An Integrated Model of Cardiac Mitochondrial Energy Metabolism and Calcium Dynamics

Abstract

We present an integrated thermokinetic model describing control of cardiac mitochondrial bioenergetics. The model describes the tricarboxylic acid (TCA) cycle, oxidative phosphorylation, and mitochondrial Ca²⁺ handling. The kinetic component of the model includes effectors of the TCA cycle enzymes regulating production of NADH and FADH₂, which in turn are used by the electron transport chain to establish a proton motive force (Δμ_H), driving the F₁F₀-ATPase. In addition, mitochondrial matrix Ca²⁺, determined by Ca²⁺ uniporter and Na⁺/Ca²⁺ exchanger activities, regulates activity of the TCA cycle enzymes isocitrate dehydrogenase and α-ketoglutarate dehydrogenase. The model is described by twelve ordinary differential equations for the time rate of change of mitochondrial membrane potential (ΔΨ_m), and matrix concentrations of Ca²⁺, NADH, ADP, and TCA cycle intermediates. The model is used to predict the response of mitochondria to changes in substrate delivery, metabolic inhibition, the rate of adenine nucleotide exchange, and Ca²⁺. The model is able to reproduce, qualitatively and semiquantitatively, experimental data concerning mitochondrial bioenergetics, Ca²⁺ dynamics, and respiratory control. Significant increases in oxygen consumption (V_O2), proton efflux, NADH, and ATP synthesis, in response to an increase in cytoplasmic Ca²⁺, are obtained when the Ca²⁺-sensitive dehydrogenases are the main rate-controlling steps of respiratory flux. These responses diminished when control is shifted downstream (e.g., the respiratory chain or adenine nucleotide translocator). The time-dependent behavior of the model, under conditions simulating an increase in workload, closely reproduces experimentally observed mitochondrial NADH dynamics in heart trabeculae subjected to changes in pacing frequency. The steady-state and time-dependent behavior of the model support the hypothesis that mitochondrial matrix Ca²⁺ plays an important role in matching energy supply with demand in cardiac myocytes.

INTRODUCTION

The high metabolic demand of the heart necessitates close matching of energy production to workload. Approximately 2% of cellular ATP is consumed on each heartbeat and almost all of this energy is replenished by mitochondrial oxidative phosphorylation under normoxic conditions (Harris and Das, 1991). Remarkably, the vital process of control and regulation of mitochondrial metabolism is not well understood.

After the development of the chemiosmotic theory of energy transduction by Mitchell (1961), several hypotheses for control of metabolism have been proposed. The first, referred to as respiratory control (Chance and Williams, 1956; Harris and Das, 1991), asserts that ADP availability to the ATP synthase is the limiting factor for mitochondrial ATP production. Increases in workload lead to a rise in cellular ADP and stimulation of oxidative phosphorylation, with the downstream “pull” on electron transport eventually causing acceleration of NADH production by the TCA cycle. The lack of a substantial increase in whole-heart ADP concentration in response to increased pacing frequency, and the observed decrease in Δμ_H produced by Ca²⁺ entry into mitochondria (Heineman and Balaban, 1990; Territo et al., 2000), have been used to argue against activation of ATP synthase based solely on its substrates (Harris and Das, 1991).

A second control hypothesis emerged from the discovery that Ca²⁺ activates mitochondrial matrix dehydrogenases (McCormack and Denton, 1984; McCormack et al., 1990). According to this hypothesis an increase in the workload of the heart is accompanied by a rise in cytosolic Ca²⁺, and the subsequent effect of matrix Ca²⁺ on the TCA cycle increases the supply of reducing equivalents (NADH, FADH₂) and an increased “push” of electrons through the respiratory chain, thereby increasing ATP production by generating more proton motive force. However, it has been argued that dehydrogenase activation by Ca²⁺ might not account quantitatively for observed changes in ATP synthesis rates (Harris and Das, 1991). Therefore, additional regulatory mechanism(s), including direct activation of the ATP synthase by Ca²⁺ (Territo et al., 2000), have been hypothesized to explain the acceleration of ATP production (Heineman and Balaban, 1990, Harris and Das, 1991). The present work addresses the hypothesis that Ca²⁺ is involved in coupling energy demand and supply in heart mitochondria. This hypothesis is tested through development and application of an integrative mathematical model that takes into account mitochondrial matrix- and membrane-based processes such as the TCA cycle, oxidative phosphorylation, and Ca²⁺ dynamics. To date, an integrated kinetic and thermodynamic model of cardiac energy metabolism incorporating mitochondrial Ca²⁺ dynamics has not been available. The present work builds a unified model of mitochondrial energy metabolism and examines its behavior with respect to changes in substrate supply, metabolic inhibition, and Ca²⁺. The results provide insight into the essential control properties of the system.

METHODS

General description

The model incorporates the main mitochondrial electrophysiological and metabolic processes and their interactions (Fig. 1). Model differential equations are given in Table 1, whereas rate equations and their behavior are presented in the Appendix. The major components of the model—mitochondrial matrix-associated processes, oxidative phosphorylation, and Ca²⁺ dynamics—are described in the following sections.

FIGURE 1.

Schematic representation of the mitochondrial electrophysiological and metabolic processes and their interactions, as described by the model. The model takes into account oxidative phosphorylation and matrix-based processes in mitochondria. The tricarboxylic (TCA) cycle of the mitochondrial matrix is fed by acetyl CoA (AcCoA), which is the point of convergence for the oxidation of fatty acids and glucose, the two main substrates of the heart. The TCA (or Krebs) cycle completes the oxidation of AcCoA to CO₂ and produces NADH and FADH₂, which provide the driving force for oxidative phosphorylation. NADH and FADH₂ are oxidized by the respiratory chain and the concomitant pumping of protons across the mitochondrial inner membrane establishes an electrochemical gradient, or proton motive force (Δμ_H), composed of an electrical gradient (ΔΨ_m) and a proton gradient (ΔpH). This proton motive force drives the phosphorylation of matrix ADP to ATP by the F₁F₀-ATPase (ATP synthase). The large ΔΨ_m of the inner membrane (−150 to −200 mV; matrix negative with respect to the cytoplasm) also governs the electrogenic transport of ions, including the cotransport of ATP and ADP by the adenine nucleotide translocator, Ca²⁺ influx via the Ca²⁺ uniporter and Ca²⁺ efflux via the Na⁺/Ca²⁺ antiporter (Magnus and Keizer, 1997). The model also accounts for the explicit dependence of the TCA cycle enzymes isocitrate dehydrogenase and α-ketoglutarate dehydrogenase on Ca²⁺. In this way, the rate of Ca²⁺ uptake by mitochondria is involved in membrane energization through the TCA cycle and oxidative phosphorylation, which are described in detail in the Appendix. Key to symbols: The concentric circles with an arrow across located at the inner mitochondrial membrane represent the ΔΨ_m. Dotted arrows indicate regulatory interactions either positive (arrowhead) or negative (−).

TABLE 1.

Model differential equations

State variable	Differential equation
[ADP]m	(1)
ΔΨm	(2)
[NADH]	(3)
[ISOC]	(4)
[αKG]	(5)
[SCoA]	(6)
[Suc]	(7)
[FUM]	(8)
[MAL]	(9)
[OAA]	(10)
[ASP]	(11)
[Ca2+]m	(12)

Mitochondrial matrix-associated processes

The tricarboxylic acid (TCA) cycle

The first phase of the model, the TCA cycle, represents the end stage for carbon substrates in the cell. The catabolic pathways for substrate oxidation (e.g., fatty acids, amino acids, and sugars) converge to produce AcCoA, the main input of the TCA cycle.

The dashed lines in Fig. 1 show the regulatory feedback loops incorporated into the model. The rates of the highly-regulated dehydrogenases of the TCA cycle, namely, citrate synthase (CS), isocitrate dehydrogenase (IDH), α-ketoglutarate dehydrogenase (KGDH) and malate dehydrogenase (MDH) and their regulatory interactions have been studied in detail and their behavior is presented in the Appendix. A fundamental feature of the present model is the explicit dependence of the dehydrogenases IDH and KGDH on mitochondrial calcium (Rutter and Denton, 1988; McCormack and Denton, 1984; Nichols et al., 1994; McCormack et al., 1990; Hansford and Zorov, 1998; also see Appendix).

Oxidative phosphorylation (OxPhos)

The OxPhos model is based on that of Magnus and Keizer (1997). The model describes NADH-driven electron transport and proton efflux, and F₁F₀-ATPase activity and its associated proton influx (Fig. 1; see also Appendix). The adenine nucleotide translocator and the proton leak are also taken into account. In general, this model follows a thermokinetic formulation. The driving forces are redox (A_res) and phosphorylation (A_F1) potentials (see Eqs. 26–34) and the proton motive force (Δμ_H). Coefficients in the equations correspond to rate constants of explicit conversion steps postulated in the Altman-King-Hill formulation of the model (Pietrobon and Caplan, 1985; Magnus and Keizer, 1997).

The F₁F₀-ATPase

The original expression for the rate of the F₁F₀-ATPase (Magnus and Keizer, 1997) was modified to account for reversibility of the enzyme when Δμ_H collapses (see Appendix). For example, in uncoupled mitochondria, hydrolysis rather than synthesis of ATP can occur and drive proton pumping.

Mitochondrial calcium dynamics

Mitochondrial Ca²⁺, [Ca²⁺]_m, is determined mainly by Ca²⁺ influx through the Ca²⁺ uniporter and the Na⁺-dependent Ca²⁺ efflux driven by the Na⁺/Ca²⁺ antiporter. The latter is assumed to be electrogenic based on the reported results of Baysal et al. (1994) (see Appendix).

Model building and numerical methods

The model was built using a modular approach. Briefly, each model component was examined separately, iteratively fine-tuning the kinetic parameters for the best fit to experimental data (Eqs. 13–39). Maple (Maple V, release 4, Waterloo, Canada) was used to study each analytical expression as a function of underlying parameters (see electronic Appendix: http://www.jhmi.edu/bor/model.htm).

Before the final model assembly, the TCA cycle and OxPhos modules were analyzed separately. The adequacy of the expressions for each of the processes involved was tested by simulating the model behavior at steady state. The system was considered to be in steady state when the magnitude of each time derivative was ≤10⁻¹³. Model parameters such as the level of AcCoA, ion leak, and levels of electron carriers, were then varied to determine their influence on important mitochondrial functions. If the results were counterintuitive, or inconsistent with basic bioenergetic principles, the model was revised at the level of the analytical expressions stated in Maple and retested. The iteration process was performed until complete and satisfactory expressions were obtained, which converged to results consistent with available experimental evidence.

Numerical integration of model equations (Eqs. 1–12, Table 1) was performed using an Adams routine (SCoP, version 3.51w, Simulation Resources) until steady-state solutions (according to the criterion described above) were obtained. Steady-state values of each state variable were then used as input to software for performing bifurcation and continuation analysis (AUTO 1997, E. Doedel, Concordia University). This software was used to determine the dependence of steady-state solution properties (type and stability) on model parameters.

Dynamic simulations

To test whether or not the global kinetic response of the model accurately reproduces the dynamic response of mitochondrial energetics to changes in workload, we simulated the time-dependent behavior of mitochondrial NADH and Ca²⁺ during changes in the myocardial pacing frequency. Forcing pulse functions (SCoP version 3.51w, Simulation Resources) for the parameters [Ca²⁺]_i and [ADP]_i were incorporated into the model, and the simulations were compared with experimental data obtained in cardiac trabeculae (Brandes and Bers, 2002). Ca²⁺ transients were simulated with periodic square pulses of [Ca²⁺]_i to 1 μM (from a basal level of 100 nM) for 400 ms duration, applied at frequencies of 0.25 Hz or 2 Hz. To simulate the increased energy consumption when pacing frequency was increased, a single [ADP]_i pulse from 50 to 150 μM was applied for the whole interval of high frequency stimulation.

RESULTS

Simulations of the response of mitochondrial energetics to changes in substrate input, levels of electron carriers, and cytoplasmic Ca²⁺ were performed to test whether the model qualitatively and semiquantitatively reproduced available experimental evidence. The overriding objective was to gain insight into the important sites and mechanisms of control of bioenergetics.

Control properties of the integrated model

Metabolic control analysis of oxidative phosphorylation has previously been used to demonstrate that control is distributed among the respiratory chain/substrate transport, the phosphorylation system, and the proton leak (Hafner et al., 1990; Murphy, 2001). This control can redistribute during electron chain inhibition, mitochondrial uncoupling, or with postischemic mitochondrial damage (Borutaite et al., 1995). Therefore, the simulations presented below examine the effects of metabolic perturbation on mitochondrial energetics to better understand control of the integrated system (Brand and Kesseler, 1995).

Influence of AcCoA on mitochondrial energetics

The contribution of substrate availability to the global response of bioenergetics was tested by varying either the earliest (AcCoA) or last (ADP) substrate (next section) of the integrated model. When the concentration of AcCoA was increased to a maximum of 10 μM, an approximate twofold increase in V_O2 was obtained; beyond this, V_O2 plateaued (Fig. 2 A). This increase in respiration rate is driven by the generation of NADH in the range of 0.001–1.5 mM (Fig. 2 A). The V_O2 registered at very low AcCoA is mainly driven by glutamate (see Appendix Eq. 24).

FIGURE 2.

Effects of increasing substrate delivery on mitochondrial energetics. AcCoA concentration is varied and the steady-state behavior of the model analyzed according to a continuation analysis (see text). The correlative changes in the steady-state behavior of V_O2 and NADH versus (A) AcCoA and (B) ΔΨ_m are depicted when AcCoA is systematically changed. The simulation was run with the following set of parameters: V_maxANT = 5.0 mM s⁻¹; ΔpH = −0.6; ρ^res = 0.048 mM; ADPi = 0.05 mM; mM s⁻¹; ρ^F1 = 0.06 mM; [Ca²⁺]_i = 0.5 μM; g_H = 0.01 mM s⁻¹ V⁻¹. The rest of the parameters are as indicated in Table 3. (C) Simulation results match experimental data obtained with pig heart mitochondria, redrawn from Territo et al. (2000).

Over the full range of the V_O2 increase, stimulation of electron transport is associated with a modest increase in ΔΨ_m (∼14 mV; Fig. 2 B). The linear flow-force relationship between V_O2 and ΔΨ_m represented in Fig. 2 B is in agreement with experimental results reported for cardiac mitochondria by Territo et al. (2000) (redrawn in Fig. 2 C).

Influence of phosphorylation potential on mitochondrial energetics

The respiratory rate (V_O2) increases as a function of the maximal rate of the adenine nucleotide translocator, ANT (Fig. 3 A) by ∼six-to-sevenfold before saturation. In keeping with the classical theory of respiratory control, V_O2 varies with matrix ADP concentrations in the mM range (Fig. 3 A, inset). At low ADP, representing state 4 respiration, the basal rate of oxygen consumption is nonzero, supported in the model by a small proton leak across the mitochondrial inner membrane. A steep response of V_O2 to small changes in ADP, as would occur during increased metabolic demand, is evident as the transition is made from state 4 to state 3.

FIGURE 3.

Relationship between membrane potential and respiratory flux upon changes in adenine nucleotide translocator activity. The changes in respiration rate, V_O2, and membrane potential, ΔΨ_m, are determined by the behavior of the whole model according to a steady-state analysis performed at different adenine nucleotide translocator maximal activities (V_maxANT) which result in different ADP_m concentrations (A, inset). The left portion of curves a and b in panels A and B, corresponds to mitochondria in state 4 (i.e., low ADP) and the increases in V_O2 result from the respiratory control exerted by increasing ADP concentrations. The simulations were run with the following parameter values: (a) ΔpH = −0.6; ρ^res = 0.00072 mM; ADP_i = 0.05 mM; mM s⁻¹; ρ^F1 = 1.216 mM; [Ca²⁺]_i = 0.1 μM; g_H = 0.01 mM s⁻¹ V⁻¹; AcCoA = 1.0 mM. V_maxANT is varied in the range 0.01–25 mM s⁻¹; (b) all parameters are the same as in a, except for ρ^res = 0.0051 mM and ρ^F1 = 0.61 mM. The remainder of the parameters are reported in Table 3. (C) Changes in the relationship between ΔΨ_m and the respiratory rate obtained at various ratios of creatine/creatine phosphate in a preparation of heart mitochondria with normal or reduced respiratory activity (left-shifted curves). Redrawn from Borutaite et al. (1995).

At this point it is worthwhile introducing the definition of “push” or “pull” conditions as those in which the control exerted on the respiratory flux correspond to processes upstream (e.g., TCA cycle) or downstream (e.g., ANT, respiration, ATPase) of NADH, respectively (see also Metabolic Control Analysis of Mitochondrial Energetics). Push is obtained when simulations are performed with a large concentration of respiratory electron carriers, ρ^RES, and low k_cat for both IDH and KGDH activities, whereas the opposite is true for pull conditions (see Fig. 5 legend). A decline in ΔΨ_m accompanies ADP stimulation of V_O2 under pull conditions (Fig. 3 B). Thus, respiration is stimulated as phosphorylation potential drops along with an increased F₁F₀ATPase activity.

FIGURE 5.

Mitochondrial Ca²⁺ level after changes in cytoplasmic Ca²⁺. (A) The behavior of the whole model is studied with the AUTO 97 software as a function of cytoplasmic Ca²⁺ concentration, [Ca²⁺]_i. The simulations were performed with the following set of parameters: (pull): ρ^RES = 0.00072 mM, ρ^F1 = 1.216 mM, s⁻¹, s⁻¹, and mM s⁻¹, and (push): ρ^RES = 0.0036 mM, ρ^F1 = 0.0522 mM, s⁻¹, s⁻¹, and mM s⁻¹. Curve a is obtained with the same parametric set as “pull” except for mM s⁻¹. Otherwise the concentration of AcCoA is 1.0 mM, and the catalytic constants of CS and MDH are s⁻¹ and s⁻¹, respectively. (B) Experimental results redrawn from McCormack et al. (1990) showing the accumulation of mitochondrial Ca²⁺ in the absence (curve I) or in the presence of Na⁺ (curve II).

Conditions which diminish the ability of the respiratory chain to oxidize NADH decrease V_O2 for any given ADP concentration (Fig. 3, A and B, trace a). This is simulated by decreasing the maximum velocity of the ANT, and reducing the concentration of respiratory electron carriers (ρ^res) by sevenfold.

The results in Fig. 3 B are in qualitative agreement with the experimental results obtained with isolated heart mitochondria (reproduced in Fig. 3 C; Borutaite et al., 1995). Under conditions of a decrease in the respiratory activity of heart mitochondria due to ischemia, the curve describing the relation between ΔΨ_m and succinate-driven respiratory rate is shifted down and to the left with respect to the behavior of normal mitochondria (Fig. 3 C).

Influence of proton leak and respiratory inhibition on mitochondrial energetics

Inhibition of respiration is simulated by decreasing the concentration of electron carriers in the mitochondrial membrane (ρ^res) (Fig. 4 A). The steepness of the V_O2 curve over the low range of ρ^res, and the electron carrier concentration at which V_O2 reaches saturation, are influenced by the magnitude of proton influx across the inner membrane, as depicted in Fig. 4 A, curves a and b. Proton influx can be varied by changing the concentration of F₁F₀-ATPase, which dissipates Δμ_H. Increasing the concentration of F₁F₀-ATPase increases V_O2 over the low ρ^res range (Fig. 4 A; curvea). As expected, with minimal ATP synthase activity, the membrane can support a higher ΔΨ_m at any V_O2 (Fig. 4 B; curve b). These modeling results resemble experimental data obtained with normal mitochondria in either state 3 or 4 upon titration with the metabolic inhibitor malonate at various creatine/creatine phosphate ratios (Fig. 4 C; Borutaite et al., 1995).

FIGURE 4.

Relationship between mitochondrial inner membrane potential and the respiratory flux. The rate of the electron transport chain and the ΔΨ_m are modulated through the concentration of electron carriers (ρ^res) (A, B). A steady-state analysis of the whole model behavior is performed. The parameters utilized in the simulations are those reported in Table 3 and as follows: P_i = 20.0 mM; ΔpH = −0.6; mM s⁻¹; ADP_i = 0.05 mM; mM s⁻¹; g_H = 0.01 mM s⁻¹ V⁻¹, [Ca²⁺]_i = 0.1 μM and AcCoA = 1.0 mM. The values of F₁F₀-ATPase concentration, ρ^F1, differ between curves, being 1.216 mM for curve a and 0.06 mM for curve b. (C) Changes in the relationship between ΔΨ_m and the respiratory rate obtained upon malonate titration at different phosphorylation potentials (different ratios of creatine to creatine phosphate) in a preparation of heart mitochondria. Redrawn from Borutaite et al. (1995).

Response of mitochondrial energetics to changes in cytoplasmic and mitochondrial Ca²⁺

There are two effects of Ca²⁺ on mitochondrial bioenergetics that are diametrically opposed: on the one hand, Ca²⁺ influx dissipates ΔΨ_m and decreases the proton motive force; on the other, Ca²⁺ stimulates the production of NADH by the TCA cycle, increasing the proton motive force and ATP production. One of the main objectives of this study is to determine the balance between these effects under different conditions. Therefore, simulations are presented using two different parameter sets that illustrate different model behaviors. A key question is whether or not the intrinsic biophysical properties of the system allow control of mitochondrial metabolism by matrix Ca²⁺.

Steady-state behavior of mitochondrial calcium

The first step toward understanding the effects of Ca²⁺ on the global behavior of the model is to simulate the steady-state response of mitochondrial matrix free Ca²⁺ ([Ca²⁺]_m) to changes in free cytosolic Ca²⁺ concentration ([Ca²⁺]_i). Inasmuch as there is little quantitative information about the relative contributions of the two major Ca²⁺ transport pathways in determining steady-state [Ca²⁺]_m, the parameter space is explored with respect to the maximal Na⁺/Ca²⁺ antiporter rate. Results of these studies show that following [Ca²⁺]_i changes in the physiological range (0.01 to 1.2 μM), [Ca²⁺]_m increases nonlinearly after crossing a certain threshold [Ca²⁺]_i, and continues to increase to more than 20 μM without saturation (Fig. 5 A). This threshold depends on the ratio of Ca²⁺ uniporter and Na⁺/Ca²⁺ antiporter rates. In fact, the rightward shift of the [Ca²⁺]_m curve (Fig. 5, trace a) is due to a large increase in the rate of the Na⁺/Ca²⁺ antiporter, which counteracts influx through the Ca²⁺ uniporter. The general profile of [Ca²⁺]_m accumulation upon increasing [Ca²⁺]_i are in agreement with experimental results obtained by several authors (Fig. 5 B) (Miyata et al., 1991; Collins et al., 2001; reviewed by McCormack et al., 1990). The right shift in the experimental curve shown in Fig. 5 B was elicited by the addition of Na⁺ (McCormack et al., 1990) supporting the involvement of the Na⁺/Ca²⁺ antiporter in this response.

As the rates of the Ca²⁺ uniporter and Na⁺/Ca²⁺ antiporter are coupled indirectly through ΔΨ_m, the steady-state [Ca²⁺]_m uptake curve (Fig. 5 A) is relatively unaltered by changing other parameters that determine push or pull conditions. The effect of [Ca²⁺]_i on V_O2, however, is sensitive to such parameters, as described in the following section.

Response of mitochondrial energetics to changes in Ca²⁺

Effects on V_O2

Activation of V_O2 by Ca²⁺ is evident for both push and pull parameter sets (Fig. 6 A). Higher basal V_O2 values at a given [Ca²⁺]_i are evident for conditions in which the ANT and the electron transport flux are the main rate-controlling steps of respiration (pull) (Table 2). However, the degree of activation by Ca²⁺ is reduced compared to the push condition (∼9.1% versus 23%, Fig. 6 A) due to the Ca²⁺ insensitivity of the main rate-controlling steps of respiration. Paralleling the increase in V_O2, the steady-state fluxes through the Ca²⁺-sensitive enzymes, KGDH and IDH, also increases as a function of Ca²⁺ (Fig. 6 B, inset).

FIGURE 6.

Respiratory flux and NADH levels upon changes in [Ca²⁺]_I. The model is analyzed as described in Fig. 5 with the same set of parameters indicated in its legend for push and pull conditions. The steady-state respiratory flux (A) and NADH concentration (B) corresponding to the same simulations presented in Fig. 5 are depicted as a function of [Ca²⁺]_i.

TABLE 2.

Control of the respiratory flux by TCA cycle enzymes and the rates of membrane-associated processes

	Parametric set, flux control coefficient
	Push	Pull
TCA cycle
CS	0.33	0.35
IDH	22.8	5.3
KGDH	70.0	0.3
MDH	3.6	13.9
Membrane-associated processes
ANT	3.8	38
Respiration	2.1	6.7
F1F0-ATPase	−0.2	2.7
Proton leak	0.1	0.3

The control exerted by the rate of each process on mitochondrial respiration is quantified around the steady state corresponding to the set of parameters indicated in the legend of Fig. 5. The flux control coefficient (in %) indicates to which extent a flux (i.e., the respiratory flux) changes upon a change on the activity of an enzyme in the pathway. The flux control coefficient was calculated from the slope of the double logarithmic plot of V_O2 as a function of the process rate under study. Under push and pull conditions, the flux control coefficients were determined at a [Ca²⁺]_i concentration of 0.1 or 0.5 μM. Inasmuch as the differences between flux control coefficients quantified at 0.1 or 0.5 μM are 1% or less, we present the data obtained with 0.1 μM.

In the cases of CS, IDH, KGDH, and MDH the k_cat is varied around the steady-state level (<5% change) to compute the control coefficient. ρ^res and ρ^F1 are varied (<10% change) to compute the control exerted by respiration itself and the F₁F₀-ATP synthase, respectively. The control exerted by the adenine nucleotide translocator and the proton leak is performed with parameters V_maxANT and g_H, respectively.

Effects on NADH and ΔΨ_m

A positive correlation between NADH and Ca²⁺ was obtained for the pull condition and the intermediate Ca²⁺ range for push (Fig. 6 B). Similar to the behavior of V_O2, higher levels of NADH are obtained when the control shifts toward the respiratory chain (pull) (Fig. 6 B). However, this condition gives a relatively smaller stimulation of NADH by Ca²⁺ (1.8-fold against 3.7-fold over the same range of [Ca²⁺]_i) (Fig. 6 B). For the push condition, ΔΨ_m is slightly increased with an increase in Ca²⁺ (Fig. 7 B)—this is possible only if the stimulatory effect of Ca²⁺ on TCA cycle, and the concomitant increase in proton pumping, exceed the depolarization of ΔΨ_m due to energy dissipation by Ca²⁺ entry. The decrease in NADH observed at high [Ca²⁺]_i in the push condition is due to saturation of the stimulatory effect on dehydrogenase activity whereas respiratory flux continues to increase (Fig. 6, A and B, inset).

FIGURE 7.

F₁F₀ ATP synthase activity and membrane potential as a function of [Ca²⁺]_i. The different curves represent the rate of the mitochondrial ATP synthase, V_ATPase (A) and ΔΨ_m (B) at steady state after an increase in [Ca²⁺]_i, as shown in Fig. 5 with the same set of parameters described in its legend (push and pull).

Effects on V_ATPase

The fundamental question of whether the sum total of the effects of Ca²⁺ on bioenergetics is able to increase the production of ATP is addressed in Fig. 7. The increase in the rate of ATP synthesis (V_ATPase) with increasing Ca²⁺ is higher under push (+18.1%) rather than pull (+5.2%) conditions, although the V_ATPase is larger over the whole range of Ca²⁺ in the pull condition. In the region of high Ca²⁺, the negative correlation between ΔΨ_m and Ca²⁺ in the push condition (Fig. 7 B) results in a negative correlation between V_ATPase and Ca²⁺ (Fig. 7 A).

The net difference between the stimulatory effect of Ca²⁺ on TCA cycle dehydrogenases and the dissipative effect of Ca²⁺ on ΔΨ_m (Fig. 7 B) in turn determines the extent of stimulation or inhibition of ATP synthesis. Maximal stimulation occurs in the ΔΨ_m range of maximal responsiveness, i.e., near the midpoint of the V_ATPase versus ΔΨ_m curve (see Appendix Fig. A6, B). If ΔΨ_m is too high or too low, the increase in ATP synthesis rate is negligible, even if respiration is stimulated strongly. The same reasoning explains why, in a condition of pull V_ATPase is stimulated marginally (+5.2% from 24 nM to 1.07 μM [Ca²⁺]_i).

Metabolic control analysis of mitochondrial energetics

A control analysis of respiration has also been undertaken to further understand the control of mitochondrial respiration prevailing under the conditions analyzed above (Figs. 5–7). Control analysis is performed around the steady states given by [Ca²⁺]_i 0.1 and 0.5 μM (Fig. 5). The respiratory flux control coefficients shown in Table 2 are calculated by varying the V_max values of the different processes (see Methods).

Under push conditions, the main rate-controlling steps of respiration are the Ca²⁺-sensitive dehydrogenases (KGDH and IDH) of the TCA cycle (Table 2). KGDH and IDH controlled 70% and 23%, respectively, of the respiratory flux when [Ca²⁺]_i stimulates the rates of respiration (Fig. 6 A), the F₁F₀-ATP synthase (Fig. 7 A) and the TCA cycle.

A large control by the respiratory chain and the ANT on the respiratory flux is obtained under pull conditions (Table 2). The latter situation corresponds closely to that described by Borutaite et al. (1995) regarding the control of respiration in heart mitochondria. Under these conditions, the processes downstream of NADH are flux-controlling (Table 2), but the ATP synthase is less stimulated by [Ca²⁺]_i (5.2%) when compared to the 18% increase registered under push conditions (Table 2). Thus, we conclude that [Ca²⁺]_i is better able to stimulate the rate of mitochondrial ATP synthesis when the TCA cycle exerts a significant control on respiration.

After a closer inspection of the initial reaction rate response of IDH and KGDH with respect to their substrates, only KGDH activity exhibits substantial elasticity to Ca²⁺ concentration within the stimulation range, in agreement with reported data (see Appendix Figs. A2 and A3; also see McCormack et al., 1990). This result explains why KGDH must exert a significant control on metabolism for respiratory flux to be stimulated by [Ca²⁺]_m (Figs. 5, and 6 A).

Time-dependent behavior of mitochondrial energetics during workload changes

The ability of the model to simulate time-dependent behavior was assessed by attempting to reproduce experimental data concerning mitochondrial NADH and Ca²⁺ dynamics obtained in heart trabeculae when challenged by changes in pacing frequency (Brandes and Bers, 1997, 2002). In these experiments, mitochondrial and cytoplasmic Ca²⁺ and NADH signals from rat heart trabeculae are simultaneously recorded during changes in the frequency of the electrical stimulation (Brandes and Bers, 2002). Increasing the pacing frequency from 0.25 Hz to 2 Hz results in the NADH concentration exhibiting an undershoot followed by a recovery phase to a new steady level (Fig. 8 A), whereas a decrease in the pacing frequency leads to an overshoot in NADH and slow relaxation of the signal back to the original baseline level.

FIGURE 8.

Time-dependent behavior of Ca²⁺ and NADH after changes in workload. (A, B) Experimental data showing the responses of NADH (A) and mitochondrial Ca²⁺ (B) to an increase in stimulation frequency from 0.25 to 2 Hz in a rat cardiac trabeculae (Reproduced from Brandes and Bers (2002) by copyright permission of the Biophysical Society). (C) Model response of mitochondrial Ca²⁺ to the pacing protocol described in Model Building and Numerical Methods, applied at low frequency (0.25 Hz) for 100 s, and then a high frequency (2 Hz) for 200 s, followed by a return to the low frequency. (D) Time course of NADH levels during changes in pacing frequency, simulated by increasing the Ca²⁺ pulse frequency combined with an increase in ADP_i. The “pull” parametric condition is similar to that defined in the legend of Fig. 5 except for ρ^RES = 0.0006, ρ^F1 = 1.787 mM, and mM s⁻¹; “pull II” is similar to “pull” except ρ^RES = 0.00072 mM and mM s⁻¹. (E) The effects of a change in Ca²⁺ pulse frequency or ADP_i applied separately.

An identical increase in pacing frequency in the mitochondrial model reveals a time course of [Ca²⁺]_m accumulation similar to the experimental results (Fig. 8 C), with [Ca²⁺]_m rising monotonically as the mitochondria function as an integrator of the pulsatile cytoplasmic Ca²⁺ transients, consistent with the previous results of Miyata et al. (1991).

Varying only the [Ca²⁺]_i transient driving function without changing ADP in the simulations results in a monotonic rise in NADH during an increase in pacing frequency (0.25 Hz–2 Hz) and a monotonic decline upon decreasing frequency (Fig. 8 E).

A pulse of ADP (50–150 μM) in the absence of an increase in [Ca²⁺]_i pacing frequency results in a rapid decrease in NADH due to stimulation of the respiratory chain (pull condition) and recovery upon returning to the basal level (Fig. 8 E).

When the change in Ca²⁺ transient frequency and the increase in ADP are combined, model NADH concentration transients closely resemble the experimental data (Fig. 8 D). The rate of recovery of NADH can be accelerated by a small (20%) increase in respiration and a fourfold increase in (Fig. 8 D; pull II). Fitting the curve of NADH recovery with a single exponential function renders t_1/2 values of 112 and 20 s for the conditions of pull and pull II, respectively. Mitochondrial Ca²⁺ relaxes with similar kinetics as NADH after the onset of high pacing frequency (t_1/2 = 127 and 33 s for pull and pull II, respectively), resembling more closely the relaxations observed by Brandes and Bers (2002).

DISCUSSION

General

The main contribution of this modeling study is to link the TCA cycle, OxPhos, and mitochondrial Ca²⁺ dynamics through a mathematical model able to describe mitochondrial energetics based on well-known bioenergetic principles, i.e., the dependence of respiration on the redox potential and Δμ_H, and the F₁F₀-ATPase on the phosphorylation potential and ΔΨ_m. The model is used to explore the interaction between mitochondrial energy metabolism and important extramitochondrial signals such as Ca²⁺ and substrate delivery. Several features distinguish this model from those presented previously: 1), the model integrates the kinetics of the TCA cycle with oxidative phosphorylation and Ca²⁺ handling (Fig. 1), 2), both thermodynamics and kinetics of the reactions are taken into account, permitting dynamic studies of enzyme kinetics and key metabolic intermediates as well as redox potential and ΔΨ_m as the steady state is approached, and 3), the model incorporates the explicit dependence of the TCA cycle dehydrogenases on Ca²⁺, allowing examination of the influence of Ca²⁺ on mitochondrial metabolism as a whole.

Metabolic control of oxidative phosphorylation

The integrative framework provided by the model makes it possible to analyze points of control of mitochondrial respiration by investigating how mitochondrial ATP production is matched to an increase in cardiac workload through increases of cytoplasmic Ca²⁺.

Stimulation of respiration by ADP after changes in cellular ATP utilization has been proposed as one of the mechanisms by which mitochondrial ATP production meets the cellular ATP demand (respiratory control; see also Chance and Williams, 1956; Harris and Das, 1991). A shift from state 4 (respiration is low, Δμ_H is high, no ATP synthesis) to state 3 (respiration is high, Δμ_H is lowered by maximal ATP synthesis) is known to be accompanied by control redistribution in isolated mitochondria (Groen et al., 1982; Brown et al., 1990; Murphy, 2001). In cells, mitochondria are most often intermediate between states 3 and 4 (Ainscow and Brand, 1999; Murphy, 2001). The model reproduces the expected relationships between ΔΨ_m, V_O2, and F₁F₀ATPase rate after shifts between states 3 and 4 when experiments simulating inhibition of the respiratory chain or the adenine nucleotide translocator are performed (Figs. 3, 4).

Control redistribution also occurs during impairment of mitochondrial function after ischemic injury. Mitochondria isolated from ischemic hearts show a decrease in respiratory capacity and a decrease in phosphorylation potential when respiring in the presence of succinate. These effects are more pronounced the longer the ischemic period, e.g., from 30 to 45 min (Borutaite et al., 1995). Such an inhibition of the respiratory chain, simulated here by reducing the concentration of electron carriers, results in a decreased respiration rate and a concomitant decrease in membrane potential (Fig. 4), consistent with experimental results (Fig. 4 C; see also Borutaite et al., 1995).

The model also reveals a steep dropoff of oxygen consumption as the concentration of electron carriers (ρ^res) reached a critical low level (Fig. 4 A). This type of threshold for disruption of mitochondrial function with respect to respiratory complex inhibition has been described previously (Murphy, 2001) and studied in vitro by titrating complex activity with irreversible inhibitors (Davey and Clark, 1996; Taylor et al., 1994; Rossignol et al., 1999). Maintenance of oxidative phosphorylation despite inhibition of a substantial fraction of the respiratory complexes may be explained by their high elasticities with respect to their substrates (Rossignol et al., 1999; Murphy, 2001). For example, partial inhibition of complex III will increase the ubiquinol/ubiquinone ratio, which would then increase the activity of uninhibited complexes. Thus, oxidative phosphorylation may be buffered against loss of respiratory chain components depending on the extent of inhibition of an individual complex required to decrease overall flux (Taylor et al., 1994). In our simulations, the threshold for collapse of respiration also depends on the inward flux of protons across the mitochondrial inner membrane (Fig. 4 A): when proton uptake is faster, the threshold ρ^res is shifted to a lower concentration range so that a greater fraction of carriers needs to be inhibited to affect respiration. It is important to note that ATP production would decrease before the decline in V_O2, as a result of the drop in ΔΨ_m at high proton uptake rates (Fig. 4 B).

Control by availability of substrate to the TCA cycle, AcCoA, may be relevant in the case of reperfusion of the heart after ischemia, inasmuch as TCA cycle flux is maintained or enhanced after brief ischemia (Weiss et al., 1993). The overall simulated flux through the TCA cycle in the range 0.05–0.1 mM s⁻¹ (10–20 μmol g dry wt⁻¹ min⁻¹), closely reproduces the global flux in whole hearts as reported in Jeffrey et al. (1999) and Williamson et al. (1976). Our results show that AcCoA levels exert control on respiratory flux, in parallel with NADH production, until saturation is reached (Fig. 2 A). As reported by Territo et al. (2001), substrate addition to isolated Ca²⁺-depleted mitochondria increases the respiration rate by ∼3.5-fold, ΔΨ_m by ∼30 mV (Fig. 2 C), and NADH by 7.7-fold. Similar trends are exhibited by V_O2, NADH, and ΔΨ_m in the simulations shown in Fig. 2.

Effects of Ca²⁺ on mitochondrial energetics

The inotropic mechanisms activated in response to an increase in cardiac workload (e.g., an increase in heart rate and/or Ca²⁺ release) also increase the time-averaged Ca²⁺ in the cytoplasm: this has been proposed as a control factor for increasing mitochondrial ATP production (Harris and Das, 1991; Territo et al., 2000, 2001).

An increase in [Ca²⁺]_i has two opposite effects on mitochondria: a stimulatory effect on the activity of the TCA cycle dehydrogenases as a consequence of higher [Ca²⁺]_m (Figs. 5 and 6), and a dissipative effect on the inner mitochondrial membrane potential due to the influx of the divalent cation. The balance between these two effects is reflected by ΔΨ_m. In isolated mitochondria, Ca²⁺ stimulation can increase NADH (McCormack et al., 1990; Hansford and Zorov, 1998; Territo et al., 2000), but the magnitude of this increase will also depend on any downstream effects on NADH oxidation. Flux through the electron transport chain will be accelerated by dissipation of Δμ_H, which could be caused by an increase in F₁F₀-ATPase velocity (due to increased ADP availability or direct stimulation of the ATPase (Territo et al., 2001)), or by non-ATPase-mediated energy dissipating pathways such as Ca²⁺ influx (relevant to the present model simulations) or enhanced proton leak. Non-ATPase energy dissipation will increase V_O2 without necessarily increasing ATP production; however, our model simulates both an increase in V_ATPase and an increase in V_O2 as a function of [Ca²⁺]_i (Figs. 6 A and 7 A), demonstrating that TCA cycle stimulation is adequate to bolster NADH levels and ΔΨ_m (Figs. 6 B and 7 B). The increase in the rate of ATP production by the mitochondria after an increase in [Ca²⁺]_i from 20 to 660 nM is ∼18%, a moderate change relative to the increase in V_O2 (usually more than 20% increase; Figs. 6 A and 7 A).

In the framework of the present model, the control of the respiratory flux at the mitochondrial level may be exerted by processes upstream and downstream of NADH (Table 2). Mitochondrial energetics is in a condition of push when respiration is controlled mainly by processes upstream of NADH, i.e., the TCA cycle. In this regard, a significant stimulation of ATP synthesis by Ca²⁺ can be achieved when respiration is controlled at the level of the Ca²⁺-sensitive dehydrogenases (Table 2: push). A drawback of the push condition is that the NADH levels produced by the model are lower than those determined experimentally (e.g., Brandes and Bers, 1997). Under pull conditions, downstream processes such as ANT or respiration itself, are the main rate-controlling steps of respiration (Table 2: pull).

Experimental data obtained for isolated rat liver cells show that control of respiration and oxidative phosphorylation is primarily downstream of NADH (Brown et al., 1990). Accordingly, rat liver mitochondria would be predominantly in the pull condition, inasmuch as 49% of the control is exerted by the processes of ATP synthesis, transport and consumption, 22% by proton cycling not coupled to ATP synthesis, and only 29% by respiration and upstream processes. Brown et al. (1990) estimated that 15–30% of the control over respiration would be exerted by processes involved in NADH generation. In our simulations, under pull conditions, the stimulation of oxidative phosphorylation by increasing cytosolic Ca²⁺ concentration is diminished (Figs. 6 and 7). Thus, the further incorporation of Ca²⁺-activated cytosolic ATPases, such as those involved in Ca²⁺ handling and muscle contraction, may be required in the future, to better couple the acceleration of V_O2 with the V_ATPase and increase the contribution of downstream pull on the system. In this regard, Territo et al. (2000) have reported that addition of Ca²⁺ to isolated mitochondria stimulated O₂ consumption as well as NADH levels. Determining ATP contents in mitochondrial preparations subjected to increasing Ca²⁺ concentrations, they inferred a stimulation of the ATP synthase activity. To account for these findings, they concluded that the F₁F₀-ATPase must be directly activated by Ca²⁺ (Territo et al., 2000, 2001). Nevertheless, it is notable that we can simulate an increase in ATPase activity upon increasing [Ca²⁺]_i without invoking other mechanisms of activation apart from the known biochemical and biophysical principles ruling mitochondrial bioenergetics. Stimulation depends only on the driving forces involved: the phosphorylation potential and the proton motive force. Obviously, inasmuch as the model simulations can reproduce different control properties of the system, they cannot provide an unequivocal quantitative answer to the relative contributions of upstream and downstream control mechanisms in living tissues. Therefore, we do not presume to strongly favor one control point over another for all cases; however, the model provides important feasibility arguments to be tested in future experimental studies.

An additional limitation of model interpretation lies in the absence of spatial organization of the mitochondria and cellular structures. For example, intracellular compartmentation of the adenine nucleotide pool has been proposed as an alternative explanation to Ca²⁺ activation in the context of matching of energy production and demand. It has been argued that cardiac contractile failure during ischemia or hypoxia still happens despite high ATP contents (70–80% of normal ATP content), leading to the hypothesis that a localized metabolic compartment may be present in which larger changes in phosphorylation potential occur. There is evidence for metabolic compartmentation (reviewed in Saks et al., 1998) and such metabolite channeling may be due to the close proximity of energy-consuming and energy-producing sites or to the actions of the creatine phosphate shuttle (Saks et al., 2001).

Similarly, the close apposition of sarcoplasmic reticular Ca²⁺ release sites and mitochondria may create a local Ca²⁺ microdomain, in which Ca²⁺ concentrations might be much higher than the bulk phase. Experimental evidence for fast uptake of mitochondrial Ca²⁺ has been reported (Rizzuto et al., 1998; Duchen et al., 1998). Our simulation results of the dynamics of Ca²⁺ uptake in response to pulses of [Ca²⁺]_i indicate that mitochondrial Ca²⁺ must essentially act as an integrator of cytosolic Ca²⁺ to reproduce intact muscle behavior (Fig. 8). This is another area in which spatial considerations and alternative Ca²⁺ uptake pathways may be incorporated in the future.

Time-dependent behavior of mitochondrial energetics

The ability of the model to simulate time-dependent behavior is indeed the most rigorous test of its validity, and emphasizes the utility of a model based on realistic kinetic properties of its components. We have been able to closely reproduce the NADH evolution in rat heart trabeculae when challenged by increases in workload through the pacing frequency of electrical stimulation (Brandes and Bers, 2002). The temporal profile of NADH evolution found experimentally can be simulated (compare Fig. 8, A and D). To reproduce the observed NADH transients, we have to incorporate an increase in ADP to represent increased cellular ATPase activity together with the increase in the pacing frequency of [Ca²⁺]_i. Thus, the model results suggest a co-participation of Ca²⁺ and ADP to explain mitochondrial NADH dynamics during an increase in workload. Additionally, we conclude that the under- and over-shoots of mitochondrial NADH during changes in pacing frequency are due to the rapid effects of ADP on respiration followed by the slower adaptation of the Ca²⁺-sensitive dehydrogenase activities. These results point out the involvement of respiratory control and activation of dehydrogenases by Ca²⁺ in the coupling between energy supply and demand in cardiac myocytes challenged by changes in workload.

Previous models of energetics

Previously, the most comprehensive metabolic models of cardiac metabolism were reported by Kohn and co-workers, to describe the energetics of the rat heart consuming different carbon substrates (Kohn and Garfinkel, 1979; 1983). Glycolysis, fatty acid oxidation, the TCA cycle, a lumped system representing mitochondrial respiration and oxidative phosphorylation, and chelation equilibria for nucleotides and organic acids were included in this model. However, these models were essentially based on mass action laws, with the assumption that most of the metabolic steps were reversible.

Other TCA cycle models were specifically developed for interpreting the data from NMR studies of whole hearts (Weiss et al., 1992, 1993; Yu et al., 1997; Tran-Dinh et al., 1996; Jeffrey et al., 1999). In such cases, the model equations were designed to fit the evolution of the ¹³C enrichment profiles of various TCA cycle intermediates, following the approach of Chance et al. (1983). The tracer equations used in these models do not address any explicit kinetic mechanism of each enzyme participating in the TCA cycle.

Other models of the TCA cycle have been developed for the amoeba Dictyostelium discoideum based on data about the temporal evolution of tracers (Wright et al., 1992; Shiraishi and Savageau, 1992). Wright and collaborators (1992) based their model on Michaelis-Menten kinetics taking into account the kinetic constants for substrates, products and some effectors obtained from the in vitro characterization of enzyme kinetics. Shiraishi and Savageau (1992), in turn, developed a model based on the power law formalism. Both models appropriately describe the behavior of the system for a certain range of parametric conditions. However, none of those models account for the effect of Ca²⁺, which is of central importance to cardiac mitochondrial energetics.

A model of pancreatic β-cell mitochondrial metabolism by Magnus and Keizer (1997), which incorporates Ca²⁺ handling by mitochondria, involves three ordinary differential equations describing ΔΨ_m, Ca²⁺, and ADP concentrations as a function of time. This model takes into account the most important transport processes of the inner mitochondrial membrane, i.e., redox-driven proton pumps, the F₁F₀-ATPase, proton leak, the adenine nucleotide transporter, the Ca²⁺ uniporter, and the mitochondrial Na⁺/Ca²⁺ exchanger. Important limitations of this model are that it does not include stimulation of TCA cycle dehydrogenases by Ca²⁺ and that NADH is a fixed parameter (Magnus, 1995; Magnus and Keizer, 1997) instead of a variable (as in the present model). Nevertheless, in agreement with experimental data, the model behavior was able to reproduce the dependence of the rate of Ca²⁺ uptake by the uniporter on cytosolic Ca²⁺ concentration and predicts a sharp threshold for uptake at cytosolic Ca²⁺ concentrations of ∼0.4–0.5 μM (Magnus and Keizer, 1997).

An oxidative phosphorylation mathematical model has been developed by Korzeniewski (1996, 1998) and applied to the analysis of how the rate of ATP production by respiration is adjusted to meet energy demand during muscle contraction (Korzeniewski, 1998). As in previous cases, this model does not account for the coupling of OxPhos to the TCA cycle and thus no explicit dependence on Ca²⁺ by TCA cycle dehydrogenases was taken into account.

CONCLUSION

In summary, the integrated thermokinetic model constructed here reproduces the expected relationships among ΔΨ_m, V_O2, NADH, and mitochondrial matrix Ca²⁺ accumulation under various conditions including simulated inhibition of respiratory chain complexes and/or adenine nucleotide translocase.

Based solely on simple bioenergetic principles, it is demonstrated that Ca²⁺ activation of the TCA cycle, NADH production, mitochondrial respiration, and ATP synthesis is possible when respiratory flux is mainly controlled by the Ca²⁺-sensitive KGDH. During changes in workload, under pull conditions, both respiratory control and stimulation by Ca²⁺ appear to be required to reproduce the temporal evolution of NADH in heart trabeculae. Under other parametric conditions, supplemental control mechanisms for oxidative phosphorylation may be necessary to account for experimental data. Increased ATP production by Ca²⁺ can only be achieved when the extent of stimulation of NADH production exceeds the energy-dissipating effect of Ca²⁺ influx on ΔΨ_m.

This comprehensive model of cardiac mitochondrial metabolism will permit testing of bioenergetic control hypotheses, provide a basis for further refinement of individual steps of oxidative phosphorylation and mitochondrial Ca²⁺ handling, and allow for future incorporation of other biochemical pathways.

GLOSSARY

[ADP]_m

Concentration of ADP in mitochondrial matrix

[Ca²⁺]_i

Ca²⁺ concentration in cytosolic compartment

[Ca²⁺]_m

Ca²⁺ concentration in mitochondrial matrix

[NADH]

Concentration of NADH in mitochondrial matrix

A_F1

Phosphorylation potential

A_res

Redox potential

αKG

α-ketoglutarate

ASP

Aspartate

CIT

Citric acid

F₁F₀-ATPase

Mitochondrial F₁F₀ ATP synthase activity

FUM

Fumarate

IDH

Isocitrate dehydrogenase

ISOC

Isocitrate

KGDH

α-ketoglutarate dehydrogenase

MAL

Malate

OAA

Oxalacetate

SCoA

Succinyl CoA

Suc

Succinate

TCA

Cycle tricarboxylic acid cycle

V_AAT

Rate of aspartate amino transferase

V_ACO

Rate of aconitase

V_ANT

Rate of the adenine nucleotide translocator (ANT)

V_ATPase

Rate of the F₁F₀ ATP synthase

V_{C_ASP}

Rate of Aspartate consumption

V_CS

Rate of citrate synthase

V_FH

Rate of fumarate hydratase

V_He

Rate of H transport across the membrane driven by NADH oxidation

V_He(F)

Rate of H transport across the membrane driven by FADH₂ oxidation

V_Hleak

H leak across the inner mitochondrial membrane

V_Hu

Rate of H uptake by mitochondrial via F1FoATPase

V_IDH

Rate of isocitrate dehydrogenase

V_KGDH

Rate of alpha ketoglutarate dehydrogenase

V_MDH

Rate of malate dehydrogenase

V_NaCa

Rate of the mitochondrial Na⁺/Ca²⁺ antiporter

V_O2

Rate of oxygen consumption

V_SDH

Rate of succinate dehydrogenase

V_SL

Rate of succinate lyase

V_uni

Rate of the Ca²⁺ uniporter in the mitochondrial membrane

ΔΨ_m

Electrical potential across the mitochondrial inner membrane

Δμ_H

Proton motive force

SUPPLEMENTAL INFORMATION

An online Appendix of the Model of mitochondrial bioenergetics in cardiac myocytes can be found by visiting BJ online at http://www.biophysj.org.

APPENDIX

The model has three main components: mitochondrial matrix processes, oxidative phosphorylation, and Ca²⁺ dynamics.

The tricarboxylic acid (TCA) cycle

In the mitochondrial matrix, we have considered the dynamics of the TCA cycle that represents the end stage for carbon-chain metabolic substrates in the cell.

The TCA cycle in our model may be split into two pathways: the tricarboxylate (from oxalacetate, OAA, to α-ketoglutarate, αKG, via citrate, CIT) and the dicarboxylate (from αKG to OAA via succinate, Suc; see also Fig. 1) (Jafri et al., 2001). In the tricarboxylate pathway AcCoA and OAA react to produce αKG, NADH, and CO₂. In turn, αKG serves as a substrate of the dicarboxylate pathway producing OAA that will resupply the first pathway. Alternatively, a shunt may directly lead from OAA into αKG through the aspartate amino transferase (AAT)-catalyzed reaction (Fig. 1).

The model includes detailed kinetics of the key regulatory enzymes of the TCA cycle with respect to their effectors such as Ca²⁺, Mg²⁺, ADP, NADH, and NAD⁺, as well as their carbon substrates (Dudycha, 2000; Jafri et al., 2001).

Unlike succinate dehydrogenase (SDH), aconitase, succinyl CoA lyase, and fumarate hydratase are considered to operate under near equilibrium conditions.

Citrate synthase (CS)

The first enzyme of the cycle is CS that catalyzes the condensation of OAA and AcCoA to give CIT. The kinetic rate law (Eq. 13) we used was modified from that derived by Dudycha (2000) based on data by Shepherd and Garland (1969):

(13)

Since the TCA cycle and the AAT shunt are closed from the point of view of carbon intermediates, a conservation equation relating all TCA metabolites is included in the model. Thus, the level of CIT is the result of the balance of all other intermediates in the cycle as follows:

(14)

with C_Kint as the total concentration of TCA cycle intermediates.

CS is studied as a function of the concentrations of its substrates, OAA and AcCoA, and displays saturating activity with respect to both (Fig. A1, A and B). Saturation is achieved at low concentrations of OAA. Low levels of this intermediate have been estimated from in vivo measurements and theory (Jeffrey et al., 1999). This result supports the operation of CS at high rate at low concentrations of its substrate OAA.

FIGURE A1.

Dependence of citrate synthase activity on oxalacetate and AcCoA. (A) The rate of CS is plotted against oxaloacetate concentration with s⁻¹ at the specified AcCoA concentrations and parameter values according to Table 3 (see Eq. 13). (B) The same parameter set is used to study CS activity as a function of AcCoA at different oxalacetate concentrations.

Aconitase (ACO)

ACO is considered to be operating near equilibrium as reported data in the literature suggest (Jeffrey et al., 1999) (Eq. 15).

(15)

The kinetic expression of Eq. 15 is based on the law of mass action and assumes that the enzyme operates reversibly near equilibrium, characterized by an equilibrium constant,

Isocitrate dehydrogenase (IDH)

IDH is one of the matrix dehydrogenases regulated by calcium. Additionally, IDH is activated by ADP and inhibited by its product, NADH (Fig. 1). Moreover, it shows a sigmoidal dependence on isocitrate, ISOC (Rutter and Denton, 1988).

(16)

IDH activity exhibits a modest response to Ca²⁺ in the concentration range described in vivo (Fig. A2, A). The rate of IDH is more sensitive to Ca²⁺ concentrations near saturation of the activity with respect to the substrate ISOC (Fig. A2, A). The quasi-exponential dependence of IDH with respect to NAD⁺ is due primarily to the strong inhibitory effect exerted by NADH on the enzymatic activity. Inasmuch as NAD⁺ and NADH are constrained by a conservation relationship, an increase in NAD⁺ implies a decrease in NADH and strong activation of enzyme activity as postulated (Fig. A2, B).

FIGURE A2.

Isocitrate dehydrogenase activity as a function of substrate and Ca²⁺ concentrations. (A) The rate of IDH is studied as a function of isocitrate and Ca²⁺ concentrations in the physiological range (see Eq. 16). The analysis was performed with the following parameter values: [NAD⁺] = 9.5 mM; [ADP]_m = 2.0 mM; mM; s⁻¹ and those described in Table 3. (B) The IDH dependence on NAD⁺ concentration is analyzed at different ADP concentrations with [ISOC] = 0.5 mM, and [Ca²⁺] = 0.5 μM. Otherwise the parameters used are as in A.

Alpha-ketoglutarate dehydrogenase (KGDH)

KGDH is another Ca²⁺-sensitive dehydrogenase participating in the TCA cycle. The rate expression for this enzyme was derived from Dudycha (2000) fitting data from Panov and Scarpa (1996). We have modified several of the parameters used by Dudycha (2000) to reproduce the experimental data from Rutter and Denton (1988).

(17)

Unlike IDH, KGDH shows a marked dependence on Ca²⁺ concentration in the physiological range (Fig. A3, A). This highlights one of the main conditions for the model to exhibit a maximal response to Ca²⁺ concentrations, KGDH needs to be the main rate-controlling step of respiratory flux. Also unlike IDH, the kinetics of KGDH are saturable with respect to NAD⁺, with the maximal activation by NAD⁺ depending on αKG levels (Fig. A3, B).

FIGURE A3.

Alpha-ketoglutarate dehydrogenase dependence on α-ketoglutarate, NAD⁺, and Ca²⁺. (A) The set of parameters used in the analysis of the rate of KGDH (V_KGDH) as a function of its substrate is as indicated in Table 3 (see also Eq. 17) with the following modifications: mM; s⁻¹; [NAD⁺] = 1.0 mM; and [Mg²⁺] = 0.4 mM. (B) The analysis of the dependence of V_KGDH on NAD⁺ concentration is performed with the following parameter values: [Mg²⁺] = 0.4 mM, and [Ca²⁺] = 0.1 μM. Otherwise the parameters used are as in A.

Succinyl CoA lyase (SL)

SL catalyzes the conversion of SCoA into Suc, involving the direct phosphorylation of GDP (Kadrmas et al., 1991). In our model we have assumed rapid equilibration of the GTP and ATP pools via a matrix nucleoside diphosphate kinase (Colomb et al., 1972). Thus, the rate equation for SL is ruled by the following expression, including mitochondrial ADP and ATP levels:

(18)

Succinate dehydrogenase (SDH)

SDH catalyzes the reduction of ubiquinone by succinate, coupled with the formation of fumarate and ubiquinol. It is the only enzyme of the TCA cycle which is bound to the inner membrane of mitochondria, directly linking the TCA cycle to the electron transport chain at complex II. The kinetic properties of SDH have been extensively studied (reviewed by Singer et al., 1973, and Hatefi and Stiggall, 1976). Its reaction rate is subject to inhibition by its product, FUM, and by OAA (Fig. 1; see also Singer et al., 1973; Hatefi and Stiggall, 1976). The rate expression is described by the following equation:

(19)

Fumarate hydratase (FH)

FH is considered to operate near equilibrium and the kinetic parameters are taken from Dudycha (2000) based on data from Teipel and Hill (1971). Its rate expression is as follows:

(20)

Malate dehydrogenase (MDH)

MDH is another highly regulated TCA cycle enzymatic activity. MDH is inhibited by its product, OAA, and modulated by pH (Dudycha, 2000). The parameters of the equation have been adjusted based on the data reported by Elduque and co-workers (1983).

(21)

with

(22)

and

(23)

Fig. A4 shows the MDH activity as a function of its substrate, malate (MAL), at different NAD⁺ concentrations. MDH displays typical Michaelis-Menten kinetics that saturate at high substrate concentrations, which are at least two orders of magnitude higher than the actual MAL levels determined in myocytes (Albe et al., 1990).

FIGURE A4.

Malate dehydrogenase activity as a function of substrate concentration. The dependence of MDH rate on its substrate malate is studied with the following parameter values: mM, s⁻¹, and those shown in Table 3 (see also Eq. 21).

Aspartate amino transferase (AAT)

AAT activity constitutes a shunt linking directly OAA to αKG. This activity is considered to be working close to equilibrium in the presence of a large excess of glutamate, GLU (Eq. 24).

(24)

The aspartate (ASP) resulting from AAT activity is assumed to be consumed in a reaction that follows a mass action law with respect to its substrate ASP, as follows:

(25)

Differential Eqs. 3–11 are assembled from this group of processes in the model. The parameters of Eqs. 13–25 for this section of the model description are described in the module: TCA cycle in Table 3.

TABLE 3.

Model parameters: TCA cycle

Symbol	Value	Units	Description	Eq.	Ref.*
	3.2	s−1	Catalytic constant of CS	13	a
	0.4	mM	Concentration of CS	13	b
	1.26 10−2	mM	Michaelis constant for AcCoA	13	a
	6.4 10−4	mM	Michaelis constant for OAA	13	a
CKint	1.0	mM	Sum of TCA cycle intermediates' concentration	14	c
	12.5	s−1	Forward rate constant of ACO	15	a
	2.22		Equilibrium constant of ACO	15	a
	1.94–16.3	s−1	Rate constant of IDH	16	a
	0.109	mM	Concentration of IDH	16	b
	0.19	mM	Inhibition constant by NADH	16	a
	6.2 10−2	mM	Activation constant by ADP	16	a
[H+]	2.5 10−5	mM	Matrix proton concentration	16	k
kh,1	8.1 10−5	mM	Ionization constant of IDH	16	a
kh,2	5.98 10−5	mM	Ionization constant of IDH	16	a
	1.52	mM	Michaelis constant for isocitrate	16	a
	0.923	mM	Michaelis constant for NAD+	16	a
	0.00141	mM	Activation constant for Ca2+	16	a
	0.15–5.0	s−1	Rate constant of KGDH	17	a
	0.5	mM	Concentration of KGDH	17	f
	1.94	mM	Michaelis constant for αKG	17	a
	38.7	mM	Michaelis constant for NAD	17	a
	0.0308	mM	Activation constant for Mg2+	17	a
	1.27 10−3	mM	Activation constant for Ca2+	17	a
nαKG	1.2		Hill coefficient of KGDH for αKG	17	f
Mg2+	0.4	mM	Mg2+ concentration in mitochondria	17	d
	0.127	mM−1 s−1	Forward rate constant of SL	18	f
	3.115		Equilibrium constant of the SL reaction	18	a
[CoA]	0.02	mM	Coenzyme A concentration	18	b
	1.0	s−1	Rate constant of SDH	19	f
	0.5	mM	SDH enzyme concentration	19	f
	3.0 10−2	mM	Michaelis constant for succinate	19	e
	1.3	mM	Inhibition constant by fumarate	19	e
	0.15	mM	Inhibition constant by oxalacetate	19	e
	0.83	s−1	Forward rate constant for FH	20	a
	1.0		Equilibrium constant of FH	20	a
kh1	1.1310−5	mM	Ionization constant of MDH	22	j
kh2	26.7	mM	Ionization constant of MDH	22	a
kh3	6.68 10−9	mM	Ionization constant of MDH	23	j
kh4	5.62 10−6	mM	Ionization constant of MDH	23	j
koffset	3.99 10−2		pH-independent term in the pH activation factor of MDH	22	a
	2.775 101	s−1	Rate constant of MDH	21	f
	0.154	mM	Total MDH enzyme concentration	21	b
	1.493	mM	Michaelis constant for malate	21	a
	3.1 10−3	mM	Inhibition constant for oxalacetate	21	a
	0.2244	mM	Michaelis constant for NAD+	21	a
	0.644	s−1	Forward rate constant of AAT	24	a
	6.6		Equilibrium constant of AAT	24	a
KC_ASP	0.01	s−1	First order rate constant of aspartate consumption	25	f
Model parameters: oxidative phosphorylation
ra	6.394 10−10	s−1	Sum of products of rate constants	26,27	g,h
rb	1.762 10−13	s−1	Sum of products of rate constants	27	g,h
rc1	2.656 10−19	s−1	Sum of products of rate constants	26	g,h
rc2	8.632 10−27	s−1	Sum of products of rate constants	26	g,h
r1	2.077 10−18		Sum of products of rate constants	26,27	g,h
r2	1.728 10−9		Sum of products of rate constants	26,27	g,h
r3	1.059 10−26		Sum of products of rate constants	26,27	g,h
ρres	0.0006–0.05	mM	Concentration of electron carriers (respiratory complexes I-III-IV)	26,27	g,h
Kres	1.35 1018		Equilibrium constant of respiration	28	g,h
ρres(F)	0.0045	mM	Concentration of electron carriers (respiratory complexes II-III-IV)
ΔΨB	0.05	V	Phase boundary potential	26,27,31	g,h
g	0.85		Correction factor for voltage	26,27	g,h
Kres(F)	5.765 1013		Equilibrium constant of FADH2 oxidation	29
[FADH2]	1.24	mM	Concentration of FADH2 (reduced)	29
[FAD]	0.01	mM	Concentration of FAD (oxidized)	29
pa	1.656 10−5	s−1	Sum of products of rate constants	31,32	g,h
pb	3.373 10−7	s−1	Sum of products of rate constants	32	g,h
pc1	9.651 10−14	s−1	Sum of products of rate constants	31	g,h
pc2	4.585 10−14	s−1	Sum of products of rate constants	31	g,h
p1	1.346 10−8		Sum of products of rate constants	31,32	g,h
p2	7.739 10−7		Sum of products of rate constants	31,32	g,h
p3	6.65 10−15		Sum of products of rate constants	31,32	g,h
ρF1	0.06–1.8	mM	Concentration of F1F0-ATPase	31,32	g,h
KF1	1.71 106		Equilibrium constant of ATP hydrolysis	33	g,h
R	8.315	V C mol−1 °K−1	Gas constant
T	310.16	°K	Mammalian body temperature
F	96480	C mol−1	Faraday constant
Pi	20.0	mM	Inorganic phosphate concentration	33	i
Cm	15.0	mM	Total sum of mito adenine nucleotides	34	d
VmaxANT	0.05–24.0	mM s−1	Maximal rate of the ANT	35	g,h
h	0.5		Fraction of ΔΨm	35	g,h
[ADP]i	0.05–0.2	mM	Cytoplasmic ADPi concentration	35
[ATP]i	6.5	mM	Cytoplasmic ATPi concentration	35	d
gH	0.01	mM s−1 V−1	Ionic conductance of the inner membrane	36	g,h
ΔpH	−0.6	pH units	pH gradient across the inner membrane	37	k
CPN	10.0	mM	Total sum of mito pyridine nucleotides	30	b
Cmito	1.812	mM V−1	Inner membrane capacitance	2	l,m
Model parameters: calcium dynamics
	0.625–1.25	μM s−1	Vmax uniport Ca2+ transport	38	g,h
[Ca2+]i	2.0 10−2–1.2	μM	Cytosolic Ca2+ concentration	38	i
ΔΨ°	0.091	Volts	Offset membrane potential	38	g,h
Kact	3.8 10−4	mM	Activation constant	38	g,h
Ktrans	0.019	mM	Kd for translocated Ca2+	38	g,h
L	110.0		Keq for conformational transitions in uniporter	38	g,h
na	2.8		Uniporter activation cooperativity	38	g,h
	0.005–0.2	mM s−1	Vmax of Na+/Ca2+ antiporter	39	g,h
b	0.5		ΔΨm dependence of Na+/Ca2+ antiporter	39	g,h
KNa	9.4	mM	Antiporter Na+ constant	39	g,h
[Na+]i	10.0	mM	Cytosolic Na+ concentration	39	g,h
KCa	3.75 10−4	mM	Antiporter Ca2+ constant	39	g,h
n	3		Na+/Ca2+ antiporter cooperativity	39	g,h
f	0.0003		Fraction of free [Ca2+]m	12	g,h

^*References: a, Dudycha et al., 2000; b, Albe et al., 1990; c, calculated from Jeffrey et al., 1999; d, Corkey et al., 1986; e, Singer, 1966; f, adjusted; g, Magnus and Keizer, 1997; h, Magnus, 1995; i, Crompton, 1999; j, calculated from data of Emyanitoff and Kelly, 1982; k, Jung et al., 1989; l, Gunter and Pfeiffer, 1990; m, Wojtczak et al., 1986.

Oxidative phosphorylation

The respiration-driven proton pump

The main reaction steps described are the O₂ consumption flux, V_O2, and its linked proton efflux, V_He. The rate of these two processes depends on Δμ_H and the redox potential as driving forces. Eqs. 26–28 describe the flow of electrons from NADH to O₂ and its associated proton flux from the matrix to the intermembrane space in mitochondria (Magnus and Keizer, 1997). In adapting the Altman-King-Hill-based model of the respiration-driven proton pump for heart mitochondria, we have considered the electron flow from NADH oxidation to be incompletely coupled to the outward flow of protons, V_He as in Magnus (1995) and Magnus and Keizer (1997).

(26)

(27)

with

(28)

Another very important modification of the previous model is that we have replaced the dependence of the respiratory flux on Δμ_H instead of just the electrical potential, ΔΨ_m. This is done so as not to ignore the transmembrane proton concentration gradient (ΔpH) component of the proton motive force (Eqs. 26, 27). Thus, we assume that most transitions between the six states in the proton pump model (Magnus, 1995; Magnus and Keizer, 1997) depend on both ΔΨ_m and ΔpH through Δμ_H.

In a modification of the original formulation, we also consider the complex II electrons input by succinate (Suc) through FADH₂ to the respiratory chain:

K_eq in Eq. 29 reads as K_res(F):

(29)

The flux of protons driven by FADH₂ oxidation (V_He(F)) has the same form as Eq. 27 except for adjustment of the redox potential (Eq. 29) and the H⁺ stoichiometry (four instead of six). A careful analysis reveals that the sensitivity of the overall O₂ consumption with complex II is very low, which leads us to consider FADH₂ as a parameter in our model.

Mitochondrial NAD⁺ is assumed to be conserved according to the following relation:

(30)

with C_PN as the total concentration of pyrimidine nucleotides.

Oxygen consumption rate (V_O2) is studied as a function of NADH and ΔΨ_m (Fig. A5). NADH exerts its influence on V_O2 through the redox potential A_res (Eqs. 26, 28), with the shape of the curve depending on ΔΨ_m (Fig. A5, A); i.e., at high ΔΨ_m, the dependence of V_O2 on NADH becomes shallower, as it becomes more difficult to pump protons against a high Δμ_H. The decrease in V_O2 at high ΔΨ_m is evident in Fig. A5, B. The midpoint of this flux-force relation is shifted to the left by an increase in the ΔpH component of Δμ_H (Fig. A5, B). Around the midpoint, the relationship between V_O2 and ΔΨ_m is almost linear, but outside this range the rate of respiration becomes independent of ΔΨ_m (Fig. A5, B).

FIGURE A5.

Dependence of the respiration rate, V_O2, on redox potential, ΔΨ_m and ΔpH. The respiration rate is studied as a function of: (A) NADH concentration in the mitochondrial matrix with ΔpH = −0.4; ρ^res = 1.2 10⁻² mM; and ΔΨ_m as indicated next to each curve (see Eq. 26); (B) ΔΨ_m at various ΔpH values as indicated and fixed NADH concentration (=0.1 mM). The other parameter values in Eq. 26 are as indicated in Table 3.

The F₁F₀-ATPase

The rate of the F₁F₀-ATPase from Magnus and Keizer (1997) was modified to account for reversibility (Eq. 31). A factor of 10² was introduced in the first term of Eq. 31 and the value of the sum of products of rate constants p_c2 was increased with respect to the original value (Magnus and Keizer, 1997; see also Table 3, this article). These modifications together with the substitution of ΔΨ_m by Δμ_H in Eqs. 31 and 32 allow reversibility of the H⁺ pump when Δμ_H falls below a certain value or at very high phosphorylation potentials.

According to the concept of respiratory control (Chance and Williams, 1956), in state 3, mitochondrial function is governed by the availability of ADP and P_i. The chemiosmotic hypothesis dictates that ΔΨ_m is lowered by a H⁺ influx, which drives the production of ATP by the F₁F₀-ATPase. Escape from respiratory control results in higher O₂ consumption, electron transfer, and substrate oxidation that is not coupled to phosphorylation of ADP, and a decrease in the P:O ratio (Mitchell, 1969).

(31)

(32)

with

(33)

Mitochondrial ATP, [ATP]_m, is assumed to be conserved according to the following relation:

(34)

with C_m as the total sum of adenine nucleotides and [ADP]_m as the mitochondrial ADP concentration.

The dependence of F₁F₀-ATPase rate on its substrate, ADP, is illustrated in Fig. A6, A. At low concentrations of ADP_m (state 4 respiration), the thermodynamics of the pump favors ATP hydrolysis (shown by convention as a negative velocity) depending on ΔΨ_m (Eq. 2). An accumulation of ADP_m tends to shift mitochondrial energetics toward state 3, and net ATP synthesis (positive velocity). Thus, the relationship between F₁F₀-ATPase activity and [ADP]_m depends on ΔΨ_m; at higher ΔΨ_m, less [ADP]_m is required for the F₁F₀-ATPase to achieve its maximal rate and the forward (synthetic) mode is favored (Fig. A6, A). Inasmuch as Δμ_H is the actual driving force for ATP synthesis, the forward mode is enhanced at any given ΔΨ_m as ΔpH increases (Fig. A6, B).

FIGURE A6.

Dependence of F₁F₀-ATP synthase activity on ADP_m, ΔΨ_m and ΔpH. The ATP synthetic (positive) or hydrolytic (negative) activity is plotted as a function of (A) ADP_m concentration with ΔpH = −0.4, and ρ^F1 = 0.525 mM, at different ΔΨ_m as indicated (see Eq. 31); (B) ΔΨ_m at the indicated ΔpH values and a constant [ADP]_m (=1.0 mM).

Adenine nucleotide translocator (ANT) and proton leak

To complete the description of the major membrane-OxPhos-associated processes, the exchange of adenine nucleotides across the mitochondrial membrane as well as a proton leak, are considered (Fig. 1).

The ANT is modeled according to a sequential mechanism of the carrier (Magnus and Keizer, 1997; Bohnensack, 1982). Oppositely oriented sites bind either ATP⁴⁻ or ADP³⁻, and both sites must be filled before the protein isomerization causes a ligand exchange across the inner membrane. V_ANT, the flux of ANT-mediated exchange between cytosolic ADP (ADP_i) and matrix ATP (ATP_m) (Eq. 35) is considered to be electrogenic and dependent on the gradients of both ATP and ADP across the inner mitochondrial membrane as follows:

(35)

The numbers (i.e., 0.45, 0.05, and 0.8) preceding [ATP]_i,m and [ADP]_i,m account for the nucleotides concentration in the correct ionization form that bind to their site in the ANT. An explicit expression of this equation, in terms of the ionic species participating in the translocation, is available in Magnus and Keizer (1997; see their Eq. 16 and Table 4).

The parameter h accounts for the fraction of ΔΨ_m responsible for the large increase in the Michaelis-Menten constant for ATP uptake by energized mitochondria (Magnus and Keizer, 1997). The maximum rate for ADPi:ADPm exchange in the absence of ATP is given by V_maxANT.

The H⁺ leakage is considered to be a linear function of the Δμ_H through a proportionality constant given by the H⁺ conductance, g_H. Although the actual relationship is roughly exponential (Brown and Brand, 1986) there is an extensive linear region, especially at high ΔΨ_m values, which justifies simplification of the rate equation, described as follows:

(36)

with

(37)

The parameters of Eqs. 26–37 for this section of the model are described in the module: Oxidative phosphorylation in Table 3. Ordinary differential Eqs. 1–3 are the ones that can be assembled from this section (Table 1).

Mitochondrial calcium dynamics

The Ca²⁺ uniporter

The mitochondrial Ca²⁺ uniporter was modeled in Magnus and Keizer (1997) and depends on the electrochemical driving force for Ca²⁺. Thus, ΔΨ_m and cytoplasmic Ca²⁺ concentration ([Ca²⁺]_i) are the primary determinants of the uniporter velocity as follows:

(38)

This equation produces current-voltage curves that fit most available electrophysiological data (Wingrove et al., 1984; Gunter and Pfeiffer, 1990; Gunter and Gunter, 1994; see also electronic Appendix: http://www.jhmi.edu/bor/model.htm).

The Na⁺/Ca²⁺ exchanger

The expression for the Na⁺/Ca²⁺ antiporter was modified from Magnus and Keizer (1997). Available experimental evidence indicates that the antiporter is able to sense both [Ca²⁺]_m and [Ca²⁺]_i based on the reversal of its activity under pathological conditions (Griffiths et al., 1998). Under ischemic conditions the Na⁺/Ca²⁺ antiporter may import Ca² while extruding Na⁺. Extrusion of Ca²⁺ from the matrix is favored under normal conditions. Thus, our rate expression accounts for Na⁺/Ca²⁺ antiporter dependence on the Ca²⁺ gradient across the inner mitochondrial membrane (Eq. 39).

(39)

The rate of the Na⁺/Ca²⁺ antiporter, V_NaCa, displays an exponential behavior as a function of [Ca²⁺]_m. As a result of the assumption of electrogenicity, V_NaCa exhibits a steeper dependence on [Ca²⁺]_m at high than at low ΔΨ_m (Fig. A7, A).

FIGURE A7.

Rate of the mitochondrial Na⁺/Ca²⁺ antiporter and Ca²⁺ uniporter. In A the rate of the Na⁺/Ca²⁺ antiporter, V_NaCa, is analyzed for its dependence on [Ca²⁺]_m at different ΔΨ_m with the following parameter values: [Ca²⁺]_i = 0.2 μM, = 0.051 mM s⁻¹ (see Eq. 39). B shows the dependence of Ca²⁺ uniporter as a function of the [Ca²⁺]_i at various ΔΨ_m with mM s⁻¹. The remainder of the parameters involved in Eqs. 38 and 39 are described in Table 3.

The flux through the Ca²⁺ uniporter, V_uni, increases exponentially with ΔΨ_m and the concentration of cytoplasmic Ca²⁺, in the physiological range (Fig. A7, B). Clearly, the ability of the mitochondria for Ca²⁺ uptake increases with the energization of the inner membrane as reflected by ΔΨ_m.

Unit conversions

We have expressed all concentrations in millimolar (mM) units. For heart cells, the mitochondrial protein density is ∼125 mg protein/ml (Randle and Tubbs, 1979). Because heart cells are compartmentalized into 58.5% and 36% cytosolic and mitochondrial volumes, respectively (Page, 1978; Katz, 1992) a factor of 0.125 converts measured nanomoles per milligram mitochondrial protein (nmol mg⁻¹ protein) to mitochondrial mM terms.

Several of the parameters shown in Table 3 were taken from Magnus and Keizer (1997). We have converted their parameter values, usually expressed in nmol mg⁻¹ protein, to mM by the conversion factor referred above.