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.


Introduction
In C 4 plants photorespiration is reduced by concentrating CO 2 around Rubisco (ribulose 1,5-bisphosphate carboxylase/oxygenase) (Edwards and Walker, 1983;Hatch, 1987;Sage, 2004). In Kranz-type C 4 plants this is achieved with a compartmentalized two-carboxylation process: (1) in the cytosol of mesophyll cells, bicarbonate (HCO 3 -) 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 CO 2 released from the decarboxylation of these acids is fixed by Rubisco.
Mesophyll conductance (g m ) describes the movement of CO 2 from stomata across the intercellular spaces to the sites of first carboxylation, which are the chloroplast stroma or mesophyll cytosol in C 3 and C 4 species, respectively (Evans and von Caemmerer, 1996). There is extensive research describing g m in C 3 species; however, C 4 -g m is poorly understood because it is difficult to estimate. Traditionally g m was assumed to be larger in C 4 compared to C 3 species, but most recent studies suggest that values for C 4 -g m correspond to higher-end C 3 -g m reports, and that C 4 -g m reacts similarly to C 3 -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 C 4 -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 13 C discrimination (Δ 13 C) (Farquhar, 1983;Farquhar and Cernusak, 2012). Historically, g m is generally assumed to be infinite when solving for ϕ from Δ 13 C; 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 C 3 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 C 4 photosynthesis as the reduction of photorespiration achieved by the CO 2 -concentrating mechanism saturates C 4 photosynthesis at ambient pCO 2 . If g m were to limit C 4 photosynthesis, it would likely only be at very low pCO 2 . However, not much is known about the variation of C 4 -g m with pCO 2 . In the C 4 grass Setaria viridis, g m derived with the 18 O discrimination (Δ 18 O) method increased as pCO 2 decreased, although the variation was not significant (Osborn et al., 2017). Some reports have shown that in C 3 species g m increases with short-term exposure to decreasing pCO 2 (Bongi and Loreto, 1989;Loreto et al., 1992;Flexas et al., 2007Flexas et al., , 2008Hassiotou et al., 2009;Bunce, 2010;Douthe et al., 2011;Tazoe et al., 2011). However, others have suggested that C 3 -g m is insensitive to changes in pCO 2 (Loreto et al., 1992;Tazoe et al., 2009). It has been hypothesized that the observed C 3 -g m response to pCO 2 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 C 4 plants, g m has been estimated with the Δ 18 O method 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 pCO 2 in the mesophyll cells (C m ) needed to simultaneously match modeled and measured rates of CO 2 assimilation and Δ 13 C when the models are parameterized with in vitro V pmax , as determined in a crude leaf extract. Values derived for g m with the Δ 18 O and in vitro V pmax methods were similar in two C 4 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 CO 2 or HCO 3 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 pCO 2 , but CA limitation is expected at low pCO 2 .
In this study, we calculated g m using the in vitro V pmax method across a range of pCO 2 in two C 4 grasses, one economically important (Zea mays) and the other the adopted model system for studying C 4 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 C 4 -g m to short-term variation in pCO 2 ; (2) evaluate the impact of disequilibrium between CO 2 and HCO 3 at a range of pCO 2 and temperatures; (3) investigate if g m represents a limitation to C 4 photosynthesis across pCO 2 ; and (4) assess the impact of finite g m on ϕ calculations.

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. (2013Ubierna et al. ( , 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 [CO 2 ] 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.

Enzyme-limited C 4 photosynthesis model for CA-limited or CA-saturated conditions
The enzyme-limited C 4 photosynthesis rate is (von Caemmerer, 2000): where: where α (= 0) is the fraction of PSII activity in the bundle-sheath cells (von Caemmerer, 2000); u oc is the ratio of O 2 and CO 2 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 O 2 partial pressure in the mesophyll (19.5 kPa, which corresponds to 21%); R d is the non-photorespiratory CO 2 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.  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 (pCO 2 in the mesophyll cells) will be discussed subsequently. CA-saturated and CA-limited models differ as follows.
If the rate of PEP regeneration is limiting, then V p is (von Caemmerer, 2000): 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: where K CA is the rate constant of CA for CO 2 , 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 13 C ( ∆ obs 13 ) is calculated as (Evans et al., 1986): where C and δ are the 12 CO 2 mol fraction and the δ 13 C of the CO 2 , respectively, in dry air in and out the chamber. The theoretical model for Δ 13 C is (Farquhar and Cernusak, 2012):

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: 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 Δ 13 C (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 CO 2 assimilation by either finite stomatal conductance (L s ), by mesophyll conductance (L m ), or by carbonic anhydrase (L CA ), we adapted to C 4 photosynthesis an approach previously used for C 3 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: Then L s , L m , and L CA were calculated as: .

Calculation of leakiness (ϕ)
The C 4 photosynthesis model (von Caemmerer, 2000) calculates ϕ as: Eqn 14 where C bs , the pCO 2 in the bundle-sheath cells, is (von Caemmerer, 2000): where O s is the O 2 partial pressure in the bundle-sheath cells.
From Δ 13 C (Eqn 9), ϕ is solved as: where b 3 (combined effects of Rubisco fractionation, and fractionations associated with respiration and photorespiration) and b 4 (combined fractionation during PEP carboxylation, hydration, and respiration) are calculated as (Farquhar, 1983;Cousins et al., 2006): 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 Δ 13 C; 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 Δ 13 C. 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 pCO 2 , ϕ was also calculated with a simplified equation derived from Δ 13 C assuming that C bs is much larger than C m and that hydration and assimilation fluxes are large (V p /V h ≈0, and

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 CO 2 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; pCO 2 , 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.

A-C i curves and observed 13 C photosynthetic discrimination
Under all leaf measurement temperatures (T L ), the rate of net photosynthesis (A) in Setaria increased with C i as the pCO 2 supplied increased from ~5 Pa to ambient air values (~35 Pa) and then leveled off ( Fig. 1A). At all pCO 2 , 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 pCO 2 before decreasing at higher pCO 2 (Fig. 1B). At ambient air pCO 2 and 25 °C, ∆ obs 13 was larger in Setaria (4.5 ± 0.1‰) than in Zea (3.1 ± 0.2‰) (Fig. 1C, D). In Zea, the ∆ obs 13 was low at ambient air pCO 2 and increased at lower or higher C i (Fig. 1D). However, in Setaria, ∆ 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 pCO 2 was above ambient (Fig. 2). As pCO 2 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 pCO 2 at 40 °C, at all pCO 2 except at ambient and the measurement just above ambient at 25 °C, and at the largest pCO 2 at 10 °C ( Fig. 2A). In Zea, CA-lim g m and CA-sat g m were significantly different (P<0.05) at all pCO 2 ≤ambient air (Fig. 2 B).
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 pCO 2 was 20 ± 8% larger than CA-sat g m at 25 °C. Because CA limitation was relevant at low pCO 2 , for subsequent analyses we use the CA-lim g m values for all species, temperatures, and pCO 2 .
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 pCO 2 by CA-lim g m at ambient pCO 2 at that temperature ( Fig. 3 D-F). At 25 °C, the increase in CA-lim g m with decreasing pCO 2 was steeper in Zea than in Setaria (Fig. 3E). In Setaria, the g m pCO 2 response was greatest at 40 °C and there was little difference between the 25 and 10 °C curves.

Limitations to photosynthesis
At elevated pCO 2 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 pCO 2 , the diffusional limitation to A increased exponentially with decreasing pCO 2 . 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 pCO 2 increased.

Leakiness (ϕ)
Values of ϕ across pCO 2 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 pCO 2 (Model 1), ϕ increased from low to high pCO 2 , 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 pCO 2 (Model 2) removed the pCO 2 response of ϕ and generally decreased ϕ at all pCO 2 in Setaria, but only at large pCO 2 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 ∆ 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 pCO 2 ≤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.
For comparison we also present ϕ at ambient pCO 2 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).

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 Δ 18 O-g m across temperatures (Ubierna et al., 2017) and across pCO 2 (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 pCO 2 (see Supplementary Fig. S3). Halving in vitro V pmax increased g m by <20% at large pCO 2 and almost doubled it at low pCO 2 and high temperature. Alternatively, doubling in vitro V pmax decreased g m by <15% at all pCO 2 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 pCO 2 . The sensitivity analysis also demonstrated that variations up to ±50% in K P , K C , or K CA resulted in negligible (when pCO 2 ≥ambient) or small (at low pCO 2 ) errors in g m calculations at any temperature (Supplementary Fig.  S3A-I) and did not affect the observed trend of g m with pCO 2 .
In C 3 plants, it has been suggested that large g m values reported for low pCO 2 might be an artifact of uncertainties in parameters such as R d , Γ*, and b3 (Gu and Sun, 2014). The simulations with different values for R d (see Supplementary  Fig. S4A, B) or b3 ( Supplementary Fig. S4C, D) resulted in variations in g m of <6% and did not affect the trend of increasing g m with decreasing pCO 2 . 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 HCO 3 and not CO 2 . However, V p is often calculated in terms of CO 2 , because the hydration of CO 2 (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 HCO 3 -. The value of HCO 3 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 pCO 2 ; however, the aim here is to compare these calculations for a range of pCO 2 .
In both species and at all temperatures, the difference between CA-sat g m and CA-lim g m was negligible for pCO 2 >ambient (Fig. 2). However, as pCO 2 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 CO 2 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 pCO 2 , 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 CO 2 availability to CA through higher g m . Osborn et al. (2017) also suggested large g m as a mechanism to increase CO 2 assimilation rate at low pCO 2 . At low pCO 2 or in species with low K CA , ignoring the hydration step results in under-estimation of g m . However, the error is insignificant at pCO 2 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 pCO 2 in species with low K CA and/or high A, such as C 4 grasses (Cousins et al., 2008), especially at high temperatures.
Values for CA-lim g m and variation with pCO 2 Across pCO 2 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 pCO 2 , indicating deactivation at low C i that did not fully recover when pCO 2 supply was returned to ambient levels (Fig. 1B). This could have introduced some bias in the CA-lim g m values calculated at high pCO 2 . Nevertheless, the CA-lim g m values were used at pCO 2 ≤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 Δ 18 O method (Barbour et al., 2016;Osborn et al., 2017;Ubierna et al., 2017;Fig. 3). In Zea, there was a good agreement between Δ 18 O-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 Δ 18 O-g m and in vitro V pmax g m across a range of pCO 2 , although both estimations of g m deviated at very low pCO 2 . In Setaria, Δ 18 O-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 pCO 2 . Increasing g m with decreasing pCO 2 has also been observed in C 3 species (Bongi and Loreto, 1989;Loreto et al., 1992;Flexas et al., 2007Flexas et al., , 2008Hassiotou 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 C 4 -g m across pCO 2 . In Osborn et al. (2017), Δ 18 O-g m values for Setaria increased with decreasing pCO 2 but the trend was not significant. In Zea, Kolbe and Cousins (2018) found a significant increase in Δ 18 O-g m with decreasing pCO 2 .
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 pCO 2 , and cancelling the observed trend in CA-lim g m would require V pmax to decrease with increasing pCO 2 ( Supplementary Fig. S6). There is evidence showing that CO 2 levels affect the phosphorylation state of PEPC and PEPCK, and therefore variation of in vivo V pmax across pCO 2 could be expected (Bailey et al., 2007). However, the CO 2 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 CO 2 is saturating and some other factor limits C 4 photosynthesis. At ambient pCO 2 and below it is generally thought, although not known, that PEPC is operating at V pmax . The fact that Δ 18 O-g m data have demonstrated a similar trend of increasing g m with decreasing pCO 2 (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 pCO 2 . Future work should investigate in vivo regulation of V pmax and its impact on g m calculations.
Limitation to photosynthesis at low pCO 2 C 4 photosynthesis saturates at ambient pCO 2 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 pCO 2 , diffusional limitations constrained CO 2 assimilation and increased exponentially with decreasing pCO 2 . As shown in Fig. 1 and Supplementary Fig. S5, in both species the CO 2 responsive part of the A-C i curve corresponded to C i below ~10 Pa. This raises the question of whether C 4 plants operate below this threshold. In laboratory experiments, high irradiance and N fertilization shifted the operational C i down to the CO 2 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 C 4 species, although under severe drought declines in A precluded C i from getting very low (Ghannoum, 2009, and references herein). Under ambient air pCO 2 , 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, CO 2 availability may limit C 4 photosynthesis.
Interestingly, Setaria and Zea displayed different behavior at low pCO 2 . At low pCO 2 , 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 pCO 2 could increase or maintain photosynthesis at low C i and could improve photosynthetic rates under situations that result in low CO 2 availability, such as drought.
In both species, the conversion of CO 2 into bicarbonate as catalysed by CA was fast enough that the hydration rate only limited A at low pCO 2 (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 pCO 2 , 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 pCO 2 to 100 μmol m -2 s -1 at the lowest pCO 2 measured. Using values from Osborn et al. (2017) at 25 °C and ambient pCO 2 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 pCO 2 , g m posed a greater limitation than CA activity. Our study confirms that g m is a major determinant of photosynthetic capacity at low pCO 2 and CA further constrains assimilation rates only at very low pCO 2 . However, the CA limitation at low pCO 2 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).
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 pCO 2 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 Δ 13 C 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 ∆ obs 13 (see Supplementary Fig. S2). Forcing the solution to satisfy both models of A and ∆ obs 13 resulted in increases in ϕ in Setaria, but not in Zea.
This can be explained through the relationship between Δ 13 C 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 Δ 13 C depending on whether ϕ is low (≤0.3) or high von Caemmerer et al., 2014). When ∆ 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 pCO 2 ) increasing C m /C a results in decreased ϕ; meanwhile the opposite is true when ∆ obs i a 13 < 4.4 2.6 -. / C C The value a s + (a m -a s )(C i /C a ) represents the intercept of the line Δ 13 C versus C m /C a when C bs and boundary layer conductance are large and ternary effects are ignored. At ambient air pCO 2 and 25 °C, ∆ 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, ∆ 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).
It is questionable that Setaria operates with ϕ=0.7, and it is seemly unreasonable that it does. Because Δ 13 C 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 Δ 13 C 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 pCO 2 , this was not the case. The simplified calculation of ϕ produced values similar to the complete model, suggesting that at ambient air pCO 2 or above, modifying parameters such as C bs , b 3 , or b 4 within their current definition did not result in large changes in Δ 13 C.
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 pCO 2 but no change at low pCO 2 (compare Models 1 and 4 in Fig. 5 and Supplementary Fig. S2). In fact, at low pCO 2 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 pCO 2 or other conditions are possible if V pr is allowed to vary, although at present V pr variation across species, temperatures, or pCO 2 is unknown. The V pr values that would be needed to remove the observed trend in g m with pCO 2 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 Δ 13 C as V pr is not often measured and is not incorporated into the Δ 13 C models.
Our calculations assume that theoretical models of photosynthesis and discrimination represent the actual photosynthetic Fig. 6. Δ 13 C (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, b 4 =-4.49‰, and b 3 =29.87‰. These values correspond to the mean values measured or calculated in Setaria at 25 °C and ambient pCO 2 . 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). 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 13 C discrimination in C 4 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 CO 2 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 b3 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 pCO 2 . 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 pCO 2 .