Computational analysis of Ca²⁺ dynamics in isolated cardiac mitochondria predicts two distinct modes of Ca²⁺ uptake

Abstract

Cardiac mitochondria can act as a significant Ca²⁺ sink and shape cytosolic Ca²⁺ signals affecting various cellular processes, such as energy metabolism and excitation–contraction coupling. However, different mitochondrial Ca²⁺ uptake mechanisms are still not well understood. In this study, we analysed recently published Ca²⁺ uptake experiments performed on isolated guinea pig cardiac mitochondria using a computer model of mitochondrial bioenergetics and cation handling. The model analyses of the data suggest that the majority of mitochondrial Ca²⁺ uptake, at physiological levels of cytosolic Ca²⁺ and Mg²⁺, occurs through a fast Ca²⁺ uptake pathway, which is neither the Ca²⁺ uniporter nor the rapid mode of Ca²⁺ uptake. This fast Ca²⁺ uptake component was explained by including a biophysical model of the ryanodine receptor (RyR) in the computer model. However, the Mg²⁺-dependent enhancement of the RyR adaptation was not evident in this RyR-type channel, in contrast to that of cardiac sarcoplasmic reticulum RyR. The extended computer model is corroborated by simulating an independent experimental dataset, featuring mitochondrial Ca²⁺ uptake, egress and sequestration. The model analyses of the two datasets validate the existence of two classes of Ca²⁺ buffers that comprise the mitochondrial Ca²⁺ sequestration system. The modelling study further indicates that the Ca²⁺ buffers respond differentially depending on the source of Ca²⁺ uptake. In particular, it suggests that the Class 1 Ca²⁺ buffering capacity is auto-regulated by the rate at which Ca²⁺ is taken up by mitochondria.

Introduction

Mitochondrial Ca²⁺ is known to play a crucial role in cardiac physiology and pathophysiology. Under physiological conditions, it regulates mitochondrial ATP production and shapes cytosolic Ca²⁺ oscillations (Dedkova & Blatter, 2008, 2013; Drago et al. 2012; Rizzuto et al. 2012); under pathophysiological conditions, it leads to excess mitochondrial reactive oxygen species generation, mitochondrial dysfunction and cell death (Celsi et al. 2009; Stowe & Camara, 2009; Kurdi & Booz, 2011; Luo & Anderson, 2013). A number of Ca²⁺ uptake pathways have been identified in the inner mitochondrial membrane (IMM) (Kirichok et al. 2004; Gunter & Sheu, 2009; Jiang et al. 2009; Ryu et al. 2011; Jean-Quartier et al. 2012). Among them, the Ca²⁺ uniporter (CU) is the most widely studied and regarded as the primary Ca²⁺ uptake pathway in mitochondria (Dash & Beard, 2008; Dash et al. 2009; Pradhan et al. 2010, 2011). However, two important observations need to be considered: (i) recent patch-clamp studies on mitoplasts have revealed a very high half-activation constant (19 mm) for the CU (Kirichok et al. 2004), and (ii) cytosolic Mg²⁺ significantly inhibits the CU at physiological concentrations (high nanomolar range) of cytosolic Ca²⁺ (Favaron & Bernardi, 1985; Pradhan et al. 2011; Boelens et al. 2013). Recent biophysical modelling work by Pradhan et al. (2011) on the kinetics of mitochondrial Ca²⁺ uptake via the CU, based on extensive available experimental data from the literature (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973; Crompton et al. 1976; Bragadin et al. 1979; Wingrove et al. 1984; McCormack et al. 1989; Szanda et al. 2009), suggests that the intrinsic K_m of Ca²⁺ for the CU is ∼3–5 μm and the intrinsic K_m of Mg²⁺ for the CU is ∼0.5–0.75 mm, implying that the apparent K_m of Ca²⁺ for the CU is ∼45–90 μm at physiological concentrations of cytosolic Mg²⁺ (∼1 mm). Such a scenario suggests that the CU might be important in sequestering Ca²⁺ inside the mitochondria when the cytosolic Ca²⁺ is very high. This begs a few questions: What happens under physiological conditions when Ca²⁺ uptake through the CU is significantly reduced due to cytosolic Mg²⁺? Is the Ca²⁺ influx alone through the CU sufficient to modulate the mitochondrial respiration rate? If not, then which pathway is responsible for Ca²⁺ uptake in mitochondria under physiological conditions?

A recent experimental study in isolated guinea pig cardiac mitochondria by Boelens et al. (2013) addressed some of these questions. They measured mitochondrial Ca²⁺ uptake in the presence of several concentrations of extra-matrix MgCl₂ in response to various concentrations of CaCl₂ boluses added to the experimental buffer. They also measured relative changes in key mitochondrial bioenergetic variables (redox state, membrane potential and respiration rate) under the same experimental conditions. It was observed that the dynamics of mitochondria Ca²⁺ uptake is very different in the presence versus absence of extra-matrix MgCl₂. We first analysed these experimental data using a recent integrated model of mitochondrial bioenergetics and cation handling by Bazil et al. (2013). However, the mitochondrial Ca²⁺ uptake experiments could not be explained using this integrated model when MgCl₂ was present in the experimental buffer. Note that this integrated model included a mechanistically detailed kinetic model of the CU that accounted for its inhibition by Mg²⁺ (Pradhan et al. 2011). Model analyses of the experiments suggested the existence of at least two different mitochondrial Ca²⁺ uptake pathways: a high affinity fast uptake pathway and a low affinity slow uptake pathway, the latter having the characteristics of the CU.

The kinetic profile of the fast Ca²⁺ uptake component was indicative of Ca²⁺-induced Ca²⁺ uptake (CICU). This suggested Ca²⁺ uptake by the ryanodine receptor (RyR)-type channel (RTC), but not the previously characterized rapid-mode Ca²⁺ uptake pathway (RaM) (Sparagna et al. 1995; Buntinas et al. 2001; Bazil & Dash, 2011). Accordingly, a new fast Ca²⁺ uptake pathway using a recent integrated model of mitochondrial bioenergetics and cation handling (Bazil et al. 2013) was refined here to explain the mitochondrial Ca²⁺ uptake measurements of Boelens et al. (2013). This new pathway was mimicked using the Keizer–Levine model of cardiac sarcoplasmic reticulum (SR) RyR with Ca²⁺-dependent adaptation (Keizer & Levine, 1996). The Keizer–Levine model was also modified to exhibit the cytosolic Mg²⁺-dependent modulation of the RyR adaptation, as previously reported by Valdivia et al. (1995). We also simplified the CU model of Pradhan et al. (2011) to have the CU Mg²⁺ binding site facing only the cytosolic side based on recent experimental evidence about the CU structure (Perocchi et al. 2010; De Stefani et al. 2011) and the finding that MgCl₂ added to experimental buffer does not increase matrix Mg²⁺ concentration ([Mg²⁺]_m) (Boelens et al. 2013).

The extended integrated mitochondrial model (shown in Fig.1A) invokes a total of only eight unknown parameters related to the CU, RTC and Ca²⁺ sequestration, which were optimized to explain a total of 30 mitochondrial Ca²⁺ uptake experiments from Boelens et al. (2013). The unified Ca²⁺ uptake pathway (CU + RTC) model along with Ca²⁺ egress and sequestration was corroborated by simulating an independent set of experiments consisting of 48 data sets by Blomeyer et al. (2013), featuring mitochondrial Ca²⁺ uptake, egress and sequestration.

Figure 1. Integrated mitochondrial model and modified model components.

A, schematic diagram of the integrated model of mitochondrial bioenergetics and Ca²⁺ handling of Bazil et al. (2013) with the revised CU model and an additional RTC model (highlighted with an orange box). B, mitochondrial Ca²⁺ sequestration system depicting how Ca²⁺ is differentially buffered depending on how it is taken up by mitochondria (see arrows). C, modified CU model for Mg²⁺ inhibition of the CU occurring only at the cytosolic side. D, Keizer–Levine model of cardiac RyR modified to include modulation of the RTC to account for the effects by cytosolic Mg²⁺. The dotted box represents Mg²⁺ binding with the C₂ state, which is assumed to be in rapid equilibrium.

Methods

Experimental data used for model parameterization and corroboration

Recently published mitochondrial Ca²⁺ uptake experiments from Boelens et al. (2013) were used to characterize the relative contribution of Ca²⁺ influx from the slow (CU) and fast (RTC) Ca²⁺ uptake pathways as well as the Ca²⁺ buffering capacity from the Class 1 and Class 2 Ca²⁺ buffers in mitochondria (Bazil et al. 2013; Blomeyer et al. 2013). Mitochondrial Ca²⁺ uptake was measured spectrofluorometrically in isolated guinea pig cardiac mitochondria suspended in an experimental buffer containing 140 mm KCl, 5 mm inorganic phosphate (Pi), 1 mm EGTA, pH 7.15, and variable MgCl₂ (0–2 mm), with different concentrations of CaCl₂ (0.0–0.6 mm) added to the experimental buffer as boluses to provide different Ca²⁺ uptake responses. The time line of the various perturbations was as follows: mitochondria (0.5 mg protein ml⁻¹) were first added to the buffer and were then energized by adding 0.5 mm pyruvic acid (PA) as substrate; different amounts of CaCl₂ were then added at t = 120 s to induce Ca²⁺ uptake by mitochondria. The experiments were conducted at room temperature.

An independent set of experiments carried out by Blomeyer et al. (2013) in isolated guinea pig cardiac mitochondria examining Ca²⁺ uptake, egress and sequestration were used to further corroborate the extended integrated mitochondrial bioenergetics and cation handling model; this model has two distinct Ca²⁺ uptake pathways, which were included to determine if the model can describe the data just as well as that modelled by Bazil et al. (2013). In Blomeyer et al. (2013), the experimental buffer was the same as that of Boelens et al. (2013), except that there was 25-fold less EGTA (40 μm) and MgCl₂ was not added to the experimental buffer. In these experiments, various concentrations of CaCl₂ (0, 10, 20, 30 and 40 μm) were added as boluses to the experimental buffer at t = 120 s (after adding mitochondria to the experimental buffer at t = 0 s and then energizing the mitochondria with substrate PA at t = 60 s) to observe the differing extra-matrix and matrix Ca²⁺ concentration ([Ca²⁺]_e and [Ca²⁺]_m) transient characteristics that were used, for the first time, to characterize the mitochondrial Ca²⁺ sequestration system (Bazil et al. 2013). The resulting novel mitochondrial Ca²⁺ sequestration model was used in the presented Ca²⁺ handling model. Various concentrations of NaCl were added to the experimental buffer after stabilization of mitochondrial Ca²⁺ uptake and [Ca²⁺]_m, and further Ca²⁺ uptake was completely blocked by ruthenium red (RR; 25 μm) to probe the kinetics of the Na⁺/Ca²⁺ exchanger (NCE). Specifically, at t = 300 s, RR was added to stop further Ca²⁺ uptake, and at t = 360 s, different amounts of NaCl were added to initiate Ca²⁺ efflux that provides the kinetics of the NCE. These experiments were also conducted at room temperature.

Mathematical model of mitochondrial bioenergetics and cation handling

The present mathematical model of mitochondrial bioenergetics and cation handling is extended from Bazil et al. (2013) and Dash & Beard (2008) (Fig.1A). Briefly, the integrated model includes the kinetics of the reactions at complexes I, III, IV and V (F₀F₁-ATPase) of the electron transport chain that comprise oxidative phosphorylation; the adenine nucleotide translocase (ANT) and phosphate-H⁺ cotransporter (PHT) for ATP/ADP exchange and phosphate transport; and the cation transporters CU, RTC, NCE, Ca²⁺/H⁺ exchanger (CHE), Na⁺/H⁺ exchanger (NHE) and K⁺/H⁺ exchanger (KHE), and the H⁺ leak across the IMM. It also includes the passive metabolite transport fluxes of adenine nucleotides and phosphates (ATP, ADP, Pi) across the outer mitochondrial membrane (OMM), the binding of protons (H⁺) and metal ions (Na⁺, K⁺, Mg²⁺ and Ca²⁺) to metabolites (ATP, ADP and Pi), as well as the newly developed model of the mitochondrial Ca²⁺ sequestration system. This Ca²⁺ sequestration model employs two classes of endogenous Ca²⁺ buffers and successfully explains the experimentally measured mitochondrial Ca²⁺ buffering capacity and [Ca²⁺]_m dynamics under various cation perturbations in isolated guinea pig cardiac mitochondria (Blomeyer et al. 2013). The first class represents the prototypical Ca²⁺ buffering systems (mostly metabolites and mobile proteins), and the second class encompasses the complex binding events associated with amorphous calcium phosphate (ACP) formation (Bazil et al. 2013) (see Fig.1B). The components added, modified or found redundant, based on the experimental data, in the original Bazil et al. (2013) model are briefly described below.

Bazil and Dash (2011) previously developed a mathematical model of Ca²⁺ uptake through the RaM that did not possess the cytosolic Ca²⁺ graded mitochondrial Ca²⁺ uptake characteristics, as observed in the experiments of Boelens et al. (2013). Also, the Ca²⁺ uptake via the RaM is usually small (Bazil & Dash, 2011; Bazil et al. 2013), and hence can be lumped with the other Ca²⁺ uptake pathways (i.e. the CU and RTC), which are individually considered here. Therefore, the RaM Ca²⁺ uptake pathway was not considered in the integrated model. Recent experimental evidence based on structural studies (Perocchi et al. 2010; De Stefani et al. 2011) suggest that the MICU1 – the CU EF hand – is orientated towards the cytosol. Moreover, it is known that Mg²⁺ saturates the Ca²⁺ binding sites of many EF hands (Grabarek, 2011). Based on these recent findings, the CU model of Pradhan et al. (2011) was modified so that Mg²⁺ inhibition of the CU occurred only via the binding of cytosolic Mg²⁺ to the CU. Our revised schema of CU inhibition by cytosolic Mg²⁺ is shown in Fig.1C. After removing the CU inhibition by matrix Mg²⁺, we re-estimated the CU model parameters reported by Pradhan et al. (2011) to satisfactorily describe all the Mg²⁺-dependent and Mg²⁺-independent Ca²⁺ uptake data via the CU (Scarpa & Graziotti, 1973; Vinogradov & Scarpa, 1973; Crompton et al. 1976; Bragadin et al. 1979; Wingrove et al. 1984; McCormack et al. 1989; Szanda et al. 2009). The revised CU flux expression, the optimized set of CU model parameters, and the CU model fitting to the available Ca²⁺ uptake data are provided in the Supplementary Material (see Fig. A.1 and Table A.5).

As described in the mitochondrial Ca²⁺ uptake experiments of Boelens et al. (2013), there was a fast CICU that was unaffected by increasing [Mg²⁺]_e (0.125–2.0 mm). Because this Ca²⁺ graded Ca²⁺ uptake response had kinetic properties analogous to that of RyR, we named this Ca²⁺ uptake pathway as RTC. We tried two widely used biophysical models of RyR to imitate this CICU pathway found in the experimental data: the Keizer–Levine (Keizer & Levine, 1996) and Tang–Othmer (Tang & Othmer, 1994) models. Of the two, the Keizer–Levine model could better explain the fast CICU phenomenon seen in the experiments of Boelens et al. (2013). Cytosolic Mg²⁺ ions are known to increase the rate of adaption and inhibit the open probability of the RyR (O'Brien, 1986; Valdivia et al. 1995). Interestingly, we also observed similar effect in the CICU kinetics when [Mg²⁺]_e was raised from 0 to 0.125 mm. However, there was no apparent change when [Mg²⁺]_e was raised further, for the extra-matrix Ca²⁺ range probed. Therefore, we modified the Keizer–Levine model to account for the cytosolic Mg²⁺-dependent changes in RyR opening and adaptation. Mathematically, this was done by adding an additional state, , in the Keizer–Levine model, which depends on [Mg²⁺]_e (Fig.1D). For simplicity, we assumed that Mg²⁺ binding with the C₂ state is in rapid equilibrium (denoted by dotted lines in Fig.1D). Effectively, it results in multiplying the transition rate from C₂ to O₁ by a function:

(1)

Here K_m is the half-maximal Mg²⁺ concentration of RyR inhibition and n_m is the Hill coefficient. Experiments conducted during the last decade suggest that cardiac mitochondria possess the RyR1 isoform or the skeletal muscle isoform (Beutner et al. 2005). Therefore, the value of K_m is chosen to be that of skeletal muscle RyR. But the value of the Hill coefficient of RyR inhibition and adaptation by Mg²⁺ ions is that of cardiac RyR, because experimental data pertaining to Mg²⁺ modulated adaptation of the mitochondrial or skeletal muscle isoform of RyR are not available. However, an effect of Mg²⁺ ions on skeletal muscle RyR adaptation is expected (Valdivia et al. 1995). Both values are listed in Table A.6.

Note that the effect of cytosolic Mg²⁺ on the RTC open probability and adaptation is minimal for the range of Ca²⁺ levels probed in the experiments of Boelens et al. (2013), as shown in Fig. A.3. Also, based on the proposed formulation on Mg²⁺ modulation of the RTC, the effect of cytosolic Mg²⁺ over the Ca²⁺-dependent adaptation of the RTC is not robust, but the RTC open probability is affected, depending on the level of cytosolic Ca²⁺, as depicted in Fig. A.3 (B–D, F–H). More robust inhibition of the RTC open probability and adaptation can be obtained through an alternative formulation for Mg²⁺ binding step in the Keizer–Levine model, as shown in Fig. A.3 (I–L). In this alternative formulation, cytosolic Mg²⁺ binds rapidly with the O₁ state instead of C₂. Mathematically, it is achieved by multiplying by . However, this type of RTC model could not explain the cardiac mitochondrial Ca²⁺ uptake experiments of Boelens et al. (2013), as depicted in Fig. A.4.

We modelled Ca²⁺ influx through the RTC using a centred Eyring energy barrier with saturable binding sites on either side of the channel; this is given by the following expression:

(2)

F is Faraday's constant (96,487 J mol⁻¹ mV⁻¹), R is the ideal gas constant (8.314 J mol⁻¹ K⁻¹), T is temperature (298 K), Z_Ca is Ca²⁺ valence (+2 unitless) and is the mitochondrial membrane potential. and govern the fraction of open RTC and are obtained from the model shown in Fig.1D (see Supplementary Material for details). X_RTC represents RTC activity and is a Ca²⁺-based saturation parameter assumed to be equal for the extra-matrix and matrix side (estimated values are listed in Table 1).

Table 1.

Model parameters estimated from high Ca²⁺ (Boelens et al. 2013) and low Ca²⁺ (Blomeyer et al. 2013) bolus experiments

Parameter	Description	Value	Unit
High CaCl2 boluses with high EGTA (1 mm)
[BCa, 1]m	Total Class 1 Ca2+ buffer concentration*	4.1	mm
KCa, 1	Ca2+ affinity for Class 1 Ca2+ buffers	2.7	μm
[BCa,2]m	Total Class 2 Ca2+ buffer concentration*	20.7	mm
KCa, 2	Ca2+ affinity for Class 2 Ca2+ buffers	1.7	μm
XRTC	Mitochondrial RTC activity	2.5	nmol (mg protein)−1 s−1
	Ca2+ affinity for RTC activation	9.1	μm
XCU	Mitochondrial CU activity	1.1 × 10−2	nmol (mg protein)–1 s−1
	Mg2+ affinity for CU inhibition	0.42	mm
Low CaCl2 boluses with low EGTA (40 μm)
XRTC	Mitochondrial RTC activity	4.0 × 10−3	nmol (mg protein)−1 s−1
XCU	Mitochondrial CU activity	1.6 × 10−2	nmol (mg protein)−1 s−1

^*Class 1 and Class 2 Ca²⁺ buffers have the same number of binding sites as initially proposed, i.e. 1 and 6, respectively, by Bazil et al. (2013).

Parameter estimation

There are eight parameters in the extended mitochondrial model that had to be estimated to sufficiently explain the experimental data from Boelens et al. (2013). These parameters are: total concentration of Class 1 Ca²⁺ buffers ([B_{Ca, 1}]_m), Ca²⁺ affinity of Class 1 Ca²⁺ buffers (K_{Ca, 1}), total concentration of Class 2 Ca²⁺ buffers ([B_Ca,2]_m), Ca²⁺ affinity of Class 2 Ca²⁺ buffers (K_{Ca, 2}), activity of the CU (X_CU), [Mg²⁺]_e affinity for the CU (), X_RTC and . These parameters were estimated using a simultaneous least-squares fitting of the model to the experimental data from Boelens et al. (2013):

(3)

Here, δ is the set consisting of the eight parameters, i runs over the total number of experiments (N_exp = 30) comprising different MgCl₂ (N_exp,Mg = 6) and CaCl₂ (N_exp,Ca = 5) perturbations, j runs over the time-points during the course of the experiment at which model outputs were computed and compared with the experimental observations, and N_data is the total number of such time-points. and are the experimentally measured [Ca²⁺]_m or [Ca²⁺]_e and model-simulated [Ca²⁺]_m or [Ca²⁺]_e, respectively, for jth time-point of the ith experiment; is the variance associated with . Minimization of the mean residual error (objection function) for optimal estimation of the model parameter set δ is carried out using a MATLAB (The MathWorks, Natick, MA, USA) based code of the FORTRAN package PIKAIA implementing Genetic Algorithm (Charbonneau, 2002). The list of estimated model parameter values is presented in Table 1. All other (fixed) parameters are listed in Table A.1–A.6 and Table C.1.

Results

We illustrate the parameterization and corroboration of the integrated model of mitochondrial bioenergetics and cation handling shown in Fig.1A using the experimental data of Boelens et al. (2013) and Blomeyer et al. (2013). Only the necessary model parameters related to the mitochondrial Ca²⁺ sequestration, CU and RTC were estimated based on the experimental data (Blomeyer et al. 2013; Boelens et al. 2013). Model simulations for the data of Boelens et al. (2013) are shown in Figs. 2 and 3. Corroboration simulations using the data of Blomeyer et al. (2013) are shown in Fig.6. Estimated parameter values used for model simulations are shown in Table 1. The experimental conditions are the same for each experiment except for the EGTA concentrations and perturbations.

Figure 2. Model simulations of experimentally measured [Ca²⁺]_e with different [Mg²⁺]_e and CaCl₂ boluses from Boelens et al. (2013).

Extra-matrix [Ca²⁺] dynamics (filled circles with error bars) in response to added CaCl₂ boluses of 0 mm (blue), 0.25 mm (green), 0.4 mm (red), 0.5 mm (cyan) and 0.6 mm (magenta) with experimental buffer [MgCl₂] = 0 mm (A) and 1 mm (C), also containing 1 mm EGTA. The dynamics of [Ca²⁺]_e are not affected in the presence (A) or absence of MgCl₂ (C). The model simulations (continuous lines) with [MgCl₂] = 0 mm (A) and 1 mm (C) are shown with the experimental data. The model simulations with [MgCl₂] = 0.25 mm are also shown (B), for which no experimental data are shown because Boelens et al. (2013) did not measure [Ca²⁺]_e with [MgCl₂] = 0.25 mm. The model simulations show that mitochondria are exposed to high [Ca²⁺]_e only for a short duration (for the duration of CaCl₂ bolus addition and Ca²⁺ chelation by EGTA). Note that such a phenomenon is not captured in the experiments due to the limitation of the data acquisition system.

Figure 3. Model simulations of experimentally measured [Ca²⁺]_m with different [Mg²⁺]_e and CaCl₂ boluses from Boelens et al. (2013).

Experimentally measured matrix [Ca²⁺] dynamics (filled circles with error bars) and model simulations (continuous lines) in response to added CaCl₂ boluses of 0 mm (blue), 0.25 mm (green), 0.4 mm (red), 0.5 mm (cyan) and 0.6 mm (magenta). The experimental buffer contained either 0 mm (A), 0.125 mm (B), 0.25 mm (C), 0.5 mm (D), 1 mm (E) or 2 mm (F) MgCl₂, also containing 1 mm EGTA. The estimated parameters corresponding to the mitochondrial Ca²⁺ sequestration system, CU and RTC that provide the model fittings to these experimental data are listed in Table 1. In a few cases, to account for experimental variability, the mitochondrial Ca²⁺ load was varied by ±10%.

Figure 6. Corroboration simulations using the updated integrated model of mitochondrial bioenergetics and cation handling.

Extra-matrix and matrix [Ca²⁺] dynamics are shown for 0 μm CaCl₂ bolus (A, B), 10 μm CaCl₂ bolus (C, D), 20 μm CaCl₂ bolus (E, F), 30 μm CaCl₂ bolus (G, H) and 40 μm CaCl₂ bolus (I, J). The experimental buffer contained 40 μm EGTA and 0 mm MgCl₂. The experimental data are shown with open circles and error bars, while the model simulations are shown with continuous lines above the data. To simulate these experimental data, only the CU and RTC activities were scaled (values shown in Table 1). Fixed model parameters are the same as listed in Table A.1–A.6 and C.1. Mitochondrial Ca²⁺ sequestration system parameters are the same as reported by Bazil et al. (2013). In a few cases, to account for experimental variability, the mitochondrial Ca²⁺ load was varied by ±10%.

Model simulations of extra-matrix and matrix [Ca²⁺]

Due to the high EGTA concentration (1 mm) in the experimental buffer of Boelens et al. (2013), [Ca²⁺]_e was relatively unchanged in the presence of different [Mg²⁺]_e. Therefore, during the estimation process of model parameters, only the [Ca²⁺]_e measurements when [Mg²⁺]_e = 0 were included in the objective function, defined by equation (3). Figure2 shows [Ca²⁺]_e data (closed circles with error bars) and model simulations (continuous lines) when [Mg²⁺]_e is equal to 0 and 1 mm, respectively. Note that Fig.2B shows only model simulations because [Ca²⁺]_e measurements at a value other than 0 and 1 mm [Mg²⁺]_e were not available from the experiments of Boelens et al. (2013). As observed in these plots, the model simulations correspond to the experimental data on [Ca²⁺]_e for different [Mg²⁺]_e reasonably well.

Figure3 shows the [Ca²⁺]_m dynamics in response to added boluses of CaCl₂ to the experimental buffer (shown in Fig.2) with different levels of [Mg²⁺]_e (in mm): 0 (A), 0.125 (B), 0.25 (C), 0.5 (D), 1.0 (E) and 2.0 (F). Model simulations (continuous lines) are shown on the top of the experimental data (closed circles with error bars). There are no apparent differences in the [Ca²⁺]_m dynamics with [Mg²⁺]_e equal to 0.125 mm and greater. The net mitochondrial Ca²⁺ uptake that was observed is through the CU and RTC (schematized in Fig.1C and D). We therefore posit that the major Ca²⁺ uptake is through the CU, which is almost completely inhibited by a [Mg²⁺]_e of only 0.125 mm and the fast Ca²⁺ uptake, which is slightly inhibited by [Mg²⁺]_e, is through the RTC. Note that the fast uptake is also inhibited by 0.125 mm [Mg²⁺]_e, but is relatively unaffected by greater increases in [Mg²⁺]_e, as depicted further by the effects of cytosolic Mg²⁺ on the RTC open probability and adaptation in Fig. A.3 (E–H).

Model-simulated Ca²⁺ fluxes and mitochondrial Ca²⁺ buffering capacity

Figure4 shows the model-simulated Ca²⁺ fluxes through the CU (J_CU) and RTC (J_RTC) for [Mg²⁺]_e = 0, 0.25 and 1 mm, which can be used to understand the relative contribution of the two Ca²⁺ uptake pathways in regulating the [Ca²⁺]_m dynamics. It is apparent from Fig.4A–C that there is a significant reduction in Ca²⁺ flux via the CU with [Mg²⁺]_e increased to 0.25 mm (B) from 0 mm (A), and which is completely inhibited when [Mg²⁺]_e is increased to 1 mm (C). On the other hand, there is marginal or no change in Ca²⁺ flux through the RTC with an increase in [Mg²⁺]_e (compare Fig.4D–F), consistent with the simulations of the Ca²⁺-dependent open probability and adaptation of the RTC in Fig. A.3 (E–H). Although not visible from the figures shown, Ca²⁺ flux through the RTC reaches zero with elapsed time even though the [Ca²⁺]_e is clamped at different concentrations depending on the initial CaCl₂ bolus (see Fig.2). The reason is the time-dependent adaption of the RTC, a property that is followed from the Keizer–Levine model (Keizer & Levine, 1996) used to imitate the fast CICU (see Fig. A.3). However small, there is a continuous uptake of Ca²⁺ through the CU even long after the RTC closes (seen in Fig.4A) and this is responsible for the slow Ca²⁺ uptake primarily visible in Fig.3 for a CaCl₂ bolus of 0.6 mm.

Figure 4. Model predicted fluxes via the CU and RTC corresponding to the experimental data of Boelens et al. (2013).

Mitochondrial Ca²⁺ uptake fluxes through the CU (A–C) and RTC (D–F) in the absence (A, D) and in the presence of 0.25 mm (B, E) and 1 mm (C, F) MgCl₂ in the experimental buffer. To avoid the sudden surges in [Ca²⁺]_e-induced Ca²⁺ uptake at the time of CaCl₂ bolus addition, the fluxes are shown from time t+0.5 s onwards (the colour scheme has the same meaning as in Fig.3). Increasing [Mg²⁺]_e completely inhibits the CU-mediated Ca²⁺ uptake (A–C), whereas Ca²⁺ uptake via the RTC is affected only marginally (D–F). The model simulations of Ca²⁺ fluxes are based on experimental conditions as outlined by Boelens et al. (2013) with the estimated model parameters as given in Table 1. Note that the ordinate limits in B, C and E, F are the same as in A, D, and hence have been omitted for clarity.

Based on the estimated model parameter values for the Class 1 and Class 2 endogenous Ca²⁺ buffers, we computed β_Ca,m (mitochondrial Ca²⁺ buffering capacity) and total mitochondrial Ca²⁺ concentration ([TCa²⁺]_m) as functions of [Ca²⁺]_m [for details on how to compute

them, see Supplementary Material or Bazil et al. (2013)]. It is apparent from Fig.5A that β_Ca,m is ∼10³ when [Ca²⁺]_m is ∼0.5 μm, but increases by two orders of magnitude to ∼10⁵ when [Ca²⁺]_m approaches 2 μm. The model predicts a decrease in β_Ca,m when [Ca²⁺]_m increases above 2 μm. Our modelling framework suggests that the ability of cardiac mitochondria to sequester total Ca²⁺ is about ∼125 mm (see Fig.5B). This estimate of total mitochondrial Ca²⁺ sequestration is equivalent to ∼125 nmol Ca²⁺ (mg protein)^–1 (assuming 1 mg of mitochondrial protein is about 1 μL of mitochondrial water space), which is the same as reported by Bazil et al. (2013).

Figure 5. Estimated mitochondrial Ca²⁺ sequestration system for high CaCl₂ bolus experiments.

Mitochondrial Ca²⁺ buffering capacity (A) and total mitochondrial Ca²⁺ concentration, i.e. the sum of matrix-free Ca²⁺ and Ca²⁺ bound with Class 1 and Class 2 Ca²⁺ buffers (B), as estimated from the experimental data of Boelens et al. (2013). Evidently, there is much less matrix Ca²⁺ bound with the Class 1 Ca²⁺ buffers, suggesting that most of the Ca²⁺ is buffered by Class 2 Ca²⁺ buffers when the majority of Ca²⁺ uptake is via the RTC. The estimated model parameters characterizing the mitochondrial Ca²⁺ sequestration system are as listed in Table 1.

Corroboration simulations

We used the integrated mitochondrial bioenergetics and cation handling model with the unified slow (CU) and fast (RTC) Ca²⁺ uptake mechanisms to also explain the independent experimental data set of Blomeyer et al. (2013). Fixed model parameters are the same as listed in Table A.1–A.6 and C.1, and mitochondrial Ca²⁺ sequestration system parameters are the same as estimated by Bazil et al. (2013); only the activities of the CU and RTC were scaled to match the experiments of Blomeyer et al. (2013) (values listed in Table 1). It is evident from the model simulations that the unified Ca²⁺ uptake pathway (CU + RTC) is able to explain the data just as well (compare Fig.6 in the present study with Fig. 3 of Bazil et al. (2013)). Of interest is that the relative fluxes through the two Ca²⁺ uptake pathways when there is a high CaCl₂ addition (0.0–0.6 mm), as in Boelens et al. (2013), versus a low CaCl₂ addition (0–40 μm), as in Blomeyer et al. (2013), are very different (respective values are listed in Table 1). Individual Ca²⁺ fluxes via the CU and RTC are shown in Fig.7A and B, respectively. Net mitochondrial Ca²⁺ buffering capacity and [TCa²⁺]_m are shown in Fig.7C and D, respectively. These simulations mostly agree with the simulations reported in Bazil et al. (2013).

Figure 7. Model predicted Ca²⁺ uptake fluxes via the CU and RTC, and mitochondrial Ca²⁺ sequestration system for the corroboration simulations.

Mitochondrial Ca²⁺ uptake rates via the CU (A) and RTC (B), mitochondrial Ca²⁺ buffering capacity (C), and total mitochondrial Ca²⁺ concentration (D), as estimated from the experimental data of Blomeyer et al. (2013). As there is no added MgCl₂ in the experimental buffer, A and B are comparable with Fig.4A and D. C and D are comparable to those reported by Bazil et al. (2013). The parameter values used for these model simulations are those described for the model corroboration simulations in Fig.6.

Discussion

We have developed here a quantitative framework for the mitochondrial Ca²⁺ uptake system using the recent experimental data of Boelens et al. (2013) in isolated guinea pig heart mitochondria. The rationale was to characterize the mitochondrial Ca²⁺ uptake mechanisms under physiological conditions of cytosolic Ca²⁺ (high nanomolar range) when the majority of mitochondrial Ca²⁺ uptake via the CU is inhibited by cytosolic Mg²⁺ ions in the sub-millimolar range (Favaron & Bernardi, 1985; Boelens et al. 2013). A recent model of mitochondrial bioenergetics and cation handling (Bazil et al. 2013) used to characterize the experiments of Blomeyer et al. (2013) was not sufficient to explain the experiments of Boelens et al. (2013), as depicted in Fig. A.2. This suggested differences in the mitochondrial Ca²⁺ uptake mechanisms in the presence of extra-matrix Mg²⁺ and addition of high CaCl₂ boluses (with 1 mm EGTA), which were absent in the experiments of Blomeyer et al. (2013) where low CaCl₂ boluses (with 40 μm EGTA) were added. Therefore, we modified/updated the mitochondrial Ca²⁺ uptake system of the recent mitochondrial cation handling model of Bazil et al. (2013) to quantitatively reproduce the mitochondrial Ca²⁺ uptake experiments of Boelens et al. (2013). Our modified model could also qualitatively reproduce the mitochondrial Ca²⁺ uptake, egress and sequestration experiments of Blomeyer et al. (2013) where Mg²⁺ was absent in the experimental buffer. The model analysis of the two experimental data sets revealed some interesting aspects of the mitochondrial Ca²⁺ uptake and sequestration system, which are discussed next.

As pointed out earlier, the mitochondrial Ca²⁺ uptake profile in Boelens et al.'s (2013) experiments with Mg²⁺ present in the experimental buffer had two components: a slow component (CU) and a fast component (RTC). We modified the mitochondrial Ca²⁺ uptake system to account for Ca²⁺ uptake from both components/pathways and estimated the associated model parameters based on least-squares fitting of the model outputs to the experimental data from Boelens et al. (2013) and Blomeyer et al. (2013). Interestingly, the CU activity estimates from the two studies are of the same orders of magnitude (see Table 1). However, the estimates of RTC activity between the two studies are two orders of magnitude different (see Table 1).

There are two possible explanations for these observed differences: (i) high Ca²⁺ addition and/or (ii) the presence of Mg²⁺ in the experimental buffer. Due to the presence of 1 mm EGTA in the study of Boelens et al. (2013), higher amounts of CaCl₂ (0.0–0.6 mm) were added in the experimental buffer to attain a [Ca²⁺]_e within the physiological range, i.e. few hundred nanomolar range. On the other hand, in the study by Blomeyer et al. (2013), the experimental buffer had a much lower amount of EGTA (40 μm), and therefore relatively low concentrations of CaCl₂ (0–40 μm) were required to attain [Ca²⁺]_e within the same physiological range. So we hypothesize that adding high [CaCl₂] triggers uptake by an additional pathway that is different from the CU, for which the RTC compensates during the parameter estimation process. Another possibility is that the presence of Mg²⁺ in the buffer, which restricts uptake via the CU, invokes an alternative Ca²⁺ uptake mechanism (e.g., RTC). Further experimentation performed with added Mg²⁺ and low EGTA may help to verify if such a distinct fast Ca²⁺ uptake is still observed.

Mitochondria possess a remarkable ability to take up and sequester large amounts of cytosolic Ca²⁺ without significant changes in the [Ca²⁺]_m – an aspect termed mitochondrial Ca²⁺ sequestration. Earlier observations indicated that β_Ca,m is constant for low [Ca²⁺]_m and decreases linearly with increases in [Ca²⁺]_m (Coll et al. 1982; Corkey et al. 1986). However, the Bazil et al. (2013) minimal model of the mitochondrial Ca²⁺ sequestration system described β_Ca,m as a bell-shaped function of [Ca²⁺]_m, i.e. an increase in β_Ca,m occurs with an increase in [Ca²⁺]_m, which then breaks down when a threshold is reached (see Figs. 5A and 7C, which describe this phenomena).

We used the same minimal model to characterize β_Ca,m for the experiments of Boelens et al. (2013). Our results show that the total β_Ca,m (Class 1 + Class 2 Ca²⁺ buffering capacity) identified is strikingly similar to that of Bazil et al. (2013) (compare Fig.5A with Fig.7C). Also, the net mitochondrial Ca²⁺ sequestered is about the same (compare Fig.5B with Fig.7D; continuous line). The only significant difference between the two models for β_Ca,m is the concentration of Class 1 Ca²⁺ buffers, which is estimated to be an order of magnitude less in our study. This is an important observation because it suggests that Class 1 Ca²⁺ buffers can actually sense the rate of mitochondrial Ca²⁺ uptake and modify their individual Ca²⁺ buffering capacity. It is noteworthy that our view of how mitochondria buffer or sequester Ca²⁺ is based on the mode by which Ca²⁺ is taken up and is consistent with recent findings (Wei et al. 2012; O-Uchi et al. 2013). Such a feature would enable the mitochondria to switch between two important functions: Ca²⁺-regulated metabolism and Ca²⁺ buffering, depending on the cytosolic Ca²⁺ levels. This observation also fits the role proposed for the CU (non-saturating slow uptake) of Ca²⁺ sequestration (Kirichok et al. 2004; O-Uchi et al. 2013) and the RTC (saturating fast uptake) during excitation–metabolism coupling (Beutner et al. 2005; O-Uchi et al. 2013).

Although a number of studies published in the last decade indicate the presence of the RyR in cardiac mitochondria (Beutner et al. 2001, 2005; Ryu et al. 2011; O-Uchi et al. 2013), still not much is known regarding the regulatory mechanisms of this ryanodine-sensitive channel in cardiac mitochondria by physiological ions (Ca²⁺, Mg²⁺, H⁺, etc.). However, considerable information is available for the cardiac or skeletal muscle isoform of the RyR. One such study by Valdivia et al. (1995) suggests that the RyR adaptation phenomenon is increased several fold in the presence of cytosolic Mg²⁺ ions. Interestingly, we first attempted to explain the CICU component seen in the Ca²⁺ uptake experiment of cardiac mitochondria of Boelens et al. (2013) using a different modification of the Keizer–Levine model that emulated the same kind of rapid adaptation of RTC by Mg²⁺ (see Fig. A.3 (I–L)), but were not successful in reproducing the mitochondrial Ca²⁺ uptake experiments (see Fig. A.4). Model analyses suggests that the mechanism of Mg²⁺-dependent modulation of the RTC adaptation is much smaller than that for cardiac SR-RyR. This is a novel finding, but future experiments would need to verify this hypothesis.

Our quantitative analysis of the Boelens et al. (2013) experimental data suggests that in the presence of cytosolic Mg²⁺, the majority of mitochondrial Ca²⁺ uptake at physiological levels of cytosolic Ca²⁺ (a few hundred nanomolar) is via a mechanism that could not be explained with a mechanistic model of the CU by Pradhan et al. (2011) that accounts for Mg²⁺ inhibition of the CU. Note that the CU model by Pradhan et al. (2011) identified the CU as a carrier, but recent experiments suggest that the CU is a low Ca²⁺ affinity ion channel (Kirichok et al. 2004; De Stefani et al. 2011). However, data important for kinetic model development of the CU (such as Mg²⁺ and Ca²⁺ dependence of the single CU gating kinetics) are not yet available. On the other hand, an extensive amount of literature data from Ca²⁺ uptake experiments on isolated mitochondria from different tissues are available. Therefore, we revisited the CU model of Pradhan et al. (2011) to identify other mechanisms of Mg²⁺ inhibition that could help describe the data from Boelens et al. (2013), but none worked properly. Apparently, increased [Mg²⁺]_e and high CaCl₂ boluses, evoked Ca²⁺ uptake via a fast Ca²⁺ uptake pathway other than the CU and RaM. The fast Ca²⁺ uptake kinetics could be explained using a biophysical model of the cardiac RyR; therefore, we refer to this as an RTC Ca²⁺ uptake pathway. Does this mean that under physiological conditions, when Ca²⁺ uptake via the CU is minimal, the Ca²⁺ signal that couples cardiac excitation–contraction with mitochondrial metabolism (Balaban, 2002) is a via a fast Ca²⁺ uptake pathway similar to the RTC? If so, then new methods need to be employed to identify and characterize this fast Ca²⁺ uptake pathway that can aid in development of novel therapeutic interventions targeting diseases linked to mitochondrial Ca²⁺ overload and bioenergetics dysfunction (Kurdi & Booz, 2011; Sharma et al. 2013).

Conclusion

Recent experimental studies have shown the existence of a ryanodine-sensitive Ca²⁺ uptake pathway in cardiac mitochondria (Beutner et al. 2001, 2005; Ryu et al. 2011; O-Uchi et al. 2013) that has been linked to cardiac energy metabolism. However, evidence accumulated during the last five decades suggests the CU as the primary Ca²⁺ uptake pathway in cardiac mitochondria (Scarpa & Graziotti, 1973; Favaron & Bernardi, 1985; McCormack et al. 1989; Kirichok et al. 2004; Rizzuto et al. 2012). This report presented the first quantitative framework that accounts for individual contributions of the CU and RTC Ca²⁺ fluxes across cardiac mitochondria in two different experimental conditions (Blomeyer et al. 2013; Boelens et al. 2013). The modelling framework conforms to recent experimental evidence that the majority of mitochondrial Ca²⁺ uptake under physiological concentrations of cytosolic Ca²⁺ and Mg²⁺ is via the RTC. It provides quantitative support to an interesting hypothesis that poses RTC as the major regulator of mitochondrial morphology and cardiac energetics (O-Uchi et al. 2013). In the future, in vivo and in situ experiments will need to be designed to challenge (prove/disprove) this hypothesis.

Glossary

ACP

amorphous calcium phosphate

ANT

adenine nucleotide translocase

β_Ca,m

mitochondrial Ca²⁺ buffering capacity

[Ca²⁺]_e

extra-matrix Ca²⁺ concentration

[Ca²⁺]_m

matrix Ca²⁺ concentration

CHE

Ca²⁺/H⁺ exchanger

CICU

Ca²⁺-induced Ca²⁺ uptake

CU

Ca²⁺ uniporter

IMM

inner mitochondrial membrane

KHE

K⁺/H⁺ exchanger

[Mg²⁺]_e

extra-matrix Mg²⁺ concentration

[Mg²⁺]_m

matrix Mg²⁺ concentration

NCE

Na⁺/Ca²⁺ exchanger

NHE

Na⁺/H⁺ exchanger

OMM

outer mitochondrial membrane

PA

pyruvic acid

PHT

phosphate-H⁺ cotransporter

RaM

rapid-mode Ca²⁺ uptake pathway

RR

ruthenium red

RTC

RyR-type channel

RyR

ryanodine receptor

SR

sarcoplasmic reticulum

[TCa²⁺]_m

total mitochondrial Ca²⁺ concentration

Key points

• Cytosolic, but not matrix, Mg²⁺ inhibits mitochondrial Ca²⁺ uptake through the Ca²⁺ uniporter (CU).

• The majority of mitochondrial Ca²⁺ uptake under physiological levels of cytosolic Ca²⁺ and Mg²⁺ is through a fast uptake pathway, namely the ryanodine receptor (RyR)-type channel (RTC), that has characteristics similar to the ryanodine receptor.

• Modulation of mitochondrial RTC adaptation and opening probability by cytosolic Mg²⁺ is not robust, in contrast to that of cardiac sarcoplasmic reticulum RyR.

• Model analysis of the mitochondrial Ca²⁺ sequestration system further validates the existence of two different classes of Ca²⁺ buffering proteins, i.e. Class 1 and Class 2.

• The Ca²⁺ buffering capacity of Class 1 protein is auto-regulated by the rate at which Ca²⁺ is taken up by cardiac mitochondria.

• The quantitative framework suggests differential roles for the two modes of Ca²⁺ uptake pathways: CU–Ca²⁺ buffering, and RTC–Ca²⁺ modulated bioenergetics.

Additional information

Competing interests

The authors have declared that no competing interests exist.

Author contributions

S.G.T. and R.K.D. designed and performed the experiments. S.G.T. analysed the data and drafted the manuscript. R.K.D., D.F.S. and A.K.S.C. critically reviewed the manuscript and contributed to important intellectual content based on the original mitochondrial studies. All authors approved the final version of the manuscript.

Funding

This work was supported by the National Institutes of Health grants R01-HL095122 and P50-GM094503.

Author's present address

S. G. Tewari: Department of Molecular & Integrative Physiology, University of Michigan, Ann Arbor, MI 48109, USA.

Translational perspective

Cardiac mitochondria play a crucial role in buffering and shaping cytosolic Ca²⁺ oscillations. They also meet the beat-to-beat energy demand of the heart by Ca²⁺-dependent stimulation of mitochondrial dehydrogenases (pyruvate dehydrogenase, α-ketoglutarate dehydrogenase, etc.). To date, a number of Ca²⁺ uptake pathways have been identified with the Ca²⁺ uniporter (CU) positioned as the major player. However, an important question: Is the CU the major Ca²⁺ uptake pathway under physiological conditions? Probably not, because physiological concentrations of cytosolic Mg²⁺ potently inhibit the CU. Our modelling analysis (present work) suggests likewise and identifies an additional uptake pathway with characteristics similar to the ryanodine receptor (termed RyR-type channel or RTC). The current work also identifies/validates the existence of two protein classes that have roles as major mitochondrial Ca²⁺ buffers. The model analysis also suggests that these buffers auto-regulate their Ca²⁺ buffering capacity depending on the pathway of Ca²⁺ uptake, i.e. CU versus RTC. The presented work proposes specific roles for the two mitochondrial Ca²⁺ uptake pathways: CU in Ca²⁺ buffering, and RTC in cardiac metabolism. Future experiments should be designed to test this hypothesis. If true, it will lead to development of specific therapeutic interventions targeting heart failure diseases associated with diastolic dysfunction and myocardial workload.

Supporting Information

The following supporting information is available in the online version of this article.

Four appendices providing details of the mitochondrial bioenergetics and cation handling model, including model equations and parameters for different components of the system. The supporting information also provides the model simulation results of the mitochondrial Ca²⁺ uniporter (CU) (Fig. A.1) and RyR-type channel (RTC) (Fig. A.3), and integrated model simulation results and their comparison with experimental data on mitochondrial Ca²⁺ uptake with only CU without RTC (Fig. A.2) and with both CU and RTC, but with Ca²⁺- and Mg²⁺-dependent fast adaptation of the RTC similar to that of sarcoplasmic reticulum ryanodine receptor (Fig. A.4).

Fig. A.1. Modified CU model fitting to the experimental data on the extra-matrix Ca²⁺ and ΔΨ dependencies of mitochondrial Ca²⁺ uptake measured in both the presence and the absence of extra-matrix Mg²⁺.

Fig. A.2. Model simulations of the Ca²⁺ uptake experiments in the absence of RTC.

Fig. A.3. Model simulations showing the Ca²⁺-dependent RTC adaptation phenomena with different cytosolic Mg²⁺ concentrations.

Fig. A.4. Model simulations of the Ca²⁺ uptake experiments with RTC model exhibiting fast adaptation in the presence of cytosolic Mg²⁺ ions.

Table A.1. Fixed parameter values in the model of mitochondrial respiratory system and oxidative phosphorylation. Modified CU model fitting to the experimental data.

Table A.2. Sodium–Hydrogen exchanger parameters

Table A.3. Calcium–Hydrogen exchanger parameters

Table A.4. Sodium–Calcium exchanger parameters

Table A.5. Calcium Uniporter parameters

Table A.6. RyR-type channel parameters

Table C.1. Additional mitochondrial buffering parameters