Mitochondrial energetic metabolism—some general principles

Authors

  • Jean-Pierre Mazat,

    Corresponding author
    1. CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
    2. Univ. Bordeaux, IBGC, UMR 5095, 33077 Bordeaux cedex, France
    • Jean-Pierre Mazat, CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
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    • Tel: +33 5 56 99 90 41; Fax: +33 5 56 99 90 40

  • Stéphane Ransac,

    1. CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
    2. Univ. Bordeaux, IBGC, UMR 5095, 33077 Bordeaux cedex, France
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  • Margit Heiske,

    1. CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
    2. Univ. Bordeaux, IBGC, UMR 5095, 33077 Bordeaux cedex, France
    3. Institut für Biologie, Theoretische Biophysik, Humboldt-Universität zu Berlin, Invalidenstraβe 42, Berlin, Germany
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  • Anne Devin,

    1. CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
    2. Univ. Bordeaux, IBGC, UMR 5095, 33077 Bordeaux cedex, France
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  • Michel Rigoulet

    1. CNRS-UMR5095, Institute of Biochemistry and Genetics of the Cell, 1 Rue Camille Saint Saëns, 33077 Bordeaux cedex, France
    2. Univ. Bordeaux, IBGC, UMR 5095, 33077 Bordeaux cedex, France
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Abstract

Summary

In nonphotosynthetic organisms, mitochondria are the power plant of the cell, emphasizing their great potentiality for adenosine triphosphate (ATP) synthesis from the redox span between nutrients and oxygen. Also of great importance is their role in the maintenance of the cell redox balance. Even though crystallographic structures of respiratory complexes, ATP synthase, and ATP/adenosine diphosphate (ADP) carrier are now quite well known, the coupling between ATP synthesis and cell redox state remains a controversial issue. In this review, we will present some of the processes that allow a modular coupling between ATP synthesis and redox state. Furthermore, we will present some theoretical approaches of this highly integrated system. © 2013 IUBMB Life, 65(3):171–179, 2013

Introduction

During aerobic growth, a great part of the adenosine triphosphate (ATP) produced in nonphotosynthetic cells originates from mitochondrial oxidative phosphorylation (OxPhos). The respiratory chain (Fig. 1) supports a sequence of redox reactions in which electrons passing along a series of enzymes located in the inner mitochondrial membrane release free energy that is used for the translocation of protons across this membrane. This proton flux from the matrix to the intermembrane space establishes a difference in proton electrochemical potential (called proton-motive force), which is used by different energy transducers and particularly by membrane bound ATP-synthase to convert adenosine diphosphate (ADP) and phosphate to ATP. This sequence of reactions called OxPhos ensures two important cell functions, the redox state maintenance (mainly nicotinamide adenine dinucleotide (NADH) reoxidation) and ATP synthesis (2).

Figure 1.

Mitochondrial OxPhos. Pumped protons are in red. Chemical protons are in brown. Electrons transfers are in blue. The scheme is designed with transfer of two electrons, even for complex IV (half reaction). PDB codes are: 2FUG and 3RKO for complex I (in blue); 3SFE for complex II (in red); 1PP9 for complex III (dimer in magenta and cyan); 1OCR for complex IV (in green); 2B4Z for cyt c (in grey); 2C3E for ADP/ATP carrier (in cyan) and Pi carrier (in purple); (1) for ATPase. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Abbreviations

ΔGox, (Joule) Gibbs energy difference of a redox reactions: ΔGox = −nF · ΔE, where n is the number of electron involved, ΔE (volt) = Eacceptor − Edonor and E = Eh,7 − RT/nF · ln[(Red)/(Ox)]; ΔGP, phosphorylation potential: ΔGP = ΔGp,o + RTln[(ATP)/(ADP)(Pi)], ΔGp,o is the free energy of ATP synthesis at standard conditions. Δ equation image, gradient of electrochemical potential of protons across a membrane: Δ equation image = F · ΔΨ − 2,3RT. ΔpH (J/mole); ΔΨ, membrane potential. Δp, proton-motive force: Δp = Δ equation image/F (volt); ROS, reactive oxygen species.

Figure 1 is an idealized representation of OxPhos. Indeed, several other proteins exist in the inner mitochondrial membrane, which can feed (usually to a lesser extent) the respiratory chain with electrons such as glycerol-3-phosphate (G3P) dehydrogenase, electron transfer flavoprotein dehydrogenase, dihydroorotate dehydrogenase, and choline dehydrogenase. In the yeast Saccharomyces cerevisiae and in some fungi, there are both external and internal NADH dehydrogenases that reduce the quinone pool without extruding protons from the matrix. For space restrictions, we will not approach the diversity of respiratory chains in this short review (3), but rather we will review the thermodynamics of the respiratory complexes, the link (stoichiometry) between the conservation of the cell redox state and ATP synthesis, the control of OxPhos in the light of the metabolic control theory (MCT), and finally we will present some OxPhos mathematical models.

Thermodynamics of Respiratory Chain Complexes

The enzymatic reactions catalyzed by the four complexes of the respiratory chain are as follows: Complex I: NADH + Q + 5H+in ↔ NAD+ + QH2 + 4H+out Complex II: Succinate + Q ↔ fumarate + QH2 Complex III: QH2 + 2 cyt cox + 2H+inQ + 2 cyt cred + 4H+out Complex IV: 2 cyt cred + 1/2 O2 + 4H+in ↔ 2 cyt cox + H2O + 2H+out I + III + IV: NADH + 1/2 O2 + 11H+in ↔ NAD+ + H2O + 10H+out II + III + IV: Succinate + 1/2 O2 + 6H+in ↔ fumarate + H2O + 6H+out

The ΔG′° of these reactions are listed in Table 1.

Table 1. Energies, standard midpoint potentials, and stoichiometry in mitochondrial respiratory chain complexes
complexElectron donorElectron acceptorPumping H+ matrix → intermembrane spaceChemical H+Total H+ transferredCharge transfer other than transferred H+Em,7 donor (mV)Em,7 acceptor (mV)ΔG0 (kJ/mol)
INADH/NAD+Q/QH241 H+ (matrix → compound)41 less q+ in matrix−320 (4)+60 (4)−73.3
IISuccinate/ fumarateQ/QH2    +30 (4)+60 (4)−5.8
IIIQH2/Q2 Cyt c (ox/red) 4 H+ (compound → intermembrane space) 2 H+ (matrix → compound)22 q+ and 2 q to intermembrane space+60 (4)+220 (4)−30.9
IV2 Cyt c (red/ox)½O2/H2O22 H+ (matrix → compound)22 less q+ in matrix 2 less q in intermembrane space+220 (4)+820 (4)−115.8
I + III + IVNADH/NAD+½O2/H2O64 H+ (compound → intermembrane space) 5 H+ (matrix → compound)101 less q+ in matrix−320 (4)+820 (4)−220.0
II + III + IVSuccinate/ fumarate½O2/H2O24 H+ (compound → intermembrane space) 4 H+ (matrix → compound)6 +30 (4)+820 (4)−152.4
          

Important differences in proton transfers exist between the mitochondrial respiratory chain complexes. First, the energy produced by redox reaction at complex II (ΔG′° = −5.8 kJ/mol, see Table 1) is too low to allow proton transfer across the membrane. Second, complexes I and IV have special protons pathways to transfer protons across the membrane, whereas complex III transfers protons according to the modified Q cycle mechanism (5, 6). Complex III is not a pump because the protons released on the intermembrane side are not the same protons taken up on the matrix side. Instead, chemical protons are taken from the matrix at the Qi site and transferred into the Q pool; other chemical protons are expelled into the intermembrane space at the Qo site from the Q pool, taking advantage of the position of Qo and Qi sites close to the intermembrane space and the matrix side, respectively. Thus, the stoichiometry at complex III is strictly 2H+/2e.

On the contrary, the coupling mechanism between redox reactions and proton transfers operating in complex I is the object of great debates, recently revived by the crystallographic structure of this complex (7, 8). Complex I has an L shape with the redox reactions in the hydrophilic arm of the L well separated from three proton-pumping antiporter-like subunits L, M, and N (Escherichia coli nomenclature) situated in the membrane (the membranous arm of the L) linked by a long α-helix parallel to the membrane. Subunits N, K, J, and A form the “heel” of the “boot” structure of complex I and are probably involved in the coupling mechanism and perhaps in a putative fourth H+ transfer (Fig. 1). The fact that only three subunits clearly involved in proton transfer are apparent in this structure suggests that the number of protons transferred per 2e could be three contrary to the well accepted values of 4 (discussed in ref.9). Regarding the number of protons pumped through complex I, thermodynamics imposes the constraint that:

equation image

where ΔEh = EhUQEhNADH (Eh = Em + RT/nF · ln([Ox]/[Red])) is the redox span of complex I reaction and Δp (mV) = Δ equation image/F (F = 96,500 coulomb/mole) is the proton-motive force (Δ equation image (kJ/mole) is the electrochemical proton gradient), n is the number of electrons transferred, and n′ is the number of protons transferred. The Eh values under state 3 are given in Table 2 and represented in states 3 and 4 in Fig. 2.

Figure 2.

Thermodynamics of proton transfers in complexes I and III. The Eh redox potentials of the couples of electron transporters are given according to the equation Eh = Em + RT/nF · ln([Ox]/[Red]). The Em midpoint potentials are represented with dashed line. Left part: at state 3, where the Δp is approximately −175 mV, the transfers of 4H+ by complex I and 2H+ by complex III are possible. Middle part: at state 4, if the Δp is approximately −220 mV, the transfers of 4H+ by complex I and 2H+ by complex III are still possible, but by enlarging the ΔEh span at the maximum width with highly reduced NAD/NADH couple and highly oxidized cytochrome c. Right part: a possible situation where complex I transfers only 3 H+ per e at state 4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 2. Mitochondrial redox ratio for substrates of complexes in mitochondrial respiratory chain and associated midpoint potentials and energies under state 3 (Δp = 175 mV)a
complexElectron donorElectron acceptorMitochondrial electron donor concentrations ratioMitochondrial electron acceptor concentrations ratioEh,7 donor (mV)Eh,7 acceptor (mV)ΔG′ (kJ/mol)
  • a

    Concentration ratios without reference have been estimated.

INADH/NAD+Q/QH20.14 (12% reduced) (11)0.76 (57% reduced) (10)−29456−67.5
IISuccinate/ fumarateQ/QH22 (67% reduced)0.76 (57% reduced) (10)2156−6.9
IIIQH2/Q2 Cytochrome c (ox/red)1.3 (57% reduced) (10)1.5 (40% reduced) (10)56231−33.8
IV2 Cyt. c (red/ox)½O2/H2O0.66 (40% reduced) (10)5.4 10−5 (100% reduced)231754−100.9
I + III + IVNADH/NAD+½O2/H2O0.14 (12% reduced) (11)5.4 10−5 (100% reduced)−267754−202.2
II + III + IVSuccinate/ fumarate½O2/H2O2 (67% reduced)5.4 10−5 (100% reduced)21754−141.6

In state 3, the mitochondrial membrane electrochemical potential is approximately −175 mV, which necessitates at least a redox input of 2 × 175 = 350 mV to transfer 2H+/e, that is, 4H+/2e by complex I.

The matrix NAD+/NADH ratio is known to be about 7.15 (11); hence, EhNADH = −294 mV (as compared to EmNADH = −320 mV ). Assuming complex I is operated at equilibrium, which means that EhUQ = +56 mV (as compared to a midpoint potential of +60 mV) that corresponds to an ubiquinone pool 57% reduced. Benard et al. (10) measured a reduction state of this pool of 60% in liver and kidney, in agreement with this value. Hence, the redox span of complex I is sufficient in all tissues to provide the energy needed for the transfer of 4H+ under state 3. These authors (10) also determined the percentage of reduced cytochrome c (about 40% in liver, kidney, and brain, i.e., Ehcytc = +231 mV and about 65% in muscle and heart, i.e., Ehcytc = +204 mV). Contrary to complex I, complex III has a well-established stoichiometry of 2H+/2e that necessitates a minimum redox span of 175 mV in state 3, which is satisfied in liver and kidney (231 − 56 = 175 mV). However, Benard et al. (10) determined that the ubiquinone pool is only 2% reduced, EhUQ = +110 mV, in heart, muscle, and brain, which does not satisfy the minimum redox span required, reinforcing the idea of several Q pools in these tissues. It also means that at state 3, complexes I and III operate at near-equilibrium. Moreover, things are more constrained in state 4, because in this case, Δp is approximately 220 mV. Figure 2 shows that taking the same central value of EhUQ = +56 mV, we need EhNADH = 384 mV (99% reduced) and EhCytc = +276 mV (10% reduced). This has been experimentally shown: Kim et al. (12) measured all the thermodynamical parameters in whole cells and found at state 4 in the presence of oligomycin with a Δp ≈ 200 mV associated with EhUQ = 70 mV and EhCytc = +280 mV, which gives 210 mV as redox span for complex III and places EhNADH at at least −330 mV, that is, 68% of reduction (NAD+/NADH = 0.46).

Thus, although only three antiporter-like subunits exist in complex I, in principle, the transfer of four protons is thermodynamically possible up to a Δp ≈ 200mV. The fourth pathway for protons could be situated at the interface linking the hydrophilic to the hydrophobic arms (13) (see also Fig. 1). However, for Δp > 200 mV, the transfer of 4 H+ will be rather difficult as stressed by Wikström and Hummer (9) and shown in Fig. 2.

The role of the Q/QH2 ratio as an adjusting variable must be mentioned. It allows distributing the full (NAD+/NADH – cyt cox/cyt cred) span between complexes I and III to permit an optimal transfer of protons.

Link Between Redox and Energy State: are the Stoichiometry Fixed?

We have shown in the previous section how OxPhos links two important cell fluxes, the ATP synthesis and NADH (and some other reduced cofactors) reoxidation. Indeed, keeping both a high phosphate potential and an intracellular redox balance is crucial for all living organisms and a necessity for sustained metabolic function. However, the specific turnovers of either NADH or ATP are not necessarily always the same. They are largely dependent on a number of variables such as cell types, metabolic activities, and phases of growth. Thus, it is necessary that the coupling between both fluxes is adaptable to different physiological situations. More precisely, it means that mechanisms must exist to vary the ratio ATP/O (ATP synthesis over oxygen consumption and thus NADH reoxidation) in response to different demands in energy and/or in reducing power. We will describe below the five mechanisms that are known to affect the ATP/O ratio and thus allow a modulation between ATP synthesis and NADH reoxydation.

The first mechanism decreasing the coupling efficiency, the passive proton leak, is a direct consequence of the nature of the energetic intermediary, the proton-motive force. Indeed, biological membranes present some proton conductance (LH), and the resulting proton (in)flux is strictly dependent on the proton-motive force (JH = LH · Δp). The membrane conductance is a specific property of the membrane itself, but is not entirely independent from the proton-motive force: at high value of this force, the proton membrane conductance increases. It means that LH is a function of Δp itself, LH = fp). This determines a non-ohmic relationship between passive proton flux (proton leak) and proton-motive force that has been observed in mitochondria of various origins (14, 15) and lead to a maximal value of Δp in which the leak is equal to the flux of proton transduction by the respiratory chain (state 4). Obviously, the size of this passive proton leak may modulate the yield of OxPhos (ATP/O). More precisely, it will decrease the rate of ATP synthesis for a given oxygen consumption because the proton flux giving rise to leak cannot be used to synthesize ATP. Passive proton leak, for a given type of mitochondria, is only dependent on the proton-motive force and consequently cannot be widely modulated.

Another mitochondrial process allows modulation of the membrane proton permeability: the proton transport catalyzed by the uncoupling proteins (UCP). The expression of these proteins is tissue specific. In the case of UCP1, which is mainly expressed in brown adipose tissue, the H+ transport function is well-established and highly regulated by guanosine triphosphate (GTP) (inhibition) and fatty acids (stimulation) (16). The participation of the other UCP's in proton transport is controversial (17). However, it has been proposed that even if they participate only marginally to H+ permeability, their effect may be crucial in decreasing reactive oxygen species (ROS) production simply by a slight decrease in a high proton-motive force (18).

Another mechanism, which can be called facilitated H+ transport (called “active proton leak” in ref.19) has been observed in yeast, where a decrease in the proton-motive force occurs because of the activity of external NADH or G3P dehydrogenases. It should be stressed here that in the yeast Saccharomyces cerevisiae, there is no complex I and the NADH and other dehydrogenases are not coupled to proton extrusion. This observed increase in proton permeability associated with a high activity of these dehydrogenases is independent of the rest of the respiratory chain and the ATP synthase proton pump. This mechanism could permit a decrease in redox pressure with little effect on ATP synthesis.

In the examples described above, the uncoupling of ATP synthesis (decrease in VATP) from the respiration (increase in VO2) is mediated by a decrease in proton-motive force (decrease in Δ equation image). There are experimental data showing that VATP can be decreased and VO2 increased without changing the Δ equation image (20). This led Azzone's group (21) to propose another mechanism causing a loss of OxPhos yield called slip or intrinsic uncoupling. It is a decrease in the efficiency of a proton pump because of a partial and variable decoupling of the chemical reaction from the proton transport, that is, a decrease in the H+/2e stoichiometry of a respiratory chain complex or an increase in the H+/ATP stoichiometry of the ATP synthase. A kinetic model for proton pump functioning has been proposed by Pietrobon and Caplan (22). This model is only possible with H+ pumps (complexes I and IV) and not with Mitchell loops (complex III, which has a fixed stoichiometry). In the case of a reversible pump, the stoichiometry (H+/ATP or H+/2e) varies oppositely when the forces inducing the slip are increased.

This was not observed in the case of Almitrine effect on ATP synthase for which an increase in H+/ATP is observed both in the ATP synthesis and in the ATP hydrolysis. It is concluded in this case that almitrine induces an actual change in the mechanistic stoichiometry of the ATPase/ATPsynthase activity (23).

The respiratory chains of bacteria and plants possess alternative oxidases in which the electron transfer is not linked to a H+ extrusion and which allows an uncoupling of ATP synthesis from reoxidation mechanisms that could decrease the ROS production (24).

Control of Oxphos

In the previous section, we analyzed the structure of the OxPhos that both reoxidizes NADH and FADH2 and generates ATP. The problem is now: what triggers the changes in the rate of OxPhos? What is the target when ATP or a readjustment of NADH/NAD ratio is necessary? The control of OxPhos was the object of a great deal of discussion with the idea in mind that a unique “rate-limiting” step should exist. However, opinions differed regarding the limiting step: cytochrome oxidase, ATP synthase, and ATP/ADP carrier, which were presented as good candidates (25), most of the time because their inhibition decreased mitochondrial respiration. This long standing riddle was solved by the Tager and coworkers' group in Amsterdam (25) and the Kunz's group in Magdeburg (26) showing that the control was not restricted to a unique step as it was supposed but was shared between many steps according to the MCT (27–30) with control coefficients all <1 and summing up to 1. The control coefficient of a flux by a reaction (with rate v) of a metabolic network is defined as the ratio δF/δv, where δF is the variation in the flux δF (of the global mitochondrial respiration for instance) because of δv, a small variation in the reaction (the cytochrome oxidase for instance): CFv = δF/δv or δF = CFv · δv

Figure 3 shows a theoretical curve of what is usually obtained with a slight decrease at the beginning according to the control coefficient (the initial slope) followed by an abrupt decrease afterward to reach zero flux when the step is completely blocked. The dashed straight line in Fig. 3 represents the case of the limiting step completely controlling the flux. The fact that the response of a global flux is not equal to the change in one (one at the time) step in the network is because of re-equilibration of the metabolites. The inhibition of a reaction δv leads to the accumulation of the substrates of the reaction that push the inhibited reaction and slows down the reactions uphill leading to a new steady-state close to the first one. It was also showed that the summation theorem (the sum of all control coefficients in a pathway sum up to 1) was also satisfied experimentally in several cases (25, 31, 32).

Figure 3.

Threshold curves. Decrease in the flux in a network at steady-state δF as a function of the inhibition of a step δv. The control coefficient is defined as CFv = δF/δv. A threshold is usually associated to low control coefficients (<0.2) with a slow linear decrease in the flux even at high inhibition of the step. At higher control coefficients (>0.2), the decrease is more gradual. A control coefficient of one means that any change in the activity of a given reaction is repercuted on the flux (to the same extend). It mainly occurs when the quantity of enzyme is low, that is, the activity is low compared with the activities of the other enzymes which appear as in excess.

Another prediction of the theory was also rapidly confirmed in the control of OxPhos: the control can vary according to different steady states and kind of mitochondria (25, 33). In ref.33, the analysis of control coefficient distribution of OxPhos in five different rat tissues under identical experimental conditions shows two tissue groups: a) the muscle and the heart, essentially controlled at the level of the respiratory chain; and b) the liver, the kidney, and the brain, controlled mainly at the phosphorylation level by ATP synthase and the phosphate carrier. As concluded by the authors, this variation in control coefficient according to the tissue origin of the mitochondria can explain part of the tissue specificity observed in mitochondrial pathologies.

Metabolic control analysis was of great help to understand the expression of mutations in mitochondrial diseases. Furthermore, the systematic study of threshold curves (34) (Fig. 3) showed that the threshold itself is dependent on the tissue and probably also on physiological conditions. The authors evidence two kinds of behavior according to two kinds of threshold curves: either a continuous slight decrease in the respiration as a function of the defect intensity associated with an appreciable control coefficient (>0.2 – 0.3; CVF = 0.4 in Fig. 3) or no effect until a high deficiency associated with a low control coefficient (<0.2) and a high threshold. Thus, the same mutation with the same residual activity in a respiratory complex may affect to a very different extent the mitochondrial ATP synthesis in different tissues as is actually observed in mitochondrial diseases. These studies based on MCT were very useful to understand the clinical presentation of mitochondrial diseases.

MCT is also useful when investigating channeling (35). The fact that the sum of control coefficient of the complexes of the respiratory chain summed up to a value >1 was taken as an evidence of a possible association of respiratory complexes in super complexes (36) and (37) in certain conditions, which was experimentally demonstrated later (38).

Models of Oxphos

There is a long tradition of modeling OxPhos to integrate all aspects, kinetic and thermodynamic, of chemiosmotic theory (2). The first approach was in the framework of nonequilibrium thermodynamic model involving a linear approximation of the coupled fluxes on the thermodynamic forces, ΔGox of the redox reactions, ΔGP, the phosphate potential, and Δp, the proton-motive force (3942). In this framework, all the states of OxPhos (state 4, state 3, uncoupled state, etc.) were described as phenomenological relations in terms of energy conversion with the use of phenomenological coefficients (Lij):

equation image
equation image

with L12 = L21, Onsager relationships, and equation image, the degree of coupling, and equation image, the phenomenological stoichiometry.

The optimal efficiency of the system and the degree of coupling in these conditions were defined and analyzed. Even though one can argue that OxPhos could be out of the linear domain around equilibrium, this description is simple and indicates the fundamental parameters involved: degree of coupling, thermodynamic forces, rates, optimal efficiency, phenomenological stoichiometry, and so on. Furthermore, as evidenced in ref.43, the linear domain might be extended away from equilibrium because of kinetics regulations,

Similar models were derived by Pietrobon and coworkers (22, 44) to describe redox-driven proton pumps and ATP synthesis in mitochondria. These models are kinetic models but with the calculation of thermodynamic parameters, evidencing the relationships between kinetics and thermodynamics.

Bohnensack was probably the first to derive a quantitative model involving nearly all the components of OxPhos. To do so, he used approximate rate laws of near equilibrium reaction detailed in ref.45 (first appendix in ref.45). With the help of this model, the Magdeburg group (26, 46) was able to demonstrate that the control of OxPhos was shared by several steps as predicted by MCT (see above).

Korzeniewski and Froncisz (47) applied, to complexes I and III of the respiratory chain, the principles of linear dependency on the thermodynamic force, that is, ΔGoxn′ Δ equation image, where n′ is the number of protons extruded by the complex. Different versions of the model, which include other type of rate equations, were applied to isolated mitochondria or to intact tissues (muscle, heart, and liver). The model was used to calculate the control coefficient in OxPhos (48), to fit threshold curves in muscle and to predict the shape of threshold curves at low oxygen pressure (49), to study the transition from rest to intensive work in muscle (50), leading to the concept of parallel activation. We used this model (51) to compare the threshold curves obtained with mitochondrial or nuclear DNA mutations.

One of the first models taking into account the organization and compartmentalization of oxidative-phosphorylation inside the cell was that of Aliev and Saks (52) describing heart bioenergetics and creatine/creatine phosphate shuttle. This model was refined by Vendelin et al. (53, 54).

More recently, Beard (55) proposed “A biophysical model of the mitochondrial respiratory system and oxidative phosphorylation” mainly applied to cardiac mitochondria. The rate equations of the complex are based on mass-action law with the introduction of the Δ equation image. More recently, the model was extended to a larger model encompassing the mitochondrial energy metabolism (56). Several models are now available, which try to represent cell energy metabolism. The model of Cortassa et al. (57) with one equation for the whole respiratory chain and the one of Holzhütter and coworkers (58) with a very detailed modeling of the respiratory complexes must be cited among others.

In the area of Systems Biology and high throughput methods in biology, the Oxphos models increase in size, now incorporating tricarboxylic acid (TCA) cycle, β-oxidation of fatty acids, and several other metabolic pathways. Because in these large models not all the kinetic parameters are known (and are different in different tissues and organisms), a detailed approach based on ordinary differential equations is difficult (59). To circumvent this problem, a new theoretical approach was developed by the Pallson's laboratory called Flux Balance Analysis and applied to mitochondrial metabolism (60–62). It uses linear programming, the stoichiometric coefficients for each reaction and biological constraints to optimize a function (objective function) supposed to represent the biological phenotype (maximizing ATP production, biomass, growth rate, etc.). This interesting method does not require the knowledge of the kinetic parameters but relies on optimized functions, which can reflect a simplified picture of the biological reality.

Conclusion

In nonphotosynthetic organisms, mitochondria are the power plant of the cell, emphasizing their great potentiality for ATP synthesis from the redox span between nutriments and oxygen. More important is probably their role in the maintenance of cell redox balance as it is well demonstrated in mitochondrial diseases where this function is also affected. These two functions are linked at several levels: in the respiratory complexes and in the global functioning of OxPhos through the ã Δ equation imageH+ (or the Δp). However, the link/links between these two important functions has/have to be modulated because of the differences in cell demand in energy and redox state maintenance.

If the coupling between respiratory chain and ATP synthesis is well understood thanks to Mitchell's theory (2), the molecular intimate mechanism through which the redox energy is used for H+ transport by the respiratory complexes and ATP synthase is still a matter of debates and investigation. The fact that the crystallographic structures are now known does not solve these problems but shifts it to a more molecular and precise level and raises new issues. This is particularly the case with the coupling in complex I with the quinone site out of the membrane. It is also the case when one compares the stoichiometry of ATPase with the number of c subunits in the Fo.

These old fundamental questions are not only a matter of basic research. Mitochondria appear yet as a great player in cell life with newly evidenced role such as in apoptosis, autophagy, calcium cell signaling, ROS production, aging, cancer, and so on. All these functions are dependent on a correct metabolic function that has to be comprehensively understood. In this matter, use of models is of great help to take into account the highly structured integration of these systems.

Acknowledgements

The authors acknowledge Dr Roger Springett for constructive comments and editing the manuscript and Dr Alain Dautant for help in Fig. 1.

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