ATP yield of plant respiration: potential, actual and unknown

Abstract

Background and Aims

The ATP yield of plant respiration (ATP/hexose unit respired) quantitatively links active heterotrophic processes with substrate consumption. Despite its importance, plant respiratory ATP yield is uncertain. The aim here was to integrate current knowledge of cellular mechanisms with inferences required to fill knowledge gaps to generate a contemporary estimate of respiratory ATP yield and identify important unknowns.

Method

A numerical balance sheet model combining respiratory carbon metabolism and electron transport pathways with uses of the resulting transmembrane electrochemical proton gradient was created and parameterized for healthy, non-photosynthesizing plant cells catabolizing sucrose or starch to produce cytosolic ATP.

Key Results

Mechanistically, the number of c subunits in the mitochondrial ATP synthase F_o sector c-ring, which is unquantified in plants, affects ATP yield. A value of 10 was (justifiably) used in the model, in which case respiration of sucrose potentially yields about 27.5 ATP/hexose (0.5 ATP/hexose more from starch). Actual ATP yield often will be smaller than its potential due to bypasses of energy-conserving reactions in the respiratory chain, even in unstressed plants. Notably, all else being optimal, if 25 % of respiratory O₂ uptake is via the alternative oxidase – a typically observed fraction – ATP yield falls 15 % below its potential.

Conclusions

Plant respiratory ATP yield is smaller than often assumed (certainly less than older textbook values of 36–38 ATP/hexose) leading to underestimation of active-process substrate requirements. This hinders understanding of ecological/evolutionary trade-offs between competing active processes and assessments of crop growth gains possible through bioengineering of processes that consume ATP. Determining the plant mitochondrial ATP synthase c-ring size, the degree of any minimally required (useful) bypasses of energy-conserving reactions in the respiratory chain, and the magnitude of any ‘leaks’ in the inner mitochondrial membrane are key research needs.

INTRODUCTION

Plant respiration produces ATP, reducing agents and carbon-skeleton intermediates needed for active transport, nutrient assimilation, synthesis of new monomers and macromolecules (especially during growth), re-synthesis of labile compounds, and metabolic repair and acclimation processes. In some circumstances it produces substantial beneficial heat. Two central questions regarding respiratory energetics, and many biological processes that depend on it, are: ‘How much ATP can respiration produce?’ and ‘How much ATP does respiration produce?’ Various answers to these questions, the first one in particular, have been suggested over several decades. By the 1970s, typical biochemistry textbook estimates of potential (maximal) ATP yields of aerobic respiration gave 36 (or 38) ATP/glucose respired as the sum of net substrate-level and oxidative phosphorylations of ADP [e.g. Lehninger, 1970 (pp. 385–386), 1975 (pp. 517, 535); Stryer, 1975 (p. 344)]. Nonetheless, those ATP yields were under downward revision before 1980 as knowledge of the mechanisms of oxidative phosphorylation increased and as the quality of experimental data improved (Hinkle and Yu, 1979). During the ensuing decades some estimates of potential ATP yield of respiration declined to 29.5 (or 31) ATP/glucose (Hinkle et al., 1991), 32 ATP/glucose (Taiz and Zeiger, 1991) and 30 (or 32) ATP/glucose (Stryer, 1995). More recent calculations further refined values of potential ATP yield by considering stoichiometric consequences of newly obtained information about the molecular structure of F-type ATP synthases responsible for oxidative phosphorylations of ADP (Stock et al., 1999). Conversely, knowledge of the stoichiometry of substrate-level ADP phosphorylations with respect to respiratory carbon metabolism reactions/pathways has not changed in over 50 years.

Despite decades of awareness that aerobic respiration of a hexose cannot produce 36 (or 38) ATP, those values still are cited. Moreover, there is an historical tendency to apply a single value of potential respiratory ATP yield across all organisms whereas the potential ATP/hexose ratio is unlikely to be a universal biological constant. This is because of differences in (possible) pathways of respiratory electron transport, the presence or absence of various NADH shuttles between cytosol and the mitochondrial matrix, and the well-established variation in the molecular structures of the F-type ATP synthases in different organisms. And notwithstanding great progress over the decades, current assessments of ATP yield of plant respiration still involve important uncertainties.

Numeration of potential and actual ATP yield of plant respiration, even if imperfect, is critical to a wide range of issues because the ATP yield is a quantitative link between rates of many active processes and concurrent rates of respiratory substrate consumption. When the ATP cost of a process (or competing processes) is known or approximated, the time integral of the rate of that process(es) can be equated with an amount of substrate consumption. That then defines the substrate supply – ultimately the amount of photosynthesis – needed to support that process(es) (Penning de Vries et al., 1974; Penning de Vries, 1975; de Visser et al., 1992; Dewar, 2000; Noguchi et al., 2001). With respect to heterotrophic ATP-consuming processes that might be added, deleted, slowed or quickened through bioengineering of crop plants, knowledge of the ATP yield of respiration would allow an assessment of the effects of that bioengineering on the amount of additional photosynthate that would be required, or the amount that could be spared and then used for other purposes including additional growth. This may be crucial to evaluating the cost–benefit ratios of opportunities for directed modification of crop-plant metabolism (Amthor et al., 2019; Reynolds et al., 2021;Garcia et al., 2023) because, for example, an assumption that respiration can produce 36 ATP/hexose when the potential might be only 27 ATP/hexose would lead to underestimation of respiratory substrate requirements of an ATP-requiring process by fully one-quarter.

To gauge current knowledge related to both of the questions above, and to quantify the importance of several unknowns, a balance sheet model was constructed from pathways and stoichiometries of carbon, electron and proton flows in plant respiration that underlie ATP production. When data required to assign values to model parameters or variables were unavailable for plants, values from other organisms were used when available. Assumptions, including simplifying ones, were made at various points to keep the analysis tractable while focusing on the most important variables governing the amounts of ATP that plant respiration can and does produce. The possible significance of some unknowns was assessed through model tests of the sensitivity of ATP yield to variation in variable values. This plant model was compared to the standard model for mammals, for which there is more complete knowledge, and for which there are differences from the plant model.

MODEL PHYSIOLOGICAL SETTING AND STRUCTURE

Notable features of plant respiration include (1) existence of alternative, parallel pathways of carbon and electron flows, (2) possible withdrawal of intermediates from many points in the catabolic respiratory pathways as inputs to concurrent anabolism and (3) potential entry of intermediates into the respiratory network at many points. The flexibility in respiratory pathways and multifaceted coupling between respiration and anabolism allows numerous substrates to be matched with myriad products (Plaxton and Podestá, 2006; Millar et al., 2011; O’Leary et al., 2019). The product of interest in this model analysis is ATP produced by glycolysis in the cytosol plus ATP produced by the tricarboxylic acid cycle (TCA) cycle and the oxidative phosphorylation system in the matrix of mitochondria that is then exported to the cytosol. The designated carbon substrates are starch and sucrose because of their widespread use by plant respiration and status as main storage and transport forms of photosynthate (Stitt and Steup, 1985; Plaxton and Podestá, 2006). In contrast, mammalian-centric, fungal-centric and bacterial-centric models of respiratory ATP yield usually take glucose as the substrate, which should be recognized when comparing a plant model to a non-plant model.

The model’s physiological setting is a plant cell that is (1) not photosynthesizing (including photosynthetically capable cells in the dark), to avoid numerous complications associated with the complex interactions and dependencies between photosynthesis, photorespiration and respiration in illuminated photosynthetic cells (e.g. Sweetlove et al., 2006); (2) not experiencing stress or metabolic limitation or imbalance that might disrupt the normal aerobic pathways of healthy-plant respiration (e.g. Che-Othman et al., 2020); (3) mature, so that it is not diverting a significant fraction of respiratory intermediates into biosynthetic pathways associated with growth, but rather oxidizing most (all) intermediates to produce ATP; and (4) not in a thermogenic mode of metabolism, when the ATP yield of respiration is, and must be, small. This model setting in mature, healthy, non-photosynthesizing and non-thermogenic cells focuses attention on the relationship between respiratory substrate oxidation and ATP production for the support of active processes, and although all plant respiratory substrate-level ATP-producing reactions can be bypassed to the plant’s benefit in specific circumstances (O’Leary et al., 2019), those circumstances are outside the model scope.

The model’s aim is to quantitatively summarize the central respiratory carbon metabolism pathways of glycolysis and the TCA cycle, the plant mitochondrial electron transport (i.e. respiratory) chain that oxidizes both NADH and succinate produced in those carbon metabolism pathways, the transmembrane electrochemical proton gradient generated by the respiratory chain (reviewed by Nicholls and Ferguson, 2013), and the use of that electrochemical proton gradient to drive oxidative phosphorylation of ADP. (The word ‘chain’ in ‘electron transport chain’ can imply a fixed linear sequence of electron transfer reactions whereas that system, particularly in plants, is better described as a pathway or network.) The model closes all metabolic cycles and balances all outputs with inputs so that the only net inputs are a carbohydrate substrate (starch and/or sucrose in the present case), ADP, P_i and O₂ while the only net outputs are ATP in the cytosol, CO₂, water and heat (the amount of heat is not quantified in the model). The ratio of model ADP and P_i input to ATP output is strictly 1:1:1 and all carbon in the carbohydrate substrate is released as CO₂. Further, all NAD⁺ reduced in the respiratory carbon metabolism pathways is regenerated through an equal amount of NADH oxidation by the respiratory chain; all ubiquinone (Q) reduced to ubiquinol (QH₂) in the respiratory chain is regenerated by an equal amount of QH₂ oxidation; and the number of protons pumped out of the mitochondrial matrix by the respiratory chain is matched by an equal number entering the matrix by any of several mechanisms, including movement through the membrane-embedded F_o sector of the mitochondrial H⁺-transporting F_oF₁-ATP synthase (EC 7.1.2.2; accepted name: H⁺-transporting two-sector ATPase) catalysing oxidative ADP phosphorylation. Activity of the F_o (but not F₁) sector of the mitochondrial ATP synthase is inhibited by oligomycin; hence it is letter ‘o’ rather than numeral ‘0’ in the sector name. (Conversely, the chloroplast two-sector ATP synthase is insensitive to oligomycin and may be referred to as F₀F₁-ATP synthase, i.e. numeral ‘0’ rather than letter ‘o’.)

Mitochondrial membranes and respiratory transmembrane electrochemical proton gradients

Mitochondria, which are central to plant respiratory ATP production, are surrounded by two membranes. The volume between the two membranes is called the intermembrane space, which includes cristae spaces (or cristae lumens sensuWolf et al., 2019; Heldt and Piechulla, 2021; Meyer et al., 2022; Klusch et al., 2023). The outer membrane is ignored in the model. It is considered freely permeable for all relevant model molecules – ADP, ATP, NADH, P_i, pyruvate and malate – via mitochondrial porin channels, which may account for more than one-third of the outer membrane’s surface area (Fuchs et al., 2020).

In contrast, the inner mitochondrial membrane, which bounds the mitochondrial matrix, is a key defining element of the model. It is a physical barrier between glycolysis in the cytosol and the TCA cycle in the mitochondrial matrix, and the model must account for any cost of metabolite transport through the inner membrane. It also is the structure supporting the respiratory chain enzymes and related proteins – about 30 % of the inner membrane’s surface area may be protein (Fuchs et al., 2020) – and is the membrane across which the electrochemical proton gradient exists (Fig. 1). That transmembrane gradient is composed of a proton concentration gradient, ΔpH, and an electrical potential difference, ΔΨ_m, between the aqueous phases separated by the membrane. In respiring mitochondria ΔΨ_m usually dominates the total electrochemical proton gradient and may typically be −150 to −200 mV (electrical potential on the matrix-side of the membrane minus electrical potential on the intermembrane space–side). The proton concentration gradient may be rather small, perhaps corresponding to a ΔpH of about 0.5 units in mammals (Nicholls and Ferguson, 2013) while in Arabidopsis protoplasts a cytoplasmic pH of 7.3 (±0.1) and mitochondrial matrix pH of 8.1 (±0.2) have been reported (Shen et al., 2013), indicating a ΔpH of about 0.8 units between cytosol and mitochondrial matrix. In addition, the plant mitochondrial intermembrane space, and in particular the cristae spaces into which the respiratory chain pumps protons, may have a lower pH than bulk cytosol, as reported for human cells (Porcelli et al., 2005), so the ΔpH across the plant inner mitochondrial membrane itself may exceed 0.8 units. In contrast, the ΔpH across an illuminated-chloroplast thylakoid membrane can be much larger, perhaps about 3 units, and accounts for the majority of the total electrochemical proton gradient there. That large transmembrane ΔpH of chloroplast thylakoids is made possible by transport of negative counterions (anions) along with protons pumped through the thylakoid membrane; a large ΔpH requires proton charge compensation by anions such as chloride, whereas the inner mitochondrial membrane is impermeable for anions (Heldt and Piechulla, 2021).

Fig. 1.

Key modelled plant respiratory components associated with the inner mitochondrial membrane (IMM) (not to scale or with details of structural geometries). N indicates the negative side of the membrane (i.e. the mitochondrial matrix) from which protons are extracted by the respiratory chain. P indicates the positive side of the membrane (i.e. crista space/intermembrane space) to which protons are released. Entities are roughly arranged from top to bottom in the order they are considered in the model. Pyruvate (C₃H₃O₃⁻), the default modelled product of glycolysis, enters the mitochondrial matrix in proton symport via the mitochondrial pyruvate carrier (MPC). As modelled, pyruvate is completely oxidized in the matrix, producing CO₂ that exits the mitochondrion as dissolved CO₂ (shown here), H₂CO₃ and/or HCO₃⁻ + H⁺. The membrane segment from complex I to complex IV represents the mitochondrial electron transport (respiratory) chain. Blue text and arrows identify the respiratory chain substrates (NADH and succinate) and pathways of electron pair (2e⁻) transfer from those substrates to ubiquinone (Q). Reduction of Q involves an overall 2e⁻ + 2 H⁺ two-step process yielding ubiquinol (QH₂) (i.e. $\mathrm{Q}+{\mathrm{e}}^{-}\to \mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e};\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}+{\mathrm{e}}^{-}+2{\mathrm{H}}^{+}\to \mathrm{Q}{\mathrm{H}}_2$). Red text and arrows identify pathways of electron and proton movement away from QH₂ to oxygen, forming water, or for some protons to their release on the P side of the IMM. Both the lipophilic Q and QH₂ are localized inside the membrane with their combined molar amount exceeding that of other respiratory chain components. Complex I oxidizes mitochondrial matrix (N side) NADH, transferring one 2e⁻ and two protons to Q and pumping four protons from the N to the P side of the membrane per NADH oxidized. Non-proton-pumping complex II (C II), a component of the TCA cycle, transfers one 2e⁻ and two protons to Q during succinate oxidation (complex II spans the membrane though not drawn that way here). In addition to complex I, a non-proton-pumping alternative NADH dehydrogenase(s) with NADH binding site on the N side of the IMM (ND_in, internal) also can oxidize mitochondrial NADH, transferring one 2e⁻ and two protons to Q. A different non-proton-pumping NADH dehydrogenase(s), this one(s) with NADH binding site on the P side of the IMM (ND_ex, external), can oxidize cytosolic (crista space) NADH, transferring one 2e⁻ and two protons to Q. The blue dashed-line box indicates a 2e⁻ + 2 H⁺ transfer from one of complex I, complex II, ND_in or ND_ex to Q at a time, not mixing of electrons or protons from different substrate molecules. Complex III removes electrons and protons from QH₂, regenerating Q and releasing two protons transferred from QH₂ to the P side of the membrane (red 2 H⁺). A Q-cycle occurs within complex III (not shown; see e.g. Nicholls and Ferguson, 2013) that extracts two (black) protons from the N side and releases them on the P side of the membrane. Thus four, rather than two, protons are released on the P side of the membrane by complex III. Electrons are transferred, one at a time, from complex III to hydrophilic cytochrome c (cyt c) molecules on the P side of the membrane, with complex IV in turn transferring those electrons to oxygen, forming water. For each 2e⁻ transferred from cyt c to oxygen, complex IV extracts four protons from the N side of the membrane and deposits two of those protons on the P side with the other two contained in the water formed. To illustrate respiratory chain activity on a per 2e⁻ basis this diagram combines one oxygen atom (written ½O₂) with one 2e⁻ and two protons, forming one water molecule, rather than portraying the actual O₂/2 H₂O redox cycle. Transfer of electrons from QH₂ to oxygen through complex III, cyt c and complex IV is called the ‘cytochrome pathway’, which, as drawn, releases six protons at the P side of the membrane. In addition to the cytochrome pathway, the non-proton-pumping alternative oxidase (AOX) can regenerate Q by transferring four electrons (two 2e⁻) and four protons from two QH₂ molecules to an O₂ molecule producing two water molecules. (As for water formation in complex IV, AOX is shown forming one water from a single oxygen atom, one 2e⁻ and two protons rather than portraying the actual O₂/2 H₂O redox cycle.) The red dashed-line box indicates a 2e⁻ + 2 H⁺ transfer from QH₂ to either complex III or AOX, not to both simultaneously. As drawn, the overall transfer of one 2e⁻ from N-side NADH to oxygen via the serial combination of complexes I, III and IV has the potential to deposit 10 protons on the P side of the membrane. In modelled steady-state respiration, three O₂ molecules that are reduced to form water enter the matrix per pyruvate catabolized. The two-sector mitochondrial F_oF₁-ATP synthase, phosphate carrier (PiC) and adenine nucleotide translocator (ANT) jointly import protons, P_i and ADP into the matrix, phosphorylate the imported ADP with the imported P_i to produce ATP and H₂O in the matrix, and export the ATP for use outside the mitochondrion. The set of transmembrane translocations facilitated by F_o, PiC and ANT shown here are associated with oxidative phosphorylation of ADP in the mitochondrial matrix; the set of transmembrane translocations associated with substrate-level ADP phosphorylation in the mitochondrial matrix is the same for PiC and ANT but not F_o (illustrated later). The modelled F_oF₁-ATP synthase transports n/3 protons from the P side to the N side of the membrane for each ADP phosphorylated where n is the number of c subunits in the F_o sector c-ring. The critical number n remains an unknown at the time of writing for the plant mitochondrial F_o sector (see text). For the typical case of matrix (N side) pH > 7.5, the ATP synthesis reaction may be $\mathrm{A}\mathrm{D}{\mathrm{P}}^{3-}+{\mathrm{P}}_{\mathrm{i}}^{2-}+{\mathrm{H}}^{+}\to \mathrm{A}\mathrm{T}{\mathrm{P}}^{4-}+{\mathrm{H}}_2\mathrm{O}$, but the associated ${\mathrm{P}}_{\mathrm{i}}^{-}\to {\mathrm{P}}_{\mathrm{i}}^{2-}+{\mathrm{H}}^{+}$ dissociation of imported P_i⁻ is omitted here for simplicity. An uncoupling protein(s) (UCP) can transport protons in a (semi-)regulated manner from the P side to the N side of the IMM under some circumstances. Passive proton ‘leak’ associated with an intrinsic proton conductivity of the IMM also might allow movement of protons from the P to the N sides of the membrane. Text with curly brackets, }, on the right identifies use (dissipation) by transport proteins (or passive membrane leaks) of the ΔpH and/or ΔΨ_m produced by the respiratory chain (see text).

The modelled transmembrane proton concentration gradient ΔpH is defined by two processes: (1) proton gradient generation via respiratory chain–driven outward proton pumping from the mitochondrial matrix to the intermembrane space and (2) proton gradient dissipation via inward proton flux from the intermembrane space to the mitochondrial matrix. Proton flux outward from the mitochondrial matrix is actively driven by three of the respiratory chain components that transfer electrons along pathways from the modelled metabolic intermediates NADH and succinate to a final acceptor, which is O₂ in the model. The inward flux, down the proton gradient from the intermembrane space to the mitochondrial matrix, can be a combination of passive and facilitated processes, including, as already noted, proton transport through the F_o sector of the ATP synthase.

The model represents steady-state respiration with a balance between outward and inward proton translocations across the inner mitochondrial membrane. The result is persistence of a stable proton gradient from the intermembrane space side of the inner mitochondrial membrane – the ‘P’ or positive side with higher [H⁺] – to the matrix side of the inner membrane – the ‘N’ or negative side with lower [H⁺]. Instead of being widely dispersed in an aqueous volume, the accumulation, lateral movement and depletion of protons in the intermembrane space, and cristae spaces in particular, may (partially) occur on the crista-side-surface of the inner mitochondrial membrane over short distances between densely packed enzyme complexes (Sjöholm et al., 2017; Toth et al., 2020) with different crista spaces within a mitochondrion perhaps acting like ‘independent bioenergetic units’ (Wolf et al., 2019).

ATP yield of plant respiration

The net number of respiratory ATP produced in the cytosol, plus ATP produced in a mitochondrion and transported to the cytosol, during complete respiratory oxidation of a unit of substrate (i.e. ATP/hexose unit respired, or simply ATP/hexose) is totalled in the model as:

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{A}\mathrm{T}\mathrm{P}}=N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{s}\right)}+N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}+N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}}\end{array}}$$ (1)

where N_ATP is net ATP produced (= net ADP phosphorylated) in the cytosol plus ATP produced in the mitochondrial matrix that is exported to the cytosol; N_{ATP(glycolysis)} is net ATP produced in cytosolic glycolysis, which is all in substrate-level ADP phosphorylation reactions; N_ATP(TCA) is ATP produced in substrate-level ADP phosphorylations in the mitochondrial matrix during TCA cycle operation, which is all then exported to the cytosol; and N_ATP(FoF1) is ATP produced in the mitochondrial matrix by the F_oF₁-ATP synthase, which is all then exported to the cytosol. The hexose substrate unit is one glucose residue in a starch molecule or either of the glucose or fructose residues in a sucrose molecule. Net ATP production represented in N_{ATP(glycolysis)} and ATP production represented in N_ATP(TCA) are stoichiometrically linked to respiratory carbon metabolism reactions that mostly have been well charted for decades and thus are known with high confidence. This contrasts with quantitatively important current uncertainties in the calculation of N_ATP(FoF1) in plants. These uncertainties are related to alternative pathways of electron flow in the plant respiratory chain and details of proton re-entry into the plant mitochondrial matrix (proton gradient dissipation) and coupling of that proton re-entry to oxidative phosphorylation of ADP. Despite uncertainties in their exact quantities, it is known that oxidative phosphorylations of ADP account for a majority of potential ATP production in the model setting; i.e. N_ATP(FoF1) > [N_{ATP(glycolysis)} + N_ATP(TCA)]. Dissipation of a part of the transmembrane ΔpH that is required for transport of ATP from the mitochondrial matrix to the cytosol (via the intermembrane space) is accounted for as a limitation on the value of N_ATP(FoF1) (see below).

In summary, four metabolic components of plant respiration are quantified in the model. (1) Starch and/or sucrose breakdown reactions followed by glycolysis and production of cytosolic pyruvate or malate, ATP and NADH. These pathways are responsible for N_{ATP(glycolysis)}. (2) Mitochondrial oxidation of imported pyruvate (or malate) coupled to the TCA cycle. These reactions convert all the substrate carbon to CO₂ and produce the ATP represented by N_ATP(TCA) plus NADH. (3) Oxidation of the TCA cycle’s succinate and all respiratory NADH and generation of the proton gradient across the inner mitochondrial membrane by the respiratory chain. (4) Dissipation of the transmembrane proton gradient, including proton transport through the F_o sector of the ATP synthase, which is stoichiometrically related to N_ATP(FoF1) in the model. These four components are distinguished in the model to organize the results; they are intimately linked in situ in various ways. Model parameters and variables are listed along with their default (i.e. model baseline) values in Table 1. The model was implemented with an R programming language script [Supplementary Data].

Table 1.

Model parameters and variables and their values for plant respiration of starch or sucrose producing ATP in cytosolic glycolysis, the mitochondrial TCA cycle and the mitochondrial oxidative phosphorylation system with the ATP produced in the mitochondrial matrix exported to the cytosol

Symbol	Default value {range allowed in model}	Definition and notes
Fixed parameters
H+Glu	1	Number of protons translocated into the mitochondrial matrix per NADH reducing equivalent shuttled into the matrix in the malate–aspartate shuttle. Not used in plants but required for model simulation of mammalian ATP production.
H+I	4	Number of protons pumped from the N side to the P side of the inner mitochondrial membrane per mitochondrial NADH oxidized by complex I in the absence of slippage (H+/NADH or H+/2e−).
H+III-IV	6	Number of protons pumped from the N side to the P side of the inner mitochondrial membrane by combined activity of complexes III and IV per electron pair (2e−) transferred from QH2 to molecular oxygen (H+/2e− or H+/QH2).
H+Pi	1	Number of protons translocated across the inner mitochondrial membrane into the matrix in symport with each Pi− entering the matrix via the mitochondrial Pi carrier.
Input variables
f I	1 {range [0,1]}	Fraction of NADH in the mitochondrial matrix that is oxidized by complex I; the fraction (1–fI) is oxidized by the non-proton-pumping rotenone-insensitive alternative matrix-facing NADH dehydrogenase (NDin).
f III-IV	1 {range [0,1]}	Fraction of QH2 in the inner mitochondrial membrane that is oxidized by complexes III and IV (i.e. the cytochrome pathway from QH2 to molecular oxygen); the fraction (1–fIII-IV) is oxidized by the mitochondrial alternative oxidase. This excludes QH2 formed in the complex III Q-cycle.
f leak	0 {range [0,<1]}	Fraction of protons actively pumped out of the mitochondrial matrix that re-enter the matrix through passive proton leaks and/or uncoupling proteins (UCPs) in the inner mitochondrial membrane. Quantitatively important in animals; possibly negligible in plants in most settings except perhaps for UCP-mediated thermogenesis in some flowers respiring fatty acids (see text).
f Pyr	1 {range [0,1]}	Fraction of PEP converted to pyruvate in cytosolic glycolysis; the fraction (1 − fPyr) is converted to cytosolic malate.
f shuttle	0 {range [0,1]}	Fraction of cytosolic glycolytic NADH reducing power that is transferred into the mitochondrial matrix in a malate–aspartate shuttle. Not thought to operate in plants but required for model simulation of mammalian ATP production.
f slip	0 {range [0,<1]}	Fraction of protons expected to be pumped across the inner mitochondrial membrane by activity of complexes I, III and/or IV, but that are not pumped because of slippage.
f slip(FoF1)	0 {range [0,<1]}	Fraction of slippage in the mitochondrial ATP synthase.
f starch	0 {range [0,1]}	Fraction of respiratory substrate composed of starch; the fraction (1 − fstarch) is composed of sucrose.
H+NDex	0 {range [0,1]}	Number of protons produced on the P side of the inner mitochondrial membrane with oxidation of each cytosolic NADH by the non-proton-pumping external rotenone-insensitive NADH dehydrogenase (NDex).
H+Pyr	1 {range [0,1]}	Number of protons pumped to the P side of the inner mitochondrial membrane by the respiratory chain that re-enter the matrix in symport with each pyruvate entering a mitochondrion. Although only values of 0 or 1 would apply to individual pyruvate molecules, if, for example half the pyruvate entering mitochondria were accompanied by protons previously pumped to the P side of the membrane by the respiratory chain and the other half were accompanied by protons resulting from CO2 hydration on the P side, H+Pyr would be 0.5.
n	10 {values range from 8 to 17 across a range of organisms; see text}	Number of c subunits in the mitochondrial FoF1-ATP synthase Fo sector c-ring; n has not been determined for plant mitochondria (see text for rationale on use of the default value of 10).
Intermediate variables (one a constant)
H+FoF1	n/3	Number of protons transported into the matrix through the Fo sector of the mitochondrial FoF1-ATP synthase per ADP phosphorylated, in the absence of slippage. This represents the intrinsic (mechanistic) H+/ATP ratio of the ATP synthase.
N Suc	2	Number of succinates oxidized by complex II in the TCA cycle per hexose respired.
Output variables (one a constant)
N ATP		Net number of ATP produced in, or translocated into, the cytosol per hexose respired.
N ATP(FoF1)		Number of ATP produced by mitochondrial FoF1-ATP synthases and then transported into the cytosol per hexose respired.
N ATP(glycolysis)	{range [0,2.5]}	Net number of ATP produced in cytosolic glycolysis per hexose respired.
N ATP(TCA)	2	Number of ATP produced in TCA cycle substrate-level phosphorylations that are then transported into the cytosol per hexose respired.
P/O		Net number of ATP produced by respiration per oxygen atom reduced by the respiratory chain. In some cases, P/O represents ATP produced in oxidative phosphorylation alone, but in the full model it is total respiratory ATP, i.e. NATP.

STARCH AND SUCROSE BREAKDOWN COUPLED TO CYTOSOLIC GLYCOLYSIS

Modelled pathways of starch and/or sucrose breakdown linked to glycolysis and the production of cytosolic pyruvate and/or malate (Fig. 2) directly account for N_{ATP(glycolysis)}. They can also produce cytosolic NADH that may contribute to N_ATP(FoF1). The pyruvate and/or malate produced are the substrate(s) for mitochondrial carbon metabolism.

Fig. 2.

Modelled carbon metabolism pathways from starch and sucrose to pyruvate and malate. (A) Plastidic starch breakdown to maltose (glucose–glucose disaccharide) that is exported to the cytosol where its metabolism releases one free glucose and transfers the other glucose unit to a glycan. The free glucose is phosphorylated, using ATP to produce glucose 6-phosphate (G-6-P) which then is converted to fructose 6-phosphate (F-6-P). The glucose transferred to a glycan is in turn phosphorolytically cleaved from the glycan as glucose 1-phosphate (G-1-P), regenerating the glycan. G-1-P is then converted to G-6-P and on to F-6-P (after Lu and Sharkey, 2006; Weise et al., 2011). Here, ‘glycan’ is generically used to indicate any number of compounds consisting of glycosidically linked monosaccharides. The overall pathway uses one ATP and produces two F-6-P per 12 carbons of input. (B) Cleavage of cytosolic sucrose (fructose–glucose disaccharide) by invertase yielding free fructose and glucose, both of which are phosphorylated using ATP to form F-6-P and G-6-P, respectively, with the G-6-P in turn converted to F-6-P. The overall pathway uses two ATP and produces two F-6-P per 12 carbons of input. (C) Glycolytic pathway from F-6-P to pyruvate or malate. Two F-6-P from (A) and/or (B) are phosphorylated, forming two fructose 1,6-bisphosphate (F-1,6-P₂) that are split into equal parts dihydroxyacetone phosphate (DHAP) and glyceraldehyde 3-phosphate (G-3-P), with all the DHAP then converted to G-3-P. The resulting four G-3-P are converted to four phosphoenolpyruvates (PEP) in a four-step process. Conversion of PEP to pyruvate phosphorylates one ADP per PEP molecule. Conversion of PEP to malate (shown in red) is via oxaloacetate (OAA), which is produced by a carboxylation reaction consuming HCO₃⁻ arising from aqueous CO₂. The OAA to malate conversion oxidizes NADH, regenerating the NAD⁺ reduced earlier in the pathway from G-3-P. The overall pathway from F-6-P to pyruvate consumes two ATP and produces four pyruvate, four NADH and eight ATP (eight substrate-level ADP phosphorylations) for a net production of six ATP per 12 carbons of input. The overall pathway from F-6-P to malate consumes two ATP, four NADH and four CO₂ and produces four malate, four NADH and four ATP (four substrate-level ADP phosphorylations) for a net production of two ATP per 12 carbons of input. The three dotted-line enclosures are used to group reaction sets (A), (B) and (C), not to indicate cellular compartments [modelled sets (B) and (C) are cytosolic]. The dashed arrows between the F-6-P product in (A) and (B) and the F-6-P substrate in (C) indicate that the reaction sets in (A) and/or (B) are input to (C). The combination of reaction sets (A) and (C) is overall metabolism from starch to pyruvate and/or malate and the combination of (B) and (C) is overall metabolism from sucrose to pyruvate and/or malate. Model ATP yields are on a per-hexose-respired basis although the pathways are drawn here on a per-disaccharide (i.e. maltose or sucrose) basis. 1,3-DiPGA, 1,3-bisphospoglycerate; glycan_{n}, glycan molecule containing n linked monosaccharides; glycan_{n+G}, the same glycan molecule with a transferred (linked) glucose from maltose; IMS, plastidic intermembrane space; MEX1, maltose exporter; 2-PGA, 2-phosphoglycerate; 3-PGA, 3-phosphoglycerate; starch_{n}, starch molecule containing n glucose units; starch_{n−2}, the same starch molecule less two adjacent glucose (i.e. one maltose) units.

Modelled starch, inside a plastid, is metabolized via the disaccharide maltose that exits the plastid stroma and enters cytosolic glycolysis as equal parts glucose and glucose 1-phosphate. Production of pyruvate or malate from that starch is summarized (omitting water) on a per hexose unit basis as:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{c}{\mathrm{h}}_{\left\{n\right\}}+2.5\mathrm{c}\mathrm{A}\mathrm{D}\mathrm{P}+2.5\mathrm{c}{\mathrm{P}}_{\mathrm{i}}+2\mathrm{c}\mathrm{N}\mathrm{A}{\mathrm{D}}^{+}\to 2\mathrm{p}\mathrm{y}\mathrm{r}\mathrm{u}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{e}+2.5\mathrm{c}\mathrm{A}\mathrm{T}\mathrm{P}+2\mathrm{c}\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}+\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{c}{\mathrm{h}}_{\{n-1\}}}\end{array}}$$ (2a)

and

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{c}{\mathrm{h}}_{\left\{n\right\}}+0.5\mathrm{c}\mathrm{A}\mathrm{D}\mathrm{P}+0.5\mathrm{c}{\mathrm{P}}_{\mathrm{i}}+2\mathrm{C}{\mathrm{O}}_2\to 2\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}+0.5\mathrm{c}\mathrm{A}\mathrm{T}\mathrm{P}+2{\mathrm{H}}^{+}+\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{c}{\mathrm{h}}_{\{n-1\}}}\end{array}}$$ (2b)

respectively, where starch_{n} is a plastidic starch molecule composed of n glucose units and starch_{n−1} is the same molecule less one glucose unit, recognizing that modelled glucose units are cleaved from starch two at a time in maltose units. The prefix ‘c’ indicates a cytosolic location for ADP, ATP, NAD⁺, NADH and P_i. The pyruvate in eqn (2a) and malate and H⁺ in eqn (2b) are also cytosolic.

The modelled disaccharide sucrose is cleaved by an invertase into fructose and glucose in the cytosol. Both products are metabolized in parallel by glycolysis. Production of pyruvate or malate from sucrose is summarized (omitting water) on a per hexose basis as:

$${\displaystyle \begin{array}{l}{\displaystyle 0.5\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{e}+2\mathrm{c}\mathrm{A}\mathrm{D}\mathrm{P}+2\mathrm{c}{\mathrm{P}}_i+2\mathrm{c}\mathrm{N}\mathrm{A}{\mathrm{D}}^{+}\to 2\mathrm{p}\mathrm{y}\mathrm{r}\mathrm{u}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{e}+2\mathrm{c}\mathrm{A}\mathrm{T}\mathrm{P}+2\mathrm{c}\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}}\end{array}}$$ (3a)

and

$${\displaystyle \begin{array}{l}{\displaystyle 0.5\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{e}+2\mathrm{C}{\mathrm{O}}_2\to 2\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}+2{\mathrm{H}}^{+}}\end{array}}$$ (3b)

respectively. As modelled, glycolysis produces more cytosolic ATP (0.5 ATP/hexose) when starch, relative to sucrose, is the initial substrate.

Integrating pathways from starch or sucrose to pyruvate or malate results in the following model relationships, with all terms on the left on a per hexose respired basis:

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{s}\right)}=0.5f_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}}+2f_{\mathrm{P}\mathrm{y}\mathrm{r}}}\end{array}}$$ (4a)

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}=2f_{\mathrm{P}\mathrm{y}\mathrm{r}}}\end{array}}$$ (4b)

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{P}\mathrm{y}\mathrm{r}}=2f_{\mathrm{P}\mathrm{y}\mathrm{r}}}\end{array}}$$ (4c)

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{M}\mathrm{a}\mathrm{l}}=2(1-f_{\mathrm{P}\mathrm{y}\mathrm{r}})}\end{array}}$$ (4d)

where f_starch is the fraction, [0,1], of respiratory substrate composed of starch, with the remainder (1 − f_starch) composed of sucrose; f_Pyr is the fraction, [0,1], of carbon in the substrate(s) that appears in cytosolic pyruvate, with the remainder (1 − f_Pyr) appearing in cytosolic malate; N_NADHc is the net number of NADH formed from NAD⁺ in the cytosol; N_Pyr is the number of pyruvate formed in the cytosol; and N_Mal is the number of malate formed in the cytosol. Thus, maximum modelled N_{ATP(glycolysis)} is 2.5 ATP/hexose, which occurs with f_starch = 1 and f_Pyr = 1 (i.e. starch → pyruvate pathway). Minimum N_{ATP(glycolysis)} is zero, which occurs with f_starch = 0 and f_Pyr = 0 (i.e. sucrose → malate pathway). Any combination of f_starch and f_Pyr can be used in the model.

Sucrose may be available as a respiratory substrate in more situations compared with starch, and the model default is f_starch = 0 (Table 1). This default is consistent with traditional respiratory ATP yield calculations based on the unphosphorylated hexose glucose as substrate because respiration of sucrose in the model (as cleaved by invertase) is quantitatively the same as respiration of glucose on a per hexose basis. Modelled respiration of starch, however, is the equivalent of respiration of equal parts unphosphorylated and phosphorylated glucose.

It may be expected that under model conditions pyruvate is the main product of glycolysis (Plaxton and Podestá, 2006) and the model default is f_Pyr = 1.

Sucrose cleavage catalysed by sucrose synthase

Modelled sucrose is cleaved by a cytosolic invertase. In plants, sucrose also can be cleaved by sucrose synthase, requiring uridine 5ʹ-diphosphate (UDP) as a co-substrate and producing fructose and UDP-glucose instead of fructose and glucose. Glycosidic bond energy in sucrose is lost in the invertase-catalysed reaction but (partially) conserved in the sucrose synthase-catalysed reaction. For sucrose synthase-initiated sucrose catabolism, the two ADP and two P_i on the input side of eqn (3a) are replaced with three ADP, two P_i and 0.5 PP_i. On the output side the two ATP are replaced with three ATP, a gain of one ATP/hexose, but with the additional 0.5 PP_i/hexose input requirement.

Plant cells can contain appreciable amounts of PP_i (Stitt, 1998; Plaxton and Podestá, 2006) but the model structure would require that all PP_i use (input) is balanced with an equal amount of PP_i production (output), which could be by oxidation of additional carbon substrate, hydrolysis of ATP (i.e. ATP → AMP + PP_i) stoichiometrically coupled with conversion of resulting AMP to ADP (i.e. ATP + AMP → 2 ADP), or other PP_i-releasing reactions. As designed, the model inputs are limited to a carbon substrate(s), ADP, P_i and O₂, and any ATP consumed in the early stage of glycolysis is supplied by ATP produced in the latter stage of glycolysis, the TCA cycle and/or the oxidative phosphorylation system. Thus, in spite of the potential importance of salvaging available PP_i as a ‘free’ resource, the modelled system is ‘closed’ in the sense that only a defined set of inputs can be consumed, and all intermediates are regenerated or appear in the designated set of outputs.

While possibilities may exist for balanced PP_i production and use in respiration (Igamberdiev and Kleczkowski, 2021), the overriding consideration during model development was that sucrose synthase may (will) be most active (1) in growing tissues where PP_i is a co-product/by-product of a range of anabolic reactions and/or (2) during anoxia, P_i deficiency or other stresses (Stitt, 1998; Plaxton and Podestá, 2006; Igamberdiev and Kleczkowski, 2011; Koch, 2013; O’Leary and Plaxton, 2016). Since the model setting is mature unstressed cells, sucrose synthase activity in this instance may be of limited significance and is ignored. It might be prominent, however, in expanded model applications associated with metabolic stresses or for modelling ‘growth respiration’.

Fructose 6-phosphate phosphorylation catalysed by PP_i-dependent phosphofructokinase

Cytosolic conversion of fructose 6-phosphate to fructose 1,6-bisphosphate can be catalysed by a PP_i-dependent phosphofructokinase instead of the modelled ATP-dependent phosphofructokinase in Fig. 2. That PP_i-dependent route reflects a second potential opportunity (in addition to sucrose synthase activity above) to salvage any available cytosolic PP_i and spare ATP used in production of fructose 1,6-bisphosphate in the early phase of glycolysis. Again, however, the model is intended to represent respiratory ATP production from a net input of only a carbon substrate(s), ADP, P_i and O₂, meaning the PP_i would need to be regenerated from additional carbon substrate oxidation or ATP hydrolysis (as above) and any PP_i produced or available for PP_i-dependent phosphofructokinase activity is outside the model scope.

MITOCHONDRIAL PYRUVATE OXIDATION AND TCA CYCLE OPERATION

Oxidation of pyruvate by the pyruvate dehydrogenase complex in the mitochondrial matrix is the modelled link between glycolysis in the cytosol and TCA cycle in the mitochondrial matrix (Fig. 3). The modelled pyruvate oxidation reaction reduces NAD⁺, releases CO₂ and generates acetyl-CoA that enters a closed-loop TCA cycle. Modelled supply of pyruvate to the mitochondrial matrix is by (1) uptake of cytosolic pyruvate and/or (2) uptake of cytosolic malate followed by its NAD-malic enzyme-catalysed decarboxylation in the mitochondrial matrix that produces pyruvate, releases CO₂ and reduces NAD⁺. Uptake of cytosolic pyruvate⁻ is facilitated by a mitochondrial pyruvate carrier in symport with a proton whereas uptake of cytosolic malate²⁻ is facilitated by a mitochondrial dicarboxylate carrier in antiport with P_i²⁻ that previously entered the mitochondrion as P_i⁻ in symport with a proton (Fig. 3).

Fig. 3.

Pyruvate and malate import into the mitochondrial matrix through the inner mitochondrial membrane (IMM), oxidation/decarboxylation of that malate and pyruvate, and closed-loop TCA cycle metabolism. P indicates the positive side of the membrane [i.e. crista space/intermembrane space (IMS)] to which protons are released by the respiratory chain proton pumps. N indicates the negative side of the membrane (i.e. the mitochondrial matrix) from which protons are extracted by the respiratory chain. Pyruvate uptake – i.e. the default modelled link between cytosolic glycolysis and the TCA cycle in the mitochondrial matrix – is via the mitochondrial pyruvate carrier (MPC) in symport with a proton. Shown in red, as an alternative to pyruvate import, is malate entry into the mitochondrial matrix via a dicarboxylate carrier (DIC) in antiport with P_i²⁻ and the NAD-malic enzyme catalysed conversion of that malate to pyruvate in the matrix, forming NADH and releasing CO₂. P_i²⁻ used in antiport with malate arises from deprotonation of P_i⁻ imported earlier in proton symport via the phosphate carrier (PiC). The deprotonation is driven by the lower proton concentration (higher pH) in the matrix. For either direct uptake of pyruvate or uptake of malate followed by its conversion to pyruvate, pyruvate in the mitochondrial matrix is decarboxylated by the pyruvate dehydrogenase complex, producing NADH and acetyl-CoA, with the latter entering the TCA cycle. Complete respiratory metabolism of each pyruvate results in (1) release of three CO₂ (all the carbon in pyruvate); (2) reduction of four NAD⁺; (3) generation of one ATP in a substrate-level ADP phosphorylation; and (4) transfer of one electron pair (2e⁻) and two protons from succinate to a lipophilic ubiquinone (Q), generating ubiquinol (QH₂) in the IMM. The 2e⁻ transfer in (4) is catalysed by succinate:ubiquinone oxidoreductase (complex II) as part of the TCA cycle. Complex II, which is anchored in the IMM, also is a component of the respiratory electron transport chain (Fig. 1) and the only direct physical connection between the TCA cycle and IMM. Under modelled conditions, TCA cycle-generated ATP is exported for use outside the mitochondrion and each substrate-level ADP phosphorylation is synchronized with import of one P_i in proton symport via PiC, import of one ADP, and export of the ATP produced (lower left). ADP and ATP are stoichiometrically exchanged across the IMM via the adenine nucleotide translocator (ANT). Proton uptake coupled to import of both pyruvate and P_i may represent minimum requirements for the respiratory chain-generated transmembrane proton concentration gradient ΔpH to sustain TCA cycle operation. In a closed system, such as might be used to measure O₂ consumption by tissue samples or mitochondrial suspensions, proton production from extra-mitochondrial CO₂ hydration might acidify the external medium and provide protons for H⁺/pyruvate⁻ symport (dashed arrows) (see LaNoue and Schoolwerth, 1979). In the model’s steady-state implicit open plant-atmosphere system, however, CO₂ is released to the atmosphere (not shown) without contributing to net proton accumulation in the IMS. It is proposed here that pyruvate uptake depends on protons delivered to the P side of the IMM by the respiratory chain proton pumps. Each molecule shown throughout this diagram is replicated (once) in the model to reflect the hexose substrate basis of ATP yield calculations, with one hexose producing either two pyruvate, two malate or one each of pyruvate and malate in glycolysis. Charge on protons and molecules crossing the IMM is indicated, but not on other molecules except NAD⁺. Curly brackets, }, and associated text indicate use (dissipation) by transport proteins of ΔpH or ΔΨ_m produced by the respiratory chain. Reactions are shown in the direction of respiration; some are reversible. CoA-SH, coenzyme A (also often abbreviated CoA); OAA, oxaloacetate; 2-OG, 2-oxoglutarate (old name: α-ketoglutarate); suc-CoA, succinyl-coenzyme A.

For each modelled pyruvate molecule metabolized in the mitochondrial matrix: (1) three CO₂ are generated, accounting for all the carbon in the pyruvate; (2) one P_i enters the mitochondrion in symport with one proton to support a substrate-level phosphorylation of ADP during the TCA cycle’s conversion of succinyl-CoA to succinate; (3) one ATP⁴⁻ is produced by that substrate-level phosphorylation in the matrix, which then exits the mitochondrion in exchange for one cytosolic ADP³⁻; (4) four NAD⁺ in the mitochondrial matrix are reduced to NADH; and (5) one electron pair (2e⁻) and two protons are transferred from mitochondrial succinate to Q, producing fumarate in the matrix and QH₂ within the inner mitochondrial membrane (Fig. 3). The ADP³⁻/ATP⁴⁻ antiport is via the mitochondrial transmembrane adenine nucleotide translocator without use of the ΔpH across the inner mitochondrial membrane but with partial dissipation of the ΔΨ_m component of the electrochemical proton gradient due to the greater negative charge on the outward-moving ATP⁴⁻ (Fig. 3, lower left). In parallel with this ADP–ATP exchange, and as a requirement for the TCA cycle’s substrate-level phosphorylation of ADP, cytosolic P_i⁻ enters the mitochondrion in an electroneutral H⁺/P_i⁻ symport that does dissipate some of the ΔpH. Thus, as modelled, steady-state TCA cycle substrate-level phosphorylation of ADP and export of the resulting ATP occurs with a 1:1:1:1 ratio of ATP export, ADP import, P_i import and proton import as: ATP⁴⁻_outward = ADP³⁻_inward + P_i⁻_inward + H⁺_inward.

Pyruvate imported from the cytosol

By accounting for two pyruvate molecules produced per hexose of respiratory substrate catabolized, the modelled complete mitochondrial oxidation of pyruvate that is directly imported from the cytosol is summarized (omitting water) per hexose respired as:

$${\displaystyle \begin{array}{l}{\displaystyle 2\mathrm{p}\mathrm{y}\mathrm{r}\mathrm{u}\mathrm{v}\mathrm{a}\mathrm{t}{\mathrm{e}}^{-}+2\mathrm{c}\mathrm{A}\mathrm{D}{\mathrm{P}}^{3-}+2{\mathrm{c}\mathrm{P}}_i^{-}+4{\mathrm{H}}_{\mathrm{P}}^{+}+8\mathrm{m}\mathrm{N}\mathrm{A}{\mathrm{D}}^{+}+2\mathrm{Q}\to 6\mathrm{C}{\mathrm{O}}_2+2\mathrm{c}\mathrm{A}\mathrm{T}{\mathrm{P}}^{4-}+4{\mathrm{H}}_{\mathrm{N}}^{+}+8\mathrm{m}\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}+2\mathrm{Q}{\mathrm{H}}_2}\end{array}}$$ (5a)

where H⁺_P is a proton on the P side of the inner mitochondrial membrane (i.e. in the intermembrane space or crista space), H⁺_N is a proton on the N side of the membrane (i.e. a proton transported from the P side of the membrane to the N side in the mitochondrial matrix), the prefix ‘m’ indicates NAD⁺ and NADH in the mitochondrial matrix, and the Q and QH₂ are localized within the inner mitochondrial membrane (Figs 1 and 3). The NADH and QH₂ produced are important to downstream electrochemical proton gradient generation whereas the 4 H⁺_P input is stoichiometrically linked to the 4 H⁺_N output as a component of proton gradient dissipation (see below).

Malate imported from the cytosol and then converted to pyruvate in the mitochondrial matrix

To summarize modelled respiration of malate imported from the cytosol, eqn (5a) is modified to include the H⁺/P_i⁻ symport leading to P_i²⁻/malate²⁻ antiport and the conversion of malate to pyruvate in the mitochondrial matrix (Fig. 3 upper left red entities). Equation (5a) is further modified to exclude the H⁺/pyruvate⁻ symport. In total, this results in (omitting water):

$${\displaystyle \begin{array}{l}{\displaystyle 2\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{t}{\mathrm{e}}^{2-}+2\mathrm{c}\mathrm{A}\mathrm{D}{\mathrm{P}}^{3-}+2{\mathrm{c}\mathrm{P}}_i^{-}+4{\mathrm{H}}_{\mathrm{P}}^{+}+10\mathrm{m}\mathrm{N}\mathrm{A}{\mathrm{D}}^{+}+2\mathrm{Q}\to 8\mathrm{C}{\mathrm{O}}_2+2\mathrm{c}\mathrm{A}\mathrm{T}{\mathrm{P}}^{4-}+4{\mathrm{H}}_{\mathrm{N}}^{+}+10\mathrm{m}\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}+2\mathrm{Q}{\mathrm{H}}_2}\end{array}}$$ (5b)

where the two malate molecules are produced in glycolysis from a hexose unit from either starch or sucrose. Relative to summary eqn (5a), eqn (5b) produces two additional mitochondrial NADH per hexose respired. Conversely, the glycolytic production of the malate that was imported produced less ATP and NADH in the cytosol relative to glycolytic production of pyruvate [eqns (4a) and (4b)].

Yield of ATP from TCA cycle substrate-level phosphorylations

The two turns of the modelled TCA cycle that occur per hexose respired – as derived from either starch or sucrose – produce two ATP molecules in substrate-level phosphorylations in the mitochondrial matrix. Those two ATP are exported to the cytosol as described above. The ATP supplied to the cytosol that originates from substrate-level ADP phosphorylations in the TCA cycle in the mitochondrial matrix is thus:

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}=2}\end{array}}$$ (6)

with either starch or sucrose as the initial substrate and either pyruvate or malate as products of glycolysis, i.e. 2 cATP⁴⁻ is on the product side of both eqns (5a) and (5b). The ΔpH ‘cost’ of exporting each ATP to the cytosol is the transport of one proton from the P side to the N side of the inner mitochondrial membrane for P_i uptake to support production of each ATP as embedded in the 1:1:1:1 stoichiometry of exported ATP with imported ADP, P_i and proton. Those protons are accounted for in the proton gradient ΔpH dissipation portion of the model below. Their use slightly reduces the potential number of ATP generated in oxidative phosphorylations than would otherwise be possible (about 0.5 ATP/hexose; see below).

The other two key model products of pyruvate and/or malate metabolism in the mitochondrial matrix are given by (per hexose):

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}=8+2(1-f_{\mathrm{P}\mathrm{y}\mathrm{r}})}\end{array}}$$ (7a)

and

$${\displaystyle \begin{array}{l}{\displaystyle N_{\mathrm{S}\mathrm{u}\mathrm{c}}=2}\end{array}}$$ (7b)

where N_NADHm is the number of NADH formed in the matrix and N_Suc is the number of succinates oxidized by complex II. N_Suc quantifies direct QH₂ production by the TCA cycle and is equal to the coefficient on QH₂ in eqns (5a) and (5b), which is always 2 for the model setting. The sum N_NADHc + N_NADHm is 10 NADH/hexose for all combinations of f_starch and f_Pyr.

PROTON GRADIENT GENERATION

Three proton-pumping enzymes – complexes I, III and IV – generate the respiratory proton gradient across the inner mitochondrial membrane (Fig. 1). The mechanisms of proton pumping differ between the three (Wikström et al., 2015). All three complexes are components of the respiratory chain, which in plants is a multi-branch pathway transferring electrons from NADH, NADPH and succinate to molecular oxygen (Braun, 2020; Meyer et al., 2022). In this model application the electron donors are NADH produced in the cytosol by glycolysis and NADH and succinate produced in the mitochondrial matrix by reactions included in Fig. 3; the model does not use NADPH. The modelled plant respiratory chain has four entry points; there are others in actual plants (Møller et al., 2021) that may be significant in non-modelled circumstances. There are two terminating branches in the model. Importantly, only one of the entry points (complex I) and one of the terminating branches (combined activity of complexes III and IV) pump protons through the inner mitochondrial membrane.

All four entry points transfer a 2e⁻ from a donor molecule to a Q molecule inside the inner mitochondrial membrane, forming a QH₂ molecule, also inside the membrane (Fig. 1). The two terminating branches oxidize QH₂ (regenerate Q) and transfer the extracted electrons to molecular oxygen, forming water. Flexibility in a plant’s partitioning of electrons among the entry points and terminating branches of the respiratory chain contributes to (1) a plant’s ability to respond to changing developmental, physiological and environmental circumstances and (2) a modeller’s uncertainty about plant respiratory ATP yield.

Respiratory chain entry point 1: oxidation of matrix NADH by complex I

Complex I is a large proton-pumping NADH:ubiquinone oxidoreductase (NADH dehydrogenase) embedded in the inner mitochondrial membrane (Meyer et al., 2022). The NADH binding site is near the end of complex I’s arm that extends a considerable distance into the matrix, potentially providing access to NADH in the matrix beyond the liquid layer adjacent to the inner membrane (Verkhovskaya and Bloch, 2013). Complex I activity is inhibited by rotenone. The accepted proton pumping stoichiometry is four protons per NADH oxidized (4 H⁺/NADH = 4 H⁺/2e⁻), though most quantitative data are from non-plants (Jones et al., 2017; Braun, 2020; Vercellino and Sazanov, 2022). The model parameter H⁺_I encodes this 4 H⁺/NADH proton pump stoichiometry of complex I (Table 1).

Respiratory chain entry point 2: oxidation of matrix succinate by complex II

Oxidation of succinate in the mitochondrial matrix is catalysed by the non-proton-pumping, membrane-bound complex II (succinate:ubiquinone oxidoreductase) as part of the TCA cycle (Fig. 3). Complex II advances the TCA cycle by converting succinate to fumarate. Although complex II does not pump protons, the QH₂ it generates can contribute to downstream respiratory chain proton pumping if complex III oxidizes that QH₂.

Respiratory chain entry point 3: oxidation of matrix NADH by a non-proton-pumping, alternative matrix-facing NADH dehydrogenase

In addition to its oxidation by complex I, mitochondrial NADH also can be oxidized by a non-proton-pumping alternative NADH dehydrogenase with an NADH-binding site in the matrix (ND_in in Fig. 1) (Møller et al., 2021). Unlike complex I, it is insensitive to rotenone. There are multiple isoforms in plants but this matrix-facing alternative NADH dehydrogenase is absent from mammals. Due to its much larger K_m for NADH compared to that of complex I it may be (most) active when the NADH/NAD⁺ ratio in the mitochondrial matrix is especially large (highly reduced) (but see Rasmusson et al., 2020). It might help regenerate NAD⁺ required for continued operation of the TCA cycle if for any reason complex I is overwhelmed or non-functional. Although this alternative NADH dehydrogenase does not pump protons, the QH₂ it generates can contribute to downstream respiratory chain proton pumping if complex III oxidizes that QH₂.

The model distinguishes this NADH dehydrogenase from a matrix-facing NADPH dehydrogenase (Møller et al., 2021). The latter is not included in the model because modelled carbon metabolism in the matrix is restricted to the TCA cycle producing NADH (Fig. 2).

Respiratory chain entry point 4: oxidation of cytosolic NADH

Cytosolic NADH can be oxidized by a non-proton-pumping inner mitochondrial membrane-bound NADH dehydrogenase with its NADH-binding site in the intermembrane space (Rasmusson et al., 2020; Møller et al., 2021). This plant NADH dehydrogenase (ND_ex in Fig. 1) is insensitive to rotenone, exists in multiple isoforms in plants and is absent from mammals. All modelled cytosolic NADH not shuttled into the mitochondrial matrix (see below) is oxidized by this external NADH dehydrogenase.

Although this external NADH dehydrogenase does not pump protons, the QH₂ it generates can contribute to downstream respiratory chain proton pumping if complex III oxidizes that QH₂. As stated above, that is also the case for QH₂ produced by complex II and/or by the rotenone-insensitive non-proton-pumping alternative matrix-facing NADH dehydrogenase ND_in. Unlike activity of complex II and ND_in, however, the ND_ex reaction initiates a transmembrane transport of electrons from the P side to the N side of the inner mitochondrial membrane, contributing to ΔΨ_m. Also, ND_ex activity produces one proton on the P side of the membrane per 2e⁻ removed from external (i.e. modelled glycolytic) NADH, thus contributing modestly to proton gradient generation (Møller et al., 2021). The model parameter H⁺_NDex is the number of protons produced on the P side of the inner mitochondrial membrane per NADH oxidized by ND_ex. The model default is H⁺_NDex = 0 to reflect a traditional view that activity of ND_ex per se does not contribute to ΔpH. Model scenarios with H⁺_NDex > 0 indicate the relative importance of a direct contribution of ND_ex to proton gradient generation and resulting respiratory ATP production.

The model distinguishes this external NADH dehydrogenase from an NADPH dehydrogenase on the outside of the inner mitochondrial membrane (Møller et al., 2021). That NADPH dehydrogenase is not included in the model because the modelled pathways of carbon metabolism outside mitochondria do not produce NADPH.

Respiratory chain terminating branch 1: cytochrome pathway

The main terminating branch of the plant respiratory chain is composed of proton-pumping complex III (quinol–cytochrome-c reductase), which oxidizes QH₂ and reduces cytochrome c, in serial combination with proton-pumping complex IV (cytochrome-c oxidase), which oxidizes cytochrome c and reduces molecular oxygen (forming water). This branch is called the ‘cytochrome pathway’ in plants and is homologous to the portion of the respiratory chain transferring electrons from QH₂ to molecular oxygen in other organisms (Millar et al., 2011). In plants, complex III exists as a dimer (complex III₂; Braun 2020). Complex IV is inhibited by cyanide.

In healthy mitochondria, the small hydrophilic haem-containing protein cytochrome c resides in the crista space, where it is bound or loosely attached to the inner mitochondrial membrane. Cytochrome c accepts electrons from complex III one at a time, with its reduction occurring in a two-step Q-cycle (Nicholls and Ferguson, 2013; Vercellino and Sazanov, 2022).

Current consensus is that the cytochrome pathway translocates six protons across the inner mitochondrial membrane for each 2e⁻ transferred from QH₂ to molecular oxygen (6 H⁺/2e⁻ = 6 H⁺/O). The model parameter H⁺_III-IV encodes this 6 H⁺/2e⁻ stoichiometry of the cytochrome pathway (Table 1).

Respiratory chain terminating branch 2: alternative oxidase

The other terminating branch of the plant respiratory chain involves a transfer of electrons and protons from QH₂ to oxygen $(2\mathrm{Q}{\mathrm{H}}_2+{\mathrm{O}}_2\to 2\mathrm{Q}+2{\mathrm{H}}_2\mathrm{O})$ catalysed by the alternative oxidoreductase (commonly called alternative oxidase). It is a non-proton-pumping single-enzyme alternative ‘pathway’ found in plants and some fungi and protists (Moore et al., 2013; Vanlerberghe et al., 2020). Five expressed alternative oxidase genes are documented in Arabidopsis. Unlike complex IV, alternative oxidase is insensitive to cyanide.

Alternative oxidase activity can have a marked negative effect on modelled plant proton gradient generation and therefore respiratory ATP production. Uncertainty about the degree of in situ alternative oxidase activity across species, organs, developmental stages, environments and circumstances – as a fraction of total oxygen reduction by the respiratory chain – is a key limitation to quantifying actual, as opposed to potential, ATP yield of plant respiration.

Although neither the rotenone-insensitive external mitochondrial NADH dehydrogenase, ND_ex, nor the alternative oxidase pump protons, their combined activity will transfer electrons from the P side to the N side of the inner mitochondrial membrane. This contributes to electrochemical proton gradient generation so that plant mitochondria oxidizing external NADH with the non-proton-pumping ND_ex in combination with the alternative oxidase were found to generate enough membrane potential to support protein import into those mitochondria (Glaser et al., 1998) – mitochondrial protein import requires both ΔΨ_m and ATP (Ford et al., 2022). Also, it was noted that the transfer of a 2e⁻ from the P side to the N side of the inner mitochondrial membrane resulting from joint activity of the rotenone-insensitive external NADH dehydrogenase and the alternative oxidase is equivalent to a two H⁺/NADH pump from the N side to the P side of the membrane (Møller et al., 2021).

Slip in the electron transfer–proton pump couplings

The 4 H⁺/2e⁻ ratio of complex I (model parameter H⁺_I) and the 6 H⁺/2e⁻ ratio of the cytochrome pathway (model parameter H⁺_III-IV) reflect current consensus for mitochondrial proton pumps fully coupled to electron transport in the respiratory chain. ‘Slip’ in the respiratory chain refers to electron transport without the attainable (maximal) concurrent proton translocation, i.e. a shortfall in actual, relative to expected or potential, H⁺/2e⁻ in complexes I, III or IV. Slip is a negative factor in proton gradient generation. It differs from ‘proton leak’, which refers to protons that have been pumped from the N side to the P side of the inner mitochondrial membrane that then passively leak back to the N side. Leak is a component of proton gradient dissipation (see below).

Some research with animal mitochondria and bacteria indicates that slip can occur in complex IV under physiological conditions (Kadenbach, 2003). Nonetheless, it is perhaps more generally thought that slip is at most very slight under conditions supporting oxidative phosphorylation (Divakaruni and Brand, 2011; Wikström and Springett, 2020). The model variable f_slip is the degree or fraction, [0,1), of slip in the respiratory chain, with 0 indicating perfect H⁺/2e⁻ coupling (the model default) and with 1 indicating complete uncoupling in all H⁺-pumping complexes, which is an extreme that will not occur in healthy mitochondria. The modelled value of f_slip, if greater than zero, is an average of slip in complexes I, III and IV weighted by the number of protons expected to be pumped by each complex. A more detailed approach would assign independent f_slip values to each complex. To my knowledge, there is no conclusive measurement data relevant to respiratory-chain slip in plants in conditions supporting oxidative phosphorylation.

Shuttling cytosolic NADH reducing power into the mitochondrial matrix

Because NADH in the mitochondrial matrix has access to proton-pumping complex I but cytosolic (intermembrane space) NADH does not, NADH in the matrix is more valuable for proton gradient generation. Although NADH cannot pass through the inner mitochondrial membrane, a malate–aspartate shuttle (Fig. 4) can translocate the reducing equivalents of cytosolic NADH into the mitochondrial matrix in some organisms. This malate–aspartate shuttle is included in models of mammalian respiratory ATP yield (Hinkle et al., 1991; Brand, 2005; Nichols and Ferguson, 2013; Mookerjee et al., 2017), but there is scant evidence that such a shuttle operates in plants to transfer reducing equivalents into the mitochondrial matrix during respiration in the model setting. Nonetheless, for application to other organisms, and to allow assessment of its hypothetical importance, this plant model can accommodate the traditional malate–aspartate shuttle through the variable f_shuttle, which is the fraction, [0,1], of cytosolic NADH (reducing equivalents) shuttled into the mitochondrial matrix. In the model, the fraction of cytosolic NADH (reducing equivalents) not shuttled into the matrix (1 − f_shuttle) is oxidized by the non-proton-pumping rotenone-insensitive NADH dehydrogenase ND_ex on the intermembrane-space side of the inner mitochondrial membrane. The plant model default is f_shuttle = 0.

Fig. 4.

Mammalian malate–aspartate shuttle translocating electrons (reducing equivalents) from NADH on the P (cytosolic or intermembrane space) side of the inner mitochondrial membrane (IMM) to NAD⁺ on the N (mitochondrial matrix) side of the membrane (Borst, 2020). The shuttle involves: (1) reduction of oxaloacetate (OAA) by NADH in the cytosol (or intermembrane space), producing malate and NAD⁺; (2) malate entry into the mitochondrial matrix in exchange for 2-oxoglutarate (2-OG; old name: α-ketoglutarate) via the oxoglutarate carrier (OGC); (3) reduction of NAD⁺ by malate in the mitochondrial matrix, producing NADH and OAA; (4) conversion of OAA and glutamate to 2-OG and aspartate in the mitochondrial matrix; (5) exchange of aspartate in the mitochondrial matrix for cytosolic (intermembrane space) glutamate via the aspartate–glutamate carrier (AGC); and (6) cytosolic (or intermembrane space) conversion of aspartate and 2-OG to glutamate and OAA. Mitochondrial uptake of glutamate through the AGC is accompanied by a proton dissipating some of the proton concentration gradient, ΔpH, and electrical potential difference, ΔΨ_m, across the inner mitochondrial membrane.

In contrast, a malate–oxaloacetate (OAA) shuttle moving NADH reducing power out of the plant mitochondrial matrix may be important under non-model conditions including (1) during photorespiration in photosynthesizing cells when NAD⁺ in the mitochondrial matrix is extensively reduced by the photorespiratory conversion of glycine to serine, (2) if/when extra-mitochondrial NADH demand is high or (3) if/when the mitochondrial NADH level is excessive for any reason (Selinski and Scheibe, 2019; Møller et al., 2021). This outward NADH shuttle is not included in the model.

Integrating the modelled components of proton gradient generation

The number of modelled protons transported outward from the N side to the P side of the inner mitochondrial membrane (H⁺_out, protons per hexose respired) is given by:

$${\displaystyle \begin{array}{l}{\displaystyle {\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{+}=(1-f_{slip})[f_{\mathrm{I}}{\mathrm{H}}_{\mathrm{I}}^{+}(N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}+f_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}})+f_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}(N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}+N_{\mathrm{S}\mathrm{u}\mathrm{c}}+N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}})]+(1-f_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}){\mathrm{H}}_{\mathrm{N}\mathrm{D}\mathrm{e}\mathrm{x}}^{+}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}}\end{array}}$$ (8)

where f_I is the fraction, [0,1], of NADH in the mitochondrial matrix that is oxidized by complex I, with the remainder oxidized by the non-proton-pumping rotenone-insensitive alternative matrix-facing NADH dehydrogenase (ND_in); and f_III-IV is the fraction, [0,1], of QH₂ oxidized via the proton-pumping cytochrome pathway, with the remainder oxidized by the alternative oxidase.

The quantity (N_NADHm + f_shuttleN_NADHc) is NADH produced per hexose respired that could be oxidized by proton-pumping complex I. In the absence of an NADH shuttle (i.e. f_shuttle = 0), which is the model default, it ranges from 8, for f_Pyr = 1, to 10, for f_Pyr = 0, and is unaffected by the value of f_starch. With all net glycolytically produced NADH shuttled into the mitochondrial matrix (f_shuttle = 1), which is not expected in plants, it equals 10 irrespective of the value of f_Pyr. The product (1 − f_shuttle) N_NADHc is the amount of NADH per hexose respired that is produced by glycolysis in the cytosol that remains in the cytosol and that is oxidized by the rotenone-insensitive external NADH dehydrogenase ND_ex. The sum of these quantities – i.e. $(N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}+f_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}})+(1-f_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}})N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}$ – is the same as the sum N_NADHc + N_NADHm, which equals 10 NADH/hexose in all possible model scenarios.

The sum (N_NADHc + N_Suc + N_NADHm) is total QH₂ production in the inner mitochondrial membrane per hexose respired, excluding that produced in the complex III Q-cycle. This complies with the model constraints that all NADH produced by the respiratory carbon metabolism pathways is oxidized by a dehydrogenase that reduces Q in the inner mitochondrial membrane and that all succinate is oxidized to fumarate in the TCA cycle. Under model conditions that sum is 12 QH₂/hexose for respiration of both starch and sucrose with glycolysis producing pyruvate or malate.

PROTON GRADIENT DISSIPATION AND ATP SYNTHASE ACTIVITY

Protons translocated from the N side to the P side of the inner mitochondrial membrane by the respiratory chain return to the N side (re-enter the mitochondrial matrix) by several mechanisms or routes. The model includes five. Most notable for respiratory ATP production is proton re-entry through the membrane-embedded F_o sector of the mitochondrial F_oF₁-ATP synthase. It is worth noting that the chemical and electrical aspects of protons re-entering the mitochondrial matrix do not always follow one another. A proton re-entering the mitochondrial matrix in symport with an anion having one negative charge affects ΔpH but does so electroneutrally without net charge transfer across the inner membrane. Examples in the model are H⁺/P_i⁻ and H⁺/pyruvate⁻ symports. Conversely, a proton re-entering the matrix passively through a membrane leak or uniport process reduces ΔpH while also transferring charge across the membrane and influencing the transmembrane potential.

Proton re-entry into the mitochondrial matrix via the F_o sector of the ATP synthase and associated stoichiometry with F₁ sector-catalysed ADP phosphorylation

Multicomponent F-type ATP synthases occur in association with inner mitochondrial membranes, bacterial plasma membranes and chloroplast thylakoid membranes. Mitochondrial H⁺-transporting F_oF₁-ATP synthases use the respiratory chain-produced electrochemical proton gradient to generate ATP from ADP and P_i in oxidative phosphorylations in the mitochondrial matrix (Walker, 2013; Zancani et al., 2020). These mitochondrial ATP synthases – sometimes called complex V by incrementing the notation of respiratory chain complexes I, II, III and IV – are dimeric complexes forming rows along mitochondrial cristae ridges. The F_o sector is an integral membrane proton channel that transports protons down their concentration gradient from the P side to the N side of the inner mitochondrial membrane through a rotating ring (motor) of multiple c subunits, or c-ring. The attached water-soluble F₁ sector, which also contains a rotary motor, is in the mitochondrial matrix where it catalyses the production of ATP from P_i and ADP (Fig. 1). One complete (360°) rotation of the F_oc-ring drives one complete rotation of the F₁ motor that generates and releases three ATP molecules (Boyer, 1997; Walker, 2013). Because a complete F_o sector c-ring rotation apparently requires transport of one proton through each c subunit, the theoretical F-type ATP synthase intrinsic H⁺/ATP ratio is n/3, where n is the number of c subunits in the c-ring, denoted c_n. Thus, n is inversely related to the amount of ATP that potentially can be produced by an F-type ATP synthase transporting a given number of protons through its F_o sector. The model represents this intrinsic H⁺/ATP ratio (= n/3) with the variable named H⁺_FoF1, which is the number of protons transported through the ATP synthase F_o sector per ATP produced by the F₁ sector in the mitochondrial matrix.

It is noteworthy that n varies across organelles, species and kingdoms. At the time of writing, reported n values determined for H⁺-transporting F-type ATP synthases ranged from eight to 17, indicating intrinsic H⁺/ATP ratios from 8/3 (about 2.67) to 17/3 (about 5.67), but I am unaware of any plant mitochondrial ATP synthase F_o sector c-ring size determination, which is an identifiable unknown in the model’s quantitative treatment of plant respiratory ATP yield. In mammalian mitochondria from hearts of Bos taurus and Sus scrofa domesticus c₈-rings were observed (Watt et al., 2010; Gu et al., 2019), but to my knowledge the size (i.e. number of c subunits) of all c rings determined to date in non-mammalian mitochondria is 10, and all of those are from unicellular organisms including the yeasts (single-celled fungi) Saccharomyces cerevisiae (Stock et al., 1999), Yarrowia lipolytica (Hahn et al., 2016) and Pichia angusta (Vinothkumar et al., 2016) and the protists Polytomella sp. (Allegretti et al., 2015), Euglena gracilis (Mühleip et al., 2019), Tetrahymena thermophila (Kock Flygaard et al., 2020) and Toxoplasma gondii (Mühleip et al., 2021). Reported non-mitochondrial ATP synthase c ring sizes include bacterial plasma-membrane ATP synthases with c₉-rings in Mycobacterium phlei and M. smegmatis (Preiss et al., 2015; Guo et al., 2021), c₁₀-rings in a thermophilic Bacillus and Escherichia coli (Mitome et al., 2004; Sobti et al., 2016), c₁₂-rings in nitrate-reducing Paracoccus denitrificans (Morales-Rios et al., 2015), c₁₃-rings in another Bacillus sp. (Matthies et al., 2009) and c₁₇-rings in Burkholderia pseudomallei (Schulz et al., 2017), whereas the thylakoid-membrane ATP synthase in the cyanobacterium Spirulina platensis has c₁₅-rings (Pogoryelov et al., 2005). In addition, Na⁺-transporting F_oF₁-ATP synthases in the anaerobic bacteria Ilyobacter tartaricus, Clostridium paradoxum and Fusobacterium nucleatum have c₁₁-rings (Stahlberg et al., 2001; Meier et al., 2006; Schulz et al., 2013).

The lack of c-ring size data from any plant mitochondrial ATP synthase requires that an assumption be made about the value of n used in the model. The consistent observation of c₁₀-rings in all non-mammalian mitochondria – albeit all from single-celled organisms – may make n = 10 the best present conjecture for plant mitochondrial ATP synthases, and that is the model default, but with acknowledged uncertainty. The importance of this uncertainty is studied by varying n in the model. In contrast to the hypothetical plant mitochondrial ATP synthase F_o sector c₁₀-ring, plant chloroplast ATP synthases from Spinacia oleracea and Pisum sativum have measured c₁₄ rings (Seelert et al., 2000; Saroussi et al., 2012).

It may be important to allow for modelled slippage in the F_oF₁-ATP synthase that partially uncouples proton transport from ADP phosphorylation. This could be uncoupling in the linked rotations of the F_o and F₁ sectors and/or uncoupling of proton transport through F_o and c-ring rotation. The model variable f_slip(FoF1) represents such uncoupling over the range [0,1), with 0 indicating full coupling and 1 indicating full uncoupling, a situation deemed impossible in the model for healthy cells. The ratio $\left({\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}/[1-f_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}]\right)$ is then the effective H⁺/ATP ratio of the ATP synthase. In experiments with a c₁₀ F_oF₁-ATP synthase from a thermophilic Bacillus bacterium the derived H⁺/ATP ratio was 3.3 (±0.1), an excellent match to the ratio n/3 (i.e. 10/3), indicating ‘chemomechanical coupling between F_o and F₁ is perfect’ (Soga et al., 2017). While this result was from a plasma-membrane rather than mitochondrial (plant or otherwise) ATP synthase, it indicates a small value for f_slip(FoF1). The model default is f_slip(FoF1) = 0, but this is a data-poor assumption.

Proton re-entry into the mitochondrial matrix during phosphate symport

Modelled oxidative and substrate-level phosphorylations of ADP in the mitochondrial matrix require a stoichiometric import of one each of extramitochondrial P_i and ADP for each ATP formed and exported to the cytosol (Figs 1 and 3). The P_i required for ADP phosphorylation in the mitochondrial matrix is transported into the mitochondrion via the P_i carrier in electroneutral symport with a proton, thus contributing to ΔpH dissipation. The model sets the H⁺/P_i symport ratio, symbolized as parameter H⁺_Pi, to unity (Table 1). This is a well-established value. As noted above, extramitochondrial ADP³⁻ enters the mitochondrial matrix in exchange for exiting ATP⁴⁻ via the adenine nucleotide translocator, dissipating ΔΨ_m but not ΔpH (Figs 1 and 3). As with the TCA cycle’s substrate-level ADP phosphorylation, modelled steady-state mitochondrial ATP production and export in oxidative phosphorylations occurs with a 1:1:1:1 ratio of ATP export, ADP import, P_i import and proton import.

Because mitochondrial ADP phosphorylation catalysed by the ATP synthase is stoichiometrically linked to H⁺/P_i symport into the matrix, the sum H⁺_FoF1 + H⁺_Pi is the effective (operational) proton requirement (H⁺/ATP) for mitochondrial ATP synthase activity in the absence of slippage. That effective H⁺/ATP ratio equals n/3 + 1, or (n + 3)/3, when H⁺_Pi = 1. Accounting for possible slippage in the ATP synthase, the effective H⁺/ATP ratio is the quantity H⁺_Pi + H⁺_FoF1/[1 − f_slip(FoF1)].

The second modelled use of P_i imported into a mitochondrion in symport with a proton is linked to malate import described above. In that case, P_i imported into the matrix is subsequently used in antiport through a dicarboxylate carrier in exchange for cytosolic (intermembrane space) malate (Fig. 3 upper left). This occurs when modelled f_Pyr < 1.

Proton re-entry into the mitochondrial matrix during pyruvate symport

Pyruvate transport from the cytosol to the mitochondrial matrix is a link between modelled glycolysis and the TCA cycle when f_Pyr > 0, which is assumed to usually be the case. Data from rat liver and yeast mitochondria indicated that pyruvate uptake occurs by proton symport (Papa and Paradies, 1974; Halestrap, 1975; Tavoulari et al., 2019). It was suggested, however, that protons accompanying pyruvate do not dissipate the respiratory chain-generated proton gradient ΔpH, but instead protons used in symport are balanced (i.e. supplied) by mitochondrial CO₂ efflux producing bicarbonate ion plus a proton $(\mathrm{C}{\mathrm{O}}_2+{\mathrm{H}}_2\mathrm{O}\to {\mathrm{H}\mathrm{C}\mathrm{O}}_3^{-}+{\mathrm{H}}^{+})$ in the intermembrane space (LaNoue and Schoolwerth, 1979; Hinkle, 2005; Fig. 3 upper right). A different view is expressed in the default steady-state model configuration: CO₂ hydration on the P side of the inner mitochondrial membrane does not provide the proton for pyruvate symport, HCO₃⁻ does not continue to accumulate in the intermembrane space and pyruvate uptake does contribute to dissipation of the respiratory proton gradient ΔpH. The model variable H⁺_Pyr is the number of protons derived from the respiratory chain-generated proton gradient that accompany pyruvate into the matrix, with a default value of 1 H⁺/pyruvate⁻ (Table 1).

Whether CO₂ chemistry in the mitochondrial intermembrane space contributes to in vivo proton supply for H⁺/pyruvate⁻ symport apparently is undetermined for plants. Complex I in plants, but not animals and fungi, has a spherical domain within the mitochondrial matrix that contains carbonic anhydrase subunits that may be involved in complex I assembly (Fromm et al., 2016; Meyer et al., 2022), and it was suggested that complex I’s carbonic anhydrase might catalyse hydration of respired CO₂ in the mitochondrial matrix as part of a mechanism for efficiently transporting CO₂ (as HCO₃⁻) from mitochondria to chloroplasts during photosynthesis (Braun, 2020) – which is outside modelled conditions. While there is indirect evidence of carbonic anhydrase activity within the matrix (Klusch et al., 2023; Møller, 2023), whether this facilitates CO₂ export from mitochondria to chloroplasts (which would be physiologically relevant only in illuminated photosynthetic cells rather than during heterotrophic metabolism) and/or whether it might have a bearing on the process of pyruvate import are yet to be established.

Proton re-entry into the mitochondrial matrix via inner mitochondrial membrane leaks or uncoupling proteins

The phospholipid bilayer fraction of the inner mitochondrial membrane is practically impermeable to protons. Moreover, a principle underlying the chemiosmotic mechanism of oxidative phosphorylation is that the entire membrane system of lipids, proteins and any other components is impervious to free diffusion of protons and other ions. If, however, the inner mitochondrial membrane has non-zero passive conductance to protons (leak), some of the protons pumped by the respiratory chain from the N side to the P side of the membrane will leak back to the N side, dissipating part of the electrochemical proton gradient in a futile proton cycle producing heat without contributing to transmembrane metabolite transport or ATP production. The model variable f_leak represents the fraction of protons pumped from the N side to the P side of the membrane that leak back to the N side, with hypothetically possible values over the range [0,1] but which will not reach 1 in healthy non-thermogenic mitochondria under respiratory conditions.

Plants and other eukaryotes have uncoupling proteins in their inner mitochondrial membranes (UCP in Fig. 1) that (may) facilitate transport of protons from the P side to the N side of the membrane in a type of regulated (protein mediated) proton leak (Woyda-Ploszczyca and Jarmuszkiewicz, 2017; Barreto et al., 2020). Such leaks are allowed in the model (but see below) and subsumed in the variable f_leak. In the absence of data for significant endogenous proton leak or UCP-mediated proton transport through a plant inner mitochondrial membrane within the model's physiological setting (see below for a non-model-setting plant respiratory UCP case), and for the purpose of calculating potential ATP yield, the model default is f_leak = 0 (Table 1).

Proton re-entry into the mitochondrial matrix during NADH shuttle-related glutamate import

The model can accommodate a malate–aspartate shuttle transferring reducing equivalents from cytosolic NADH to NAD⁺ in the mitochondrial matrix (see above; recall that this shuttle is not thought to be used in plants to support respiration under modelled conditions). The shuttle involves (Glu²⁻ + H⁺)_inward/Asp²⁻_outward antiport (Fig. 4), dissipating some of the proton gradient in models of mammalian respiration (Hinkle et al., 1991; Brand, 2005). The present model’s parameter H⁺_Glu quantifies the proton requirement (H⁺/NADH) associated with the malate–aspartate shuttle. Its value is fixed at H⁺_Glu = 1 (Table 1), meaning one proton is transported from the P side to the N side of the inner mitochondrial membrane for each NADH reducing equivalent shuttled into a mitochondrion during (mammalian) respiration.

Integrating the modelled components of proton gradient (ΔpH) dissipation

During modelled steady-state respiration, total inward proton flux from the P side to the N side of the inner mitochondrial membrane (H⁺_in, protons per hexose respired) equals H⁺_out. The modelled value of H⁺_in also can be obtained by summing the model’s five ΔpH dissipation components, which is required to calculate ATP production by the F_oF₁-ATP synthase, as follows:

$${\displaystyle \begin{array}{lll}& {\mathrm{H}}_{\mathrm{i}\mathrm{n}}^{+}=({\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+\frac{{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}{[1-f_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}]})N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}+{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}(N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}+N_{\mathrm{M}\mathrm{a}\mathrm{l}})+& {\mathrm{H}}_{\mathrm{P}\mathrm{y}\mathrm{r}}^{+}{N}_{\mathrm{P}\mathrm{y}\mathrm{r}}+f_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}{\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{+}+{\mathrm{H}}_{\mathrm{G}\mathrm{l}\mathrm{u}}^{+}{f}_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}\end{array}}$$ (9)

Under the steady-state constraint H⁺_in = H⁺_out, and after rearrangement of eqn (9) with respect to N_ATP(FoF1), the number of ATP produced by the ATP synthase per hexose respired is:

$${\displaystyle \begin{array}{lll}& N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}=& \frac{({\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{+}-[{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}(N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}+N_{\mathrm{M}\mathrm{a}\mathrm{l}})+{\mathrm{H}}_{\mathrm{P}\mathrm{y}\mathrm{r}}^{+}{N}_{\mathrm{P}\mathrm{y}\mathrm{r}}+f_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}{\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{+}+{\mathrm{H}}_{\mathrm{G}\mathrm{l}\mathrm{u}}^{+}{f}_{\mathrm{s}\mathrm{h}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}])}{({\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+\frac{{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}{[1-f_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}\left(\mathrm{F}\mathrm{o}\mathrm{F}1\right)}]})}\end{array}}$$ (10)

where the numerator is the number of protons pumped by the respiratory chain to the P side of the inner mitochondrial membrane that remain there after all modelled uses (including leaks) of those protons are accounted for except (1) protons moving back to the N side of the membrane through the ATP synthase F_o sector and (2) protons re-entering the matrix in symport with P_i that is stoichiometrically linked to ATP production by the ATP synthase F₁ sector. Those remaining modelled protons are all used to drive the ATP synthase F_o sector motor and to transport P_i into the mitochondrial matrix (via H⁺/P_i symport) as substrate to the ATP synthase F₁ sector’s ADP phosphorylation reactions. For N_ATP(FoF1) to be positive, which will be the case for model conditions, f_leak must be less than the quantity 1 − [H⁺_Pi(N_ATP(TCA) + N_Mal) + H⁺_PyrN_Pyr + H⁺_Gluf_shuttleN_NADHc]/H⁺_out.

P/O RATIO

For each hexose unit in substrate completely respired in the model setting, 12 oxygen atoms (as six O₂ molecules) are reduced by complex IV and/or alternative oxidase. The ratio of net ATP formed to oxygen atoms reduced is the P/O ratio (or ATP/O ratio or ADP/O ratio). In this model, it is based on all (net) ATP formation in both substrate-level and oxidative phosphorylations, with all that ATP supplied to the cytosol, and is equal to N_ATP/12. This is equivalent to N_ATP/QH₂ where QH₂ is total QH₂ formed per hexose respired, excluding QH₂ formed in the complex III Q-cycle (footnotes to Table 2).

Table 2.

Modelled plant respiratory proton gradient (ΔpH) generation (H⁺_out, protons per hexose respired), proton gradient dissipation (H⁺_in, protons/hexose) and net total supply of ATP to the cytosol (N_ATP, ATP/hexose) during respiration of sucrose or starch for multiple model input variable value combinations

Scenario*	f starch	f Pyr	H+Pyr	f I	f III-IV	H+NDex	f leak	n	H+FoF1	H+out (= H+in)	N ATP(glycolysis)	N ATP(TCA)	N ATP(FoF1)	N ATP	P/O
A.1	0	1	1	1	1	0	0	10	3.33	104	2.0	2	23.1	27.1	2.26
A.2	1	1	1	1	1	0	0	10	3.33	104	2.5	2	23.1	27.6	2.30
A.3	0	0	NA†	1	1	NA†	0	10	3.33	112	0.0	2	24.9	26.9	2.24
A.4	1	0	NA	1	1	NA	0	10	3.33	112	0.5	2	24.9	27.4	2.29
A.5	0	1	0	1	1	0	0	10	3.33	104	2.0	2	23.5	27.5	2.29
A.6	1	1	0	1	1	0	0	10	3.33	104	2.5	2	23.5	28.0	2.34
A.7	0	1	1	1	1	1	0	10	3.33	106	2.0	2	23.5	27.5	2.29
A.8	0	1	0	1	1	1	0	10	3.33	106	2.0	2	24.0	28.0	2.33
A.9	0	1	1	1	1	0	0	9	3.00	104	2.0	2	25.0	29.0	2.42
A.10	0	0	NA	1	1	NA	0	9	3.00	112	0.0	2	27.0	29.0	2.42
A.11	0	1	0	1	1	1	0	9	3.00	106	2.0	2	26.0	30.0	2.50
A.12	0	1	1	1	1	0	0	8	2.67	104	2.0	2	27.3	31.3	2.61
A.13	0	1	0	1	1	1	0	8	2.67	106	2.0	2	28.4	32.4	2.70
A.14	0	1	1	1	1	0	0	12	4.00	104	2.0	2	20.0	24.0	2.00
A.15	0	1	1	1	0.75	0	0	10	3.33	86	2.0	2	18.9	22.9	1.91
A.16	0	1	1	1	0.50	0	0	10	3.33	68	2.0	2	14.8	18.8	1.56
A.17	0	1	1	1	0	0	0	10	3.33	32	2.0	2	6.5	10.5	0.87
A.18	0	1	1	0	1	0	0	10	3.33	72	2.0	2	15.7	19.7	1.64
A.19	0	1	1	0.75	0.75	0	0	10	3.33	78	2.0	2	17.1	21.1	1.76
A.20	0	1	1	1	1	0	0.25	10	3.33	104	2.0	2	17.1	21.1	1.76

Columns headed from ‘f_starch’ to ‘n’ are model input variables (defined in Table 1). Model parameter values of H⁺_I = 4, H⁺_III-IV = 6 and H⁺_Pi = 1 were used in all scenarios as were the default variable values (see Table 1) f_slip = 0, f_slip(FoF1) = 0 and f_shuttle = 0, in which case the value of H⁺_Glu is not applicable. H⁺_FoF1 is the modelled intrinsic H⁺/ATP ratio of the mitochondrial F_oF₁-ATP synthase given by n/3. H⁺_out is the number of protons pumped by the respiratory chain from the N side to the P side of the inner mitochondrial membrane per hexose respired. H⁺_in (= H⁺_out and sharing a single column with H⁺_out above) is the number of protons translocated ‘back’ through any of several routes from the P side to the N side of the inner mitochondrial membrane per hexose respired. During modelled steady-state respiration H⁺_in = H⁺_out. N_{ATP(glycolysis)} is the net number of ATP produced in the cytosol by glycolysis per hexose respired. N_ATP(TCA) is the number of ATP produced in substrate-level phosphorylations in the mitochondrial TCA cycle per hexose respired (= 2 in all scenarios), with all of them transported to the cytosol. N_ATP(FoF1) is the number of ATP produced by the mitochondrial F_oF₁-ATP synthase per hexose respired, with all of them transported to the cytosol. For all scenarios six CO₂ are released and 12 QH₂ are produced per hexose respired (excluding QH₂ formed in the Q-cycle around complex III). All 12 QH₂ are oxidized by complex III (in combination with complex IV activity) and/or the alternative oxidase, producing 12 H₂O from 12 O atoms (six O₂ molecules). P/O is the net total number of ATP molecules produced per oxygen atom (not O₂ molecule) reduced by complex IV and/or the alternative oxidase and is given by N_ATP/QH₂ (= N_ATP/12) for respiration of starch or sucrose.

^*Scenarios: (A.1) model baseline scenario using default values for all variables; (A.2) same as A.1 but with f_starch = 1 (respiration of starch); (A.3) same as A.1 but with f_Pyr = 0 (glycolysis producing malate instead of pyruvate); (A.4) same as A.1 but with both f_starch = 1 and f_Pyr = 0; (A.5) same as A.1 but with H⁺_Pyr = 0 (pyruvate uptake without any dissipation of ΔpH generated by the respiratory chain); (A.6) same as A.1 but with f_starch = 1 and H⁺_Pyr = 0; (A.7) same as A.1 but with H⁺_NDex = 1, representing a contribution of the external rotenone-insensitive NADH dehydrogenase to generation of the electrochemical proton gradient; (A.8) same as A.7 but with H⁺_Pyr = 0, producing the maximum modelled N_ATP for respiration of sucrose with n = 10; (A.9) same as A.1 but with n = 9, producing H⁺_FoF1 = 3 H⁺/ATP, a value consistent with much experimental data from isolated plant mitochondria; (A.10) same as A.3 but with n = 9; (A.11) same as A.8 but with n = 9, producing the maximum modelled N_ATP for respiration of sucrose with n = 9; (A.12) same as A.1 but with n = 8 (H⁺_FoF1 = 2.67 H⁺/ATP), the smallest F_o sector c-ring size measured to date in any organism; (A.13) same as A.12 but with H⁺_Pyr = 0 and H⁺_NDex = 1, producing the maximum modelled N_ATP for respiration of sucrose with n = 8; (A14) same as A.1 but with n = 12 (H⁺_FoF1 = 4 H⁺/ATP); (A.15) same as A.1 but with f_III-IV = 0.75, simulating oxidation by the alternative oxidase of 25 % of the QH₂ produced by the respiratory chain to approximate the median of literature values in Table A1; (A.16) same as A.15 but with f_III-IV = 0.50 (half of QH₂ oxidized by the alternative oxidase); (A.17) same as A.15 but with f_III-IV = 0 (no cytochrome pathway activity); (A.18) same as A.1 but with f_I = 0 (complete bypass of complex I); (A.19) same as A.1 but with both f_I = 0.75 and f_III-IV = 0.75, simulating oxidation of 25 % of NADH in the mitochondrial matrix by the rotenone-insensitive alternative matrix-facing (internal) NADH dehydrogenase (i.e. bypassing complex I) in combination with oxidation by the alternative oxidase of 25 % of the QH₂ produced by the respiratory chain, resulting in a 25 % reduction in proton gradient generation (H⁺_out); (A.20) same as A.1 but with f_leak = 0.25.

^†NA – not applicable. When f_Pyr = 0 only malate is formed in glycolysis and values of H⁺_Pyr and H⁺_NDex do not enter into calculations of ATP yield. The R script used to implement the model (Supplementary Data) does not allow ‘NA’ entries so values of 0 or 1 are used for H⁺_Pyr and H⁺_NDex in the R script in such cases, with no effect on results.

Alternatively, P/O often is calculated for oxidative phosphorylation alone or for ATP production by isolated (in vitro) mitochondria respiring individual substrates such as internal (matrix) NADH, external NADH or succinate (Taiz and Zeiger, 1991; Hinkle, 2005). P/O ratios for mitochondria respiring specific substrates (e.g. mitochondria oxidizing succinate alone) derived from submodels of the plant model are presented in Appendix 1.

POTENTIAL ATP YIELD FROM RESPIRATION OF SUCROSE OR STARCH

The modelled baseline or default scenario for potential (maximal) ATP yield of plant respiration, N_ATP, in healthy, mature, non-photosynthesizing and non-thermogenic cells – with all the ATP produced in, or transported to, the cytosol – was constructed with the intent to reflect a classical view of glycolysis coupled to the TCA cycle and a respiratory chain without bypasses of energy-conserving reactions while expressing an intrinsic F_oF₁-ATP synthase H⁺/ATP ratio based on F_o sector c₁₀ rings. Specifically, the baseline scenario (Table 2 scenario A.1): (1) metabolized sucrose to pyruvate in cytosolic glycolysis (f_starch = 0, f_Pyr = 1); (2) transported the pyruvate into a mitochondrion in symport with a proton from the electrochemical proton gradient (H⁺_Pyr = 1) where it was decarboxylated, forming acetyl-CoA that was oxidized in the TCA cycle; (3) oxidized all cytosolic NADH produced by glycolysis with the non-proton-pumping rotenone-insensitive external mitochondrial NADH dehydrogenase ND_ex (f_shuttle = 0); (4) oxidized all mitochondrial matrix NADH with complex I (f_I = 1); (5) transferred all electrons from QH₂ to oxygen via the cytochrome pathway (f_III-IV = 1); (6) evaded all slips and leaks (f_slip = 0, f_slip(FoF1) = 0, f_leak = 0); and (7) used a c₁₀-ring in the ATP synthase F_o sector (n = 10; H⁺/ATP = 10/3). This baseline scenario for respiration of sucrose yielded a net 27.1 ATP/hexose unit respired, with 85 % of the net ATP generated by the F_oF₁-ATP synthase (calculated prior to rounding digits in Table 2). Using starch rather than sucrose as substrate increased N_ATP by 0.5 ATP/hexose [see eqn (4a)] to 27.6 ATP/hexose (Table 2 scenario A.2), with 84 % of the net ATP generated by the F_oF₁-ATP synthase. Both scenarios A.1 and A.2 – the baseline scenarios for respiration of sucrose and starch, respectively – resulted in H⁺_out = H⁺_in = 104 H⁺/hexose.

With glycolysis producing malate instead of pyruvate (f_Pyr = 0) and with that malate imported into mitochondria and decarboxylated to form pyruvate, but with all other variables at default values, H⁺_out (= H⁺_in) increased 7.7 % to 112 H⁺/hexose for respiration of sucrose or starch (Table 2 scenarios A.3 and A.4, respectively). Despite the increased H⁺_out and a resulting 8 % increase in N_ATP(FoF1), baseline N_ATP declined 0.154 ATP/hexose for respiration of sucrose or starch. That reduction in N_ATP was due to reduced net glycolytic ATP formation, N_{ATP(glycolysis)}, of 2 ATP/hexose [compare eqn (2a) with (2b) and eqn (3a) with (3b)]. For scenario A.3, almost 93 % of net ATP production was from the ATP synthase, compared to 85 % for scenario A.1.

Scenarios A.1 and A.2 dissipate some of the respiratory chain-generated proton gradient ΔpH during pyruvate transport (H⁺/pyruvate⁻ symport) into mitochondria, but it is common to ignore this (possible) use of the proton gradient in calculating ATP yields of respiration (e.g. Hinkle et al., 1991; Brand, 2005; Hinkle, 2005). When eliminating that modelled proton gradient dissipation component by setting H⁺_Pyr = 0, N_ATP was 27.5 and 28.0 ATP/hexose for respiration of sucrose and starch (Table 2 scenarios A.5 and A.6, respectively), a 1.7 % increase relative to the respective baseline scenarios yielding 27.1 and 27.6 ATP/hexose.

The baseline scenario A.1 treats complex II, ND_in and ND_ex (in combination with H⁺_NDex = 0) equivalently with respect to proton gradient generation and ATP production. For scenario A.1 modified with H⁺_NDex = 1, H⁺_out increased 1.9 % to 106 H⁺/hexose and ATP yield increased 1.7 % to 27.5 ATP/hexose (Table 2 scenario A.7). With respect to ATP production that result is the same as scenario A.5, where scenario A.5 uses two fewer protons than scenario A.1 by assuming that the H⁺/pyruvate⁻ symport does not affect the electrochemical proton gradient whereas scenario A.7 increases H⁺_out by two protons due to H⁺_NDex equalling one rather than zero. Based on the mechanisms represented, the model scenario A.7 (27.5 ATP/hexose, or about 0.5 ATP/hexose greater than the baseline scenario) might be taken as the more complete (definitive) estimate of potential ATP yield from plant respiration of sucrose with c₁₀-rings in the mitochondrial ATP synthase.

Across the four scenarios A.1, A.3, A.5 and A.7, which are all generally favourable for ATP production by respiration of sucrose associated with F_o sector c₁₀-rings, ATP yield covers the range 26.9–27.5 ATP/hexose. (Modelled ATP yield from starch is always 0.5 ATP/hexose greater than from sucrose.) All these N_ATP values fall below the historical 29.5–38 ATP/glucose range of values cited in the Introduction. Model P/O ratios for scenarios A.1, A.3, A.5 and A.7 cover the range 2.24–2.29 (Table 2) compared to P/O = 3 for hypothetical respiration generating 36 ATP/hexose or P/O = 2.5 for respiration yielding 30 ATP/hexose.

‘Cost’ of exporting TCA cycle-generated ATP from the mitochondrial matrix to cytosol

Modelled N_ATP(TCA) is 2 ATP/hexose for all scenarios. Based on the 1:1:1:1 stoichiometry above for ATP⁴⁻ export with import of ADP³⁻, P_i⁻ and H⁺, two protons (plus some of the ΔΨ_m) are dissipated from the electrochemical proton gradient for export of those two TCA cycle-generated ATP. Two protons are about 1.8–1.9 % of the H⁺_out range of 104–112 H⁺/hexose in scenarios A.1–A.7 (Table 2). With model default values of H⁺_FoF1 = 10/3, f_slip(FoF1) = 0 and H⁺_Pi = 1, those two protons could produce 2/(13/3), or about 0.46, ‘additional’ ATP/hexose in oxidative phosphorylations. With H⁺_FoF1 = 3 rather than 10/3 (see below) the potential ‘loss’ to oxidative phosphorylation of the two protons associated with exporting the two ATP from the TCA cycle in the mitochondrial matrix to the cytosol increases to 0.5 ATP/hexose. In either case, substrate-level ADP phosphorylations in the TCA cycle make a net positive contribution to N_ATP of at least 1.5 ATP/hexose. This is not an assertion that the TCA cycle generates only 1.5 (or 1.54) ATP/hexose that can be supplied to the cytosol. Two ATP are produced and exported intact, but those substrate-level phosphorylations are linked to proton gradient dissipation and as a result N_ATP(FoF1) is affected.

The mitochondrial ATP synthase c-ring size is an important unknown relative to potential ATP yield of plant respiration

The model default assumptions that the plant mitochondrial ATP synthase F_o sector c-ring contains 10 c subunits and that the c-ring size, along with the molecular structure of F₁, determines the intrinsic H⁺/ATP ratio of the ATP synthase are of consequence. For example, reducing the c-ring size from 10 to nine c subunits increases modelled N_ATP for respiration of sucrose 7.1 %, from 27.1 in the baseline scenario A.1 to 29.0 ATP/hexose in scenario A.9 (Table 2) matching Amthor (2000). A c₉-ring implies an intrinsic ATP synthase H⁺/ATP ratio of 3, instead of 10/3, and that value of 3 is consistent with many in vitro measurements of ATP production by isolated mitochondria and was the basis of numerous previous calculations of potential respiratory ATP yield (e.g. Hinkle et al., 1991; Taiz and Zeiger, 1991; Amthor, 2000). Moreover, H⁺/ATP = 3 was recently considered accurate for plants [e.g. Møller et al., 2021 (p. 936): ‘one ATP is formed for every three H⁺ that flow through the [ATP synthase]’] and other organisms [e.g. Nelson and Cox, 2021 (p. 682): ‘The most widely accepted experimental value for number of protons required to drive the synthesis of an ATP molecule is 4, of which 1 is used in transporting P_i, ATP and ADP across the mitochondrial membrane’—giving $4-1=3{\mathrm{H}}^{+}/\mathrm{A}\mathrm{T}\mathrm{P}$ for the ATP synthase reaction itself].

As noted above, with n = 10 (and H⁺_Pyr = 1 and H⁺_NDex = 0), N_ATP is slightly reduced when glycolysis produces malate (scenario A.3, f_Pyr = 0) rather than pyruvate (scenario A.1, f_Pyr = 1), but with n = 9 (leading to H⁺_FoF1 = 3) and other variables unchanged, N_ATP is 29.0 ATP/hexose for all values of f_Pyr (compare scenarios A.9 and A.10 representing the endpoints of f_Pyr across the interval [0,1]). This perhaps unintuitive outcome is due to the greater ‘metabolic value’ of each proton moving through the ATP synthase with smaller n, in combination with the greater H⁺_out when glycolysis produces malate. ATP yield with n = 9 would still be greater for glycolysis producing pyruvate relative to malate if either pyruvate uptake by mitochondria does not dissipate ΔpH (i.e. H⁺_Pyr = 0) and/or the external rotenone-insensitive NADH dehydrogenase contributes to generation of the electrochemical proton gradient (i.e. H⁺_NDex > 0). In the case of f_Pyr = 1, H⁺_Pyr = 0 and H⁺_NDex = 1, N_ATP increases to 30.0 ATP/hexose for respiration of sucrose if H⁺_FoF1 = 3 (Table 2, scenario A.11). Compared to the about 2 ATP/hexose increase (7 %) in modelled potential ATP yield with n = 9 rather than 10, changes in the variables f_starch, H⁺_Pyr, f_Pyr and H⁺_NDex individually have only modest effects on ATP yield (Fig. 5 left-side bars) and in the aggregate represent ± 2 % uncertainty in the magnitude of potential ATP yield of plant respiration for a given value of n under the constraint that n determines H⁺_FoF1.

Fig. 5.

Modelled baseline scenario for ATP yield of plant respiration of sucrose with F_oF₁-ATP synthase F_o sector c₁₀-rings (solid horizontal line = 27.1 ATP/hexose; Table 2 scenario A.1) relative to the obsolete textbook 38 ATP/hexose (upper dashed horizontal line), the obsolete textbook 36 ATP/hexose (second dashed horizontal line), the plant model maximum value for respiration of sucrose with c₉-rings (third dashed horizontal line = 30 ATP/hexose; Table 2 scenario A.11) and the plant model maximum value for respiration of sucrose with c₁₀-rings (lower dashed horizontal line = 28 ATP/hexose; Table 2 scenario A.8). Vertical bars show effects of individual variable values on ATP yield relative to the baseline scenario A.1. First bar from left (f_starch = 1): ATP yield increase (0.5 ATP/hexose) when changing substrate from sucrose to starch. Second bar (H⁺_Pyr = 0): ATP yield increase when eliminating the respiratory proton concentration gradient (ΔpH) requirement for pyruvate transport into mitochondria. Third bar (f_Pyr = 0): ATP yield decrease when glycolysis produces malate instead of pyruvate. Fourth bar (H⁺_NDex = 1): ATP yield increase when the external non-proton-pumping rotenone-insensitive NADH dehydrogenase (ND_ex) contributes to electrochemical proton gradient generation when oxidizing external (cytosolic) NADH. Fifth bar (f_III-IV = 0): ATP yield decrease when the cytochrome pathway in the respiratory chain is fully bypassed by the alternative oxidase. Sixth bar (f_I = 0): ATP yield decrease when complex I is fully bypassed by activity of the non-proton-pumping rotenone-insensitive alternative matrix-facing NADH dehydrogenase (ND_in). Seventh bar (f_leak = 0.25): ATP yield decrease when 25 % of protons pumped from the N side to the P side of the inner mitochondrial membrane leak back to the N side without contributing to ATP synthase activity or transmembrane metabolite transport. Eighth bar (n = 8 to 12): increase in ATP yield when the F_o sector c-ring size is decreased from 10 to 8 (top portion of bar) and decrease in ATP yield when the c-ring size is increased from 10 to 12 (bottom portion). The short horizontal line intersecting the fifth bar indicates ATP yield decrease when 25 % of respiratory O₂ uptake is by the alternative oxidase (i.e. f_III-IV = 0.75; scenario A.15), a fraction consistent with measured in vivo values (Table A1).

To further explore effects of n on ATP yield in the context of a model relating H⁺_FoF1 to c-ring size, n was additionally reduced in the baseline scenario A.1 for respiration of sucrose from the default of 10 to eight (Table 2 scenario A.12), which reflects the c₈-rings measured in mammalian heart mitochondria cited above. That reduction in n increases ATP yield of respiration of sucrose to 31.3 ATP/hexose, 15.5 % above the baseline scenario A.1. I am unaware of evidence of c rings containing fewer than eight c subunits, so this 31.3 ATP/hexose might reflect an upper limit for achievable (intrinsic) ATP yield of plant respiration of sucrose initiated by its cleavage into fructose and glucose by an invertase and with H⁺_Pyr = 1 and H⁺_NDex = 0. Setting instead H⁺_Pyr = 0 and H⁺_NDex = 1, N_ATP increases further to 32.4 ATP/hexose (Table 2 scenario A.13).

In the absence of observed plant mitochondrial c-ring size(s), it remains possible that there could be more than 10 c subunits in the F_o sector of the plant mitochondrial ATP synthase. Relative to the model baseline scenario A.1, increasing c-ring size from 10 to 12 c subunits decreases ATP yield 11.4 %, to 24.0 ATP/hexose for respiration of sucrose (Table 2 scenario A.14). A c₁₂-ring implies H⁺/ATP = 4, a value considerably greater than estimates derived from measured respiration by isolated plant mitochondria of seemingly high quality, many of which indicate that plant mitochondrial H⁺_FoF1 is close to 3, which is indirect evidence that n is nine or 10, and not eight, 11 or 12 (or even larger). In principle, one conclusive determination of a plant mitochondrial ATP synthase c-ring size could resolve calculation of the intrinsic H⁺/ATP ratio (assuming one plant species represents all others), but it is yet unknown whether such a determination will show agreement between an n-based H⁺/ATP ratio and experimental data.

ACTUAL ATP YIELD OFTEN FALLS SHORT OF ITS POTENTIAL

Potential (maximal) ATP yields of modelled respiration depend on strong coupling of (1) oxidation of NADH and succinate to proton gradient generation and (2) proton gradient dissipation to oxidative phosphorylation. In plants, activity of the alternative oxidase may cause notable uncoupling in (1), and so too might oxidation of mitochondrial NADH by the rotenone-insensitive alternative matrix-facing NADH dehydrogenase. In animals, proton leaks are known to cause significant in vitro uncoupling in (2) (Brand, 2005). These and other factors can produce a significant shortfall in ATP yield relative to a potential amount.

Pervasive alternative oxidase activity often prevents maximal values of plant respiratory ATP yield

The alternative oxidase can provide significant benefit during photosynthesis (in photosynthetically capable cells) and periods of stress and stress recovery (Del-Saz et al., 2018; Selinski et al., 2018; Vanlerberghe et al., 2020; Oh et al., 2022), circumstances that are outside the model’s scope. Nevertheless, even in unstressed non-photosynthesizing cells the flow of electrons from QH₂ to oxygen via the alternative oxidase can be a large fraction of total QH₂ oxidation, limiting proton gradient generation per unit of substrate respired. The fraction of respiratory O₂ taken up by the alternative oxidase, often denoted τ_a in the alternative oxidase literature, is represented in the model by the difference (1 − f_III-IV). This can be non-intrusively measured (i.e. without use of metabolic inhibitors) in vivo by an oxygen isotope fractionation technique applicable in laboratory (e.g. Ribas-Carbo et al., 1995; Guy and Vanlerberghe, 2005; Cheah et al., 2014) and field settings (e.g. Kornfeld et al., 2013; Henriksson et al., 2019). Maximal ATP yield of plant respiration requires f_III-IV = 1 (i.e. τ_a = 0) whereas in vivo values of f_III-IV in unstressed leaves and roots estimated with the oxygen isotope fractionation technique, as compiled in Table A1, had a range from under 0.5 to fully 1, with 25th, 50th and 75th percentile values of 0.64, 0.75 and 0.85, respectively. Assigning a generally applicable value to f_III-IV under model conditions is therefore problematic because little is known of in situ (versus in vivo) alternative oxidase activity and because available data cover that large range. Perhaps f_III-IV = 0.75 ± 0.10 would be reasonable for modelling in the absence of measurement data for specific circumstances. It should be noted that f_III-IV generally may be larger in growing cells than mature cells and that some of the data in Table A1, for roots in particular, may have been from a mixture of growing and mature cells.

Relative to the modelled baseline scenario A.1 for respiration of sucrose with f_III-IV = 1 [no alternative oxidase activity, which may occur (Noguchi et al., 2001; Guy and Vanlerberghe, 2005) but perhaps rarely], reducing f_III-IV to 0.75, 0.50 and the extreme case of 0.00 (no cytochrome pathway activity) with all else unchanged reduces ATP yield more than 15, 30 and 60 %, from 27.1 to 22.9, 18.8 and 10.5 ATP/hexose, respectively (Table 2 scenarios A.15, A.16 and A.17; Fig. 5). For the case of f_III-IV = 0 (in combination with f_I = 1, representing mitochondrial NADH oxidation solely by complex I), 62 % of net ATP production is still from F_oF₁-ATP synthase activity. For comparison to a complete lack of cytochrome pathway activity but with complex I oxidizing all matrix NADH (i.e. scenario A.17), the ATP yield of sucrose respiration with a complete bypass of complex I (f_I = 0) but no alternative oxidase activity (f_III-IV = 1) is 19.7 ATP/hexose (Table 2 scenario A.18), about 88 % greater than the scenario A.17 ATP yield.

Despite the potentially large negative effect of alternative oxidase activity on ATP yield in unstressed plants, it is probably improper to characterize it as (entirely) wasteful. Electrons can leak from at least 11 metabolic points in a (mammalian) respiring mitochondrion and produce damaging reactive oxygen species (Brand, 2016; Kuksal et al., 2017). Alternative oxidase activity may limit over-reduction of the respiratory system in circumstances that favour large QH₂/Q ratios, thus reducing electron leaks and reactive oxygen species production (Møller, 2001; Jayawardhane et al., 2020). Persistent alternative oxidase activity (capacity) might therefore deliver a beneficial combination of limiting reactive oxygen species production by preventing an overly reduced respiratory system (i.e. large QH₂/Q ratio) while simultaneously providing capacity for rapid on-demand carbon processing and ATP production when QH₂ is rapidly oxidized, which itself limits the QH₂/Q ratio (Millar et al., 1998). This might produce enough benefit to compensate for the lost ATP under favourable conditions, though a net benefit remains unproven (Vanlerberghe et al., 2020). Nonetheless, from the perspective of respiratory ATP yield, alternative oxidase activity (i.e. f_III-IV < 1) prevents attainment of the potential ATP yield, and the present model does not have a mechanism to quantify (any) net beneficial effects of f_III-IV < 1. In addition, there does not yet appear to be a robust mechanistic way to set a lower limit on minimally ‘required’ or useful (if any) alternative oxidase activity in unstressed cells, though it might be relatable to QH₂/Q ratio. Thus, while acknowledging that alternative oxidase probably functions in many unstressed plant cells to limit production of damaging reactive oxygen species, an important question remains about how much activity is required in that role. If there is a significant unnecessary (unbeneficial) fraction of alternative oxidase activity in crop plants, and it could be reduced or eliminated by bioengineering, crop yield might benefit through increased respiratory ATP yield and associated savings in respiratory substrate that could be used instead for additional growth (Amthor et al., 2019).

How does uncertainty about alternative oxidase activity compare to uncertainty about the plant mitochondrial ATP synthase F_o sector c-ring size n with respect to ATP yield?

The above discussion points to likely values of nine or 10 for n but with no clear resolution to date, reflecting an uncertainty of 10–11 % in the value of n. Conversely, it is suggested above that f_III-IV has an observed range of about 0.5–1.0, which might be restated as 0.75 ± 0.25, or 33 % uncertainty (i.e. 0.25/0.75) in f_III-IV. Relative to the baseline scenario A.1, a modest reduction in f_III-IV from 1 to 0.88 – i.e. 12 % of QH₂ instead of none being reduced by the alternative oxidase – lowers ATP yield 7.4 % relative to its potential, to the value 25.1 ATP/hexose (not shown, scenario X.1 in the R script in Supplementary Data), whereas decreasing n from 10 to nine (i.e. reducing H⁺_FoF1 from 10/3 to 3) increases potential ATP yield 7.1 % to the value 29.0 ATP/hexose (scenario A.9). This indicates that reducing f_III-IV from unity to about 0.88 is as significant, in a negative way, as changing n from 10 to nine is in a positive way. But a 0.88 value for f_III-IV is above the 75th percentile value in Table A1, i.e. relatively rare with most observations < 0.88, indicating that the need to quantify any minimal alternative oxidase requirement may be more important than establishing c-ring size for assessing actual ATP yields of plant respiration. [To determine the potential ATP yield, n (or H⁺_FoF1) must be known and f_III-IV will be 1.] Moreover, while one definitive cryogenic electron microscopy image of a plant mitochondrial ATP synthase c ring might resolve the uncertainty about intrinsic H⁺/ATP of the ATP synthase, determining a minimally required in situ alternative oxidase activity (i.e. maximal attainable and sustainable f_III-IV) will certainly be more involved and will almost certainly be variable across different situations.

Will the alternative oxidase be active in the absence of a complex I bypass?

Whether the plant cytochrome pathway is significantly bypassed by alternative oxidase activity (i.e. f_III-IV < 1) while complex I remains fully engaged (i.e. f_I = 1), or vice versa, is perhaps an open question. Davies et al. (2018) suggested that nearly 60 % of complex I enzymes in plant mitochondria are physically associated with complex III dimers (complex III₂) and that this I₁III₂ supercomplex arrangement – Braun (2020) proposed a I₂III₂ association – might facilitate efficient electron transfer from mitochondrial NADH through complex I directly to complex III. This conceivably would reduce alternative oxidase’s access to QH₂ produced by complex I. Further, it was suggested that I₂III₂IV₂ supercomplexes are physiologically most relevant in plants and that complex II does not appear to associate with complexes I or III (Braun, 2020). This could indicate a correlation between f_I and f_III-IV, and if both are set to a model value of 0.75, reflecting a coordinated 25 % reduction in proton pumping by the entire respiratory chain (Table 2 scenario A.19), modelled ATP yield falls 22 % relative to the baseline scenario for respiration of sucrose. In that reduced proton pumping scenario 81 % of net ATP production is from the ATP synthase.

Speculatively, based on possible I₂III₂ or I₂III₂IV₂ (or I₁III₂IV₁) supercomplexes, the alternative oxidase might have better access to QH₂ generated by the non-proton-pumping rotenone-insensitive NADH dehydrogenases (ND_in and ND_ex) and/or complex II compared to QH₂ generated by complex I. In the modelled baseline scenario A.1 with f_Pyr = 1, 33 % (4/12) of QH₂ is produced by non-proton-pumping dehydrogenases (i.e. ND_ex and complex II; ND_in is not engaged in that scenario for potential ATP yield) and if all that QH₂ is oxidized by alternative oxidase, f_III-IV would be 0.67, a value broadly consistent with some in vivo measurements (Table A1). This itself is insufficient evidence, however, that ND_in is normally inactive and/or that QH₂ produced by complex II and ND_ex is preferentially oxidized by alternative oxidase. Also, for the baseline scenario modified so that glycolysis produces only malate instead of pyruvate (f_Pyr = 0; scenario A.3), although unlikely to occur in vivo during aerobic respiration, only 17 % (2/12) of QH₂ is produced by a non-proton-pumping dehydrogenase. When that is translated into f_III-IV = 0.83 by the reasoning above, it too would be consistent with some in vivo measurements.

As noted, alternative oxidase activity may provide significant benefit in circumstances outside the model scope, such as during photosynthesis and periods of stress and stress recovery. In some stress settings, non-proton-pumping NADH dehydrogenases and the alternative oxidase may be jointly induced, and it was suggested that an ND_in and/or ND_ex may form complexes with alternative oxidase (Møller et al., 2021). In particular, transgenic Arabidopsis experiments were interpreted to indicate that an ND_ex and the alternative oxidase ‘forms a complete, functional, nonphosphorylating pathway of electron transport, whose operation enhances tolerance to environmental stress’ (Sweetman et al., 2019, p. 774). And as cited above, combined activity of ND_ex and alternative oxidase was reported to produce sufficient electrochemical gradient to power protein uptake into mitochondria, which requires ΔΨ_m. Although outside the present model’s scope, flexibility in expression and activity of various NADH (and NADPH) dehydrogenases aside from complex I, with or without coordinated expression/activity of the alternative oxidase, potentially provides considerable flexibility in respiratory ATP production in photosynthesising and/or stressed plant cells (and see e.g. Bykova et al., 2014).

Proton leaks through the inner mitochondrial membrane are important in animals – what of plants?

Proton leaks (proton cycles) are quantitatively important to respiratory proton gradient dissipation in animals (Brand and Nicholls, 2011; Divakaruni and Brand, 2011). Proton cycles facilitated by uncoupling protein 1 in mammalian brown adipose (fatty) tissue drive thermogenesis (Nicholls, 2021). While this thermogenic function of leaks in endotherms is clear, leaks also prevail in ectotherms, and even outside brown adipose tissue roughly 20–30 % of protons pumped to the P side of the inner membrane of in vitro animal mitochondria appear to leak back to the N side (Brand, 2005). This may imply a vital non-thermogenic function of proton leaks that is generally acknowledged to be reduced (or eliminated) production of damaging reactive oxygen species in animal mitochondria (Kuksal et al., 2017), a function also proposed for alternative oxidase activity in plants (Møller, 2001; Jayawardhane et al., 2020; Vanlerberghe et al., 2020).

The endogenous (non-brown adipose tissue) proton leak in animals is not facilitated by uncoupling protein 1, but another inner mitochondrial membrane protein(s) is probably involved (Brand, 2005). The highly abundant mitochondrial adenine nucleotide translocator is a specific candidate (Brand et al., 2005; Kuksal et al., 2017). When ATP production slows, the adenine nucleotide translocator can shift away from ADP–ATP exchange toward translocation of protons down their concentration gradient from the P to N sides of the inner mitochondrial membrane, indicating competition for (i.e. overlapping) translocation pathways for protons and ADP/ATP, but also indicating that conditions favouring ATP production prevent proton flux (leak) through the translocator (Bertholet et al., 2019).

There also is an identified function for mitochondrial proton leaks in non-thermogenic plant cells during C3 photosynthesis (which, again, is outside the model scope). Uncoupling protein 1 can support rapid oxidation of NADH (i.e. regeneration of NAD⁺) in the plant mitochondrial matrix to facilitate NAD⁺-dependent photorespiratory mitochondrial glycine oxidation needed for rapid photosynthetic carbon flux, but without the need for concomitant ADP phosphorylation by the mitochondrial ATP synthase that might be ADP-limited (Sweetlove et al., 2006). In addition, a mitochondrial uncoupling protein might facilitate thermogenesis in some plant flowers respiring lipids (Ito and Seymour, 2005), in contrast to the better-known alternative oxidase-supported thermogenesis in plants based on carbohydrate oxidation (Moore et al., 2013). To my knowledge, however, there is a lack of data indicating a significant endogenous mitochondrial proton leak in unstressed, non-photosynthesizing and non-thermogenic plant cells.

The model default value of f_leak = 0 is thus based on lack of data from plants indicating (1) significant passive proton leaks through the inner mitochondrial membrane, (2) uncoupling protein 1-facilitated proton translocation outside of photorespiratory conditions or (3) adenine nucleotide translocator-facilitated proton translocation during ATP-generating respiration. In addition, while proton leaks may function to limit reactive oxygen species production in animals, the alternative oxidase may perform that function in plants, perhaps obviating the need for proton cycles in plants by limiting excessive proton pumping in the first place. But as an example of an effect of a hypothetical proton leak on ATP yield of plant respiration, modifying the baseline scenario A.1 to include f_leak = 0.25, a value typical for animals, reduces ATP yield about 22 %, from 27.1 to 21.1 ATP/hexose (Table 2 scenario A.20). Unsurprisingly, that ATP yield is identical to scenario A.18, but by a different mechanism. Scenario A.19 (f_I = f_III-IV = 0.75) reduces proton pumping to the P side of the inner mitochondrial membrane by 25 % (H⁺_out = 78 H⁺/hexose = H⁺_in) whereas the leak-based scenario A.20 pumps the full proton number (H⁺_out = 104 H⁺/hexose = H⁺_in) but then cycles 25 % of those protons (26 protons) back to the N side of the membrane in a futile proton cycle with the remaining 75 % of the pumped protons (78 protons) available to drive transport processes and the ATP synthase. In both scenarios A.19 and A.20, 81 % of net ATP production is by the ATP synthase. The significance of a slip rather than a leak, in particular slip in the ATP synthase, was quantified for f_slip(FoF1) = 0.25, which reduced ATP yield about 17 % relative to the baseline scenario A.1 (not shown, scenario X.2 in the R script in Supplementary Data).

Mathematically, any f_leak value in the interval [0,1] is possible, but values approaching 1 would greatly limit both oxidative and substrate-level ADP phosphorylations, and perhaps mitochondrial pyruvate and/or malate uptake, and would be indicative of unhealthy cells (unless thermogenesis is the respiratory ‘goal’).

COMPARING THE PLANT MODEL TO THE STANDARD MAMMALIAN MODEL

Fundamental respiratory metabolism research often focuses on mammals (and yeasts). Hence estimates of potential respiratory ATP yield frequently are based on mammalian data. To confirm that the plant model can accommodate other organisms with appropriate parameterization, it was compared to three previous analyses (models) of potential respiratory ATP yield in mammalian cells spanning a 27-year publication period (Hinkle et al., 1991; Brand, 2005; Mookerjee et al., 2017).

All three mammalian analyses included glucose as a substrate that was catabolized by cytosolic glycolysis producing pyruvate that was in turn completely oxidized in the mitochondrial matrix by the pyruvate dehydrogenase complex in combination with the TCA cycle. The plant model defaults of f_starch = 0 (sucrose as substrate) and f_Pyr = 1 (glycolysis producing pyruvate) mimic the mammalian models’ carbon metabolism pathways, with the plant model’s cleavage of sucrose into fructose and glucose by invertase being metabolically equivalent to two glucose molecules and therefore a match to the mammalian models on a per hexose basis. The mammalian analyses all assumed that mitochondrial uptake of pyruvate does not dissipate the proton gradient ΔpH, which is represented by H⁺_Pyr = 0 in the plant model. The plant model default of H⁺_Pyr = 1, representing use of the ΔpH by the H⁺/pyruvate⁻ symport, is therefore less efficient for ATP production than the mammalian model assumption. The mammalian models of potential ATP yield all excluded bypasses of proton pumps in the respiratory chain and ignored slips and leaks, consistent with plant model default variable values of f_I = 1, f_III-IV = 1, f_slip = 0, f_slip(FoF1) = 0 and f_leak = 0.

The three mammalian analyses each included two scenarios for respiratory oxidation of the NADH produced by glycolysis in the cytosol: (1) use of the glycerol 3-phosphate shuttle [e.g. Nicholls and Ferguson, 2013 (pp. 273–274)] in which electrons from glycolytically produced NADH in the cytosol enter the respiratory chain without proton pumping ‘between’ complexes I and III from the P side of the inner mitochondrial membrane, and (2) use of the malate–aspartate shuttle to transfer NADH reducing equivalents from the cytosol to the mitochondrial matrix (Fig. 4). The glycerol 3-phosphate shuttle as used in the mammalian models is quantitatively equivalent to the plant model’s default oxidation of cytosolic NADH by the non-proton-pumping rotenone-insensitive external mitochondrial NADH dehydrogenase (ND_ex in Fig. 1) as represented by f_shuttle = 0 in combination with H⁺_NDex = 0. The mammalian malate–aspartate shuttle is represented in the plant model by f_shuttle = 1 in combination with H⁺_Glu = 1.

The only difference among the three mammalian analyses was the H⁺/ATP ratio of the mitochondrial F_oF₁-ATP synthase, which in the plant model is determined by the F_o sector c-ring size. The first mammalian analysis (Hinkle et al., 1991) used the equivalent of c₉-rings by using an ATP synthase H⁺/ATP ratio of 3. That was based on experimental estimates of P/O ratios rather than enzyme structural characteristics at a time before determinations of the F_oc-ring size made n-based H⁺/ATP values calculable. The second analysis (Brand, 2005) explicitly used c₁₀-rings to mechanistically evaluate H⁺/ATP as 10/3. That was before c₈-rings were discovered in mammalian hearts and when all mitochondrial F_o sector c-rings were assumed to be composed of 10 c subunits based on the observations of Stock et al. (1999) for yeast. This, importantly, is the current approach used in the plant model to assign a default value of 10 to n – i.e. c₁₀-rings observed in mitochondria from yeasts and other unicellular organisms are assumed to reflect the value in plants in the absence of any direct observations from plant mitochondrial ATP synthases. The third mammalian analysis (Mookerjee et al., 2017) used c₈-rings based on observed c₈-rings in Bos taurus mitochondria (Watt et al., 2010). Thus, to simulate the three mammalian analyses, the plant model used n values of 9, 10 and 8, respectively.

With these input-variable values, the plant model exactly matched the three analyses of potential ATP yield of mammalian respiration (Table 3). Notable features of the three mammalian analyses were (1) as noted, the only model variable that changed value over 27 years was the ATP synthase H⁺/ATP ratio, most recently calculated mechanistically from F_o sector c-ring size; (2) the c₈-ring model produced about 15.6 % more ATP than the preceding c₁₀-ring model, again emphasizing the importance of determining the plant mitochondrial ATP synthase F_o sector c-ring size for quantitative understanding of the potential (and actual) ATP yield of plant respiration; and (3) engagement of the malate–aspartate shuttle increased ATP yield slightly more than 5 % relative to the glycerol 3-phosphate shuttle in all cases.

Table 3.

Potential (maximal) ATP yield of complete respiratory oxidation of glucose (glucose → CO₂) calculated by the plant model parametrized for mammals and compared to three previous mammalian-based analyses that each considered two cytosolic NADH ‘shuttle’ scenarios

Scenario	Process to oxidize glycolytic NADH (shuttle type)	ATP synthase Fo sector c-ring size n	ATP/glucose produced: this study	ATP/glucose produced: previous study	ATP gain due to malate–aspartate shuttle relative to glycerol 3-phosphate shuttle	Previous study source
H.G	Glycerol 3-phosphate shuttle*	9 (equivalent value†)	29.5	29.5	–	Hinkle et al. (1991)
B.G		10	27.5	27.5	–	Brand (2005)
M.G		8	31.8	31.8	–	Mookerjee et al. (2017)
H.M	Malate-aspartate shuttle‡	9 (equivalent value)	31.0	31.0	5.08 %	Hinkle et al. (1991)
B.M		10	28.9	28.9	5.03 %	Brand (2005)
M.M		8	33.5	33.5	5.14 %	Mookerjee et al. (2017)

The plant model was parametrized for mammalian respiration of glucose by using default variable values (Table 1) with these changes: (1) c-ring size n was set as shown to match each previous mammalian study, (2) H⁺_Pyr = 0 to simulate pyruvate uptake by mitochondria without use of ΔpH generated by the respiratory chain as assumed in the mammalian models and (3) for the malate–aspartate shuttle scenarios (H.M, B.M and M.M), f_shuttle = 1 in combination with H⁺_Glu = 1 (see text). Results are listed in chronological order of publication of the mammalian studies within each of the NADH shuttle scenarios. Values are rounded to three digits; unrounded plant model values exactly matched the earlier publications in all cases.

^*The glycerol 3-phosphate shuttle may be most important in mammalian skeletal muscle, brown adipose tissue and parts of the brain.

^† Hinkle et al. (1991) specified H⁺/ATP = 3 for the ATP synthase rather than specifying n, and H⁺/ATP = 3 when n = 9 in the model.

^‡The malate–aspartate shuttle is active in mammalian heart, liver, kidney and parts of the brain (neurons).

The ability of the plant model to replicate the mammalian model does not mean that potential plant and mammalian respiratory ATP yields are the same or arise from identical processes. On the contrary, plant and mammalian respiratory systems differ in several ways, and the unknown plant mitochondrial ATP synthase F_o sector c-ring size prevents an unambiguous current comparison to mammalian systems. Moreover, for estimation of actual rather than potential ATP yield in plants the model would logically simulate activity of the alternative oxidase and/or the rotenone-insensitive alternative matrix-facing NADH dehydrogenase, whereas it would simulate inner mitochondrial membrane leaks in mammals. Nonetheless, the two systems can be quantitatively compared using appropriate variable values in the plant model framework, but the reverse is untrue because the mammalian models lack, for example, the alternative oxidase and the non-proton-pumping rotenone-insensitive alternative matrix-facing NADH dehydrogenase. The greater complexity of the plant respiratory system requires a more complicated model than those used for mammals respiring glucose.

PERSPECTIVE

Respiratory ATP production is a fundamental quantitative link between rates (or amounts) of energy-requiring heterotrophic processes and rates (or amounts) of substrate oxidation. Estimates of the potential ATP yield of respiration are updated as knowledge of the stoichiometries of respiratory reactions and pathways improves, but uncertainties currently remain. Instead of being a universal biological constant across life forms, it is apparent that ATP yields of respiration, both potential and actual, can differ between groups of organisms both in the (net) amount of ATP produced and the set of reactions delivering that production.

This model-based assessment of potential ATP yield of respiration in unstressed non-photosynthesizing mature plant cells incorporates a mechanistic relationship between the H⁺/ATP ratio of the mitochondrial F_oF₁-ATP synthase and the number, n, of c subunits of the F_o sector c-ring. That relationship dictates that the model variable H⁺_FoF1 (i.e. H⁺/ATP ratio of the ATP synthase) is equal to n/3 H⁺/ATP. Unfortunately, n is unestablished in plant mitochondria. The baseline modelling assumption that n is 10 in plant mitochondria is an extrapolation of mitochondrial data from several unicellular organisms that have F_o sector c₁₀-rings, which is inconclusive evidence for that value in plant mitochondria. After all, it was determined that n is eight in mammalian heart mitochondria. Nonetheless when n is set to 10 in the plant model, and therefore H⁺_FoF1 is 10/3, calculated potential ATP yield of respiration covers the range 26.9–28.0 ATP/hexose for respiration of sucrose (27.4–28.5 ATP/hexose for starch respiration) across a set of model scenarios.

Although n may be 10 in plants, as used in the model’s baseline scenario, there are considerable experimental measurements of concomitant rates of in vitro oxygen uptake and ADP phosphorylation by plant mitochondria indicating that the plant mitochondrial ATP synthase H⁺/ATP ratio is functionally about three (e.g. Møller et al., 2021), which would correspond to n = 9 if the H⁺/ATP ratio is precisely determined by n. When n is set to nine instead of 10 in the plant model and all else is unchanged, potential respiratory ATP yield increases about 2 ATP/hexose, raising the range of potential ATP yield to about 29.0–30.0 ATP/hexose for respiration of sucrose (29.5–30.5 ATP/hexose for starch respiration).

If c₁₀-rings exist in plant mitochondrial ATP synthases, as observed in several other organisms, but experimental data indicate H⁺/ATP = 3, then H⁺/ATP may be only loosely coupled to c subunit number and/or analysis of experimental data is imprecise enough that H⁺/ATP ratios of 3.00 and 3.33 cannot be clearly distinguished (and see related discussion in Hinkle, 2005). Further, if the plant mitochondrial ATP synthase F_o sector c-ring is found to contain 10 c subunits, and experimental data indicating that H⁺/ATP = 3 rather than 10/3 are deemed accurate, a rethinking of the fundamental relationship between n and H⁺/ATP in this and other models of potential ATP yield of aerobic respiration will be in order. In that case, the variable n might be disregarded as a model input and an experimentally determined value(s) of H⁺_FoF1 might be used as direct model input rather than being a calculated intermediate variable.

In contrast to the modest, but non-trivial, uncertainty in modelled potential ATP yield of plant respiration resulting from the possibility that n might be either nine or 10 in plants – i.e. uncertainty between the range 27–28 and the range 29–30 ATP/hexose for respiration of sucrose, or about 7 % of the modelled values – uncertainty in actual ATP yield due to other factors is relatively large, even in the modelled setting of mature, unstressed non-photosynthesizing plant cells. In particular, in vivo f_III-IV varies over the range of about 0.5–1.0 in healthy plants (Table A1), which itself would widen the range in possible ATP yield in plants with c₁₀-rings to about 19–28 ATP/hexose for respiration of sucrose. If in addition to observed alternative oxidase activity, the rotenone-insensitive alternative matrix-facing NADH dehydrogenase is also active, the lower end of that uncertainty range would decline further. Thus, there is a need to understand not only the extent of actual and minimally ‘required’ alternative oxidase activity, but also to unravel (any) coordination between induction and activity of the alternative oxidase with alternative NADH dehydrogenases that bypass complex I and that therefore potentially produce a complete non-proton-pumping electron transport chain from NADH to oxygen. Added to that is a need to understand any coordination between complex II and the alternative oxidase that would generate a complete non-proton-pumping electron transport chain from succinate to oxygen within the model’s physiological setting. Uncertainty in a few other model variables including f_starch, f_Pyr, H⁺_Pyr and f_slip is perhaps less serious or has limited effects on calculated ATP yield, but f_leak is poorly understood in plants and has potential for a relatively large impact on ATP yield.

Uncertainty aside, actual in vivo ATP yield of respiration in plants (and other organisms) is expected often to be less than the potential, and while the model can in principle mimic actual ATP yield, providing accurate inputs remains a challenge. The largest shortfall in actual ATP yield relative to its potential is typically likely to be the result of alternative oxidase activity bypassing complexes III and IV and thus meaningfully limiting generation of the transmembrane electrochemical proton gradient. While a wide range of values are reported for different species in different environments, a central tendency for QH₂ oxidation by the alternative oxidase relative to total QH₂ oxidation (i.e. 1 − f_III-IV, or τ_a) in unstressed cells is roughly in the range 20–30 %. This itself would reduce proton pumping by about 14–21 % relative to its potential. In animals, which lack the alternative oxidase, the largest contributor to ATP yield shortfall appears to be an endogenous proton leak through the inner mitochondrial membrane that typically dissipates, through a proton cycle, about 20–30 % of the respiratory proton gradient that is generated (Brand, 2005). Alternative oxidase activity in plants and the endogenous proton leak in animals both probably function to beneficially reduce production of damaging reactive oxygen species when the respiratory chain is highly reduced, i.e. when the QH₂/Q ratio is large. A cost of both these mechanisms that reduce reactive oxygen production – if indeed selection has occurred for that function – is reduced ATP yield. Fortunately, non-intrusive in vivo measurement of isotopic discrimination during respiratory oxygen uptake can determine to a fair degree activity of the alternative oxidase relative to the cytochrome pathway in plants. Hence the key model variable f_III-IV need not be an unknown. This is an important factor for reducing uncertainty about actual ATP yield of respiration in plants. Unfortunately, robust methods for non-intrusive in vivo or in situ quantification of other bypasses of energy-conserving reactions, or the magnitude of any proton leaks or slips in the respiratory chain of plants or other organisms presently are unavailable. As a result, in vivo or in situ values of several model variables remain unknown, and perhaps for the present unknowable.

CONCLUSIONS

This model analysis highlights the importance of measuring the number of subunits in the F_o sector c-ring in the plant mitochondrial F_oF₁-ATP synthase to the quantification of potential ATP yield of plant respiration, which may lie between about 27 and 30 ATP/hexose for respiration of sucrose and slightly more for respiration of starch. Actual ATP yield may typically be less than its potential, perhaps only rarely attaining 90 % or more of the potential and often considerably less than that depending on the degree of alternative oxidase activity and perhaps engagement of complex I bypasses and other inefficiencies, including slips and leaks in the oxidative phosphorylation system. That is, a stoichiometrically balanced model of plant respiratory reactions can approximate the potential (maximal) ATP yield well, but actual in vivo/in situ yields remain variable and uncertain. Research relevant to closing the gap between actual and potential respiratory ATP yield in crop plants includes determining whether some fraction of current alternative oxidase activity is not useful and then if elimination of (some of) that fraction is possible through bioengineering (Amthor et al., 2019). This in essence is an issue of uncertainty about the necessity of alternative oxidase activity across a range of circumstances. Similarly, if a complex I bypass is generally active, yet non-essential, reducing its activity could also close the gap between potential and actual ATP yield, and so too could reducing or eliminating unneeded slips or leaks. Whether there is near-term scope for improving the ATP yield of crop-plant respiration and therefore increasing potential crop productivity through its modification depends on both the magnitude and causes of any differences between actual and potential ATP yield and if the causes can be eliminated. Presently, those are largely open questions (Reynolds et al., 2021; Garcia et al., 2023).

SUPPLEMENTARY DATA

Supplementary data are available at Annals of Botany online and consist of the following. The model was implemented in an R programming language script available as a text file online at https://academic.oup.com/aob. All parameters, functions and variables are defined in comments in the script, which contains all model parameter values, input variable values and R instructions arranged to reproduce results in Tables 2 and 3 and Appendix 1. The R script uses only base R (i.e. does not require any extensions or external packages) and can be copied into any R session for direct execution without editing; it also can be edited with any text editor to facilitate easy batch processing of an unlimited number of modelled scenarios at one time.

APPENDIX 1

Potential (maximal) P/O ratios of segments of plant respiration associated with the respiratory chain in isolated mitochondria can be calculated with subsets of the model. The following idealized examples, unless otherwise stated, are based on f_I = 1, f_III-IV = 1, H⁺_I = 4, H⁺_III-IV = 6, f_leak = 0, f_slip = 0, f_slip(FoF1) = 0, f_shuttle = 0, H⁺_Pi = 1 and H⁺_FoF1 = n/3 with all ATP formed in the mitochondrial matrix exported from the organelle via ATP⁴⁻_outward/ADP³⁻_inward exchange. The number of oxygen atoms reduced, symbolized O_substrate, equals the number of 2e⁻ entering (the portion of) the respiratory chain modelled, with the variable ‘substrate’ indicating the molecule being respired. In experiments, f_III-IV may be ‘set to’ one with the use of an alternative oxidase inhibitor such as n-propyl gallate and f_I may be ‘set to’ zero with the use of rotenone [see Jacoby et al. (2015) for these and other experimental considerations]. The P/O calculations that follow are included in the supplemental R programming language script [Supplementary Data].

Pyruvate

The modelled P/O ratio of mitochondria supplied with external pyruvate (N_Pyr = 2 for consistency with the full model) and with a fully functioning TCA cycle (Fig. 3) is given by:

$${\displaystyle \begin{array}{lll}& \mathrm{P}/{\mathrm{O}}_{\mathrm{P}\mathrm{y}\mathrm{r}}=[N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}& +\frac{([{\mathrm{H}}_{\mathrm{I}}^{+}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}+{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}(N_{\mathrm{S}\mathrm{u}\mathrm{c}}+N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}})]-[{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}{N}_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}+{\mathrm{H}}_{\mathrm{P}\mathrm{y}\mathrm{r}}^{+}{N}_{\mathrm{P}\mathrm{y}\mathrm{r}}])}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}]/{\mathrm{O}}_{\mathrm{P}\mathrm{y}\mathrm{r}}\end{array}}$$ (A1)

where N_ATP(TCA) = 2, N_NADHm = 8, N_Suc = 2 and O_Pyr = 10 oxygen atoms reduced (= N_NADHm + N_Suc) for each pair of pyruvate molecules respired. With n = 10 and H⁺_Pyr = 1 (model defaults), P/O_Pyr = 2.23, and with n = 10 but H⁺_Pyr = 0 to simulate pyruvate uptake without dissipation of ΔpH, P/O_Pyr = 2.28. With n = 9 to simulate an experimentally based H⁺_FoF1 value of 3 H⁺/ATP in combination with H⁺_Pyr = 1, P/O_Pyr = 2.40. For n = 9 and H⁺_Pyr = 0, P/O_Pyr = 2.45.

Malate

The modelled P/O ratio of mitochondria supplied with external malate (N_Mal = 2 for consistency with the full model) that is converted to pyruvate in the mitochondrial matrix followed by pyruvate oxidation and a functioning TCA cycle (Fig. 3 with malate import) is given by:

$${\displaystyle \begin{array}{lll}& {\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{M}\mathrm{a}\mathrm{l}}=[N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}}& +\frac{([{\mathrm{H}}_{\mathrm{I}}^{+}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}+{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}(N_{\mathrm{S}\mathrm{u}\mathrm{c}}+N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}})]-{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}[N_{\mathrm{M}\mathrm{a}\mathrm{l}}+N_{\mathrm{A}\mathrm{T}\mathrm{P}\left(\mathrm{T}\mathrm{C}\mathrm{A}\right)}])}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}]/{\mathrm{O}}_{\mathrm{M}\mathrm{a}\mathrm{l}}\end{array}}$$ (A2a)

where N_ATP(TCA) = 2, N_NADHm = 10, N_Suc = 2 and O_Mal = 12 oxygen atoms reduced (= N_NADHm + N_Suc) for each pair of malate molecules respired. With n = 10, P/O_Mal = 2.24, and with n = 9, P/O_Mal = 2.42. A key difference between malate and pyruvate supplied to mitochondria is an additional NADH formed in the mitochondrial matrix per malate taken up and therefore an additional oxygen atom is reduced per malate relative to pyruvate.

P/O_Mal of 2.4 was reported for Spinacea oleracea leaf mitochondria (Fig. 2 in Douce et al., 1977), a value consistent with H⁺_FoF1 = 3 H⁺/ATP (i.e. n = 9) rather than H⁺_FoF1 = 10/3 H⁺/ATP (i.e. n = 10). Pisum sativum leaf mitochondria supplied with malate in combination with pyruvate produced P/O = 2.6 (Day et al., 1985), exceeding all modelled values as parameterized above for either pyruvate or malate, but closer to modelled values for H⁺_FoF1 = 3 H⁺/ATP rather than 10/3 H⁺/ATP. (Experimental P/O ratios cited herein are examples only, not an exhaustive summary or review of plant-based data.)

Equation (A2a) is based on oxidation of all pyruvate formed by NAD-malic enzyme (malate + NAD⁺ → pyruvate + NADH + CO₂), associated acetyl-CoA production and entry of OAA into a steady-state TCA cycle. Pyruvate oxidation requires thiamine pyrophosphate, which may be limited in isolated plant mitochondria (Palmer et al., 1982) and therefore often added as a reagent in experiments with isolated mitochondria (Jacoby et al., 2015). When mitochondria respiring malate are deficient in thiamine pyrophosphate, both pyruvate and OAA may accumulate and the TCA cycle will cease, with the only production of NADH being from activities of NAD-malic enzyme and malate dehydrogenase (malate + NAD⁺ ↔ OAA + NADH; note the reversibility). If the NADH produced is not oxidized by complex I and/or the internal non-proton-pumping alternative NADH dehydrogenase (ND_in), for example because ADP supply is exhausted and electron transport therefore slows, the resulting high NADH/NAD⁺ ratio may eventually drive malate dehydrogenase activity in the direction of malate production from OAA (Palmer et al., 1982; Hagedorn et al., 2004). Also, a large (increasing) NADH/NAD⁺ ratio in the matrix, from any cause, can stimulate ND_in activity (e.g. see model of Hagendorn et al., 2004). For malate oxidation in mitochondria without oxidation of the pyruvate produced or condensation of the OAA produced with acetyl-CoA (e.g. Palmer et al., 1982), the modelled P/O ratio (and with N_Mal = 2 for consistency with the full model) is:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{M}\mathrm{a}\mathrm{l}}=\left[\frac{([f_{\mathrm{I}}{\mathrm{H}}_{\mathrm{I}}^{+}+{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}]N_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{m}}-{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}{N}_{\mathrm{M}\mathrm{a}\mathrm{l}})}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right]/{\mathrm{O}}_{\mathrm{M}\mathrm{a}\mathrm{l}}}\end{array}}$$ (A2b)

where N_NADHm = 2 and O_Mal = 2 oxygen atoms reduced (= N_NADHm) per pair of malate molecules oxidized. With n = 10 and f_I = 1 (all NADH oxidized by complex I), P/O_Mal = 2.08, and with n = 10 but f_I = 0.5 (half the NADH oxidized by ND_in), P/O_Mal drops to 1.62. With n = 9 and f_I = 1, P/O_Mal = 2.25, and with n = 9 but f_I = 0.5, P/O_Mal drops to 1.75. To the extent that NADH accumulates in the matrix, rather than being oxidized at the rate of its formation, malate oxidation and oxygen consumption will depart from the above N_Mal:O_Mal 1:1 stoichiometry. Further, if NADH accumulates in the matrix the value of f_I probably would decline.

Glycine

In photorespiring cells the mitochondrial glycine decarboxylase complex produces one NADH (reduces one NAD⁺) per pair of glycine molecules oxidized in the matrix. Though photorespiration is outside the model scope, glycine supplied to mitochondria isolated from photosynthetic cells and studied under non-photorespiratory conditions can produce NADH in the mitochondrial matrix that is then oxidized by complex I with the P/O ratio of glycine oxidation defined in terms of model parameters given by:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{G}\mathrm{l}\mathrm{y}}=\left(\frac{{\mathrm{H}}_{\mathrm{I}}^{+}+{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right)/{\mathrm{O}}_{\mathrm{G}\mathrm{l}\mathrm{y}}}\end{array}}$$ (A3a)

where one oxygen atom is reduced per pair of glycine molecules oxidized (O_Gly = 1) and no other compound(s) are oxidized by the respiratory chain. With n = 10, P/O_Gly = 2.31, and with n = 9, P/O_Gly = 2.50. With n = 8 to simulate a potential lower bound on mitochondrial c-ring size, P/O_Gly = 2.73.

P/O_Gly of 2.5 was reported for Spinacea oleracea leaf mitochondria (Fig. 4 in Douce et al., 1977) and P/O_Gly of 2.6 was reported for Pisum sativum leaf mitochondria (Day et al., 1985). Both values are more consistent with H⁺_FoF1 = 3 H⁺/ATP rather than 10/3 H⁺/ATP.

Equation (A3a) is based on oxidation of NADH by complex I in the mitochondrial matrix with the resulting QH₂ oxidized by the cytochrome pathway. If complex I is inhibited by rotenone, NADH oxidation can take place via the rotenone-insensitive alternative matrix-facing NADH dehydrogenase with the QH₂ produced oxidized by the cytochrome pathway. In that case, with all other model variables unchanged, P/O_Gly is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{G}\mathrm{l}\mathrm{y}}=\left(\frac{{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right)/{\mathrm{O}}_{\mathrm{G}\mathrm{l}\mathrm{y}}}\end{array}}$$ (A3b)

where again O_Gly = 1. With n = 10, P/O_Gly = 1.38; with n = 9, P/O_Gly = 1.50; and with n = 8, P/O_Gly = 1.64.

In the presence of rotenone, P/O_Gly of 1.5–1.6 was reported for Spinacea oleracea leaf mitochondria (Fig. 5 in Douce et al., 1977), a range more consistent with H⁺_FoF1 = 3 H⁺/ATP rather than 10/3 H⁺/ATP.

Succinate

The modelled P/O ratio of succinate oxidation is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{S}\mathrm{u}\mathrm{c}}=\left(\frac{{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right)/{\mathrm{O}}_{\mathrm{S}\mathrm{u}\mathrm{c}}}\end{array}}$$ (A4)

where one oxygen atom is reduced per succinate molecule oxidized (O_Suc = 1), no other compound(s) are oxidized by the respiratory chain and no substrate-level ADP phosphorylations occur. With n = 10, P/O_Suc = 1.38, and with n = 9, P/O_Suc = 1.50.

P/O_Suc of 1.4 was reported for Spinacia oleracea leaf mitochondria (Fig. 2 in Douce et al., 1977) and P/O_Suc of 1.45–1.50 was reported for Solanum tuberosum tuber mitochondria (Fig. 4 in Neuburger et al., 1982). Those values are consistent with both H⁺_FoF1 = 10/3 H⁺/ATP and H⁺_FoF1 = 3 H⁺/ATP, but the P/O range of 1.45–1.50, perhaps reflecting better mitochondrial preparations, is more consistent with H⁺_FoF1 = 3 H⁺/ATP.

External (cytosolic) NADH

The modelled P/O ratio of external (cytosolic) NADH oxidation (N_NADHc = 2 for consistency with the full model producing pyruvate in the cytosol) is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}=\left(\frac{{\mathrm{H}}_{\mathrm{I}\mathrm{I}\mathrm{I}-\mathrm{I}\mathrm{V}}^{+}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}+{\mathrm{H}}_{\mathrm{N}\mathrm{D}\mathrm{e}\mathrm{x}}^{+}{N}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right)/{\mathrm{O}}_{\mathrm{N}\mathrm{A}\mathrm{D}\mathrm{H}\mathrm{c}}}\end{array}}$$ (A5)

where O_NADHc = 2 oxygen atoms reduced (= N_NADHc), no other compound(s) are oxidized by the respiratory chain and no substrate-level ADP phosphorylations occur. For n = 10 and with H⁺_NDex = 0, reflecting a view that the external rotenone-insensitive NADH dehydrogenase does not contribute to generation of the proton gradient, P/O_NADHc = 1.38, whereas with n = 10 and H⁺_NDex = 1, P/O_NADHc = 1.62. With n = 9 and H⁺_NDex = 0, P/O_NADHc = 1.50, and with n = 9 and H⁺_NDex = 1, P/O_NADHc = 1.75. Modelled P/O_NADHc is the same as P/O_Suc when H⁺_NDex is zero but larger than P/O_Suc when H⁺_NDex is greater than zero.

P/O_NADHc of 1.30–1.35 was reported for Spinacia oleracea leaf mitochondria (Fig. 2 in Douce et al., 1977) and 1.40–1.45 was reported for Solanum tuberosum tuber mitochondria (Fig. 4 in Neuburger et al., 1982), values broadly consistent with H⁺_FoF1 = 10/3 H⁺/ATP in combination with H⁺_NDex = 0. Conversely, P/O_NADHc of 1.5–1.6 was reported for Pisum sativum leaf mitochondria (Day et al., 1985), which is consistent with the model for n = 10 in combination with H⁺_NDex = 1 or for n = 9 in combination with H⁺_NDex = 0.

Ascorbate

Although not included in the model, experimentally added ascorbate, via its reduction of experimentally added N,N,Nʹ,Nʹ-tetramethyl-p-phenylenediamine dihydrochloride, can reduce oxidized cytochrome c on the P (external) side of the inner mitochondrial membrane of isolated mitochondria. Complex IV can then oxidize the reduced cytochrome c by extracting from it two electrons and combining them with two protons extracted from the N side of the membrane and one atom of oxygen to form water (on a per 2e⁻ basis). In parallel, complex IV pumps two protons from the N side to the P side of the membrane per 2e⁻ extracted from cytochrome c. In total, two electrons are removed from the P side of the membrane, four protons are removed from the N side (including/counting the two protons consumed in water formation) and two protons are added (pumped) to the P side per ascorbate oxidized. That is equivalent to translocation of four protons from the N side to the P side of the membrane per ascorbate. The modelled P/O ratio of such ascorbate oxidation is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{P}/{\mathrm{O}}_{\mathrm{A}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{e}}=\left(\frac{4}{{\mathrm{H}}_{\mathrm{P}\mathrm{i}}^{+}+{\mathrm{H}}_{\mathrm{F}\mathrm{o}\mathrm{F}1}^{+}}\right)/{\mathrm{O}}_{\mathrm{A}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{e}}}\end{array}}$$ (A6)

where one oxygen atom is reduced per ascorbate molecule oxidized (O_Ascorbate = 1), no other compound(s) are oxidized by the respiratory chain and no substrate-level ADP phosphorylations occur. With n = 10, P/O_Ascorbate = 0.92; with n = 9, P/O_Ascorbate = 1; and with n = 8, P/O_Ascorbate = 1.09. Hinkle et al. (1991) reported P/O_Ascorbate of 0.98 ± 0.09 for rat liver mitochondria, smaller than the model value for n = 8 (H⁺_FoF1 = 8/3) corresponding to the c-ring size reported for mammals (Watt et al., 2010; Gu et al., 2019).

P/O measurement accuracy

In a detailed review of experimentally derived P/O values for animal (mostly rat liver) mitochondria, Hinkle (2005) cited several methodological difficulties and oversights and concluded that most reports of P/O ratios greater than 2.5 for NADH-linked substrates (producing NADH in the mitochondrial matrix) were overestimations. Similarly, he suggested that most experimental P/O ratios greater than 1.5 for succinate oxidation were overestimates. Those ‘limiting’ values of 2.5 and 1.5 were targeted because they are consistent with H⁺_FoF1 = 3 H⁺/ATP, and Hinkle further suggested that animal H⁺_FoF1 might be 10/3 H⁺/ATP (i.e. n = 10), meaning that the respective mechanistic (ideal, maximal) animal P/O ratios might then be less than 2.5 and 1.5 for NADH-linked substrates and succinate, respectively. Those suggestions were made before c₈-rings were measured in mammalian heart mitochondrial F_oF₁-ATP synthases. If the ATP synthase H⁺/ATP ratio is given by n/3, mechanistic P/O ratios corresponding to ATP synthases with c₈-rings are 2.73 and 1.64 for NADH-linked substrates and succinate, respectively.

APPENDIX 2

Values of the model variable f_III-IV (= 1 − τ_a) estimated from published in vivo oxygen isotope discrimination by respiration in unstressed leaves and roots are variable across species and experiments (Millar et al., 1998; Millenaar et al., 2000; Gonzàlez‐Meler et al., 2001; Noguchi et al., 2001; Guy and Vanlerberghe, 2005; Ribas-Carbo et al., 2005; Florez‐Sarasa et al., 2007; Rachmilevitch et al., 2007; Kornfeld et al., 2013; Cheah et al., 2014; Florez-Sarasa et al., 2014; Del‐Saz et al., 2016; Florez‐Sarasa et al., 2016; Henriksson et al., 2019). This is summarized in Table A1, which is intended to be a broad sampling of relevant measurement data to inform estimates of f_III-IV for modelling rather than an exhaustive compilation or review. When comparisons between growing and mature plant organs or tissues were possible, f_III-IV was generally observed to be larger (i.e. less involvement of the alternative oxidase in respiratory oxygen reduction) in growing tissues than in the mature tissues. Because mature cells are the subject of the model, it is therefore important to note that unless otherwise stated the f_III-IV values in Table A1 that are derived from measurements of root respiration may reflect a mixture of growing and mature cells.