The energy budget in C4 photosynthesis: insights from a cell‐type‐specific electron transport model

Summary Extra ATP required in C4 photosynthesis for the CO2‐concentrating mechanism probably comes from cyclic electron transport (CET). As metabolic ATP : NADPH requirements in mesophyll (M) and bundle‐sheath (BS) cells differ among C4 subtypes, the subtypes may differ in the extent to which CET operates in these cells. We present an analytical model for cell‐type‐specific CET and linear electron transport. Modelled NADPH and ATP production were compared with requirements. For malic‐enzyme (ME) subtypes, c. 50% of electron flux is CET, occurring predominantly in BS cells for standard NADP‐ME species, but in a ratio of c. 6 : 4 in BS : M cells for NAD‐ME species. Some C4 acids follow a secondary decarboxylation route, which is obligatory, in the form of ‘aspartate‐malate’, for the NADP‐ME subtype, but facultative, in the form of phosphoenolpyruvate‐carboxykinase (PEP‐CK), for the NAD‐ME subtype. The percentage for secondary decarboxylation is c. 25% and that for 3‐phosphoglycerate reduction in BS cells is c. 40%; but these values vary with species. The ‘pure’ PEP‐CK type is unrealistic because its is impossible to fulfil ATP : NADPH requirements in BS cells. The standard PEP‐CK subtype requires negligible CET, and thus has the highest intrinsic quantum yields and deserves further studies in the context of improving canopy productivity.


Fig. S1
The decarboxylation mechanism and minimum cell-type specific energy requirements of three standard subtypes of C4 photosynthesis Methods S1 The analytical model for cell-type specific electron transport Methods S2 FST codes of our model for NADPH and ATP production and quantum yield Notes S1 The effect of structure parameters on modelled fractions of CET in NAD-ME species Notes S2 The effect of structure parameters on the estimated requirement of the "aspartatemalate" mechanism as the secondary decarboxylating pathway in NADP-ME species

Notes S3
The effect of structure parameters on the estimated requirement of the "PEP-CK" mechanism as the secondary decarboxylating pathway in NADP-ME and NAD-ME species Notes S4 The effect of light extinction coefficient k on the estimated requirement of the secondary decarboxylating pathway in NADP-ME and NAD-ME species Table S1 The effect of structure parameters on modelled fractions of CET in NAD-ME species Table S2 The effect of structure parameters on the modelled requirement of the "aspartate-malate" mechanism as the secondary decarboxylating pathway in NADP-ME species Table S3 The effect of structure parameters on the modelled requirement of the "PEP-CK" mechanism as the secondary decarboxylating pathway Table S4 The effect of light extinction coefficient k on the modelled requirement of the secondary decarboxylating pathway 3 Methods S1 The analytical model for cell-type specific electron transport Here we describe the model on energy production, in a step-wise manner. First, basic model equations for ATP and NADPH production and quantum yield for CO2 assimilation (CO2) are given for the case where only CO2 fixation is considered, particularly for NADP-ME and NAD-ME subtypes. Special cases of a low ATP:NADPH requirement as occurring in the standard PEP-CK subtype as well as in the hypothetical "pure" PEP-CK type are then modelled.
The model was further extended to account for the effects of photorespiration and alternative electron and ATP sinks on the cell energy budgets. How the model was adjusted to accommodate mixed decarboxylating types involving PEP-CK and to deal with other cases is also described. Finally, all model versions are summarised, and our model is compared with other existing models.
The model was presented in such a progressive manner with increasing complexity, for three reasons: (i) to reflect how the model was developed, (ii) for the purpose of clarifying: a complex version would have been hard to conceive and to understand without an earlier simpler version, and (iii) to have better insights about the importance of individual processes (for example, the minor contribution of photorespiration and alternative energy sinks indicated in the main text would not have been revealed if they were already included right at the beginning of the model).
where 2 and 3 are mol NADPH and ATP, respectively, required per C3 cycle, a is additional mol NADPH required in the PEP-CK subtype for reducing OAA into malate (a has to be set to 0 when eqn 12 is applied for NADP-ME or NAD-ME subtypes), is mol ATP required per mol C4 carboxylation (= 2 mol for NADP-ME and NAD-ME subtypes, and = 2a mol for the PEP-CK subtype), and  is leakiness. The actual gross CO2 assimilation rate (Ag) is co-limited by NADPH and ATP supply, as our model is formulated to ensure that Quantum yield for CO2 assimilation under limiting light (CO2) can be expressed as:

Specific cases of low ATP:NADPH requirement as occurring in the standard PEP-CK subtype and in the "pure" PEP-CK type
Compared with NADP-ME and NAD-ME subtypes (which require 5:2 for the ATP:NADPH when there is no leakiness), the standard PEP-CK subtype has a different ratio of ATP:NADPH requirement. The minimum value is (3+2a) mol ATP and (2+a) mol NADPH per mol CO2 assimilation, where a = 0.286 or 0.250 depending on whether ATP produced per oxidation of NADH in the respiratory electron transport (n) is 2.5 or 3.0 [a = 1/(1+n), see Introduction and ]. If leakage occurs, these requirements are somewhat higher. To run the model for the PEP-CK subtype, care needs to be taken to account for the higher rate of LET needed to meet a higher NADPH requirement. The NADPH requirement per CO2 fixed is ) 1 ( 2    a ; so, the , and the parameter p in eqn (A1) needs to be adjusted to: where (2+fQ)/h is HLET/h, ATP produced per electron transferred by LET (Yin et al. 2004). In case that fQ = 1, eqn (15) predicts that p = 3 if a = 0, as has so far been the case applied to NADP-ME and NAD-ME subtypes. To account for this, the denominator of the right side of eqn (A1), w, which is (1+) for NADP-ME and NAD-ME subtypes (where  = 2), needs to be adjusted to the following for the Such an adjustment would allow a balanced ATP and NADPH production, ensuring that CO2 in terms of ATP requirements are equal to CO2 in terms of NADPH requirements.
For the hypothetical "pure" PEP-CK type without using mitochondrial electron transport to provide ATP, the ATP to fuel PEP-CK has to come from chloroplastic electron transport. For this hypothetical type, eqn (15) and eqn (16) still apply, on the condition that a in eqns (15) and (16) is set to 0 (to accommodate that no NADPH from M chloroplasts is used for ATP production from NADH oxidation in BS mitochondria) and  (mol ATP required per mol C4 carboxylation) is adjusted to 1 (1 mol ATP required for PEP-CK to directly decarboxylate 1 mol OAA into CO2 and PEP). So, the minimum value is (3+1) mol ATP and 2 mol NADPH per mol CO2 assimilation, and its ATP:NADPH ratio is 4:2, higher than that for the standard PEP-CK subtype, but lower than that for the two ME subtypes. If considering leakiness, the ATP:NADPH ratio for the "pure" PEP-CK type is (4+):2.

Accounting for photorespiration and alternative electron and ATP sinks
In this section, our model is extended to quantify the effects of photorespiration and other electron-and ATP-consuming processes on the energy budget and quantum yield. Nitrate reduction and starch synthesis are considered as two major alternative processes utilising chloroplastic electrons and ATP. The Mehler reaction is not considered here as this reaction is negligible under conditions where photosynthesis is limited by electron transport. Sucrose synthesis, which occurs in the M cytosol, will not be considered either as it consumes no additional NADPH or ATP (Amthor 2010). The malate valve may act on the NADPH and ATP balance (Kramer & Evans 2011), but reductants exported from chloroplasts by this valve may be used for nitrate reduction; so any operation of this mechanism is lumped to nitrate reduction, as energy costs for the latter process are more quantifiable. Note that fractions of the whole-chain 8 electrons consumed by all these alternative processes are lumped into the term fpseudo for the "pseudocyclic" form in our previous whole-leaf model (Yin & Struik 2012).
Per mol RuBP oxygenation 2 mol NADPH and 3.5 mol ATP are consumed, 0.5 mol ATP more than per mol RuBP carboxylation (Farquhar et al. 1980). Per mol nitrate reduction 10 mol electrons and 1 mol ATP are consumed (Noctor & Foyer 1998). If starch is considered as the end product of photosynthesis, there is a cost of 2/12 (= 0.167) mol ATP per mol CO2 for polymerising one mol hexose into starch (Noctor & Foyer 1998;Amthor 2010). So, parameter p in eqn (A1), equivalent to eqn (15) above, needs to be further adjusted to: Parameter w in eqn (A1), equivalent to eqn (16) above, needs to be adjusted to: where o/c is the RuBP oxygenation to RuBP carboxylation ratio, n/c is the nitrate reduction to RuBP carboxylation ratio, r/c is the day respiration to RuBP carboxylation ratio, and cstarch is the ATP cost for starch synthesis (= 0.167). Gross CO2 assimilation rate, equivalent to eqns (12) and (13), also need to be adjusted: Equations (18) and (20) assume (i) that the rate of starch synthesis (in mol CO2 m -2 s -1 ) is equal to (Vc -0.5Vo -Rd), where Vc, Vo and Rd correspond to rates of RuBP carboxylation, RuBP oxygenation and day respiration, respectively, and (ii) that per mol RuBP oxygenation 0.5 mol CO2 is released. All these equations ensure again that the whole-leaf CO2 in terms of ATP requirement and the whole-leaf CO2 in terms of NADPH requirement are equal.

Adjusting the model to accommodate energy production in mixed types involving PEP-CK
Unlike the mixed "NADP-ME + aspartate-malate" mechanism where the total whole-leaf energy requirement per CO2 fixed stays the same as that of the NADP-ME pathway, the total ATP requirement in the mixed types having PEP-CK changes compared with the primary NADP-ME or NAD-ME pathway (Table 4). However, only parameter  (mol ATP required per mol C4 carboxylation) in eqns (18) and (20) needs to be adjusted from 2 for NAD(P)-ME subtypes to (1+) for the mixed types, where be the fraction of OAA following the primary NADP-ME (or 9 NAD-ME) route and the remaining (1-) be the fraction following the secondary "PEP-CK" mechanism. Because the model for calculating the fractions of NADPH or ATP produced in BS cells and the model for the fraction of ATP required in BS cells both need parameter , this would need an iterative approach to solve . However, using a range of pre-set values for  showed that the calculated fractions of NADPH or ATP produced in BS cells changed little with , meaning that the fractions of NADPH or ATP produced in BS cells were largely determined by other parameters such as , fbsCHL and fbsPSI (see Table M2 for their definition). This insensitivity simplifies the analysis for the mixed type involving PEP-CK.

Considering other possible values of h, fQ and HCET
Stoichiometric coefficients related to ATP production are uncertain. For example, the H + :ATP ratio (h) is often also believed to be 14/3 or 4.67 (Kramer & Evans 2011), based on the structural data that the H + -driven turbine of the chloroplast ATPase has 14 subunits. If h = 4.67, then the ATP:NADPH ratio generated by LET in combination with the Q cycle is lower than the required 1.5 for the C3 cycle, and the shortfall in ATP must come from a higher fraction for CET. A similar case requiring more CET is when the Q cycle is partially operated (fQ < 1) while h is 4. In either case, parameter w in eqn (18) then needs to be completed with another term: where 3 is the number of ATP, and 4 is the number of linear electrons required to produce 2 mol NADPH, required for 1 mol CO2 assimilation by the Calvin cycle, so the whole term is the shortfall of ATP required by both the Calvin cycle and the photorespiratory cycle per mol CO2 assimilated if h > 4 or if fQ < 1. Actually this extension also applies to the case where h < 4, like in earlier days when h was considered to be 3 (Furbank et al. 1990). In such a case, there may be an overproduction of ATP by LET so that a lower CET would be required. If h is as low as 3, then the value of fQ may also need to be lowered (i.e., a zero or partial Q cycle), especially for the PEP-CK types, to avoid too much overproduction of ATP by LET. Otherwise, our model will predict an unrealistic value of parameters u and v, which would suggest, as discussed in the main text, an impossibility of certain (sets of) input parameter values.
Another uncertain parameter is HCET. As stated in the main text, HCET could become higher than 1+ fQ, our default expression for HCET, because two extra H + are generated if CET runs in the NAD(P)H dehydrogenase (NDH)-dependent pathway (Kramer & Evans 2011). Our model can accommodate this possibility if HCET is changed to be expressed as 1 + fQ + 2fNDH, where fNDH is the fraction of the whole-leaf CET that follows the NDH pathway.
Our model ensures an equal whole-leaf CO2 in terms of ATP and NADPH requirement for all these possibilities. However, these possibilities are no longer further discussed in the main text and elsewhere of the Supporting Information, where we stay with the most likely, simple scenario that h = 4, fQ = 1, HCET = 1+fQ, and HLET = 2+fQ (Yin & Struik 2012).

Summary of various model versions
Model was presented above with seemingly increasing complexity to show a step-wise approach to its development.
where n is relevant to the standard PEP-CK subtype, referring to mol ATP produced per oxidation of NADH in mitochondrial electron transport chain (n = 2.5 or 3; see the main text).
For the model version without considering photorespiration and alternative electron and ATP sinks, o/c, n/c, r/c and cstarch in eqns (17-21) just need to be set to 0. Algorithms and solutions in Derivations A-C stay the same for all model versions.

Comparison of our model with existing C4 models
Several models have been published for C4 cell-type specific processes (e.g. Wang et al. 2014), thereby going beyond those classical models (Farquhar 1983;Furbank et al. 1990;von Caemmerer & Furbank 1999) and a recent model of Yin & Struik (2012) for whole-leaf C4 11 photosynthesis. In particular, papers of Bellasio & Griffiths (2014), Bellasio & Lundgren (2016) and Bellasio (2017) also address the bioenergetics detailed in our model. Therefore, it is necessary to compare our model with these existing models.
As stated in Introduction, the model of Wang et al. (2014) for numerical simulation of various C4 subtypes or mixed types does not incorporate CET; so the difference between this numerical simulation model and our analytical model is obvious.
The model of Bellasio & Griffiths (2014) did incorporate CET as well as detailed stoichiometric algorithms for cell-type specific metabolic processes. However, both ATP production and metabolic processes were tailored for maize, a consummate NADP-ME species.
Its submodel for ATP supply, considering light penetration in dependence of anatomical traits, was extended to analyse whether sufficient ATP could be produced in BS cells in the context of the evolutionary continuum from C3 to C4 (Bellasio & Lundgren 2016). The model of Bellasio & Griffiths (2014) and Bellasio & Lundgren (2016) mainly quantified the relative ATP production in the different cell types, expressed as BS:M ratio (Jatp,BS/Jatp,M).
In contrast, our model presented here suits for predicting cell-type specific NADPH as well as ATP production for all various C4 subtypes or mixed types. From our model eqns (1) and (3), Jatp,BS/Jatp,M can be formulated: Substituting eqns (7-10) into eqn (24) gives: In eqn (25) If u = 1, eqn (26) becomes: This is exactly eqn (2) of Bellasio & Lundgren (2016). It is clear that their model is a special case of our model when absorbed light by M cells is used only for LET (u = 1). As shown in the main text, this assumption holds approximately only for the NADP-ME subtype and does not hold for NAD-ME and PEP-CK subtypes. In the model of Bellasio & Griffiths (2014), Jatp,BS/Jatp,M was further simplified to 2aBS/aM (their eqn 3), resulting from additional assumptions that v = 0 and CET/LET = 2. Again the assumption that v = 0 (absorbed light in BS cells is used only for CET) holds approximately only for the NADP-ME subtype. From our discussion in the main text, it is also hard to exactly reconcile the assumption that CET/LET = 2.
Instead of fixing them to approximate values, parameters u, v, and  (the fraction of PSI used for CET) in our model were solved analytically from current understanding of the most likely stoichiometry of C4 physiology (see below for Derivations A and B), conditional on experimentally measurable parameters such as [CHL], fbsCHL and fbsPSI ( Table 2). The algorithms in these derivations were carefully formulated to simultaneously account for (i) energy lost due to leakiness in addition to alternative electron and ATP sinks, and (ii) the balanced production of NADPH and ATP that co-limit the photosynthetic rate. Neither of the latter two aspects was considered explicitly in the model of Bellasio and colleagues. Because of the coherent analytical algorithms conditional on C4 physiology and some input parameter values, our model allows the solving of the physiologically plausible range of the variation in other parameters as shown for  (the fraction of PSII that is in BS cells) in various C4 types (Tables 3, 5 and 6 in the main text).
This feature of our model in combining C4 physiology and analytical mathematics also allows us, as shown in the main text, to identify knowledge gaps that could be used to design new experimental studies.
The model of Bellasio & Griffiths (2014) has detailed cell-type specific stoichiometries for maize metabolic processes that consume NADPH and ATP, and this was extended for various photosynthetic types C3, C2, C2+C4, and C4 including the three subtypes and mixed types (Bellasio 2017). Our algorithms for these stoichiometries on NADPH and ATP demands by M and BS cells in various C4 types are in an intermediate detail between those of Bellasio and colleagues and the classical C4 model, and are summarised as a table, Table 4. This allowed us to analytically solve the required fraction of 3-PGA reduction in each cell type and the fraction for 13 the primary and secondary decarboxylation (see the main text). Furthermore, Bellasio & Griffiths (2014) and Bellasio & Lundgren (2016) numerically modelled the relative BS/M light capture (aBS/aM) from underlying absorption and scattering coefficients, thereby generating light penetration profiles dependent on light spectrum (Bellasio & Griffiths 2014). In comparison, we used a simpler descriptive approach using light extinction coefficient k based on the experimental observation of Evans (1995) (Derivation C). A sensitivity analysis of our model with respect to the value of k will be given in Notes S4 (see later).

Derivation A Deriving equations that express v and u as a function of other parameters
Let T be the total amount of PSII per unit leaf area in BS and M cells; then T will be the amount of PSII in BS cells (where  is the fraction of PSII in BS cells) and (1-)T will be the amount of PSII in M cells (Table M1). Note that has the same meaning as used in the classical C4-photosynthesis model of von Caemmerer & Furbank (1999).
To account for the difference in the electron transport efficiency between PSI and PSII (Yin et al. 2004), the amount of PSI in BS cells has to be (2LL/1LL)T to enable an equal electron transport flux passing through PSI and PSII for LET; similarly the amount of PSI in M cells has to be (2LL/1LL)(1-)T (Table M1).
Let the total amount of PSI per unit leaf area used for CET be Cx; its fraction in BS is , and the remaining fraction, 1- , is in M (Table M1). We now need to solve Cx, based on ATP requirement for C4 physiology. ATP produced from total LET should be It is recognised that NADPH and ATP and their ratio generated from LET if h = 4 and if the Q cycle is fully operated exactly match the requirement for the C3 cycle. Thus, additional ATP requirement for the C4 cycle must be satisfied from CET which does not generate NADPH. Let p 14 be mol ATP required by the C3 cycle that is satisfied from LET, and let w be mol ATP requirement by the C4 cycle from chloroplastic electron transport, per mol CO2 fixation, then the following can be written: For NADP-ME and NAD-ME subtypes, w should be equal to (1+), where is mol ATP required for PEP regeneration and  is leakiness (see above for the discussion on w for the PEP-CK subtype). This results in: The total amount of light absorbed by BS (Iabs,BS) can be written, based on Table M1, as: The amount of light absorbed by BS that is used for LET (Iabs,BS,LET) can be written, based on Table M1, as:  Table M1, as: The amount of light absorbed by M that is used for LET (Iabs,M,LET) can be written, based on Table M1, as: The parameter u, by definition, is the ratio of Iabs,M,LET to Iabs,M, which can be solved from the above two equations as:

Derivation B Deriving equations to solve for  and kBS/kM
Although parameter  (the fraction of Cx in BS cells) cannot be measured directly, the fraction of the whole-leaf PSI that lies in BS cells, fbsPSI, can be experimentally measured (Ghannoum et al. 2005;Majeran et al. 2005). fbsPSI can be written, by definition and according to Table M1, as: From this, Cx could be solved as: According to the definitions and the information in Table M1, kBS and kM can be written as: The ratio of the two can be written: This equation can be re-formulated as: The Iabs,BS/Iabs,M ratio required for calculating b and c will be given in Derivation C.
There are two equations, eqn (A2) and eqn (B2), calculating Cx, and the two must be equal, that is: Solving for kBS/kM and re-arranging all the terms give: The two kBS/kM, given by eqn (B6) and eqn (B8), must be equal, that is: This gives the solution to : , and eg cj   C . In eqn (B10), the matching of  < fbsPSI or  < fbsPSI with the two solutions was determined from the fact that the Cx/T ratio has to be positive. According to eqn (B2), the positive Cx/T ratio requires that  has to be > fbsPSI if  <  Table 2 of the main text showed that  < fbsPSI in two NADP-ME and two NAD-ME species.
After  is solved, kBS/kM can be solved from either eqn (B6) or eqn (B8). Then the Cx/T ratio can be calculated, and a whole-leaf PSI:PSII ratio can be calculated thereof (Table M1): Or substituting eqn (B2) into the above equation gives:

Derivation C Calculating aM, aBS and their ratio
Vertical light-absorption profile inside a leaf can be modelled at a different level of sophistication, and it is modelled here in a simplest possible way, based on the observation that the light profile obeys the Beer-Lamberts law with the cumulative chlorophyll contents (Evans 1995). Because this part of our model is independent of other parts, the algorithms could be replaced with more sophisticated ones if such sophisticated algorithms are deemed necessary to meet different goals of modelling analysis.
Often the fraction of the whole-leaf chlorophyll in BS cells, fbsCHL, can be experimentally measured (Ghannoum et al. 2005). It is therefore possible to derive the aBS/aM ratio, in relation to the anatomical distribution of M and BS cells (Fig. 1). The chlorophyll content in each of the BS, M1, M2 and M3 sections can be easily calculated from whole-leaf chlorophyll content, fbsCHL, and schematic anatomical parameters as described in Fig. 1 where k is the extinction coefficient, and CHLM1, CHLM2, CHLBS, and CHLM3 are chlorophyll contents of M1, M2, BS and M3 sections, respectively, and the size for each of these sections is calculated from structural parameters m and nBS (Fig. 1) represents the whole-leaf absorptance, which can be used to fit the value of k. The aBS/aM ratio, which is also the Iabs,BS/Iabs,M ratio, can be expressed as:

Fraction of total PSII that is in BS cells -AREAbs
Relative area of the BS section in Fig. 1 -AreaBST Area ratio of the BS section to the total in Fig. 1 -AREAm Relative area of the M sections in Fig. 1 atpadd ATP produced from additional LET due to extra NADPH required in the standard PEP-CK subtype and due to alternative esinks mol ATP (mol CO2) -1 ATPPEP Chloroplastic ATP required for PEP regeneration in C4 cycle (for types other than the standard PEP-CK subtype) Chlorophyll content in the M1 section in Fig. 1 mol CHL m -2 CHL2MAB Chlorophyll content in the M3 section in Fig. 1 mol CHL m -2 CHL2MAD Chlorophyll content in the M2 section in Fig. 1 mol CHL m -2 CHLbs  Chlorophyll content in the BS section in Fig. 1 mol CHL m -2 CHLleaf [CHL] Leaf chlorophyll content mol CHL m -2 CHLm Chlorophyll content in the M sections in Fig. 1  A code to indicate C4 types: 1. to represent the pure PEP-CK type, 0. to represent the mixed "ME + PEP-CK" types, and -1. to represent any other types

Impact of uncertainties in some input parameters on model results
The default values of structure parameters in Table 1 (m = 0.55, nBS = 0.6) were determined based on the literature (Christin et al. 2013;Griffiths et al. 2013;Bellasio & Lundgren 2016) as an average of C4 species. According to our simple scheme in Fig. 1, these default values yield a BS:(BS+M) area ratio of 0.27 as an average of diverse C4 species (Griffiths et al. 2013).
However, significant variation of this ratio exists among species (Hattersley 1984), with the NAD-ME species having higher ratios than the NADP-ME species. For example, this ratio was 0.29 and 0.39 for two NAD-ME species P. miliaceum and P. coloratum, respectively, versus 0.21 and for 0.23 for Zea mays and S. bicolor, respectively (Hattersley 1984). This difference was largely due to the higher mesophyll area per vein (determined by parameter m in Fig. 1) in NAD-ME than in NADP-ME and PEP-CK subtypes (Hattersley 1984). Such a difference between subtypes may be counteracted, to some extent, by the fact that the arrangement of BS chloroplasts is centripetal in NAD-ME, and centrifugal in NADP-ME and PEP-CK species (von Caemmerer & Furbank 2003). This may support using a common default m:(1-m) as the interveinal mesophyll:vein surface area ratio (Fig. 1) (Table 1) also affects the relative absorptance between M and BS cells, and therefore, a sensitivity analysis with respect to k is necessary. All the modelled results of sensitivity analysis shown below are for the case in the presence of photorespiration and alternative energy-using sinks.
Notes S1 The effect of structure parameters on modelled fraction of CET in each cell type for the NAD-ME species We have shown in the main text that using the default values of m or nBS, the predicted PSI:PSII ratio and fCET in BS cells were undoubtedly higher than in M cells in the two NADP-ME species; but they were also higher in the two NAD-ME species (Table 2). This latter prediction differs from the statement of Takabayashi et al. (2005) that the activity of CET and the PSI:PSII ratio should be higher in M cells than in BS cells of NAD-ME species, opposite to that shown for the NADP-ME species. We examine to what extent our result varies with different sets of structural parameter values (Table S1).
The modelled values of both PSI:PSII ratio and fCET were still higher in BS than in M cells of the two NAD-ME species (Table S1), irrespective of the changes in either m or nBS for a higher BS:(BS+M) area ratio that is often observed for NAD-ME species. So, our conclusion with regard to differences from Takabayashi et al. (2005) still holds.

Notes S2
The effect of structure parameters on the estimated requirement of the "aspartatemalate" mechanism as the secondary decarboxylating pathway in NADP-ME species We have shown in the main text the estimated (the required fraction of C4 acids that follow the primary decarboxylating pathway) if the "aspartate-malate" mechanism acts as the secondary decarboxylating route in the two NADP-ME species. The solved value of was 0.68-0.73 ( shown here, we set changes of ± 18% for m, while changes in nBS remained ± 40% (Table S2).
The calculated fnadph,BS varied little, but fatp,BS did vary significantly, with a change in structural parameters. As a consequence, the solved  for ATP and  varied as well (Table S2).
However, solved  for ATP was always lower than solved  for NADPH (except one extreme case where they were nearly equal for Cenchrus ciliaris with the BS:(BS+M) area ratio set high for the NADP-ME type), suggesting that a secondary decarboxylating pathway is generally required for NADP-ME species. But the quantitative extent for this secondary pathway had a wide range of variation, depending more on m than on nBS, even wider than what was discussed in the main text for reported ranges of variation in the fraction for the initial carbon label partitioned to aspartate (1-), i.e. from ca 25% in maize (Hatch 1971) to ca 50% for Flaveria bidentis (Meister et al. 1996).
Therefore, whether the "aspartate-malate" mechanism as the secondary decarboxylating pathway is required, and if so, to what extent it operates, may depend on species within the NADP-ME subtype.

Table S2
The modelled fraction of total NADPH or ATP that is produced in BS cells (fnadph,BS or fatp,BS), calculated  (the fraction of required NADPH or ATP for the reduction phase of the Calvin circle that is consumed in BS cells) and the required fraction () of C4 acids that follow the primary decarboxylation if the "aspartate-malate" mechanism acts as the secondary decarboxylating pathway in two NADP-ME species, in response to a certain % change of parameters m or nBS. Values for the default case are also shown in Table 5.

Notes S3
The effect of structure parameters on the estimated requirement of the "PEP-CK" mechanism as the secondary decarboxylating pathway in NADP-ME and NAD-ME species We have shown in the main text that the PEP-CK mechanism alone acts as the secondary decarboxylation route, hardly for the NADP-ME subtype but well for the NAD-ME subtype. We now examine whether this conclusion can be affected by the uncertainties in input structural parameters m and nBS. Again, we used the criteria that the predicted u or v must be within the physiologically relevant range (0 ≤ u ≤ 1 and 0 ≤ v ≤ 1 ) to define the range of variation in m and nBS. It turned out that parameter m cannot be decreased by more than 2.5% and nBS cannot be increased by more than 13% in the NADP-ME species when involving the PEP-CK as the secondary decarboxylation, as the modelled u would otherwise be > 1. This little allowed decrease in m reflects the stiff system once involving PEP-CK, echoing what has been shown in the main text that the allowable range of parameter  in the PEP-CK subtype is very narrow (Table 3). However, the NAD-ME species were not very sensitive as shown in Table S1 where m could be allowed to vary by ± 40% and beyond. So, our model predicts a higher phenotypic plasticity of structural parameters in NAD-ME than NADP-ME subtypes. Given this high sensitivity in the NADP-ME species, we varied nBS by ± 13% and set the lower limit of m as -2.5% while leaving its upper limit still as +18% of its default as in Table S2. The results are given in the upper and lower parts of Table S3 for the NADP-ME and NAD-ME species, respectively.
Using the changes made for parameter m or nBS, the solved  in the NADP-ME species was always above 1 (the upper part of Table S3), which is physiologically impossible as discussed in the main text. This confirms that PEP-CK alone cannot act as the secondary decarboxylation pathway; it either does not exist, or co-acts with the "aspartate-malate" mechanism, in these species.
The solved  values, using the changes made for parameter m or nBS, were all physiologically sensible for the NAD-ME species (the lower part of Table S3). But  depends on the structural parameters: more on m than on nBS. When m was increased by 18%, the obtained  was 0.93-0.95, suggesting the required PEP-CK as secondary decarboxylation pathway was quite small. However, this decreased m corresponds to the BS:(BS+M) area ratio of 0.21 (Table S3), a low value hardly found for an NAD-ME species (Hattersley 1984). Table S3 The modelled fraction of total NADPH or ATP that is produced in BS cells (fnadph,BS or fatp,BS), calculated  (the fraction of required NADPH or ATP for the reduction phase of the Calvin circle that is consumed in BS cells) and the required fraction () of C4 acids that follow the primary decarboxylation if the "PEP-CK" mechanism acts as the secondary decarboxylating pathway in two NADP-ME species (the upper part of this table) and in two NAD-ME species (the lower part of the table), in response to a certain percent change of input parameters m or nBS. Values for the default case are also shown in Table 5.

Notes S4
The effect of light extinction coefficient k on the estimated requirement of the secondary decarboxylating pathway in NADP-ME and NAD-ME species The default k value of 0.005 m 2 (mol CHL) -1 (Table 1) was obtained from fitting the light absorptance model, eqns (C1-C4), to have a good modelled whole-leaf absorptance. The value of k has an effect not only on the whole-leaf absorptance but also on the relative absoprtance of light between M and BS cells, thereby, affecting the fraction of NADPH or ATP that is produced in BS cells. Also, Bellasio & Griffiths (2014) showed that "blue light was strongly absorbed (steep profile), green light was weakly absorbed (gradual profile), while red light had an intermediately profile of light penetration", indicating that value of k depends on the light spectrum. Given these uncertainties of the k value, we conducted a sensitivity analysis by varying k by ± 40% of its default value, to check to what extent the key result of this paper on the 30 Table S4 The modelled fraction of total NADPH or ATP that is produced in BS cells (fnadph,BS or fatp,BS), calculated  (the fraction of the required NADPH or ATP for the reduction phase of the Calvin circle that is consumed in BS cells) and the required fraction () of C4 acids that follow the primary decarboxylation, when the secondary decarboxylating pathway was the "aspartate-malate" mechanism in two NADP-ME species (the upper part of the Table) and the "PEP-CK" mechanism in two NAD-ME species (the lower part of the Table), in response to a certain percent change in the value of the input parameter light extinction coefficient k. Values for the default case are also shown in Table 5. distribution of 3-PGA distribution and the required fraction of the secondary decarboxylating pathway would vary in NADP-ME and NAD-ME species.
As expected, with a change in k, the modelled whole-leaf absorptance varied significantly, but the relative absorptance between M and BS cells varied less (results not shown) because the latter depended more on the relative M:BS distribution of chlorophyll. As a result, the modelled fraction of total NADPH or ATP that is produced in BS cells (fnadph,BS or fatp,BS), the calculated  (the fraction of required NADPH or ATP for the reduction phase of the Calvin circle that is consumed in BS cells) and the required fraction () of C4 acids that follow the primary decarboxylation, all only varied marginally in response to k (Table S4). Similar marginal impacts were modelled for some intermediate variables (results not shown). Therefore, our key quantitative estimates in the main text on the distribution of the 3-PGA reduction and the requirement of the secondary decarboxylation are conservative to an uncertainty in light extinction coefficient k. Needless to say, k has no impact on the calculation of the whole-leaf fCET and CO2.