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.

Update of a C₄ photosynthesis model developed by von Caemmerer and Furbank to include temperature dependencies and equations 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 C₄ crops have stimulated considerable interest in the C₄ photosynthetic pathway which is characterized by a high photosynthetic rate and high nitrogen and water use efficiency relative to plants with the C₃ 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., 2018, 2019).

C₄ photosynthesis requires the coordinated functioning of mesophyll and bundle sheath cells of leaves, and is characterized by a CO₂-concentrating mechanism which allows Rubisco, located in the bundle sheath cells, to operate at high CO₂ partial pressures. This overcomes the low affinity Rubisco has for CO₂ and largely inhibits its oxygenation reaction, reducing photorespiration rates. In the mesophyll, CO₂ is initially fixed by phosphoenolpyruvate (PEP) carboxylase (PEPC) into C₄ acids, which are then decarboxylated in the bundle sheath to supply CO₂ for Rubisco. Both the structure of the bundle sheath wall (which has a low permeability to CO₂) and the relative biochemical capacities of the C₃ cycle in the bundle sheath and C₄ acid cycle (which operates across the mesophyll–bundle sheath interface) contribute to the high CO₂ partial pressure in the bundle sheath. The biochemistry of the C₄ 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 C₄ 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 CO₂ partial pressures and their dependence on bundle sheath conductance. Many of the gas exchange characteristics of C₄ 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.

C₄ models have not been used as frequently as the C₃ 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, 2017; Zhou et al., 2019).

Here an update of the C₄ 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 C₄ monocot species Setaria viridis (green foxtail millet), an NADP-ME type which is closely related to agronomically important C₄ 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 C₄ photosynthesis model presented here has a similar structure to the widely used model of C₃ photosynthesis by Farquhar et al. (1980). That is C₄ 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 CO₂ 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 C₄ photosynthesis. After diffusion of CO₂ across the mesophyll cell interface, CO₂ is converted to HCO₃^– by carbonic anhydrase (CA), which is fixed by PEPC into C₄ acids, which diffuse to and are decarboxylated in the bundle sheath. Rubisco and the complete C₃ photosynthetic pathway are located in the bundle sheath cells, bounded by a relatively gas-tight cell wall such that the C₃ cycle relies almost entirely on C₄ acid decarboxylation as its source of CO₂.

Fig. 1.

Schematic representing the main features of the C₄ photosynthetic pathway. CO₂ diffuses into the mesophyll where it is converted to HCO₃^– and fixed by PEPC at the rate V_p. In the steady state, C₄ acid decarboxylation occurs at the same rate. CO₂ 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, CO₂ is released at half the oxygenation rate (V_o). CO₂ is also released by respiration (R_m, R_s) in mesophyll and bundle sheath cells, respectively. Electron transport components are not shown.

The net rate of CO₂ fixation for C₄ 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 CO₂ assimilation, A, can be given by

$${\displaystyle \begin{array}{l}{\displaystyle A=V_{\mathrm{c}}-0.5V_{\mathrm{o}}-R_{\mathrm{d}}}\end{array}}$$ (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 CO₂ released in the bundle sheath, R_d is described by its mesophyll and bundle sheath components

$${\displaystyle \begin{array}{l}{\displaystyle R_{\mathrm{d}}=R_{\mathrm{m}}+R_{\mathrm{s}}.}\end{array}}$$ (2)

CO₂ assimilation rate, A, can also be written in terms of the mesophyll reactions as

$${\displaystyle \begin{array}{l}{\displaystyle A=V_{\mathrm{p}}-L-R_{\mathrm{m}},}\end{array}}$$ (3)

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 CO₂ 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 C₄ acid decarboxylation are equal.

The leak rate, L, is given by

$${\displaystyle \begin{array}{l}{\displaystyle L=g_{\mathrm{b}\mathrm{s}}(C_{\mathrm{s}}-C_{\mathrm{m}})}\end{array}}$$ (4)

where g_bs is the conductance to CO₂ leakage and is determined by the properties of the bundle sheath cell wall; C_s and C_m are the bundle sheath and mesophyll CO₂ partial pressures, respectively. It is assumed that there is a negligible amount of HCO₃^– leakage from the bundle sheath since the HCO₃^– 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 C₄ cycle consumes additional energy during the regeneration of PEP, and leakage of CO₂ from the bundle sheath is an energy cost to the leaf. This represents a compromise between retaining CO₂, allowing efflux of O₂, and permitting metabolites to diffuse in and out at rates fast enough to support the rate of CO₂ fixation (Hatch and Osmond, 1976; Raven, 1977). The CO₂ leak rate depends upon the balance between the rates of PEP carboxylation and Rubisco activity and the conductance of the bundle sheath to CO₂.

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 C₄ cycle

$${\displaystyle \begin{array}{l}{\displaystyle \varphi =L/V_{\mathrm{p}}.}\end{array}}$$ (5)

A related term ‘overcycling’ has also been used (Jenkins, 1989; Furbank et al., 1990). Overcycling defines the leak rate as a fraction of CO₂ assimilation rate and gives the fraction by which the flux through the C₄ acid cycle has to exceed the net CO₂ assimilation rate

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{O}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}=L/A=\left[V_{\mathrm{p}}(A+R_{\mathrm{m}})\right]/A.}\end{array}}$$ (6)

C₄ 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 C₄ 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 C₃ 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

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{c}}=\frac{C_{\mathrm{s}}{V}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}}{C_{\mathrm{s}}+K_{\mathrm{c}}(1+O_{\mathrm{s}}/K_{\mathrm{o}})}}\end{array}}$$ (7)

where O_s is the O₂ partial pressure in the bundle sheath. Following the oxygenation of 1 mol of RuBP, 0.5 mol of CO₂ is evolved in the photorespiratory pathway and the ratio of oxygenation to carboxylation can be expressed as

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{o}}/V_{\mathrm{c}}=2{\mathrm{\Gamma }}_{\ast }/C_{\mathrm{s}}}\end{array}}$$ (8)

where Γ _* is the CO₂ compensation point in a C₃ plant in the absence of other mitochondrial respiration, and

$${\displaystyle \begin{array}{l}{\displaystyle {\mathrm{\Gamma }}_{\ast }=0.5\left[V_{\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{x}}{K}_c/V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}{K}_{\mathrm{o}}\right]O_{\mathrm{s}}={\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}},}\end{array}}$$ (9)

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 O₂ partial pressure in the bundle sheath may vary.

The Rubisco-limited rate of CO₂ assimilation can be derived from Equations 1, 7, 8, and 9.

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{c}}=\frac{(C_{\mathrm{s}}-{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}})V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}}{C_{\mathrm{s}}+K_{\mathrm{c}}(1+O_{\mathrm{s}}/K_{\mathrm{o}})}-R_{\mathrm{d}}.}\end{array}}$$ (10)

To derive an overall expression for the CO₂ assimilation rate as a function of mesophyll CO₂ and O₂ 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:

$${\displaystyle \begin{array}{l}{\displaystyle C_s=\frac{\gamma \ast O_s+K_c(1+O_s/K_o)((A+R_d)/V_{cmax})}{1-(A+R_d)/V_{cmax}}}\end{array}}$$ (11)

If V_cmax could be estimated accurately from biochemical measurements together with A, it would provide a means of estimating bundle sheath CO₂ partial pressure. One can also obtain an expression for C_s from Equations 3 and 4:

$${\displaystyle \begin{array}{l}{\displaystyle C_{\mathrm{s}}=C_{\mathrm{m}}+\frac{V_{\mathrm{p}}-A-R_{\mathrm{m}}}{g_{\mathrm{b}\mathrm{s}}}.}\end{array}}$$ (12)

PSII activity and O₂ evolution in the bundle sheath vary widely amongst C₄ 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 O₂ partial pressure of the bundle sheath (Raven, 1977; Berry and Farquhar, 1978). Following Berry and Farquhar (1978), we assume that the net O₂ evolution, E_o, in the bundle sheath cells equals its leakage, L_o, out of the bundle sheath, that is

$${\displaystyle \begin{array}{l}{\displaystyle E_{\mathrm{o}}=L_{\mathrm{o}}=g_{\mathrm{o}}(O_{\mathrm{s}}-O_{\mathrm{m}}).}\end{array}}$$ (13)

The conductance to leakage of O₂ across the bundle sheath, g_o, can be related to the conductance to CO₂ by way of the ratio of diffusivities and solubilities by

$${\displaystyle \begin{array}{l}{\displaystyle g_{\mathrm{o}}=g_{\mathrm{b}\mathrm{s}}{D}_{{\mathrm{O}}_2}{S}_{{\mathrm{O}}_2}/\left(D_{\mathrm{C}{\mathrm{O}}_2}{S}_{\mathrm{C}{\mathrm{O}}_2}\right),}\end{array}}$$ (14)

where D_O2 and D_CO2 are the diffusivities for O₂ and CO₂ in water, respectively, and S_O2 and S_CO2 are the respective Henry constants such that

$${\displaystyle \begin{array}{l}{\displaystyle g_{\mathrm{o}}=a_{\mathrm{o}}{g}_{\mathrm{b}\mathrm{s}}}\end{array}}$$ (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 O₂ evolution occurring in the bundle sheath, then O_s:

Table 1.

Photosynthetic parameters used in the model: when available, values for Setaria viridis have been chosen.

Parameter	Value at 25 °C	Definition	Activation energy E (kJmol–1)	Reference
V cmax	40 µmol m–2 s–1 or variable	Maximum Rubisco activity	78a	Boyd et al. (2015)
K c	1210 µbar	Michaelis constant of Rubisco for CO2	64.2	Boyd et al. (2015)
K o	292 mbar	Michaelis constant of Rubisco for O2	10.5	Boyd et al. (2015)
γ *	=0.0003817 (0.5/1310)	0.5/(Sc/o) half the reciprocal of Rubisco specificity	31.1	Boyd et al. (2015)
V pmax	200 µmol m–2 s–1 or variable	Maximum PEPC activity	50.1b	Boyd et al. (2015)
V pr	80 µmol m–2 s–1 or variable	PEP regeneration rate
K p	82 µbar	Michaelis constant of PEPC for CO2	38.3b	DiMario and Cousins (2019)
g bs	0.003 mol m–2 s–1 bar–1	Bundle sheath conductance to CO2	Constant with temperature	Alonso-Cantabrana et al. (2018)
a o	0.047	Ratio of solubility and diffusivity of O2 to CO2	1.63	Farquhar (1983); Yin et al. (2016)
g o	aogbs	Bundle sheath conductance to O2		Farquhar (1983)
R d	0.01×Vcmax	Leaf mitochondrial respiration	66.4	Farquhar et al. (1980)
R m	0.5 Rd	Mesophyll mitochondrial respiration
α	0<α>1	Fraction of PSII activity in the bundle sheath
x	0.4	Partitioning factor of electron transport rate
J max	248 µmol electrons m–2 s–1 or variable	Maximal linear electron transport rate
T o	43 °C	T optimum	$J_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(T_{\mathrm{L}}\right)={\mathrm{J}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(T_{\mathrm{o}}\right)e^{-{\left(\frac{T_{\mathrm{L}}-T_{\mathrm{o}}}{\mathrm{\Omega }}\right)}^2}$	June et al. (2004); Yamori et al. (2010)
Ω	26	Ω is the difference in temperature from To at which J falls to e–1 (0.37)		Yamori et al. (2010)
Jmax@ To	400 µmol electrons m–2 s–1 or variable
h	4	Number of protons per ATP
f cyc	0.3 or variable	Fraction of cyclic electron transport		Yin and Struik (2012)
z		Ratio of the rate of ATP production to linear electron transport (JATP/J)		Yin and Struik (2012)
g m	1 mol m–2 s–1 bar–1	Mesophyll conductance to CO2	49.8c	Ubierna et al. (2017)

^a Values of parameters where activation energies have been given can be calculated at any temperature from the following equation: $\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\left(25{}^{\circ }\mathrm{C}\right)\mathrm{e}\mathrm{x}\mathrm{p}\{\left(T25\right)E/[298R(273+T)]\}$, where R (8.314J K^–1 mol^–1) is the universal gas constant and T is temperature in °C.

^b Data from Boyd et al. (2015) have been refitted with the simple Arrhenius function in ^a.

^c Data from Ubierna et al. (2017) have been refitted with the simple Arrhenius function in ^a including data from 10 °C to 35 °C.

$${\displaystyle \begin{array}{l}{\displaystyle O_{\mathrm{s}}=\frac{\mathrm{\alpha }A}{a_{\mathrm{o}}{g}_{\mathrm{b}\mathrm{s}}}+O_{\mathrm{m}}.}\end{array}}$$ (16)

Like Berry and Farquhar (1978), it is assumed that a steady-state balance exists between the rate of PEP carboxylation and the release of C₄ 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 CO₂ by CA. As PEPC utilizes HCO₃^– rather than CO₂, hydration of CO₂ is really the first step in carbon fixation in C₄ species (Hatch and Burnell, 1990).

When CO₂ is limiting, the rate of PEP carboxylation is given by a Michaelis–Menten equation

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{p}}=\frac{C_{\mathrm{m}}{V}_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}}{C_{\mathrm{m}}+K_{\mathrm{p}}}}\end{array}}$$ (17)

where V_pmax is the maximum PEP carboxylation rate, and K_p is the Michaelis–Menten constant for CO₂. 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

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{p}}=V_{\mathrm{p}\mathrm{r}}}\end{array}}$$ (18)

where V_pr is a constant (Peisker, 1986; Peisker and Henderson, 1992) and

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{p}}=\mathrm{m}\mathrm{i}\mathrm{n}\{\frac{C_{\mathrm{m}}{V}_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}}{C_{\mathrm{m}}+K_{\mathrm{p}}},V_{\mathrm{p}\mathrm{r}}\}}\end{array}}$$ (19)

To obtain an overall rate equation for CO₂ assimilation as a function of the mesophyll CO₂ and O₂ partial pressures (C_m and O_m), one combines Equations 10, 12, and 16. The resulting expression is a quadratic of the form

$${\displaystyle \begin{array}{l}{\displaystyle a{A}_{\mathrm{c}}^2+b{A}_{\mathrm{c}}+c=0,}\end{array}}$$ (20)

where

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{c}}=(-b+\sqrt{b^2-4ac})/\left(2a\right)}\end{array}}$$ (21)

$${\displaystyle \begin{array}{l}{\displaystyle a=\frac{\mathrm{\alpha }}{a_{\mathrm{o}}}\frac{K_{\mathrm{c}}}{K_{\mathrm{o}}}}\end{array}}$$ (22)

$${\displaystyle \begin{array}{l}{\displaystyle b=-\{(V_{\mathrm{p}}-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_{\mathrm{m}})+(V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}-R_{\mathrm{d}})+g_{\mathrm{b}\mathrm{s}}(K_{\mathrm{c}}[1+O_m/K_o])+\frac{\mathrm{\alpha }}{a_{\mathrm{o}}}({\mathrm{\gamma }}_{\ast }{V}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}+R_{\mathrm{d}}{K}_{\mathrm{c}}/K_{\mathrm{o}})\}}\end{array}}$$ (23)

$${\displaystyle \begin{array}{l}{\displaystyle c=(V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}-R_{\mathrm{d}})(V_{\mathrm{p}}-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_{\mathrm{m}})-(V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}{g}_{\mathrm{b}\mathrm{s}}{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{m}}+R_{\mathrm{d}}{g}_{\mathrm{b}\mathrm{s}}{K}_c[1+O_m/K_o])}\end{array}}$$ (24)

Equation 21 can be approximated by:

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{c}}=\mathrm{m}\mathrm{i}\mathrm{n}\{(V_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}-R_{\mathrm{d}}),(V_{\mathrm{p}}-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_{\mathrm{m}})\}}\end{array}}$$ (25)

where min {} stands for minimum of.

At low CO₂ partial pressures, the CO₂ assimilation rate can be approximated by

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{c}}=\frac{C_{\mathrm{m}}{V}_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}}{C_{\mathrm{m}}+K_{\mathrm{p}}}-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_{\mathrm{m}}}\end{array}}$$ (26)

Under these conditions, A_c is linearly related to the maximum PEPC activity, V_pmax. The product g_bsC_m is the inward diffusion of CO₂ 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 CO₂ partial pressures, CO₂ 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 C₃ 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:

$${\displaystyle \begin{array}{l}{\displaystyle RateofATPconsumption=2V_{\mathrm{p}}+(3+7{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/C_{\mathrm{s}})V_c}\end{array}}$$ (27)

where $\left(7{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/C_{\mathrm{s}}\right)V_{\mathrm{c}}$ is the energy requirement due to photorespiration (since $V_{\mathrm{o}}/V_{\mathrm{c}}=2{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/C_{\mathrm{s}}$) (Berry and Farquhar, 1978). In the PCK-type C₄ 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 C₄ 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 C₄ 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 C₃ cycle:

$${\displaystyle \begin{array}{l}{\displaystyle RateofNADPconsumption=(2+4{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/C_{\mathrm{s}})V_{\mathrm{c}}.}\end{array}}$$ (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 CO₂ 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 C₄ species with different C₄ 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 C₄ and C₃ 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

$${\displaystyle \begin{array}{l}{\displaystyle J=J_{\mathrm{m}}+J_{\mathrm{s}}}\end{array}}$$ (29)

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

$${\displaystyle \begin{array}{l}{\displaystyle J_{{\mathrm{H}}^{+}}=3J+2J_{\mathrm{c}\mathrm{y}\mathrm{c}}=\frac{3-f_{\mathrm{c}\mathrm{y}\mathrm{c}}}{1-f_{\mathrm{c}\mathrm{y}\mathrm{c}}}J}\end{array}}$$ (30)

If J₁ is the electron flow out of PSI and f_cyc is the fraction of J₁ that precedes via cyclic electron flow, then $J_1=J+J_{\mathrm{c}\mathrm{y}\mathrm{c}}=J/(1-f_{\mathrm{c}\mathrm{y}\mathrm{c}})$. For details, see Fig. 2. The rate of ATP production is given by

Fig. 2.

Scheme for linear and cyclic electron transport in the light reactions of photosynthesis. The arrows indicate electron transport. Thick curved arrows show the number of protons produced per electron transported. J denotes linear electron transport through PSII, J₁ is electron transport through PSI, and J_cyc is the rate of cyclic electron transport. f_cyc denotes the fraction of J₁ 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).

$${\displaystyle \begin{array}{l}{\displaystyle J_{\mathrm{A}\mathrm{T}\mathrm{P}}=J_{{\mathrm{H}}^{+}}/h=\frac{3-f_{\mathrm{c}\mathrm{y}\mathrm{c}}}{h(1-f_{\mathrm{c}\mathrm{y}\mathrm{c}})}J=zJ}\end{array}}$$ (31)

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

$${\displaystyle \begin{array}{l}{\displaystyle J_{\mathrm{m}}=\frac{2}{z}{V}_{\mathrm{p}}}\end{array}}$$ (32)

Where z=0.75 when there is no cyclic electron flow and z=1.25 when f_cyc=0.5.

$${\displaystyle \begin{array}{l}{\displaystyle J_{\mathrm{s}}=\frac{3}{z}(1+7{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/3C_{\mathrm{s}})V_{\mathrm{c}}.}\end{array}}$$ (33)

The relationship between the electron transport, J, and the absorbed irradiance that is used here is the same as that used previously where

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{\theta }{J}^2-J(I_2+J_{\mathrm{m}\mathrm{a}\mathrm{x}})+I_2{J}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0}\end{array}}$$ (34)

I₂ 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 C₃ species (Evans, 1989) and is similar for C₄ species (Sonawane et al., 2018). I₂ is related to incident irradiance by

$${\displaystyle \begin{array}{l}{\displaystyle I_2=I\times \mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\rho \left(1-f\right)}\end{array}}$$ (35)

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 C₃ species. However, since an increase in cyclic electron flow is commonly observed in C₄ species, it has here been linked to the amount of cyclic electron flow occurring and $\mathrm{\rho }=(1-f_{\mathrm{c}\mathrm{y}\mathrm{c}})/(2-f_{\mathrm{c}\mathrm{y}\mathrm{c}})$ 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 transport-limited quadratic expression. From Equations 3 and 1, one can derive two equations for an electron transport-limited CO₂ assimilation rate

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{j}}=\frac{z}{2}xJ-g_{\mathrm{b}\mathrm{s}}(C_{\mathrm{s}}-C_{\mathrm{m}})-R_{\mathrm{m}}}\end{array}}$$ (36)

and

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{j}}=\frac{(1-{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/C_{\mathrm{s}})\frac{z}{3}(1-x)J}{(1+7{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}/3C_{\mathrm{s}})}-R_{\mathrm{d}}}\end{array}}$$ (37)

Equation 37 can be solved for the bundle sheath CO₂ partial pressure and

$${\displaystyle \begin{array}{l}{\displaystyle C_{\mathrm{s}}=\frac{\left({\mathrm{\gamma }}_{\ast }{O}_{\mathrm{s}}\right)[7/3(A_j+R_d)+z(1-x)J/3]}{z(1-x)J/3-(A_{\mathrm{j}}+R_{\mathrm{d}})}.}\end{array}}$$ (38)

Combining Equations 16, 36, and 38 then yields a quadratic expression of the form

$${\displaystyle \begin{array}{l}{\displaystyle a{A}_{\mathrm{j}}^2+b{A}_{\mathrm{j}}+c=0,}\end{array}}$$ (39)

where

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{j}}=(-b+\sqrt{b^2-4ac})/\left(2a\right)}\end{array}}$$ (40)

And

$${\displaystyle \begin{array}{l}{\displaystyle a=1-\frac{7{\mathrm{\gamma }}_{\ast }\mathrm{\alpha }}{3\ast a_{\mathrm{o}}}}\end{array}}$$ (41)

$${\displaystyle \begin{array}{l}{\displaystyle b=-\{(\frac{z}{2}xJ-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_m)+(\frac{z}{3}[1-x]J-R_{\mathrm{d}})+g_{\mathrm{b}\mathrm{s}}\left(\frac{7{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{m}}}{3}\right)+\frac{\mathrm{\alpha }{\mathrm{\gamma }}_{\ast }}{a_{\mathrm{o}}}(\frac{z}{3}[1-x]J+\frac{7}{3}{R}_{\mathrm{d}})\}}\end{array}}$$ (42)

$${\displaystyle \begin{array}{l}{\displaystyle c=\left\{\right(\frac{z}{2}xJ-R_m+g_{{}_{bs}}{C}_m)(\frac{z}{3}[1-x]J-R_d)-g_{bs}\gamma \ast O_m(\frac{z}{3}[1-x]J+\frac{7R_d}{3}\left)\right\}}\end{array}}$$ (43)

Equation (40) can be approximated by

$${\displaystyle \begin{array}{l}{\displaystyle A_{\mathrm{j}}=\mathrm{m}\mathrm{i}\mathrm{n}\{(\frac{z}{2}xJ-R_{\mathrm{m}}+g_{\mathrm{b}\mathrm{s}}{C}_{\mathrm{m}}),(\frac{z}{3}[1-x]J-R_d)\}}\end{array}}$$ (44)

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 C₄ model and

$${\displaystyle \begin{array}{l}{\displaystyle A=\mathrm{m}\mathrm{i}\mathrm{n}\{A_{\mathrm{c}},A_{\mathrm{j}}\}.}\end{array}}$$ (45)

Peisker and Henderson (1992) 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 C₃ 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, 2001). In the case of C₄ 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 C₄ dicot (Trevanion et al., 1997; Furbank et al., 2001; Pengelly et al., 2012). C₄ transgenic plants with altered RuPB regeneration capacity have not yet been reported on.

An update on the parameterization of the C₄ 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 CO₂ diffusion parameters. Yin et al. (2011) estimated changes in g_bs with N nutrition. Variation in g_bs affects the coordination of the C₃ and C₄ cycle and leakiness (von Caemmerer, 2000). It is, however, important to note that the kinetic constants of Rubisco from C₄ species differ from those of C₃ species and vary amongst the different C₄ decarboxylation types (Badger et al., 1974; Jordan and Ogren, 1981, 1983; Ghannoum et al., 2005; Sharwood et al., 2016).

The C₄ monocot species S. viridis (green foxtail millet), an NADP-ME type which is closely related to agronomically important C₄ 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 CO₂, 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 CO₂, and K_p is converted from measured values of K_m for HCO₃^–. 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 CO₂ 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 C₄ species, as was observed for mesophyll conductance in different C₃ 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 CO₂ 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 CO₂ assimilation rate in a number of C₄ grasses, used this function to fit the saturated rate of CO₂ assimilation measured at high irradiance. The tobacco values chosen here fit within the range values reported for these C₄ grasses (Sonawane et al., 2017)

Calculating the electron transport required to sustain CO₂ assimilation

von Caemmerer and Farquhar (1981) suggested that measurements of the CO₂ assimilation rate can be used to calculate the actual electron transport rate, J_a, needed to support the CO₂ 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:

$${\displaystyle \begin{array}{l}{\displaystyle J_{\mathrm{a}}=(-b+\sqrt{b^2-4ac})/\left(2a\right)}\end{array}}$$ (46)

$${\displaystyle \begin{array}{l}{\displaystyle a=z^2\frac{x}{2}\frac{1-x}{3}}\end{array}}$$ (47)

$${\displaystyle \begin{array}{l}{\displaystyle b=\frac{z(1-x)}{3}[g_{\mathrm{b}\mathrm{s}}(C_{\mathrm{m}}-{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{m}})-R_{\mathrm{m}}-A(1-{\mathrm{\gamma }}_{\ast }\mathrm{\alpha })]-\frac{zx}{2}(A+R_{\mathrm{d}})}\end{array}}$$ (48)

$${\displaystyle \begin{array}{l}{\displaystyle c=(A+R_{\mathrm{d}})[R_{\mathrm{m}}-g_{bs}(C_{\mathrm{m}}+{\mathrm{\gamma }}_{\ast }{O}_{\mathrm{m}}7/3)+A(1-{\mathrm{\gamma }}_{\ast }\mathrm{\alpha }7/3)]}\end{array}}$$ (49)

These equations were introduced by Ubierna et al. (2013) for linear electron flow only (z=0.75) and the assumption of 3 H⁺=/ATP.

An Excel spreadsheet with the C₄ model equations discussed above is available at Dryad Digital Repository, https://doi.org/10.5061/dryad.zcrjdfnc3 (von Caemmerer, 2021).

Model evaluation

Modelled CO₂ response of CO₂ assimilation

In C₃ species, CO₂ 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 C₃ photosynthesis (Farquhar et al., 1980) with the current C₄ model presented here. In the C₃ model, the enzyme-limited rate is dominated by Rubisco and its kinetic parameters at low CO₂, and the electron transport capacity limits at high CO₂ (Fig. 3A). In the C₄ model, it is also possible to distinguish an enzyme-limited CO₂ 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 CO₂ and Rubisco at high CO₂. The electron transport-limited rate can also determine the CO₂ assimilation rate at high CO₂ (Fig. 3B). Thus, it is more difficult to identify biochemical limitations to C₄ photosynthesis.

Fig. 3.

A comparison of modelled rates of the CO₂ assimilation rate as functions of partial pressures of CO₂ for C₃ (A) and C₄ photosynthesis (B). (A) Modelled rate of CO₂ assimilation as a function of chloroplast CO₂ partial pressure for the C₃ photosynthetic pathway at 25 °C. The Rubisco-limited (RuBP-saturated) rate of CO₂ assimilation has a dashed line extension at high CO₂. The electron transport-limited (RuBP regeneration) rate of CO₂ assimilation has a dotted line extension at low CO₂. The solid curve represents the minimum rate that is the actual rate of CO₂ assimilation. A possible triose phosphate limitation at high CO₂ is not shown. (after von Caemmerer, 2000). (B) Modelled rate of CO₂ assimilation as a function of mesophyll cytosolic CO₂ partial pressure for the C₄ photosynthetic pathway at 25 °C and an irradiance of 2000 µmol m^–2 s^–1. The enzyme-limited CO₂ assimilation rate (PEPC limitation at low CO₂ and Rubisco limitation at high CO₂ shows the Rubisco limited rate as a dashed line extension. The electron transport-limited (RuBP and PEP regeneration) rate of CO₂ assimilation has a dotted line extension at low CO₂. The solid curve represents the minimum rate that is the actual rate of CO₂ assimilation. Parameters used are given in Table 1.

Usually good correlations are found between in vitro Rubisco activity and the CO₂-saturated rate of CO₂ assimilation rate at high CO₂ (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 CO₂ 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 CO₂ assimilation at high CO₂. Transgenic plants with reduced Rubisco content showed a clear decline in CO₂ assimilation rate at high CO₂ (von Caemmerer et al., 1997; Pengelly et al., 2012). Recent photosynthetic engineering that increased Rubisco content in maize leaves resulted in an increase in CO₂-saturated CO₂ 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 CO₂ assimilation at high CO₂ 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 C₄ cycle has been curtailed by molecular manipulation, it is clear that this also limits CO₂ assimilation at high CO₂ (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 b₆f complex had increased CO₂ assimilation rates at ambient and high CO₂, confirming that electron transport capacity can limit the CO₂ assimilation rate. The fact that electron transport rate limits the CO₂ assimilation rate at high CO₂ means that a reduction in irradiance is also predicted to primarily affect the CO₂-saturated rate of CO₂ assimilation rather than the initial slope of the CO₂ 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 CO₂ response curve: the mesophyll conductance to CO₂ diffusion from the intercellular airspace to the mesophyll cytosol g_m; the rate of CO₂ hydration by CA; and the rate of PEP carboxylation. It is thought that most C₄ 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 CO₂ response is observed (von Caemmerer et al., 2004; Studer et al., 2014; Osborn et al., 2017).

The initial C₄ photosynthesis models did not consider a diffusion limitation between the intercellular airspace and the mesophyll cytosol. In C₄ 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 C₃ species such as combined measurements of gas exchange and chlorophyll fluorescence, or measurements of ¹³C isotope discrimination cannot be used in C₄ species; however, a new technique has been developed to measure mesophjyll conductance in C₄ species using C¹⁸O¹⁶O 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 CO₂ partial pressure from intercellular airspace, C_i to that of the mesophyll, C_m is related in the following equation

$${\displaystyle \begin{array}{l}{\displaystyle A=g_{\mathrm{m}}(C_{\mathrm{i}}-C_{\mathrm{m}}).}\end{array}}$$ (50)

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 CO₂ response curve, one can use Equation 26 and, ignoring the term g_bsC_m and combining it with Equation 50, a quadratic similar to the one given for C₃ leaves is obtained (von Caemmerer and Evans, 1991; von Caemmerer, 2000).

$${\displaystyle \begin{array}{l}{\displaystyle A^2-A(g_{\mathrm{m}}[C_i+K_p]+V_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}-R_{\mathrm{m}})+g_{\mathrm{m}}(V_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}{C}_{\mathrm{i}}-R_{\mathrm{m}}[C_i+K_p])=0}\end{array}}$$ (51)

The first derivative with respect to C_i at C_i=0 is given by

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{d}A}{\mathrm{d}C\mathrm{i}}=\frac{g_{\mathrm{m}}{V}_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}}{g_{\mathrm{m}}{K}_{\mathrm{p}}+V_{\mathrm{p}\mathrm{m}\mathrm{a}\mathrm{x}}}}\end{array}}$$ (52)

Pfeffer and Peisker (1988 and 1998) 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 C¹⁸O¹⁶O isotope discrimination; however, the uncertainties surrounding estimates of K_p discussed above need to be considered.

Fig. 4.

The effect of mesophyll conductance, g_m, on the initial slope of the CO₂ response curve. (A) Modelled rate of CO₂ assimilation as a function of intercellular CO₂ partial pressure for the C₄ 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.

Strong correlations between leaf nitrogen, CO₂ 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 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 C₄ biochemistry with gas exchange.

Modelled light response of CO₂assimilationIt is well recognized that the light response of C₄ photosynthesis frequently does not saturate (Cousins et al., 2006; Leakey et al., 2006). Figure 5 shows typical modelled light response curves of the CO₂ assimilation rate at several mesophyll CO₂ partial pressures. In the current parameterization, the CO₂ 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 C₃ photosynthetic model remains empirical and the partition partitioning of electron transport between the C₄ and C₃ 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 C₃ and C₄ cycle (Peisker, 1988; von Caemmerer and Furbank, 1999; von Caemmerer, 2000). It is noteworthy that the fraction of electron transport allocated to the C₄ cycle, x, equals 0.4 over a wide range of irradiances but drops at very low irradiance. Under low light, the bundle sheath CO₂ partial pressures are close to the mesophyll CO₂ partial pressure, and electron transport is required for recycling of photorespiratory CO₂. 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 CO₂ decreases (Jordan and Ogren, 1984; Sharwood et al., 2016).

Fig. 5.

The effect of irradiance on the modelled rate of CO₂ assimilation. (A) Light response of the CO₂ assimilation rate at mesophyll cytosolic CO₂ partial pressures, C_m, indicated in the figure. The C₄ photosynthesis model predicts electron transport limitations at all irradiances at C_m values >150 µbar. At lower C_m, CO₂ assimilation rates are enzyme limited at high irradiance. The model was parameterized at 25 °C with values given in Table 1. (B) CO₂ assimilation rate as a function of intercellular CO₂ at the irradiances indicated. The model was parameterized at 25 °C with values given in Table 1.

In C₃ 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 C₄ photosynthesis, this quantitative link between cytochrome f content and electron transport capacity also needs to be investigated.

Modelled temperature response of CO₂ assimilation rate

C₄ plants have higher CO₂ assimilation rates at high temperatures and higher photosynthetic temperature optima than their C₃ counterparts largely because of the elimination of photorespiratory CO₂ losses (Berry and Björkman, 1980; Long, 1999). The temperature response of electron transport is not well characterized in C₄ species. With the parameterization used here, the CO₂ 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 C₄ species at ambient CO₂ (Fig. 6). There is some evidence that this is not unreasonable. Figure 7 shows a comparison of a temperature response of the CO₂ assimilation rate of wild-type and transgenic F. bidentis with reduced Rubisco content (Kubien et al., 2003). In Fig. 7B, the CO₂ 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 CO₂ assimilation rate such as electron transport capacity. Temperature optima of the CO₂ 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).

Fig. 6.

Modelled CO₂ 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 CO₂ assimilation rate was modelled at an irradiance of 2000 µmol m^–2 s^–1 and an intercellular CO₂, C_i, of 150 µbar. Other parameters are as given in Table 1.

Fig. 7.

(A) Temperature responses of the CO₂ assimilation rate in Flaveria bidentis wild-type and anti-Rubisco plants. Photosynthesis was measured at different leaf temperatures and ambient CO₂ of 370 µbar and 200 mbar O₂, and an irradiance of 1500 µmol m^–2 s^–1. Each point represents the mean (±SE) of measurements on five different leaves. (B) Temperature dependence of the in vitro and in vivo k_cat for Rubisco in wild-type and anti-Rubisco F. bidentis. 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)

In Fig. 8A, the CO₂ response curves have been modelled for different leaf temperatures. Figure 8A shows that at low temperature the CO₂ response is enzyme limited at all C_i; as temperature is increased, the transition from enzyme-limited CO₂ 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).

Fig. 8.

(A) Modelled rate of CO₂ assimilation as a function of intercellular CO₂ partial pressure, C_i, for the C₄ 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 CO₂ 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.

A note on leakiness

The bundle sheath resistance or, its inverse, the bundle sheath conductance to CO₂ diffusion are key parameters that together with relative capacities for the C₄ cycle and Rubisco and electron transport capacity determine the effectiveness of the CO₂ concentration mechanism. This is often quantified by a term called leakiness (ϕ), which is defined as the ratio of the rates of CO₂ leakage out of the bundle sheath over the rate of CO₂ supply to the bundle sheath (Equation 5). Carbon isotope discrimination can be used to determine leakiness (Farquhar, 1983). Combined measurements of gas exchange and carbon isotope discrimination have been used to assess leakiness under different environmental conditions (Henderson et al., 1992; 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 C₄ 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 down-regulated. However, when CO₂ 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 C₄ cycle using the equation below.

$${\displaystyle \begin{array}{l}{\displaystyle V_{\mathrm{p}}=\frac{A+R_{\mathrm{m}}}{1-\varphi }.}\end{array}}$$ (53)

The leak rate can then be calculated from Equation 5. With the assumption of a bundle sheath conductance, bundle sheath CO₂ partial pressure can then be estimated from Equation 4 (Pengelly et al., 2012).

Modelling different decarboxylation types

The biochemistry of the C₄ 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 C₄ 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 C₄ 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 C₄ 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 b₆f complex which has shown to be a good correlator of C₃ photosynthetic electron transport.

Data availability

Data outlining C₄ model calculations are available at the Dryad Digital Repository https://doi.org/10.5061/dryad.zcrjdfnc3; von Caemmerer, (2021).