The model is derived from stoichiometric constraints on the ATP, NADH, chloroplastic and cytosolic NADPH, CO2 and O2 fluxes that arise from biosynthesis, maintenance and photosynthesis, under the assumption that metabolic intermediates – not products – are in steady-state. In other words, we assume that metabolic carbon flow and the associated adenylate and reductant cycling and gas exchange are determined entirely by demand for anabolic products and energy carriers. Those demands are treated as input parameters or are calculated from the photosynthesis model of Farquhar et al. (1980). This assumption allows us to create and solve a set of steady-state supply/demand equations for adenylates and reductants. We then calculate fluxes of CO2 and O2 from the solution. A heuristic diagram of the model, meant to aid the reader in following the derivation below, is given in Fig. 1.
We assume that all reduced carbon comes either from sucrose, starch, or triose phosphates (TP), but that photosynthetically derived TP is the preferred C source for anabolic demands because of the ATP cost of accessing stored carbohydrates. Carbon flow from these sources is separated into five streams. The first stream, termed catabolic carbon flow, is the rate of carbon flow to CO2 via glycolysis and the tricarboxylic acid (TCA) cycle that is not attributable to anabolic carbon processing; we use the symbol Vcat to denote this stream. The second stream is flow through the chloroplastic oxidative pentose phosphate pathway (OPPP) to CO2 at a rate Vopp. The third is flow through the cytosolic OPPP to CO2 at a rate Vopc. The fourth is flow into anabolic products at a rate Vana and the fifth is flow to CO2 as a by-product of flows into anabolic products, at a rate Vby. This fifth stream represents CO2 lost in pyruvate decarboxylation or in intermediate steps of the TCA cycle when carbon atoms remaining in decarboxylated compounds end up in anabolic products rather than being fully oxidised to CO2.
We define five terms representing energy carrier supply or demand not arising from photosynthesis or photorespiration. Four of these terms arise from anabolic carbon flow (biosynthesis): Bt, Bn, Bp and Bc, which denote, respectively, ATP demand, chloroplastic NADPH demand, cytosolic NADPH demand and NADH supply. Because, and on balance, more NADH is usually produced than consumed during anabolism, Bn is expressed as a supply, so that it is typically positive. This may appear confusing, but it makes the solution easier to interpret. Bt includes ATP costs of de novo biosynthesis, turnover biosynthesis and, where relevant, polymerisation. A fifth term, M, representing ‘maintenance’ ATP demands, includes repolymerisation of existing amino acids and maintenance of ion gradients. In the Supporting Information, we provide a brief justification of numerical values assigned to these terms in our simulations, based on major biosynthetic and maintenance demands estimated from leaf composition data.
These anabolic and maintenance supply/demand terms are combined with similar terms that account for ATP synthesis and NADH oxidation in mitochondria, ATP and photoreductant supply and demand associated with photosynthesis and photorespiration, and export of redox equivalents from chloroplasts. All of these terms are brought together to create steady-state supply/demand equations (flux-balance equations) for energy carriers. The simultaneous solution of these equations yields expressions for net non-photorespiratory CO2 release and O2 consumption. Expressions also arise for the maximum and actual P : O ratios for oxidative phosphorylation, from which we can infer the relative involvement of non-phosphorylating pathways in mitochondrial electron transport). We derive the flux-balance equations below.
NADH/FADH2. NADH equivalents arise from several sources. First, every 3-C triose-phosphate (TP) molecule entering this stream generates one NADH in glycolysis, one in PDH, and four equivalents in TCA (3 NADH + 1 FADH2) (FAD accepts electrons from succinate in the sixth step in the TCA cycle; this is discussed under P : O Ratio below). The rate of catabolic production of NADH equivalents is thus 2Vcat (2 NADH equivalents per source C atom). Export of excess photoreductant from chloroplasts via the Mal/OAA shuttle also supplies NADH, at a rate Vpx (discussed under Chloroplastic NADPH below). Biosynthetic demand for carbon skeletons drives carbon flow through some steps of glycolysis and the TCA cycle, generating NADH at the rate Bn. NADH is consumed by oxidation at the mitochondrial electron transport chain (mETC). Finally, the rate of NADH consumption equals 2Ro, where Ro is the rate of O2 reduction by the mETC.
In photorespiratory conditions, NADH is also generated via Gly decarboxylation in mitochondria at a rate 0.5φVc, where Vc is the rate of RuBP carboxylation and φ is the ratio of oxygenation to carboxylation rate. The fate of this NADH is important but uncertain. An equal amount of NADH is needed to reduce hydroxypyruvate to glycerate in peroxisomes, and since NADH can be shuttled between organelles and the cytosol via the Mal/OAA shuttle (Heineke et al. 1991; Reumann, Heupel & Heldt 1994; Hoefnagel et al. 1998), it seems inconsistent to exclude such shuttling from a model based on stoichiometric flux balance. However, the few available data (Krömer & Heldt 1991b; Hanning & Heldt 1993) indicate that only 25–50% of photorespiratory NADH is exported to peroxisomes; the remainder remains in mitochondria and can contribute electrons to the mETC. We address this issue by assuming a fraction fm (nominally 0.5) of photorespiratory NADH remains in mitochondria. The limiting case of unlimited NADH transfer between mitochondria and peroxisomes can be assessed by setting fm = 0. The steady state supply/demand equation for NADH equivalents is then found by setting source terms (at left) equal to sink terms (at right):
Chloroplastic NADPH. There are two sources and four sinks for chloroplastic reductant. Linear photosynthetic electron transport at a rate Ja generates NADPH at a rate Ja/2. The chloroplastic oxidative pentose phosphate pathway (OPPP) generates 6 NADPH and 3 CO2 per TP, so the NADPH production rate is 2Vopp. NADPH is consumed by the Calvin cycle at the rate (2 + 2φ)Vc (FCB). Anabolism also consumes NADPH in chloroplasts, at the rate Bp. Chloroplasts must also export reducing equivalents to peroxisomes at a rate 0.5fmφVc to make up for the fraction of photorespiratory NADH that remains in mitochondria. Finally, at high light, excess photoreductant can be exported via the Mal/OAA shuttle, at a rate Vpx. The steady state supply/demand equation for chloroplastic NADPH is then
where ω = 2 + 2φ + 0.5fmφ. Note that Eqn A2 is a generalisation of the FCB model. Under RuBP limited conditions, FCB predicts Vc by setting photoreductant demand (2 + 2φ)Vc equal to supply (Ja/2). Equation 2 extends that model to account for anabolic reductant demands (Bp), catabolic reductant supply (Vopp) and reductant export (Vpx). The latter two variables are undetermined, so we require two additional constraints to solve Eqn A2. Chloroplastic glucose-6-phosphate dehydrogenase (G6PDH), the key regulatory enzyme in the OPPP, is strongly redox inhibited in the light (Buchanan 1980). The precise light dependence of this inhibition in higher plants is unknown, but in single-celled algae studied by Farr et al. (1994), inhibition was evident at 5 µE m−2 s−1 and complete by 10–20 µE m−2 s−1. We assume that the chloroplastic OPPP is downregulated in proportion to increasing thylakoid electron transport rate, so that reductant supply never falls below anabolic demand. This implies:
where Ja has been replaced by J, potential electron transport rate, because Ja = J at low light. Equation A3 implies that CO2 release in the chloroplastic OPPP is fully inhibited when J ≥ 2Bp; however, it does not preclude activity of reversible components of the OPPP in the light for the production of carbon skeletons (Hauschild & von Schaewen 2003). It also implies that NADPH generation in this narrow range of low light (0 < J < 2Bp) serves only to supplant OPPP activity, so that no net carboxylation of RuBP occurs in this range. Whether this actually occurs is unknown; it is a prediction of the model.
Photoreductant supply can exceed anabolic and Calvin cycle demand at high light, in which case chloroplasts may export excess reductant to maintain flux balance. The potential excess rate of NADPH generation is found by applying Vopp = Vpx = 0 and Ja = J to Eqn A2 to give J/2 + 2Vopp – Bp – ωVc. It is unclear how much of this excess actually generates NADPH, because energy can be dissipated by other mechanisms (e.g. the xanthophyll cycle and fluorescence). We assume a fraction (fx) of excess reducing potential generates excess NADPH that is exported; another constraint on fx emerges below, in Flux balance constraints on P : O ratio and photoreductant export. The export term Vpx in Eqn A1 is then given by:
Finally, Ja must satisfy Eqn A2, so
We consider only one source and one sink for cytosolic NADPH. The cytosolic OPPP generates NADPH at a rate 2Vopc, and anabolic demand consumes cytosolic NADPH at the rate Bc. We assume these two processes are perfectly coupled; i.e. the cytosolic OPPP operates only to satisfy anabolic demands for NADPH in the cytosol, and that it does not contribute significant reductant flows to the mETC. Thus,
All of the anabolic NADPH demands accounted for in this paper are localised within chloroplasts, so Vopc plays no role in the simulations shown here. It is included only for completeness.
We consider three sources of ATP. Catabolic carbon flow through glycolysis and the TCA cycle generates three ATP per triose phosphate (one per C), so the rate of ATP generation equals the rate of carbon flow, or Vcat. Oxidative phosphorylation yields ATP at a rate given by the product of the O reduction rate (2Ro) and the P : O ratio (po, the ratio of ADP phosphorylation rate to O reduction rate). po is discussed in detail below. Photophosphorylation generates ATP at a rate 0.5peJa, where pe is the number of ADP phosphorylated per two electrons flowing through the photosynthetic electron transport chain (often denoted ‘ATP : 2e-’). pe may vary and can be enhanced by cyclic electron transport around PSI; in the absence of cyclic flow it is probably between 3/2 (1.5) and 9/7 (∼1.286). These values assume either 12 or 14 protons, respectively, per 3 ATP, and 12 protons per four electrons (Seelert et al. 2000; Allen 2003). We will use the larger value for pe by default, but will compare the resulting predictions with simulations based on pe = 1.286.
ATP is consumed by biosynthesis and maintenance at the rates Bt and M, respectively. The Calvin cycle and photorespiration together consume ATP at a rate (3 + 3.5φ)Vc (FCB). ATP is also needed to store or export excess photosynthate or to access stored carbohydrate. Starch synthesis consumes one ATP per 6-C sugar, as does sucrose synthesis and export in apoplastic phloem loaders. Starch breakdown consumes two ATP per 6-C sugar. Sucrose breakdown consumes one or two ATP equivalents, via the sucrose synthase and invertase routes, respectively; we assume the latter. Denoting the combined net rate of sucrose export and starch synthesis as S (note S = Vc – 0.5Vo – Vana, the difference between photosynthetic yield and anabolic carbon flow), the net ATP demand is tS, where t = −1/3 if S < 0 and +1/6 if S > 0 (note tS > 0 in either case). The steady state supply/demand equation for ATP is then:
P : O ratio (po)
The stoichiometry of ATP production and e- flow to O2 in mitochondria depends on two factors: where electrons are donated to the mETC, and the activities of non-phosphorylating pathways. Electrons donated at Complex I yield a maximum of 2.3–3 ATP per O reduced, whereas electrons donated elsewhere net up to 1.4–2 ATP per O reduced. The range of quoted values reflects disagreement in the literature (Lee et al. 1996; Gnaiger, Méndez & Hand 2000; Hinkle 2005). Lesser ratios are considered ‘mechanistic’ inasmuch as they are derived from experiments on isolated mitochondria, whereas greater ratios are predicted by theory. We use intermediate values (1.5 and 2.5, or 3/2 and 5/2, respectively), and assume that cytosol-derived NADH is shuttled to the matrix via the Mal/OAA shuttle and donates its electrons at Complex I. The maximum P : O ratio then depends only on the proportional contribution of FADH2. (FADH2 is not labile, and it donates its electrons at Complex II in the mitochondrial inner membrane. Transport of those electrons to cytochrome oxidase can transport a maximum of 6H+– 4 fewer than possible for electrons from NADH oxidation at Complex I. FADH2 therefore supports a lower rate of ATP production.) The maximum P : O ratio, pom, is a reductant flow weighted average of these two values, where the individual flows are the positive terms in Eqn A1 (2Vcat, Bn, Vpx and 0.5fmφVc). In purely catabolic oxidation of carbohydrates, one of six reducing equivalents is FADH2, so the maximum P : O for Vcat is (1/6) · (3/2) + (5/6) · (5/2), or 7/3. Since oxaloacetic acid (OAA) is the only TCA intermediate downstream of FADH2 that is consumed in anabolic processes that we account for in this paper, and since we assume OAA is generated by PEP carboxylation for that purpose, we can consider all redox equivalents generated as a by-product of anabolic carbon flow as NADH. Therefore the larger P : O-value (5/2) applies to Bn. The same is true for redox equivalents exported from chloroplasts and for NADH generated in mitochondria by photorespiratory Gly conversion, so the 5/2-value applies to Vpx and 0.5fmφVc as well. The effective maximum P : O ratio is then:
Flux balance constraints on P : O ratio and photoreductant export
If reductant supply to the mETC exceeds ADP supply, flux balance can be maintained in one of two ways: by reducing NADH and ATP supply from catabolic substrate oxidation or by reducing the P : O ratio, perhaps by increased engagement of AOX or uncoupling protein. There is evidence that AOX activity is strongly enhanced by stress (Finnegan, Soole & Umbach 2004), including water stress (Bartoli et al. 2005; Ribas-Carbo et al. 2005), low temperatures (Matos et al. 2007; Mizuno et al. 2007) and excess light (Noguchi et al. 2005; Yoshida, Terashima & Noguchi 2007), and that it serves as a supplementary electron sink, probably to regenerate NAD+ and/or to prevent generation of reactive oxygen species (Amirsadeghi et al. 2006). A core hypothesis of our model is that po is downregulated to maintain flux balance, first by reducing TCA cycle activity, and then by downregulation of po. We implement this hypothesis with a conditional statement: if flux balance requires Vcat < 0 in Eqn A7, then Vcat is zero and flux balance is maintained by reducing po below pom. This conditional statement is formally expressed by Eqn A13 below.
However, although reducing po to zero can permit a continued increase in oxidation of redox equivalents by the mETC, it cannot ensure adenylate balance if ATP supply continues to increase – as, for example, may occur in chloroplasts at high light. This requires an additional constraint on adenylate balance, beyond those already explicit in Eqn A7. The only arbitrary term in Eqn A7 is the parameter fx, the fraction of excess thylakoid reducing potential that is dissipated via NADPH export. Photoreductant export affects adenylate balance because photosynthetic e- transport is inextricably coupled to photophosphorylation. Thus, when po = 0, adenylate balance places a constraint on thylakoid electron transport rate, Ja:
which is found by setting po = Vcat = 0 in Eqn A7. Substituting A4 into A5, setting Vopp = 0, applying the result to A9 and solving for fx gives
This produces negative or undefined values at low light (e.g. when J ≤ 2Bp), but in this case Vpx is zero, so the value of fx is irrelevant in such cases.
Non-photorespiratory CO2 release (Rc) arises in four ways: by chloroplastic and cytosolic OPPP activity at rates Vopp and Vopc, respectively; by catabolic substrate oxidation at a rate Vcat, and as a by-product of anabolic carbon flow, at a rate Vby. Rc is therefore given by
(We use the notation Rc to distinguish it from Rn and Rd[the conventional notations for mitochondrial respiration rate at night and in the daytime, respectively,], because Eqn A11 is meant to apply either in light or in darkness.) The four terms on the right-hand side of Eqn A11 must be supplied to calculate Rc. Vby is a model input, Vopc is given by Eqn A6, and Vopp is given by Eqns A2–A4, leaving only Vcat. To constrain Vcat, we will solve Eqns A1 & A7 for Vcat by eliminating Ro. We will then use the conditional assumption outlined above under Flux balance constraints to eliminate the P : O ratio. Eliminating Ro from Eqns A1 & A7 gives Vcat as
The requirement that Vcat ≥ 0 and the assumption that po = pom if Vcat > 0 leads to
pom is eliminated from Eqn A14 by combining Eqns A8 & A14 and solving for Vcat to give
The factor 3/17 arises from 2(7/3) + 1, which is the denominator after pom has been eliminated (7/3 is the maximum P : O ratio for reductant arising from catabolic substrate oxidation; see P : O ratio above). The remaining terms in Eqn A15, and thus in Eqn A11 for Rc, are all given by equations above: Vpx is given by Eqns A4 & A10, Ja by Eqn A5 and Vc by Eqns A18 & A19 below.
Expressions for the rate of O2 consumption by mitochondria (Ro), the actual P : O ratio (po) and the net rates of CO2 assimilation (A) and oxygen evolution (Ao) can also be obtained from the equations above, when combined with the FCB model. Ro is found by solving Eqn A1:
po is found by solving Eqn A7 for po and substituting Eqn A16 for Ro:
Note that this equals pom if Vcat > 0; otherwise, it equals the value of the overall P : O ratio needed to ensure flux balance for both ATP and NADH. From FCB, net CO2 assimilation rate is Vc – 0.5Vo – Rc = (1 – 0.5φ)Vc − Rc. The rate of RuBP carboxylation (Vc) is the lesser of the RuBP-saturated and -limited rates (Wc and J′, respectively). Wc is given by Eqn 9 in FCB:
where Kc and Ko are the Michaelis constants for RuBP carboxylation and oxygenation, respectively, and ci and O are intercellular CO2 and ambient O2 partial pressures, respectively. To find J′, we begin by considering Vc above the light level causing full inhibition of chloroplastic OPPP (Vopp = 0) but below the level producing excess photoreductant (Vpx = 0). Applying Vopp = Vpx = 0 to Eqn A2 gives Vc = (J/2–Bp)/ω. Generalising this to include the low-light regime in which Vopp > 0 but Vc = 0, we have
We calculate the minimum of Wc and J′ hyperbolically to reflect co-limitation near the transition (Farquhar et al. 1980): θvVc2 – (Wc + J′)Vc + WcJ′ = 0, where θv is a dimensionless parameter (θv = 1 creates a sharp transition; we used θv = 0.999). From FCB, φ, the ratio of RuBP oxygenation rate to carboxylation rate, is 0.21(O/Ko)/(ci/Kc). Noting from FCB that φ = 2Γ*/ci, where Γ* is the photorespiratory CO2 compensation point, we have:
This differs from the FCB model by two terms, both affecting the RuBP-limited phase: the offset of J by 2Bp, and the factor 0.5fm multiplied by Γ* in the denominator. If fm is on the order of 0.5, as has been reported previously (Krömer & Heldt 1991b), then its effect on the shape of the A vs. ci curve is negligible. The effect of the Bp term depends on NADPH demand, including the ferredoxin needed to reduce NO2- to NH4+ in N assimilation. Net photosynthetic O2 evolution arises from water splitting at PSII and photorespiration. For each RuBP oxygenated, 1.5 O2 are consumed, so the consumption rate is 1.5Vo (1.5φVc). The rate of O2 evolution from water splitting equals 1/4 of the actual electron transport rate, Ja. Thus, noting that O2 consumption by the mETC is Ro, the net O2 evolution rate (Ao) is:
Values are needed for ci, O, Kc, Ko, Vm and J. We used values for Kc and Ko at 25 °C from Bernacchi et al. (2001). We calculated J from incident PPFD (I) and maximum potential e- transport rate (Jm) as the lesser root of θjJ2 – (Jm + fI)J + fJmI (Farquhar & Wong 1984), where θj = 0.86 and f (0.3 e-/hν) is the product of absorbance and maximum quantum yield. Jm was taken as 2.1 Vm (Wullschleger 1993). We estimated Vm from leaf N content, based on the data used to estimate anabolic demands (see Supporting Information). ci, O and I were varied among simulations. These parameter values are given in Table 3.