Analysis of cardiac mitochondrial Na⁺–Ca²⁺ exchanger kinetics with a biophysical model of mitochondrial Ca²⁺ handing suggests a 3: 1 stoichiometry

Abstract

Calcium is a key ion and is known to mediate signalling pathways between cytosol and mitochondria and modulate mitochondrial energy metabolism. To gain a quantitative, biophysical understanding of mitochondrial Ca²⁺ regulation, we developed a thermodynamically balanced model of mitochondrial Ca²⁺ handling and bioenergetics by integrating kinetic models of mitochondrial Ca²⁺ uniporter (CU), Na⁺–Ca²⁺ exchanger (NCE), and Na⁺–H⁺ exchanger (NHE) into an existing computational model of mitochondrial oxidative phosphorylation. Kinetic flux expressions for the CU, NCE and NHE were developed and individually parameterized based on independent data sets on flux rates measured in purified mitochondria. While available data support a wide range of possible values for the overall activity of the CU in cardiac and liver mitochondria, even at the highest estimated values, the Ca²⁺ current through the CU does not have a significant effect on mitochondrial membrane potential. This integrated model was then used to analyse additional data on the dynamics and steady-states of mitochondrial Ca²⁺ governed by mitochondrial CU and NCE. Our analysis of the data on the time course of matrix free [Ca²⁺] in respiring mitochondria purified from rabbit heart with addition of different levels of Na⁺ to the external buffer medium (with the CU blocked) with two separate models – one with a 2: 1 stoichiometry and the other with a 3: 1 stoichiometry for the NCE – supports the hypothesis that the NCE is electrogenic with a stoichiometry of 3: 1. This hypothesis was further tested by simulating an additional independent data set on the steady-state variations of matrix free [Ca²⁺] with respect to the variations in external free [Ca²⁺] in purified respiring mitochondria from rat heart to show that only the 3: 1 stoichiometry model predictions are consistent with the data. Based on these analyses, it is concluded that the mitochondrial NCE is electrogenic with a stoichiometry of 3: 1.

Calcium ion has multiple roles in mitochondrial function. Alteration of mitochondrial Ca²⁺ homeostasis can lead to mitochondrial dysfunction and cellular injury (Gunter et al. 1994; Bernardi, 1999; Duchen, 2000; Brookes et al. 2004; O'Rourke et al. 2005). Therefore, a quantitative, biophysical understanding of mitochondrial Ca²⁺ handling is critical to advance our current qualitative understanding of the role of Ca²⁺ in mitochondrial function/dysfunction.

Mitochondrial Ca²⁺ is regulated by a series of uniporters/channels and antiporters/exchangers located on the inner mitochondrial membrane (IMM). It is primarily governed by a kinetic balance between the uptake of Ca²⁺ via the Ca²⁺ uniporter (CU) and the extrusion of Ca²⁺ from the matrix through the Na⁺–Ca²⁺ exchanger (NCE) (Gunter & Pfeiffer, 1990; Gunter et al. 1994; Bernardi, 1999). In addition, there may be a Na⁺-independent Ca²⁺ exchange (NICE) pathway for extrusion of Ca²⁺ from the matrix, possibly a Ca²⁺–H⁺ exchanger (CHE) (Gunter et al. 1983; Brand, 1985a; Wingrove & Gunter, 1986a) that may have low activity in cardiac mitochondria at physiological levels of ΔpH across the IMM (Rizzuto et al. 1987).

Despite the importance of mitochondrial Ca²⁺ to cell physiology and pathophysiology, there are still significant gaps in our understanding of the functional properties of the mitochondrial Ca²⁺ transport systems. One of the fundamental issues is the stoichiometry of the mitochondrial NCE. The earliest studies by Crompton et al. (1976, 1977) using purified cardiac mitochondria revealed that Na⁺-dependent Ca²⁺ extrusion is rapid in respiring mitochondria and that the respiration-dependent trans-IMM [Ca²⁺] gradient is abolished by uncouplers. These results suggest a contribution of membrane potential to an electrogenic nNa⁺–Ca²⁺ exchange, such as a 3Na⁺–Ca²⁺ exchange that is apparent for sarcolemmal NCE (Mullins, 1979). Subsequently, Brand (1985b) showed that addition of an ionophore to respiring cardiac mitochondria did not disturb the steady-state trans-IMM [Ca²⁺] distribution maintained by the NCE and concluded that the antiport supports an electroneutral 2Na⁺–Ca²⁺ exchange. This conclusion was further supported by the reconstitution studies of Li et al. (1992) who reported that a purified NCE from heart mitochondria supports a Ca²⁺-dependent extrusion of Na⁺ that was not affected by uncouplers. Such evidence led to an apparent consensus that the mitochondrial NCE is electroneutral.

Recently, in a series of reports (Baysal et al. 1991, 1994; Jung et al. 1995), Jung and co-workers re-examined the nature of nNa⁺–Ca²⁺ exchange in the heart mitochondria. In one report (Baysal et al. 1991), it was observed that the antiporter is regulated by matrix pH and that the high rates of nNa⁺–Ca²⁺ exchange can be sustained when ΔpH (and hence [Na⁺] gradient) across the IMM approaches zero. This led to a null-point study (Baysal et al. 1994) in which it was established that the trans-IMM [Ca²⁺] gradients maintained by the NCE are too large to be sustained by a passive electroneutral exchange of one Ca²⁺ for two Na⁺. Later, Jung et al. (1995) re-examined the experimental protocol of Brand (1985b) using fluorescent probes to monitor the trans-IMM pH and [Ca²⁺] gradients. They reported that when pH gradient in respiring mitochondria is collapsed by addition of nigericin, a large [Ca²⁺]_e: [Ca²⁺]_x gradient is obtained. Addition of an uncoupler to dissipate membrane potential (ΔΨ) abolished this gradient. These studies led to the conclusion that the nNa⁺–Ca²⁺ exchange in cardiac mitochondria is not electroneutral and that there is a contribution of ΔΨ to the observed trans-IMM [Ca²⁺] gradient. However, a recent study by Paucek & Jaburek (2004) reported that proteoliposomes reconstituted with the purified mitochondrial Na⁺–Ca²⁺ antiporter from beef heart supports a passive electroneutral exchange of one Ca²⁺ for two Na⁺. Thus, the stoichiometry of the mitochondrial NCE remains unresolved.

Here we introduce a biophysical model of mitochondrial Ca²⁺ handling integrated into a previously developed computational model of mitochondrial bioenergetics (Beard, 2005). The integrated model is applied to analyse available experimental data (Cox & Matlib, 1993) on the kinetics of mitochondrial Na⁺–Ca²⁺ exchange and to characterize the stoichiometry of the exchange. Kinetic flux expressions for the CU, NCE and NHE were developed and individually parameterized based on independent data sets on cation fluxes measured in purified mitochondria. These models represent improvements over available models in the literature (Magnus & Keizer, 1997; Cortassa et al. 2003, 2006; Nguyen & Jafri, 2005; Nguyen et al. 2007) as the models are thermodynamically balanced and satisfactorily explain a large number of independent measurements. Our analysis of the data on the kinetics of mitochondrial NCE with two separate models – one with a 2: 1 stoichiometry and the other with a 3: 1 stoichiometry for the NCE – supports the hypothesis that the NCE is electrogenic with a stoichiometry of 3: 1. This hypothesis is further verified and confirmed by simulating an additional independent data set (McCormack et al. 1989) showing the steady-state trans-matrix distribution of free [Ca²⁺] in rat heart mitochondria in the presence of extra-matrix Ca²⁺ and Na⁺ and/or Mg²⁺. This integrated model of mitochondrial bioenergetics and cation handling can be used to analyse and understand the integrated roles of the CU, NCE and NHE in regulating trans-IMM Ca²⁺, Na⁺ and H⁺ fluxes, and determine the consequences of these fluxes on mitochondrial NADH redox states and respiration.

Methods

The dynamics of mitochondrial Ca²⁺ is primarily regulated by the fluxes through the Ca²⁺ uniporter (CU), Na⁺–Ca²⁺ exchanger (NCE) and Na⁺–H⁺ exchanger (NHE) (Gunter & Pfeiffer, 1990; Gunter et al. 1994; Bernardi, 1999). The activity of the Ca²⁺–H⁺ exchanger (CHE) is believed to be low in cardiac mitochondria at physiological levels of ΔpH across the IMM (Rizzuto et al. 1987), so here the contribution of the exchanger to the dynamics of mitochondrial Ca²⁺ is neglected. The computational model of mitochondrial bioenergetics and Ca²⁺ handling is constructed here by developing and individually parameterizing kinetic models of cation fluxes through the CU, NCE and NHE, and then integrating the models into our previously developed biophysical model of mitochondrial electron transport system and oxidative phosphorylation (Beard, 2005, 2006; Huang et al. 2007; Wu et al. 2007a,b). The major components of the integrated model are shown in the scheme in Fig. 1. The cation transport flux expressions and dynamic mass balance equations are presented below. The previous computational model of mitochondrial bioenergetic system with necessary modifications for the purpose of the present study is briefly presented in the online Supplementary Material.

Figure 1. Major components of the integrated model of mitochondrial calcium handling and bioenergetics.

The model includes mitochondrial respiratory system and oxidative phosphorylation (Beard, 2005) and accounts for mitochondrial Na⁺–Ca²⁺ dynamics (corresponding transporters are highlighted). Specifically, the model includes reactions at complex I, III, IV and V of the electron transport systems; substrate transporters (ANT and PHT), cation transporters (CU, NCE, NHE and KHE), and passive K⁺ and H⁺ permeation across the inner mitochondrial membrane; and passive substrate transport fluxes of adenines and phosphate across the outer mitochondrial membrane. The flux through the TCA cycle producing NADH is expressed in terms of a phenomenological dehydrogenase flux (Beard, 2005, 2006; Wu et al. 2007a). ANT, adenine nucleotide translocase; PHT, phosphate–hydrogen cotransporter; CU, Ca²⁺ uniporter; NCE, Na⁺–Ca²⁺ exchanger; NHE, Na⁺–H⁺ exchanger; and KHE, K⁺–H⁺ exchanger.

Throughout this article, the subscripts ‘x’, ‘i’, and ‘e’ refer to the ‘matrix’, ‘intermembrane space’, and ‘extra-mitochondrial’ domains, respectively; F, R and T denote the Faraday's constant, ideal gas constant and absolute temperature, respectively; ΔΨ denotes the membrane potential (positive when expressed from outside to inside); and z_Ca = 2 is the valence of Ca²⁺. The species concentrations are expressed in the units of molar (m) – specifically moles per litre of the compartment water volume; the fluxes are expressed in the units of mass per unit time per unit mitochondrial volume (e.g. mol s⁻¹ (l mitochondria)⁻¹, or alternatively mol s⁻¹ (mg mitochondrial protein)⁻¹, where 1 mg mitochondrial protein is approximately equal to 3.67 × 10⁻⁶ l of mitochondria). Unless otherwise mentioned, the unit ‘mg’ stands for ‘mg mitochondrial protein’. The fluxes are positive when expressed from outside to inside.

Cation transport flux expressions

Characterization of Ca²⁺ influx

Ca²⁺ is taken up in energized mitochondria via the CU (Gunter & Pfeiffer, 1990; Gunter et al. 1994; Bernardi, 1999), which is believed to be a highly selective channel for Ca²⁺ (Kirichok et al. 2004). It is driven by the electrochemical gradient of Ca²⁺ across the IMM. It also depends on the kinetics of Ca²⁺ binding to the CU. So to develop a kinetic model for the CU-mediated Ca²⁺ influx, a hybrid model combining the Michaelis–Menten equation (Segel, 1993) for carrier-mediated facilitated transport of Ca²⁺ and the Goldman–Hodgkin–Katz (GHK) equation (Keener & Sneyd, 1998) for electrodiffusion of the CU–2Ca²⁺ complex is proposed. The model is based on a single-step binding kinetics, but accounts for cooperativity in Ca²⁺ binding to the CU (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973); this means the binding of the first Ca²⁺ ion to the CU favours binding of the second Ca²⁺ ion, and that the dissociation constant for the first binding step (K₁) is large and the second binding step (K₂) is small, but the dissociation constant for the overall binding (K₂ = K₁K₂) is finite. The model is thermodynamically balanced, since it considers reversible flux through the CU (Beard & Qian, 2007). The proposed CU flux expression is given by

(1)

where X_CU (mol mg⁻¹ s⁻¹) is the CU activity and K_Ca,CU (m) is the Michaelis–Menten constant for the binding of Ca²⁺ to the CU; n_CU is an exponent characterizing the non-linear dependency of the CU flux on the IMM potential ΔΨ. The exponent corresponding to the standard GHK equation is 1 (Keener & Sneyd, 1998). Here n_CU is a phenomenological factor that allows us to fit the observed kinetic data. The three free parameters X_CU, K_Ca,CU and n_CU are characterized based on a number of independently published data sets on initial (pseudo-steady) flux rates via the CU with varying levels of external buffer Ca²⁺ and IMM potential ΔΨ measured in purified mitochondria from rat heart and rat liver (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973; Wingrove et al. 1984; Gunter & Pfeiffer, 1990; Gunter et al. 1994) (Table 1 and Fig. 2). In contrast to the previous models of the CU (Magnus & Keizer, 1997; Cortassa et al. 2003, 2006), this model satisfies the thermodynamic criterion that the CU flux is zero at the Nerst equilibrium potential for Ca²⁺. Though the Jafri et al. (Nguyen & Jafri, 2005; Nguyen et al. 2007) model of the CU is thermodynamically balanced, it does not account for cooperativity.

Table 1.

Parameter values in the model of mitochondrial Ca²⁺ handling (activities in mmol mg⁻¹ s⁻¹ can be converted to mmol l⁻¹ s⁻¹ by using the conversion factor 1 mg mitochondrial protein = 3.67 μl mitochondria)

Parameter	Description	Values	Units	Reference
zCa	Valence of Ca2+	2.0	unitless	standard
KCa,CU	MM binding constant of Ca2+ for the Ca2+ uniporter	90.0 (rat heart)	μm	a
		48.0 (rat liver)	μm	b
XCU	Ca2+ uniporter activity	1.5 × 10−2 (rat heart)	nmol mg−1 s−1	a
		1.35 × 10−2 (rat liver)	nmol mg−1 s−1	b
		2.025 (rat liver)	nmol mg−1 s−1	c
nCU	Exponent for the IMM ΔΨ-dependence of Ca2+ uniporter	2.7	unitless	c
KCa,NCE	MM binding constant of Ca2+ for the Na+–Ca2+ exchanger	2.1 (bovine heart)	μm	d
KNa,NCE	MM binding constant of Ca2+ for the Na+–Ca2+ exchanger	8.2 (bovine heart)	mm	d
		3.45 (rabbit heart; with 2: 1 NCE stoichiometry)	mm	e
		2.4 (rabbit heart; with 3: 1 NCE stoichiometry)	mm	e
XNCE	Na+–Ca2+ exchanger activity	88 for 2: 1 NCE stoichiometry	nmol (mg NCE protein)−1 s−1	d
		1.41 for 3: 1 NCE stoichiometry	nmol (mg NCE protein)−1 s−1	d
		1.76 × 10−3 for 2: 1 NCE stoichiometry	nmol mg s	e
		3.375 × 10−5 for 3: 1 NCE stoichiometry	nmol mg−1 s−1	e
KNa,NHE	MM binding constant of Na+ for the Na+–H+ exchanger	22.0 (rat heart/liver)	mm	f
KiH,NHE	Inhibition constant for matrix H+ for the Na+–H+ exchanger	10−7.0 (rat heart/liver)	m	f
XNHE	Na+–H+ exchanger activity	12.0 (rat heart liver−1)	nmol mg−1 s−1	f
		18.0 (rat heart)	nmol mg−1 s−1	f
βCa	Matrix Ca2+ buffering capacity	10.0	unitless	g
KEGTA.Ca	EGTA–Ca dissociation constant	10−6.7	m	h

MM stands for Michaelis–Menten. ^aEstimated from Scarpa & Graziotti (1973); ^bEstimated from Vinogradov & Scarpa (1973); ^cEstimated from Wingrove et al. (1984); ^dEstimated from Paucek & Jaburek (2004); ^eEstimated from Cox & Matlib (1993); ^fEstimated from Kapus et al. (1988, 1989); ^gTaken from Jafri and Co. (Nguyen & Jafri, 2005; Nguyen et al. 2007); ^hTaken from references (Portzehl et al. 1964; Harafuji & Ogawa, 1980; Fabiato, 1988).

Figure 2. Fitting of Ca²⁺ uniporter (CU) model (lines) to data (points) on CU fluxes.

A and B, the initial rates of Ca²⁺ uptake (points) were measured with varying external buffer [Ca²⁺] in purified respiring mitochondria from rat liver (A) (Vinogradov & Scarpa, 1973) and rat heart (B) (Scarpa & Graziotti, 1973). Also shown are the model-simulated CU fluxes at 5 different levels of IMM ΔΨ (lines) in which the model is fitted to the data with IMM ΔΨ = 190 mV. The estimated values of the kinetic parameters were: A, K_Ca,CU = 48 μm and X_CU = 1.35 × 10⁻² nmol mg⁻¹ s⁻¹ for rat liver mitochondria, and B, K_Ca,CU = 90 μm and X_CU = 1.5 × 10⁻² nmol mg⁻¹ s⁻¹ for rat heart mitochondria. C, the fitting of the same kinetic model to another data set is shown where the initial rates of Ca²⁺ uptake were measured in purified rat liver mitochondria with varying IMM ΔΨ for 3 different levels of extramitochondrial buffer [Ca²⁺] (Wingrove et al. 1984; Gunter & Pfeiffer, 1990; Gunter et al. 1994). To fit the model to this data set, X_CU is adjusted to 2.025 nmol mg⁻¹ s⁻¹ while keeping K_Ca,CU fixed at 48 μm. D, model-simulated CU fluxes are shown for purified rat heart mitochondria with varying IMM ΔΨ for 4 different levels of extramitochondrial buffer [Ca²⁺] with K_Ca,CU = 90 μm and X_CU = 2.25 nmol mg⁻¹ s⁻¹.

Characterization of Na⁺-dependent Ca²⁺ efflux

Ca²⁺ is primarily extruded from respiring and non-respiring mitochondria in exchange for Na⁺ via the NCE (Na⁺-dependent efflux pathway) (Crompton et al. 1976; Crompton & Heid, 1978; Wingrove & Gunter, 1986b; Rizzuto et al. 1987; Cox & Matlib, 1993; Paucek & Jaburek, 2004). As mentioned before, the stoichiometry of the mitochondrial NCE is not well established. A number of reports indicate that the NCE is electroneutral: exchange of 2 Na⁺ for 1 Ca²⁺ (Brand, 1985b; Li et al. 1992; Paucek & Jaburek, 2004). Other reports indicate an electrogenic 3Na⁺–Ca²⁺ exchange (Baysal et al. 1991, 1994; Jung et al. 1995). To differentiate between these two possibilities, two kinetic models for the NCE are formulated here, one with a 2: 1 and the other with a 3: 1 stoichiometry. Both the models are based on a reversible, rapid-equilibrium, random-order bi-bi kinetic mechanism (Segel, 1993) for binding of Na⁺ and Ca²⁺ to the NCE. For simplicity, it is also assumed that the binding of the n Na⁺ ions (n = 2 or 3) to the NCE occurs in a single step (following the same logic used for the CU model). This assumption results in a model that both fits and can be parameterized by the available experimental data on the kinetics of the NCE. The derived flux expressions are thermodynamically balanced (Beard & Qian, 2007); the flux depends on membrane potential ΔΨ for a 3Na⁺–Ca²⁺ exchange. The proposed NCE flux expressions are

(2A)

for a 2Na⁺ -Ca⁺ exchange, and

(2B)

for a 3Na⁺–Ca²⁺ exchange. Here X_NCE (mol mg⁻¹ s⁻¹) is the NCE activity and K_Ca,NCE (m) and K_Na,NCE (m) are the Michaelis–Menten constants for the binding of Ca²⁺ and Na⁺ to the NCE. The three free parameters X_NCE, K_Ca,NCE and K_Na,NCE are characterized here for both 2: 1 and 3: 1 NCE stoichiometries based on published experimental data on initial (pseudo-steady) flux rates via the NCE measured in proteoliposomes reconstituted with the purified mitochondrial Na⁺–Ca²⁺ antiporter from bovine heart with varying levels of internal Ca²⁺ and external Na⁺ (Paucek & Jaburek, 2004) (Table 1 and Fig. 3). The NCE flux rates were expressed in the units of mol (mg NCE protein)⁻¹ s⁻¹. In contrast to the previous models of the NCE (Magnus & Keizer, 1997; Cortassa et al. 2003, 2006), the 3: 1 NCE stoichiometry model in eqn (2B) satisfies the physical criterion that the NCE flux is zero at the Nerst equilibrium potential for 3Na⁺ and Ca²⁺. The 3: 1 NCE stoichiometry model in eqn (2B) is similar to the NCE model of Nguyen et al. (2007). Note that with positive membrane potential ΔΨ (outside minus inside potential), the flux is always in the direction of Na⁺ import and Ca²⁺ export for the concentrations used here for the 3: 1 NCE stoichiometry model. In contrast, with excess extra-mitochondrial Ca²⁺, the 2: 1 NCE stoichiometry model can operate in the reverse direction.

Figure 3. Fitting of Na⁺–Ca²⁺ exchanger (NCE) model (lines) to data (points) on NCE fluxes.

The initial rates of Na⁺ efflux–Ca²⁺ influx (A and C) and Na⁺ influx–Ca²⁺ efflux (B and D) were measured with varying external buffer [Ca²⁺] and [Na⁺] (fixed matrix [Na⁺] = 25 mm and [Ca²⁺] = 18 μm), respectively, using proteoliposomes reconstituted with the purified mitochondrial Na⁺–Ca²⁺ antiporter from bovine heart (Paucek & Jaburek, 2004). The NCE model was based on a 2Na⁺–Ca²⁺ stoichiometry (electroneutral exchange) in the upper panels (A and B) and a 3Na⁺–Ca²⁺ stoichiometry (electrogenic exchange) in the lower panels (C and D). For a 3Na⁺–Ca²⁺ electrogenic exchange, the model-simulated NCE fluxes at various values of IMM ΔΨ are also shown; the model is fitted to the data with IMM ΔΨ = –190 mV (C) or IMM ΔΨ = +190 mV (D). The estimated values of the kinetic parameters are K_Na,NCE = 8.2 mm, K_Ca,NCE = 2.1 μm and X_NCE = 88 nmol Ca²⁺ mg⁻¹ s⁻¹ (176 nmol Na⁺ mg⁻¹ s⁻¹) for a 2Na⁺–Ca²⁺ exchange (A and B) and X_NCE = 1.41 nmol Ca²⁺ mg⁻¹ s⁻¹ (4.23 nmol Na⁺ mg⁻¹ s⁻¹) for a 3Na⁺–Ca²⁺ exchange (C and D). Note that for a 3Na⁺–Ca²⁺ electrogenic exchange, the Na⁺ efflux–Ca²⁺ influx with imposed external Ca²⁺ and Na⁺ influx–Ca²⁺ efflux with imposed external Na⁺ requires the opposite signed IMM ΔΨ.

Characterization of Na⁺ efflux

The Na⁺, which enters the matrix in exchange for Ca²⁺ via the NCE, is extruded from the matrix in exchange for H⁺ via the NHE (Crompton & Heid, 1978; Kapus et al. 1988, 1989; Bernardi, 1999). The function of the NHE is to maintain a low [Na⁺] (2–5 mm) in respiring mitochondria to favour the influx of Na⁺ and efflux of Ca²⁺ via the NCE (Crompton & Heid, 1978; Jung et al. 1992). So the mitochondrial NHE plays a role in regulating the mitochondrial Na⁺–Ca²⁺ cycle, and hence mitochondrial Ca²⁺ dynamics. Like sarcolemmal NHE, mitochondrial NHE is also electroneutral, and hence the flux through the NHE does not depend on membrane potential ΔΨ. However, the activity of the NHE is inhibited by alkaline matrix pH (Kapus et al. 1988, 1989; Bernardi, 1999). So the NHE flux expression is formulated here by accounting for (1) a reversible, rapid-equilibrium, random-order bi-bi kinetic mechanism for binding of Na⁺ and H⁺ to the NHE, (2) inhibition of NHE by alkaline matrix pH, and (3) negligibly small binding constant of H⁺ compared to that of Na⁺ for the NHE (i.e. K_H,NHE ≪ K_Na,NHE). The resulting NHE flux expression is given by

(3)

where X_NHE (mol mg⁻¹ s⁻¹) is the NHE activity and K_Na,NaH (m) is the Michaelis–Menten constant for the binding of Na⁺ to the NHE. The factor X_NHE·[H⁺]_x²/(Ki_H,NHE·(Ki_H,NHE + [H⁺]_x)) with Ki_H,NHE = 10⁻⁷m represents the inhibition of the NHE by alkaline matrix pH (pH_x > 7). The two free parameters X_NHE and K_Na,NaH are characterized here based on published experimental data on initial (pseudo-steady) flux rates via the NHE with varying levels of external buffer pH, matrix pH and external Na⁺ in purified mitochondria from rat heart and rat liver (Kapus et al. 1988, 1989) (Table 1 and Fig. 4). The form of the inhibition factor in eqn (3) is actually determined based on the data on matrix pH dependency of the NHE flux (Fig. 4C and D). Note that the bracketed term in eqn (3) is similar to that in eqn (2A), except that the bracketed term in eqn (3) is based on a NHE stoichiometry of 1: 1 (i.e. exchange of 1 Na⁺ for 1 H⁺) with the assumption that K_H,NHE ≪ K_Na,NHE.

Figure 4. Fitting of Na⁺–H⁺ exchanger (NHE) model (lines) to data (points) on NHE fluxes.

The initial rates of H⁺ efflux or Na⁺ influx were measured: A, with varying external buffer [Na⁺] and fixed external buffer pH = 7.0 and matrix pH = 6.95 in purified mitochondria from rat heart (the filled circles are the measured H⁺ efflux rates while the open circles are the calculated H⁺ efflux rates from the measured rates of decreased matrix pH with a fixed matrix buffering capacity of 36 nmol H⁺ mg⁻¹ (pH unit)⁻¹) (Kapus et al. 1989), and B, with varying external buffer pH and fixed external buffer [Na⁺] = 9.6 mm and 15 mm and matrix pH = 7.0 in purified mitochondria from rat liver (Kapus et al. 1988). The model fits to these data sets with K_Na,NHE = 22 mm and X_NHE = 12 nmol mg⁻¹ s⁻¹ (for both Na⁺ influx and H⁺ efflux). C and D, the fitting of the same kinetic model to another data set is shown where the initial rates of decrease in matrix pH or H⁺ efflux in rat heart mitochondria were measured with varying matrix pH and fixed external buffer pH = 7.0 and [Na⁺] = 50 mm (Kapus et al. 1989). For the kinetic model to fit this data set, the X_NHE value is adjusted to 18 nmol mg⁻¹ s⁻¹ while keeping the K_Na,NHE value fixed at 22 mm.

Kinetics of Ca²⁺ binding to EGTA

In purified mitochondrial studies, EGTA is often used to buffer excess Ca²⁺ in the external buffer medium. Various models have been proposed in the literature to describe the kinetics of EGTA–Ca²⁺ binding (Portzehl et al. 1964; Harafuji & Ogawa, 1980; Miller & Smith, 1984; Smith et al. 1984; Fabiato, 1988). Here, we express the fractional binding of Ca²⁺ to the EGTA using the non-cooperative Hill equation:

(4)

where K_EGTA.Ca is the EGTA–Ca²⁺ apparent dissociation constant; K_EGTA.Ca depends on temperature, ionic strength, pH and other metal ion concentrations (e.g. Mg²⁺, Mn²⁺, Na⁺, K⁺) in the medium which compete with Ca²⁺ for binding to the EGTA (Portzehl et al. 1964). At physiological levels of pH and with other cations in the external buffer medium, the estimate K_EGTA.Ca = 10^−6.7m is consistent with the available estimates from the literature (Portzehl et al. 1964; Harafuji & Ogawa, 1980; Smith et al. 1984; Fabiato, 1988).

The total Ca²⁺ in the external buffer medium can then be expressed as

(5)

The volume of distribution for Ca²⁺ in the external buffer medium can be derived by taking the derivative of [Ca²⁺]_e,tot with respect to [Ca²⁺]_e and then multiplying it by the external buffer volume (W_e), expressed as the litres of external buffer water per litre of mitochondria:

(6)

The bracketed term characterizes the EGTA-dependent buffering capacity for Ca²⁺ (see below).

Dynamic mass balance equations for the integrated model

Matrix free [Ca²⁺] in respiring mitochondria is primarily affected by three fluxes: (1) Ca²⁺ influx through the CU; (2) Ca²⁺ efflux through the NCE; and (3) Ca²⁺ buffering in the matrix. This is under the assumption that the flux through the CHE (NICE) is negligible at physiological levels of ΔpH across the IMM. Matrix [Na⁺] is mainly affected by two fluxes: (1) Na⁺ influx via the NCE and (2) Na⁺ efflux via the NHE. Furthermore, in purified mitochondrial studies, with the external buffer containing EGTA, the external buffer free [Ca²⁺] depends on the kinetics of Ca²⁺ binding to the EGTA. To track matrix [Ca²⁺] and [Na⁺], our model of the mitochondrial respiratory system and oxidative phosphorylation (see Supplementary Material) is modified by adding dynamic mass balance equations governing the free [Ca²⁺] and [Na⁺]:

(7A)

(7B)

(7C)

(7D)

In addition, the matrix [H⁺] and IMM ΔΨ equations are updated as follows:

(7E)

(7F)

In the above equations, β_Ca and β_H are the matrix buffering capacities for Ca²⁺ and H⁺; n_NCE is the stoichiometry of Na⁺ in NCE (i.e. n_NCE = 2 for a 2Na⁺–Ca²⁺ exchange and n_NCE = 3 for a 3Na⁺–Ca²⁺ exchange); n_F1F0 = 3 is the stoichiometry of proton pumping at complex V (F₁F₀-ATPase); C_IMM denote the effective IMM capacitance; W_x denotes the litres of mitochondrial water volume per litre of mitochondrial volume and W_e denotes the litres of external water volume per litre of mitochondrial volume; W_e,Ca denote the effective volume or volume of distribution for Ca²⁺ in the external buffer medium accounting for the binding space of EGTA for Ca²⁺ (as derived above). The external water volume relative to the mitochondrial volume (W_e) is calculated based on the amount of mitochondrial protein added to the external buffer medium in a particular experimental protocol.

The dynamic mass balance eqns (7) along with the model equations presented in the Supplementary Material constitute the integrated model of mitochondrial bioenergetics and Ca²⁺ handling. The model parameter values are presented in Table 1 and in Table S1 of the Supplementary Material. The integrated model is implemented here in MatLab (The MathWorks Inc.; http://www.mathworks.com) and is solved using the ODE15S solver, a robust, implicit integrator suitable for the numerical solutions of stiff, initial value ODE problems. The model is available for download at http://www.bbc.mcw.edu/Computation.

Integrated model analysis of the experimental data on mitochondrial free [Ca²⁺]

Application to characterize the kinetics and stoichiometry of the mitochondrial NCE

The utility of the integrated model of mitochondrial bioenergetics and Ca²⁺ handling is illustrated by first applying the model to characterize the kinetics and stoichiometry of the mitochondrial NCE. Cox & Matlib (1993) reported spectrofluorometrically measured matrix free Ca²⁺ time course following addition of Na⁺ to the suspensions of respiring mitochondria purified from rabbit heart with the CU inhibited with ruthenium red. To distinguish between a 2: 1 and 3: 1 NCE stoichiometry, we analysed the dynamic data with the integrated model incorporating both NCE stoichiometries. The binding constants for Ca²⁺ and Na⁺ for the antiporter (K_Ca,NCE and K_Na,NCE) and the overall activity of the antiporter (X_NCE) were varied to obtain the best possible match of the model predictions to the dynamic observations, characterized by the least-square mean residual error, defined below by eqn (8). For this analysis, the activity of the CU was reduced by 100 times compared to the normal value in Table 1 to mimic the inhibitory effect of the ruthenium red. Note that since the external buffer contains EGTA to chelate the extruded Ca²⁺ from the matrix and the buffer volume is ∼390 times higher than the matrix volume (see below), the free Ca²⁺ in the buffer as a result of extrusion of Ca²⁺ from the matrix would not be high enough to activate the CU and to have a measurable Ca²⁺ influx through the CU. Thus, by inhibiting the CU by 10 times or 100 times will not influence the simulated dynamics of matrix free [Ca²⁺] significantly. The matrix Ca²⁺ buffering capacity was fixed at 10 (Nguyen & Jafri, 2005; Nguyen et al. 2007). The parameters characterizing the mitochondrial NHE were fixed as in Table 1, and that of mitochondrial bioenergetics were fixed as in Table S1.

In the experiment of Cox & Matlib (1993), the external buffer contained 50 μm of EGTA to chelate the extruded Ca²⁺ from the matrix. However, the total [Ca²⁺] (free + EGTA-bound) in the buffer was not reported. Therefore, we explored a range of reasonable values of total buffer [Ca²⁺] (1–40 μm) to characterize the sensitivity of the kinetics of the NCE to the buffer [Ca²⁺], as described in the Results section. The initial matrix free [Ca²⁺] was set at 1.27 μm to match the measured initial conditions, and the initial matrix [Na⁺] was set at 2.5 mm, corresponding to the standard level of endogenous Na⁺. The buffer pH, [K⁺] and [Pi] levels were set at 7.2, 180 mm and 2.5 mm, respectively, according to the experimental protocol (Table S1). All measurements were made using 1.4 mg of mitochondrial protein in an external buffer medium of 2 ml. Since 1 mg of mitochondrial protein corresponds to approximately 3.67 μl of mitochondria (Vinnakota & Bassingthwaighte, 2004; Huang et al. 2007), the relative external buffer volume (W_e) was set to 2.0/(1.4 × 3.67 × 10⁻³) = 390 ml of external buffer water per ml of mitochondria.

To complement the integrated model analysis of the dynamic data of Cox & Matlib (1993) and confirm the stoichiometry of the mitochondrial NCE, we performed an alternative analysis in which both 2: 1 and 3: 1 NCE models eqns (2A) and (2B) were first individually fitted to the kinetic data on the measured rates of decrease of matrix free [Ca²⁺] immediately following addition of Na⁺ and then used the fitted NCE model parameter values to simulate the dynamic data with the help of the integrated model incorporating both the NCE models. For this analysis, the matrix free [Ca²⁺], buffer free [Ca²⁺] and matrix [Na⁺] were set at 1.27 μm, 0.15 μm and 0.5 mm, respectively. The buffer [Na⁺] was varied as per the experimental protocol. The buffer free [Ca²⁺] of ∼0.15 μm corresponds to a buffer total [Ca²⁺] of ∼20 μm (based on EGTA–Ca²⁺ binding model); these estimates are approximate levels of buffer Ca²⁺ with a steady-state level of 0.1 μm of matrix free Ca²⁺ with a presumed 2: 1 NCE model (see Results section for details). The estimate of 0.5 mm of matrix [Na⁺] would be the approximate level of matrix Na⁺ within 1 min of the stabilization period after addition of substrates but before addition of Na⁺ to the mitochondrial preparation with the mitochondrial NHE functioning.

Further validation of the integrated model by independent experimental data

In this section, the integrated model of mitochondrial bioenergetics and Ca²⁺ handling was further validated and the stoichiometry of the mitochondrial NCE was further evaluated by applying the model to analyse an independent experimental data set from McCormack et al. (1989). The study reports spectrofluorometrically measured steady-state variations of matrix free [Ca²⁺] with respect to the variations in external buffer free [Ca²⁺] in respiring rat heart mitochondria in the presence of extra-matrix Na⁺ (10 mm) and/or Mg²⁺ (2 mm), or an uncoupler (1 μm FCCP). All measurements were made using 1 mg mitochondrial protein suspended (respiring) in 1 ml buffer medium maintained at pH = 7.3, [K⁺] = 180 mm, and [Pi] = 5 mm. This corresponds to a relative external buffer volume (W_e) of 272 ml of water per ml of mitochondria. The buffer free [Ca²⁺] was maintained at a desired level with addition of calculated amount of EGTA and CaCl₂ in the external buffer medium.

With these experimental conditions, the integrated model incorporating both 2: 1 and 3: 1 NCE stoichiometries was fitted to the data by fixing the binding constants K_Ca,CU, K_Na,NCE and K_Ca,NCE as estimated before, but allowing the activity parameters X_CU and X_NCE to vary. The activity of the NCE (X_NCE) was fixed at the same level for all data curves, while the activity of the CU (X_CU) was allowed to vary over the data curves to implicitly account for the inhibition of the CU by extramitochondrial free [Mg²⁺], which is not explicitly accounted for in the present model of the CU. The matrix Ca²⁺ buffering capacity and parameters characterizing the mitochondrial NHE were fixed as in Table 1.

Statistical method of optimization and parameter estimation

The NCE model parameters θ = (X_NCE, K_Ca,NCE, K_Na,NCE) characterizing the experimental data of Cox & Matlib (1993) on the kinetics of mitochondrial NCE were estimated by simultaneous least-squares fitting of the model-simulated outputs to the dynamic experimental data:

(8)

where ${\left[{\text{Ca}}^{2+}\right]}_{\text{x},i,j}^{\text{data}}$ and ${\left[{\text{Ca}}^{2+}\right]}_{\text{x},j}^{\text{model}}$ (t_i, θ) are the experimental data and model-simulated outputs on matrix free [Ca²⁺] at time t_i in jth Na⁺-addition experiment; ${\left[{\text{Ca}}^{2+}\right]}_{\text{x},0}^{\text{data}}$ is the measured initial matrix free [Ca²⁺] (to normalize concentrations); N_j is the number of dynamic measurements (time points) on jth Na⁺-addition experiment and min E(θ) is the minimization of the mean residual error E(θ). The minimization of mean residual error (objective function) for optimal estimation of NCE model parameters θ = (X_NCE, K_Ca,NCE, K_Na,NCE) is carried out using the FMINCON optimizer in MatLab (The MathWorks Inc.; http://www.mathworks.com). The robustness of model fitting to data for a particular NCE model is assessed based on the value of mean residual error (8) at optimal parameter estimates (least-square error).

Results

Parameterization of the individual model components

This section demonstrates the parameterization and independent validation of the individual model components of the mitochondrial Ca²⁺ handling system. Simulations of the individually parameterized kinetic models of the Ca²⁺ uniporter (CU), Na⁺–Ca²⁺ exchanger (NCE) and Na⁺–H⁺ exchanger (NHE), and their comparisons to the experimental data on cation fluxes are illustrated in Figs 2, 3 and 4, respectively. The simulations are shown as continuous lines while the data are shown as points. The data are from independent studies and are based on measurements of initial flux rates via the cation transporters in respiring/non-respiring mitochondria (from heart or liver) or in reconstituted transporters under a variety of experimental conditions. Details for each experiment and simulation are described below. The estimated parameter values are summarized in Table 1. Unless stated otherwise, these parameter values are used later in integrated model simulations of the mitochondrial Ca²⁺ handling system.

Ca²⁺ uniporter model

Figure 2 shows simulations of mitochondrial Ca²⁺ uptake via the CU in purified energized (respiring) mitochondria with varying levels of external buffer Ca²⁺ and IMM potential ΔΨ. Here, the CU model is compared to three independent experimental data sets. Specifically, the CU model was used to simulate the experimental data of Scarpa and co-workers (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973) (Fig. 2A and B) and Gunter and co-workers (Wingrove et al. 1984; Gunter & Pfeiffer, 1990; Gunter et al. 1994) (Fig. 2C). In the experiments of Scarpa and co-workers, the initial rates of Ca²⁺ uptake were measured in respiring mitochondria purified from rat liver (Fig. 2A) (Vinogradov & Scarpa, 1973) and rat heart (Fig. 2B) (Scarpa & Graziotti, 1973) following addition of varying amounts of Ca²⁺ (CaCl₂) to the external buffer medium. For these analyses, IMM ΔΨ was held fixed at 190 mV, a typical value for state 2 and state 4 respiration in cardiac mitochondria.

In the experiments of Gunter and co-workers, the initial rates of Ca²⁺ uptake were measured in respiring mitochondria purified from rat liver (Fig. 2C) as a function of IMM ΔΨ for three different levels of external buffer Ca²⁺ ([Ca²⁺]_e = 0.5, 1.0 and 1.5 μm); ΔΨ was varied by adding malonate to the external buffer medium (Wingrove et al. 1984). The corresponding model simulations of ΔΨ-dependent Ca²⁺ influx in cardiac mitochondria with varying levels of external buffer Ca²⁺ (i.e. [Ca²⁺]_e = 25, 50, 100 and 250 μm) are shown in Fig. 2D in which ΔΨ was ramped from –50 mV to 200 mV, a simulation protocol similar to the experimental protocol of Kirichok et al. (2004) in which Ca²⁺ currents were measured using patch-clamp techniques in single mitoplasts from cardiac mitochondria.

As illustrated in Fig. 2, our kinetic model of the CU is able to satisfactorily describe all three independent data sets. The fitting of the model to the data from liver mitochondria (Vinogradov & Scarpa, 1973) gives K_Ca,CU = 48 μm and X_CU = 1.35 × 10⁻² nmol mg⁻¹ s⁻¹, while that from cardiac mitochondria (Scarpa & Graziotti, 1973) provides K_Ca,CU = 90 μm and X_CU = 1.5 × 10⁻² nmol mg⁻¹ s⁻¹ (Fig. 2A and B), given the exponent n_CU = 2.7. These analyses suggest that the CU activities are similar in both liver and cardiac mitochondria under similar experimental conditions, while the Michaelis–Menten (MM) constant for Ca²⁺ binding to the CU in cardiac mitochondria is about twice that of the value in liver mitochondria. By keeping the MM constant fixed at K_Ca,CU = 48 μm and n_CU = 2.7 and adjusting the CU activity parameter X_CU, our CU model is able to simultaneously fit all three data curves in Fig. 2C from liver mitochondria (Wingrove et al. 1984), but provide an estimate X_CU = 2.025 nmol mg⁻¹ s⁻¹ which is about two-orders of magnitude higher (∼150 times) than the estimated X_CU = 1.35 × 10⁻² nmol mg⁻¹ s⁻¹ obtained from the liver mitochondria data in Fig. 2A (Vinogradov & Scarpa, 1973). The difference may be attributed to the fact that the data were from different mitochondrial preparations and different experimental protocols; the external buffer medium in the experiments of Vinogradov & Scarpa (1973) contained 5 mm of Mg²⁺ which is known to inhibit the CU activity. Though the activity of the CU in cardiac and liver mitochondria estimated from these data varies widely, model simulations show that even at the highest estimated value (X_CU = 2.025 nmol mg⁻¹ s⁻¹), the flux through the CU is only a small fraction (∼10⁻⁵) of the resting (state 2 and state 4) electron transport system flux. Thus, we do not expect the Ca²⁺ current to significantly depolarize mitochondria in vivo under physiological conditions.

Na⁺–Ca²⁺ exchanger model

Figure 3 illustrates the model simulations of experimental data from Paucek & Jaburek (2004) in which the initial rates of Na⁺ efflux (Ca²⁺ influx) (negative IMM ΔΨ with outside negatively charged and inside positively charged) and Na⁺ influx (Ca²⁺ efflux) (positive IMM ΔΨ with outside positively charged and inside negatively charged) were measured, respectively, with varying external [Ca²⁺] and [Na⁺] in proteoliposomes reconstituted with the purified mitochondrial Na⁺–Ca²⁺ antiporter from the bovine hearts. The simulations from 2Na⁺–Ca²⁺ electroneutral and 3Na⁺–Ca²⁺ electrogenic exchange models are shown in the upper panels (A and B) and lower panels (C and D), respectively. In the case of 3Na⁺–Ca²⁺ electrogenic exchange, the exchange fluxes also depend on ΔΨ, which is illustrated in Fig. 3C and D. The simulations show that the exchange fluxes are higher in the energized mitochondria (higher ΔΨ) than the depolarized mitochondria (lower ΔΨ). The model was fitted to the data with the standard state 2 and state 4 IMM ΔΨ of 190 mV.

The data from Paucek & Jaburek (2004) suggest a 2: 1 stoichiometry for the mitochondrial NCE, since the rates of Na⁺ efflux are twice the rates of Ca²⁺ influx with imposed external Ca²⁺. It is, however, not clear whether Ca²⁺ influx rates were measured or calculated based on a presumed 2: 1 NCE stoichiometry. So to allow for the possibility of a 3: 1 NCE stoichiometry, the data on Ca²⁺ fluxes were excluded from the analysis.

As shown in Fig. 3, both of our 2: 1 and 3: 1 NCE stoichiometry models are able to fit to the data on Na⁺ fluxes reasonably well, though the fitting is slightly better for the 2: 1 model, which is expected given that the data suggest a 2: 1 NCE stoichiometry. In this study, the dependency of the NCE fluxes on the IMM ΔΨ is not reported. In addition, the data are only on the initial or pseudo-steady flux rates and no information about the dynamics of the NCE flux is available. So these data are not suitable to characterize the stoichiometry of the NCE. However, these data are useful to estimate the MM binding constants of Na⁺ and Ca²⁺ for the NCE (K_Na,NCE and K_Ca,NCE). The fittings from both the 2: 1 and 3: 1 models provide similar estimates for the binding constants: K_Na,NCE = 8.2 mm and K_Ca,NCE = 2.1 μm. However, the estimates of the NCE activity parameter were different: X_NCE = 88 nmol Ca²⁺ (mg NCE protein)⁻¹ s⁻¹ (176 nmol Na⁺ (mg NCE protein)⁻¹ s⁻¹) for the 2Na⁺–Ca²⁺ model, and X_NCE = 1.41 nmol Ca²⁺ (mg NCE protein)⁻¹ s⁻¹ (4.23 nmol Na⁺ (mg NCE protein)⁻¹ s⁻¹) for the 3Na⁺–Ca²⁺ model. (For the same data, the activity of the 3: 1 model is smaller than that of the 2: 1 model because the 3Na⁺–Ca²⁺ exchange is driven by the IMM ΔΨ.) Since these experiments were conducted using NCE reconstituted in liposomes, these estimated activities cannot be related to activity in the intact mitochondrion. Therefore, the activity in the integrated system is estimated based on a different data set, as described in the next section.

Na⁺–H⁺ exchanger model

Figure 4 demonstrates the model simulations of the experimental data from two different studies of Kapus et al. (1988, 1989) in which the initial rates of H⁺ efflux (Na⁺ influx) and/or decrease in matrix pH were measured in purified mitochondria from rat heart and rat liver with varying external buffer [Na⁺], external buffer pH, and matrix pH. The data in Fig. 4A represent the measured rate of H⁺ efflux (filled circles) or decrease in matrix pH times matrix pH buffering capacity (open circles) in purified rat heart mitochondria with varying buffer [Na⁺] and fixed buffer pH of 7.0 and initial matrix pH of 6.95 (Kapus et al. 1989). The model simulations are shown for three different levels of buffer pH (pH_e = 6.5, 7.0 and 7.6). The data in Fig. 4B represent the measurements of H⁺ efflux in rat liver mitochondria with varying buffer pH, fixed initial matrix pH of 6.95, and fixed external buffer [Na⁺] of 9.6 mm and 15 mm (Kapus et al. 1988). The model simulations are shown for three different levels of buffer [Na⁺] ([Na⁺]_e = 9.6, 15 and 30 mm). The data in Fig. 4C and D represent the measured rates of H⁺ efflux (calculated from the measured rates of decrease in matrix pH times matrix pH buffering capacity) in rat heart mitochondria following addition of 50 mm of [Na⁺] to the external buffer medium with varying matrix pH and fixed buffer pH of 7.0.

As depicted in Fig. 4, our two-parameter kinetic model of the mitochondrial NHE is able to satisfactorily describe all the data sets presented above. To fit the NHE model to the data set in Fig. 4C and D, it was necessary to account for the inhibition at high (alkaline) matrix pH (pH_x > 7) on the NHE, which is also evident from the work of Kapus et al. (1988, 1989). The inhibition factor was found to be of the form [H⁺]_x²/(Ki_H,NHE·(Ki_H,NHE + [H⁺]_x)) in which the matrix H⁺ inhibition constant for the NHE (Ki_H,NHE) is fixed at 10⁻⁷m (see eqn (3)). As seen in Fig. 4C and D, a high pH_x > 7 reduces the flux of the electroneutral Na⁺–H⁺ exchange; at pH_x > 7.5, the H⁺ efflux becomes negligible. The NHE model fits to the data sets in Fig. 4A and B with K_Na,NHE = 22 mm and X_NHE = 12 nmol mg⁻¹ s⁻¹. To fit the same model to the data sets in Fig. 4C and D, it was necessary to adjust the value of X_NHE to 18 nmol mg⁻¹ s⁻¹ while keeping the value of K_Na,NHE fixed at 22 mm. Consistent with the experimental observations, our NHE model simulations show that the H⁺ efflux (Na⁺ influx) increases with the increase in external buffer [Na⁺] and pH and decrease in matrix pH.

The next section shows how integrating these individual model components of the mitochondrial Ca²⁺ handling system into our computational model of mitochondrial respiratory system and oxidative phosphorylation (Beard, 2005, 2006; Huang et al. 2007; Wu et al. 2007a,b) is useful in analysing independent experimental data sets on mitochondrial [Ca²⁺] dynamics from the literature and characterizing the stoichiometry of the mitochondrial NCE. The model simulations also suggest potential new experiments to be performed to further characterize the stoichiometry of the mitochondrial NCE.

Integrated model analysis of the kinetics and stoichiometry of the mitochondrial Na⁺–Ca²⁺ exchanger

The integrated model of mitochondrial bioenergetics and Ca²⁺ handling was next applied to analyse the experimental data of Cox & Matlib (1993) on the kinetics of mitochondrial NCE in rabbit heart (with the CU blocked by ruthenium red) as well as to characterize the stoichiometry of the NCE. Results of these analyses are shown in Fig. 5 (and also in Figs S1 and S2 in the Supplementary Material).

Figure 5. Characterization of the kinetics and stoichiometry of the mitochondrial Na⁺–Ca²⁺ exchanger (NCE).

A and B, comparison of model predictions (lines) to measurements (points) of the time courses of Ca²⁺ decay in purified respiring mitochondria from rabbit heart with addition of varying levels of Na⁺ to the external buffer medium with the activity of CU blocked with ruthenium red (Cox & Matlib, 1993). The buffer medium contained 50 μm of EGTA to buffer the extruded Ca²⁺. The kinetic model of the NCE is based on a 2Na⁺–Ca²⁺ stoichiometry (electroneutral exchange) in A and a 3Na⁺–Ca²⁺ stoichiometry (electrogenic exchange) in B. The continous lines are the model-fitted curves with a total of 1 μm of Ca²⁺ in the buffer (negligible compared to the total EGTA of 50 μm) and the dashed lines are the model-simulated profiles with a total of 40 μm of Ca²⁺ in the buffer (comparable with the total EGTA of 50 μm). The model fitting provides estimates K_Na,NCE = 3.45 mm, K_Ca,NCE = 2.1 μm and X_NCE = 1.76 × 10⁻³ nmol Ca²⁺ mg⁻¹ s⁻¹ (3.52 × 10⁻³ nmol Na⁺ mg⁻¹ s⁻¹) for a 2Na⁺–Ca²⁺ exchange and K_Na,NCE = 2.4 mm, K_Ca,NCE = 2.1 μm and X_NCE = 3.375 × 10⁻⁵ nmol Ca²⁺ mg⁻¹ s⁻¹ (10.125 × 10⁻⁵ nmol Na⁺ mg⁻¹ s⁻¹) for a 3Na⁺–Ca²⁺ exchange. C and D, comparison of the regions of model predictions (shaded area) to the measurements (points) of the initial rates of decrease of matrix free [Ca²⁺] (C), and matrix free [Ca²⁺] after 3 min of addition of Na⁺ (D), both as functions of external buffer [Na⁺] with the total external buffer [Ca²⁺] varying from 1 μm to 40 μm. The model predictions are shown for both the 2Na⁺–Ca²⁺ and 3Na⁺–Ca²⁺ stoichiometry models with model parameters as described above.

Sensitivity analysis

Figure 5 (upper panels, A and B) shows the Na⁺ dose-dependent responses of matrix free [Ca²⁺] in purified respiring cardiac mitochondria with the CU blocked. The filled circles are the measured matrix free [Ca²⁺] data (Fig. 1 of Cox & Matlib (1993)). The continuous and dashed lines represent the model simulations in the presence of 1 μm and 40 μm of total Ca²⁺ (free + EGTA-bound) in the external buffer medium. To characterize the stoichiometry of the NCE, the model analysis of the data was carried out using both the 2Na⁺–Ca²⁺ (A) and 3Na⁺–Ca²⁺ (B) stoichiometry models. The fitting of the models to the data was carried out with 1 μm of total external buffer Ca²⁺ (negligible compared to the total EGTA of 50 μm) and simulation of the models was carried out with 40 μm of total external buffer Ca²⁺ (comparable with the total EGTA of 50 μm) with the estimated NCE model parameter values. The corresponding model simulations of the dynamics of trans-matrix NCE and NHE fluxes and intra-matrix Na⁺ concentrations are shown in Fig. S1 of the Supplementary Material.

The effect of external buffer Na⁺ on the rate of decrease and magnitude of matrix free Ca²⁺ is shown in Fig. 5 (lower panels, C and D). The simulation and experimental protocols are exactly the same as that in Fig. 5A and B. Specifically, Fig. 5C and D shows the comparison of the regions of model predictions (shaded areas) to the experimental data (points) (Fig. 2 of Cox & Matlib (1993)) on the rate of decrease of matrix free [Ca²⁺] immediately following addition of Na⁺ (C), and matrix free [Ca²⁺] resulting 3 min after addition of Na⁺ (D), both as functions of external buffer [Na⁺] with the total external buffer [Ca²⁺] varying from 1 μm to 40 μm. In plot (C) and (D), the regions of model predictions are shown for both 2Na⁺–Ca²⁺ (A) and 3Na⁺–Ca²⁺ (B) stoichiometry models with the NCE model parameter values estimated from the fitting of the models to the data in Fig. 5A and B with a total external buffer [Ca²⁺] of 1 μm.

The two values of 1 μm and 40 μm used for total buffer Ca²⁺ correspond to a reasonable range of total buffer Ca²⁺ considering the total EGTA used in the external buffer medium to chelate the extruded Ca²⁺ from the matrix is 50 μm. At the low extreme of total buffer Ca²⁺, either model is able to reasonably reproduce the measured data on the dynamics of matrix free Ca²⁺ (continuous lines). However, fitting of the 3: 1 model is more robust. While the 2: 1 model deviates from the data, the 3: 1 model accurately predicts the data over the entire duration of the measurements. The mean residual error at the fitted optimal parameter estimates (least-square error) from the 2: 1 model was approximately 1 × 10⁻², while that from the 3: 1 model was approximately 3.5 × 10⁻³. Furthermore, the model simulated initial rates of decrease of matrix free Ca²⁺ (J_NCE/W_x.β_Ca) using the optimal parameter estimates from the 2: 1 model deviate considerably from the data at higher values of extramitochondrial buffer Na⁺, while the model predictions for the same from the 3: 1 model match the data reasonably well over the entire range of buffer Na⁺. At higher assumed values of total extramitochondrial buffer Ca²⁺, only the 3: 1 model is able to explain the data. Therefore, the 2: 1 model predicts that this experiment is extremely sensitive to total buffer Ca²⁺, while the 3: 1 model predicts that it is not.

The model simulations of matrix free [Ca²⁺] are not in steady state at the beginning of the simulations with the 2: 1 NCE stoichiometry model in the presence of excess extra-matrix buffer Ca²⁺ (Fig. 5A, dotted lines). This is not consistent with the experimental data of Cox & Matlib (1993). This is because with the 2: 1 model in the presence of excess buffer Ca²⁺, the NCE operates in the reverse direction and Ca²⁺ enters the matrix via the NCE in the exchange of Na⁺, assuming the initial matrix [Na⁺] is non-zero. (See Fig. S1(A), dotted lines, of the Supplementary Material for the simulations of NCE fluxes.)

It is further depicted from Fig. 5A and B that neither of the models stabilize even after 200 s of extra-matrix Na⁺ addition. With the 2: 1 NCE stoichiometry model, all simulations of intra-matrix [Ca²⁺] tend to reach a single steady state that is proportional to extra-matrix [Ca²⁺] (since [Ca²⁺]_x/[Ca²⁺]_e = [Na⁺]_x²/[Na⁺]_e² = [H⁺]_x²/[H⁺]_e² at steady state). However, the time intra-matrix [Ca²⁺] takes to reach the steady state depends on the amount of Na⁺ added. In contrast, with the 3: 1 NCE stoichiometry model, virtually all intra-matrix Ca²⁺ is eventually extruded to the buffer medium due to the electrogenic nature of the NCE. The model simulations with the 3: 1 NCE stoichiometry match the data for the entire duration of the experiment, while the simulations with the 2: 1 NCE stoichiometry deviate significantly from the data.

The NCE model parameter values that best explained the dynamic data of Cox & Matlib (1993) were K_Na,NCE = 3.45 mm, K_Ca,NCE = 2.1 μm and X_NCE = 1.76 × 10⁻³ nmol Ca²⁺ mg⁻¹ s⁻¹ (3.52 × 10⁻³ nmol Na⁺ mg⁻¹ s⁻¹) for the 2Na⁺–Ca²⁺ electroneutral exchange model, and K_Na,NCE = 2.4 mm, K_Ca,NCE = 2.1 μm and X_NCE = 3.38 × 10⁻⁵ nmol Ca²⁺ mg⁻¹ s⁻¹ (10.14 × 10⁻⁵ nmol Na⁺ mg⁻¹ s⁻¹) for the 3Na⁺–Ca²⁺ electrogenic exchange model. The estimate of K_Ca,NCE was almost the same from both the models as the mean residual error was relatively insensitive to the kinetic parameter K_Ca,NCE. The estimate K_Na,NCE = 2.4 mm from the 3Na⁺–Ca²⁺ model agrees with the value K_Na,NCE = 2.6 ± 0.15 mm reported in Cox and Matlib, while the estimate K_Na,NCE = 3.45 mm from the 2Na⁺–Ca²⁺ model was significantly higher. Thus, the binding constants K_Na,NCE and K_Ca,NCE obtained here from the dynamic data of Cox & Matlib are of the same order of magnitude as that obtained earlier from the kinetic data of Paucek & Jaburek (2004) (Fig. 3).

Alternative analysis

From the kinetic and dynamic data of Cox & Matlib (1993), it is apparent that the matrix free [Ca²⁺] reaches a steady-state level of 0.1–0.2 μm within 3 min of addition of 5–10 mm of [Na⁺] to the external buffer medium. With the 2: 1 NCE stoichiometry model with the CU blocked (J_CU = 0), this steady state implies J_NCE = 0 and J_NHE = 0, or equivalently, [Ca²⁺]_x/[Ca²⁺]_e = [Na⁺]_x²/[Na⁺]_e² = [H⁺]_x²/[H⁺]_e² ≈ 0.65 (assuming a 0.1 trans-matrix pH gradient), or equivalently, [Ca²⁺]_x ≈ 0.65[Ca²⁺]_e. This indicate that with a 2: 1 NCE stoichiometry, the external buffer medium in the experiment of Cox and Matlib would contain about 0.15–0.3 μm of free [Ca²⁺], or equivalently, about 20–30 μm of total [Ca²⁺] (free + EGTA-bound). Therefore, we performed further analysis of the data of Cox & Matlib with this assumed level of free and total external buffer [Ca²⁺]. Results for this integrated model analysis of the data are shown in Fig. S2 of the Supplementary Material.

The fittings of the individual 2: 1 and 3: 1 NCE models (eqns (2A) and (2B)) to the kinetic data on measured initial rates of decrease of matrix free [Ca²⁺] (J_NCE/W_x.β_Ca) are shown in Fig. S2(C). The integrated model simulations of the dynamic and steady-state kinetic data on matrix free [Ca²⁺] using the fitted 2: 1 and 3: 1 NCE model parameters are shown in Fig. S2(A and B) and S2(D). The NCE model parameter values obtained from these fittings were K_Na,NCE = 2.4 mm, K_Ca,NCE = 2.1 μm and X_NCE = 1.32 × 10⁻³ nmol Ca²⁺ mg⁻¹ s⁻¹ (2.64 × 10⁻³ nmol Na⁺ mg⁻¹ s⁻¹) for a 2: 1 model and K_Na,NCE = 2.4 mm, K_Ca,NCE = 2.1 μm and X_NCE = 3.375 × 10⁻⁵ nmol Ca²⁺ mg⁻¹ s⁻¹ (10.125 × 10⁻⁵ nmol Na⁺ mg⁻¹ s⁻¹) for a 3: 1 model. Note that the estimates of the NCE model parameters are the same as those obtained from Fig. 5 for the 3: 1 model, while the estimates of both K_Na,NCE and X_NCE are reduced for the 2: 1 model. Also the estimates of K_Na,NCE are the same for both the models and agree with the value K_Na,NCE = 2.6 ± 0.15 mm reported in Cox & Matlib (1993). As seen in Fig. S2(C), comparisons of the individual 2: 1 and 3: 1 models to the kinetic data on initial NCE fluxes are excellent; however, while the integrated model with 2: 1 NCE stoichiometry deviates considerably from the dynamic data, the integrated model with 3: 1 NCE stoichiometry is more consistent with the dynamic data throughout the measurement period.

In summary, both 2: 1 and 3: 1 NCE stoichiometry models can fit to the experimental data of Cox & Matlib (1993) on the magnitude and initial rate of decrease of matrix free [Ca²⁺] reasonably well by suitably choosing the NCE model parameters (K_Na,NCE, K_Ca,NCE and X_NCE) with a low total Ca²⁺ in the external buffer medium (e.g. with [Ca²⁺]_tot = 1–10 μm, negligible compared to the total EGTA of 50 μm in the buffer). However, the fittings using the 3: 1 NCE model are consistently better than those obtained using the 2: 1 NCE model. More significantly, the model predictions of the matrix free [Ca²⁺] and the initial rates of Ca²⁺ efflux from the matrix using the 2: 1 NCE model were sensitive to the amounts of total buffer Ca²⁺, while those of the 3: 1 NCE model were not. Therefore, this analysis suggests that the experimental data of Cox and Matlib would be significantly sensitive to an unreported experimental variable (i.e. total external buffer Ca²⁺) if the NCE has a 2: 1 stoichiometry. Small variations in total external buffer Ca²⁺ would lead to larger differences in the measured matrix free [Ca²⁺]. In contrast, with 3: 1 stoichiometry, these results would be insensitive to this variable.

Further validation of the integrated model by independent experimental data

The integrated model of mitochondrial bioenergetics and Ca²⁺ handling and the proposed 3: 1 stoichiometry for the NCE were further validated by analysing an independent experimental data set from McCormack et al. (1989) involving the integrated functions of mitochondrial CU, NCE and NHE. Results of these analyses are summarized in Fig. 6. Specifically, Fig. 6 shows the integrated model simulations (lines) of the experimental data (points ± s.d.) on the steady-state distributions of Ca²⁺ across the IMM in rat heart in the presence of extra-matrix Ca²⁺ (variable) and Na⁺ (10 mm) and/or Mg²⁺ (2 mm), or an uncoupler (1 μm FCCP). To simulate the effects of FCCP, the IMM potential ΔΨ was set to zero (uncoupled mitochondria). The external buffer free [Ca²⁺] was maintained at a desired level with the addition of calculated amounts of EGTA and CaCl₂ to the buffer medium. The integrated model incorporates both 2: 1 and 3: 1 NCE stoichiometry models as well as two different models for the CU, one without inhibition of the CU by matrix free Ca²⁺ and the other with inhibition of the CU by matrix free Ca²⁺. Simulations using the 2: 1 model are shown in the left panels (A and C), while those using the 3: 1 model are shown in the right panels (B and D). Simulations without inhibition of the CU by matrix free Ca²⁺ are shown in the upper panels (A and B), while those with inhibition of the CU by matrix free Ca²⁺ are shown in the lower panels (C and D).

Figure 6. Independent validation of the integrated model of mitochondrial Ca²⁺ handling.

Comparison of model predictions (lines) to the experimental data (points ± s.d.) of McCormack et al. (1989) on steady-state relationships between matrix and external buffer free [Ca²⁺] in purified respiring rat heart mitochondria in the presence of extramitochondrial Na⁺ and/or Mg⁺ or an uncoupler (FCCP). The buffer free [Ca²⁺] was maintained at a desired level with the addition of calculated amount of EGTA and CaCl₂ to the external buffer medium; • with no further addition, ♦ with 10 mm of NaCl, ▪ with 2 mm of MgCl₂, ▴ with 10 mm of NaCl and 2 mm of MgCl₂, and ○ with 1 μm of FCCP. The integrated model was fitted to the data by fixing the Michaelis–Menten binding constants in the models of CU and NCE as estimated before (i.e. K_Ca,CU = 90 μm, K_Na,NCE = 2.4 mm and K_Ca,NCE = 2.1 μm), but allowing the activity parameters X_CU and X_NCE to vary. However, the activity of NCE (X_NCE) was fixed at the same level for all data curves, while the activity of CU (X_CU) was allowed to vary over the data curves (to implicitly account for the inhibition of CU by extramitochondrial free [Mg²⁺]). The left panels (A and C) are for 2: 1 NCE stoichiometry model, while the right panels (B and D) is for 3: 1 NCE stoichiometry model. The upper panels (A and B) is with the CU model without accounting for the inhibition of CU by matrix free [Ca²⁺], while the lower panels (C and D) is with the CU model with accounting for the inhibition of CU by matrix free [Ca²⁺] to better fit the integrated model to the steady-state distribution of Ca²⁺ across the mitochondrial inner membrane. The activity parameter values that were used to fit the model to the data are: A and B, X_NCE′ = 4·X_NCE and X_CU′ = 10·X_CU (•, ♦), X_CU′ = X_CU/12.5 (▪), and X_CU′ = X_CU (▴), and C and D, X_NCE′ = 2.5·X_NCE and X_CU′ = 20·X_CU·K_i/(K_i + [Ca²⁺]_x) (•, ♦), X_CU′ = (X_CU/5.5)·K_i/(K_i + [Ca²⁺]_x) (▪), and X_CU′ = (X_CU/0.6)·K_i/(K_i + [Ca²⁺]_x) (▴), where X_NCE = X_NCE = 3.375 × 10⁻⁵ nmol mg⁻¹ s⁻¹, X_CU = 1.5 × 10⁻² nmol mg⁻¹ s⁻¹, and K_i = 1 μm.

As depicted in Fig. 6, our integrated model is able to satisfactorily simulate the independent experimental data of McCormack et al. (1989). As with the analysis of Cox & Matlib (1993) data, the model simulations of these additional data using the 3: 1 NCE model are also consistently better than those obtained using the 2: 1 NCE model. To simulate these data, the binding constants in the models of CU and NCE were fixed as estimated before (i.e. K_Ca,CU = 90 μm, K_Na,NCE = 2.4 mm K_Ca,NCE = 2.1 μm), but allowing the activity parameters X_CU and X_NCE to vary. The activity of the NCE (X_NCE) was fixed at the same level for all data curves, while the activity of the CU (X_CU) was allowed to vary over the data curves to implicitly account for the inhibition CU by extra-matrix free [Mg²⁺], which is not explicitly accounted for in the present model of the CU. As depicted in Fig. 6A and B, we were not able to fit the steady-state trans-membrane Ca²⁺ distribution data over the entire range of external buffer free [Ca²⁺]. However, by accounting for a specialized inhibition mechanism of the CU by matrix free [Ca²⁺], we were able to fit the model to the data well, as shown in Fig. 6C and D). The activity parameter values that were obtained to fit the model to the data are: A and B: X_NCE′ = 4·X_NCE and X_CU′ = 10·X_CU (control data and data with Na⁺ addition: •, ♦), X_CU′ = X_CU/12.5 (data with Mg²⁺ addition: ▪), and X_CU′ = X_CU (data with both Na⁺ and Mg²⁺ addition: ▴); and C and D: X_NCE′ = 2.5·X_NCE and X_CU′ = 20·X_CU·K_i/(K_i + [Ca²⁺]_x) (•, ♦), X_CU′ = (X_CU/5.5)·K_i/(K_i + [Ca²⁺]_x) (▪), and X_CU′ = (X_CU/0.6)·K_i/(K_i + [Ca²⁺]_x) (▴), where X_NCE = X_NCE = 3.375 × 10⁻⁵ nmol mg⁻¹ s⁻¹, X_CU = 1.5 × 10⁻² nmol mg⁻¹ s⁻¹, and K_i = 1 μm. All the models were able to simulate the effects of FCCP on steady-state (equilibrated) trans-matrix free Ca²⁺ distribution. These analyses further substantiate the validity of the integrated model and support the hypothesis that the stoichiometry of the mitochondrial NCE is 3: 1.

Discussion

Salient features of the kinetic models of the Na⁺–Ca²⁺ transport system

One of the major contributions of the current study is the characterization of the mitochondrial Na⁺–Ca²⁺ transport system with the help of a system of biophysical models of the mitochondrial Ca²⁺ uniporter (CU), Na⁺–Ca²⁺ exchanger (NCE) and Na⁺–H⁺ exchanger (NHE). Our efforts to characterize these cation transporters (CU, NCE and NHE) differ from the previous attempts (Magnus & Keizer, 1997; Cortassa et al. 2003; Nguyen & Jafri, 2005; Cortassa et al. 2006; Nguyen et al. 2007) in that our kinetic models for the fluxes are mechanistic, thermodynamically balanced, and individually parameterized based on a large number of independently published data sets on initial (pseudo-steady) flux rates through the cation transporters, measured in respiring/non-respiring mitochondria. The flux expressions account for the specific effects of inhibitors that are experimentally established, such as the inhibition of the NHE activity by high matrix pH (Kapus et al. 1988, 1989). The kinetic models of the CU, NCE and NHE are further validated by integrating these models into our previously developed, broadly validated model of mitochondrial respiratory system and oxidative phosphorylation (Beard, 2005, 2006; Wu et al. 2007a) to characterize the integrated roles of the CU, NCE and NHE in regulating mitochondrial [Ca²⁺] dynamics in purified cardiac mitochondria.

The CU model introduced here is able to fit to the experimental data of Gunter and co-workers (Wingrove et al. 1984; Gunter & Pfeiffer, 1990; Gunter et al. 1994) on Ca²⁺ influx as a function of IMM potential ΔΨ (Fig. 2C) without introducing non-physical assumptions of previous models (Gunter & Pfeiffer, 1990; Magnus & Keizer, 1997; Cortassa et al. 2003; Cortassa et al. 2006). Specifically, previous models have introduced an offset potential ΔΨ* (≈ 90 mV) and flux expressions that appropriate for potential measured relative to this offset potential. These models were justified based on the explanation that the electrical potential across the CU may not fall to zero concomitantly with the bulk IMM potential, perhaps because of fixed charges producing electric field gradients localized to the CU. However, such models cannot be reconciled with measurements of bulk Ca²⁺ movement between matrix and extramitochondrial space. The current model of the CU is able to account for the observed data based on a mechanistic formulation that is thermodynamically feasible. In doing so the singularity that occurs at ΔΨ = ΔΨ* in previous models does not exist in the current model.

Our CU model simulations show that as IMM depolarizes, mitochondrial Ca²⁺ uptake as well as maximal uptake velocity and saturating Ca²⁺ concentration decreases. This is consistent with experimental observations (Fig. 2C). As shown in Fig. 2D, the Ca²⁺ uptake in cardiac mitochondria saturates beyond [Ca²⁺]_e = 90 μm (K_Ca,CU = 90 μm and X_CU = 2.25 nmol mg⁻¹ s⁻¹). Though precise experimental data were not available to validate these simulations, the Ca²⁺ uptake kinetics are similar to those of the Ca²⁺ currents observed in the studies of Kirichok et al. (2004) using patch-clamp techniques in single myoplasts from cardiac mitochondria. However, their reported value of K_Ca,CU = 19 mm is significantly higher compared to the values obtained here (i.e. K_Ca,CU = 90 μm for cardiac mitochondria and K_Ca,CU = 48 μm for liver mitochondria). The lower K_m values and saturation points are biologically reasonable. Therefore, we were unable to successfully compare our model simulations relating the Ca²⁺ influx and IMM ΔΨ to their data relating the Ca²⁺ current and IMM ΔΨ. Since the data of Scarpa and co-workers (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973) and Gunter and co-workers (Wingrove et al. 1984; Gunter & Pfeiffer, 1990; Gunter et al. 1994) were obtained from the intact mitochondria, we chose these data to characterize the kinetics of the CU instead of the data of Kirichok et al. (2004).

While the available experimental data support a wide range of possible values for the overall activity of the CU in cardiac and liver mitochondria, our model simulations show that even at the highest estimated values, Ca²⁺ current via the CU does not have a significant effect on the IMM ΔΨ. Within the physiological range of ΔΨ, Ca²⁺ uptake via the CU is predicted to vary approximately linearly with the IMM ΔΨ at a physiological level of extramitochondrial Ca²⁺ (fixed). The Ca²⁺ uptake in state 2 and state 4 respiration (with ΔΨ = 190 mV) is predicted to be about 50% higher than that in state 3 respiration (with ΔΨ = 160 mV) at about 1 μm of extramitochondrial Ca²⁺, as depicted in Fig. 2C in liver mitochondria. Under all conditions, our model predicts Ca²⁺ current to be less than 0.001% of the total electron transport chain flux. Therefore, we predict that Ca²⁺ will not significantly depolarize IMM ΔΨ in vivo under physiological conditions.

Kinetics and stoichiometry of the mitochondrial Na⁺–Ca²⁺ exchanger

Another significant contribution of the present modelling efforts to cardiac cell physiology is the characterization of the kinetics and stoichiometry of the mitochondrial NCE which is not well established. To do this, the integrated model of mitochondrial bioenergetics and Ca²⁺ handling was applied to analyse the experimental data of Cox & Matlib (1993) on the kinetics of mitochondrial NCE (i.e. the dynamics of matrix free [Ca²⁺] following extra-matrix Na⁺ additions with the CU blocked) from rabbit heart (Fig. 5 and Figs S1 and S2 of the Supplementary Material). This integrated model analysis reveals that an assumed 3: 1 stoichiometry for the NCE is more consistent with the observed data than the 2: 1 stoichiometry. However, further experimentation may be required to reach a concrete conclusion on the NCE stoichiometry. Reproducing the experiments of Cox & Matlib at known and varying levels of total external buffer Ca²⁺ with enough replicates for statistical analysis would be valuable to concisely verify the proposed 3: 1 stoichiometry for the NCE.

Analyses also suggest that for a 2Na⁺–Ca²⁺ electroneutral exchange with the CU blocked, the trans-membrane gradient of free [Ca²⁺] at steady state would be negligible with the intra-matrix and extra-matrix free [Ca²⁺] reaching a ‘single’ equilibrium steady state depending on the concentrations of intra-matrix and extra-matrix Na⁺ and H⁺ (pH). The following relationship would indeed hold at the steady state: [Ca²⁺]_x/[Ca²⁺]_e = [Na⁺]_x²/[Na⁺]_e² = [H⁺]_x²/[H⁺]_e². However, for a 3Na⁺–Ca²⁺ electrogenic exchange with the CU blocked, the trans-membrane gradient of free [Ca²⁺] at steady state would be significantly high and nearly all the Ca²⁺ from the matrix would be eventually extruded to the extra-matrix buffer space. Since Cox & Matlib (1993) reported only the first 3 min of data from their experiments, the steady state was not evident. Therefore, recording matrix [Ca²⁺] for a longer time would provide better kinetic data for characterizing the kinetics and stoichiometry of the mitochondrial NCE.

However, even without these additional data, we were able to validate both our integrated model of mitochondrial bioenergetics and Ca²⁺ handling and 3: 1 NCE stoichiometry, based on analysis of an additional independent data set from the experimental work of McCormack et al. (1989). As in the case of the analysis of the data from Cox & Matlib (1993), model simulations of these additional data set using the 3: 1 NCE model are consistently better than those obtained using the 2: 1 NCE model. The model analyses also showed a matrix free Ca²⁺-mediated regulation of the CU flux in regulating the matrix free [Ca²⁺] in respiring cardiac mitochondria. Based on these analyses, we conclude that the mitochondrial NCE is electrogenic with a stoichiometry of 3: 1, and the matrix free [Ca²⁺] is regulated by a Ca²⁺-induced Ca²⁺ regulation mechanism.

Supplemental material

Online supplemental material for this paper can be accessed at: http://jp.physoc.org/cgi/content/full/jphysiol.2008.151977/DC1 and http://www.blackwell-synergy.com/doi/suppl/10.1113/jphysiol.2008.151977