The response of mesophyll conductance to short-term variation in CO₂ in the C₄ plants Setaria viridis and Zea mays

In two C₄ species, mesophyll conductance increases with short-term exposure to decreasing pCO₂ and limits photosynthetic capacity below ambient levels, whilst carbonic anhydrase imposes a further limitation only at very low pCO₂.

Abstract

Mesophyll conductance (g_m) limits rates of C₃ photosynthesis but little is known about its role in C₄ photosynthesis. If g_m were to limit C₄ photosynthesis, it would likely be at low CO₂ concentrations (pCO₂). However, data on C₄-g_m across ranges of pCO₂ are scarce. We describe the response of C₄-g_m to short-term variation in pCO₂, at three temperatures in Setaria viridis, and at 25 °C in Zea mays. Additionally, we quantified the effect of finite g_m calculations of leakiness (ϕ) and the potential limitations to photosynthesis imposed by stomata, mesophyll, and carbonic anhydrase (CA) across pCO₂. In both species, g_m increased with decreasing pCO₂. Including a finite g_m resulted in either no change or increased ϕ compared with values calculated with infinite g_m depending on whether the observed ¹³C discrimination was high (Setaria) or low (Zea). Post-transitional regulation of the maximal PEP carboxylation rate and PEP regeneration limitation could influence estimates of g_m and ϕ. At pCO₂ below ambient, the photosynthetic rate was limited by CO₂ availability. In this case, the limitation imposed by the mesophyll was similar or slightly lower than stomata limitation. At very low pCO₂, CA further constrained photosynthesis. High g_m could increase CO₂ assimilation at low pCO₂ and improve photosynthetic efficiency under situations when CO₂ is limited, such as drought.

Introduction

In C₄ plants photorespiration is reduced by concentrating CO₂ around Rubisco (ribulose 1,5-bisphosphate carboxylase/oxygenase) (Edwards and Walker, 1983; Hatch, 1987; Sage, 2004). In Kranz-type C₄ plants this is achieved with a compartmentalized two-carboxylation process: (1) in the cytosol of mesophyll cells, bicarbonate (HCO₃^–) and phosphoenolpyruvate are fixed into four-carbon acids by phosphoenolpyruvate carboxylase (PEPC) (Hatch et al., 1967); and (2) in chloroplasts of the bundle-sheath cells the concentrated CO₂ released from the decarboxylation of these acids is fixed by Rubisco.

Mesophyll conductance (g_m) describes the movement of CO₂ from stomata across the intercellular spaces to the sites of first carboxylation, which are the chloroplast stroma or mesophyll cytosol in C₃ and C₄ species, respectively (Evans and von Caemmerer, 1996). There is extensive research describing g_m in C₃ species; however, C₄-g_m is poorly understood because it is difficult to estimate. Traditionally g_m was assumed to be larger in C₄ compared to C₃ species, but most recent studies suggest that values for C₄-g_m correspond to higher-end C₃-g_m reports, and that C₄-g_m reacts similarly to C₃-g_m with regards to variation in factors such as leaf age and temperature (Barbour et al., 2016; Osborn et al., 2017; Ubierna et al., 2017). If C₄-g_m is lower than previously thought, that could affect derivations of other key parameters such as leakiness (ϕ, the proportion of C fixed by PEPC that subsequently leaks out the bundle-sheath cells). Leakiness cannot be directly measured and is commonly estimated from observations and models of ¹³C discrimination (Δ¹³C) (Farquhar, 1983; Farquhar and Cernusak, 2012). Historically, g_m is generally assumed to be infinite when solving for ϕ from Δ¹³C; however, this simplification and estimates of ϕ would be compromised if g_m is finite and low.

Mesophyll conductance has long been recognized as a significant limitation for C₃ photosynthesis (Evans, 1983; Evans et al., 1986; Evans and Terashima, 1988), limiting photosynthesis as much as stomatal conductance (Warren, 2008). It is unclear if g_m limits C₄ photosynthesis as the reduction of photorespiration achieved by the CO₂-concentrating mechanism saturates C₄ photosynthesis at ambient pCO₂. If g_m were to limit C₄ photosynthesis, it would likely only be at very low pCO₂. However, not much is known about the variation of C₄-g_m with pCO₂. In the C₄ grass Setaria viridis, g_m derived with the ¹⁸O discrimination (Δ¹⁸O) method increased as pCO₂ decreased, although the variation was not significant (Osborn et al., 2017). Some reports have shown that in C₃ species g_m increases with short-term exposure to decreasing pCO₂ (Bongi and Loreto, 1989; Loreto et al., 1992; Flexas et al., 2007, 2008; Hassiotou et al., 2009; Bunce, 2010; Douthe et al., 2011; Tazoe et al., 2011). However, others have suggested that C₃-g_m is insensitive to changes in pCO₂ (Loreto et al., 1992; Tazoe et al., 2009). It has been hypothesized that the observed C₃-g_m response to pCO₂ might result from a significant chloroplast resistance (Tholen and Zhu, 2011; Tholen et al., 2012) or artifacts in the calculations (Gu and Sun, 2014).

In C₄ plants, g_m has been estimated with the Δ¹⁸O method (Gillon and Yakir, 2000a, 2000b; Barbour et al., 2016; Osborn et al., 2017; Ubierna et al., 2017) and the in vitro maximal PEP carboxylation rate (V_pmax) method (Ubierna et al., 2017). The latter method solves for the pCO₂ in the mesophyll cells (C_m) needed to simultaneously match modeled and measured rates of CO₂ assimilation and Δ¹³C when the models are parameterized with in vitro V_pmax, as determined in a crude leaf extract. Values derived for g_m with the Δ¹⁸O and in vitro V_pmax methods were similar in two C₄ species measured over a range of temperatures (Ubierna et al., 2017). The in vitro V_pmax method also allows the implementation of two modeling alternatives: carbonic anhydrase (CA)-saturated and CA-limited. They differ in the calculation of PEP carboxylation rate as a function of CO₂ or HCO₃^– for the CA-saturated and -limited scenarios, respectively. Ubierna et al. (2017) found no difference between CA-limited and CA-saturated estimates of g_m at ambient pCO₂, but CA limitation is expected at low pCO₂.

In this study, we calculated g_m using the in vitro V_pmax method across a range of pCO₂ in two C₄ grasses, one economically important (Zea mays) and the other the adopted model system for studying C₄ photosynthesis (S. viridis). Measurements were performed at three temperatures (10, 25, and 40 °C) in Setaria and at 25 °C in Zea. Our objectives were to: (1) describe the response of C₄-g_m to short-term variation in pCO₂; (2) evaluate the impact of disequilibrium between CO₂ and HCO₃^– at a range of pCO₂ and temperatures; (3) investigate if g_m represents a limitation to C₄ photosynthesis across pCO₂; and (4) assess the impact of finite g_m on ϕ calculations.

Materials and methods

Plant material

Seeds of Z. mays (var. Trucker’s Favorite, Victory Seed Company, Oregon, USA) were grown in a greenhouse supplemented with artificial lighting at the School of Biological Sciences at Washington State University, Pullman, WA (USA) during August to October 2011. Seeds of S. viridis (A-010) were grown in a controlled environment growth chamber (Enconair Ecological GC-16) in 2013. Plants used for measurements were 4 and 6 weeks old for Zea and Setaria, respectively. Zea was fertilized with 17-3-6 NPK and weekly additions of 4 g l^–1 solution of 10% Fe-DPTA (Sprint 330, Becker Underwood, IA, USA). Setaria was treated weekly with Peters 20-20-20 (J. R. Peters, Inc., Allentown, PA, USA). For all plants, the photon flux density was ≥500 μmol m^–2 s^–1, the day length was 14 h, and the temperature was 25–28/20–25 °C for day/night.

Coupled gas exchange and isoflux measurements

The system used for measurements has been described in detail in Ubierna et al. (2013, 2017). Briefly, a LI-6400XT open gas exchange system assembled with a 6400-22L conifer chamber fitted with a LI-6400–18 RGB light source (Li-Cor, Lincoln, NE, USA) was coupled with a tunable-diode laser absorption spectroscope (TDLAS, TGA 200A, Campbell Scientific, Inc. Logan, UT, USA). The entire gas exchange system was placed in a growing cabinet (Percival Scientific, Perry, IA), where the temperature was varied to match leaf temperature (T_L) settings. The TDLAS data were calibrated with the concentration series method (Tazoe et al., 2011; Ubierna et al., 2013) using two calibration gases, one measured at different [CO₂] that spanned the gas exchange reference and sample lines. Each measurement cycle included five to seven TDLAS sequences of zero air, calibration gases, reference, and sample lines measured for 40 s each. Data from the last three sequences were averaged and used for calculations.

Young fully-expanded leaves of Setaria and Zea were acclimated for ~1 h with chamber conditions of C_a (ambient CO₂ supply to the chamber) ≈ 35 Pa, 21% O₂, and photosynthetically active radiation (PAR) =2000 μmol m^–2 s^–1. Then, C_a was varied in steps, and gas and ¹³C isotopic exchange were measured simultaneously. In Setaria (n=4) C_a was set at 5, 7, 10, 12, 14, 19, 28, 38, 56, and 93 Pa, and measurements were performed at T_L=10, 25 and 40 °C. In Zea (n=3), C_a was set at 9, 14, 19, 35, 56, 84, and 112 Pa, and T_L=25 °C. In both species the measurements were performed in the sequence ambient – low – ambient – high pCO₂.

Enzyme-limited C₄ photosynthesis model for CA-limited or CA-saturated conditions

The enzyme-limited C₄ photosynthesis rate is (von Caemmerer, 2000):

$$A=\frac{-b-\sqrt{b^2-4ac}}{2a},$$ Eqn 1

where:

$$a=1-\frac{\alpha }{u_{\text{oc}}}\frac{K_c}{K_o},$$ Eqn 2

$$b=-\left[V_p-R_m+g_{bs}{C}_m+V_{cmax}-R_d+g_{bs}{K}_c\left(1+\frac{O_{\text{m}}}{K_o}\right)+\frac{\alpha }{u_{\text{oc}}}\left({\text{γ}}^{*}{V}_{cmax}+R_d\frac{K_c}{K_o}\right)\right],$$ Eqn 3

$$c=\left(V_{cmax}-R_d\right)\left(V_p-R_m+g_{bs}{C}_m\right)-V_{cmax}{g}_{bs}{\text{γ}}^{*}{O}_m-R_d{g}_{bs}{K}_c\left(1+\frac{O_{\text{m}}}{K_o}\right),$$ Eqn 4

where α (= 0) is the fraction of PSII activity in the bundle-sheath cells (von Caemmerer, 2000); u_oc is the ratio of O₂ and CO₂ diffusivities and solubilities, 0.047 at 25 °C but variable with temperature (Yin et al., 2016); g_bs is the bundle-sheath conductance, 0.0164 μmol m^–2 s^–1 Pa^-1 (Ubierna et al., 2013) or variable; O_m is the O₂ partial pressure in the mesophyll (19.5 kPa, which corresponds to 21%); R_d is the non-photorespiratory CO₂ released in the dark, assumed to equal measured rates of dark respiration after 30 min of dark adaptation, which at 25 °C were 1.89 and 1.06 μmol m^–2 s^–1 in Zea and Setaria, respectively, but were also measured at each temperature; R_m is the mesophyll mitochondrial respiration rate, R_m=0.5R_d (von Caemmerer, 2000); γ* is half of the reciprocal of Rubisco specificity, and equals 0.5/S_C/O (von Caemmerer, 2000), where S_C/O is the Rubisco CO₂/O₂ specificity. K_C and K_O are the Michaelis–Menten constants of Rubisco for CO₂ and O₂, respectively. S_C/O, K_C, and K_O were determined in vitro at 25 °C in Zea (S_C/O=2147 Pa Pa^–1, K_C=96 Pa, K_O=49 683 Pa; R.A. Boyd, Washington State University, pers. comm.) and Setaria (S_C/O=1310 Pa Pa^–1, K_C=121 Pa, K_O=29 200 Pa; Boyd et al., 2015). Their values at different temperatures were obtained using the temperature functions of Boyd et al. (2015). For V_cmax (maximal Rubisco carboxylation rate) we used in vivo values calculated as described in Ubierna et al. (2017) or as specified otherwise. The calculation of C_m (pCO₂ in the mesophyll cells) will be discussed subsequently.

CA-saturated and CA-limited models differ as follows.

(1) The calculation of PEP carboxylation rate (V_p):

$$V_p=\left\{\begin{array}{c}CA saturated\to \frac{C_m{V}_{pmax}}{C_m+K_p} \\ CA limited\to \frac{\left[HC{O}_3^{-}\right]V_{pmax}}{\left[HC{O}_3^{-}\right]+K_p} \end{array}\right\},$$ Eqn 5

where the maximal PEP carboxylation rate (V_pmax) was measured in vitro at 25 °C in Zea (184 μmol m^–2 s^–1, R. A. Boyd, pers. comm.) and in Setaria (450 μmol m^–2 s^–1, Boyd et al. 2015) and varied with temperature as described in Boyd et al. (2015). For all species, the Michaelis–Menten constant of PEPC for CO₂ (K_P) was modeled with the temperature response and value at 25 °C (60.5 μM HCO₃^–) from Boyd et al. (2015). The [HCO₃^–] was calculated as previously discussed (Jenkins et al., 1989; Hatch and Burnell, 1990; Boyd et al., 2015): for details see Ubierna et al. (2017).

If the rate of PEP regeneration is limiting, then V_p is (von Caemmerer, 2000):

$$V_p=\text{min}\left(V_{\text{p}} calculated with Eqn 5, V_{pr}\right),$$ Eqn 6

where V_pr is the PEP regeneration rate (Peisker, 1986; Peisker and Henderson, 1992). We arbitrarily set V_pr to 64 and 59 μmol m^–2 s^–1 in Setaria and Zea, respectively, which corresponded to twice the maximum measured net assimilation rate, A.

(2) The calculation of the ratio V_p/V_h, where V_h is hydration rate:

$$\frac{V_p}{V_h}=\left\{\begin{array}{c}CA saturated\to 0 \\ CA limited\to \frac{V_p}{C_{\text{m}}{K}_{CA}} \end{array}\right\},$$ Eqn 7

where K_CA is the rate constant of CA for CO₂, that at 25 °C was 65.5 and 124 μmol m^–2 s^–1 Pa^–1 in Zea and Setaria, respectively (R.A. Boyd pers. comm., Boyd et al., 2015), varying with temperature as described in Boyd et al. (2015).

Measurements and models of discrimination

The observed photosynthetic discrimination against ¹³C (${\Delta }_{obs}^{13}$) is calculated as (Evans et al., 1986):

$${\Delta }_{obs}^{13}=\frac{\frac{C_{in}}{C_{in}-C_{out}}\left({\delta }_{out}-{\delta }_{in}\right)}{1+{\delta }_{out}-\frac{C_{in}}{C_{in}-C_{out}}\left({\delta }_{out}-{\delta }_{in}\right)},$$ Eqn 8

where C and δ are the ¹²CO₂ mol fraction and the δ¹³C of the CO₂, respectively, in dry air in and out the chamber.

The theoretical model for Δ¹³C is (Farquhar and Cernusak, 2012):

$${\Delta }_{theo}^{13}=\frac{1}{1-t}\left[a_b\frac{C_a-C_L}{C_a}+a_s\frac{C_L-C_i}{C_a}\right]+\frac{1+t}{1-t}\left[a_m\frac{C_i-C_m}{C_a}+\frac{b_4+\varphi \left(\frac{b_3{C}_{bs}}{C_{bs}-C_m} - s\right)}{1+\frac{\varphi C_m}{C_{bs}-C_m}}\frac{C_m}{C_a}\right].$$ Eqn 9

Values and calculations of the variables included in this equation have been discussed before (i.e. Ubierna et al., 2017) and can also be found in Supplementary Methods S1 at JXB online.

Calculation of mesophyll conductance (g_m)

Following Fick’s law of diffusion:

$$g_{\text{m}}=\frac{A}{C_{\text{i}}-C_{\text{m}}},$$ Eqn 10

where the C_m is calculated for two case scenarios, CA-saturated and CA–limited, resulting in CA-sat g_m and CA-lim g_m values. In both cases, C_m is derived with the in vitro V_pmax method as the C_m that needs to be combined with in vitro V_pmax to match measurements and predictions of A and Δ¹³C (Eqns 1, 9); details on these calculations have been provided in Ubierna et al. (2017). The CA-sat and CA-lim options are introduced through the calculation of V_p and V_p/V_h (Eqns 5–7).

Limitations to photosynthesis

To calculate the limitation on CO₂ assimilation by either finite stomatal conductance (L_s), by mesophyll conductance (L_m), or by carbonic anhydrase (L_CA), we adapted to C₄ photosynthesis an approach previously used for C₃ photosynthesis. This compares A when all conductances are finite with the A estimated assuming that the conductance related with the limitation of interest is infinite (Farquhar and Sharkey, 1982; Warren et al., 2003). In all cases A was calculated with Eqn 1 and assuming:

• (a) A_all (expected A with all limitations, ≈ measured photosynthetic rate): finite g_s and g_m, CA-lim model.

• (b) A_s (expected A if there were no stomatal limitations): infinite g_s (C_i=C_a), finite g_m, CA-lim model.

• (c) A_m (expected A if there were no mesophyll limitations): infinite g_m (C_m=C_i), g_s as measured, CA-lim model.

• (d) A_CA (expected A if there were no CA limitations): g_s as measured, g_m finite, CA-sat model.

Then L_s, L_m, and L_CA were calculated as:

$$L_{\text{s}}=100\times \frac{A_{\text{s}}-A_{\text{all}}}{A_{\text{s}}},$$ Eqn 11

$$L_{\text{m}}=100\times \frac{A_{\text{m}}-A_{\text{all}}}{A_{\text{m}}},$$ Eqn 12

$$L_{\text{CA}}=100\times \frac{A_{\text{CA}}-A_{\text{all}}}{A_{\text{CA}}}.$$ Eqn 13

Calculation of leakiness (ϕ)

The C₄ photosynthesis model (von Caemmerer, 2000) calculates ϕ as:

$$\varphi =\frac{g_{\text{bs}}\left(C_{\text{bs}}-C_{\text{m}}\right)}{V_{\text{p}}},$$ Eqn 14

where C_bs, the pCO₂ in the bundle-sheath cells, is (von Caemmerer, 2000):

$$C_{\text{bs}}=\frac{{\text{γ}}^{*}{O}_{\text{s}}+K_{\text{C}}\left(1+\frac{O_{\text{s}}}{K_{\text{o}}}\right)\left(\frac{A+R_{\text{d}}}{V_{\text{cmax}}}\right)}{1-\frac{A+R_{\text{d}}}{V_{\text{cmax}}}}=C_{\text{m}}+\frac{V_{\text{p}}-A-R_{\text{m}}}{g_{\text{bs}}},$$ Eqn 15

where O_s is the O₂ partial pressure in the bundle-sheath cells.

From Δ¹³C (Eqn 9), ϕ is solved as:

$${\varphi }_1=\frac{C_{\text{bs}}-C_{\text{m}}}{C_{\text{m}}}\times \frac{{\Delta }_{obs}^{13}\left(1-t\right)C_{\text{a}} - \overline{a}\left(C_{\text{a}}-C_i\right) - a_{\text{m}}\left(C_{\text{i}}-C_{\text{m}}\right)\left(1+t\right) - b_4\left(1+t\right)C_{\text{m}}}{\left(1+t\right)\left[b_3{C}_{\text{bs}} - s\left(C_{\text{bs}}-C_{\text{m}}\right) + a_{\text{m}}\left(C_{\text{i}}-C_{\text{m}}\right)\right] + \overline{a}\left(C_{\text{a}}-C_{\text{i}}\right) - C_{\text{a}}{\Delta }_{obs}^{13}\left(1-t\right)}$$ Eqn 16

where b₃ (combined effects of Rubisco fractionation, and fractionations associated with respiration and photorespiration) and b₄ (combined fractionation during PEP carboxylation, hydration, and respiration) are calculated as (Farquhar, 1983; Cousins et al., 2006):

$$b_3\cong b_3^{´}-\frac{e{R}_{\text{d}}}{V_{\text{c}}}-\frac{0.5f{V}_{\text{o}}}{V_{\text{c}}},$$ Eqn 17

$$b_4=b_4^{´}\left(1-\frac{V_{\text{p}}}{V_{\text{h}}}\right)+(e_{\text{s}}+h)\frac{V_{\text{p}}}{V_{\text{h}}}-\frac{e{R}_{\text{m}}}{V_{\text{p}}}.$$ Eqn 18

A description of other variables included in Eqns 16–19 can be found in Supplementary Methods S1.

To evaluate the effect of g_m on calculations of ϕ we implemented four model scenarios, which differed in values for g_m, calculation of V_p, or constrains imposed. Model 1 used in vitro V_pmax and g_m finite and equal to the values for CA-lim g_m presented in the Results; Model 2 used in vivo V_pmax and g_m infinite; Model 3 was the same as Model 1 but the solution was only constrained by A and not Δ¹³C; and Model 4 was the same as Model 1 but with V_p calculated with Eqn 6, which introduces a PEP regeneration limitation. The in vitro V_pmax method calculates g_m by solving the system of two equations formed by the models of A and Δ¹³C. Therefore, once a solution is found, ϕ values calculated with either Eqn 14 or 16 are identical. This is the case for Models 1, 2, and 4; however, in Model 3, which is constrained only by A, ϕ was obtained only with Eqn 14. All four modeling scenarios described above used the CA-limited calculations (Eqns 5–7).

At ambient pCO₂, ϕ was also calculated with a simplified equation derived from Δ¹³C assuming that C_bs is much larger than C_m and that hydration and assimilation fluxes are large (V_p/V_h≈0, and V_o≈0, where V_o is oxygenation rate):

$${\varphi }_2= \frac{\frac{{\Delta }_{\text{obs}}\left(1-t\right)C_{\text{a}} - \overline{a}\left(C_{\text{a}}-C_{\text{i}}\right)}{1+t} - a_{\text{m}}{C}_{\text{i}} + C_{\text{m}}\left(a_{\text{m}}-\overline{b_4}\right)}{C_{\text{m}}\left(\overline{b_3}-s\right)},$$ Eqn 19

where $\overline{b_3}$ and $\overline{b_4}$ are (von Caemmerer et al., 2014):

$$\overline{b_3}=b_3^{´ }-\frac{e{R}_{\text{d}}}{A+R_{\text{d}}}+\frac{e0.5R_{\text{d}}}{A+0.5R_{\text{d}}},$$ Eqn 20

$$\overline{b_4}=b_4^{´ }-\frac{e0.5R_{\text{d}}}{A+0.5R_{\text{d}}}.$$ Eqn 21

Statistical analyses

Statistical analyses were performed using SAS v9.4 (SAS Institute Inc., Cary, NC, USA). Differences between CA-lim g_m and CA-sat g_m were investigated using t-tests (H_o: CA-lim g_m/CA-sat g_m=1). The effect of CO₂ supply on CA-lim g_m was analysed using repeated measurements ANOVA. Data were log-transformed to meet normality criteria. In Setaria we used PROC MIXED with: plant as the repeated measurement; pCO₂, temperature, and their interaction as fixed effects; a covariance structure of compound symmetry; and we applied Kenward–Roger’s approximation to correct the denominator degrees of freedom (Arnau et al., 2009). In Zea, we used PROC ANOVA with the statement REPEAT.

Results

A-C_i curves and observed ¹³C photosynthetic discrimination

Under all leaf measurement temperatures (T_L), the rate of net photosynthesis (A) in Setaria increased with C_i as the pCO₂ supplied increased from ~5 Pa to ambient air values (~35 Pa) and then leveled off (Fig. 1A). At all pCO₂, increasing T_L resulted in larger A (Fig. 1A). In Zea, A also increased with increasing C_i and reached a maximum at ambient air pCO₂ before decreasing at higher pCO₂ (Fig. 1B).

Fig. 1.

Responses of (A, B) photosynthetic rate (A) and (C, D) observed ¹³C photosynthetic discrimination (${\Delta }_{obs}^{13}$) to variation in the CO₂ partial pressure inside the leaf (C_i) in Setaria viridis (circles) and Zea mays (squares). In Setaria, three leaf temperatures (T_L) were measured: 40, 25, and 10 °C, as indicated in the key. Measurements in Zea were at T_L=25 °C. Values are means ±SE; n=4 in Setaria and n=3 in Zea.

At ambient air pCO₂ and 25 °C, ${\Delta }_{obs}^{13}$ was larger in Setaria (4.5 ± 0.1‰) than in Zea (3.1 ± 0.2‰) (Fig. 1C, D). In Zea, the ${\Delta }_{obs}^{13}$ was low at ambient air pCO₂ and increased at lower or higher C_i (Fig. 1D). However, in Setaria, ${\Delta }_{obs}^{13}$ remained constant with C_i when T_L=25 °C, but decreased as C_i increased both at 40 and 10 °C (Fig. 1C).

Mesophyll conductance calculated assuming CA-saturated or CA-limited conditions

For both species and at all temperatures, the ratio CA-lim g_m/CA-sat g_m ≈ 1 when pCO₂ was above ambient (Fig. 2). As pCO₂ decreased, CA-lim g_m became larger than CA-sat g_m; the differences increased with temperature and were larger in Zea than in Setaria. In Setaria, CA-lim g_m and CA-sat g_m were significantly different (P<0.05) at all pCO₂ at 40 °C, at all pCO₂ except at ambient and the measurement just above ambient at 25 °C, and at the largest pCO₂ at 10 °C (Fig. 2A). In Zea, CA-lim g_m and CA-sat g_m were significantly different (P<0.05) at all pCO₂≤ambient air (Fig. 2B).

Fig. 2.

The ratio of carbonic anhydrase-limited mesophyll conductance (CA-lim g_m) to CA-saturated g_m (CA-sat g_m) at different pCO₂ inside the leaf (C_i) in (A) Setaria viridis (circles) and (B) Zea mays (squares). Setaria was measured at three leaf temperatures (T_L): 40, 25, and 10 °C, as indicated in the key. Zea was measured at T_L=25 °C. Values are means ±SE; n=4 in Setaria and n=3 in Zea. An asterisk inside a symbol indicates CA-lim g_m/CA-sat g_m ≠ 1 with P<0.05. The arrows indicate the values at ambient pCO₂ and at 40 °C (black arrow), 25 °C (grey arrow), and 10 °C (dashed arrow).

In Setaria, the under-estimation of g_m by ignoring the CA limitation was very small (maximum of 5%, CA-lim g_m/CA-sat g_m<1.1; Fig. 2A). However, in Zea, the CA-lim g_m calculated at the lowest pCO₂ was 20 ± 8% larger than CA-sat g_m at 25 °C. Because CA limitation was relevant at low pCO₂, for subsequent analyses we use the CA-lim g_m values for all species, temperatures, and pCO₂.

CO₂ response of mesophyll conductance

The CA-lim g_m significantly increased as pCO₂ decreased in Setaria at all temperatures (P<0.0001) and in Zea at 25 °C (P<0.0004) (Fig. 3). At ambient pCO₂ and 25 °C, CA-lim g_m values (mean±SE) were 2.00 ± 0.10 μmol m^–2 s^–1 Pa^–1 in Setaria, and 2.43 ± 0.13 μmol m^–2 s^–1 Pa^–1 in Zea. At the lowest pCO₂ measured (~5–9 Pa) and 25 °C, the CA-lim g_m increased to 6.30 ± 0.32 and 16.20 ± 5.74 μmol m^–2 s^–1 Pa^–1 in Setaria and Zea, respectively. Values for C_m across C_i can be found in Supplementary Fig. S1.

Fig. 3.

The response of carbonic anhydrase-limited mesophyll conductance (CA-lim g_m) to changes in pCO₂ inside the leaf (C_i) in (A, C) Setaria viridis (circles) and (B) Zea mays (white squares). Setaria was measured at three leaf temperatures, as indicated at the top of the figure. Zea was measured at T_L=25 °C. For comparison, the available literature reports for Δ¹⁸O-g_m for different species and temperatures are included, as indicated in the keys: Ubierna et al. (2017)Setaria measured at T_L=40 °C, T_L=25 °C, and T_L=10 °C; Osborn et al. (2017)Setaria measured at T_L=25 °C; Barbour et al. (2016)Setaria measured with block temperature of 30 °C; Ubierna et al. (2017)Zea measured at T_L=25 °C; Barbour et al. (2016)Zea measured with block temperature of 30 °C. For all species and temperatures CA-lim g_m significantly varied with pCO₂. (D–F) The CO₂ response of normalized g_m, calculated by dividing individual values by the g_m at ambient pCO₂ at each temperature. Values are means ±SE; n=4 in Setaria, n=3 in Zea. The arrows indicate the values at ambient pCO₂: black, Setaria; grey, Zea.

To compare the magnitude of the change in CA-lim g_m across species and temperatures, CA-lim g_m was normalized by dividing each value at a given temperature and pCO₂ by CA-lim g_m at ambient pCO₂ at that temperature (Fig. 3D–F). At 25 °C, the increase in CA-lim g_m with decreasing pCO₂ was steeper in Zea than in Setaria (Fig. 3E). In Setaria, the g_mpCO₂ response was greatest at 40 °C and there was little difference between the 25 and 10 °C curves.

Limitations to photosynthesis

At elevated pCO₂ assimilation rate was not limited by diffusion or substrate availability, as indicated by L_s, L_m, and L_CA ≈ 0% for both species and all temperatures (Fig. 4). However, below ambient pCO₂, the diffusional limitation to A increased exponentially with decreasing pCO₂. The data in Fig. 4 show the different limitations as a function of the amount of substrate available: C_a, C_i, and C_m for L_s, L_m, and L_CA, respectively. In Setaria, diffusional limitations were lower at 10 °C than at any other temperature. Comparing Zea and Setaria at 25 °C, they had similar L_s but L_m was larger in Setaria than in Zea. For example, when C_a=9 Pa, L_s=23% and 19% in Setaria and Zea, respectively. The corresponding C_i at this C_a was 5 Pa for both species, whereas L_m was almost double in Setaria (23%) compared to Zea (12%) (Fig. 4C, D). In both species, L_CA was small in comparison with L_s and L_m, and rapidly decreased below 5% as pCO₂ increased.

Fig. 4.

Diffusional limitation to photosynthetic rate (A) imposed by stomatal resistance (L_s, Eqn 11, solid line), mesophyll resistance (L_m, Eqn 12, dashed line), and carbonic anhydrase (L_CA, Eqn 13, dotted line) as a function of the CO₂ supply (pCO₂) available for each (C_a, C_i, and C_m for L_s, L_m, and L_CA, respectively). (A) Setaria viridis at T_L=40 °C, (B) Setaria viridis at T_L=10 °C, (C) Setaria viridis at T_L=25 °C, and (D) Zea mays at T_L=25 °C.

Leakiness (ϕ)

Values of ϕ across pCO₂ for Setaria and Zea at 25 °C calculated under different modeling assumptions are shown in Fig. 5. When g_m was finite and variable with pCO₂ (Model 1), ϕ increased from low to high pCO₂, with a range of 0.16–0.59 in Zea and 0.45–0.76 in Setaria. Assuming that g_m was infinite and V_pmax variable with pCO₂ (Model 2) removed the pCO₂ response of ϕ and generally decreased ϕ at all pCO₂ in Setaria, but only at large pCO₂ in Zea. Model 3 resulted in nearly identical ϕ to Model 2 using the same finite g_m as Model 1 but with the solution constrained by only the photosynthesis model. However, this scenario failed to predict ${\Delta }_{obs}^{13}$ (see Supplementary Fig. S2). Imposing a PEP regeneration rate (V_pr) limitation of 64 and 59 μmol m^–2 s^–1 in Setaria and Zea, respectively (Model 4), decreased ϕ compared to the results with Model 1 in Setaria but resulted in no change in Zea. Interestingly, at pCO₂ ≤ambient air, values for ϕ were similar across models in Zea, but they differed in Setaria. The values of V_pmax, V_cmax, V_p, V_c, C_bs, and g_bs used in these four models are reported in Supplementary Fig. S2.

Fig. 5.

Effect of different parameterizations of models of photosynthesis in the calculation of leakiness (ϕ) in (A) Setaria virids and (B) Zea mays at 25 °C and over a range of pCO₂ inside the leaf intercellular spaces (C_i). Model 1 (solid black line) uses in vitro V_pmax and g_m finite and equal to the values presented in Fig. 3; Model 2 (dashed grey line) uses in vivo V_pmax (which is variable with pCO₂, see Supplementary Fig. S2) and g_m infinite; Model 3 (dotted grey line) uses the same as Model 1 but the solution was only constrained by A and not Δ¹³C; Model 4 (dashed black line) uses the same as Model 1 but introducing V_pr (= 64 and 59 μmol m^–2 s^–1 in Setaria and Zea, respectively) in the calculation of V_p (Eqn 6). The rest of the variables included in these models were calculated as explained in the Methods section: values for some of them can be found in Supplementary Fig. S2. In Models 1, 2, and 4, ϕ was calculated with Eqns 14 or 16 (same result) and in Model 3, ϕ was calculated with Eqn 14. The symbols indicate the value of ϕ at ambient air pCO₂ calculated with the simplified Eqn 19 assuming either g_m finite (solid symbols) or infinite (clear symbols). Values are means ±SE; n=4 in Setaria, n=3 in Zea.

For comparison we also present ϕ at ambient pCO₂ calculated with the simplified Eqn 19 and assuming either g_m finite or infinite. For both species, ϕ calculated with Eqn 19 was not different to values obtained with the complete Eqn 16 when g_m was finite (compare black lines and black symbols in Fig. 5) and when g_m was infinite (compare grey dashed line and clear symbols).

Discussion

Calculation of mesophyll conductance and model parameterization

Mesophyll conductance (g_m) was derived with the in vitro V_pmax method (Ubierna et al., 2017). Estimations of g_m with this method were similar to Δ¹⁸O-g_m across temperatures (Ubierna et al., 2017) and across pCO₂ (Kolbe and Cousins, 2018). Potential errors in g_m originating from inaccurate model parameterization of the in vitro V_pmax method were tested with a sensitivity analysis using Setaria data at three temperatures and across pCO₂ (see Supplementary Fig. S3). Halving in vitro V_pmax increased g_m by <20% at large pCO₂ and almost doubled it at low pCO₂ and high temperature. Alternatively, doubling in vitro V_pmax decreased g_m by <15% at all pCO₂ and temperatures ( Supplementary Fig. S3J–L). This demonstrates that uncertainties in in vitro V_pmax affect absolute values of g_m, but not the trend of increasing g_m with decreasing pCO₂. The sensitivity analysis also demonstrated that variations up to ±50% in K_P, K_C, or K_CA resulted in negligible (when pCO₂ ≥ambient) or small (at low pCO₂) errors in g_m calculations at any temperature (Supplementary Fig. S3A–I) and did not affect the observed trend of g_m with pCO₂.

In C₃ plants, it has been suggested that large g_m values reported for low pCO₂ might be an artifact of uncertainties in parameters such as R_d, Γ*, and b′₃ (Gu and Sun, 2014). The simulations with different values for R_d (see Supplementary Fig. S4A, B) or b′₃ (Supplementary Fig. S4C, D) resulted in variations in g_m of <6% and did not affect the trend of increasing g_m with decreasing pCO₂. Ubierna et al. (2017) demonstrated that g_m is largely independent of values of g_bs or V_cmax and this is also illustrated in Supplementary Fig. S2.

CA-limited versus CA-saturated models to estimate g_m

The substrate for the initial carboxylation by PEPC is HCO₃^– and not CO₂. However, V_p is often calculated in terms of CO₂, because the hydration of CO₂ (V_h) generally happens very fast when catalysed by CA (Stryer, 1988). We refer to this case as the CA-saturated model. In contrast, the CA-limited model calculates V_p as a function of HCO₃^–. The value of HCO₃^– is calculated with C_m, V_h, V_pmax, and a series of rate constants (see Ubierna et al., 2017, for details). Producing the same V_p with the CA-limited and the CA-saturated calculations requires larger C_m for the former than the latter, and the difference could potentially be large if V_h is low. Subsequently, neglecting the hydration step, as in the CA-saturated calculations, can result in under-estimation of C_m and g_m. The terminology CA-saturated or -limited refers to the modeling of V_p and how this affects the calculated C_m value, but it does not imply different roles of CA in the photosynthetic process. Ubierna et al. (2017) found no difference between CA-sat g_m and CA-lim g_m at ambient pCO₂; however, the aim here is to compare these calculations for a range of pCO₂.

In both species and at all temperatures, the difference between CA-sat g_m and CA-lim g_m was negligible for pCO₂ >ambient (Fig. 2). However, as pCO₂ decreased, CA-lim g_m became larger than CA-sat g_m, especially at high temperatures and in Zea. In this species ignoring the hydration step resulted in under-estimating g_m by as much as 20%, whereas in Setaria the under-estimation was <5%.

The larger differences at high temperatures can be explained by the temperature response of K_CA, which increases from 10 to 30 °C but plateaus above that (Boyd et al., 2015). Species differences can be explained by different K_CA values and CO₂ availability to CA. Firstly, K_CA in Setaria (124 μmol m^–2 s^–1 Pa^–1) was double the value for Zea (65.5 μmol m^–2 s^–1 Pa^–1). Below ambient pCO₂, Setaria and Zea had similar A, g_s, and C_i. Sustaining similar A in these two species requires larger C_m in Zea than in Setaria because of the lower in vitro V_pmax value in the former (184 μmol m^–2 s^–1) versus the latter (450 μmol m^–2 s^–1). Therefore, in Zea the lower K_CA and in vitro V_pmax was counterbalanced by increased CO₂ availability to CA through higher g_m. Osborn et al. (2017) also suggested large g_m as a mechanism to increase CO₂ assimilation rate at low pCO₂.

At low pCO₂ or in species with low K_CA, ignoring the hydration step results in under-estimation of g_m. However, the error is insignificant at pCO₂ above ambient or in species with large K_CA, such as Setaria. The hydration step should be included for accurate determination of g_m at low pCO₂ in species with low K_CA and/or high A, such as C₄ grasses (Cousins et al., 2008), especially at high temperatures.

Values for CA-lim g_m and variation with pCO₂

Across pCO₂ and temperatures, CA-lim g_m ranged from 0.6 ± 0.1 to 9.3 ± 1.5 μmol m^–2 s^–1 Pa^–1 in Setaria, and 0.6 ± 0.1 to 16.2 ± 5.7 μmol m^–2 s^–1 Pa^–1 in Zea (Fig. 3). In Zea, photosynthetic rate declined above ambient pCO₂, indicating deactivation at low C_i that did not fully recover when pCO₂ supply was returned to ambient levels (Fig. 1B). This could have introduced some bias in the CA-lim g_m values calculated at high pCO₂. Nevertheless, the CA-lim g_m values were used at pCO₂ ≤ambient, because above ambient, photosynthesis was not restricted by diffusional limitations (Fig. 4).

To validate CA-lim g_m values, they were compared with literature reports for the same species obtained with the alternative Δ¹⁸O method (Barbour et al., 2016; Osborn et al., 2017; Ubierna et al., 2017; Fig. 3). In Zea, there was a good agreement between Δ¹⁸O-g_m (Barbour et al., 2016; Ubierna et al., 2017) and CA-lim g_m (Fig. 3B). A recent study in Zea by Kolbe and Cousins (2018) also found agreement between Δ¹⁸O-g_m and in vitro V_pmaxg_m across a range of pCO₂, although both estimations of g_m deviated at very low pCO₂. In Setaria, Δ¹⁸O-g_m (Barbour et al., 2016; Osborn et al., 2017; Ubierna et al., 2017) was larger than our CA-lim g_m results (Fig. 3A–C). This discrepancy could have originated if in vitro V_pmax was over-estimated, and more studies exploring g_m variation and assessing the impacts of the method are needed.

In Zea at 25 °C and in Setaria at three temperatures, the CA lim-g_m increased with short-term exposure to decreasing pCO₂. Increasing g_m with decreasing pCO₂ has also been observed in C₃ species (Bongi and Loreto, 1989; Loreto et al., 1992; Flexas et al., 2007, 2008; Hassiotou et al., 2009; Bunce, 2010; Douthe et al., 2011; Tazoe et al., 2011), although there are also a few studies that have concluded there is no change (Loreto et al., 1992; Tazoe et al., 2009). There are only two studies that have presented C₄-g_m across pCO₂. In Osborn et al. (2017), Δ¹⁸O-g_m values for Setaria increased with decreasing pCO₂ but the trend was not significant. In Zea, Kolbe and Cousins (2018) found a significant increase in Δ¹⁸O-g_m with decreasing pCO₂.

The initial slope of an A-C_i curve can be modified with either C_m (g_m) or V_pmax (see Supplementary Fig. S5). Therefore, there may be a value for V_pmax that would cancel out the trend in CA-lim g_m. However, this is not the case if V_pmax is independent of pCO₂, and cancelling the observed trend in CA-lim g_m would require V_pmax to decrease with increasing pCO₂ (Supplementary Fig. S6). There is evidence showing that CO₂ levels affect the phosphorylation state of PEPC and PEPCK, and therefore variation of in vivo V_pmax across pCO₂ could be expected (Bailey et al., 2007). However, the CO₂ response of photosynthetic rate was found to be no different between wild-type and transgenic plants with low PEPC phosphorylation (Furumoto et al., 2007). Much of the post-translational modifications that presumably lower V_pmax would probably occur when CO₂ is saturating and some other factor limits C₄ photosynthesis. At ambient pCO₂ and below it is generally thought, although not known, that PEPC is operating at V_pmax. The fact that Δ¹⁸O-g_m data have demonstrated a similar trend of increasing g_m with decreasing pCO₂ (Kolbe and Cousins, 2018) points to a constant V_pmax value. Nevertheless, if fast in vivo regulation of V_pmax occurs it could alter values and trends in g_m. In reality, there might be a combination of both fluctuations in g_m and V_pmax in response to short-term variation in pCO₂. Future work should investigate in vivo regulation of V_pmax and its impact on g_m calculations.

Limitation to photosynthesis at low pCO₂

C₄ photosynthesis saturates at ambient pCO₂ and A was not limited by diffusion, as indicated by L_s, L_m, and L_CA ≈ 0% for both species and all temperatures (Fig. 4). However, below ambient air pCO₂, diffusional limitations constrained CO₂ assimilation and increased exponentially with decreasing pCO₂. As shown in Fig. 1 and Supplementary Fig. S5, in both species the CO₂ responsive part of the A-C_i curve corresponded to C_i below ~10 Pa. This raises the question of whether C₄ plants operate below this threshold. In laboratory experiments, high irradiance and N fertilization shifted the operational C_i down to the CO₂ responsive part of the A-C_i curve (Ghannoum et al., 1997; Ghannoum and Conroy, 1998). Additionally, moderate water stress decreased C_i in several C₄ species, although under severe drought declines in A precluded C_i from getting very low (Ghannoum, 2009, and references herein). Under ambient air pCO₂, C_i<11 Pa were reported for Zea grown in FACE-type experiments (Leakey et al., 2004; Markelz et al., 2011), and Sorghum bicolor grown in an open field reached C_i/C_a=0.2 after two consecutive water-stress cycles (Steduto et al., 1997). Therefore, under certain growth conditions, CO₂ availability may limit C₄ photosynthesis.

Interestingly, Setaria and Zea displayed different behavior at low pCO₂. At low pCO₂, Zea was more efficient because it achieved high A despite lower V_pmax and K_CA by decreasing diffusional limitations and sustaining greater C_m with high g_m. The high g_m at low pCO₂ could increase or maintain photosynthesis at low C_i and could improve photosynthetic rates under situations that result in low CO₂ availability, such as drought.

In both species, the conversion of CO₂ into bicarbonate as catalysed by CA was fast enough that the hydration rate only limited A at low pCO₂ (L_CA=6–16% for C_m<4 Pa, Fig. 4). Such low C_m is unlikely to occur, even under drought conditions. At these very low pCO₂, the hydration rate (V_h) was comparable to rates in CA-depleted transgenic plants (Supplementary Fig. S7). For example, in Setaria at 25 °C, V_h decreased from 581 μmol m^–2 s^–1 at ambient pCO₂ to 100 μmol m^–2 s^–1 at the lowest pCO₂ measured. Using values from Osborn et al. (2017) at 25 °C and ambient pCO₂ to calculate V_h as C_m×K_CA resulted in 1215 and 142 μmol m^–2 s^–1 for the wild type and CA-depleted transgenic, respectively. Osborn et al. (2017) concluded that in Setaria at low pCO₂, g_m posed a greater limitation than CA activity. Our study confirms that g_m is a major determinant of photosynthetic capacity at low pCO₂ and CA further constrains assimilation rates only at very low pCO₂. However, the CA limitation at low pCO₂ will be exacerbated at higher temperatures as the hydration rate is less able to keep up with the increase in PEPC activity (Boyd et al., 2015).

Leakiness (ϕ)

Leakiness is often estimated from comparing models and measurements of Δ¹³C assuming g_m is infinite (Pengelly et al., 2010; Ubierna et al., 2011, 2013) or large (Kromdijk et al., 2010). Values of ϕ vary by as much as 0.04–0.9 (for a compilation of values and review of methods see Kromdijk et al., 2014), although for most species under most conditions ϕ=0.2–0.3 (Cousins et al., 2006; Kromdijk et al., 2010; Pengelly et al., 2010; Ubierna et al., 2013; Bellasio and Griffiths, 2014).

In our study, considering g_m to be finite had a different effect on the calculation of ϕ for Setaria and Zea. At ambient air pCO₂ and 25 °C, both Setaria and Zea had similar g_m (2.00 and 2.43 μmol m^–2 s^–1 Pa^–1, respectively). However, while ϕ in Zea was the same whether g_m was finite or infinite, in Setaria, accounting for a finite g_m doubled ϕ (Fig. 5, compare Models 1 and 2). This high ϕ in Setaria was driven by constrains imposed by the Δ¹³C model rather than the photosynthesis model. This is illustrated by the comparison of Models 2 and 3 (Fig 5). Both models predicted the same A and ϕ, but Model 2 used g_m finite (and in vitro V_pmax) and Model 3 assumed g_m infinite (and in vivo V_pmax). However, Model 3 failed to predict ${\Delta }_{obs}^{13}$ (see Supplementary Fig. S2). Forcing the solution to satisfy both models of A and ${\Delta }_{obs}^{13}$ resulted in increases in ϕ in Setaria, but not in Zea.

This can be explained through the relationship between Δ¹³C and C_m/C_a, which is illustrated in Fig. 6 for different values of ϕ. Increasing C_m/C_a results in either increased or decreased Δ¹³C depending on whether ϕ is low (≤0.3) or high (Henderson et al., 1992; von Caemmerer et al., 2014). When ${\Delta }_{obs}^{13}$ > a_s + (a_m – a_s)(C_i/C_a) ( = 4.4–2.6 C_i/C_a ≈ 2.9‰ in our data set at 25 ºC and ambient pCO₂) increasing C_m/C_a results in decreased ϕ; meanwhile the opposite is true when ${\Delta }_{obs}^{13}$ The value a_s + (a_m – a_s)(C_i/C_a) represents the intercept of the line Δ¹³C versus C_m/C_a when C_bs and boundary layer conductance are large and ternary effects are ignored. At ambient air pCO₂ and 25 °C, ${\Delta }_{obs}^{13}$=3.1‰ in Zea. Therefore, varying C_m/C_a resulted in minimal changes in ϕ (compare black triangle and circle in Fig. 6). However, in Setaria, ${\Delta }_{obs}^{13}$=4.5‰ and therefore low C_m/Ca translated into large ϕ (compare grey triangle and circle in Fig. 6). The photosynthesis model demonstrated that this increase in ϕ was achieved by increased V_p and g_bs (see Supplementary Fig. S2).

Fig. 6.

Δ¹³C (Eqn 9) as a function of C_m/C_a for different ϕ values (indicated by the numbers at the end of each line). For calculations we used values of 37, 36, 20, and 1364 Pa for C_a, C_L, C_i, and C_bs, respectively; t=0.0058, b₄=–4.49‰, and b₃=29.87‰. These values correspond to the mean values measured or calculated in Setaria at 25 °C and ambient pCO₂. Black symbols represent data for Zea and grey symbols for Setaria. For both species, ϕ was calculated assuming either g_m infinite (triangles) or g_m=2.00 and 2.43 μmol m^–2 s^–1 Pa^–1 in Setaria and Zea, respectively (circles).

It is questionable that Setaria operates with ϕ=0.7, and it is seemly unreasonable that it does. Because Δ¹³C is mostly determined by C_m/C_a and ϕ, low C_m/C_a forces the increase in ϕ. But are there any other parameters in the discrimination equation that could be manipulated in order to predict large Δ¹³C with low C_m/C_a without large ϕ? Calculations of ϕ with the complete (Eqn 16) and simplified (Eqn 19) models suggest that, at least at ambient pCO₂, this was not the case. The simplified calculation of ϕ produced values similar to the complete model, suggesting that at ambient air pCO₂ or above, modifying parameters such as C_bs, b₃, or b₄ within their current definition did not result in large changes in Δ¹³C.

In addition to the possible post-translational regulation of V_pmax, PEP regeneration (V_pr) may also influence V_p (Eqn 6) and estimates of ϕ. In our calculations, V_pr=64 μmol m^–2 s^–1 decreased ϕ in Setaria by 0.3 and resulted in slightly larger g_m values at high pCO₂ but no change at low pCO₂ (compare Models 1 and 4 in Fig. 5 and Supplementary Fig. S2). In fact, at low pCO₂ it is expected that V_pr would not limit V_p and would have no effect on estimates of g_m or ϕ under these conditions. Changes in ϕ in response to pCO₂ or other conditions are possible if V_pr is allowed to vary, although at present V_pr variation across species, temperatures, or pCO₂ is unknown. The V_pr values that would be needed to remove the observed trend in g_m with pCO₂ are shown in Supplementary Fig. S8. Introducing a value for V_pr implies decoupling V_p from C_m (g_m). In other words, the required V_p value to support the measured A could be achieved by choosing the adequate V_pr rather than by varying C_m. This would also further complicate estimations of ϕ from Δ¹³C as V_pr is not often measured and is not incorporated into the Δ¹³C models.

Our calculations assume that theoretical models of photosynthesis and discrimination represent the actual photosynthetic process; any inaccuracy in the models will introduce error in the calculated g_m. We have evaluated one common modelling simplification, the effect of CA limitation, and also the impact of uncertainty on input parameters. Additionally, we have used two contrasting species to illustrate the sensitivity of ϕ to g_m. Although a complete analysis of ϕ is beyond the scope of this work, this should be undertaken in future studies together with investigations on PEP regeneration limitations. Other future foci for research include: investigating in vivo and in vitro V_pmax values and variation across species and environmental conditions; and compiling leaf structure, CA, aquaporins, or other data that could reveal potential mechanisms behind observed g_m patterns.

Supplementary data

Supplementary data are available at JXB online.

Methods S1. Model of ¹³C discrimination in C₄ species.

Table S1. Gas exchange values for C_i and A, and calculated values for C_m and CA-lim g_m in Setaria viridis and Zea mays at 25 °C and variable CO₂ supply.

Fig. S1. C_m across C_i in Setaria viridis at three temperatures, and in Zea mays at 25 °C.

Fig. S2. Description of the models used to evaluate the effect of g_m in calculations of ϕ.

Fig. S3. Sensitivity of calculations of CA-lim g_m in Setaria viridis to uncertainty in input parameters.

Fig. S4. Impact of R_d and b′₃ in the calculation of CA-lim g_m in Setaria viridis at 25 °C.

Fig. S5. Measured versus modeled response of A to C_i at 25 °C in Setaria viridis and Zea mays for different values of V_pmax and g_m.

Fig. S6. Values for in vivo V_pmax across C_i in Setaria viridis calculated when CA-lim g_m is constant with pCO₂.

Fig. S7. V_h across C_i in Setaria viridis at three temperatures.

Fig. S8. Values for V_pr across C_i in Setaria viridis calculated when CA-lim g_m is constant with pCO₂.