Precipitation of Inorganic Salts in Mitochondrial Matrix

In the mitochondrial matrix, there are insoluble, osmotically inactive complexes that maintain a constant pH and calcium concentration. In the present paper, we examine the properties of insoluble calcium and magnesium salts, such as phosphates, carbonates and polyphosphates, which might play this role. We find that non-stoichiometric, magnesium-rich carbonated apatite, with very low crystallinity, precipitates in the matrix under physiological conditions. Precipitated salt acts as pH buffer, and, hence, can contribute in maintaining ATP production in ischemic conditions, which delays irreversible damage to heart and brain cells after stroke.


Introduction
Mitochondria are responsible for the adenosine triphosphate (ATP) production in eukaryotic organisms. They are also involved in Ca 2+ signalling, lipid metabolism, heat production, reactive oxygen species production and apoptosis [1]. It is becoming increasingly apparent that mitochondria can accumulate, store and release a large amount of calcium ions under a variety of physiological and pathological conditions such as epilepsy [2], ischemia [3,4] and concussive brain injury [5].
The mitochondrial matrix is surrounded by two membranes. The outer membrane acts as a molecular sieve, while the inner membrane is ion-selective. The major function of the mitochondrial inner membrane is the formation of the electrochemical proton gradient, known as proton motive force [6], utilizing the energy produced by burning fats and sugars. The proton motive force is then used by ATP synthase to synthetize ATP from adenosine diphosphate (ADP) and inorganic phosphate. In the mitochondrial inner membrane there are numerous ion channels and ion exchangers which are involved in the transport of calcium, magnesium, potassium, sodium, protons, phosphates and other ions. The most relevant transporters in the mitochondrial inner membrane are depicted in Scheme 1. Due to the very high electric potential difference (inside negative), the mitochondrial matrix tends to accumulate calcium ions, which form a complex of calcium phosphates and carbonates. Of interest in this report are free calcium and magnesium ions, and the composition of the solid aggregates precipitating in the matrix.
The free concentration of Ca 2+ and Mg 2+ depends on complex chemistry which is not easy to decode because of thermodynamic and kinetic factors. The capacity of isolated mitochondria to accumulate calcium up to total matrix concentration of 0.5 M to 0.8 M, has been established more than Scheme 1. Ion channels, exchangers, pumps and oxidative phosphorylation chain of the inner mitochondrial membrane: green-uniporters, blue-exchangers, yellow-four complexes of oxidative phosphorylation chain, and red-ATP synthasome. There are more than one type of mitoBK Ca , mitoKv or mitoCl channels.
In mitochondria calcium has important physiological role in stimulation of ATP synthesis via activation of several Ca 2+ -sensitive enzymes (pyruvate-, α-ketoglutarate-, isocitrate dehydrogenases), ATP synthase and adenine nucleotide translocase [12,13]. Matrix enzymes are activated by calcium of concentration in the micromolar range, i.e., [Ca 2+ ] matrix = 0.5 to 2 µM [14]. Calcium can be accumulated in or released from the mitochondrion, depending on the proton motive force of the mitochondrion and cytoplasmic free calcium concentration.
In the presence of phosphates, calcium appears to form an insoluble and osmotically inactive complex, which buffer calcium concentration [7]. Amorphous granules containing calcium and phosphorus were identified within the mitochondria of osteoclasts [15], chondrocytes [16,17], osteoblasts [18], osteocytes [19][20][21], calcifying cartilage [22], and mineralizing bone [23][24][25]. It is impossible to investigate the nature of the calcium phosphates in the matrix directly, due to their instant dissociation when mitochondria are disrupted and due to their ability to transform during fixation or drying for chemical analysis [26]. Additionally, it is reasonable to presume that these complexes in the matrix are inert, to some extend at least. Even more importantly, the composition of mitochondrial granules is still largely unknown [27] and it is based on several hypotheses. Thomas and Greenawalt [28] suggested that inorganic granules found in mitochondria are a colloidal, sub-crystalline precursor of calcium-deficient hydroxyapatite. Lehninger [8] and later Nicholls and Chalmers [29] assumed the formation of amorphous tricalcium phosphate, while Kristian et al. [27] suggested the formation of brushite and non-stoichiometric amorphous calcium phosphate with Ca/P ratio equal 1.47. Dołowy, in the most recent hypothesis [30], assumed that brushite turns into isoclasite via all intermediate forms of calcium phosphate minerals forming a polymer-like structure.
The inorganic cations present in the mitochondrial matrix include Na + , K + , Ca 2+ and Mg 2+ . All sodium and potassium salts are soluble. Calcium and magnesium, on the other hand, can form a variety of insoluble salts. The solubility products of calcium and magnesium hydroxides are relatively high (K Ca(OH)2 = 6.5 × 10 −6 and K Mg(OH)2 = 5.6 × 10 −12 ), therefore in considered system Ca(OH) 2 and Mg(OH) 2 will dissociate completely. In this work, the chemical equilibria of numerous compounds, including calcium and magnesium carbonates, phosphates and polyphosphates are intentionally examined. Based on the solubility products and other properties of these salts, we identify the reason for the pH buffering effect, i.e., the calcium hydroxide buffer effect in the mitochondrial matrix.

Mitochondrial pH, Ionic Concentrations and Activities
One must realize the meaning of pH in the systems with very low volumes. The amount of protons can be calculated as n H+ = N A × V × 10 −pH , where N A = 6.02 × 10 −23 is the Avogadro number and V represents volume. The mitochondrion is an organelle with a diameter of c.a. 1 µm, hence the volume of~1 fL. The mitochondrial matrix occupies 90% of mitochondrial volume. With pH mat = 8.2, one obtains 3.4 protons in the matrix. This indicates that water is too weakly dissociated to be a direct donor or acceptor of protons at very low biological volumes (even if one can still define the pH formally) and that protons can be shuttled to reaction centers by other biomolecules, whose concentrations are not constrained by the solubility product of water [31]. These protons may be provided by carbonic or phosphoric acid or dissolution of their salts.
The values of pH and the concentrations of inorganic ions, measured in mitochondria using various methods, are listed in Table 1. The mitochondrion is a complex organelle in which pH and concentrations of ions fluctuate in response to metabolic processes and perturbations of the cytosolic environment [38]. Mitochondrial pH is reported to be in the range of 7.2 to 8.4 with an average value of 7.8 (see Table 1). However, when mitochondria are loaded with succinate, the pH can reach values close to 9 [39]. The concentration of calcium in mitochondria usually varies between 0.17 to 5 µM (see Table 1). However, in the case of mitochondria of firing axon, it can reach values as high as 24 µM [40]. Furthermore, the concentrations of ions are interdependent, e.g., calcium effluxes from mitochondria can depend on the concentration of sodium [41].
The driving force for the transition of dissolved matter from a supersaturated solution to the equilibrium solid phase, is given by the change of standard free energy ∆G 0 = −RTlnS , where S = Q/K sp is supersaturation defined as the ratio of the ion activity product (Q) of constituent ions in solution over its equilibrium solubility product (K sp ). If S > 1, the solution is supersaturated, and precipitation may occur. If S < 1, the solution is undersaturated, and there will be no precipitation.
The activity product of a salt A x B y is defined as: In ideal solution, the activity coefficients of all ions equal one. However, in the concentrated solutions γ i can significantly vary from unity and strongly depends on the ionic strength of the solution defined as: where c i is the concentration of i-th ion and z i represents its charge. It is impossible to precisely calculate the ionic strength of the mitochondrial matrix. Taking into account just the inorganic ions present in the matrix (see Table 1) and assuming similar concentration of inorganic ions, one can assume I of the matrix to be in the range of 0.4 to 1 M. The extended Debye-Hückel can predict the activity coefficient in a solution with an ionic strength up to 100 mM and Davies equation is valid up to an ionic strength of 500 mM [42]. In this work, we use the modified Davies  εRT , F is the Faraday constant (96,488.46 C/mol), R is the ideal gas constant (8.3143 J/mol/K), T is the temperature (310 K), e 0 stands for the electronic charge (1.602 × 10 −19 C), ε = ε 0 ε r is the permittivity of the medium (ε 0 = 8.854 × 10 −12 F/m, ε r = 80 for water), and a i represents the radius of hydrated ion (values according to Kielland [70]). As presented in Figure 1, in the ionic strengths I > 1 mM, the ideal solution assumption (γ i = 1) is not valid. In strong electrolytes, I > 200 mM, the activity coefficients (γ i 1) remain almost constant. In further calculations, the activity coefficients obtained for I = 0.7 M will be used. fluorescent protein; HEK-293-human embryonic kidney cells; Fura-2-calcium indicator: C29H22N3O14K5; HeLa-human cervix-carcinoma cells; Indo-1-calcium indicator: C32H31N3O12; MDCK-Madin-Darby Canine Kidney; PBFI-potassium-binding benzofuran isophthalate; SBFIsodium-binding benzofuran isophthalate; SNARF-seminaphthorhodafluors fluorescence; YC2-Yellow Cameleon protein; YFP-yellow fluorescent protein; ai represents the radius of hydrated ion (values according to Kielland [70]).
As presented in Figure 1, in the ionic strengths I > 1 mM, the ideal solution assumption (γi = 1) is not valid. In strong electrolytes, I > 200 mM, the activity coefficients (γi ≠ 1) remain almost constant. In further calculations, the activity coefficients obtained for I = 0.7 M will be used.

Precipitation of Carbonates
Calcium and magnesium carbonates may form in the mitochondrial matrix due to the presence of carbon dioxide produced in citric acid cycle. Although the measurements of partial pressure of CO2 inside mitochondria is not yet possible with existing methods, it might be assumed to be equal or slightly higher than the partial pressure in blood. The partial pressure of carbon dioxide measured in human mixed venous blood is pCO2 = 6.1 kPa [71] to 6.3 kPa [72], however during exercises it can raise to 7.8 kPa and after crossing lactate acidosis threshold it can reach values as high as 10.4 kPa [72].
The amount of gas dissolved in a solution is proportional to its partial pressure in the gas phase [73]. The proportionality factor is called the Henry's law constant. Nevertheless, it should be kept in mind that its value still depends on certain parameters, e.g., temperature and the ionic strength of the solution. The Henry's law constant for carbon dioxide dissolved in water at 25 °C equals KH = 3.4·10 −4 mol·m −3 ·Pa −1 [74]. It lowers with the increasing temperature reaching 2.5 × 10 −4 mol·m −3 ·Pa −1 at 37 °C [75].

Precipitation of Carbonates
Calcium and magnesium carbonates may form in the mitochondrial matrix due to the presence of carbon dioxide produced in citric acid cycle. Although the measurements of partial pressure of CO 2 inside mitochondria is not yet possible with existing methods, it might be assumed to be equal or slightly higher than the partial pressure in blood. The partial pressure of carbon dioxide measured in human mixed venous blood is pCO 2 = 6.1 kPa [71] to 6.3 kPa [72]. However, during exercises, it can rise to 7.8 kPa and after crossing the lactate acidosis threshold, it can reach values as high as 10.4 kPa [72].
The amount of gas dissolved in a solution is proportional to its partial pressure in the gas phase [73]. The proportionality factor is called the Henry's law constant. Nevertheless, it should be kept in mind that its value still depends on certain parameters, e.g., temperature and the ionic strength of the solution. The Henry's law constant for carbon dioxide dissolved in water at 25 • C equals K H = 3.4 × 10 −4 mol·m −3 ·Pa −1 [74]. It lowers with the increasing temperature reaching 2.5 × 10 −4 mol·m −3 ·Pa −1 at 37 • C [75].
The hydration of carbon dioxide to form carbonic acid is slow to attain equilibrium below pH = 8 in pure systems. In biological systems the hydration of carbon dioxide is catalysed by carbonic anhydrase (a Zn-containing enzyme). Only a small portion of the aqueous carbon dioxide exists in its hydrated form as H 2 CO 3 . Hence, the relation between the carbonic species is usually expressed as: where the equilibrium constants in pure water solutions at 25 • C equal K 1 = 4.5 × 10 −7 , K 2 = 4.7 × 10 −11 [76]. These constants are temperature-dependent. Several different empirical equations have been developed to describe the relation between the equilibrium constants and temperature, including ones determined by Harned and co-workers [77][78][79], Ryzhenko [80], Millero and co-workers [81][82][83], as well as Plummer and Busenberg [75]. The values calculated using the latter at 37 • C, i.e., K 1 = 4.96 × 10 −7 and K 2 = 5.73 × 10 −11 are used in this work.
It should be noted, that mentioned equations are valid only for the solutions of carbon dioxide in pure water. The presence of other ions in the system lead to the increase of K 1 and K 2 , e.g., values of equilibrium constants measured in sea water at 40 • C can reach K 1 = 2.1 × 10 −6 and K 2 = 2.3 × 10 −10 [83], comparing with calculated K 1 = 5.1 × 10 −7 and K 2 = 6.0 × 10 −11 . In biological systems they can vary significantly and are treated not as true thermodynamic constants, but rather as apparent dissociation constants determined experimentally from the measurements of pH, pCO 2 and dissolved CO 2 [84]. Several studies [85][86][87][88][89] revealed the variability of K 1 in human plasma in the range of 4.78 × 10 −7 to 1.58 × 10 −6 . A case study of a diabetic child in ketoacidosis showed a change of K 1 from 3.24 × 10 −6 to 9.54 × 10 −7 after just 7 h of treatment [ Although the first attempts to quantify the solubility of calcium carbonates in aqueous systems date back to the middle of the 19 th century [91,92] and despite the many efforts with current advanced ionic interaction models available, there are still no universally accepted values of the solubility constants of the calcium carbonate forms [93]. Values of the solubility products for various forms of CaCO 3 and their activity products are presented in Table 2. Using these values, one can calculate the equilibrium concentration of free calcium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Not all of the salts listed above can be precipitated directly from water solutions. Barringtonite is formed as the result of cold meteoric water percolating through olivine basalt and leaching magnesium from it [104]. Artinite is found along with hydromagnesite and other minerals as a low-temperature alteration product, as veinlets or crusts in serpentinized ultrabasic rocks [103]. Pokrovskite occurs within ultramafic bodies of dunite or serpentinite [105], and can be formed as an intermediate product in the thermal decomposition of artinite [106]. Shelkovite is a substance of anthropogenic origin, i.e., a product of burning coal mine dump [103].
Formation of the most stable anhydrous magnesite (MgCO 3 ) is strongly kinetically inhibited because of the high hydration energy of Mg 2+ [107,108]. The precipitation of magnesite is apparently inhibited at temperatures of less than 80 • C [109][110][111]. Therefore, various hydrous magnesium carbonates form instead. In water solutions, landsfordite, nesquehonite and hydromagnesite can precipitate, depending on the temperature. Lansfordite decomposes into nesquehonite at temperatures above 10 • C [112,113]. Most commonly, only the mineral nesquehonite can be precipitated from aqueous solutions at 25 • C and partial pressure of CO 2 close to the ambient pressure or below [100,[112][113][114][115][116]. At higher temperatures, i.e., above approximately 40 • C, various basic Mg-carbonates are usually formed by precipitation, mostly in the form of hydromagnesite [112,114,116,117].
The transformation from nesquehonite to hydromagnesite is unlikely to follow a single simple sequential reaction pathway [107] and a range of intermediate phases may form during this transformation [99]. Giorgiosite is a very poorly described basic magnesium carbonate [103,125] and it is considered as a metastable phase between nesquehonite and hydromagnesite [103]. Protohydromagnesite, a phase similar to dypingite, appears only as a metastable intermediate phase [114,126]. One intermediate phase is of particular interest, namely dypingite, a mineral that is known to be produced by cyanobacteria from alkaline wetlands [127]. Octahydrate Mg 5 (CO 3 ) 4 (OH) 2 ·8H 2 O is often considered a form of dypingite [120].
The reactions leading to the formation of the relevant magnesium carbonates together with the values of respective solubility products at 25 • C are summarized in Table 3. Several values obtained at 35 • C are also included in Table 3. Using these values, one can calculate the equilibrium concentration of free magnesium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Table 3. The precipitation of insoluble magnesium carbonates. The activity products (Q) and the solubility products K sp (at 25 • C) are obtained from the literature.

Formula
Activity Product (Q) Solubility Product, K sp at 25 Hydromagnesite * Values used in the calculations presented in this work; † K sp obtained at 35 • C.

Precipitation of Orthophosphates
Orthophosphate salts containing the PO 4 3− group are distinguished from metaphosphates and pyrophosphates that contain PO 3 − and P 2 O 7 4− groups, respectively. The PO 4 3− group takes part in ATP synthesis and occurs inside the mitochondrion in relatively high concentrations. Hence, the formation of metaphosphates and pyrophosphates will not be taken into consideration. The concentration of phosphate ions is kept constant by the phosphate carrier while the relationship between them is given as: where K 1 = 7.41 × 10 −3 , K 2 = 6.31 × 10 −8 and K 3 = 4.07 × 10 −13 [128].
Some of the calcium salts mentioned above, including tetracalcium phosphate, oxyapatite, α-tricalcium phosphate and β-tricalcium phosphate, cannot be precipitated from aqueous solutions [131]. Anhydrous monocalcium phosphate and anhydrous dicalcium phosphate are stable only in temperatures above 100 • C, while monocalcium phosphate monohydrate is stable only at pH < 2 [131]. Energy dispersive x-ray spectra from mitochondrial precipitates do not show the presence of fluor [27]. Hence, the formation of fluoroapatite can be ruled out as well.
The formulas of the calcium phosphates that might precipitate in the mitochondrion matrix, together with the values of respective solubility products at 37 • C are summarized in Table 4 (some of the literature values have been recalculated to correspond to the activity product given in the table). To calculate the equilibrium concentration of chloroapatite and carbonated apatite, the concentrations of chlorides and carbonates have to be provided. The free chloride concentration in the mitochondrial matrix equals [Cl − ] = 4.2 mM [69]. The concentration of carbonate ions [CO 3 2− ] are calculated using Equation (4), assuming that partial pressure of carbon dioxide equals pCO 2 = 7 kPa.
It has been shown that increased levels of ammonia disturb mitochondrial function, induce oxidative stress, causes mitochondrial membrane potential collapse and negatively affects several key enzymes responsible for energy metabolism [145][146][147][148]. Although the mitochondria are responsible for the production of NH 4 + ions during the oxidation of glutamate [149], in healthy mitochondria the concentration of ammonia is very low. Hence, the formation of struvite can be ruled out. Furthermore, in the biological precipitation of struvite, Ca 2+ ions make struvite precipitation difficult and even inhibit it [150,151]. The reactions leading to the formation of the relevant magnesium phosphates together with the values of respective solubility products at 38 • C are summarized in Table 5. Using these values, the equilibrium concentration of phosphates [ P i ] eq , required for the formation of each magnesium orthophosphate can be calculated.

Precipitation of Polyphosphates
It has been reported that PO 4 3− can be concentrated and stored by biochemical polymerization into polyphosphate (polyP) ions [157]. They are negatively charged polyanions (P n O 3n+1 ) (n+2)− with great affinity for calcium and other multivalent cations [158]. PolyP is a macroergic compound so that the energy of phosphoanhydride bond hydrolysis is the same as in ATP [159]. Polyphosphates have been found in all organisms ranging from bacteria to mammals [160]. The synthesis of polyP is mainly realized by the polyphosphate kinase, which catalyses the reaction ATP + [PO 3 − ] n ↔ ADP + [PO 3 − ] n+1 . PolyP can play a role in the energetic metabolism of mitochondria [161], e.g., it has been shown that, in human cell lines, specific reduction of mitochondrial polyP under expression of yeast exopolyphosphatase PPX1 significantly modulates mitochondrial bioenergetics and transport [159]. Polyphosphates (including pyrophosphate) have also a significant role in the regulation of the intensity of several biosynthetic processes in mitochondrial and cell energy metabolism [162,163]. PolyP can form several insoluble salts. Bobtelsky and Kertes [164], have analysed the properties of pyrophosphates (P 2 O 7 ) 4− and tripolyphosphates (P 3 O 10 ) 5− and the precipitation of calcium, magnesium, barium and strontium salts of these ions. With the addition of calcium to the respective polyP solution, the insoluble Ca 2 P 2 O 7 and Ca 5 (P 3 O 10 ) 2 will form, while the addition of magnesium leads to the precipitation of Na 2 Mg 3 (P 2 O 7 ) 2 and NaMg 2 P 3 O 10 , respectively. Interestingly, the redissolution of these salts will take place if the excess of the polyphosphate ions is added to the solution [164].
Polyphosphate chelation of calcium reduces the free calcium concentration [165] and produces a neutral, amorphous, [Ca(PO 3 ) 2 ] n complex. Discrete granules containing calcium-polyP complexes have been observed e.g., in acidocalcisomes, the electron-dense acidic organelles that have been conserved from bacteria to humans [166]. PolyP are noted to be highly inhibitory to calcium phosphate nucleation and precipitation [167], e.g., they inhibit crystalline calcium hydroxyapatite nucleation and growth from solution [168]. However, in basic pH solutions calcium polyphosphate glass releases hydrated calcium and polyphosphate ions into solution and, through hydrolytic degradation of polyP, calcium polyphosphate eventually transforms into hydroxyapatite [169]. The relationships between polyphosphate chemistry and apatite biomineralization as well as the possible role of polyP in mitochondria have been discussed in detail by Omelon et al. [165,170].
The important polyP forms in living cells are specific complexes with polyhydroxybutyrate (PHB) and Ca 2+ , which have been found in the membranes of many organisms, including membranes of mitochondria of animal cells [171,172]. The PHB forms a helix with a lipophilic shell of methyl groups and polar lining of ester carbonyl oxygens surrounding a core helix of polyP with Ca 2+ bridging the two polymers. These helices can form voltage-dependent, Ca 2+ -selective channels, or be components of ion-conducting proteins: namely, the human erythrocyte Ca 2+ -ATPase pump and the Streptomyces lividans potassium channel [173].
The polyP pool in mitochondria likely consists of two parts, i.e., one part localized directly in the membranes as a polyP-PHB-Ca 2+ complex, and the other part present in the intermembrane space and not bound to PHB, though its association with cations, including Ca 2+ , is not improbable [159]. It was determined by subfractionation of isolated mitochondria by mild osmotic treatment, that intermembrane space contained~90% of total mitochondrial polyP [174]. This suggests that polyphosphates are formed outside the mitochondrial matrix. Furthermore, the formation of calcium polyP inside the matrix seems less likely than calcium orthophosphates, because the Ca/P ratio of this complex is less than 1, while the mitochondrial Ca/P ratio is between 1 and 2.

Precipitation of Calcium Carbonates
Using the values presented in Table 2, one can calculate the equilibrium concentration of free calcium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Such calculations are shown in Figure 2.

Precipitation of Calcium Carbonates
Using the values presented in Table 2, one can calculate the equilibrium concentration of free calcium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Such calculations are shown in Figure 2. The more stable crystalline forms are generally those with lower solubility product, hence lower equilibrium concentration of calcium. However, the formation and transformation of solid phases during the spontaneous precipitation of calcium carbonate from alkaline, carbonate rich solutions is a highly complicated, and in general, time-dependent process. The local conditions of temperature, concentration and impurities can have a profound effect on the qualitative nature of the phases formed, as well as on the quantitative kinetics of individual processes.
The Ostwald-Lussac Law of Stages states that in the pathway to the final crystalline state will pass through all less stable states in order of increasing stability [175]. The formation of metastable phases is a quite common phenomenon during spontaneous precipitation of supersaturated solutions [98,176,177]. During the precipitation of more stable anhydrous forms, amorphous calcium carbonate, calcium carbonate hexahydrate and calcium monohydrocalcite appear as intermediate phases of importance and consequence. At high supersaturation, the first-formed phase is ACC, a rather unstable precursor consisting of spherical particles of calcium carbonate [94,98]. ACC changes to more stable forms via dissolution and precipitation. Solubility product of ACC is higher than for other forms of CaCO3, hence any solution in equilibrium with the amorphous phase will be highly supersaturated with respect to the crystalline phases, leading to nucleation controlled growth [178]. In other words, ACC serves as precursor for other polymorphs, the nucleation of the polymorphous The more stable crystalline forms are generally those with a lower solubility product, hence lower equilibrium concentration of calcium. However, the formation and transformation of solid phases during the spontaneous precipitation of calcium carbonate from alkaline, carbonate-rich solutions is a highly complicated and time-dependent process. The local conditions of temperature, concentration and impurities can have a profound effect on the qualitative nature of the phases formed, as well as on the quantitative kinetics of individual processes.
The Ostwald-Lussac Law of Stages states that, in the pathway to the final crystalline state, will pass through all less stable states in order of increasing stability [175]. The formation of metastable phases is a quite common phenomenon during spontaneous precipitation of supersaturated solutions [98,176,177]. During the precipitation of more stable anhydrous forms, amorphous calcium carbonate, calcium carbonate hexahydrate and calcium monohydrocalcite appear as intermediate phases of importance and consequence. At high supersaturation, the first-formed phase is ACC, a rather unstable precursor consisting of spherical particles of calcium carbonate [94,98]. ACC changes to more stable forms via dissolution and precipitation. The solubility product of ACC is higher than for other forms of CaCO 3 . Hence, any solution in equilibrium with the amorphous phase will be highly supersaturated with respect to the crystalline phases, which leads to nucleation-controlled growth [178]. In other words, ACC serves as a precursor for other polymorphs, the nucleation of the polymorphous crystals occurs in the vicinity of the clouded precursor and the crystals grow by consuming the precursors surrounding the crystals [179].
If magnesium ions are present in the reaction system, they become incorporated into the ACC structure. The crystalline phase, which forms from the amorphous precursor, is controlled by the magnesium content of the precursor. Pure crystallises to vaterite [180], ACC containing 10% Mg crystallises to calcite [181], ACC with 30% crystallises to monohydrocalcite [178,182] and ACC with 50% Mg crystallises to protodolomite/dolomite (at temperatures higher than 60 • C) [183].
High concentrations of orthophosphates prevent the crystallization of the more stable anhydrous forms of CaCO 3 , which leads to the occurrence of ikaite and monohydrocalcite in the system [94,98]. Ikaite, however, for its formation requires near-freezing conditions and is unstable in the temperatures above 25 • C [184]. Soluble acidic proteins occurring in the mitochondrial matrix might also have a strong inhibiting effect on the nucleation of anhydrous polymorphs, e.g., it has been shown that poly-l-aspartic acid completely inhibits the formation of varietite [185]. Na + and Cl − ions exert a weak or no influence on the precipitation of carbonates [186,187].
The presence of both magnesium and orthophosphate ions in mitochondrial matrix suggests that monohydrocalcite might be the phase occurring inside mitochondria. However, the solubility calculations presented in Figure 2, clearly show that monohydrocalcite can form only if matrix pH is greater than 7.8 (for [Ca 2+ ] = 5 µM) or even 8.6 (for [Ca 2+ ] = 0.17 µM). In lower pH, neither monohydrocalcite, nor any other calcium carbonate will precipitate.

Precipitation of Magnesium Carbonates
Using the solubility products presented in Table 3, one can calculate the equilibrium concentration of free magnesium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Solubility calculations, presented in Figure 3, clearly show that nesquehonite is the most stable magnesium carbonate in given conditions. It is also the only magnesium carbonate that can precipitate in the mitochondrial matrix, but only if the matrix pH is greater than 7.5 (for [Mg 2+ ] = 1.5 mM) or even 7.9 (for [Mg 2+ ] = 0.35 mM). In lower pH, neither nesquehonite, nor any other magnesium carbonate will precipitate.
temperatures above 25 °C [184]. Soluble acidic proteins occurring in mitochondrial matrix might also have a strong inhibiting effect on the nucleation of anhydrous polymorphs, e.g., it has been shown that poly-L-aspartic acid completely inhibits the formation of varietite [185]. Na + and Cl − ions exert a weak or no influence on the precipitation of carbonates [186,187].
Presence of both magnesium and orthophosphate ions in mitochondrial matrix suggests that monohydrocalcite might be the phase occurring inside mitochondria. However, the solubility calculations presented in Figure 2, clearly show that monohydrocalcite can form only if matrix pH is greater than 7.8 (for [Ca 2+ ] = 5 μM) or even 8.6 (for [Ca 2+ ] = 0.17 μM). In lower pH, neither monohydrocalcite, nor any other calcium carbonate will precipitate.

Precipitation of Magnesium Carbonates
Using the solubility products presented in Table 3, one can calculate the equilibrium concentration of free magnesium in the solutions of respective salts as functions of pH and partial pressure of carbon dioxide. Solubility calculations, presented in Figure 3, clearly show that nesquehonite is the most stable magnesium carbonate in given conditions. It is also the only magnesium carbonate that can precipitate in the mitochondrial matrix, but only if the matrix pH is greater than 7.5 (for [Mg 2+ ] = 1.5 mM) or even 7.9 (for [Mg 2+ ] = 0.35 mM). In lower pH, neither nesquehonite, nor any other magnesium carbonate will precipitate.

Precipitation of Calcium Orthophospates
The total concentration of phosphates [∑Pi]eq, required for the formation of each salt at constant free calcium concentration [Ca 2+ ] = 0.17μM and [Ca 2+ ] = 5 μM, is presented in Figure 4. Salts with the lowest [∑Pi]eq are thermodynamically most stable and therefore dominate in the system, i.e., whenever the total concentration of free phosphates exceeds the lowest [∑Pi]eq, formation of the

Precipitation of Calcium Orthophospates
The total concentration of phosphates [ P i ] eq , required for the formation of each salt at constant free calcium concentration [Ca 2+ ] = 0.17 µM and [Ca 2+ ] = 5 µM, is presented in Figure 4. Salts with the lowest [ P i ] eq are thermodynamically most stable and therefore dominate in the system, i.e., whenever the total concentration of free phosphates exceeds the lowest [ P i ] eq , formation of the respective salt will occur, keeping [ P i ] at a constant level. As shown in Figure 4, [30]. The calculated values of [ P i ] mat compared with the [ P i ] eq of the respective salts (see Figure 4b) clearly show that if the calcium concentration is elevated, the solution of the mitochondrion matrix (pH = 7.4 to 8.2) is supersaturated with respect to up to three salts. i.e., hydroxyapatite, chloroapatite and carbonated apatite. Other salts (OCP, brushite, CDHA and ACP) having [ P i ] eq > [ P i ] mat will dissolve, hence cannot form inside mitochondria.
Calcium apatites (hydroxyapatite, chloroapatite, fluoroapatite and carbonated apatite) are members of a large family of more than 75 chemically different apatites, i.e., compounds with the chemical formula: A 2 B 3 (XO 4 ) 3 Y, where A and B are bivalent cations, and XO 4 and Y are trivalent and monovalent anions, respectively [188]. The high-symmetry members of the apatite family crystallise in the hexagonal system.
Calcium apatites are very often non-stoichiometric, due to the substitution of OH − in hydroxyapatite with Cl − , F − and CO 3 2− . Therefore, the formulas for carbonated apatite and chloroapatite are often given as Ca 10 (PO 4 ) 6 (CO 3 ) x (OH) 2−2x and Ca 10 (PO 4 ) 6 Cl x (OH) 2−x , respectively [139,140]. The apatite structure allows varied substitutions of calcium and phosphate ions to take place without a significant alteration in its basic structure, e.g., one of the basic phosphorite minerals, known as francolite, has a variable chemical composition, which can be represented by (Ca,Mg,Sr,Na) 10 (PO 4 ,SO 4 ,CO 3 ) 6 F 2+x , where 0 < x < 1 [189,190].  [30]. The calculated values of [∑Pi]mat compared with the [∑Pi]eq of the respective salts (see Figure 4b) clearly show that if the calcium concentration is elevated, the solution of mitochondrion matrix (pH = 7.4 to 8.2) is supersaturated with respect to up to three salts. i.e., hydroxyapatite, chloroapatite and carbonated apatite. Other salts (OCP, brushite, CDHA and ACP) having [∑Pi]eq > [∑Pi]mat will dissolve, hence cannot form inside mitochondria. Calcium apatites (hydroxyapatite, chloroapatite, fluoroapatite and carbonated apatite) are members of a large family of more than 75 chemically different apatites, i.e., compounds with the chemical formula: A2B3(XO4)3Y, where A and B are bivalent cations, and XO4 and Y are trivalent and monovalent anions, respectively [188]. The high-symmetry members of the apatite family crystallise in the hexagonal system. Calcium apatites are very often non-stoichiometric, due to the substitution of OH − in hydroxyapatite with Cl − , F − and CO3 2− , therefore the formulas for carbonated apatite and chloroapatite are often given as Ca10(PO4)6(CO3)x(OH)2 − 2x and Ca10(PO4)6Clx(OH)2−x, respectively [139,140]. The apatite structure allows varied substitutions of calcium and phosphate ions to take place without a significant alteration in its basic structure, e.g., one of the basic phosphorite minerals, known as francolite, has a variable chemical composition which can be represented by (Ca,Mg,Sr,Na)10(PO4,SO4,CO3)6F2+x, where 0 < x < 1 [189,190].
Formation of CO3 2− -substituted hydroxyapatite is common in biological and geological systems and this substitution has been well documented in several studies [191][192][193]. Carbonate ion in the structure of biological apatites can substitute either the OH − ion (type A carbonated apatite) or PO4ion (type B carbonated apatite) [194] and these two locations of carbonate species can be distinguished using spectroscopic methods [195,196]. The carbonate group distorts the apatite lattice and causes the resulting solid to be considerably more soluble than hydroxyapatite [197]. The substitution of carbonate for phosphate would also reduce the crystallinity and precipitation rate of HA [198]. Another specificity of biological apatites is the presence of hydrogen phosphate (HPO4 2-) ions in PO4 3sites [199][200][201]. These two types of bivalent ions substituting for PO4 3-have been shown to correspond to the formation of calcium deficient apatites, i.e., the loss of a negative charge due to these substitutions is compensated by the creation of cationic vacancies and anionic vacancies in the OH − site [202]. The formula for biological apatites taking into account the possible existence of both type A and B substitutions reads [203,204]: Ca10-x(PO4)6-x(HPO4 or CO3)x(OH or ½CO3)2-x with 0 < x < 2.
(6) Formation of CO 3 2− -substituted hydroxyapatite is common in biological and geological systems and this substitution has been well documented in several studies [191][192][193]. Carbonate ion in the structure of biological apatites can substitute either the OH − ion (type A carbonated apatite) or PO 4 3− ion (type B carbonated apatite) [194] and these two locations of carbonate species can be distinguished using spectroscopic methods [195,196]. The carbonate group distorts the apatite lattice and causes the resulting solid to be considerably more soluble than hydroxyapatite [197]. The substitution of carbonate for phosphate would also reduce the crystallinity and precipitation rate of HA [198]. Another specificity of biological apatites is the presence of hydrogen phosphate (HPO 4 2− ) ions in PO 4 3− sites [199][200][201].
These two types of bivalent ions substituting for PO 4 3− have been shown to correspond to the formation of calcium deficient apatites, i.e., the loss of a negative charge due to these substitutions is compensated by the creation of cationic vacancies and anionic vacancies in the OH − site [202]. The formula for biological apatites taking into account the possible existence of both type A and B substitutions reads [203,204]: This general formula can be used, e.g., to represent the bone composition at all ages in many vertebrates (x = 1.7) or a typical composition of human tooth enamel (x = 0.6) [203]. The Ca/P ratio of carbonated apatite vary from 2.0 (x = 2 and all type B substitutions with CO 3 2− ) down to 1.2 (x = 2 and all type B substitutions with HPO 4 2− ) or even lower if cationic substitutions (2Na + or Mg 2+ for Ca 2+ ) occur. Precipitated apatite crystals generally exhibit a structured hydrated surface layer containing mainly bivalent cations and anions. This feature should not be confused with the Stern double layer that forms on most mineral surfaces. The substitution possibilities in the hydrated layer are not well known but seem to be greater than in the core of the apatite structure [203]. Magnesium ions for example, which hardly penetrate the apatite lattice, are easily and reversibly incorporated into the hydrated layer in large amounts [205]. Ion substitutions in the hydrated layer considerably modify its structure even though these alterations seem reversible in most cases [206].

Precipitation of Magnesium Orthophospates
The solubility calculations for magnesium orthophosphates at low magnesium concentration [Mg 2+ ] = 0.35 mM (see Figure 5a) clearly show that solution of mitochondrial matrix is undersatured with the respect to all magnesium phosphate salts. Hence, they cannot precipitate. At high magnesium concentrations [Mg 2+ ] = 1.5 mM, the precipitation of babierrite is possible if pH > 7.9 (see Figure 5b). In lower pH, neither babierrite, nor any other magnesium phosphate will precipitate.
Precipitated apatite crystals generally exhibit a structured hydrated surface layer containing mainly bivalent cations and anions. This feature should not be confused with the Stern double layer that forms on most mineral surfaces. The substitution possibilities in the hydrated layer are not well known but seem to be greater than in the core of the apatite structure [203]. Magnesium ions for example, which hardly penetrate the apatite lattice, are easily and reversibly incorporated into the hydrated layer in large amounts [205]. Ion substitutions in the hydrated layer considerably modify its structure although these alterations seem reversible in most cases [206].

Precipitation of Magnesium Orthophospates
The solubility calculations for magnesium orthophosphates at low magnesium concentration [Mg 2+ ] = 0.35 mM (see Figure 5a) clearly show that solution of mitochondrial matrix is undersatured with the respect to all magnesium phosphate salts, hence they cannot precipitate. At high magnesium concentrations [Mg 2+ ] = 1.5 mM the precipitation of babierrite is possible if pH > 7.9 (see Figure 5b). In lower pH, neither babierrite, nor any other magnesium phosphate will precipitate.

Discussion
The results of solubility calculations (presented in  show that four different salts can form in the mitochondrion, depending on the changes in ionic concentrations induced by the metabolic processes in mitochondria and by the concentration changes in cytosol. Babierrite can precipitate at high magnesium concentration and pH > 9; nesquehonite precipitates at pH > 7.5 (or pH > 7.8 if calcium concentration is low); monohydrocalcite precipitates at high Ca 2+ concentrations and pH > 7.8; carbonated apatite at high calcium concentrations precipitates in whole mitochondrial pH range, and dissolves when mitochondrial [Ca 2+ ] is low. However, these calculations do not take into account the influence of magnesium ions on precipitation of calcium salts and vice versa. They also neglect the influence of phosphate ions on the precipitation of carbonate salts and the influence of carbonate ions on the precipitation of phosphate salts. These relations are discussed below.

Discussion
The results of solubility calculations (presented in  show that four different salts can form in the mitochondrion, depending on the changes in ionic concentrations induced by the metabolic processes in mitochondria and by the concentration changes in cytosol. Babierrite can precipitate at high magnesium concentration and pH > 9. Nesquehonite precipitates at pH > 7.5 (or pH > 7.8 if calcium concentration is low). Monohydrocalcite precipitates at high Ca 2+ concentrations and pH > 7.8. Carbonated apatite at high calcium concentrations precipitates in a whole mitochondrial pH range, and dissolves when mitochondrial [Ca 2+ ] is low. However, these calculations do not take into account the influence of magnesium ions on precipitation of calcium salts and vice versa. They also neglect the influence of phosphate ions on the precipitation of carbonate salts and the influence of carbonate ions on the precipitation of phosphate salts. These relations are discussed below.
The hydration shell of magnesium is more strongly bound than the hydration shell of calcium [108,212]. Hence, the energy required to desolvate magnesium is higher than that of calcium. That explains the high temperatures required to crystallize anhydrous Mg-Ca carbonates, i.e., temperature of 60 • C is required for the formation of dolomite [183] and repeated trials to precipitate it under laboratory conditions at room temperature were unsuccessful [213,214]. Another anhydrous carbonate with two cations, namely eitelite with the formula Na 2 Mg(CO 3 ) 2 , can also be precipitated only at temperatures above 60 • C [215].
The hydrated nature of monohydrocalcite means that full dehydration of Mg 2+ is not required before incorporation into the crystal lattice. Therefore, it will more likely form than the anhydrous calcium carbonate phases [178]. Monohydrocalcite has a high capacity to accommodate magnesium within its structure. The coordination number of Ca 2+ in monohydrocalcite is 8 [216,217] and Mg 2+ does not take a coordination number other than 6. Therefore, Mg 2+ is not incorporated into the monohydrocalcite structure, directly but forms discrete hydrated magnesium carbonate insertions. According to the solubility calculations presented in Figures 2 and 3, the solution in mitochondrial matrix is saturated with respect to both monohydrocalcite and nesquehonite, if pH > 8. It is in agreement with the observations of Nishiyama et al. [182], who have shown that in Ca 2+ /Mg 2+ /CO 3 2− systems nesquehonite with very low crystallinity is expected to be formed with monohydrocalcite and the amorphous material. The hydrous magnesium carbonates surrounding monohydrocalcite play a protective role in preventing its dehydration to anhydrous calcium carbonates.
High-Mg monohydrocalcite consists of individual nanometer-sized crystals (<35 nm) with a significant part of its structural H 2 O being associated with magnesium [178]. X-ray diffraction (XRD) experiments show that monohydrocalcite exhibits low crystallinity and small particle size when the ratio of magnesium to calcium in the solid is higher than 0.4, i.e., the increase of magnesium content lead to broad X-ray diffraction peaks with low intensity. Monohydrocalcite with Mg/Ca ratio higher than 0.5 exhibits the properties characteristic for amorphous material (no XRD peaks are observed) [182]. This corresponds well with the results of the electron microscopy measurements of brain and liver mitochondria that show a lack of crystal structures in the matrix [27]. The value of solubility of amorphous Mg-rich monohydrocalcite is slightly higher than in the case of pure monohydrocalcite [182]. Hence, in order to precipitate, it requires the solution to be more alkaline than the mitochondrial matrix.

Carbonated Apatite in Physiological Solutions
The studies on slow spontaneous precipitation of calcium carbonates and calcium phosphates under physiological conditions [218], show that calcium carbonate precipitation is prevented by the presence of phosphate ions at concentrations too low for the precipitation of calcium phosphate. Such precipitation is prevented, when the phosphate/bicarbonate concentration ratio is higher than about l/300. Above this ratio, the competition of the phosphate ions with the carbonate ions is sufficient to inhibit the nucleation and crystal growth of calcium carbonates, while the phosphate concentration is still too low for the nucleation of phosphates [218]. Pyrophosphate and hexametaphosphate ions inhibit the precipitation of calcium carbonate in a similar way. Hence, the composition of most biological fluids does not allow the deposition of calcium carbonate and it is suggested that such deposition can occur only under the influence of metabolic processes which locally raise the bicarbonate and lower the phosphate concentration [218].
Carbonate ions may affect the solubility of phosphate salts in several ways [191]. Ion pair complexes between calcium and bicarbonate and carbonate will form and the presence of ion pairs and reduction of free Ca 2+ would increase the soluble concentration of phosphate. However, the calculations indicate that ion pairs will be the predominant calcium-containing species at pH greater than 8.5 and at high carbonate levels (0.05 M) [219]. The carbonate ions may also compete with phosphate for sites on the growing crystallites. Nucleation and crystal growth may be hindered by the extra time and energy required to eliminate carbonate from phosphate sites and vice versa. When crystal growth does take place, carbonate can readily substitute for phosphate in the apatite lattice, as discussed earlier. Chemical analyses reveal that the apatitic tissues always contain smaller or larger amounts of carbonate which does not crystallize separately as carbonate [197].
Precipitation of babierrite is also inhibited, because of the presence of carbonate and calcium ions. Due to the chemical similarity between Mg 2+ and Ca 2+ , magnesium co-precipitates with calcium phosphates instead, inducing the formation of an amorphous phase rather than a stable crystalline carbonated apatite [191,220]. Magnesium ions stabilize the amorphous precipitates of calcium carbonate phosphate. Apatitic precipitates are more poorly-crystallized when formed in the presence of magnesium ions than when formed in their absence, and the crystallinity of the precipitates formed in the presence of magnesium decreases with rising [Mg 2+ ]. The Ca/P ratios of the amorphous precipitates are often significantly lower when formed in the presence of Mg 2+ ions, and the equilibrium concentrations of calcium and phosphates are higher when the precipitates are amorphous, than when they are crystalline [221].
Biological calcium apatites show a broad range of chemical compositions but also structure. It has been shown that hydroxyapatite precipitates obtained in the conditions where pH is kept constant show low crystallinity [222]. In primitive organisms, calcium apatites are generally amorphous, while in more elaborated organisms, especially in vertebrates, calcium apatites can crystallise [203]. Magnesium has been shown to disturb the crystallization of calcium apatites from solution when used at concentrations sufficient to be a major competitor for calcium [223]. The growth of apatite crystals is impeded by the Mg 2+ ions, because they block the active growth sites and interrupt the crystal structure [224].
Organic compounds present in the mitochondrial matrix have a similar effect on phosphate crystals formation. Namely, the nucleotide diphosphates and triphosphates, including ADP and ATP, as well as low molecular weight metabolites containing two attached ester phosphate groups such as citrate also have the ability to inhibit the crystallization of calcium phosphates and may function as in vivo mineralization inhibitors [225,226]. Adsorption of organic compounds on the newly-formed nuclei results in a small crystal size and could be a factor in producing less crystalline phase [198]. Although apatite is usually considered a crystallographic structure, we use the term amorphous carbonated apatite to distinguish it from the amorphous calcium phosphate with the formula Ca x H y (PO 4 ) z .
A characteristic property of apatite compounds is the ability to incorporate ions and individual molecules during its synthesis. One of the most interesting examples is the molecular oxygen, which forms inside the structure through the decomposition of unstable precursor: peroxide ions and hydrogen peroxide molecules trapped in the apatite structure [227]. Mitochondria are involved in the cell's response to oxidative stress, hence the ability of apatite to entrap H 2 O 2 and convert it to O 2 might have had a crucial role in the evolution of mitochondria, especially in the early stages before the ability to create enzyme glutathione peroxidase has been developed.
Apatite can incorporate a large variety of foreign ions, including sodium, potassium, strontium, barium, manganese, zinc, iron, cobalt, nickel and even rare earth elements (see [228] for a detailed review). This property is used by a few organisms to eliminate toxic elements and survive in polluted environments, e.g., in gastropods Littorina Littorea, apatites allow the accumulation and biochemical deactivation of excess quantities of cadmium and other potentially harmful cations [229,230]. Small organic molecules, such as formate ions and glycine molecules, can also be incorporated in apatites [202,203]. It has been shown that amorphous granular aggregates forming in mitochondria include glycoproteins in their structure [231].

Summary and Conclusions
The solubility calculations for calcium and magnesium salts, including carbonates and phosphates, led to the identification of four salts that may form in the mitochondrial matrix: monohydrocalcite, nesquehonite, carbonated apatite and babierrite. Further analysis of inhibition properties of several inorganic and organic ions present in the mitochondrial matrix led to a conclusion, that non-stoichiometric magnesium-rich carbonated apatite precipitates in the matrix under physiological conditions.
The formation of crystalline apatites is inhibited by the presence of magnesium ions, ATP, ADP and citrates. Therefore, amorphous and low-crystalline carbonated apatite is precipitated. Low crystallinity of the precipitated material can be attributed to numerous ionic substitutions in the apatite structure. The general chemical formula of this salt can be expressed as: (Ca,Mg) 10−x (PO 4 ) 6−x (HPO 4 ,CO 3 ) x (OH, 1 2 CO 3 ,Cl) 2−x with 0 < x < 2.
Numerous ions and molecules can also be incorporated in the structure of apatite, which leads to a further decrease of the crystallinity of precipitated material. The ability to incorporate hydrogen peroxide and decompose it to molecular oxygen might have an auxiliary role in the response to oxidative stress.
Precipitation and dissolution of carbonated apatite inside the matrix, affects the mitochondrial ionic concentrations, and therefore indirectly influences the action of several protein entities in the inner mitochondrial membrane including e.g., ATP synthase, phosphate translocase, calcium uniporter (MCU), mitoK ATP and mitoK Ca channels as well as the K + /H + , Na + /H + and Ca 2+ /3Na + exchangers.
When the oxidative phosphorylation chain is active, the membrane potential on the inner mitochondrial membrane is highly negative, which causes the influx of calcium to the matrix through MCU and leads to the precipitation of carbonated apatite. The dominant forms of carbonic and phosphoric acid in the mitochondrial pH range are HCO 3 − and HPO 4 2− , respectively.
Hence, precipitation of carbonated apatite leads to the production of protons, e.g.,: 10Ca 2+ + HCO 3 − + 6HPO 4 2− → Ca 10 (PO 4 ) 6 CO 3 + 7H + These protons are then pumped outside the mitochondrial matrix by the oxidative phosphorylation chain, which allows the production of ATP. In anaerobic conditions, i.e., during hypoxia, the oxidative phosphorylation chain stops. Hypoxia can be caused e.g., by ischemia (low blood flow) or carbonate monoxide poisoning (oxygen in hemoglobin replaced by CO). There are three phases of ATP production during ischemia [232]. At first, up to 16 min ATP is produced due to glycolysis cytoplasm acidifies but there is no increase in sodium (5 mM) or calcium concentration (140 nM) in the cytoplasm, i.e., Na-K-ATPase of the cytoplasmic membrane is fully active. In the second phase, as hypothesised by Dołowy [30], during hypoxia, the proton motive force is replaced by the calcium-motive force caused by the dissolution of calcium phosphates, efflux of calcium ions and alkalization of the matrix due to the following processes: Ca 10 (PO 4 ) 6 CO 3 + 7H 2 O → 10Ca 2+ ↑ + CO 2 ↑ + 6HPO 4 2− + 8OH − Ca 10−x (PO 4 ) 6−x (CO 3 ) x (OH) 2−x + 6H 2 O → (10 − x)Ca 2+ ↑ + xCO 2 ↑ + (6 − x)HPO 4 2− + 8OH − (9) where ↑ represents ions or molecules leaving the mitochondrial matrix. Phosphate ions forming due to the dissolution of carbonated apatite and lower ATP/ADP ratio are the two factors responsible for lowering the free energy of the ATP synthesis, hence allowing its production in hypoxic conditions. During the second phase which lasts up to 30 min, sodium concentration in the cytoplasm is slowly rising to 30 mM and calcium concentration to 500 nM level. The mitochondrion can survive, as long as it can keep the pH of the matrix constant by the precipitation and dissolution of carbonated apatite. Then during the third phase, when all available salt dissolves, further influx of protons leads to the acidification of the matrix, hence lowering the proton motive force below the level necessary for ATP production. When matrix acidifies Na/H and K/H exchangers cannot prevent influx of osmotically active cations into the matrix and mitochondria swells. Low ATP levels in the cytoplasm lead to opening of the mitoK ATP channel. Upon reperfusion sharp increase in mitochondrial electric potential leads to further influx of osmotically active cations into the mitochondrial matrix and formation of the permeability transition pore [233,234].
The build-up of amorphous carbonated apatite should increase the maximum time of hypoxia that the cell can survive. It explains the increased life-time observed in the heart cells preconditioned by short increases of the calcium concentration [235,236]. Calcium preconditioning increases the cytoplasmic calcium ion concentration. Therefore Ca 2+ from the cytoplasm is more rapidly transported into the matrix, which leads to the precipitation of carbonated apatite. A better understanding of the relationship between calcium storage and the mechanisms of ischemic preconditioning may lead to the development of new stroke-protective drugs.
Author Contributions: J.J.J. conceived, designed and performed all the calculations, analyzed the data, reviewed the available literature, wrote the paper, and supervised the overall research. R.F., K.D. and A.L. discussed the research. A.L. acquired funding. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.