Updating the steady-state model of C4 photosynthesis

Abstract C4 plants play a key role in world agriculture. For example, C4 crops such as maize and sorghum are major contributors to food production in both developed and developing countries, and the C4 grasses sugarcane, miscanthus, and switchgrass are major plant sources of bioenergy. In the challenge to manipulate and enhance C4 photosynthesis, steady-state models of leaf photosynthesis provide an important tool for gas exchange analysis and thought experiments that can explore photosynthetic pathway changes. Here a previous C4 photosynthetic model developed by von Caemmerer and Furbank has been updated with new kinetic parameterization and temperature dependencies added. The parameterization was derived from experiments on the C4 monocot, Setaria viridis, which for the first time provides a cohesive parameterization. Mesophyll conductance and its temperature dependence have also been included, as this is an important step in the quantitative correlation between the initial slope of the CO2 response curve of CO2 assimilation and in vitro phosphoenolpyruvate carboxylase activity. Furthermore, the equations for chloroplast electron transport have been updated to include cyclic electron transport flow, and equations have been added to calculate the electron transport rate from measured CO2 assimilation rates.


Introduction
To meet the challenge of increasing crop yield for a growing world population, it has become apparent that photosynthetic efficiency and capacity must be increased per unit leaf area to improve yield potential (Long et al., 2015). High yields from C 4 crops have stimulated considerable interest in the C 4 photosynthetic pathway which is characterized by a high photosynthetic rate and high nitrogen and water use efficiency relative to plants with the C 3 photosynthetic pathway (Mitchell and Sheehy, 2006). In the challenge to increase photosynthetic rate per leaf area; steady-state models of leaf photosynthesis provide an important tool for gas exchange analysis and thought experiments that can explore photosynthetic pathway changes (von Caemmerer, 2003;von Caemmerer and Evans, 2010;Price et al., 2011;Long et al., 2015;von Caemmerer and Furbank, 2016). The mathematical simplicity of these leaf-level models has facilitated incorporation into higher order canopy, crop, and earth system models (Yin and Struik, 2009;Rogers et al., 2017;Wu et al., 2018Wu et al., , 2019. C 4 photosynthesis requires the coordinated functioning of mesophyll and bundle sheath cells of leaves, and is This paper is available online free of all access charges (see https://academic.oup.com/jxb/pages/openaccess for further details) characterized by a CO 2 -concentrating mechanism which allows Rubisco, located in the bundle sheath cells, to operate at high CO 2 partial pressures. This overcomes the low affinity Rubisco has for CO 2 and largely inhibits its oxygenation reaction, reducing photorespiration rates. In the mesophyll, CO 2 is initially fixed by phosphoenolpyruvate (PEP) carboxylase (PEPC) into C 4 acids, which are then decarboxylated in the bundle sheath to supply CO 2 for Rubisco. Both the structure of the bundle sheath wall (which has a low permeability to CO 2 ) and the relative biochemical capacities of the C 3 cycle in the bundle sheath and C 4 acid cycle (which operates across the mesophyll-bundle sheath interface) contribute to the high CO 2 partial pressure in the bundle sheath. The biochemistry of the C 4 photosynthetic pathway is not unique, and three main biochemical subtypes are recognized on the basis of the predominant decarboxylating enzymes NADP-ME (NADP-dependent malic enzyme), NAD-ME (NAD-dependent malic enzyme), or PCK (PEP carboxykinase) (Hatch, 1987).
The first models to capture the C 4 photosynthetic biochemistry were designed by Berry and Farquhar (1978) and Peisker (1979). The Berry and Farquhar model did not provide analytical solutions but was able to predict high bundle sheath CO 2 partial pressures and their dependence on bundle sheath conductance. Many of the gas exchange characteristics of C 4 photosynthesis observed with intact leaves could be predicted by these models. Collatz et al. (1992) and von Caemmerer and Furbank (1999) have revised and expanded these original models with analytical solutions. C 4 models have not been used as frequently as the C 3 models, so fewer data relating leaf biochemistry to gas exchange are available in the literature. Massad et al. (2007) have parameterized the model of von Caemmerer (2000) for Zea mays and developed the first temperature dependencies for key parameters. Fitting routines have also been developed (Bellasio et al., 2016(Bellasio et al., , 2017Zhou et al., 2019).
Here an update of the C 4 photosynthetic model of von Caemmerer and Furbank (1999) and von Caemmerer (2000) is provided with new parameterization and temperature dependencies derived from experiments on the C 4 monocot species Setaria viridis (green foxtail millet), an NADP-ME type which is closely related to agronomically important C 4 crops. It has become a popular model species due to its rapid generation time, small stature, high seed production, diploid status, and a small sequenced and publicly available genome, and it can be readily transformed (Doust, 2007;Brutnell et al., 2010;Li and Brutnell, 2011;Osborn et al., 2017;Alonso-Cantabrana et al., 2018;Ermakova et al., 2019).

The basic model equations
The C 4 photosynthesis model presented here has a similar structure to the widely used model of C 3 photosynthesis by Farquhar et al. (1980). That is C 4 photosynthesis can be limited either by the enzymatic rates of PEPC and Rubisco or by irradiance and the capacity of chloroplast electron transport which supports the regeneration of PEP and ribulose bisphosphate (RuBP), and two sets of rate equations are given for these two scenarios. The actual CO 2 assimilation rate is then the minimum of the enzyme-limited or electron transport-limited rate. Figure 1 shows a schematic representation of the proposed carbon fluxes in C 4 photosynthesis. After diffusion of CO 2 across the mesophyll cell interface, CO 2 is converted to HCO 3 by carbonic anhydrase (CA), which is fixed by PEPC into C 4 acids, which diffuse to and are decarboxylated in the bundle sheath. Rubisco and the complete C 3 photosynthetic pathway are located in the bundle sheath cells, bounded by a relatively gas-tight cell wall such that the C 3 cycle relies almost entirely on C 4 acid decarboxylation as its source of CO 2 .
The net rate of CO 2 fixation for C 4 photosynthesis can be given by two equations. The first describes Rubisco carboxylation in the bundle sheath. Since all carbon fixed into sugars ultimately must be fixed by Rubisco, overall CO 2 assimilation, A, can be given by (1) where V c and V o are the rates of Rubisco carboxylation and oxygenation and R d is the rate of mitochondrial respiration not associated with photorespiration.
Mitochondrial respiration may occur in the mesophyll as well as in the bundle sheath. As Rubisco may more readily refix CO 2 released in the bundle sheath, R d is described by its mesophyll and bundle sheath components (2) Fig. 1. Schematic representing the main features of the C 4 photosynthetic pathway. CO 2 diffuses into the mesophyll where it is converted to HCO 3 and fixed by PEPC at the rate V p . In the steady state, C 4 acid decarboxylation occurs at the same rate. CO 2 released in the bundle sheath either leaks out of the bundle sheath at (L) or is fixed by Rubisco (V c ). In the photosynthetic carbon oxidation cycle, CO 2 is released at half the oxygenation rate (V o ). CO 2 is also released by respiration (R m , R s ) in mesophyll and bundle sheath cells, respectively. Electron transport components are not shown. CO 2 assimilation rate, A, can also be written in terms of the mesophyll reactions as where V p is the rate of PEP carboxylation, R m is the mitochondrial respiration occurring in the mesophyll, and L is the rate of CO 2 leakage from the bundle sheath to the mesophyll (Fig.  1). This assumes that in the steady state the rate of PEP carboxylation and the rate of C 4 acid decarboxylation are equal. The leak rate, L, is given by where g bs is the conductance to CO 2 leakage and is determined by the properties of the bundle sheath cell wall; C s and C m are the bundle sheath and mesophyll CO 2 partial pressures, respectively. It is assumed that there is a negligible amount of HCO 3 leakage from the bundle sheath since the HCO 3 pool should be small due to the absence of CA activity in the cytosol of bundle sheath cells (Farquhar, 1983;Jenkins et al., 1989;Ludwig et al., 1998).
The C 4 cycle consumes additional energy during the regeneration of PEP, and leakage of CO 2 from the bundle sheath is an energy cost to the leaf. This represents a compromise between retaining CO 2 , allowing efflux of O 2 , and permitting metabolites to diffuse in and out at rates fast enough to support the rate of CO 2 fixation (Hatch and Osmond, 1976;Raven, 1977). The CO 2 leak rate depends upon the balance between the rates of PEP carboxylation and Rubisco activity and the conductance of the bundle sheath to CO 2 .
Leakiness (ϕ), a term coined by Farquhar (1983), defines leakage as a fraction of the rate of PEP carboxylation and thus describes the efficiency of the C 4 cycle A related term 'overcycling' has also been used (Jenkins, 1989;Furbank et al., 1990). Overcycling defines the leak rate as a fraction of CO 2 assimilation rate and gives the fraction by which the flux through the C 4 acid cycle has to exceed the net CO 2 assimilation rate C 4 photosynthesis can be either limited by the enzymatic rates of PEPC and Rubisco or by the irradiance and the capacity of chloroplast electron transport which supports the regeneration of PEP and RuBP.

Enzyme-limited rate equations
Many important features of the C 4 model can be examined with the enzyme-limited rates, which are presumed to be appropriate under conditions of high irradiance. As is the case in C 3 models of photosynthesis (Farquhar et al., 1980;Farquhar and von Caemmerer, 1982;von Caemmerer, 2000), Rubisco carboxylation at high irradiance can be described by its RuBP-saturated rate where O s is the O 2 partial pressure in the bundle sheath. Following the oxygenation of 1 mol of RuBP, 0.5 mol of CO 2 is evolved in the photorespiratory pathway and the ratio of oxygenation to carboxylation can be expressed as where Γ * is the CO 2 compensation point in a C 3 plant in the absence of other mitochondrial respiration, and where the term in the bracket is the reciprocal of Rubisco specificity, S c/o (Farquhar et al., 1980). In what follows, the third expression in Equation 9 is used for Γ * since the O 2 partial pressure in the bundle sheath may vary. The Rubisco-limited rate of CO 2 assimilation can be derived from Equations 1, 7, 8, and 9.
To derive an overall expression for the CO 2 assimilation rate as a function of mesophyll CO 2 and O 2 partial pressure, C m and O m , one needs to derive an expression for C s and O s . Equation 10 can be used to derive an expression for C s : If V cmax could be estimated accurately from biochemical measurements together with A, it would provide a means of estimating bundle sheath CO 2 partial pressure. One can also obtain an expression for C s from Equations 3 and 4: PSII activity and O 2 evolution in the bundle sheath vary widely amongst C 4 species. Some NADP-ME species such as Z. mays and Sorghum bicolor have little or none, whereas NADP-ME dicots and NAD-ME and PCK species can have high PSII activity (Chapman et al., 1980;Hatch, 1987;Pfundel and Pfeffer, 1997). In S. viridis, the amount of PSII activity depends on the growth light environment (Ermakova et al., 2021). Because the bundle sheath is a fairly gas-tight compartment, this has implications for the steady-state O 2 partial pressure of the bundle sheath (Raven, 1977;Berry and Farquhar, 1978). Following Berry and Farquhar (1978), we assume that the net O 2 evolution, E o , in the bundle sheath cells equals its leakage, L o , out of the bundle sheath, that is The conductance to leakage of O 2 across the bundle sheath, g o , can be related to the conductance to CO 2 by way of the ratio of diffusivities and solubilities by where D O2 and D CO2 are the diffusivities for O 2 and CO 2 in water, respectively, and S O2 and S CO2 are the respective Henry constants such that g o = a o g bs (15) where a o =0.047 at 25 ° C (Berry and Farquhar, 1978;Farquhar, 1983). Yin et al. (2016) have shown that a o has only a small temperature dependency (Table 1). If E o =αA, where α (0<α>1) denotes the fraction of O 2 evolution occurring in the bundle sheath, then O s : Like Berry and Farquhar (1978), it is assumed that a steady-state balance exists between the rate of PEP carboxylation and the release of C 4 acids in the bundle sheath. Furthermore, it is assumed that PEP carboxylation provides the rate-limiting step and not, for example, the rate of hydration of CO 2 by CA. As PEPC utilizes HCO 3 rather than CO 2 , hydration of CO 2 is really the first step in carbon fixation in C 4 species (Hatch and Burnell, 1990).
When CO 2 is limiting, the rate of PEP carboxylation is given by a Michaelis-Menten equation where V pmax is the maximum PEP carboxylation rate, and K p is the Michaelis-Menten constant for CO 2 . This assumes that the substrate PEP is saturating under these conditions. When the rate of PEP regeneration is limiting, for example by the capacity of pyruvate orthophosphate dikinase (PPDK), then where V pr is a constant (Peisker, 1986;Peisker and Henderson, 1992) and To obtain an overall rate equation for CO 2 assimilation as a function of the mesophyll CO 2 and O 2 partial pressures (C m and O m ), one combines Equations 10, 12, and 16. The resulting expression is a quadratic of the form where (24) Equation 21 can be approximated by: where min {} stands for minimum of. At low CO 2 partial pressures, the CO 2 assimilation rate can be approximated by Under these conditions, A c is linearly related to the maximum PEPC activity, V pmax . The product g bs C m is the inward diffusion of CO 2 into the bundle sheath and, because g bs is low (0.003 mol m -2 s -1 ), the flux is only 0.3 µmol m -2 s -1 at a C m of 100 µbar and can thus be ignored. At high CO 2 partial pressures, CO 2 assimilation rate is given by either the maximal Rubisco activity, V cmax or the rate of PEP regeneration (V pr ), although it is not possible to easily distinguish these limitations in practice.

Light-and electron transport-limited rate equations
The energy requirements for the regeneration of RuBP in the bundle sheath are the same as in a C 3 leaf (Farquhar et al., 1980;von Caemmerer, 2000). There is, however, the additional cost of 2 mol ATP for the regeneration of 1 mol of PEP from pyruvate in the mesophyll such that: (Berry and Farquhar, 1978). In the PCK-type C 4 species, some of the ATP for PEP regeneration may come from the mitochondria such that the photosynthetic requirement may be less (for reviews, see von Caemmerer and Furbank, 1999;Furbank, 2011;Yin and Struik, 2021).
There is no net NADPH requirement by the C 4 cycle itself, although, for example in NADP-ME species, NADPH consumed in the production of malate from oxaloacetic acid (OAA) in the mesophyll is released in the bundle sheath during decarboxylation (Hatch and Osmond, 1976). This may have implications on the behaviour of C 4 photosynthesis under fluctuating light environments (Krall and Pearcy, 1993;Kubásek et al., 2013). The rate of NADP consumption is given by the requirement of the C 3 cycle: (28) It is important to note that in most situations, C s is probably sufficiently large that the photorespiratory term in Equations 27 and 28 can be ignored, but it does become relevant at low mesophyll CO 2 partial pressures, or at very low light (Siebke et al., 1997).
NADPH and ATP are produced by chloroplast electron transport. The reduction of NADP + to NADPH+H + requires the transfer of two electrons through the whole-chain electron transport which in turn requires two photons each at PSII and PSI. The generation of ATP can be coupled to the proton production via whole-chain electron transport, or ATP can be generated via cyclic electron transport around PSI.
PSII activity in the bundle sheath varies amongst C 4 species with different C 4 decarboxylation types. Presumably, when PSII is deficient or absent from the bundle sheath chloroplasts, some ATP is generated via cyclic photophosphorylation and 50% of the NADPH required for the reduction of 3-phosphoglyceric acid (PGA) is derived from NADPH generated by NADP + -ME (Chapman et al., 1980). The remainder of the PGA must be exported to the mesophyll chloroplast where it is reduced and then returned to the bundle sheath (Hatch and Osmond, 1976). Measurements of metabolite pools of Amaranthus edulis, an NAD + -ME species, having PSII activity in the bundle sheath, suggest that it may also export a part of the PGA to the mesophyll for reduction (Leegood and von Caemmerer, 1988). It appears therefore that energy production and consumption is shared between mesophyll and bundle sheath cells more generally across decarboxylation types (von Caemmerer and Furbank, 2016).
A very simple approach was taken in the basic photosynthesis model. The potential electron transport is modelled as a whole, allocating a different fraction of it to the C 4 and C 3 cycle rather than compartmenting it to mesophyll and bundle sheath chloroplasts (von Caemmerer and Furbank, 1999;von Caemmerer, 2000).
That is the potential whole-chain linear electron transport and J m = xJ and J s = (1-x)J, where 0<x>1. Because at most two out of five ATPs are required in the mesophyll, x equals ~0.4 (Equation 27). This partitioning approach of electron transport has been adopted by subsequent users of the model (Massad et al., 2007;Yin and Struik, 2009;Kromdijk et al., 2010;Ubierna et al., 2011;Yin et al., 2011;Zhou et al., 2019). Peisker (1988) has modelled the optimization of x at low light in some detail. See also figure 4.22 in von Caemmerer (2000). New information exists for the calculation of the ATP requirement. Following Furbank et al. (1990), von Caemmerer and Furbank assumed a stoichiometry of 3 H + per ATP produced and the operation of a Q-cycle (von Caemmerer and Furbank, 1999;von Caemmerer, 2000).
Current models of rotational catalysis predict that the H + / ATP ratio is identical to the stoichiometric ratio of c-subunits to β-subunits which is c/β=4.7 for spinach chloroplasts (Vollmar et al., 2009). However, measured values are closer to 4 for the chloroplast enzyme (Petersen et al., 2012). If 4 H + are required per ATP generated, it seems necessary to also have a functional Q-cycle which yields 3 H + per linear electron flow. The proton production during cyclic electron flow is only 2 H + per electron so that the overall proton production per electron is dependent on the balance of linear to cyclic electron flow (Yin and Struik, 2012).
Following the derivations by Yin and Struik (2012), the rate of proton production from linear and cyclic electron flow is If J 1 is the electron flow out of PSI and f cyc is the fraction of J 1 that precedes via cyclic electron flow, then J 1 = J + J cyc = J/ 1 − f cyc . For details, see Fig. 2. The rate of ATP production is given by Where h is the number of protons required per ATP generated, which here is assumed to be 4, and z relates linear electron flow J to the rate of ATP production (Yin and Struik, 2012). It follows that Where z=0.75 when there is no cyclic electron flow and z=1.25 when f cyc =0.5.
The relationship between the electron transport, J, and the absorbed irradiance that is used here is the same as that used previously where I 2 is the photosynthetically useful light absorbed by PSII, J max is the maximum electron transport, and θ is an empirical curvature factor. 0.7 is a good average value for C 3 species (Evans, 1989) and is similar for C 4 species (Sonawane et al., 2018). I 2 is related to incident irradiance by In sunlight, the absorptance of leaves is commonly ~0.85 and f is to correct for spectral quality of the light (~0.15; Evans, 1987). Ögren and Evans (1993) give a detailed discussion of the parameters of Equations 34 and 35. The parameter ρ gives the fraction of light absorbed by PSII rather than PSI. It is frequently assumed to be 0.5 for C 3 species. However, since an increase in cyclic electron flow is commonly observed in C 4 species, it has here been linked to the amount of cyclic electron flow occurring and ρ = 1 − f cyc / 2 − f cyc which equals point 0.5 if there is not cyclic occurring and 0.41 with 30% cyclic electron flow (Yin and Struik, 2012).
Putting it all together gives a light-and electron transportlimited quadratic expression. From Equations 3 and 1, one can derive two equations for an electron transport-limited CO 2 assimilation rate and Equation 37 can be solved for the bundle sheath CO 2 partial pressure and J denotes linear electron transport through PSII, J 1 is electron transport through PSI, and J cyc is the rate of cyclic electron transport. f cyc denotes the fraction of J 1 that flows via the cyclic mode. The diagram has been adapted from Yin et al. (2004). Since J=J1-Jcyc=J1(1-fcyc), J1=J/(1-fcyc).
Combining Equations 16, 36, and 38 then yields a quadratic expression of the form where And (40) can be approximated by where min {} stands for minimum of. Sometimes, when the equations are used to fit gas exchange measurements, it is sufficient to use Equation 44.

Summary of equations
Equations 21 and 40 are the two basic equations of the C 4 model and Peisker and  pointed out that either the enzyme activity or the substrate regeneration rate can limit both Rubisco and PEPC reactions and that in theory four types of combinations of rate limitations are possible. In the way the electron transport-limited equations are presented here, it is assumed that light or the electron transport capacity limit both PEP and RuBP regeneration rates simultaneously. In the model of C 3 photosynthesis by Farquhar et al. (1980) and von Caemmerer and Farquhar (1981), it was assumed that the limitation of RuBP regeneration could be adequately modelled by an electron transport limitation without consideration of limitations by other PCR cycle enzymes. This is probably the case in most instances; however; in transgenic studies, care needs to be taken. Transgenic tobacco with reduced sedoheptulose-1,7-bisphosphatase (SBPase) regeneration of RuBP has been shown to be the more limiting step (Harrison et al., 1998(Harrison et al., , 2001. In the case of C 4 photosynthesis, the possibility that PEP regeneration may also be limited by the enzyme activity of enzymes such as PPDK and ME at high irradiance has also been found in transgenic studies with Flaveria bidentis, a C 4 dicot (Trevanion et al., 1997;Furbank et al., 2001;Pengelly et al., 2012). C 4 transgenic plants with altered RuPB regeneration capacity have not yet been reported on.

An update on the parameterization of the C 4 photosynthesis model
This model is built on the same principal as the model of Farquhar et al. and many of the model's parameters can be assigned a priori, and this is indicated in Table 1, leaving only key variables such as V cmax , V pmax , V pr , and J max to be assigned. These parameters vary with leaf age and environmental variables such as nitrogen nutrition and light environment (von Caemmerer, 2000). Mesophyll conductance, g m , has been shown to decrease with leaf age (Barbour et al., 2016). It is unclear how bundle sheath conductance, g bs , varies with leaf age as it is linked to both anatomy and CO 2 diffusion parameters. Yin et al. (2011) estimated changes in g bs with N nutrition. Variation in g bs affects the coordination of the C 3 and C 4 cycle and leakiness (von Caemmerer, 2000). It is, however, important to note that the kinetic constants of Rubisco from C 4 species differ from those of C 3 species and vary amongst the different C 4 decarboxylation types (Badger et al., 1974;Ogren, 1981, 1983;Ghannoum et al., 2005;Sharwood et al., 2016). The C 4 monocot species S. viridis (green foxtail millet), an NADP-ME type which is closely related to agronomically important C 4 crops including Setaria italica (foxtail millet), Z. mays (maize), S. bicolor (sorghum), and Saccharum officinarum (sugarcane) has been suggested as a new model species (Brutnell et al., 2010). It has become a popular model species due to its rapid generation time, small stature, high seed production, diploid status, and small sequenced and publicly available genome, and it can be readily transformed (Doust, 2007;Brutnell et al., 2010;Li and Brutnell, 2011;Osborn et al., 2017;Alonso-Cantabrana et al., 2018). This has led to excellent biochemical characterization of S. viridis PEPC and Rubisco (Boyd et al., 2015;DiMario and Cousins, 2019). Most parameters were taken from Boyd et al. (2015), but the Michaelis-Menten constant for CO 2 , K p , was updated with more recent measurements by Di Mario and Cousins (2019). It is important to note that PEPC fixes bicarbonate rather than CO 2 , and K p is converted from measured values of K m for HCO 3 -. Here a cytosolic pH of 7.2 and pKa of 6.12 were assumed (Hatch and Burnell, 1990). The precise value of the cytosolic pH is unknown and if a pH of 7.4 is assumed, K p decreases from 82 µbar to 50 µbar. PEPC S. viridis RNAi lines have been used to characterize bundle sheath conductance to CO 2 diffusion (g bs ) and its temperature dependence (Alonso-Cantabrana et al., 2018). The authors observed little temperature dependence in g bs , so none was included here, but it is worth noting that significant temperature dependencies have been measured for A. edulis (Kiirats et al., 2002) and for Z. mays (Yin et al., 2016), thus there are likely to be differences in the temperature dependence for different C 4 species, as was observed for mesophyll conductance in different C 3 species (von Caemmerer and Evans, 2015). The parameterization here has for the first time provided a cohesive parameter set and the associated temperature functions (Table 1).
It is best to estimate the temperature response of J max from a series of light response curves made at high CO 2 and different temperatures, as was done by Massad et al. (2007) for Z. mays. They used an Arrhenius function for their parameterization. The simpler temperature function suggested by June et al. (2004) is used here to parameterize the temperature dependence of electron transport (Table 1), and the parameterization for tobacco has been used (Yamori et al., 2010) since these experiments still need to be done for S. viridis. Sonawane et al. (2017), who characterized the temperature response of the CO 2 assimilation rate in a number of C 4 grasses, used this function to fit the saturated rate of CO 2 assimilation measured at high irradiance. The tobacco values chosen here fit within the range values reported for these C 4 grasses (Sonawane et al., 2017) Calculating the electron transport required to sustain CO 2 assimilation von Caemmerer and Farquhar (1981) suggested that measurements of the CO 2 assimilation rate can be used to calculate the actual electron transport rate, J a , needed to support the CO 2 assimilation rate. Equation 39 can be used in the same way. Using Equation 39 to solve for J a , this results in the following quadratic equation:

Modelled CO 2 response of CO 2 assimilation
In C 3 species, CO 2 response curves are widely used to assess photosynthetic capacity (von Caemmerer and Farquhar, 1981;von Caemmerer, 2000;Ainsworth and Rogers, 2007;Sharkey et al., 2007). Figure 3 compares the model output of the Farquhar, von Caemmerer, and Berry model of C 3 photosynthesis (Farquhar et al., 1980) with the current C 4 model presented here. In the C 3 model, the enzyme-limited rate is dominated by Rubisco and its kinetic parameters at low CO 2 , and the electron transport capacity limits at high CO 2 (Fig. 3A). In the C 4 model, it is also possible to distinguish an enzyme-limited CO 2 assimilation rate at high light (Equations 20-25) and an electron transport-limited rate (Equations 37-44). However, the enzyme-limited rate is determined by PEPC at low CO 2 and Rubisco at high CO 2 . The electron transport-limited rate can also determine the CO 2 assimilation rate at high CO 2 (Fig. 3B). Thus, it is more difficult to identify biochemical limitations to C 4 photosynthesis.
Usually good correlations are found between in vitro Rubisco activity and the CO 2 -saturated rate of CO 2 assimilation rate at high CO 2 (Usuda, 1984;Usuda et al., 1984;Sonawane et al., 2017). The relationship should be almost one to one as Rubisco operates close to its saturated rate in vivo. In the study of F. bidentis, transgenics with varying reductions in Rubisco content show a slight curve in the relationship between Rubisco content and CO 2 assimilation rate, hinting at a possible electron transport limitation in wild-type plants (Furbank et al., 1996;von Caemmerer et al., 1997). These studies have also provided evidence that Rubisco limits CO 2 assimilation at high CO 2 . Transgenic plants with reduced Rubisco content showed a clear decline in CO 2 assimilation rate at high CO 2 (von Pengelly et al., 2012). Recent photosynthetic engineering that increased Rubisco content in maize leaves resulted in an increase in CO 2 -saturated CO 2 assimilation rate (Salesse-Smith et al., 2018).
Here the model has been tuned in such a way that at 25 °C the electron transport rate is limiting CO 2 assimilation at high CO 2 and high irradiance. This balance can of course vary with growth conditions or species, but there is no straightforward technique to determine the limitation. Furthermore, the assumption has been made that the electron transport capacity and PEP and RuBP regeneration generally co-limit. In transgenic studies where regeneration of the C 4 cycle has been curtailed by molecular manipulation, it is clear that this also limits CO 2 assimilation at high CO 2 (Trevanion et al., 1997;Pengelly et al., 2012). In a study by Ermakova et al. (2019), transgenic S. viridis with overexpression of the Rieske iron-sulfur protein in the cytochrome b 6 f complex had increased CO 2 assimilation rates at ambient and high CO 2 , confirming that electron transport capacity can limit the CO 2 assimilation rate. The fact that electron transport rate limits the CO 2 assimilation rate at high CO 2 means that a reduction in irradiance is also predicted to primarily affect the CO 2 -saturated rate of CO 2 assimilation rather than the initial slope of the CO 2 response curve, except at low irradiance (Leegood and von Caemmerer, 1989;Pfeffer and Peisker, 1998).
There are three possible limitations to the initial slope of the CO 2 response curve: the mesophyll conductance to CO 2 diffusion from the intercellular airspace to the mesophyll cytosol g m ; the rate of CO 2 hydration by CA; and the rate of PEP carboxylation. It is thought that most C 4 leaves have sufficient CA for it not to be rate limiting (Hatch and Burnell, 1990;Cousins et al., 2008). However, studies with transgenic or mutant plants in F. bidentis, Z. mays, and S. viridis have shown that when CA activity is greatly reduced, a reduction in initial slope of the CO 2 response is observed (von Caemmerer et al., 2004;Studer et al., 2014;Osborn et al., 2017).
The initial C 4 photosynthesis models did not consider a diffusion limitation between the intercellular airspace and the mesophyll cytosol. In C 4 species, mesophyll conductance, g m , is likely to be proportional to mesophyll surface area exposed to intercellular airspace (Evans and von Caemmerer, 1996). The standard techniques used to quantify mesophyll conductance in C 3 species such as combined measurements of gas exchange and chlorophyll fluorescence, or measurements of 13 C isotope discrimination cannot be used in C 4 species; however, a new technique has been developed to measure mesophjyll conductance in C 4 species using C 18 O 16 O isotope discrimination (Gillon and Yakir, 2000;Barbour et al., 2016;Osborn et al., 2017;Ogée et al., 2018), and here the temperature dependence of g m measured for S. viridis has been used for parameterization of the model (Table 1; Ubierna et al., 2017).
The drop in CO 2 partial pressure from intercellular airspace, C i to that of the mesophyll, C m is related in the following equation Incorporating Equation 50 into Equation 21 results in a cubic expression which is not easily solved. It can be incorporated into Equation 40, giving a slightly more complex quadratic. In the case of the initial slope of the CO 2 response curve, one can use Equation 26 and, ignoring the term g bs C m and combining it with Equation 50, a quadratic similar to the one given for C 3 leaves is obtained (von Caemmerer and Evans, 1991;von Caemmerer, 2000).
(51) The first derivative with respect to C i at C i =0 is given by Pfeffer andPeisker (1988 and used this equation together with measurements of PEPC activity and initial slope (dA/dC i ) to estimate g m in plants grown under different light intensities. Figure 4 shows the effect that inclusion of mesophyll conductance has on the initial slope. Equation 52 was used by Ubierna et al. (2017) to estimate g m from in vitro measurements of PEPC activity, V pmax , and they found good agreement with estimates of g m from measurements of C 18 O 16 O isotope discrimination; however, the uncertainties surrounding estimates of K p discussed above need to be considered. Strong correlations between leaf nitrogen, CO 2 assimilation rate, and PEPC activity have been observed in several studies (Usuda, 1984;Wong et al., 1985;Sage and Pearcy, 1987;Meinzer and Zhu, 1998); however, it has been more difficult to provide quantitative correlations. Without the inclusion of a mesophyll conductance, estimates of V pmax from the initial slope are often less than what is measured in vitro. For example, if the initial slope of the lower curve in Fig. 4A is used to estimate V pmax with an infinitely large g m , the predicted V pmax Fig. 3. A comparison of modelled rates of the CO 2 assimilation rate as functions of partial pressures of CO 2 for C 3 (A) and C 4 photosynthesis (B). (A) Modelled rate of CO 2 assimilation as a function of chloroplast CO 2 partial pressure for the C 3 photosynthetic pathway at 25 °C. The Rubisco-limited (RuBP-saturated) rate of CO 2 assimilation has a dashed line extension at high CO 2 . The electron transport-limited (RuBP regeneration) rate of CO 2 assimilation has a dotted line extension at low CO 2 . The solid curve represents the minimum rate that is the actual rate of CO 2 assimilation. A possible triose phosphate limitation at high CO 2 is not shown. (after von Caemmerer, 2000). (B) Modelled rate of CO 2 assimilation as a function of mesophyll cytosolic CO 2 partial pressure for the C 4 photosynthetic pathway at 25 °C and an irradiance of 2000 µmol m -2 s -1 . The enzyme-limited CO 2 assimilation rate (PEPC limitation at low CO 2 and Rubisco limitation at high CO 2 shows the Rubisco limited rate as a dashed line extension. The electron transportlimited (RuBP and PEP regeneration) rate of CO 2 assimilation has a dotted line extension at low CO 2 . The solid curve represents the minimum rate that is the actual rate of CO 2 assimilation. Parameters used are given in Table 1. is 58 µmol m -2 s -1 , whereas it has here been modelled with a V pmax of 200 µmol m -2 s -1 and g m =1 mol m -2 s -1 bar -1 . Hence mesophyll conductance is an important parameter in linking C 4 biochemistry with gas exchange.

Modelled light response of CO2 assimilation
It is well recognized that the light response of C 4 photosynthesis frequently does not saturate (Cousins et al., 2006;Leakey et al., 2006). Figure 5 shows typical modelled light response curves of the CO 2 assimilation rate at several mesophyll CO 2 partial pressures. In the current parameterization, the CO 2 assimilation rate is electron transport limited at all irradiances above a C m of 150 µbar at 25 °C. The shapes of the curves are determined by Equation 34 which as for the C 3 photosynthetic model remains empirical and the partition partitioning of electron transport between the C 4 and C 3 cycle has been set at x=0.4 (Equation 29). Furbank and von Caemmerer gave a detailed discussion about the optimal partitioning of electron transport capacity between the C 3 and C 4 cycle (Peisker, 1988;von Caemmerer and Furbank, 1999;von Caemmerer, 2000). It is noteworthy that the fraction of electron transport allocated to the C 4 cycle, x, equals 0.4 over a wide range of irradiances but drops at very low irradiance. Under low light, the bundle sheath CO 2 partial pressures are close to the mesophyll CO 2 partial pressure, and electron transport is required for recycling of photorespiratory CO 2 . The optimal partitioning increases from 0.404 to 0.417 if oxygen is evolved in the bundle sheath (α=1). It also declines slightly with increasing temperature as Rubisco specificity for CO 2 decreases (Jordan and Ogren, 1984;Sharwood et al., 2016).
In C 3 species, a close link has been established between chloroplast electron transport capacity and electron transport chain intermediates such as cytochrome f (Yamori et al., 2010). In C 4 photosynthesis, this quantitative link between cytochrome f content and electron transport capacity also needs to be investigated.
Modelled temperature response of CO 2 assimilation rate C 4 plants have higher CO 2 assimilation rates at high temperatures and higher photosynthetic temperature optima than their C 3 counterparts largely because of the elimination of photorespiratory CO 2 losses (Berry and Björkman, 1980;Long, 1999). The temperature response of electron transport is not well characterized in C 4 species. With the parameterization used here, the CO 2 assimilation rate is electron transport limited above 25 °C, and enzyme limited below a C i of 150 µbar at high light, which corresponds to the operating C i of many C 4 species at ambient CO 2 (Fig. 6). There is some evidence that this is not unreasonable. Figure 7 shows a comparison of a temperature response of the CO 2 assimilation rate of wild-type and transgenic F. bidentis with reduced Rubisco content (Kubien et al., 2003). In Fig. 7B, the CO 2 assimilation rate is expressed on a Rubisco site basis (in vivo k cat ) and the temperature response of Rubisco in vitro activity is compared. For Flaveria with a reduced amount of Rubisco there is a match between in vivo and in vitro k cat up to ~30 °C, whereas for the wild type in vivo k cat is less than the in vitro Rubisco k cat around 20 °C, indicating other limitations to the CO 2 assimilation rate such as electron transport capacity. Temperature optima of the CO 2 assimilation rate are dependent on growth environment (Berry and Björkman, 1980;Dwyer et al., 2007). The modelling suggests that the temperature optimum is most probably determined by the properties of the electron transport capacity (Fig. 6).
In Fig. 8A, the CO 2 response curves have been modelled for different leaf temperatures. Figure 8A shows that at low temperature the CO 2 response is enzyme limited at all C i ; as Fig. 4. The effect of mesophyll conductance, g m , on the initial slope of the CO 2 response curve. (A) Modelled rate of CO 2 assimilation as a function of intercellular CO 2 partial pressure for the C 4 photosynthetic pathway at 25 °C and an irradiance of 2000 µmol m -2 s -1 modelled with g m =1 mol m -2 s -1 bar -1 or an infinite g m . (B) Initial slope (Equation 52) as a function of mesophyll conductance. Model parameters are those given in Table 1. temperature is increased, the transition from enzyme-limited CO 2 assimilation rate to electron transport-limited rate occurs at progressively lower C i . There is an increase in initial slope with increasing temperature which is caused by the temperature response of the mesophyll conductance and the different temperature dependencies of maximal PEPC carboxylation and the Michaelis-Menten constant (V pmax and K p ) (Fig. 8A, B; Table 1). These model predictions fit well with experimental observations by Sonawane et al. (2017).

A note on leakiness
The bundle sheath resistance or, its inverse, the bundle sheath conductance to CO 2 diffusion are key parameters that together with relative capacities for the C 4 cycle and Rubisco and electron transport capacity determine the effectiveness of the CO 2 concentration mechanism. This is often quantified by a term called leakiness (ϕ), which is defined as the ratio of the rates of CO 2 leakage out of the bundle sheath over the rate of CO 2 supply to the bundle sheath (Equation 5). Carbon isotope discrimination can be used to determine leakiness (Farquhar, 1983). Combined Fig. 5. The effect of irradiance on the modelled rate of CO 2 assimilation. (A) Light response of the CO 2 assimilation rate at mesophyll cytosolic CO 2 partial pressures, C m , indicated in the figure. The C 4 photosynthesis model predicts electron transport limitations at all irradiances at C m values >150 µbar. At lower C m , CO 2 assimilation rates are enzyme limited at high irradiance. The model was parameterized at 25 °C with values given in Table 1. (B) CO 2 assimilation rate as a function of intercellular CO 2 at the irradiances indicated. The model was parameterized at 25 °C with values given in Table 1. Fig. 6. Modelled CO 2 assimilation rate as a function of leaf temperature. The dotted line and its extension line show the enzyme-limited rate, and the dashed line and its extension line show the electron transport-limited rate. The CO 2 assimilation rate was modelled at an irradiance of 2000 µmol m -2 s -1 and an intercellular CO 2 , C i , of 150 µbar. Other parameters are as given in Table 1. The in vitro data reflect the activity of the fully carbamylated enzyme; in vivo k cat is estimated as gross photosynthesis divided by the number of Rubisco catalytic sites. Each value represents the mean (±SE) of four measurements. The data are redrawn from figs 1 and 4 of Kubien et al. (2003) measurements of gas exchange and carbon isotope discrimination have been used to assess leakiness under different environmental conditions von Caemmerer and Furbank, 2003;Kromdijk et al., 2010;Pengelly et al., 2010;Ubierna et al., 2011;King et al., 2012;Sun et al., 2012). It is tempting to predict leakiness from the C 4 photosynthesis model but, because it is a flux model, little can be said about the rate of the component that is not limiting, and leakiness estimates are not realistic as non-rate-limiting steps are likely to be downregulated. However, when CO 2 assimilation rates and leakiness are known from combined measurements of gas exchange and carbon isotope discrimination, the model can be used to calculate the rate of the C 4 cycle using the equation below.
The leak rate can then be calculated from Equation 5. With the assumption of a bundle sheath conductance, bundle sheath CO 2 partial pressure can then be estimated from Equation 4 (Pengelly et al., 2012).

Modelling different decarboxylation types
The biochemistry of the C 4 photosynthetic pathway is not unique, and three main biochemical subtypes are recognized on the basis of the predominant decarboxylating enzyme: NADP-ME, NAD-ME, or PCK (Hatch, 1987). How this affects modelling of C 4 photosynthesis was discussed by von Caemmerer and Furbank (1999) and von Caemmerer (2000).
Here the model has been parameterized for the NADP-ME subtype where we assume no oxygen evolution in the bundle sheath (α=0, Table 1). Both NAD-ME and PCK subtypes have some PSII activity in the bundles sheath, and this needs to be considered. Rubisco and PEPC kinetic properties have also been shown to differ between C 4 species and subtypes (Ghannoum et al., 2005;Sharwood et al., 2016;DiMario et al., 2021); however, at present, there are no complete parameter sets that can be used. For PCK subtypes, the ATP requirement for PEP regeneration is reduced, which requires a different equation for the ATP requirement (von Caemmerer and Furbank, 1999;Furbank, 2011;Yin and Struik, 2021).

Conclusion
The steady-state C 4 photosynthesis model has been updated and parameterized with the in vitro kinetic constants for Rubisco and PEPC, and values for mesophyll and bundle sheath conductance and their temperature dependencies. Furthermore, electron transport rate equations have been updated to include cyclic electron transport flow. Now it is important to compare gas exchange measurements and biochemical measurements to confirm the quantitative relationships predicted by the model and assess variation of these parameters with environmental variation. In particular, a parameterization of the temperature response of the electron transport rate is needed and information on how it relates to thylakoid electron transport components such as the b 6 f complex which has shown to be a good correlator of C 3 photosynthetic electron transport. Modelled rate of CO 2 assimilation as a function of intercellular CO 2 partial pressure, C i , for the C 4 photosynthetic pathway at three leaf temperatures of 15, 25, and 35 °C and an irradiance of 2000 µmol m -2 s -1 . At 15 °C, the CO 2 assimilation rate is enzyme limited at all C i . For 25 °C and 35 °C, the arrow indicates the transition from enzyme limitation at low C i to electron transport limitation at high C i . Parameters used are given in Table 1.
(B) Intial slope (dA/dC i ) calculated from Equation 52 as a function of leaf temperature. Parameters used are given in Table 1.