right]}_p$, and bone Mg²⁺ content are almost in line with the experimentally reported changes. However, while the experimental study reported ${\left[{Mg}^{2+}\right]}_p$ to decrease by 6% at 50% $I_{Mg}$ restriction, the model predicted a decrease of 17%. The model might have underestimated the adaptive increase in intestinal Mg²⁺ absorption at moderate Mg²⁺ deficiency in rats, since the intestinal absorption parameters were obtained from a study conducted on humans. Nevertheless, the predicted ${\left[{Mg}^{2+}\right]}_p$ at 50% $I_{Mg}$ restriction (0.54 mM) is within the normal range (0.45–0.85 mM). In addition, the predicted change in ${\left[{Ca}^{2+}\right]}_p$ content differs significantly from the experimental values. The experimental studies^9,25,26 reported ${\left[{Ca}^{2+}\right]}_p$ to decrease by 5% at 50% $I_{Mg}$ restriction, and increase by 5% and 9% at 75% and 90% $I_{Mg}$ restrictions, respectively. By contrast, our model predicted ${\left[{Ca}^{2+}\right]}_p$ to decrease by 10%, 20%, and 23% after 6 months of 50%, 75%, and 90% $I_{Mg}$ restrictions, respectively. Severe dietary Mg²⁺ deficiency causes hypocalcemia in most species (including humans^27,28) with the exception of rats and mice, where hypercalcemia develops.²⁹ The exact reasons for this are not clearly understood but could be due to the reduction in osteoblastic and osteocytic activity in the presence of hypomagnesemia, which significantly lowers the rate of bone formation.^29,30,31 Further investigation is required to understand why severe dietary Mg²⁺ deficiency results in hypercalcemia in rodents and hypocalcemia in other species.

At 50% $I_{Mg}$ restriction, ${\left[{Mg}^{2+}\right]}_p$ was predicted to decrease to 0.54 mM from the baseline concentration of 0.65 mM, and ${\left[{Ca}^{2+}\right]}_p$ decreased to 1.12 mM from the baseline concentration of 1.25 mM, thus triggering increased PTH secretion (Figure 6A). By contrast, at 75% and 90% $I_{Mg}$ restrictions, ${\left[{Mg}^{2+}\right]}_p$ was predicted to decrease to 0.31 and 0.17 mM, respectively, which are significantly below 0.4 mM. These very low plasma Mg²⁺ concentrations inhibit PTH secretion (based on Equation 5). Similar observations were reported in dietary Mg²⁺ restriction experiments conducted on humans.³² Intestinal Mg²⁺ absorption was predicted to decrease proportionally with the $I_{Mg}$ restrictions (Figure 6B). Since 45% of intestinal Ca²⁺ absorption is regulated by ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, it was predicted to decrease by 13%, 21%, and 22%, respectively (Figure 6B). Urinary Ca²⁺ and Mg²⁺ excretions decreased proportionally with decrease in ${\left[{Ca}^{2+}\right]}_p$ and ${\left[{Mg}^{2+}\right]}_p$ (Figure 6C).

Dietary Mg²⁺ reduction causes significant bone loss.^9,25,26,33 Bone loss is characterized by a decrease in bone mineral density (i.e., decrease in Ca²⁺, Mg²⁺, and other mineral content in the bone). Our model predicted bone Mg²⁺ content to decrease by 9%, 17%, and 39%, respectively, following 6 months of 50%, 75%, and 90% $I_{Mg}$ restrictions (Figure 6A), which are in line with the experimental values.^9,25,26 Several experimental and clinical studies have shown that Mg²⁺ deficiency promotes osteoporosis (summarized in³⁴). Due to this bone loss, the exchange of Mg²⁺ between the bone and plasma drops significantly (Figures 6B and 6C). The model predicted almost no change in bone Ca²⁺, in line with the experimental results.(Figure 6A).

Taken together, model results reveal the mechanisms by which sufficiently large deficiency in dietary Mg²⁺ causes dysregulation in Ca²⁺ and Mg²⁺ homeostasis.

Effect of low/high dietary Ca²⁺ in the presence of low dietary Mg²⁺

Since, as noted previously, a large portion of the population consumes Mg²⁺ lower than the recommended level, we investigated the effect that low dietary Ca²⁺ or Ca²⁺ supplementation has on Ca²⁺ and Mg²⁺ homeostasis in the presence of 60% dietary Mg²⁺ ($I_{Mg}$) restriction. To simulate low and high dietary Ca²⁺, we decreased and increased $I_{Ca}$ by 50%, respectively. Figure 7 shows the predicted fractional changes in steady state plasma concentrations of PTH, 1,25(OH)₂D₃, Mg²⁺, and Ca²⁺ and the steady state Mg²⁺ and Ca²⁺ fluxes from the steady state values at 60% $I_{Mg}$ restriction (shown by the gray line at 0) for each set of simulations.

Figure 7.

Dietary Ca²⁺ deficiency or supplementation in the presence of dietary Mg²⁺ deficiency

Predicted fractional change in steady state plasma concentrations and fluxes at (i) 60% dietary Mg²⁺ ($I_{Mg}$) restriction combined with 50% dietary Ca²⁺ ($I_{Ca}$) restriction and (ii) 60% $I_{Mg}$ restriction combined with 50% $I_{Ca}$ increase from the steady state values at 60% $I_{Mg}$ restriction (shown by the gray line at 0).

(A–C) (A) Fractional change in steady state plasma concentrations of PTH, 1,25(OH)₂D₃, Ca²⁺, and Mg²⁺. (B) Fractional change in steady state Mg²⁺ and Ca²⁺ fluxes into plasma. (C) Fractional change in steady state Mg²⁺ and Ca²⁺ fluxes out of plasma.

At 60% $I_{Mg}$ restriction and baseline $I_{Ca}$, ${\left[{Mg}^{2+}\right]}_p$ decreased by 31% to 0.45 mM from the baseline concentration of 0.65 mM and ${\left[{Ca}^{2+}\right]}_p$ decreased by 13% to 1.08 mM from the baseline concentration of 1.25 mM. Restricting $I_{Ca}$ by 50% further lowered ${\left[{Ca}^{2+}\right]}_p$ to 0.88 mM, resulting in hypocalcemia (Figure 7A). This enhanced PTH secretion (Equation 2). The decreased ${\left[{Ca}^{2+}\right]}_p$ and increased ${\left[PTH\right]}_p$ enhanced 1,25(OH)₂D₃ synthesis (Equation 7). The increased ${\left[PTH\right]}_p$ and ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ increased resorption of Mg²⁺ from the slow bone pool by 13% (Equation 22). In addition, the increased ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ enhanced intestinal absorption of Mg²⁺ raising ${\left[{Mg}^{2+}\right]}_p$ to 0.49 mM (Figure 7A). Thus, lowering dietary Ca²⁺ intake during dietary Mg²⁺ deficiency improved plasma Mg²⁺ concentration and ameliorated hypomagnesemia.³⁵

The opposite changes were observed when $I_{Ca}$ was increased by 50%. ${\left[{Ca}^{2+}\right]}_p$ increased to 1.21 mM which inhibited PTH secretion and 1,25(OH)₂D₃ synthesis (Figure 7A). This in turn inhibited resorption of Mg²⁺ from the slow bone pool and intestinal Mg²⁺ absorption. Consequently, ${\left[{Mg}^{2+}\right]}_p$ decreased further from 0.45 mM to 0.41 mM (Figure 7A). Thus, higher dietary Ca²⁺ intake combined with dietary Mg²⁺ deficiency exacerbated hypomagnesemia.^36,37,38,39

To understand these changes in the cellular level, note that when $I_{Ca}$ is decreased, ${\left[{Ca}^{2+}\right]}_p$ decreases, which stimulates the calcium-sensing receptors (CaSR) on the parathyroid glands and PTH secretion increases. Now Ca²⁺ is an inhibitor and PTH is an activator of 1α-hydroxylase, the enzyme responsible for converting 25(OH)D to 1,25(OH)₂D₃. Thus, the decreased plasma Ca²⁺ and increased plasma PTH increase the synthesis of 1,25(OH)₂D₃. The increased PTH and 1,25(OH)₂D₃ increase production of release receptor activator of nuclear factor kappa-B ligand (RANKL) and osteoprotegerin (OPG), leading to increased osteoclast formation and activity and hence increased bone resorption. Along the renal thick ascending limb, increased PTH decreases claudin 14 and activated CaSR increases claudin 14 expression; these two opposing responses slightly increase claudin 16/19 expression and hence paracellular transport of Mg²⁺. Along the renal distal convoluted tubule, PTH increases Mg²⁺ uptake through receptor-mediated cAMP release and activation of protein kinase A and 1,25(OH)₂D₃ increases Mg²⁺ uptake through calbindin-D. All these responses combine to increase plasma Mg²⁺ concentration. The opposite responses occur when $I_{Ca}$ is increased. Taken together, reduced dietary Ca²⁺ intake in the presence of Mg²⁺ deficiency helped to improve plasma Mg²⁺ concentration, whereas, increased dietary Ca²⁺ intake caused further decline in plasma Mg²⁺ concentration.

Effect of change of inactive vitamin D3 (25(OH)D)

Low plasma level of 25(OH)D, the substrate for 1,25(OH)₂D₃ (Figure 8), is commonly observed in chronic liver disease and chronic kidney disease, with the degree of 25(OH)D deficiency increasing with the severity and progression of disease.^{40,41,42,43,44} Since 1,25(OH)₂D₃ plays an important role in Mg²⁺ and Ca²⁺ homeostasis, we studied the effect of different levels of 25(OH)D deficiency on Mg²⁺ and Ca²⁺ homeostasis. To accomplish that, we progressively decreased the parameter ${\left[25\left(OH\right)D\right]}_p$ up to 100%. The predicted normalized steady state plasma concentrations of PTH, 1,25(OH)₂D₃, Mg²⁺, and Ca²⁺ and the normalized steady state Mg²⁺ and Ca²⁺ fluxes are shown in Figure 9.

Figure 8.

Schematic of the regulation of different forms of vitamin D₃ by Mg²⁺, Ca²⁺, PTH, and 1,25(OH)₂D₃

Arrows denote activation and closed circles denote inhibition.

Figure 9.

Inactive vitamin D3 (25(OH)D) deficiency

Predicted normalized steady state plasma concentrations and fluxes at decreased ${\left[25\left(OH\right)D\right]}_p$

(A–C) All y axis values are normalized to the baseline values. (A) Normalized steady state plasma concentrations of PTH, 1,25(OH)₂D₃, Ca²⁺, and Mg²⁺. (B) Normalized steady state Mg²⁺ and Ca²⁺ fluxes into plasma. (C) Normalized steady state Mg²⁺ and Ca²⁺ fluxes out of plasma.

As ${\left[25\left(OH\right)D\right]}_p$ was decreased, 1,25(OH)₂D₃ synthesis in the kidneys decreased, resulting in lower ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ (Equation 7), which became zero when ${\left[25\left(OH\right)D\right]}_p$ = 0 (Figure 9A). Now, ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ directly and indirectly regulates the following: (1) PTH synthesis, (2) intestinal absorption of Ca²⁺ and Mg²⁺, (3) renal Ca²⁺ and Mg²⁺ reabsorption, and (4) bone resorption. Let us first assess the impact on intestinal absorption of Mg²⁺ and Ca²⁺. ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ regulates 45% of intestinal Ca²⁺ absorption³ but only 12% of intestinal Mg²⁺ absorption (Equation 19). Hence, the decrease in intestinal Ca²⁺ absorption⁴⁵ was significantly higher compared to Mg²⁺ (Figure 9B). Consequently, ${\left[{Ca}^{2+}\right]}_p$ decreased by 14% to 1.08 mM (below the normal range of 1.1–1.3 mM) at 50% inhibition of ${\left[25\left(OH\right)D\right]}_p$, and further by 54% to 0.58 mM, which was significantly below the normal range (Figure 9A), at full inhibition. Several experimental and clinical studies have reported hypocalcemia in the presence of low serum 25(OH)D concentration.^45,46,47 By contrast, ${\left[{Mg}^{2+}\right]}_p$ did not change significantly until ${\left[25\left(OH\right)D\right]}_p$ was inhibited by over 60%, at which point ${\left[{Mg}^{2+}\right]}_p$ was about 2.4% lower than baseline (Figure 9A). At full inhibition, ${\left[{Mg}^{2+}\right]}_p$ dropped by 11% to 0.58 mM, within the normal range (normal range is 0.45–0.85 mM) (Figure 9A). The lower ${\left[{Ca}^{2+}\right]}_p$ and ${\left[{Mg}^{2+}\right]}_p$ enhanced PTH secretion to maintain levels of these two divalent cations (Equation 2). In addition, the inhibitory effect of ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ on PTH synthesis in the parathyroid gland (Equation 1) was attenuated, which further increased PTH secretion. Thus, ${\left[PTH\right]}_p$ increased by 29% and 193%, respectively, at 50% and 100% inhibitions (Figure 9A).⁴⁸ Now, the higher ${\left[PTH\right]}_p$ increased Ca²⁺ and Mg²⁺ reabsorption along the thick ascending limb and distal convoluted tubule. In addition, the fall in ${\left[{Ca}^{2+}\right]}_p$ and ${\left[{Mg}^{2+}\right]}_p$ also decreased the Ca²⁺ and Mg²⁺ loads filtered by the kidneys. These two factors contributed to reducing urinary excretion of Ca²⁺ and Mg²⁺ to preserve plasma levels of these cations. Ca²⁺ and Mg²⁺ excretions decreased by 13% and 8% respectively, following a 50% inhibition of ${\left[25\left(OH\right)D\right]}_p$, and by 26% and 13% respectively, when ${\left[25\left(OH\right)D\right]}_p$ was fully inhibited (Figure 9C).

Following the significant decrease in ${\left[{Ca}^{2+}\right]}_p$, the exchange of Ca²⁺ between bone and plasma decreased significantly (Figures 9B and 9C). The decrease in Mg²⁺ exchange was comparatively lower (Figures 9B and 9C). Ca²⁺ content in the fast bone pool was predicted to decrease by 13% and 47% respectively, whereas the Mg²⁺ content decreased by 1.2% and 8.5%, respectively, at 50% and 100% ${\left[25\left(OH\right)D\right]}_p$ inhibition (Figure 9A). Now in the slow bone pool, Ca²⁺ content decreased by 4.9% and 24% respectively (Figure 9A). By contrast, our model predicted the Mg²⁺ content in the slow bone pool to remain almost unchanged (Figure 9A).

These results indicated that 1,25(OH)₂D₃ plays a major role in maintaining Ca²⁺ homeostasis as its deficiency can cause severe hypocalcemia.⁴⁹ Mg²⁺ homeostasis on the other hand is not severely impacted by deficiency of 1,25(OH)₂D₃.⁴⁹

Recall that the model predicted that a decrease in plasma Ca²⁺ in response to dietary Ca²⁺ restriction would increase plasma Mg²⁺ level (section: Effect of low/high dietary Ca2+ in the presence of low dietary Mg2+). However, a decrease in plasma Ca²⁺ level in response to 25(OH)D inhibition resulted in a decrease in plasma Mg²⁺ level (section: Effect of change of inactive vitamin D3 (25(OH)D)). Why does a decrease in plasma Ca²⁺ level cause opposite changes in plasma Mg²⁺ level in these two scenarios? Figure 10 summarizes the responses in these two scenarios which help answer this question. The difference lies in the change in plasma 1,25(OH)₂D₃. During dietary Ca²⁺ restriction, plasma 1,25(OH)₂D₃ increases which together with the increased plasma PTH increase Mg²⁺ reabsorption along the distal convoluted tubule and bone resorption. By contrast, during 25(OH)D inhibition, plasma 1,25(OH)₂D₃ decreases significantly, which inhibits Mg²⁺ reabsorption along the distal convoluted tubule and bone resorption. Thus, plasma Mg²⁺ increases during dietary Ca²⁺ restriction and decreases during 25(OH)D inhibition.

Figure 10.

Summary of changes in response to dietary Ca²⁺ restriction and 25(OH)D inhibition

Downward red arrows indicate decrease and upward green arrows indicate increase. Arrows with the green plus sign (+) indicate activation and arrows with the red minus sign (−) indicate inhibition. TAL, thick ascending limb; DCT, distal convoluted tubule.

Discussion

Mg²⁺ balance is primarily maintained by the absorption of Mg²⁺ in the intestine and kidney. Typically, approximately 30–50% of ingested Mg²⁺ is absorbed by the intestine. The refining and processing of food in modern society has resulted in a substantial loss of the naturally occurring Mg²⁺. For example, the refining and processing of wheat to flour, rice to polished rice, and corn to starch depletes Mg²⁺ by >80%.⁵⁰ Thus, the consumption of modern processed food may partially explain why a significant segment of the population has an Mg²⁺ intake that falls below the recommended dietary amounts. The intestine absorbs a fraction of Mg²⁺ that is inversely proportional to intake.¹⁸ This may have an unfortunate clinical consequence of prolonging the time for the treatment of Mg²⁺ deficiency with oral supplements to be effective. In the kidneys, the proximal tubule accounts for 15–25% of Mg²⁺ reabsorption, the thick ascending limb for 60–70%, and the distal convoluted tubule for ∼10%.⁵¹ Intestinal and renal Mg²⁺ transport occurs through both paracellular and transcellular pathways. Approximately 90% of Mg²⁺ absorption in the intestine and kidneys occurs passively through the paracellular pathway. Active reabsorption of Mg²⁺ occurs transcellularly. Although it accounts for only a small fraction of the total intestinal and kidney absorption, active Mg²⁺ transport is regulated and therefore fine tunes intestinal and renal Mg²⁺ excretion.

Hypomagnesemia is not uncommon, especially among hospitalized patients (up to 12%).⁵² Hypomagnesemia can be caused by decreased intake and absorption, or increased losses and redistribution. We considered the physiological implications of low Mg²⁺ intake (model predictions summarized in Figure 11). Model simulations indicated that severe Mg²⁺ deficiency leads to hypocalcemia and bone loss (Figure 6). Indeed, Mg²⁺ deficiency has been implicated as a risk factor for osteoporosis. Epidemiological studies^53,54,55 have demonstrated a positive correlation between dietary Mg²⁺ intake and bone density and/or an increased rate of bone loss with low dietary Mg²⁺ intake. In mouse studies,²⁹ Mg²⁺ depletion has also been reported to induce impaired bone growth, decreased osteoblast number, increased osteoclast number, and loss of trabecular bone with stimulation of cytokine activity in bone.

Figure 11.

Summarized predicted changes in plasma concentrations of Mg²⁺, Ca²⁺, PTH, and 1,25(OH)₂D₃ in the presence of (i) dietary Mg²⁺ deficiency, (ii) low/high dietary Ca²⁺ in the presence of dietary Mg²⁺ deficiency, and (iii) 25(OH)D deficiency

Arrow lengths are representative of the extent of change. Downward red arrows indicate decrease and upward green arrows indicate increase.

Besides a low Mg²⁺ diet, low Mg²⁺ input can also be caused by decreased absorption. Gastrointestinal diseases that reduce the transit time of intestinal fluid or interfere with absorption can also cause hypomagnesemia. Examples of such gastrointestinal diseases include severe diarrhea, steatorrhea, malabsorption syndromes, and short-bowel syndrome. The capacity of the intestine to absorb dietary Mg²⁺ also declines with aging.⁵⁶ As such, aging is a major risk factor for Mg²⁺ deficiency.

The causes of renal Mg²⁺ loss can be further divided into those due to increased flow, for example in case of polyuria, and those due to decreased tubular reabsorption. The causes of decreased tubular reabsorption of Mg²⁺ can, in turn, be classified according to the location in the nephron at which Mg²⁺ transport is perturbed. Because the thick ascending limb and the distal convoluted tubule are the major sites of renal Mg²⁺ reabsorption, most causes affect these regions of the kidney. For instance, Type 1 Bartter’s syndrome and Gitelman’s syndrome, which inhibit Na⁺ transporters along the thick ascending limb and distal convoluted tubule, respectively, cause increased Mg²⁺ excretion.^57,58

The body’s handling of Ca²⁺ and Mg²⁺ is coupled in the kidneys and via their regulation by PTH and vitamin D. Approximately half of the world’s population has inadequate access to dietary Ca²⁺.⁵⁹ Inadequate Ca²⁺ intake is linked not only to poor bone health but to other negative health outcomes, including pregnancy complications, cancers, and cardiovascular disease. As such, Ca²⁺ supplementation is often recommended to vulnerable subpopulations such as pregnant women. We conducted model simulations to investigate the combined physiological implications of low Mg²⁺ intake combined with either low or high Ca²⁺ diet (model predictions summarized in Figure 11). Model predictions indicated that reduced dietary Ca²⁺ intake may improve serum Mg²⁺ levels; although plasma Ca²⁺, which is suppressed in Mg²⁺ deficiency, would be even lower as expected (Figure 7). In contrast, increased dietary Ca²⁺ intake raises serum Ca²⁺ levels, which inhibits PTH secretion and 1,25(OH)₂D₃ synthesis, suppressing the resorption of Mg²⁺ from the slow bone pool, intestinal Mg²⁺ absorption, and renal reabsorption. Thus, higher dietary Ca²⁺ intake may exacerbate hypomagnesemia (Figure 7).

Mg²⁺ and Ca²⁺ homeostasis is altered in chronic kidney disease, even though the kidneys undergo adaptations such that hypermagnesemia and hypocalcemia are not observed until advanced chronic kidney disease. Mg²⁺ deficiency can be associated with abnormal vitamin D function. Indeed, chronic kidney disease is one of the main conditions associated with low 25(OH)D serum levels, with the vast majority of patients with chronic kidney disease exhibiting vitamin D insufficiency (>80% of cases^60,61). The known causes and risk factors for vitamin D insufficiency include age,⁶² female sex,⁶⁰ proteinuria,⁶² diabetes,⁶³ and impaired 25(OH)D tubular reabsorption.⁶⁴

Our simulations results (summarized in Figure 11) suggested that deficiency of 1,25(OH)₂D₃ has a critical role in maintaining Ca²⁺ homeostasis as its deficiency can cause severe hypocalcemia.⁴⁹ In contrast, deficiency of 1,25(OH)₂D₃ was not predicted to severely impact Mg²⁺ homeostasis, even though low 1,25(OH)₂D₃ levels may reduce intestinal Mg²⁺ absorption in patients with chronic kidney disease. Interestingly, patients with advanced chronic kidney disease (estimated glomerular filtration rate (eGFR) <30 mL/min) were observed to develop hypermagnesemia not hypomagnesemia. That is due to the drastically reduced filtered Mg²⁺ and thus Mg²⁺ excretion, despite an increase in fractional excretion of Mg²⁺. While our simulations of 1,25(OH)₂D₃ deficiency was motivated by the impacts of chronic kidney disease on Ca²⁺ and Mg²⁺ homeostasis, they did not represent impairment in kidney function and thus predicted a drop in plasma Mg²⁺ in advanced chronic kidney disease rather than hypermagnesemia (Figure 9).

In summary, we have developed a computational model of Mg²⁺ and Ca²⁺ homeostasis and their regulation by the calciotropic hormones, PTH and 1,25(OH)₂D₃, in a male rat. The model was used to understand the underlying mechanisms involved in regulating Mg²⁺ and Ca²⁺ balance during dietary Mg²⁺ deficiency, low/high dietary Ca²⁺ with Mg²⁺ deficiency, and vitamin D₃ deficiency.

Limitations of the study

Substantial efforts have been invested in understanding the sex differences in Ca²⁺ regulation and balance.^{4,65,66,67,68} In contrast, much less is known about the sex differences in Mg²⁺ homeostasis, with no sex differences reported in serum Mg²⁺ levels.^69,70 That said, urinary Mg²⁺ excretion appears to be higher in men,⁷¹ an observation that may stem, at least in part, from differences in kidney structure and function between the two sexes. In rodent studies, Veiras et al. characterized major differences in transport capacities across tubular nephron segments in male and female rat kidneys.⁷² Specifically, in the proximal tubule, female rats exhibit heightened NHE3 phosphorylation and relocation to the base of microvilli, where activity is reduced compared to male rats. Consequently, the proximal tubule of female rats reabsorbs a notably smaller portion of filtered Na⁺ in comparison to male rats. Because proximal tubular Mg²⁺ transport is linked to Na⁺ transport, a modeling study⁷³ predicted that the proximal tubule of female rats also reabsorbs a smaller fraction of filtered Mg²⁺ compared to males. The present model is based primarily on a male rat. A worthwhile extension is to develop sex-specific models for whole-body Ca²⁺ and Mg²⁺ regulation, in health and diseases (e.g., chronic kidney disease). In addition, age is also an important factor in Mg²⁺ homeostasis. Aging is often associated with Mg²⁺ deficiency which can result from reduced intestinal absorption, increased urinary excretion due to reduction in kidney function, or inadequate dietary Mg²⁺ intake.⁵⁶ Developing age-specific models will help us better understand the mechanisms involved in Mg²⁺ dyshomeostasis in old age. Another limitation of this model is the data used to estimate the parameters; for instance, parameters representing fractional intestinal absorption of Mg²⁺ were obtained from studies conducted on humans. Also, the model describes transport and regulation mechanisms at the cellular level in the kidneys and intestine by means of simplified relationships, such as first-order kinetics and Michaelis-Menten equations. We have developed detailed epithelial cell-based models of Ca²⁺ and Mg²⁺ transport in a rat kidney^65,73; these could be integrated in the present mathematical model in the future. In addition, Mg²⁺ plays an important role in bone remodeling. Mg²⁺ increases osteoblast proliferation and its deficiency causes increased osteoclast formation and release of inflammatory cytokines leading to significant bone loss.^74,75 Developing a detailed model of bone remodeling by considering the impact of Mg²⁺ on bone resorption and formation may shed light into why severe dietary Mg²⁺ deficiency results in hypercalcemia in rodents and hypocalcemia in other species.

Resource availability

Lead contact

Further information and requests for resources should be directed to the lead contact, Pritha Dutta (p7dutta@uwaterloo.ca).

Materials availability

This study did not generate new unique reagents or other new materials.

Data and code availability

The code generated in this study can be accessed at https://github.com/Pritha17/Magnesium_calcium_homeostasis.⁷⁶

Acknowledgments

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 Canadian Institutes of Health Research (CIHR) Project grant (TNC-174963) to A.T.L.

Author contributions

Conceptualization, P.D. and A.T.L.; methodology, P.D. and A.T.L.; software, validation, formal analysis, and investigation, P.D.; resources, A.T.L.; data curation, P.D.; writing – original draft, P.D. and A.T.L; writing – review and editing, P.D. and A.T.L.; visualization, P.D.; supervision, P.D. and A.T.L.; funding acquisition, A.T.L.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
Computer code	This study	https://github.com/Pritha17/Magnesium_calcium_homeostasis (https://doi.org/10.5281/zenodo.13787641)

Method details

The Mg²⁺ homeostasis model consists of five compartments: plasma, intestine, kidney, parathyroid gland, and bone. The model equations in each compartment are described in the following sections. Model parameters are listed in Table 2. For model equations and parameters related to Ca²⁺ homeostasis refer to Refs.^3,4

Parathyroid gland and parathyroid hormone

PTH secretion and plasma Ca²⁺ and Mg²⁺ levels are regulated by a feedback loop. As plasma Ca²⁺ and Mg²⁺ levels drop, it signals the parathyroid glands to secrete more PTH. PTH stimulates Ca²⁺ and Mg²⁺ reabsorption in the kidney, Ca²⁺ and Mg²⁺ absorption in the intestine, and bone resorption. All these actions increase plasma Ca²⁺ and Mg²⁺ concentrations. The increased levels of these two cations then serve as a negative feedback signal to the parathyroid glands to decrease PTH secretion.

Parathyroid cells sense extracellular Ca²⁺ and Mg²⁺ concentrations through the calcium-sensing receptors (CaSR) on their cell surface.⁷⁷ Ca²⁺ is the main agonist of CaSR and small changes in plasma Ca²⁺ concentration can induce rapid secretion of PTH.⁷⁸ At equimolar concentrations, Mg²⁺ is 1/2 to 2/3 as potent as Ca²⁺ in activating CaSR.^79,80 Mg²⁺ has different impacts on PTH secretion depending on the plasma Ca²⁺ concentration. The combined effect of Ca²⁺ and Mg²⁺ on PTH secretion, as reported in an in vitro study,⁸¹ is represented by model equations as described below.

Change in PTH concentration in the parathyroid gland (PTH_g) is given by

$$\begin{array}{c}\frac{d\left[{PTH}_g\right]}{dt}=\frac{k_{prod}^{PTHg}}{1+{\gamma }_{prod}^{1,25{\left(OH\right)}_2{D}_3}{\left[1,25{\left(OH\right)}_2{D}_3\right]}_p}-\left(k_{\mathit{deg}}^{PTHg}+F\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)\right)\left[{PTH}_g\right]\end{array}$$ (Equation 1)

where $k_{prod}^{PTHg}$ denotes the basal rate of PTH production in the parathyroid gland, ${\gamma }_{prod}^{1,25{\left(OH\right)}_2{D}_3}$ denotes the inhibition of PTH synthesis by ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$, and $k_{\mathit{deg}}^{PTHg}$ denotes the rate constant for PTH_g degradation. The first term on the right-hand-side of Equation 1 denotes the inhibition of PTH_g synthesis by ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ and is adopted from ref.³ This term is based on the findings of Silver et al.⁸² who found an exponentially decreasing relationship between the mRNA expression of the PTH precursor in parathyroid glands and the injected quantity of 1,25(OH)₂D₃. $F\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)$ models the exocytosis of PTH_g regulated by plasma Ca²⁺ and Mg²⁺ and is defined as

$$\begin{array}{c}F\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)=h\left({\left[{Mg}^{2+}\right]}_p\right)\times F1\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)+(1-h\left({\left[{Mg}^{2+}\right]}_p\right))\times F2\left({\left[{Mg}^{2+}\right]}_p\right)\end{array}$$ (Equation 2)

where $h\left({\left[{Mg}^{2+}\right]}_p\right)$ controls the weightage given to each function and depends on the plasma Mg²⁺ concentration. The function $h\left({\left[{Mg}^{2+}\right]}_p\right)$ is defined as

$$\begin{array}{c}h\left({\left[{Mg}^{2+}\right]}_p\right)=\frac{{\left(\frac{{\left[{Mg}^{2+}\right]}_p}{{\left[{Mg}^{2+}\right]}_{thres-PTH}}\right)}^{50}}{{1+\left(\frac{{\left[{Mg}^{2+}\right]}_p}{{\left[{Mg}^{2+}\right]}_{thres-PTH}}\right)}^{50}}\end{array}$$ (Equation 3)

where ${\left[{Mg}^{2+}\right]}_{thres-PTH}$ = 0.4 mM. The function $F1\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)$ is defined as

$$\begin{array}{c}F1\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)={\left(\frac{{\gamma }_{Ca}}{{\left[{Ca}^{2+}\right]}_p}\right)}^{{\gamma }_p}\left({\beta }_{exo}^{PTHg}-\frac{1}{1+{\left(\frac{C_m}{{\left[{Mg}^{2+}\right]}_p}\right)}^2}\right)\text{.}\end{array}$$ (Equation 4)

The above equation is formulated based on observations reported in ref.⁸¹:

• 1.Very high (>=5 mM) ${\left[{Mg}^{2+}\right]}_p$ inhibits PTH secretion at all ${\left[{Ca}^{2+}\right]}_p$.
• 2.At low or moderately low ${\left[{Ca}^{2+}\right]}_p$ (0.8-1 mM), ${\left[{Mg}^{2+}\right]}_p$ (0.5-2 mM) stimulates PTH secretion.
• 3.At normal ${\left[{Ca}^{2+}\right]}_p$ (1.2-1.25 mM), ${\left[{Mg}^{2+}\right]}_p$ (0.5-2 mM) does not have significant effect on PTH secretion; however, if ${\left[{Mg}^{2+}\right]}_p$ is very high (=5 mM), it inhibits PTH secretion.
• 4.At high ${\left[{Ca}^{2+}\right]}_p$ (=1.5 mM), ${\left[{Mg}^{2+}\right]}_p$ inhibits PTH secretion.

The parameter, ${\beta }_{exo}^{PTHg}$, denotes the maximal rate of PTH secretion from the parathyroid gland. ${\left(\frac{{\gamma }_{Ca}}{{\left[{Ca}^{2+}\right]}_p}\right)}^{{\gamma }_p}$ controls the maximum effect of ${\left[{Mg}^{2+}\right]}_p$ on PTH secretion at a specific ${\left[{Ca}^{2+}\right]}_p$, such that the lower the plasma Ca²⁺ concentration, the higher the maximum effect of ${\left[{Mg}^{2+}\right]}_p$ on PTH secretion. $C_m$ controls the slope of the curve. The lower the Ca²⁺ concentration, the steeper the slope, a relation that reflects a significant effect of ${\left[{Mg}^{2+}\right]}_p$ on PTH secretion. The parameters ${\beta }_{exo}^{PTHg}$, ${\gamma }_{Ca}$, ${\gamma }_p$, and $C_m$ were estimated by fitting to experimental data reported in ref.⁸¹ and the comparison between model results and experimental data is given in Figure 2. All parameter descriptions and values are listed in Table 2.

Figure 2.

Comparison between experimental and simulated changes in PTH secretion at different plasma Ca²⁺ and Mg²⁺ concentrations

The lsqcurvefit function of MATLAB, which is a non-linear least-square solver, was used to fit our model simulations to experimental values.

The last term in Equation 2, $F2\left({\left[{Mg}^{2+}\right]}_p\right)$, is defined as

$$\begin{array}{c}F2\left({\left[{Mg}^{2+}\right]}_p\right)=\frac{1}{1+\frac{K_{low-Mg}}{{\left[{Mg}^{2+}\right]}_p}}\text{.}\end{array}$$ (Equation 5)

Equation 5 captures the following observation from ref.⁸³: very low plasma ${\left[{Mg}^{2+}\right]}_p$ (<0.4 mM) inhibits PTH secretion at all ${\left[{Ca}^{2+}\right]}_p$.

The rate of change in plasma PTH concentration (PTH_p) is given by

$$\begin{array}{c}\frac{d\left[{PTH}_p\right]}{dt}=F\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)\frac{V_g}{V_p}\left[{PTH}_g\right]-k_{\mathit{deg}}^{{PTH}_p}\left[{PTH}_p\right]\end{array}$$ (Equation 6)

where $k_{\mathit{deg}}^{PTHp}$ denotes the rate of degradation of PTH_p. V_g and V_p denote the volumes of the parathyroid gland and plasma, respectively. The ratio $\frac{V_g}{V_p}$ takes into account the dilution of PTH in plasma.

Plasma 1,25(OH)₂D₃

Dietary vitamin D₃ is first converted to 25-hydroxy vitamin D₃ (25(OH)D) by 25-hydroxylase in the liver.⁸⁴ 25(OH)D is then carried to the kidney, where it is either converted to the active form of vitamin D₃, 1,25-dihydroxy vitamin D₃ (1,25(OH)₂D₃), by 1α-hydroxylase or the inactive form, 24,25-dihydroxy vitamin D₃ (24,25(OH)₂D₃), by 24-hydroxylase.⁸⁴ Some of the 1,25(OH)₂D₃ is converted to the inactive form, 1,24,25(OH)₂D₃, by 24-hydroxylase.⁸⁴ The conversion from the inactive 25(OH)D to the active 1,25(OH)₂D₃ is regulated by Mg²⁺, Ca²⁺, PTH, and 1,25(OH)₂D₃ (Figure 8).

The plasma concentration of active 1,25(OH)₂D₃ is given by

$$\frac{d{\left[1,25{\left(OH\right)}_2{D}_3\right]}_p}{dt}=\left[k_{conv}^{\mathit{min}}+{\delta }_{conv}^{\mathit{max}}\times f_{PTH-act}\times f_{CaSR-act}\times f_{1,25{\left(OH\right)}_2{D}_3-act}\times f_{Mg-act}\right]{\left[25\left(OH\right)D\right]}_p-\left(k_{\mathit{deg}}^{1,25{\left(OH\right)}_2{D}_3}\times \left(f_{PTH-inact}+f_{Mg-inact}\right)\right){\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$$ (Equation 7)

where $k_{conv}^{\mathit{min}}$ denotes the minimum production rate constant of 1,25(OH)₂D₃, ${\delta }_{conv}^{\mathit{max}}$ denotes the maximum increase in 1,25(OH)₂D₃ production rate, and $k_{\mathit{deg}}^{1,25{\left(OH\right)}_2{D}_3}$ denotes the degradation rate constant of 1,25(OH)₂D₃. PTH promotes the production of 1,25(OH)₂D₃ which is represented by $f_{PTH-act}=\frac{{\left({\left[PTH\right]}_p\right)}^{n_{conv}}}{{\left(K_{conv}^{PTH}\right)}^{n_{conv}}+{\left({\left[PTH\right]}_p\right)}^{n_{conv}}}$. CaSR in the proximal tubule of the kidney inhibits production of 1,25(OH)₂D₃, which is represented by $f_{CaSR-act}=\frac{1}{1+{\gamma }_{conv}^{Ca}{\left[{Ca}^{2+}\right]}_p}$. 1,25(OH)₂D₃ has a self-inhibitory effect on its own production, which is represented by $f_{1,25{\left(OH\right)}_2{D}_3-act}=\frac{1}{1+{\gamma }_{conv}^{1,25{\left(OH\right)}_2{D}_3}{\left[1,25{\left(OH\right)}_2{D}_3\right]}_p}$. The regulation of 1,25(OH)₂D₃ production by Mg²⁺ is represented by the following equation²³:

$$\begin{array}{c}{f}_{Mg-act}=h_m\times f_{Mg-act}^1+\left(1-h_m\right)\times f_{Mg-act}^2\end{array}$$ (Equation 8)

where $h_m$ controls the weightage given to each function and depends on the plasma Mg²⁺ concentration and is defined as $h_m=\frac{1}{{1+\left(\frac{{\left[{Mg}^{2+}\right]}_p}{{\left[{Mg}^{2+}\right]}_{thres-1,25{\left(OH\right)}_2{D}_3}}\right)}^{50}}$. The parameter ${\left[{Mg}^{2+}\right]}_{thres-1,25{\left(OH\right)}_2{D}_3}$ = 2.4 mM. The terms $f_{Mg-act}^1$ and $f_{Mg-act}^2$ are defined as

$$\begin{array}{c}{f}_{Mg-act}^1={\delta }_{Mg-act}\times \frac{{\left({\left[{Mg}^{2+}\right]}_p\right)}^4}{{\left(K_{Mg-act}^1\right)}^4+{\left({\left[{Mg}^{2+}\right]}_p\right)}^4}\end{array}$$ (Equation 9)

$$\begin{array}{c}{f}_{Mg-act}^2={\delta }_{Mg-act}\times \frac{{\left(K_{Mg-act}^2\right)}^4}{{\left(K_{Mg-act}^2\right)}^4+{\left({\left[{Mg}^{2+}\right]}_p\right)}^4}\end{array}$$ (Equation 10)

The parameters ${\delta }_{Mg-act}$, $K_{Mg-act}^1$, and $K_{Mg-act}^2$ were estimated by fitting to experimental data reported in ref.²³ The comparison between model results and experimental data is given in Figure 3.

Figure 3.

Comparison between experimental and simulated changes in ${\left[1,25{\left(OH\right)}_2{D}_3\right]}_p$ at different plasma Mg²⁺ concentrations

The lsqcurvefit function of MATLAB was used to fit our model simulations to experimental values.

The degradation of 1,25(OH)₂D₃ is mediated by 24-hydroxylase. PTH has a negative effect on this enzyme, whereas Mg²⁺ has a positive effect (Figure 10). The effect of PTH is represented by $f_{PTH-inact}=\frac{1}{1+{\gamma }_{inact}^{PTH}{\left[PTH\right]}_p}$ and the effect of Mg²⁺ is represented by $f_{Mg-inact}=\frac{{\left({\left[{Mg}^{2+}\right]}_p\right)}^4}{{\left(K_{D3}\right)}^4+{\left({\left[{Mg}^{2+}\right]}_p\right)}^4}$. Model parameters are given in Table 2.

Proximal tubule of the kidney

About 15-20% of the filtered Mg²⁺ is reabsorbed paracellularly along the proximal tubule. PTH indirectly inhibits Mg²⁺ reabsorption in the proximal tubule by inhibiting the activity of NHE3. Since Na⁺ reabsorption is accompanied by water reabsorption, less water is reabsorbed which reduces the lumen-to-interstitium Mg²⁺ concentration gradient; this results in decreased paracellular reabsorption of Mg²⁺. We model the fractional reabsorption of Mg²⁺ in the proximal tubule as:

$$\begin{array}{c}{\lambda }_{Mg-PT}={\lambda }_{Mg-PT}^0+\frac{{\delta }_{Mg-PT}^{\mathit{max}}}{1+{\left(\frac{{\left[PTH\right]}_p}{{PTH}_{ref}}\right)}^{n_{PT}}}\text{.}\end{array}$$ (Equation 11)

${\lambda }_{Mg-PT}^0$, which denotes the minimal fractional reabsorption of Mg²⁺ in the proximal tubule, is assumed to be 0.185 and ${\delta }_{Mg-PT}^{\mathit{max}}$, which denotes the maximal stimulation of Mg²⁺ reabsorption in the proximal tubule by PTH, is assumed to be 0.015. These parameters yield a maximum value of ${\lambda }_{Mg-PT}$ of 0.20.

Thick ascending limb of the kidney

Along the cortical thick ascending limb, about 60-70% of the filtered Mg²⁺ is reabsorbed, again via the paracellular pathway. In this segment Mg²⁺ reabsorption is upregulated by PTH and downregulated by CaSR through claudin 14 ⁸⁵. Claudin 14 inhibits claudins 16 and 19, which regulate paracellular permeability of Ca²⁺ and Mg²⁺ along the thick ascending limb. Activation of PTH1R (PTH receptor on the basolateral membrane) decreases claudin 14 expression, whereas activation of CaSR increases claudin 14 expression.⁸⁵ The fractional reabsorption of Mg²⁺ along the thick ascending limb is modelled as

$$\begin{array}{c}{\lambda }_{Mg-TAL}={\lambda }_{Mg-TAL}^0+{\delta }_{TAL,CASR}\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)+{\delta }_{TAL,PTH}\left(PTH\right)\end{array}$$ (Equation 12)

where

$$\begin{array}{c}{\delta }_{TAL,CASR}\left({\left[{Ca}^{2+}\right]}_p,{\left[{Mg}^{2+}\right]}_p\right)=\frac{{\delta }_{Mg-CaSR}^{\mathit{max}}}{\left(1+{\left(\frac{{\left[{Ca}^{2+}\right]}_p}{{Ca}_{ref}}\right)}^{n_{TAL}}\right)\left(1+0.6{\left(\frac{{\left[{Mg}^{2+}\right]}_p}{{Mg}_{ref}}\right)}^{n_{TAL}}\right)}\end{array}$$ (Equation 13)

and

$$\begin{array}{c}{\delta }_{TAL,PTH}\left(PTH\right)=\frac{{\delta }_{Mg-PTH}^{\mathit{max}}{\left[PTH\right]}_p}{{\left[PTH\right]}_p+K_{TAL}^{PTH}}\text{.}\end{array}$$ (Equation 14)

${\lambda }_{Mg-TAL}^0$ denotes the minimal fractional reabsorption of Mg²⁺ in this segment and is set to be 0.66.; ${\delta }_{Mg-CaSR}^{\mathit{max}}$, which denotes the maximal stimulation of Mg²⁺ reabsorption by CaSR, is taken to be 0.028; and ${\delta }_{Mg-PTH}^{\mathit{max}}$, which denotes the maximal stimulation of Mg²⁺ reabsorption by PTH, is set to be 0.012. Together these parameters yield a maximum value of ${\lambda }_{Mg-TAL}$ of 0.7.

Distal convoluted tubule of the kidney

Mg²⁺ is reabsorbed transcellularly along the distal convoluted tubule mediated by TRPM6/7 on the apical membrane and Na-Mg exchanger (solute carrier family 41 member 1 (SLC41A1) and/or cyclin M2 (CNNM2)) on the basolateral membrane, with a fractional reabsorption rate of 5-10%. In this segment, Mg²⁺ reabsorption is upregulated by PTH and 1,25(OH)₂D₃. PTH regulates Mg²⁺ uptake through receptor-mediated cAMP release and activation of protein kinase A and 1,25(OH)₂D₃ regulates Mg²⁺ uptake through calbindin-D.⁸⁶ We assume that the contribution of PTH is greater than that of 1,25(OH)₂D₃.³

$$\begin{array}{c}{\lambda }_{Mg-DCT}={\lambda }_{Mg-DCT}^0+{\delta }_{DCT}\left(PTH,D_3\right)\end{array}$$ (Equation 15)

where

$$\begin{array}{c}{\delta }_{DCT}\left(PTH,D_3\right)={\delta }_{Mg-DCT}^{\mathit{max}}\left(0.8\times \frac{{\left[PTH\right]}_p}{{\left[PTH\right]}_p+K_{DCT}^{PTH}}+0.2\times \frac{{\left[1,25{\left(OH\right)}_2{D}_3\right]}_p}{{\left[1,25{\left(OH\right)}_2{D}_3\right]}_p+K_{DCT}^{1,25{\left(OH\right)}_2{D}_3}}\right)\end{array}$$ (Equation 16)

${\lambda }_{Mg-DCT}^0$, which denotes the minimal fractional reabsorption of Mg²⁺ in this segment, is assumed to be 0.08 and ${\delta }_{Mg-DCT}^{\mathit{max}}$, which denotes the maximal stimulation of Mg²⁺ reabsorption by PTH and 1,25(OH)₂D₃, as 0.02. Thus, the maximum value of ${\lambda }_{Mg-DCT}$ is 0.10.

Finally, the total renal reabsorption of Mg²⁺ is defined as

$$\begin{array}{c}{\lambda }_{Mg-reab}={\lambda }_{Mg-PT}+{\lambda }_{Mg-TAL}+{\lambda }_{Mg-DCT}\text{.}\end{array}$$ (Equation 17)

The urinary excretion of Mg²⁺ is defined as

$$\begin{array}{c}{\lambda }_{Mg-urine}={\mathrm{\Phi }}_{GFR}\times {\left[{Mg}^{2+}\right]}_p\times \left(1-{\lambda }_{Mg-reab}\right)\end{array}$$ (Equation 18)

where ${\mathrm{\Phi }}_{GFR}$ denotes the glomerular filtration rate (GFR).

Model parameters are given in Table 2. The fractional reabsorption of Ca²⁺ along the proximal tubule, thick ascending limb, and distal convoluted tubule is modeled similar to ref.³

Intestine

Up to 70% of dietary Mg²⁺ is absorbed in the colon. Intestinal Mg²⁺ absorption has a biphasic, non-linear relationship with luminal Mg²⁺ concentration. Studies suggest that there are at least two intestinal transport systems for Mg²⁺: one dependent on 1,25(OH)₂D₃ and the other independent of 1,25(OH)₂D₃.^18,87

Mg²⁺ absorption by the intestine consists of a non-saturable paracellular component and a saturable transcellular component.^18,87 Paracellular Mg²⁺ absorption is responsible for 11% of intestinal Mg²⁺ uptake.¹⁸ Thus, the equation for fractional absorption of Mg²⁺ along the intestine is formulated as follows. The intestine can absorb at most 70% of the ingested Mg²⁺. We assume that 12% of the ingested Mg²⁺ is absorbed through 1,25(OH)₂D₃ regulation, 11% is absorbed paracellularly, and the rest (47%) is absorbed transcellularly. Thus, fractional absorption of Mg²⁺ along the intestine is given by

$$\begin{array}{c}{\lambda }_{Mg-intestine}=I_{Mg}\left(0.11+0.47\times \frac{V_{active}}{K_{active}+I_{Mg}}+0.12\times f_{1,25{\left(OH\right)}_2{D}_3}^{intestine}\right)\end{array}$$ (Equation 19)

where $f_{1,25{\left(OH\right)}_2{D}_3}^{intestine}=\frac{{\left({\left[1,25{\left(OH\right)}_2{D}_3\right]}_p\right)}^2}{{\left({\left[1,25{\left(OH\right)}_2{D}_3\right]}_p\right)}^2+{\left(K_{abs}^{1,25{\left(OH\right)}_2{D}_3}\right)}^2}$ represents the stimulation of Mg²⁺ by 1,25(OH)₂D₃. $I_{Mg}$ denotes dietary Mg²⁺ intake, $V_{active}$ denotes the maximal rate of active absorption of Mg²⁺, $K_{active}$ denotes the stimulation of active Mg²⁺ absorption by dietary Mg²⁺ intake, and $K_{abs}^{1,25{\left(OH\right)}_2{D}_3}$ denotes the stimulation of Mg²⁺ absorption by 1,25(OH)₂D₃.

Bones

Of the total body magnesium, about 50–60% is found in the bones where it accounts for about 1% of bone ash.^19,88 One third of the bone Mg²⁺ is surface limited and easily exchangeable with plasma (referred to as the fast bone pool) for maintaining a normal extracellular Mg²⁺ concentration.^19,88 The remainder is complexed with the crystalline structure of bone mineral within the hydroxyapatite lattice (referred to as the slow bone pool), which may be released during bone resorption.

The change in the amount of Mg²⁺ in the readily exchangeable fast bone pool ($N_{{Mg}_f}$) is defined as

$$\begin{array}{c}\frac{d{N}_{{Mg}_f}}{dt}=k_{p-f}^{Mg}{\left[{Mg}^{2+}\right]}_p{V}_p-k_{f-p}^{Mg}{N}_{{Mg}_f}-{\tau }_{ac}{N}_{{Mg}_f}\end{array}$$ (Equation 20)

where $k_{p-f}^{Mg}$ denotes the rate of Mg²⁺ uptake from the plasma by the fast bone pool, $k_{f-p}^{Mg}$ denotes the rate of Mg²⁺ release from the fast bone pool to the plasma, and ${\tau }_{ac}$ denotes the rate of accretion into the slow bone pool.

The amount of Mg²⁺ in the slow bone pool ($N_{{Mg}_s}$) varies with time as

$$\begin{array}{c}\frac{d{N}_{{Mg}_s}}{dt}={\gamma }_{ac}^{Mg}{N}_{{Mg}_f}-{\tau }_{res}\left(PTH,1,25{\left(OH\right)}_2{D}_3\right)\text{.}\end{array}$$ (Equation 21)

Bone resorption rate (${\tau }_{res}\left(PTH,1,25{\left(OH\right)}_2{D}_3\right)$) is given by³

$$\begin{array}{c}{\tau }_{res}\left(PTH,1,25{\left(OH\right)}_2{D}_3\right)={\tau }_{res}^{\mathit{min}}+{\delta }_{res}^{\mathit{max}}\left(0.2\times f_{PTH}^{res}+0.8\times f_{1,25{\left(OH\right)}_2{D}_3}^{res}\right)\end{array}$$ (Equation 22)

where $f_{PTH}^{res}=\frac{{\left({\left[PTH\right]}_p\right)}^2}{{\left(K_{res}^{PTHp}\right)}^2+{\left({\left[PTH\right]}_p\right)}^2}$ represents the effect of PTH on bone resorption, and $f_{1,25{\left(OH\right)}_2{D}_3}^{res}=\frac{{\left({\left[1,25{\left(OH\right)}_2{D}_3\right]}_p\right)}^2}{{\left(K_{res}^{1,25{\left(OH\right)}_2{D}_3}\right)}^2+{\left({\left[1,25{\left(OH\right)}_2{D}_3\right]}_p\right)}^2}$ represents the effect of 1,25(OH)₂D₃ on bone resorption. PTH indirectly stimulates osteoclasts (bone cells responsible for resorption) by binding to receptors on osteoblasts (bone-forming cells), which then release receptor activator of nuclear factor kappa-B ligand (RANKL) and osteoprotegerin (OPG) that stimulate osteoclast formation and activity.⁸⁹ In addition, 1,25(OH)₂D₃ enhances bone resorption by promoting the differentiation of osteoclast precursors into mature osteoclasts by increasing the expression of RANKL.⁹⁰

Plasma magnesium

The rate of change of plasma Mg²⁺ concentration is modeled by

$$\frac{d{\left[{Mg}^{2+}\right]}_p}{dt}=\frac{\left(1-{\kappa }_{b-Mg}\right)}{V_p}\left({\lambda }_{Mg-intestine}+{\tau }_{res}\left(PTH,1,25{\left(OH\right)}_2{D}_3\right)+k_{f-p}^{Mg}{N}_{{Mg}_f}-k_{p-f}^{Mg}{\left[{Mg}^{2+}\right]}_p-{\lambda }_{Mg-urine}\right)$$ (Equation 23)

where ${\kappa }_{b-Mg}$ denotes the fraction of magnesium bound to proteins. All model parameters and descriptions are given in Table 2.

Quantification and statistical analysis

The model was implemented in MATLAB. The lsqcurvefit function of MATLAB, which is a non-linear least-square solver, was used to fit our model simulations to experimental values and this is mentioned in the legends of Figures 2 and 3.

Published: September 30, 2024