2.1 Accommodating photosynthetic insensitivity to CO₂ and O₂

The canonical FvCB model predicts that A will always increase with increasing CO₂ level, despite a lower increase in the W_j-limited range than in the W_c-limited range. However, many (e.g., Sharkey, 1985a) showed that A can be insensitive to changes in the CO₂ partial pressure within the high CO₂ range, in particular in combination with high irradiance, or low O₂ partial pressure, or at low temperature. Sharkey (1985b) hypothesized that this insensitivity was due to the limitation set by the rate at which TP are utilized in the synthesis of sucrose or starch. As the use of TP is stoichiometrically exchanged with the release of phosphate (P_i) during sucrose or starch synthesis, a limitation in TP utilization (TPU) could result in a limitation to photophosphorylation, and, thus, to RuBP regeneration. So, in addition to what has been assumed about the control of RuBP regeneration by electron transport in the canonical FvCB model, RuBP regeneration can be limited further by other components as in the Calvin–Benson cycle and beyond. If TPU limits, the equation for A, equivalent to Equation (5), in the FvCB model, should be extended as:

(6a)

where W_p is the rate of carboxylation set by TPU limitation.

The carboxylation of one mol RuBP results in two mol TP but the Calvin–Benson cycle stoichiometry suggests that only one-sixth of the TP is used for sucrose or starch synthesis, whereas the remaining five-sixths of the TP are drawn back into the cycle to contribute to the regeneration of RuBP (Taiz & Zeiger, 2002). Thus, the P_i consumption by sucrose or starch synthesis is 2V_c/6 = V_c/3. Considering the carbon loss in the photorespiratory cycle, the net P_i consumption would be (1–0.5ϕ)V_c/3, and this must be equal to the release of P_i via TPU if P_i is limiting. Let T_p be the rate of TPU, then one can write:

(6b)

Substituting Equation (6b) into Equation (6a) gives the net CO₂-assimilation rate limited by TPU, A_p, as:

(6c)

This is the simple equation given first by Sharkey (1985b), which suggests that if TPU is limiting, A is no longer sensitive to changes in CO₂ or O₂ partial pressure, or in irradiance. It sets an upper limit to net assimilation rate.

2.2 Accommodating the reversed sensitivity to CO₂ and O₂

It has been frequently observed that A even declines with increasing CO₂ partial pressure within the high CO₂ range, particularly under low O₂ conditions (e.g., von Caemmerer & Farquhar, 1981). Similarly, increasing O₂ has been observed to stimulate CO₂ assimilation under high CO₂ conditions (Harley and Sharkey (1991). These reversed sensitivities to CO₂ and O₂ cannot be explained by the simple model, Equation (6c).

Sharkey and Vassey (1989) proposed that the reverse sensitivity was caused by inhibition of starch synthesis capacity, and in turn caused reduced stromal phosphoglucoisomerase activity resulting from metabolites interfering with its activity. An alternative explanation was proposed by Harley and Sharkey (1991) that a fraction of the glycolate carbon, which leaves the chloroplast and is recycled to glycerate in the photorespiratory cycle, does not return to the chloroplast, but after converting to glycine, is diverted from the photorespiratory cycle and used elsewhere for amino acid synthesis. Thus, the P_i normally used in converting glycerate to 3-PGA is made available for phosphorylation instead, thereby, stimulating RuBP regeneration. Based on this hypothesis, Harley and Sharkey (1991) used three values for the fraction (0.0, 0.5, and 1.0) to fit data and showed how the curvature of photosynthetic CO₂-response curves (A–C_i curves) had varying extents of the reversed CO₂ and O₂ sensitivity.

Based on the analysis by Harley and Sharkey (1991), von Caemmerer (2000) formalized the model by using α as the fraction of the glycolate carbon that is not returned to the chloroplast. As one oxygenation produces 0.5 glycerate, which consumes one P_i, the rate of P_i consumption, which usually is (1 − 0.5ϕ)V_c/3, should be decreased by αV_o/2, or αϕV_c/2. Thus, the net P_i consumption in this case would be [(1 − 0.5ϕ)/3 − αϕ/2]V_c. In analogy to Equation (6b), W_p as the rate of carboxylation set by TPU limitation becomes:

(7a)

The model for the net CO₂-assimilation rate, A_p, becomes:

(7b)

where 0 ≤ α ≤ 1. If α = 0, Equation (7b) becomes Equation (6c), representing the case that glycolate carbon maximally returns to the chloroplast (i.e., ¾ of the glycolate carbon is recycled as glycerate; the other ¼ is lost as CO₂ as the result of glycine decarboxylation). Harley and Sharkey (1991) showed that for the same value of α, TPU starts to limit A at a lowering CO₂ level with increasing irradiance, with decreasing O₂ level, and with decreasing temperature. The reverse sensitivity that can occur based on Equation (7b) is frequently observed but occasionally the reverse sensitivity is greater than what can be accounted for by Equation (7b). It is likely that both the incomplete photorespiratory cycle explanation and the starch inhibition explanation (Sharkey & Vassey, 1989) can be valid, although in our experience the incomplete photorespiratory cycle phenomenon is more common.

2.3 Implications of TPU limitation in modelling leaf photosynthesis

Ellsworth, Crous, Lambers, and Cooke (2015) showed that TPU limitations to photosynthetic capacity are common in woody species grown in the field. However, TPU might not be the most important limitation under current climatic growth conditions, as evidenced by Kumarathunge, Medlyn, Drake, Rogers, and Tjoelker (2019), who reported that only ca 30% of A–C_i curves showed an obvious TPU limitation in a global data representing 141 species. Irrespective of its uncertain importance under field conditions, the inclusion of TPU limitation in models is important for elucidating the basic principles of photosynthetic mechanisms. In cases where TPU is actually limiting, the canonical, two-limitation FvCB model would underestimate J_max (when fitting to A–C_i curves) and V_cmax or J_max (when fitting to light response curves) because the maximum photosynthetic rate would be wrongly attributed to being limited by electron transport or by Rubisco activity.

It is important to note that TPU limitation is a form of very short-term sink–source disequilibrium (McClain and Sharkey (2019). It concerns the ability to remove TP quickly from the Calvin–Benson cycle. The half-life time of the cycle intermediates can be shorter than 1 s, while some larger pools still have a half-life time of <1 min. This means that TPU limitation can build up and disappear quickly. As discussed by Sharkey (2019), when plants are put into TPU limited conditions for hours or days, the TPU limitation is observable at first; but then other components like electron transport are regulated to a level that TPU is no longer “apparently” limiting (e.g., Pammenter, Loreto, & Sharkey, 1993). Furthermore, over a longer time, a larger sink can remove short-term TPU limitation. Kaschuk et al. (2012) showed that nodulated soybean plants had 14%–31% higher rates of photosynthesis and accumulated less starch in the leaves than nitrogen-fertilized plants, supporting that rhizobial symbiosis could stimulate photosynthesis due to the removal of carbon sink limitation by nodule activities.

Conversely, a small sink, especially when combined with a large source, can cause TPU limitation. Fabre et al. (2019) reported the occurrence of TPU limitation in panicle-pruned rice plants, especially those grown under 800 μmol mol⁻¹ CO₂. This reduction was associated with sucrose accumulation in the flag leaf resulting from the sink limitation. The photosynthetic stimulation by the elevated-CO₂ was lower in pruned plants compared with control plants, and this response to CO₂ in relation to sink size was also found when comparing various rice genotypes having contrasting leaf:panicle size ratios or source:sink ratios (Fabre et al., 2020). A recent review by Dingkuhn et al. (2020) even found the evidence from broader ranges of genotypes that stronger elevated-CO₂ responsiveness in wild relatives and old cultivars of crops is related to sink strength as a result of adaptive plasticity involving branching. Perhaps, the most important result in recent work of Fabre et al. is that TPU, thus net CO₂ assimilation rate, declines increasingly with time after the midday in a diurnal cycle. These findings suggest that not only TPU limitation in regulating photosynthesis should be considered, but also a shorter time-step would be needed to account for diurnal variations in sink feedback limitation to photosynthesis, in dynamic crop models for projecting the CO₂-fertilization effect on crop production.

3.1 Accommodating a balanced ATP:NADPH ratio

In the canonical FvCB model, there are two different equations, Equations (3a) and (3b), for the same electron transport-limited carboxylation rate, W_j. By comparison of the two equations, one can immediately recognize that the value of W_j determined by the ATP supply is more limiting than that determined by the NADPH supply. The two equations were used largely in a random manner in the literature before 2000. To eliminate the “random” application of the FvCB model, Yin et al. (2004) developed a generalized model that covers, among others, the two forms of the FvCB model for the electron transport-limited rate.

It is apparent that, according to the stoichiometric coefficients accepted in 1980, the LET produces an ATP:NADPH ratio of 1.333 [resulting from (2/3):(1/2), see Equation (3a) vs. Equation (3b)], well below 1.5 as required by the Calvin–Benson cycle, with ATP in deficit relative to NADPH. Chloroplasts engage several mechanisms that could remove the disparity in terms of requirement for the correct ATP:NADPH ratio (Allen, 2003; Baker, Harbinson, & Kramer, 2007; Farquhar & von Caemmerer, 1982). First, instead of going to the end electron acceptor NADP⁺, a fraction of electrons passing PSI may follow a cyclic electron pathway (f_cyc) (Figure 2). The cyclic electron transport (CET) does not produce NADPH, but passes through the “coupling” sites of ATP synthase (Allen, 2003), thereby being able to increase the ATP:NADPH ratio. Second, part of the noncyclic electrons may be used to support processes like the Mehler ascorbate peroxidase reaction or nitrate reduction, where O₂ directly or nitrate indirectly act as the electron acceptors, respectively (Figure 2). Every one mol O₂ uptake in the Mehler ascorbate peroxidase reaction is accompanied by one mol O₂ production from the splitting of water at PSII; so this reaction consumes four electrons per O₂ but requires no ATP (Asada, 1999). The first step of the reduction of nitrate into nitrite takes place in the cytosol but may use reducing power generated in the chloroplast (e.g., via the malate shuttle) and the subsequent steps in converting nitrite to ammonia and to glutamate take place in the chloroplast stroma, using the reduced ferredoxin (Noctor & Foyer, 1998). One mol nitrate reduction requires 10 mol electrons and only one mol ATP (Noctor & Foyer, 1998). Thus, both the Mehler ascorbate peroxidase reaction and nitrate reduction can help to adjust the ATP:NADPH ratio as required by the Calvin–Benson and the photorespiratory cycles. There are some other minor processes like sulphur assimilation and fatty acid biosynthesis that might use chloroplastic electrons but these are quantitatively less significant. For the convenience of modelling, the noncyclic electron transport in support of the Mehler ascorbate peroxidase reaction, nitrate reduction and any other minor processes is collectively named as the pseudocyclic category, and this fraction is denoted as f_pseudo (Yin et al., 2004). Therefore, the fraction for LET (i.e., the fraction of the total electron flux passing PSI that is to support the Calvin–Benson and the photorespiratory cycles) is (1 − f_cyc − f_pseudo) (Figure 2). Yin et al. (2004) derive a relationship for fractions of various electron transport pathways that must be met in order to ensure that the produced ATP:NADPH ratio is compatible with the required ratio by the Calvin–Benson and the photorespiratory cycles:

(8)

where h is the number of H⁺ required per ATP synthesis and f_Q is the fraction of electrons at the plastoquinone that follows the Q cycle (Figure 2).

In the presence of CET, the PSI electron flux (J₁) is higher than the PSII electron flux (J₂): J₁ = J₂/(1 − f_cyc) (Yin et al., 2004). The LET in support of the Calvin–Benson and the photorespiratory cycle is (1 − f_cyc − f_pseudo)J₁ (Figure 2). Combining these equations with Equation (3a) of the FvCB model gives:

(9a)

Substituting Equation (8) to Equation (9a) gives:

(9b)

The two forms of electron transport-limited part of the canonical FvCB model are special cases of this extended model. If f_pseudo = 0, Equation (9a) becomes Equation (3a); in such a case, the whole PSII electron flux equals the LET (J₂ = J). If f_Q = 0, h = 3, and f_cyc = 0, Equation (9b) becomes Equation (3b). So, the canonical FvCB model implies no operation of the Q cycle and a requirement of three H⁺ per ATP synthesis (the H⁺:ATP ratio h = 3).

However, the contemporary belief is that the Q cycle may operate obligatorily (f_Q = 1; e.g., Sacksteder, Kanazawa, Jacoby, & Kramer, 2000), and this cycle will effectively double the stoichiometry of the H⁺ translocation through the cytochrome b₆f complex from one H⁺ to two H⁺ per electron passed therein (Figure 2). So, plus one H⁺ pumped from splitting the water molecule through the PSII complex, a total of three H⁺ (instead of two) produced per electron are transferred along the whole-chain if the Q cycle operates (von Caemmerer, 2000; also see Figure 2). Also, the H⁺:ATP ratio is probably either 4 based on thermodynamic experiments (Petersen, Förster, Turina, & Gräber, 2012; Steigmiller, Turina, & Gräber, 2008) or 4.67 (=14/3) from the structural data for the c14 rotor ring of the H⁺ translocating chloroplast ATP synthase (Seelert et al., 2000; Hahn, Vonck, Mills, Meier, & Kühlbrandt, 2018), instead of 3 used in the FvCB model. If f_Q = 1, h = 4, and f_cyc = 0, then the produced ATP:NADPH ratio from the noncyclic electron pathway is 1.5, exactly matching the ratio required by the Calvin–Benson cycle and Equation (9b) becomes Equation (2.22) in the book of von Caemmerer (2000) for this scenario, that is,

(9c)

If f_Q = 1, h = 4.67, and f_cyc = 0, the ATP:NADPH ratio from the noncyclic electron pathway is 1.286 (even lower than 1.333 assumed in the canonical FvCB model), and the value has often been cited in the recent literature to stress the surplus of the reducing power which might be exported to cytosol (e.g., Lim et al., 2020). For such a case, Equation (9b) becomes:

(9d)

Clearly, the model of Extension 2 represents the generalized algorithm for various scenarios with regard to the H⁺:electron and the H⁺:ATP ratios. Equation (9b) actually contains the ATP production factor (z) per electron transferred through PSII when CET occurs simultaneously (also see Equation (B8b) in the Appendix B of Yin et al., 2004):

(9e)

This ATP:electron ratio factor z is 2/3 in Equation (3b) of the canonical FvCB model, 3/4 in the case of Equation (9c), and 9/14 in the case of Equation (9d). The z factor also predicts that for a given set of f_Q and h, the ATP:electron ratio increases expectedly with increasing f_cyc (see later for C₄ photosynthesis). Given that the Q cycle may not necessarily switch absolutely on (f_Q = 1) and off (f_Q = 0) but run partially (Cornic, Bukhov, Wiese, Bligny, & Heber, 2000), the model allows such scenarios with 0 ≤ f_Q ≤ 1. As noted by Yin et al. (2004), the model assumes that the Q cycle, either obligatorily or partially operated, is impartial to cyclic and noncyclic electrons (Allen, 2003).

3.2 Quantum efficiency of electron transport when cyclic and noncyclic pathways co-occur

When CET and noncyclic (including linear and pseudocyclic) electron transport run simultaneously, a higher electron flux is expected in PSI than in PSII. This means that the fraction of light energy partitioned to PSI and PSII may not be 0.5 each as set by Equation (4) in the canonical FvCB model, but higher than 0.5 for PSI. On the other hand, the partitioning factor must also depend on the photochemical efficiency of the two photosystems, with partitioning in favour of the less efficient PSII in the absence of CET, given that the photochemical efficiency of PSII (Φ₂) is lower than that of PSI (Φ₁) (e.g., Hogewoning et al., 2012). Yin et al. (2004) developed an analytical equation for describing the parameter α_2(LL), the quantum efficiency of PSII electron transport (under limiting light, LL) on the basis of absorbed photons by both photosystems:

(10a)

The fraction of absorbed light partitioned to PSII, ρ₂, can be formulated as:

(10b)

Equations (10a) and (10b) both suit for limiting light conditions, as well as for nonlimiting light conditions (if the subscript _(LL) is removed) as long as A is limited by electron transport. The model for describing J₂ as a function of the full range of absorbed irradiance (I_abs) can be formulated in analogy to Equation (4) as:

(11)

where J_2max is the maximum value of the potential J₂ under saturating irradiance, to differentiate it from J_max in Equation (4) that stands for the maximum rate of the potential LET.

3.3 Quantum yield of CO₂ uptake and of O₂ evolution

It is convenient to derive the expression for quantum yield of CO₂ uptake (Φ_CO2(LL)), from Equations (5), (9a), (10a) and (11) in terms of NADPH supply:

(12a)

Likewise, Φ_CO2(LL) can also be expressed in terms of ATP supply:

(12b)

Equivalent equations based on the canonical FvCB model are:

(12c)

(12d)

The FvCB model assumes that ρ₂ = 0.5. Comparison of Equations (12a) and (12c) immediately identifies that the f factor in the FvCB model, representing the fraction of I_abs unavailable for Calvin–Benson and photorespiratory cycles, can be expressed as: f = 1 − Φ_2LL[1 − f_pseudo/(1 − f_cyc)]. In other words, the factor f actually lumps multiple components, including the non-photochemical loss of PSII (Φ_2LL known not to be higher than 0.85, Björkman & Demmig, 1987), cyclic electron transport (f_cyc), and pseudocyclic electron components (f_pseudo) that support alternative metabolic processes. Much literature after Farquhar et al. (1980) often only refers to f to correct for spectral quality of the light (e.g., von Caemmerer, 2000). While this definition of f reflects the often-reported wavelength dependent photosystems' photochemical efficiencies and absorption by carotenoids and nonphotosynthetic pigments (e.g., Evans, 1987; Hogewoning et al., 2012), it is more difficult to reconcile well with the insights from the extended model.

Photosynthetic quantum yield can also be expressed in terms of O₂ evolution (Φ_O2(LL)). The electron requirement in support of both Calvin–Benson and photorespiratory cycles leads to O₂ evolution at PSII from the splitting of H₂O; so, the total O₂ evolved can be expressed as (1 + 2Γ_*/C_c)V_c. The O₂ uptake by photorespiration consists of (i) one mol O₂ consumed per mol RuBP oxygenation, and (ii) a further one mol O₂ consumed in the conversion of one mol glycolate to one mol glyoxylate by glycolate oxidase in the peroxisome, producing one mol hydrogen peroxide (H₂O₂) which is immediately destroyed by the action of catalase into one mol H₂O and 0.5 mol O₂ (Figure 1b). So the total O₂ uptake associated with the photorespiratory pathway is 1.5 mol O₂ per mol RuBP oxygenated, which can be expressed as 1.5V_o = (3Γ_*/C_c)V_c. Taking these together, the Rubisco-linked net O₂ evolution is (1 − Γ_*/C_c)V_c, which is the same as for CO₂ uptake (von Caemmerer, 2000).

The Mehler ascorbate peroxidase reaction consumes noncyclic electrons, but its stoichiometry is that for every mol O₂ directly reduced in this reaction, 0.5 mol O₂ is released by superoxide dismutase and 0.5 mol O₂ is evolved through the splitting of H₂O at PSII such that the reaction results in no net O₂ exchange (Asada, 1999). In contrast, processes like nitrate reduction, also consuming noncyclic electrons, do result in O₂ evolution. Thus, if photosynthetic quantum yield is expressed in terms of O₂ evolution (Φ_O2(LL)), we can break down f_pseudo in Equation (12a) into two components: one for the Mehler ascorbate peroxidase reaction and one for other basal components, and the latter is no longer needed in the equation for Φ_O2(LL). As the Mehler ascorbate peroxidase reaction acts as a photoprotection mechanism when absorbed light energy exceeds the enzymatic capacity of downstream metabolism (Ort & Baker, 2002), this reaction may be negligible under strictly limiting light conditions. Thus, quantum yield of O₂ evolution for the limiting light conditions becomes:

(12e)

3.4 Using the quantum yield model to infer hard-to-measure parameters

The unique feature of Equation (12e) based on the extended model for W_j is that f_cyc is in the model for describing NADPH-dependent quantum yield, in contrast to the conventional belief that CET can generate additional ATP and must appear only in equations for the ATP-dependent quantum yield. Parameter f_cyc does appear in Equation (12b) for the ATP-dependent quantum yield, but Equation (12b) includes uncertain parameters f_Q and h in addition to f_cyc. Relying on this unique feature and the generally conserved PSII:PSI efficiency ratio, Yin, Harbinson, and Struik (2006) showed that a hard-to-measure parameter f_cyc can be calculated from Equation (12e), based on measurable parameters Φ_O2LL and Φ_2LL under nonphotorespiratory conditions:

(13a)

A typical Φ_2LL based on chlorophyll fluorescence measurements is 0.8 and a typical Φ_1LL based on P700 absorption measurements is close to 1.0 or slightly lower (Genty & Harbinson, 1996); so, the Φ_2LL:Φ_1LL ratio is ca 0.85. Φ_O2LL of C₃ photosynthesis in the absence of photorespiration is ca 0.105 (Björkman & Demmig, 1987). The solved f_cyc from Equation (13a) is then ca 0.06. This cannot be considered as an absolute estimate, but suggests that very little CET is needed for C₃ photosynthesis, in line with previous reports (e.g., Avenson et al., 2005).

Once f_cyc is known, one can calculate another hard-to-measure light-partitioning parameter ρ₂ from Equation (10b). The obtained ρ₂ is ca 0.53, close to the assumed value 0.5 in the canonical FvCB model. This indicates that the requirement for a higher partitioning to the less efficient PSII is to some extent balanced by the requirement for a higher partitioning to PSI to run CET. Equation (10b) suggests that ρ₂ equals exactly 0.5 only if the fraction for the noncyclic electron flow, 1 − f_cyc, is equal to the Φ_2LL:Φ_1LL ratio.

By dividing Equation (12a) by Equation (12e) that assumes no Mehler ascorbate peroxidase reaction for the limiting light condition, one can solve for basal f_pseudo from the Φ_CO2LL:Φ_O2LL ratio:

(13b)

Unlike Equation (13a), Equation (13b) applies to both nonphotorespiratory and photorespiratory conditions. A typical value of Φ_CO2LL of C₃ photosynthesis under limiting light in the absence of photorespiration is ca 0.093 (Long, Postl, & Bolhár-Nordenkampf, 1993). This gives an estimate of f_pseudo being ca 0.10. The Φ_CO2LL:Φ_O2LL ratio is also known as the assimilatory quotient, and the value of its complement, (1 − the ratio), indicates the extent to which electrons are used in support of the processes like nitrogen assimilation (Bloom, Caldwell, Finazzo, Warner, & Weissbart, 1989; Skillman, 2008).

Once f_cyc and f_pseudo are known, likely combinations of f_Q and h can be solved from Equation (8) for C₃ photosynthesis. Using the above estimates of f_cyc and f_pseudo for the nonphotorespiratory conditions, the solved h is ca 3.1 if f_Q = 0 and is ca 4.67 if f_Q = 1. The latter combination is very close to the contemporary belief that the operation of the Q cycle is obligatory (Sacksteder et al., 2000) and the structural data that chloroplast ATP synthase requires 4.67 c subunits or protons to produce one ATP (Seelert et al., 2000; Hahn et al., 2018). However, like the canonical FvCB model, Equation (8) does not account for small amounts of ATP required for starch synthesis and nitrogen assimilation. As ATP for these processes most likely come from chloroplasts (Noctor & Foyer, 1998), then the calculated h would approach 4. Energy requirements for nitrogen assimilation will further be discussed next.

4.1 The general model of extension 3 integrating nitrogen and C₁ metabolisms

Busch et al. (2018) used α_G and α_S to denote the fractions of glycolate carbon taken out from the photorespiratory pathway as glycine and serine, respectively. Likewise, Busch (2020) used α_T to denote the fraction of glycolate carbon taken out from the photorespiratory pathway as CH₂-THF. As shown in Figure 3, the glycolate carbon exported in the form of the three-carbon molecule serine has to be less than or equal to the remaining carbon after the glycine export, glycine decarboxylation, and CH₂-THF export: α_S ≤ (1 − α_G) − α_T, where refers to the glycolate: serine carbon ratio (Figure 1b), and refers to half of glycine carbon lost during its decarboxylation. This relation can be converted into α_G + 2α_T + α_S ≤ 1, thereby reflecting that the total proportion of glycolate carbon exports cannot exceed 1. Of course, none of α_G, α_T and α_S can be lower than 0. In analogy to the derivation of Equation (7a) by Harley and Sharkey (1991), the rate of P_i consumption, which usually is (1 − 0.5ϕ)V_c/3, should be decreased by (α_G + 2α_T + α_S)ϕV_c/2, and the net P_i consumption would be [(1 − 0.5ϕ)/3 − (α_G + 2α_T + α_S)ϕ/2]V_c. Thus, W_p as the rate of carboxylation set by TPU limitation in this case becomes:

(14)

Equation (14) becomes Equation (7a) if α_S = 0 and α_T = 0.

While W_c remains unchanged as Equation (2a), the rate of carboxylation as determined by electron transport, W_j, will be affected as the potential electron transport rate J now has to support both carbon and nitrogen assimilation. Photorespiratory carbon entering the C₁ metabolism, in contrast, causes a net release of electrons, as the reaction catalysed by GDC releases electrons and the exit of carbon from the photorespiratory pathway saves electrons downstream that would otherwise be consumed for converting serine to glycerate in the peroxisome and for reducing this glycerate-derived 3-PGA in the chloroplast (Figures 1b and 3). These together bring the equation for electron transport-determined carboxylation rate in terms of NADPH supply to:

(15a)

The denominator can be obtained by summing up all the electron requirements for individual steps, deducted by electron equivalents of the NADH release as a result of glycine decarboxylation, indicated in Figure 3. Likewise, photorespiratory carbon export via the C₁ metabolism saves ATP that would otherwise be used for the phosphorylation of glycerate to 3-PGA and for the subsequent phosphorylation of this 3-PGA (Figures 1b and 3); thus, one can formulate the equation for W_j in terms of ATP supply:

(15b)

where J_atp in the numerator is the total ATP production rate from chloroplastic electron transport (which is not expressed in J like Equation (15a), given the uncertainties discussed earlier in Extension 2). The denominator in Equation (15b) can also be obtained by summing up all the ATP requirements indicated in Figure 3.

Traditionally, the proportion of glycolate carbon that does not return to chloroplasts (α) is relevant only for the TPU-limited carboxylation rate W_p (see Equations (7a) and (7b)). Equations (15a) and (15b) suggest that the proportion parameters (α_G, α_T and α_S) affect not only W_p but also W_j. The export of carbon as CH₂-THF always increases W_j. Glycine and serine export associated with de novo N assimilation decreases W_j in terms of NADPH requirement whereas it increases W_j in terms of ATP requirement. This suggests that photorespiration-associated N assimilation can help alleviate the deficit of ATP relative to NADPH (see earlier discussions).

In the case of glycine being diverted from the photorespiratory pathway, the amount of CO₂ released per oxygenation should be decreased by α_G (Busch et al., 2018). In contrast, as shown in Figure 3, every carbon exported as CH₂-THF from the pathway results in one carbon lost from glycine decarboxylation (Busch, 2020). Therefore, it is necessary to revise Equation (1) to:

(16a)

And Equation (6a) becomes:

(16b)

where Γ _*GT = [0.5(1 − α_G) + α_T]O/S_c/o, or Γ_*GT = (1 − α_G + 2α_T)Γ_*. It follows that the CO₂ compensation point in the absence of day respiration is no longer constant at given temperature and O₂ partial pressure, but decreases with increasing the fraction of glycine and increases with increasing the fraction of CH₂-THF diverted from the photorespiratory pathway. Therefore, equations for the net CO₂-assimilation rate corresponding to the three limitations become:

(16c)

(16d)

(16e)

Applying quantitative isotopic techniques to sunflower leaves, Abadie, Boex-Fontvieille, Carroll, and Tcherkez (2016) showed that the stoichiometric ratio of O₂ fixation by Rubisco to CO₂ production by GDC increased from 2.0 (the theoretical value used in the canonical FvCB model) at very low-photorespiration gas mixtures, to 2.05 for the normal ambient condition, and to 2.09 to high-photorespiration gas mixtures. As the export of carbon in the form of CH₂-THF would make this ratio lower than 2.0, the observed ratio being ≥2.0 suggests that the export of carbon from the photorespiratory pathway via this form may be less important than the export via glycine. If the value of >2.0 is due to glycine export alone, then α_G can be estimated to be 0.0 for the conditions with little photorespiration, 0.024 for the ambient condition, and a maximum value of 0.043 for the conditions of high-photorespiration gas mixture. Using modelling to fit Equations (16c)–(16e) to A–C_i curves, Busch et al. (2018) estimated α_G of the ambient condition to be 0.026 for plants fed with NH₄⁺-N, 0.103 for plants fed with NO₃⁻-N, and 0.077 for control plants. These all indicate that α_G is not zero as implicitly assumed in the canonical FvCB model. This means that even under Rubisco limitation where W_c is not changed by any amino acid export, A could still be increased due to a slight decrease in the CO₂-compensation point if glycine is removed from the photorespiratory pathway (see Equation (16c)). Under the TPU limitation where carbon uptake is limited by the rate at which carbohydrates can be metabolized, A could be further increased by short-circuiting carbon flux to glycine, serine, and CH₂-THF via the photorespiratory pathway (Figure 4). Only the NADPH-dependent electron transport-limited rate is decreased due to the electron consumption by the de novo nitrogen assimilation (if the potential electron transport rate J remains the same; but see later discussion).

In addition to exploring the ratio of O₂ fixation by Rubisco to CO₂ production by GDC to estimate α_G, Busch et al. (2018) showed that α_G and α_S could be roughly estimated from model fitting to A–C_i curves. There is currently no information available about the possible value for the fraction of glycolate carbon diverted via the C₁ metabolism (Busch, 2020). Therefore, hereafter we mainly discuss the relations with regard to the amino-acid exports.

4.2 Relationships with the previous two extensions

It is clear, based on the model of Busch et al. (2018), that the parameter α in the model of Harley and Sharkey (1991) deals with the carbon side of the amino acid export but not the electron requirement for NO₃⁻ assimilation. In addition, the energy associated with the changed RuBP regeneration and NH₄⁺-recycling as a result of amino acid export was not considered in Harley & Sharkey's model. Also, the decrease of CO₂-compensation point in the absence of R_d as a result of the glycine exit is not explicitly included in the model although this was discussed by Harley and Sharkey (1991). Busch et al. (2018) treated amino acid exit from the photorespiratory pathway differently, depending on whether it is glycine or serine that is exited, whereas Harley and Sharkey (1991) only assumed the glycine exit. It is clear from Equation (16e) that if it is only glycine that exits, the model under a TPU-limitation is:

(17a)

If the CO₂-compensation point is to be maintained as in the canonical FvCB model, it would be internally consistent to assume that it is only serine, instead of glycine, being exported. Then, based on Equation (16e), the model for A under a TPU-limitation should become:

(17b)

with the bound that 0 ≤ α_S ≤ 0.75. This model is supported by the above calculation that α_G was maximally only 0.043 based on isotopic measurements of Abadie et al. (2016) as well as by the modelling result of Busch et al. (2018) and measurements of Abadie, Bathellier, and Tcherkez (2018) that α_S was often much higher than α_G, reflecting high demands for serine due to its important role in the one-carbon metabolism and as precursor for several other amino acids and phospholipids (Ros et al., 2014). Previous parameterization of Equation (7b) from fitting to A–C_i curves with a moderate reverse sensitivity to C_i increases showed that the estimated α was as high as 0.77 (Busch et al., 2018), partly being the artefact of ignoring the decrease of CO₂-compensation point by Equation (7b), thereby exaggerating the actual fraction of glycolate carbon not returned to the chloroplast. This fraction would decrease by 25% if Equation (17b) is used. In fact, using the same total fraction of glycolate carbon not returned to the chloroplast, Equations (17a) and (17b) generates nearly identical curves as the full TPU-limitation model Equation (16e) without the α_T terms (Figure 4), whereas Equation (7b) generates much lower values. As Figure 4 demonstrates, there is little signal to differentiate α_G and α_S by conventional gas exchange (McClain & Sharkey, 2019), but only the sum of the two can be reliably estimated. Therefore, if α_G and α_S are to be estimated one cannot rely on Equation (16e) alone, but needs to consider at the very minimum the full range of A–C_i response fitted with Equation (16b) and include measurements of the compensation point, which is affected by α_G but not by α_S.

It is also possible to connect the model of Busch et al. (2018) with the model of Yin et al. (2004). As stated earlier, parameter f_pseudo in the model of Yin et al. (2004) can largely reflect the proportion of electrons for supporting nitrogen assimilation, especially under electron transport-limited conditions. Thus, one can equate Equation (16d) without the α_T terms to A_j formulated from Equation (9a):

(18a)

Note that J on the left side of the equation must be equal to J₂ on the right side, as they both represent the rate of whole-chain electron transport in support of the Calvin–Benson cycle, the photorespiratory pathway and nitrogen assimilation (in this context, J in the model of Busch et al., 2018 actually differs from J in the canonical FvCB model). Solving for f_pseudo gives:

(18b)

As stated earlier, f_cyc for C₃ photosynthesis is negligible (set to nil here). The modelling by Busch et al. (2018) showed that for the ambient-air condition, α_G was ca 0.10 and α_S was ca 0.15 for plants fed with NO₃⁻-N. Assuming ϕ = 0.3 for the ambient condition, then 0.058 for f_pseudo can be calculated from Equation (18b). This value would become even lower if there are small amounts of CET. For nonphotorespiratory conditions (ϕ = 0), Equation (18b) gives that f_pseudo = 0.

Equation (18b) also reveals that surprisingly f_pseudo does not increase monotonically with increasing ϕ if ϕ goes to a very high value (Figure 5a). The decline of f_pseudo beyond a threshold ϕ occurs only in the presence of α_G; and the higher is α_G, the lower is the threshold ϕ. However, f_pseudo always increases monotonically with increasing ϕ in the absence of α_G, regardless of values of α_S. All these responses are because α_G, not α_S, causes a decrease in CO₂-compensation point, and this positive impact on A becomes increasingly important under high photorespiratory states (high ϕ values) that mathematically require a low f_pseudo to enable the left and right sides of Equation (18a) in balance. For the same reason, although f_pseudo generally increases with increasing α_G or α_S, its response to α_G is stronger than to α_S at a low ϕ (Figure 5b), is comparable at an intermediate ϕ corresponding to ambient-air conditions (Figure 5c), and is weaker than to α_S at a high ϕ (Figure 5d).

It is noteworthy that f_pseudo calculated from Equation (18b) refers to the electron fraction responsible for supporting N assimilation only as result of amino acid export from the photorespiratory pathway. Therefore, the calculated f_pseudo depends on the amount of photorespiration as shown in Figure 5. In contrast, f_pseudo as one parameter in the model of Yin et al. (2004) for electron-transport-limited conditions lumps electron requirements for: (i) N assimilation of both via the photorespiratory pathway and not via this pathway and (ii) metabolic processes other than N assimilation that utilize chloroplastic electrons. As stated earlier, f_pseudo of ca 0.10 was estimated from the assimilatory quotient for nonphotorespiratory conditions. The higher f_pseudo estimated from the assimilatory quotient suggests that either not all nitrogen is assimilated via the photorespiratory pathway or/and processes other than N assimilation consumes chloroplastic electrons. Furthermore, the model of Busch et al. (2018) only applies to the case where it is NO₃⁻-N that enters the leaf. However, it cannot be ruled out that nitrogen enters the leaf in the form of NH₄⁺-N (Eichelmann, Oja, Peterson, & Laisk, 2011), and for such a case the stoichiometric coefficients of Equation (15a) has to be re-formulated whereas the model of Yin et al. (2004) remains the same but with a lower value of f_pseudo.

6.1 The standard model for C₄ photosynthesis

Berry and Farquhar (1978) presented a first model for C₄ photosynthesis, which covered the CCM and the basis of high nitrogen use efficiency. The leakiness (ϕ_L) as the ratio of the CO₂ retro-leakage (L) to the rate of PEP carboxylation (V_p), was introduced in a model that included carbon isotope discrimination (Farquhar, 1983). Based on these earlier models, von Caemmerer and Furbank (1999) described a model, which is now considered as the standard C₄ model that predicts net CO₂-assimilation rate (A) as a function of mesophyll cytosol CO₂ partial pressure (C_m). Several equations relevant to the C₄ photosynthesis are:

(22a)

(22b)

(22c)

(22d)

where R_m is the respiration in the M cell (usually assumed to be 0.5R_d), g_bs is bundle-sheath conductance, C_c and O_c are the partial pressure of CO₂ and O₂ at the active sites of Rubisco, respectively, α_bs is the fraction of O₂ evolution (or of PSII) in the BS cells, u_oc is the coefficient that lumps diffusivities of O₂ and CO₂ in water and their respective Henry constants, O_m is the partial pressure of O₂ at the mesophyll cytosol, K_p and V_pmax are the Michaelis–Menten constant and the maximum carboxylation rate of PEPc, respectively, x is the fraction of ATP consumed by the CCM cycle, φ is the mol chloroplastic ATP required for the CCM cycle, and J_atp is the rate of ATP production by chloroplastic electron transport. The original model of von Caemmerer and Furbank (1999) did not use J_atp, but the rate of electron transport (J). Because it is ATP, not electrons, that are allocated between the CCM cycle and the Calvin–Benson cycle, according to the predefined stoichiometric fraction x, it is more appropriate to use J_atp in Equation (22d) (Yin et al., 2011) and J_atp can be linked with electron transport rate via the ATP production factor z (see Equation (9e)). Equation (22d) for V_p contains either the PEPc activity-limited rate or the electron transport-limited rate, in analogy to the equations for V_c. The rate of CO₂-assimilation (A) based on V_c is the same for C₃ photosynthesis and can be collectively expressed as:

(22e)

where γ_* = 0.5/S_c/o, and x₁ = V_cmax, x₂ = K_mC/K_mO, and x₃ = K_mC for the Rubisco-limited rate. For the RuBP regeneration-limited rate, x₁ = (1 − x)J_atp/3, x₂ = 7γ_*/3, and x₃ = 0 if ATP supply is limiting. von Caemmerer and Furbank (1999) provided a solution to the combined Equations (22a)–(22e) that expresses A as a quadratic function of C_m. As C_m is unknown generally, one may add an equation (C_m = C_i − A/g_m) in order to express A as a function of C_i. Yin et al. (2011) provided the analytical solution to these, which became cubic if PEPc activity limits V_p.

Unlike in C₃ leaves, the initial carbon fixation in C₄ leaves is catalysed by PEPc in the cytosol and therefore g_m does not involve CO₂ diffusion from the cytosol to the chloroplast. Accordingly, estimates of g_m in C₄ leaves are somewhat higher than those in C₃ leaves (e.g. Barbour, Evans, Simonin, & von Caemmerer, 2016), meaning g_m appears to be less limiting to C₄ photosynthesis as it is to C₃ photosynthesis. However, g_bs, which determines the amount of CO₂ leakage (see Equation (22b)), is fundamentally important for the CCM, and thus, for determining C₄ photosynthesis. So far there is no method that can directly estimate g_bs. Its indirect estimate, mostly based on model fitting to gas exchange data (He & Edwards, 1996) and sometimes combined with chlorophyll fluorescence or ¹³C discrimination measurements, suggests a value between 1.0 and 10.0 mmol m⁻² s⁻¹ bar^-1 (Yin et al., 2011), ca two- or three-order of magnitude smaller than g_m. Like g_m, g_bs varies with leaf age or N content (Yin et al., 2011), temperature (Alonso-Cantabrana et al., 2018; Kiirats, Lea, Franceschi, & Edwards, 2002; Yin, van der Putten, Driever, & Struik, 2016), and growth light conditions (Bellasio & Griffiths, 2014; Kromdijk, Griffiths, & Schepers, 2010; Ubierna, Sun, Kramer, & Cousins, 2013). Danila et al. (2021) showed that suberization of the BS lamellae is required for a low g_bs to minimize leakage. As g_bs is a lumped model parameter, its value may also depend on other anatomical characteristics (like BS cell wall thickness, plasmodesmata density, bundle sheath surface area-to-leaf area ratio, intervein spacing, sheath layers) as well as biochemical characteristics (like the location of decarboxylation). Further research is needed to clarify how these characteristics influence g_bs.

6.2 Energetic aspects of C₄ photosynthesis

Although energy production or consumption can be cell-type specific (Yin & Struik, 2018, 2021), the model of von Caemmerer and Furbank (1999) for C₄ photosynthesis assumed that energy is shared between M and BS cells, and used x to allocate J_atp to the CCM cycle (see Equation (22d)) and thus, 1 − x to the Calvin–Benson cycle (see Equation (22e)). The default value for x is 0.4, arising from φ/(φ + 3), where φ and 3 are ATP required for the CCM cycle and the Calvin–Benson cycle, respectively. For most C₄ species, φ = 2; so x = 0.4 (von Caemmerer & Furbank, 1999; but see discussion later). Thus, the RuBP regeneration-limited form of Equation (22e) is expressed in terms of ATP supply. As with the C₃ model, it is metabolically important to keep ATP and NADPH in balance (Foyer et al., 2012; Kramer & Evans, 2011); so, one may argue that ATP and NADPH co-determine the RuBP regeneration. For Equation (22e) if NADPH supply is limiting, one can write, according to Equation (9a), that x₁ = [1 − f_pseudo/(1 − f_cyc)]J₂/4, x₂ = 2γ_*, and x₃ = 0. Based on this NADPH-determined model, Yin and Struik (2012) stated that the photosynthetic quantum yield models for C₄ photosynthesis are the same as for C₃ photosynthesis, that is, Equation (12a) or Equation (12e), reflecting that there is no net NADPH requirement for the C₄ cycle (but again, see discussion later). Similarly, Equations (13a), (13b) and (10b) for calculating f_cyc, f_pseudo and ρ₂, respectively, also suit for C₄ photosynthesis.

As discussed earlier for C₃ photosynthesis, one can rely on the unique feature of the NADPH-dependent equation for quantum yield to infer possible values of f_cyc from measurements on quantum yields. Φ_O2LL of C₄ photosynthesis (virtually without photorespiration) is ca 0.069 (Björkman & Demmig1987), considerably lower than its counterpart value of C₃ photosynthesis in the absence of photorespiration. Using Equation (13a), Yin and Struik (2012) solved f_cyc, which was ca 0.45, considerably higher than the f_cyc of C₃ photosynthesis. This suggests that CET is essential for C₄ photosynthesis, required for generating ATP required for the operation of the CCM cycle.

Once f_cyc is known, ρ₂ can be calculated from Equation (10b). The obtained ρ₂ is ca 0.4 (Yin & Struik, 2012). This differs from Equation (4), where the energy partitioning factor of 0.5 is also used for C₄ photosynthesis (von Caemmerer, 2000, 2013; von Caemmerer & Furbank, 1999). When f_cyc is known, f_pseudo can also be estimated from the assimilatory quotient (see Equation (13b)) and is ca 0.07 (Yin & Struik, 2012).

The equation equivalent to Equation (8) for C₃ photosynthesis, for the fraction of LET that keeps NADPH and ATP balance as required by C₄ metabolism, can be formulated as (see Yin & Struik, 2012 for its derivation):

(23a)

where ϕ_L is leakiness (0 ≤ ϕ_L ≤ 1). Compared with Equation (8), Equation (23a) has an extra factor (1 − x)/(1 + xϕ_L). This suggests that compared with C₃ photosynthesis, the LET of C₄ photosynthesis is decreased at least by this factor to accommodate the required increase in CET. One can solve Equation (23a) for leakiness:

(23b)

Given the above indicative values of f_cyc and f_pseudo based on quantum yield data, one can use Equation (23b) to explore likely values of uncertain parameters f_Q and h that can give a realistic estimate of leakiness. Using either obligatory or no operation of the Q cycle (f_Q = 1 or 0) and three likely values of h (3, 4 and 4.67, see earlier), Yin and Struik (2012) showed that only the combination that f_Q = 1 and h = 4 can give a realistic value of ϕ_L (Figure 7). The obligatory Q cycle has long been recognized for C₄ photosynthesis (Furbank, Jenkins, & Hatch, 1990). But whether the H⁺:ATP ratio (h) is 3, 4 or 4.67 is uncertain. The model results shown in Figure 7 support thermodynamic experiments (Petersen et al., 2012; Steigmiller et al., 2008) showing that h is 4.

The model discussed so far, for both C₃ and C₄ photosynthesis, assumes that CET, when combined with the Q cycle, generates two H⁺ per electron (Figure 2). However, CET may follow the NAD(P)H dehydrogenase (NDH)-dependent pathway (Ishikawa et al., 2016; Strand, Fisher, & Kramer, 2017; Yamori, Sakata, Suzuki, Shikanai, & Makino, 2011). When this pathway is operating, CET generates four H⁺ per electron and Kramer and Evans (2011) indicated that very likely this pathway is active in C₄ plants. Let f_NDH be the fraction of CET that follows the NDH-dependent pathway. Then, the ATP production factor z as in Equation (9e) for such a case is (Yin & Struik, 2021):

(24a)

Equation (24a) becomes Equation (9e) if f_NDH = 0. Again, the uncertainty with regard to the value of f_NDH has no impact on the model for the NADPH-dependent quantum yield, so the above estimation of f_cyc using the NADPH-dependent quantum yield model is still valid. Yin and Struik (2012) showed that this highly efficient H⁺-translocating pathway of CET cannot be obligatory as this would result in unrealistic high estimates of leakiness. Here we try to assess the extent to which CET should be this highly efficient pathway if h is 4.67 (=14/3, Seelert et al., 2000; again recently, Hahn et al., 2018). This can be achieved by equating Equation (24a) with h = 14/3 to Equation (9e) with h = 4, and then solving for f_NDH:

(24b)

This gives that f_NDH is ca 0.47 if f_Q = 1 and f_cyc = 0.45, meaning that about half of the total CET have to follow this highly efficient pathway in order to meet the high ATP requirement in C₄ photosynthesis, if the H⁺ requirement per ATP synthesis is as high as 4.67. This suggests a method to estimate f_NDH, as this parameter has been estimated only by trial and error (Bellasio & Farquhar, 2019).

Combining h = 4 and f_NDH = 0 or h = 4.67 and f_NDH = 0.47 suggests that the ATP production factor per PSII electron transport (z) is ca 1.16. This differs from the standard C₄ model of von Caemmerer and Furbank (1999), in which J_atp is set to equal PSII electron transport rate. The standard model assumes: (i) the absence of CET and (ii) and h = 3. Equation (9e) suggests that these assumptions combined with an obligatory Q cycle make z = 1.

6.3 Accommodating the C₄ species mixed with PEP-CK

It is important to point out that the above results of energetics are valid only for NADP-ME or NAD-ME subtypes of C₄ photosynthesis, although the standard model has been wrongly applied in some reports to the PEP-CK subtype. As stated earlier, the value of 0.4 for x stems from that the parameter φ in Equation (22d) is 2, referring to two mol ATP required per CCM cycle for regenerating PEP by pyruvate phosphate dikinase (PPDK) in the M cell (Hatch, 1987; Kanai & Edwards, 1999). This high ATP requirement is reflected in measured quantum yields in species of the malic-enzyme subtypes, from which the model derived f_cyc was high (ca 0.45). In the PEP-CK subtype, however, part of the oxaloacetates produced by the initial PEP carboxylation step move to and are decarboxylated in the BS cytosol by PEP-CK (Hatch, 1987). This decarboxylation reaction also generates PEP (requiring only one molecule of ATP per reaction), thereby partly bypassing the expensive step of PEP regeneration by PPDK. The remaining oxaloacetates are reduced to malate in the M cells, which move to and are decarboxylated in BS mitochondria. This decarboxylation also releases NADH, which drives mitochondrial electron transport to provide ATP for fuelling PEP-CK possibly (Kanai & Edwards, 1999), thereby further decreasing the chloroplastic ATP requirement. Given that the pure PEP-CK type hardly exists in nature and species having PEP-CK are often mixed with other decarboxylation types (Furbank, 2011; Wang et al., 2014), Yin and Struik (2021) presented a model for the electron transport-limited rate in all C₄ subtypes including their mixed types.

In this model, Equations (22a)–(22e) still apply, but with:

(25a)

and x₂ and x₃ are as defined earlier (i.e., x₂ = 2γ_*, and x₃ = 0). In Equation (25a), parameter a is the fraction of oxaloacetates that are reduced, using NADPH (equivalent to 2 electrons) from M chloroplasts, to malate moving to the BS mitochondria. To accommodate various C₄ types, two further adjustments are needed. Firstly, the chloroplastic ATP requirement for the CCM cycle (φ) should be changed from 2 for the malic-enzyme subtypes to:

(25b)

where n is the number of ATP produced per NADH oxidation from mitochondrial electron transport (n = 2.5 ~ 3.0; Taiz & Zeiger, 2002), the coefficient 1 represents one molecule ATP fewer required for per PEP regenerated by PEP-CK than by PPDK, and so, the term (n + 1)a represents ATP saved from engaging the PEP-CK mechanism, relative to the malic-enzyme mechanisms (Yin & Struik, 2021). Secondly, for the types involving PEP-CK, x for Equation (22d) is changed to:

(25c)

These equations have taken into account the required balance of NH₂-groups between M and BS cells. The analysis of Yin and Struik (2021) suggested that 0 ≤ a ≤ 0.36 ~ 0.40, and if a = 0, the model returns to the equations discussed earlier for the malic-enzyme subtypes. The model predicts that the additional cost with a mol NADPH requirement per mol CO₂ assimilated is overcompensated by the decreased chloroplastic ATP requirement for the CCM cycle, thereby predicting a higher Φ_CO2 in species involving the PEP-CK activity. However, the observed little advantage in Φ_CO2 of the PEP-CK over the NADP-ME species (Ehleringer & Pearcy, 1983) suggests the need of more studies to understand whether the energetic advantages are cancelled out by leakiness in the PEP-CK types.