- © 2018 American Society of Plant Biologists.All Rights Reserved.

## Abstract

Carbonic anhydrase (CA) activity in leaves catalyzes the ^{18}O exchange between CO_{2} and water during photosynthesis. This feature has been used to estimate the mesophyll conductance to CO_{2} (*g*_{m}) from measurements of online C^{18}OO photosynthetic discrimination (∆^{18}O). Based on CA assays on leaf extracts, it has been argued that CO_{2} in mesophyll cells should be in isotopic equilibrium with water in most C_{3} species as well as many C_{4} dicot species. However, this seems incompatible with ∆^{18}O data that would indicate a much lower degree of equilibration, especially in C_{4} plants under high light intensity. This apparent contradiction is resolved here using a new model of C_{3} and C_{4} photosynthetic discrimination that includes competition between CO_{2} hydration and carboxylation and the contribution of respiratory fluxes. This new modeling framework is used to revisit previously published data sets on C_{3} and C_{4} species, including CA-deficient plants. We conclude that (1) newly ∆^{18}O-derived *g*_{m} values are usually close but significantly higher (typically 20% and up to 50%) than those derived assuming full equilibration and (2) despite the uncertainty associated with the respiration rate in light, or the water isotope gradient between mesophyll and bundle sheath cells, robust estimates of ∆^{18}O-derived *g*_{m} can be achieved in both C_{3} and C_{4} plants.

Carbonic anhydrases (CAs) are a group of zinc metalloenzymes that catalyze the interconversion of CO_{2} into bicarbonate with great efficiency (Moroney et al., 2001; Rowlett, 2010). In the mesophyll cells of C_{3} plants, CA is most abundant in the stroma (Badger and Price, 1994), but CAs also are found in other compartments of the mesophyll such as the cytosol, the mitochondria, or the plasma membrane (Fabre et al., 2007; DiMario et al., 2016). Chloroplastic CA in C_{3} plants was first assumed to be required to provide CO_{2} to Rubisco in the stroma, given the alkalinity of this compartment (Badger and Price, 1994), but results of studies on antisense tobacco (*Nicotiana tabacum*) plants have not been conclusive (Price et al., 1994). In C_{4} plants, CA is most abundant in the cytosol (Badger and Price, 1994), primarily because it is needed to increase the supply of bicarbonate to phospho*enol*pyruvate carboxylase (PEPC; Hatch and Burnell, 1990; Badger and Price, 1994). This idea was confirmed by experiments conducted on CA-deficient mutants, which showed that cytosolic CA was required to maintain high photosynthetic rate in *Flaveria bidentis* (von Caemmerer et al., 2004) and *Zea mays*, at least in low-CO_{2} environments (Studer et al., 2014).

Irrespective of the exact functional role of CA in plants, CA catalyzes the ^{18}O exchange between CO_{2} and water, so that CO_{2} is commonly assumed to be near isotopic equilibrium with water in CA-containing mesophyll compartments. Consequently, CO_{2} partial pressure at the site of CA activity inside the mesophyll cells (*p*_{CA}) can be derived from online ^{18}O photosynthetic discrimination (Δ^{18}O) measurements, provided that the ^{18}O/^{16}O ratio of leaf water at this CA site can be estimated (Gillon and Yakir, 2000b). However, sometimes CO_{2} might not be in complete equilibration with water, especially when CO_{2} uptake rates are high, as this would result in relatively brief residence times for CO_{2} molecules inside the mesophyll. This led Gillon and Yakir (2000b) to propose a formulation for Δ^{18}O that incorporates the degree of CO_{2}-H_{2}O isotopic equilibrium, θ (0 ≤ θ ≤ 1). Online Δ^{18}O measurements and estimates of θ from in vitro CA assays indicated that, for three C_{3} species, *p*_{CA} always lay somewhere between the CO_{2} partial pressure in the substomatal cavity (*p*_{i}) and that at the sites of carboxylation in the chloroplast stroma (*p*_{c}), estimated separately from online ^{13}C photosynthetic discrimination (Δ^{13}C) measurements (Gillon and Yakir, 2000b). This finding was in line with the hypothesis that the outer limit of CA activity in C_{3} plants was located at the chloroplast surface, and thus before the carboxylation site within the chloroplast stroma. This approach was later used in a follow-up study to estimate mesophyll conductance (*g*_{m}) and θ in a number of C_{3} plants but also in C_{4} plants, which generally exhibited lower θ values than C_{3} plants (Gillon and Yakir, 2001).

In other studies (Gillon and Yakir, 2000a; Cousins et al., 2006b), a slightly different approach was adopted. *g*_{m} was derived from Δ^{18}O measurements performed under low-light conditions (i.e. when the residence time of CO_{2} inside the mesophyll was expected to be long enough to allow full equilibration [θ = 1]). Under higher light intensities, *g*_{m} was assumed constant (for a given species), and θ was then estimated from the Δ^{18}O data and usually found to be quite low, down to 0.1 (Gillon and Yakir, 2000a; Cousins et al., 2006b). Cousins et al. (2006b) noted that these Δ^{18}O-derived θ values seemed incompatible with in vitro leaf CA assays that, instead, indicated full equilibration at all light conditions. These discrepancies between in vitro and Δ^{18}O-derived θ estimates were hypothesized to arise from a spatial mismatch between the CA site and the evaporation site in C_{4} plants and an isotopically heterogenous leaf water composition in the cytoplasm of mesophyll cells (Cousins et al., 2006b).

Although it is not possible to completely rule out these hypotheses, there is growing evidence that water isotope gradients do not develop within the cytoplasm and, rather, remain confined to the vascular tissues of the leaf (Holloway-Phillips et al., 2016). Furthermore, in vitro CA assays conducted in all recent studies, using pH and CO_{2} concentrations close to those experienced in folio, seem to indicate that CO_{2} should always be near full isotopic equilibration with leaf water (Cousins et al., 2007; Studer et al., 2014; Barbour et al., 2016; Ubierna et al., 2017). Consequently, a common practice now is to assume θ = 1 when estimating *g*_{m} from online Δ^{18}O measurements (Barbour et al., 2016; Ubierna et al., 2017) and also to perform a sensitivity analysis to predict how *g*_{m} would be affected had θ been set to lower values, noting that lower θ values always lead to higher *g*_{m} values (Barbour et al., 2016; Ubierna et al., 2017). Most recently, alternative estimates of *g*_{m} in C_{4} plants using in vitro maximal phospho*enol*pyruvate (PEP) carboxylation rate measurements seem to support Δ^{18}O-derived *g*_{m} estimates assuming θ = 1 (Ubierna et al., 2017).

This article reexamines the relationship between CA activity and isotopic equilibration during photosynthesis. To address this overlooked issue, we propose a steady-state modeling framework of Δ^{18}O for both C_{3} and C_{4} plants. This new model explicitly accounts for the competition between CO_{2} hydration and carboxylation, providing the possibility for incomplete CO_{2}-H_{2}O equilibration to occur inside the leaf. In addition, the new model accounts for the physical separation between mesophyll and bundle sheath cells in C_{4} species and for the contribution of respiratory fluxes. Several factors motivated the derivation of this new model. First, as we will explain later, the current model describing the degree of θ (Gillon and Yakir, 2000b) is based on several assumptions that cannot be applied to steady-state leaf gas-exchange measurements, thus preventing any conclusion to be drawn on whether isotopic equilibrium is reached based on in vitro CA activity assays. Additionally, the current practice of setting θ to unity or lower does not allow the study of the functional link between CA activity and Δ^{18}O and how it varies between C_{3} and C_{4} species. A steady-state formulation of Δ^{18}O by C_{3} plants that includes the competition between carboxylation and CA-catalyzed CO_{2} hydration was proposed already by Farquhar and Lloyd (1993). This formulation constitutes the basis of our derivation that we extended to C_{3} and C_{4} photosynthesis pathways and mesophyll compartmentalization. The new model also applies to conditions of high leaf-to-air vapor pressure deficit that require ternary corrections on the CO_{2} and C^{18}OO assimilation rates (von Caemmerer and Farquhar, 1981; Farquhar and Cernusak, 2012). With this new modeling framework, we revisit a number of previously published data sets for C_{3} and C_{4} species, including CA-deficient mutants, and illustrate how to reconcile in vitro CA assays with online Δ^{18}O measurements while, at the same time, estimating *g*_{m} from Δ^{18}O data.

## THEORY

### The Gas-Exchange View

Throughout this article, we will assume that CO_{2} or C^{18}OO gradients within the intercellular air space are negligible, and we will use the terms intercellular air space and stomatal cavity air space interchangeably. Under the assumption of a well-identified CA site inside the mesophyll cells upstream of any carboxylation site (Fig. 1, gas-exchange view), the net leaf CO_{2} flux can be written as the product of a conductance *g*_{m} for CO_{2} diffusion from the intercellular air space to the CA site and the CO_{2} drawdown along the same path: *A* = *g*_{m}(*p*_{i} − *p*_{CA})/*P*, where *P* is atmospheric pressure and *p*_{i} and *p*_{CA} are the CO_{2} partial pressures in the intercellular air space and at the CA site, respectively. Similarly, the net C^{18}OO flux can be defined as , where *a*_{w} is the fractionation factor during CO_{2} dissolution and diffusion from the substomatal cavity to the CA site and *R*_{i} and *R*_{CA} represent the ^{18}O/^{16}O ratios of CO_{2} in the substomatal cavity and at the CA site. The fractionation factor *a*_{w} is quite small and usually taken as +0.8‰ at 25°C (Farquhar and Lloyd, 1993). These two flux-gradient relationships can be combined and rearranged as follows:

where Δ_{ci} = *R*_{CA}/*R*_{i} − 1, ε_{ci} = *p*_{CA}/(*p*_{i} − *p*_{CA}), Δ_{i} = *R*_{i}/*R*_{A} − 1, and *R*_{A} represents the ^{18}O/^{16}O ratio of the net CO_{2} flux (=0.5^{18}*A*/*A*); that is, ∆_{i} represents ∆^{18}O, expressed relative to *R*_{i} and not relative to the ^{18}O/^{16}O ratio of the CO_{2} in the air surrounding the leaf (*R*_{a}).

Our current theoretical understanding of the C^{18}OO photosynthetic discrimination has been drawn on the assumption that the CA site is located in a leaf water compartment with a homogenous ^{18}O/^{16}O ratio that includes the evaporation site. This assumption is in accordance with recent studies showing that leaf water isotopic gradients seem to be limited to a small region around the leaf veins (Holloway-Phillips et al., 2016). In this case, the ^{18}O/^{16}O ratio of the water in the CA-containing compartment can be approximated as the ^{18}O/^{16}O ratio at the evaporation site within the mesophyll (noted *R*_{es} hereafter) and, thus, estimated from water vapor isotope and leaf gas-exchange measurements (Cernusak et al., 2004; Farquhar et al., 2007). From these estimates of *R*_{es}, we can calculate the ^{18}O/^{16}O ratio of CO_{2} in full isotopic equilibrium with leaf water at the CA site (noted Δ_{ei} and expressed relative to *R*_{i}):

where α_{wc} denotes the temperature-dependent equilibrium isotopic fractionation between CO_{2} and water (Brenninkmeijer et al., 1983).

If we assume that CO_{2} is fully equilibrated with leaf water at the CA site, then ∆_{ci} = ∆_{ei} and we can estimate ε_{ci} from measurements of *R*_{es} and *R*_{A} using Equation 1 (this requires knowledge of ∆_{ia} = *R*_{i}/*R*_{a} − 1, which can be estimated from measurements of ∆_{A} alongside CO_{2} and water vapor fluxes; see “Materials and Methods”).

However, because the residence time of CO_{2} inside the mesophyll can be somewhat shorter than the time required for full isotopic equilibration with leaf water, Δ_{ci} differs from Δ_{ei}. The proportion of CO_{2} in isotopic equilibrium with leaf water can be defined as (Gillon and Yakir, 2000a, 2000b):

where Δ_{ci0} represents the value of Δ_{ci} in the absence of any CA activity (or, more correctly, of any CO_{2}-H_{2}O oxygen isotope exchange). The latter is usually derived using an approach similar to that described for ^{13}C photosynthetic discrimination (Gillon and Yakir, 2000b), assuming no isotope fractionation during carboxylation by PEPC (EC 4.1.1.31) or Rubisco or during respiration. The exact expression (see Appendix C, Eq. C28) shows that Δ_{ci0} depends on ε_{ci} and, thus, *p*_{CA}. This comes to assume that, despite the (putative) absence of CA activity, the carboxylation site coincides with the (true) CA site. This can be problematic, especially in C_{4} plants. However, in most cases, θ is expected to be close to unity and the exact knowledge of Δ_{ci0} becomes less critical.

In fact, the knowledge of Δ_{ci0} is only required to compute θ. Because Gillon and Yakir (2000a, 2000b) proposed an independent expression for θ (see below) in terms of the residence time τ_{res} (s) of CO_{2} within the leaf mesophyll and the CA-catalyzed CO_{2}-H_{2}O isotopic exchange rate *k*_{iso} (s^{−1}), they required knowledge of Δ_{ci0} to compute Δ_{ci} and, thus, ε_{ci} and *g*_{m}. Another approach, adopted by Farquhar and Lloyd (1993), provided a direct expression for ∆_{ci} in terms of CA activity and carboxylation and respiratory fluxes. This expression, combined with Equation 1, can be used to retrieve ε_{ci} and *g*_{m} without the need to estimate the degree of equilibration θ, as demonstrated in some follow-up applications (Flanagan et al., 1994; Williams et al., 1996). These two approaches are reviewed below.

### The Biochemical View

To derive an expression for θ, Gillon and Yakir (2000a, 2000b) revisited the work of Mills and Urey (1940), who showed that the ^{18}O/^{16}O ratio of CO_{2} in closed aqueous solutions rapidly follows an ordinary differential equation, which can be rewritten with the current notations as follows:

where *k*_{iso} (s^{−1}) is the CO_{2}-H_{2}O isotopic exchange rate. The leaf mesophyll is not a closed system, but Gillon and Yakir (2000b) assumed that Equation 4 would adequately describe the dynamics of Δ_{ci}. This is justified only if the CO_{2}-H_{2}O isotopic exchange rate is much greater than any C^{18}OO carboxylation flux, which is unlikely under high light intensity or for CA-deficient leaves. Despite these caveats, they proposed estimating θ (Eq. 3) by integrating Equation 4 between time *t* = 0 and *t* = τ_{res} and assuming that Δ_{ci0} precisely represents the value of Δ_{ci} at time *t* = 0 (Gillon and Yakir, 2000b):

This derivation is problematic, as it uses a non-steady-state formulation (integrated over the residence time τ_{res}) to describe steady-state gas-exchange dynamics. Additionally, stating that Δ_{ci0} precisely represents the value of Δ_{ci} at time *t* = 0 assumes that the leaf has been (initially) filled with unlabeled CO_{2}, which is not realistic even with a fluctuating environment, because CA activity continuously resets Δ_{ci}. Yet, Equation 5 has been used in several studies to link CA activity to ∆^{18}O data (Gillon and Yakir, 2000a, 2000b, 2001; Cousins et al., 2006a, 2006b, 2007). To do so, the exchange rate constant *k*_{iso} appearing in Equation 5 usually is taken as one-third of the CA-catalyzed CO_{2} hydration rate *k*_{h} and the residence time τ_{res} is taken as the ratio of the total amount of CO_{2} inside the leaf to the one-way flux of CO_{2} from the atmosphere into the leaf. However, the ratio *k*_{iso}/*k*_{h} equals one-third only in acidic conditions (see Appendix B, Eq. B8), and this definition of the residence time implicitly redefines the system boundaries to include not only the CA-containing leaf compartment but also other leaf compartments, including the intercellular air space. In this case, *k*_{iso} should be replaced by a more complex expression that depends not only on pH but also on the volumes of the gas and liquid phases and the (total) transfer coefficient between these two phases, including *g*_{m} (see Appendix B, Eq. B11). Finally, Equation 5 does not account for the competition between CO_{2} hydration and carboxylation or for the contribution of respiratory fluxes.

For all these reasons, we adopted another approach that leads to a direct relationship between Δ_{ci} and leaf CA activity at steady state while simultaneously accounting for competition between hydration and carboxylation and for respiratory fluxes. The model of Farquhar and Lloyd (1993) forms the basis of this new approach but is modified to account for the spatial separation of hydration and carboxylation sites, and their difference in leaf water isotopic composition, especially important in C_{4} species.

The CO_{2} gas-exchange rate *A* is the net result of CO_{2} hydration, carboxylation, and respiration rates (Fig. 1, biochemical view). At steady state, isotopic equilibrium may not be reached, even at the CA site, if CO_{2} carboxylation is large. Using the resistance scheme illustrated in Figure 1, and assuming no isotopic fractionation during carboxylation by Rubisco or PEPC, the isotope ratio of the net CO_{2} flux (see Appendix C for a derivation) is determined as follows:

where *R*_{eq} = *R*_{es}α_{wc}, ϕ_{r} is the fraction of respired CO_{2} not recycled by the chloroplast stroma (C_{3} plant) or not produced in the bundle sheath (C_{4} plant), and *F*_{r} is the ratio of the respiratory flux to the net flux: *F*_{r} = (*V*_{r} + 0.5*V*_{o})/*A* (all other symbols are defined in the legend of Fig. 1).

Several lines of evidence (see Appendix C and also Farquhar and Cernusak [2012]) indicate that respired CO_{2} should be fully equilibrated with mitochondrial water, suggesting *R*_{mi} = *R*_{eq} and , where *R*_{x} is the isotope ratio of the water in the bundle sheath cells. Following arguments in favor of a strong homogeneity of water isotope ratios between the cytosol and the chloroplast of single cells, we further assume that *R*_{m} and *R*_{c} should be closely related in C_{3} species and equal to the CO_{2} isotope ratio at the CA site (*R*_{CA}). In C_{4} species, we argue (see Appendix C) that *R*_{c} (the ^{18}O/^{16}O ratio of CO_{2} in the bundle sheath) should be closely related to *R*_{m}/α_{cb} ≈ *R*_{CA}/α_{cb}, where α_{cb} is the isotope fractionation between CO_{2} and bicarbonate (around 1.0095 at 25°C). With these simplifications, Equation 6 can be rewritten (see Appendix C):

where ρ = ρ_{i}(1 + ε_{ci})/ε_{ci}, ρ_{i} = *A*/(*k*_{CA}*p*_{i}), *k*_{CA} is the measured leaf CA activity rate [expressed in µmol(CO_{2}) m^{−2} s^{−1} Pa^{−1}], *a*_{cb} = α_{cb} − 1, and ∆_{eq} = (*R*_{es}/*R*_{x} − 1)α_{cw}*R*_{x}/*R*_{i} ≈ *R*_{es}/*R*_{x} – 1. Equation 7 can be combined with Equation 1 to eliminate ∆_{ci} and estimate ε_{ci} (then *p*_{CA} and *g*_{m}) from measurements of *k*_{CA}, Δ_{i}, Δ_{ei}, and water vapor and CO_{2} fluxes, provided that the respiratory terms (*F*_{r}, *V*_{r}/*A*, and ϕ_{r}) are known. Noting that *F*_{r} also depends on ε_{ci} (i.e. it can be expressed as a function of ε_{ci}, *V*_{r}/*A*, and Γ*/*C*_{i}, where *C*_{i} = *p*_{i}/*P* and Γ* is the CO_{2} compensation point in the absence of day respiration; see Eq. C17), this requires solving a quadratic (for C_{3} plants) or a cubic (for C_{4} plants) equation in ε_{ci} (Eqs. C18 and C23, respectively; see Appendix C for a full derivation). If respiratory terms are negligible (*F*_{r} = 0), then the solution for C_{3} plants is similar to that proposed by Farquhar and Lloyd (1993). However, if respiratory terms are not negligible, the situation is different because, here, we assume that CO_{2} respired by C_{3} leaves is in equilibrium with mitochondrial (and thus cytoplasmic) water, while Farquhar and Lloyd (1993) did not (see Appendix C).

In the following, we solve Equation 7 for ε_{ci} using published data sets of *k*_{CA}, Δ_{i}, Δ_{ei}, and water vapor and CO_{2} fluxes (Cousins et al., 2006b, 2007; Barbour et al., 2016). For this, we set ϕ_{r} = 0.5 and compute Γ* as a function of leaf temperature (Bernacchi et al., 2001). We then explore the possibility of using our new equation to estimate *g*_{m} in C_{3} and C_{4} species and estimate its sensitivity to the respiratory terms (*V*_{r}/*A*) or the water isotope gradients between mesophyll and bundle sheath cells.

For the sake of comparison with previous work, we also compute a degree of equilibration, as defined by Equation 3. For this, we estimated ∆_{ci0} by taking the limiting case of Equation 6 when *k*_{CA} tends to zero (i.e. *V*_{hm} = *V*_{hc} = 0) and assuming that, in the absence of CA activity, the respiratory isotope ratios were equal to *R*_{a}. Noting that *V*_{c}/*A* = 1 + *F*_{r}, this gives the following equation for both C_{3} and C_{4} plants:

where *R*_{c0} denotes *R*_{c} in the absence of CA activity. Combined with flux-gradient relationships such as that in Equation 1 (valid regardless of the CA activity), we obtain an expression for the ratio *R*_{c0}/*R*_{a} (see Appendix C, Eq. C28), from which we can derive ∆_{ci0} (Eq. C29) and thus θ (Eq. 3).

## RESULTS AND DISCUSSION

We first revisited the data from Cousins et al. (2006b), who measured online discrimination and the CA activity of C_{3} and C_{4} plants exposed to different light levels (see Table II in Cousins et al., 2006b).

We estimated the effect of assuming or not assuming full CO_{2}-H_{2}O equilibration and increasing the respiratory fraction (*V*_{r}/*A*) on the light response of *p*_{CA}/*p*_{a}, *g*_{m}, and θ in *F. bidentis* leaves (Fig. 2). We see that the assumption of full equilibration is almost valid at low light but, as incident light increases, the degree of equilibration decreases slowly (Fig. 2C), although not as sharply as the original θ values of Cousins et al. (2006b). This decrease in θ is slower when the respiratory fraction is high, as a consequence of the assumption that respired CO_{2} is fully equilibrated. More interestingly, the retrieved *g*_{m} responds very little to the increase in incident light, especially when compared with stomatal conductance (Fig. 2B). The new estimates of *g*_{m} also are much lower (around 0.4 mol m^{−2} s^{−1}) than the original value of 1 mol m^{−2} s^{−1} estimated by Cousins et al. (2006b) and only slightly higher (by around +15%) than the values we estimated assuming full equilibration (θ = 1).

At first sight, it may seem surprising that, even for the lowest light level, our estimates of *g*_{m} are much lower than the original estimate of Cousins et al. (2006b), despite the fact that, in both cases, full isotopic equilibration is reached (an assumption in the case of Cousins et al. [2006b] and a prediction in this study). This apparent contradiction arises from the ternary corrections. Cousins et al. (2006b) applied ternary corrections to estimate *p*_{i} but not to interpret C^{18}OO discrimination data, as was common practice at the time. Farquhar and Cernusak (2012) have since shown that such a practice can lead to erroneous *g*_{m} estimates. Indeed, when ternary corrections are only applied to compute *p*_{i}, then the solution of Equation 9 at low irradiance leads to the exact original *g*_{m} value (1 mol m^{−2} s^{−1}) but much lower *g*_{m} values with increasing light (Supplemental Fig. S1). On the other hand, not applying ternary corrections at all leads to *g*_{m} values almost identical to those shown in Figure 2 (Supplemental Fig. S2), a result also predicted by Farquhar and Cernusak (2012).

The same analysis also was performed on the tobacco leaf data sets of Cousins et al. (2006b), and similar results were obtained (Fig. 3). The degree of equilibration decreased slowly with an increase in incident light but not as sharply as in the original publication (Fig. 3C), while the new estimates of *g*_{m} were lower than estimated originally but slightly higher than the values obtained assuming full equilibration (+8%–20%, depending on irradiance) and with less sensitivity to light levels than stomatal conductance (Fig. 3B). Compared with the results shown in Figure 2, the sensitivity of *g*_{m} to the respiratory fraction also is much lower. This is because, for C_{3} species, *V*_{r}/*A* only affects *F*_{r} with little influence on ∆_{i} as long as ρ is small, while *V*_{r}/*A* appears in two other terms in the C^{18}OO discrimination model for C_{4} species (Eq. 7).

The above analysis demonstrates that, to explain the data from Cousins et al. (2006b), there is no need to evoke a spatial separation of the CA site and the evaporation site, nor an isotope heterogeneity of leaf water in the cytosol of mesophyll cells. By simply accounting for ternary corrections, competition between CO_{2} hydration and carboxylation, and the contribution of respiratory fluxes (Eq. 7), it is possible to reconcile the in vitro CA assays and the Δ^{18}O measurements.

Our new estimates of *g*_{m} are lower than previous estimates, even when *V*_{r}/*A* = 0 (Figs. 2 and 3). This is especially the case for *F. bidentis*, where *g*_{m} is only slightly higher than the maximum stomatal conductance for CO_{2} (*g*_{sc}; Fig. 2). As briefly explained above, this occurs because the estimation of *g*_{m} as originally performed did not account for ternary corrections when interpreting isotopic discrimination. The difference between original and revised *g*_{m} values is much lower when revisiting data sets where ternary corrections were fully accounted for, such as those from Barbour et al. (2016). In this case, our new estimates of *g*_{m} tend to agree well with the original estimates but show consistently higher values (typically around 20% and up to 50% or more in some cases) than those estimated assuming full equilibration, and the sensitivity of *g*_{m} and θ to the respiratory fraction *V*_{r}/*A* is again very small in C_{3} species and marginally small in C_{4} species (Fig. 4).

These results show that the degree of equilibration is expected to be near unity in all species (Fig. 4), and especially in C_{3} plants, partially justifying a posteriori the assumption made by Barbour et al. (2016). However, accounting for incomplete equilibration between CO_{2} and leaf water led to ∆^{18}O-derived *g*_{m} values that are significantly higher than those obtained assuming full isotopic equilibration (Figs. 2–4). Barbour et al. (2016) noticed that, in some C_{3} plants, the ∆^{18}O-derived *g*_{m} assuming full equilibration were sometimes of a magnitude similar to that of the *g*_{m} estimated from ∆^{13}C discrimination. This was the case most notably in mature wheat (*Triticum aestivum*) leaves and seemed incompatible with the idea that the CA site was located at the chloroplast surface and, thus, upstream of the carboxylation site. Here, we show that accounting for incomplete equilibration increases the difference between ∆^{13}C- and ∆^{18}O-derived *g*_{m} even in wheat (0.63 versus 0.75 mol m^{−2} s^{−1} for mature leaves). Furthermore, our modeling framework partly explains that the difference between ∆^{13}C- and ∆^{18}O-derived *g*_{m} should not be so large because the CA site is now defined as the mean location of CA activity (Eq. C14) rather than its outer limit, as defined originally by Gillon and Yakir (2000b).

Ubierna et al. (2017) showed that PEPC-derived *g*_{m} for C_{4} plants agreed well with Δ^{18}O-derived *g*_{m} assuming θ = 1. Their reported PEPC-derived *g*_{m} values for *Z. mays* and *Setaria viridis* agree well also with the Δ^{18}O-derived *g*_{m} reported by Barbour et al. (2016) assuming θ = 1, despite possible differences in plant treatments and growth conditions between the two studies. For *Z. mays* and *S. viridis*, Barbour et al. (2016) report *g*_{m} values of 0.5 and 1.1 mol m^{−2} s^{−1}, respectively, at around 30°C, which is slightly lower but in relatively good agreement with the PEPC-derived *g*_{m} estimates of Ubierna et al. (2017), of around 0.6 and 1.3 mol m^{−2} s^{−1}, respectively (see Fig. 2 of Ubierna et al. [2017]). Our reanalysis shows that accounting for competition between CO_{2} hydration and carboxylation would reconcile the two approaches even more, by leading to ∆^{18}O-derived *g*_{m} values of 0.55 to 0.65 and 1 to 1.3 mol m^{−2} s^{−1} for *S. viridis* and *Z. mays*, respectively (Fig. 4).

Another interesting data set to revisit is that of Cousins et al. (2007) on mutants of *Amaranthus edulis* that exhibited a reduced PEPC activity but a CA activity similar to that of the wild type. A reanalysis of their data set using Equation 7 is presented in Figure 5. In the original analysis, *g*_{m} was set to a constant value for all plants and the degree of equilibration was derived without fully accounting for ternary effects. This led to a rapid decrease in Δ^{18}O-derived θ in response to increasing PEPC activity (Fig. 5). This feature seemed in contradiction with the observed in vitro CA activities that were similar among the different PEPC mutants (*k*_{CA} = 60 ± 10 µmol m^{−2} s^{−1} Pa^{−1}). Reanalyzing their data set with Equation 7 led to quite different results, with a degree of equilibration much closer to unity, even in the wild type, and much smaller values of *g*_{m} that increased with PEPC activity (Fig. 5). Again, these new ∆^{18}O-derived *g*_{m} estimates are slightly higher (up to +20%), than those estimated using full equilibration (Fig. 5).

The results shown in Figure 5 also may help explain, at least qualitatively, the data from Stimler et al. (2011), who reported differences in Δ^{18}O-derived θ between C_{3} and C_{4} species, despite no difference in CA activity between the two plant groups (estimated for the first time simultaneously on the same leaves, using carbonyl sulfide [COS] gas-exchange measurements). To reconcile the COS-derived CA activities with the Δ^{18}O-derived θ values, Stimler et al. (2011) suggested that Equation 5 should be revisited. To explain the lower Δ^{18}O values of C_{4} plants relative to those of C_{3} plants, they used Equation 5 and hypothesized a reduction of *k*_{iso} of about 17% (Stimler et al., 2011). This reduction of *k*_{iso} was attributed to PEPC activity that would deplete the bicarbonate pool of C_{4} species to a point where it would affect CA activity (towards CO_{2} but not COS). Indeed, a depletion of bicarbonate would deplete *p*_{CA} because the ratio of CO_{2} to bicarbonate is fixed by pH, and this should lead to a decrease in the residence time of CO_{2} and thus θ, to some extent. However, as explained above, Equation 4 is ill designed to describe steady-state gas-exchange data. Our new formulation (Eq. 7), on the other hand, is more suitable because it explicitly accounts for the competition between hydration and carboxylation rates while satisfying the steady-state mass balance. The results shown in Figure 5C clearly demonstrate that differences in Δ^{18}O (from 207‰ in the homozygous mutant to 16‰ in the wild type) are compatible with nearly full isotopic equilibration (θ ≈ 1) or with undetectable changes in CA activity deduced from other gas-exchange techniques.

In fact, CA activity (*k*_{CA}) and the degree of equilibration (θ) are not intuitively related, because large changes in *k*_{CA} do not necessarily lead to large changes in θ (and *g*_{m}). This is demonstrated in Figure 6, which revisits data from Cousins et al. (2006b) on wild-type and CA-deficient *F. bidentis* plants grown (and measured) in ambient CO_{2}. Despite *k*_{CA} values as low as 5 µmol m^{−2} s^{−1} Pa^{−1} and ρ values above unity in some CA-deficient plants, the results using Equation 7 indicate that *g*_{m} remains relatively constant, with values of wild-type plants and CA-deficient mutants being similar (Fig. 6B). Again, these *g*_{m} values are higher than those estimated assuming full equilibration (Fig. 6). The degree of equilibration θ also stays relatively constant, between around 0.7 and 0.8 from low to high CA activity (Fig. 6C). In fact, in the data sets revisited here, the degree of equilibration θ will usually approach unity when ρ is below 0.01, irrespective of whether it is a C_{3} or C_{4} species (Fig. 7). That θ is below 0.9 in Figure 6 is primarily because ρ is not very low, even in the wild type (mean value, 0.064).

## CONCLUSION

All the results presented here indicate that ∆^{18}O-derived *g*_{m} values can be estimated robustly at steady state by considering the competition between CO_{2} hydration and carboxylation, which determines the incomplete CO_{2}-H_{2}O equilibration inside the leaf. Even though CO_{2} is in nearly full equilibration with leaf water in most cases, the newly derived *g*_{m} values are consistently higher (typically around 20% and up to 50% or more in some cases) than those estimated assuming full equilibration. However, the physical meaning of this Δ^{18}O-derived *g*_{m}, and its significance for CO_{2} assimilation, are still difficult to grasp, particularly for C_{3} plants that exhibit CA activity in different mesophyll compartments. For both C_{3} and C_{4} plants, the contribution of the respiratory fluxes to the overall net C^{18}OO discrimination (Fig. 1) complicates the classical view of *g*_{m} as a pure diffusional property of the leaf mesophyll, a problem that also arises when interpreting ∆^{13}C discrimination data (Tholen et al., 2012). The new model formulation presented in this study, by accounting for the compartmentalization of leaf water and CO_{2} hydration, carboxylation, and respiration sites, is an attempt to bring more physical meaning to this leaf parameter. However, the gas-exchange and biochemical views schematically presented in Figure 1 are still far from being fully reconciled. Clearly, a more explicit representation of CO_{2} and C^{18}OO transport in the mesophyll and their exchange in the different compartments (cytosol, chloroplasts, mitochondria, etc.), with an explicit representation of their respective volumes, enzymatic activities, and transfer resistances, is required to fully interpret Δ^{18}O (and Δ^{13}C) data in terms of the diffusional properties of the cell components.

## MATERIALS AND METHODS

### Literature Data

For the purpose of this study, three published data sets have been revisited (Cousins et al., 2006b, 2007; Barbour et al., 2016). These data sets have been selected because they were the ones that gathered measurements of CA activity (*k*_{CA}) using the ^{18}O-exchange method (expected to provide more meaningful CA activities in vivo; see next section), isotopic discrimination (∆_{A}), leaf water isotope composition (*R*_{es}), water vapor (*E*) and CO_{2} (*A*) fluxes, and stomatal (*g*_{sc}) and boundary layer (*g*_{bc}) conductances for CO_{2}.

Gas-exchange and isotope data were available for all individual measurements, except for the data sets of Cousins et al. (2006b, 2007), where separate values of *R*_{a} could not be retrieved and only mean values of *R*_{es}, already expressed relative to *R*_{a} (i.e. ∆_{ea} = *R*_{es}α_{wc}/*R*_{a} − 1), could be assigned from the published tables. In addition, for consistency with the values of Barbour et al. (2016), these mean values of ∆_{ea} from Cousins et al. (2006b, 2007) were corrected using a fractionation factor for the diffusion of water vapor in still air of 28‰ (Merlivat, 1978) instead of 32‰ (Cappa et al., 2003). Finally, in Barbour et al. (2016), *k*_{CA} values for *Gossypium hirsutum* (cotton), *Triticum aestivum* (wheat), and *Zea mays* (corn) were not reported and were assumed here to be equal to those of tobacco plants (cotton and wheat) or taken from Cousins et al. (2006b) for corn.

### Estimating in Vivo CO_{2} Hydration Rates *k*_{CA} from in Vitro CA Assays

In all the studies that we revisited, CA activity was estimated by measuring the rate of ^{18}O loss of a subsaturating, labeled C^{18}O_{2}-buffered solution (Silverman, 1973; Badger and Price, 1994). The uncatalyzed rate (*k*_{uncat,assay}) was first measured, and then leaf extracts were added to the solution to record the catalyzed rate (*k*_{cat,assay}). CA activity [in units of mol(CO_{2}) m^{−2} s^{−1} Pa^{−1}] was then converted to its expected in vivo value (von Caemmerer et al., 2004):

where *k*_{uncat,invivo} (s^{−1}) is the uncatalyzed CO_{2} hydration rate under the conditions in vivo (i.e. at physiological pH), *K*_{H} (mol m^{−3} Pa^{−1}) is the solubility of CO_{2} in water, *V*_{assay} (m^{3}) is the volume of the assay solution, and *S*_{leaf} (m^{2}) is the leaf area of the added leaf extracts in this volume. Compared with the pH method used in older studies (Gillon and Yakir, 2000a, 2000b, 2001), the CA assay using labeled CO_{2} is less sensitive to the buffer solution used (Hatch and Burnell, 1990). More importantly, because the CO_{2}-H_{2}O isotopic exchange rate is somewhat slower than the hydration rate (Mills and Urey, 1940), measurements can be performed routinely at 25°C and near physiological pH and CO_{2} concentrations, which is now the reason why this assay is preferred over the pH assay (von Caemmerer et al., 2004; Cousins et al., 2006b; Kodama et al., 2011; Studer et al., 2014; Barbour et al., 2016). A pH and CO_{2} concentration correction still needs to be applied, which is done using Equation 9. However, implicit to Equation 9 is the assumption that the catalyzed and uncatalyzed rates respond similarly to pH, so that *k*_{uncat,invivo}/*k*_{uncat,assay} equals *k*_{cat,invivo}/*k*_{cat,assay}, where *k*_{cat,invivo} would be the expected catalyzed rate in vivo (i.e. at physiological pH). According to Rowlett et al. (2002), the pH dependence of *k*_{cat} in wild-type *Arabidopsis thaliana* is well approximated by 1/(1 + 10^{7.2 −} * ^{pH}*). The pH response of

*k*

_{uncat}usually used for CA assays is

*k*

_{uncat}(

*pH*) = 0.038 + 6.22/10

^{11 –}

*(von Caemmerer et al., 2004). A modification of Equation 9 was then applied here:*

^{pH}where *k*_{CA,orig} is the original (reported) CA activity. For C_{3} plants, Equation 10 does not modify the reported CA activity because the compartment that contains the most CA is the chloroplast stroma, whose pH is very close to the pH of the assay, typically around 8 (von Caemmerer et al., 2004). On the other hand, in C_{4} plants, *pH*_{invivo} is expected to be close to the pH of the cytosol and, thus, more acidic, around 7.4. In this case, Equation 10 leads to CA activity levels of C_{4} plants that are lower by about 20% than those reported in the literature.

### Data Analysis

We define *g*_{tc} as follows: *g*_{tc} = 1/(1/*g*_{bc} + 1/*g*_{sc}). From *p*_{a}, *A*, *E*, and *g*_{tc}, we computed *p*_{i} according to von Caemmerer and Farquhar (1981):

where *t*′ = 0.5*E*/*g*_{tc} is the ternary correction factor.

From *p*_{i} and *p*_{a}, we computed ε_{ia} = *p*_{i}/(*p*_{a} − *p*_{i}). Assuming no ternary effect (*t*′ = 0), the CO_{2} isotope ratio in the intercellular air space, expressed relative to the ratio in the outside air, is derived as follows (see Appendix A for a derivation):

where represents the weighted-mean isotope fractionation factor during CO_{2} diffusion through the leaf boundary layer and the stomata. Including ternary effects (*t*′ ≠ 0) leads to (see Appendix A for a derivation):

where (Farquhar and Cernusak, 2012). From Δ_{ia}, we then computed the following:

and

Values of *p*_{i}, *k*_{CA}, ∆_{i}, ∆_{ia}, and ∆_{ei} were used to compute ε_{ci} using Equation C18 (C_{3} plants) or Equation C23 (C_{4} plants), from which we could compute *p*_{CA} = *p*_{i}ε_{ci}/(1 + ε_{ci}) and *g*_{m} = *AP*/(*p*_{i} − *p*_{CA}). We finally computed the ratio *R*_{CA0}/*R*_{a} (see Eq. C28), from which we derived ∆_{ci0} [=*R*_{CA0}/*R*_{a}/(1 + ∆_{ia}) − 1] and thus θ.

### Supplemental Data

The following supplemental materials are available.

**Supplemental Figure S1.**Ternary corrections applied to Figure 2.**Supplemental Figure S2.**No ternary correction applied to Figure 2 when computing both*p*_{i}and*p*_{CA}.**Supplemental Figure S3.**Isotope difference between CO_{2}in equilibrium with the water at the evaporation site and in the intercellular air space.**Supplemental Figure S4.**Differences between the intercellular CO_{2}in equilibrium with the water at the evaporation site and in the intercellular air space.

## Acknowledgments

We thank Margaret Barbour and Asaph Cousins for kindly gathering and sharing with us the raw data corresponding to the published work that we revisited in this article. We also thank the editor, Graham Farquhar, and the two reviewers, Lucas Cernusak and Nerea Ubierna, for their very constructive comments that helped us to greatly improve the final version of this article.

## APPENDIX A: TERNARY EFFECTS AND C^{18}OO PHOTOSYNTHETIC DISCRIMINATION

### Derivation of Equation 12

Neglecting ternary effects, the net CO_{2} flux into the leaf also can be expressed as *A*′ = *g*_{tc}/*P* (*p*_{a} – *p*_{i}) (with the prime indicating that ternary effects are neglected). Writing a similar equation for the C^{18}OO flux, the ratio *R*_{A} = ^{18}*A*′/*A*′ (we do not have a prime on *R*_{A} because it is a measured quantity that does not depend on whether ternary effects are accounted for or not) can be expressed as:

where denotes *R*_{i} when ternary effects are neglected. Using Δ_{A} = *R*_{a}/*R*_{A} − 1 and defining , Equation A1 then becomes:

which can be easily rearranged into Equation 12 in the main text.

### Derivation of Equation 13

If we now account for ternary effects, Equation A1 needs to be rewritten. As demonstrated by Farquhar and Cernusak (2012), this leads to (see their last equation on page 1223):

This can be rewritten as:

which easily leads to Equation 13 in the main text.

## APPENDIX B: DYNAMICS OF ^{18}O EXCHANGE DURING CO_{2} HYDRATION AND BICARBONATE DEHYDRATION IN CLOSED AND OPEN SYSTEMS

### Rationale

The dynamics of ^{18}O exchange between CO_{2} and water in closed solutions has been described previously (Mills and Urey, 1940; Uchikawa and Zeebe, 2012). However, because these studies were designed primarily to estimate the CA-catalyzed hydration rate by means of ^{18}O labelling techniques, kinetic and equilibrium isotopic effects were ignored as a first approximation (e.g. no distinction was made between hydration rate constants for CO_{2} and C^{18}OO). In this situation, the isotope ratio of dissolved CO_{2} at equilibrium with the surrounding water (*R*_{c,eq}) would simply equal that of the water (*R*_{w}). In practice, we know this is not the case, as an equilibrium fractionation of 1.0413 at 25°C exists between aqueous CO_{2} and water (Beck et al., 2005; Zeebe, 2007). A brief description of the kinetic isotope effects during CO_{2} hydration and bicarbonate dehydration is presented below.

### Kinetic Isotope Effects during CO_{2} Hydration and Bicarbonate Dehydration

Using the same notation as in previous studies (Mills and Urey, 1940; Uchikawa and Zeebe, 2012), that is, [66] for C^{16}O_{2} concentration, [68] for C^{16}O^{18}O concentration, [6] for H_{2}^{16}O concentration, [668] for H_{2}C^{16}O_{2}^{18}O concentration…, and neglecting doubly labeled species at natural abundance, only three hydration-dehydration reactions need to be considered:

The rates of change of [666], [68], and [668] are then:

The factor 2/3 in the middle equation comes from the fact that the ^{18}O in [668] also can be transferred to the water molecule, assuming that the three oxygen atoms in carbonic acid are distributed stochastically (but see Zeebe, 2014).

At equilibrium, these rates approach zero and we have:

where the subscript eq indicates that it is the equilibrium concentration. Noting that *R*_{c,eq} = [68]_{eq}/(2[66]_{eq}) and *R*_{w} = [8]/[6], taking the ratio of the second and third equalities in Equation B3 leads to:

Now, inserting Equation B4 and the second equality in Equation B3 into Equation B2 leads to:

where we have defined *R*_{c} = [68]/(2[66]), *R*_{b} = [668]/(3[666]), and:

where *R*_{b,eq} is the value of *R*_{b} at equilibrium. Following Uchikawa and Zeebe (2012), we will assume that carbonic acid and bicarbonate are isotopically inseparable and that their oxygen isotopic ratios are equal. In this case, we have:

According to Beck et al. (2005), at 25°C, this ratio is equal to 1.0413/1.0315 = 1.0095. Note that, as long as H_{2}CO_{3} and HCO_{3}^{−} (and H_{2}O and OH^{−}) are isotopically inseparable, Equation B5 remains valid even when CA activity is low and CO_{2} hydroxylation dominates ^{16}*k*_{h} (and ^{18}*k*_{h}…).

### Analysis of the Dynamics of the System

Equation B5 describes the rate of change of C^{18}O^{16}O and H_{2}C^{18}O^{16}O_{2} in a closed aqueous solution and can be used to estimate how fast it takes to reach isotopic steady state (i.e. *R*_{c} = *R*_{c,eq} and *R*_{b} = *R*_{b,eq}) in the case of a labeling experiment (i.e. a step change in ^{18}O in either the dissolved CO_{2} or bicarbonate pool). The time needed to recover steady state will be dictated by the dynamics of the coupled differential equations in Equation B5. The mathematical analysis of the dynamics of such coupled equations has been done elsewhere, in the case of multiply labeled species (more appropriate for highly enriched labeling) but ignoring kinetic and equilibrium isotopic effects (Mills and Urey, 1940; Silverman, 1982; Uchikawa and Zeebe, 2012). In this case, the coupled differential equations resemble Equation B5 but with atom fractions instead of isotope ratios and no fractionation factors (the α’s). The analysis of the dynamics of this coupled system shows that both *R*_{c} and *R*_{b} rapidly follow a single exponential decay function with a characteristic time scale τ given by (Mills and Urey, 1940; Silverman, 1982; Uchikawa and Zeebe, 2012):

where *S* = [H_{2}CO_{3}] + [HCO_{3}^{−}] + [CO_{3}^{2−}] and α_{cb} is provided by Equation B6. To a good approximation, we can assume that the ratio [CO_{2}]/*S* is close to its equilibrium value (this assumption is actually required to derive Eq. B8), and in this case, τ^{−1} (corresponding to *k*_{iso} in Eq. 4) is only a function of temperature and pH.

For pH < 4.5, the ratio [CO_{2}]/*S* becomes large and the square root term in Equation B8 can be approximated as α_{cb}[CO_{2}]/*S* + 1/3, so that τ^{−1} (=*k*_{iso}) approaches ^{18}*k*_{h}[*H*_{2}*O*]/3 at any temperature. For pH = 7.4, a more realistic value for the cytoplasm of mesophyll cells (Jenkins et al., 1989), τ^{−1} is much smaller and approaches 0.028^{18}*k*_{h}[*H*_{2}*O*] (i.e. it takes 10 times longer to reach the isotopic equilibrium at pH 7.4 than it takes at pH 4.5). This feature has been used to precisely measure ^{18}*k*_{h} and, thus, CA activity in vivo using ^{18}O labeling techniques and graphical estimation of the decay rate τ (Silverman, 1982). This is the technique used currently to measure leaf CA activity in vitro (Badger and Price, 1989; von Caemmerer et al., 2004), although for historical reasons (i.e. by analogy with the pH method), not one but two decay rates τ are measured, before (τ_{1}) and after (τ_{2}) the addition of leaf extracts in the reactor, and CA activity then is computed as (τ_{1}/τ_{2})*k*_{h,uncat} (see also Eq. 10), where *k*_{h,uncat} is the expected uncatalyzed hydration rate at 0.1 m ionic strength (von Caemmerer et al., 2004).

Mathematically, Equation B8 can be simplified to a good approximation to:

This is because from Equation B5, we can see that:

and if we further assume *dR*_{b}/*dt* ≈ *dR*_{c}/*dt*, then we end up with a first-order ordinary differential equation whose decay rate is given by Equation B9.

When the labeling is performed in vivo, the approximation of a uniform solution is not valid anymore. Gerster (1971a, 1971b) developed equations to include an extra gas phase where gaseous CO_{2} can exchange with the solution at a rate characterized by a transfer coefficient. In a leaf, this transfer coefficient is similar to a total leaf conductance and, therefore, includes *g*_{m}. Other authors also have considered a compartmentalized solution by means of a biological membrane (Silverman et al., 1981). In both cases, the mass balance of the different isotope species in each phase leads to a system of differential equations that resembles Equation B5 but with a third differential equation for the gas phase (or the second liquid compartment) and extra terms that involve the transfer coefficient between the gas and liquid phases (or through the biological membrane). Measurements of the decay rates of multiply labeled CO_{2} species then are required to estimate simultaneously the hydration rate and the transfer coefficient. In the case of the liquid-gas phase example, the decay rates for the C^{18}O_{2} fraction and the ^{18}O in the CO_{2} atom fraction are (Gerster, 1971a):

where *V*_{G} and *V*_{L} are the volumes of the gas and liquid phases, *B* is the dimensionless CO_{2} solubility, π_{t} (m s^{−1}) is the transfer coefficient between the gas and liquid phases, and *S*_{G} is the surface area of the gas-liquid interface. We can see easily that, in the absence of a gas phase (*V*_{G} = 0), Equation B11 simplifies to Equation B9 (with the notation *k*_{h} = ^{18}*k*_{h}[*H*_{2}*O*]). The introduction of a gas phase makes the system dynamics slower, to an extent that will increase with a larger volume of the gas phase *V*_{G} and a smaller liquid volume *V*_{L} and transfer coefficient π_{t}. Because the latter is expected to covary with total leaf conductance, which includes stomatal and mesophyll components, τ^{−1} = *k*_{iso} is a function not only of temperature and pH but also of *g*_{sc} and *g*_{m}.

### Application to Model the ^{18}O Discrimination during Leaf Photosynthesis

The rates of change of [68] and [668] (right-hand side of Equation B5) can be used to compute the *steady-state* mass balance of ^{18}O in CO_{2} and bicarbonate in folio. However, in this case, other CO_{2} fluxes (carboxylation, respiration, and atmosphere) competing with CA activity will maintain the CO_{2} and bicarbonate slightly out of isotopic equilibrium (i.e. *R*_{c} ≠ *R*_{c,eq} and *R*_{b} ≠ *R*_{b,eq}), even at steady state, and this degree of disequilibrium will vary depending on the competition between hydration and carboxylation (term ρ in Farquhar and Lloyd, 1993). This situation is described in Appendix C.

## APPENDIX C: DERIVING A NEW MODEL FOR C^{18}OO PHOTOSYNTHETIC DISCRIMINATION IN C_{3} AND C_{4} SPECIES

### Rationale

As mentioned already, the ^{18}O photosynthetic discrimination model proposed by Farquhar and Lloyd (1993; denoted FL93 hereafter) was developed for C_{3} plants only. It also neglects CA activity in the cytosol, and there is growing evidence that CA is present and rather abundant in the cytosol and plasmalemma of the mesophyll cells (Fabre et al., 2007). Additionally, the derivation of the FL93 model was never presented, which renders the identification of the different assumptions difficult. In the following, we will derive a new model for C^{18}OO discrimination during photosynthesis that is applicable to both C_{3} and C_{4} plants and takes into account CA activity in the cytosol.

### Mass Balance Equations

Let us consider the C flux diagram shown in Figure 1. At steady state, there is no accumulation of CO_{2} in the cytoplasm of mesophyll cells, so that the CO_{2} flux *A* entering the cytosol must balance the CO_{2} flux out:

Similarly, there is no accumulation of bicarbonate in the cytoplasm of mesophyll cells:

and no accumulation of CO_{2} in the chloroplasts of C_{3} plants or the bundle sheath cells of C_{4} plants:

where *V*_{p} is assumed to represent well the rate of CO_{2} that is released from C_{4} acids. The mass balance of bicarbonate in the chloroplasts of C_{3} plants leads to *V*_{hc} = *V*_{dc}, so that, for both C_{3} and C_{4} plants, the sum of Equations C1 to C3 gives *A* = *V*_{c} − 0.5*V*_{o} − *V*_{r}.

Similar mass balances can be written for the ^{18}O in CO_{2} and bicarbonate, where the net leaf photosynthetic uptake of C^{18}OO, ^{18}*A*, must balance all the C^{18}OO flux out of the cytoplasm of the mesophyll cells. Using Equation B5 above to explicate the hydration-dehydration terms, and writing *V*_{hm} = *k*_{hm}*C*_{m} and *V*_{dm} = *k*_{dm}*B*_{m}, this gives:

where ^{18}*k*_{hm} is the hydration rate for C^{18}OO (Zeebe, 2014), *a*_{ch} and *a*_{sh} are fractionation factors during CO_{2} diffusion across the chloroplast and bundle sheath cells, respectively, is the equilibrium oxygen isotope fractionation factor between CO_{2} and carbonic acid, and *R*_{m} and are the oxygen isotope composition of dissolved CO_{2} and bicarbonate in the cytoplasm of mesophyll cells. The factor 2 on the right-hand side comes from the fact that, by definition of the isotope ratios, [C^{18}OO] = 2[CO_{2}]*R*.

Similarly, the mass balance of HCO_{2}^{18}O^{−} in the cytoplasm of mesophyll cells gives (see Eq. B5):

where *R*_{w} is the oxygen isotope ratio of the water at the site of hydration in the cytoplasm of mesophyll cells (*R*_{w} ≈ *R*_{es}) and *b*_{PEPC} is the oxygen isotope fractionation factor during PEP carboxylation by PEPC.

For C_{3} plants, a similar equation can be derived for the mass balance of HCO_{2}^{18}O^{−} in the chloroplasts of mesophyll cells. If we assume that *R*_{w}, the isotope ratio of water at the hydration sites in the cytosol, represents well the isotope ratio of chloroplastic water, then the mass balance of HCO_{2}^{18}O^{−} in the chloroplasts of mesophyll cells gives: .

Finally, the budget of ^{18}O in CO_{2} in the chloroplasts of C_{3} plants or in the bundle sheath cells of C_{4} plants leads to:

where *b* is the isotope fractionation during Rubisco-catalyzed carboxylation and *R*_{p} corresponds to the isotope ratio of the CO_{2} released from C_{4} acids in the bundle sheath of C_{4} plants.

### Model Simplifications for C_{3} and C_{4} Plants

The value of *b* is unknown, but noting that the oxygen atoms of CO_{2} do not bind to the active sites of Rubisco, we will assume that oxygen isotope effects are small and *b* = 0 (Farquhar and Lloyd, 1993). In addition, in the net reaction HCO_{3}^{−} + PEP → CO_{2} + P_{i} + pyruvate, out of the three oxygens from the bicarbonate, one is lost to phosphate (P_{i}) and the other two are released as CO_{2} without binding to any of the active sites of the different enzymes involved (PEPC, MDH, NAD-ME, or NADP-ME). Therefore, we should not expect a strong oxygen isotope effect through this net reaction, and *b*_{PEPC} = 0 and seems a fair assumption. Then, combining Equations C4 to C6 leads to:

where we have defined (and kept the fractionation factor *b* for reasons explained below). Equation C7 corresponds to Equation 6 in the main text (because *R*_{A} = 0.5^{18}*A*/*A* and assuming *b* = 0).

The pH of mammalian mitochondria has been found to be rather alkaline, with a resting pH around 8 (Llopis et al., 1998), and it was suggested that the same situation occurred in plants (Tholen and Zhu, 2011). An alkaline pH favors high CA activity, whose *pK*_{a} is usually found around 7.2 (Rowlett et al., 2002), and the expression of CA in mitochondria also has been demonstrated, at least for C_{3} plants (Fabre et al., 2007) but also algae (Giordano et al., 2003). To our knowledge, the existence of mitochondrial CA in C_{4} plants has never been shown, but given the relatively small efflux of CO_{2} from the mitochondria, we will assume that respired CO_{2} is in full isotopic equilibrium with mitochondrial water in both mesophyll and bundle sheath cells and that mitochondrial water has the same isotopic composition as cytosolic water: and , where *R*_{x} is the oxygen isotope ratio of the water in the cytoplasm of bundle sheath cells. Farquhar and Cernusak (2012) revisited ^{18}O discrimination measurements on *Ricinius communis* performed in the dark (Cernusak et al., 2004) and concluded that respired CO_{2} by this plant was in full isotopic equilibrium with leaf water at the evaporation site. This result provides indirect evidence that *R*_{mi} = *R*_{eq} is a good approximation, although the presence of CA in the cytosol and the plasmalemma (Fabre et al., 2007) could be responsible for the reset of *R*_{mi} to *R*_{eq} during the diffusion of respired CO_{2} out of the leaf.

### Model Simplifications Specific to C_{3} Plants

A last simplification may arise regarding the distinction made between *R*_{m} and *R*_{c}. Could these isotope ratios be equal? In C_{4} plants, it seems quite unlikely, because they represent CO_{2} pools physically well separated between the mesophyll and the bundle sheath. On the other hand, given the rather small oxygen isotope fractionation by Rubisco (*b* ≈ 0) and CO_{2} diffusion (*a*_{w} ≈ 0.8‰), we could expect in C_{3} plants the CO_{2} isotope ratio in the cytosol and chloroplasts of individual mesophyll cells to be closely related, even if different from the surrounding water. If *R*_{m} = *R*_{c} in C_{3} plants, then Equation C7 simplifies even further and leads to:

which can be rearranged:

where *V*_{r} + 0.5*V*_{o} = *V*_{c}Γ/*C*_{c} and ρ_{c} = *V*_{c}/(^{18}*k*_{hm}*C*_{m} + ^{18}*k*_{hc}*C*_{c}). If we assume *R*_{mi} ≈ *R*_{eq}, then Equation C9 becomes:

where we have defined ρ* = ρ_{c}(1 − Γ/*C*_{c})/(1 + 3ρ_{c}Γ/*C*_{c}) and *b*′ = *b*/[1−Γ/*C*_{c}(1 + *b*)]. Dividing Equation C10 by *R*_{i} leads to:

By eliminating ∆_{ci} in Equation C11 using Equation 1 (thus defining the CA site such that *R*_{CA} = *R*_{c}) and neglecting second-order terms related to *b*′ (i.e. *b*∆_{i}, *ba*_{w},…), we obtain an equation that relates the observed variables ∆_{i} and ∆_{ei} to ε_{ci} and ρ*:

Equation C12 can be rearranged to express ∆_{i} as a function of ε_{ci}, ρ*, and ∆_{ei}:

If we assume Γ = 0, then ρ* = ρ_{c} and *b*′ = *b* and Equation C13 reduces to the one reported by Farquhar and Lloyd (1993), with the only differences that their Equation 34 contained an (obvious) typo in the expressions of ε_{ci} and 1 + ε_{ci} and that all variables here are expressed relative to the intracellular air space partial pressure (*p*_{i}) and isotope ratio (*R*_{i}) rather than to those in the outside air. Later publications (Flanagan et al., 1994) corrected the typo but introduced another one by writing 3ρ*ε_{ci} instead of 3ρ*(1 + ε_{ci}) in the denominator. Thus, we felt it important to give the exact expression here.

The similarity between Equation C13 and the formulation proposed by Farquhar and Lloyd (1993) is surprising because those authors had not made the distinction between *C*_{m} and *C*_{c}, nor had they accounted for CO_{2} hydration in the cytosol. An important implication is that ρ_{c} is not simply related to a single CO_{2} partial pressure anymore. If we define *p*_{CA} such that ^{18}*k*_{hm}*C*_{m} + ^{18}*k*_{hc}*C*_{c} = *k*_{CA}*p*_{CA}, then Equation C13 can be used to retrieve *p*_{CA} from online CO^{18}O discrimination measurements, but this will correspond to a partial pressure that lies between *PC*_{m} and *PC*_{c}, depending on the relative activity of cytosolic versus chloroplastic CA. Indeed, noting that *k*_{CA}, as estimated from the CA assay, should correspond to (^{18}*k*_{hm} + ^{18}*k*_{hc})/*P*, we should have:

In other words, the CA site is not defined here as the outer limit of CA activity (Gillon and Yakir, 2000b) but as the mean point of CA activity within the mesophyll cells. Given the more alkaline pH and higher CA concentration in the chloroplast stroma compared with the cytosol (pH 7.4 and [CA] around 0.1 mm for the cytosol and pH 8 and [CA] around 0.3 mm for the stroma; Tholen and Zhu, 2011), however, we should expect ^{18}*k*_{hc} >> ^{18}*k*_{hm}, *k*_{CA} ≈ ^{18}*k*_{hc}/*P*, and *p*_{CA} ≈ *PC*_{c}.

Another difference with the formulation of Farquhar and Lloyd (1993), and an extra complication, comes from the respiratory term. We argue here that *R*_{mi} should be closely related to *R*_{eq} rather than *R*_{c}. In contrast, in the original FL93 model that includes respiratory terms, *R*_{mi} was expressed as *R*_{c}(1 + Δ_{mc}), so that Equation C9 could simplify to an equation similar to Equation C13 with the correspondence ρ* → ρ_{c}(1 − Γ/*C*_{c}) and:

Here, we assume *R*_{mi} ≈ *R*_{eq} and ρ* keeps its original definition: ρ* = ρ_{c}(1 − Γ/*C*_{c})/(1 + 3ρ_{c}Γ/*C*_{c}). The latter also can be expressed as ρ/(1 + 3ρ*F*_{r}), where ρ = ρ_{c}(1 − Γ/*C*_{c}) = *A*/(*k*_{CA}*p*_{CA}) and *F*_{r} is the ratio of the respiratory flux to the net flux: *F*_{r} = (*V*_{r} + 0.5*V*_{o})/*A*. In other words, the respiratory terms tend to reduce ρ*, bringing the CO_{2} closer to full equilibration. With these notations and now assuming *b* = 0, Equation C11 becomes:

Noting that ρ = *A*/(*k*_{CA}*p*_{CA}) also can be reexpressed as ρ_{i}(1 + ε_{ci})/ε_{ci}, where ρ_{i} = *A*/(*k*_{CA}*p*_{i}), Equation C16 can be combined with Equation 1 to eliminate ∆_{ci} and estimate ε_{ci} (then *p*_{CA} and *g*_{m}) from measurements of *p*_{i}, ρ_{i}, *R*_{A}, and *R*_{es}. However, a complication arises from the fact that *F*_{r} depends on Γ/*C*_{c} that we do not know. Thus, we need to make assumptions on the compensation point Γ and the CO_{2} mixing ratio in the chloroplast *C*_{c}. As explained above, *p*_{CA} should relate closely to *PC*_{c} and, at least for the respiratory terms, which are only correction factors, we could assume that they are equal: *C*_{c} ≈ *p*_{CA}/*P* = *C*_{i}ε_{ci}/(1 + ε_{ci}). We can further split the compensation point into photorespiration (Γ*) and nonphotorespiration components, leading to:

Combining Equations C16 and C17 with Equation 1 is a bit tedious but leads to a cubic equation that degenerates into a quadratic equation, whose positive solution is ε_{ci}:

with:

and:

Equation C18 has two unknown parameters, the compensation point Γ* and the ratio *V*_{r}/*A*. The compensation point can be estimated from leaf temperature, using literature data (von Caemmerer et al., 2009), and a sensitivity analysis can be performed on *V*_{r}/*A*. Likely values for *V*_{r}/*A* should lie within the range 0.05 to 0.25, but we explored here a larger range, from zero to 0.6. From the values of ε_{ci} and *p*_{i}, we can calculate *p*_{CA} and then *g*_{m}.

### Summary of C_{3} Model Simplifications

To summarize, for C_{3} plants, (1) if the respiratory terms are known (*F*_{r} or at least *V*_{r}/*A*), (2) if the respired CO_{2} has an isotope ratio in equilibrium with leaf water at the evaporation site (*R*_{mi} ≈ *R*_{eq}), and (3) if the CO_{2} in the cytosol and the chloroplasts have similar isotope ratios (*R*_{m} ≈ *R*_{c}), then we can compute a *g*_{m} from ^{18}O discrimination measurements that will correspond to the CO_{2} transfer resistance between the intercellular air space (partial pressure *p*_{i}) and a location between the cytosol and the chloroplast (partial pressure *p*_{CA}). The definition of *p*_{CA} in C_{3} plants (Eq. C14) is such that it should correspond more closely to the CO_{2} partial pressure in the chloroplast (*p*_{CA} ≈ *PC*_{c}).

### Model Simplifications Specific to C_{4} plants (a Correction to this section has been posted, see at the end of Appendix C)

For C_{4} plants, the situation is somewhat simpler, because CA activity is located only in the mesophyll (*p*_{CA} = *PC*_{m}). On the other hand, the physical separation of the mesophyll and the bundle sheath cells makes it difficult to assume *R*_{m} = *R*_{c}. If *R*_{mi} = *R*_{eq} and in C_{4} plants, then Equation C7 can be rewritten:

The main CO_{2} source in the bundle sheath comes from the decarboxylation of the C_{4} acid with an isotope ratio (see discussion above). A possible approximation then would be to assume that *R*_{c} is close to this isotope ratio. Because the CA activity is high in the mesophyll cells, we also should expect the CO_{2} and bicarbonate in this compartment to be close to isotopic equilibrium: , where α_{cb} is the isotopic fractionation between CO_{2} and bicarbonate (around 1.0095 at 25°C; see Appendix B). In other words, *R*_{m}/α_{cb} seems a reasonable approximation for *R*_{c}. Equation C21 then can be simplified to:

where *a*_{cb} = α_{cb} − 1. Combining Equation C22 with Equations C17 and 1 is a bit tedious but leads to a quadratic equation that degenerates into a cubic equation in ε_{ci}:

with:

and where we have defined:

We can verify that, when ϕ_{r} = 0 and *a*_{cb} = 0, Equation C23 simplifies to Equation C18 (with *b* = 0), because then the expression for *R*_{A} is the same for C_{3} and C_{4} plants (i.e. Eqs. C10 and C22 are the same). In this study, we set ϕ_{r} = 0.5 and *a*_{cb} = 9.5‰ and made the approximation , where *R*_{trans} represents the isotope ratio of the transpired water vapor, a good proxy for the source (xylem) water (Song et al., 2015) and, thus, for bundle sheath water. When not available, *R*_{trans} was taken as the isotopic ratio of irrigation water. As was done for C_{3} plants, a sensitivity analysis was performed on *V*_{r}/*A*, with values in the range 0 to 0.6. From the values of ε_{ci} and *p*_{i}, we could calculate *p*_{CA} and then *g*_{m}.

In the majority (i.e. 86%) of the situations that we tested, the cubic equation (Eq. C23) has three real roots, and the most positive solution is always the plausible one (i.e. the only one greater than the value of ε_{ci} computed assuming full equilibrium, the other two solutions being close to zero or negative). However, we also found combinations (i.e. 14%) where Equation C23 has only one real and two complex solutions. In these situations, the single real solution is taken, except in 2% of the cases (i.e. three individual *Z. mays* leaves out of 147 measurements) where this single real solution was unrealistic (i.e. negative ε_{ci} and *g*_{m}), so that no solution could be found.

These problematic situations may arise because of uncertainties in CA activity measurements. Indeed, because CA activity measurements are all performed on leaf extracts, it is possible that they are not fully representative of the in vivo activity at the time of the leaf C^{18}OO discrimination measurements. In fact, the CA activity of corn leaves was not measured by Barbour et al. (2016), and we had to estimate its value from the literature. We set *k*_{CA} to 34 µmol m^{−2} s^{−1} Pa^{−1} according to Cousins et al. (2006b) but noticed also that Studer et al. (2014) had measured CA activity in *Z. mays* about 2 to 3 times higher. We thus explored the effect of an increase of *k*_{CA} on the estimation of ε_{ci} from Equation C23. We found that, for the problematic cases mentioned above, a 6.5-fold increase of *k*_{CA} was required to start finding a plausible solution to Equation C23. To our knowledge, a *k*_{CA} value of 220 µmol m^{−2} s^{−1} Pa^{−1} has never been reported, especially in C_{4} plants. Therefore, the uncertainty on *k*_{CA} cannot be the only reason why Equation C23 sometimes has no solution.

The problem also may have theoretical origins. For example, the assumption that the air in the substomatal cavity is saturated in water vapor was challenged recently by Cernusak et al. (2018), who collected field data where the δ^{18}O of CO_{2} in the intercellular air space did not lie between the δ^{18}O of CO_{2} in the air and that in equilibrium with the evaporation site (i.e. ∆_{ei} < 0 while ∆_{ia} ≥ 0). They explained this behavior by letting the air in the substomatal cavity be subsaturated in water vapor (which implies recalculating the stomatal conductance *g*_{sw} and the CO_{2} partial pressure *p*_{i} from the leaf gas-exchange data) and finding the substomatal vapor pressure that led to ∆_{ci} = ∆_{ei} (i.e. full equilibrium). In all the data sets that are revisited here, the situation described by Cernusak et al. (2018) was not found (i.e. we always had ∆_{ei} > 0 and ∆_{ia} ≥ 0; Supplemental Fig. S3). We also recalculated the vapor pressure deficit that would be needed to reach full equilibrium (∆_{ci} = ∆_{ei}) and did not find departures of more than 0.3%, leading to absolute changes in *p*_{i}/*P* and in *g*_{sw} of less than 1.5 µmol mol^{−1} and 8 mmol m^{−2} s^{−1} (Supplemental Fig. S4). Thus, subsaturated air in the substomatal cavity does not seem to be the reason why sometimes Equation C23 does not have a plausible solution.

Another uncertain parameter is leaf temperature. Leaf temperature is commonly monitored using a fine-wire thermocouple appraised against the leaf lower surface. However, because the contact between the thermocouple junction and the leaf surface is not perfect and there is heat conduction by thermocouple wires, the leaf temperature reading is a mixture between leaf and air temperatures and, therefore, overestimates leaf temperature when the leaf is transpiring. We thus explored the effect of an overestimation of leaf temperature on the retrieval of ε_{ci} from Equation C23. This required recalculating the stomatal conductance *g*_{sw} and the CO_{2} partial pressure *p*_{i} from the leaf gas-exchange data and the isotopic composition of leaf water at the evaporation site (*R*_{es}). We found that a 1K overestimation of leaf temperature was enough to always find a plausible solution to Equation C23, leading to higher stomatal conductance (sometimes up to 50 mmol m^{−2} s^{−1}) and lower *g*_{m} (typically by 200–300 mmol m^{−2} s^{−1}). Because an overestimation of leaf temperature by 1K is very possible, we recommend systematically performing a sensitivity analysis of the solution to leaf temperature (and maybe also *k*_{CA}) in order to determine the robustness of the results.

### Summary of C_{4} Model Simplifications

To summarize, for C_{4} plants, (1) if the respiratory terms (ϕ_{r} and *V*_{r}/*A*) are known, (2) if respired CO_{2} is in isotopic equilibrium with mitochondrial water, (3) if mitochondrial water has the isotopic composition of the evaporation site in the mesophyll (*R*_{mi} ≈ *R*_{eq}) and the transpired vapor in the bundle sheath (), and (4) if the CO_{2} in the bundle sheath cytosol is in isotopic equilibrium with the bicarbonate in the mesophyll (*R*_{c} ≈ *R*_{m}/α_{cb}), then we can compute a *g*_{m} from ^{18}O discrimination measurements that will correspond to the CO_{2} transfer resistance between the intercellular air space (partial pressure *p*_{i}) and the CO_{2} hydration site in the mesophyll cytosol (partial pressure *p*_{CA} ≈ *PC*_{m}).

### Limiting Case in the Absence of CA Activity

As explained in the main text, in the limiting case where *k*_{CA} tends to zero, the isotope ratio at the carboxylation site (*R*_{c0}) becomes very simply related to *R*_{A} and *R*_{a} (Eq. 8). However, because *R*_{A} is a measured quantity, we cannot use it to estimate a hypothetical *R*_{c0} corresponding to the situation when *k*_{CA} would tend to zero. Thus, we need to find an expression for *R*_{c0} independent of *R*_{A}.

By combining the two flux-gradient relationships that led to Equation 1, we can easily show that, whether *k*_{CA} tends to zero or not, we also have:

where *R*_{i0} and *R*_{CA0} denote *R*_{i} and *R*_{CA} in absence of CA activity. Using flux-gradient relationships between *p*_{i} and *p*_{a}, while accounting for ternary effects in the gas phase, Farquhar and Cernusak (2012) also showed that (see their Eq. 14 on page 1223):

Like Equation C26, this equation is valid irrespective of the level of CA activity. By inserting Equation 8 into Equation C26 and assuming that *R*_{CA0} = *R*_{c0} in the absence of CA activity, we obtain an expression for *R*_{i0}*p*_{i} that we can inject into Equation C27 together with Equation 8 to obtain an expression that relates *R*_{CA0}/*R*_{a} to *p*_{a}, *p*_{i}, and *p*_{CA}:

From this expression, Δ_{ci0} is computed as:

If we further assume that *F*_{r} = 0, we obtain the same expression as Ubierna et al. (2017) in their Equation 14.

Note that *F*_{r} and *R*_{CA0}/*R*_{a} are computed using the value of *p*_{CA} deduced from ^{18}O discrimination measurements (i.e. in the presence of CA activity). This is problematic, especially in C_{4} plants, where the carboxylation and CA sites are physically well separated. This demonstrates the limited meaningfulness of the degree of equilibration θ, as defined by Equation 3.

### Correction to Appendix C (February 2019)

Note that a correction to Appendix C, including a new derivation for C_{4} species, is available as a new Supplemental file on the Plant Physiology website. In particular Eqs. (C23) and (C24) have been updated. As shown in this new Supplemental file, the results of this new derivation do not change the conclusions of the paper.

## APPENDIX D: LIST OF SYMBOLS AND ACRONYMS

## Footnotes

↵1 This work was funded by the Agence Nationale de la Recherche (award no. ANR-13-BS06-0005-01 [project ORCA]) and the European Union’s Seventh Framework Programme (FP7/2007-2013; grant agreements nos. 338264 [project SOLCA], 289582 [project 3to4], and 618105 [ERA-Net Plus project MODCARBOSTRESS]).

↵[OPEN] Articles can be viewed without a subscription.

- Received July 26, 2017.
- Accepted July 26, 2018.
- Published August 13, 2018.