Estimating Mesophyll Conductance from Measurements of C¹⁸OO Photosynthetic Discrimination and Carbonic Anhydrase Activity^,

A new model based on the relationship between C¹⁸OO photosynthetic discrimination and carbonic anhydrase activity estimates mesophyll conductance in C₃ and C₄ plants.

Abstract

Carbonic anhydrase (CA) activity in leaves catalyzes the ¹⁸O exchange between CO₂ and water during photosynthesis. This feature has been used to estimate the mesophyll conductance to CO₂ (g_m) from measurements of online C¹⁸OO photosynthetic discrimination (∆¹⁸O). Based on CA assays on leaf extracts, it has been argued that CO₂ in mesophyll cells should be in isotopic equilibrium with water in most C₃ species as well as many C₄ dicot species. However, this seems incompatible with ∆¹⁸O data that would indicate a much lower degree of equilibration, especially in C₄ plants under high light intensity. This apparent contradiction is resolved here using a new model of C₃ and C₄ photosynthetic discrimination that includes competition between CO₂ hydration and carboxylation and the contribution of respiratory fluxes. This new modeling framework is used to revisit previously published data sets on C₃ and C₄ species, including CA-deficient plants. We conclude that (1) newly ∆¹⁸O-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 ∆¹⁸O-derived g_m can be achieved in both C₃ and C₄ plants.

Carbonic anhydrases (CAs) are a group of zinc metalloenzymes that catalyze the interconversion of CO₂ into bicarbonate with great efficiency (Moroney et al., 2001; Rowlett, 2010). In the mesophyll cells of C₃ 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 C₃ plants was first assumed to be required to provide CO₂ 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 C₄ plants, CA is most abundant in the cytosol (Badger and Price, 1994), primarily because it is needed to increase the supply of bicarbonate to phosphoenolpyruvate 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-CO₂ environments (Studer et al., 2014).

Irrespective of the exact functional role of CA in plants, CA catalyzes the ¹⁸O exchange between CO₂ and water, so that CO₂ is commonly assumed to be near isotopic equilibrium with water in CA-containing mesophyll compartments. Consequently, CO₂ partial pressure at the site of CA activity inside the mesophyll cells (p_CA) can be derived from online ¹⁸O photosynthetic discrimination (Δ¹⁸O) measurements, provided that the ¹⁸O/¹⁶O ratio of leaf water at this CA site can be estimated (Gillon and Yakir, 2000b). However, sometimes CO₂ might not be in complete equilibration with water, especially when CO₂ uptake rates are high, as this would result in relatively brief residence times for CO₂ molecules inside the mesophyll. This led Gillon and Yakir (2000b) to propose a formulation for Δ¹⁸O that incorporates the degree of CO₂-H₂O isotopic equilibrium, θ (0 ≤ θ ≤ 1). Online Δ¹⁸O measurements and estimates of θ from in vitro CA assays indicated that, for three C₃ species, p_CA always lay somewhere between the CO₂ 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 ¹³C photosynthetic discrimination (Δ¹³C) measurements (Gillon and Yakir, 2000b). This finding was in line with the hypothesis that the outer limit of CA activity in C₃ 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 C₃ plants but also in C₄ plants, which generally exhibited lower θ values than C₃ 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 Δ¹⁸O measurements performed under low-light conditions (i.e. when the residence time of CO₂ 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 Δ¹⁸O 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 Δ¹⁸O-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 Δ¹⁸O-derived θ estimates were hypothesized to arise from a spatial mismatch between the CA site and the evaporation site in C₄ 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 CO₂ concentrations close to those experienced in folio, seem to indicate that CO₂ 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 Δ¹⁸O 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 C₄ plants using in vitro maximal phosphoenolpyruvate (PEP) carboxylation rate measurements seem to support Δ¹⁸O-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 Δ¹⁸O for both C₃ and C₄ plants. This new model explicitly accounts for the competition between CO₂ hydration and carboxylation, providing the possibility for incomplete CO₂-H₂O equilibration to occur inside the leaf. In addition, the new model accounts for the physical separation between mesophyll and bundle sheath cells in C₄ 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 Δ¹⁸O and how it varies between C₃ and C₄ species. A steady-state formulation of Δ¹⁸O by C₃ plants that includes the competition between carboxylation and CA-catalyzed CO₂ hydration was proposed already by Farquhar and Lloyd (1993). This formulation constitutes the basis of our derivation that we extended to C₃ and C₄ 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 CO₂ and C¹⁸OO 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 C₃ and C₄ species, including CA-deficient mutants, and illustrate how to reconcile in vitro CA assays with online Δ¹⁸O measurements while, at the same time, estimating g_m from Δ¹⁸O data.

THEORY

The Gas-Exchange View

Throughout this article, we will assume that CO₂ or C¹⁸OO 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 CO₂ flux can be written as the product of a conductance g_m for CO₂ diffusion from the intercellular air space to the CA site and the CO₂ 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 CO₂ partial pressures in the intercellular air space and at the CA site, respectively. Similarly, the net C¹⁸OO flux can be defined as , where a_w is the fractionation factor during CO₂ dissolution and diffusion from the substomatal cavity to the CA site and R_i and R_CA represent the ¹⁸O/¹⁶O ratios of CO₂ 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:

Figure 1.

Resistance scheme of CO₂ and C¹⁸OO fluxes in C₃ and C₄ plants. From a gas-exchange point of view, the net CO₂ photosynthetic rate in the leaf (A) can be seen as driven by the CO₂ gradient between the intercellular air space (partial pressure p_i, isotope ratio R_i) and the leaf interior that, for C¹⁸OO, should correspond to the outer limit of CA activity (partial pressure p_CA, isotope ratio R_CA). From a biochemical point of view, we distinguish C₃ and C₄ photosynthetic pathways. For C₃ plants, the net CO₂ flux A is feeding CO₂ entirely to the cytoplasm of mesophyll cells, while photorespiration (V_o) and mitochondrial respiration (V_r) are feeding CO₂ to the cytoplasm only partially (fraction ϕ_r, isotope ratio R_mi) and the other fraction is recycled directly by the chloroplast. For C₄ plants, A also is feeding CO₂ entirely to the cytoplasm of mesophyll cells, but Rubisco-related photorespiration occurs only in the bundle sheath cells and mitochondrial respiration occurs in both mesophyll cells (fraction ϕ_r, isotope ratio R_mi) and bundle sheath cells (isotope ratio R′_mi). The CO₂ in the cytoplasm of mesophyll cells (mixing ratio C_m, isotope ratio R_m) can be hydrated (rate V_hm), and the bicarbonate (concentration B_m, isotope ratio R′_m) can be dehydrated (rate V_dm). In C₄ plants, bicarbonate also is consumed through PEPC activity (rate V_p). In C₃ plants, the CO₂ in the chloroplasts (mixing ratio C_c, isotope ratio R_c) can be hydrated (rate V_hc) or consumed by Rubisco (rate V_c), and the bicarbonate (concentration B_c, isotope ratio R′_c) can be dehydrated (rate V_dc). In C₄ plants, the CO₂ in the bundle sheath cells can only be consumed by Rubisco (rate V_c). The exact correspondence between p_CA and C_m or C_c will vary between C₃ and C₄ plants (see text), and the associated Δ¹⁸O-derived mesophyll conductance to CO₂ (g_m) is not simply a transfer resistance but also may incorporate a biochemical component.

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 ¹⁸O/¹⁶O ratio of the net CO₂ flux (=0.5¹⁸A/A); that is, ∆_i represents ∆¹⁸O, expressed relative to R_i and not relative to the ¹⁸O/¹⁶O ratio of the CO₂ in the air surrounding the leaf (R_a).

Our current theoretical understanding of the C¹⁸OO photosynthetic discrimination has been drawn on the assumption that the CA site is located in a leaf water compartment with a homogenous ¹⁸O/¹⁶O 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 ¹⁸O/¹⁶O ratio of the water in the CA-containing compartment can be approximated as the ¹⁸O/¹⁶O 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 ¹⁸O/¹⁶O ratio of CO₂ 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 CO₂ and water (Brenninkmeijer et al., 1983).

If we assume that CO₂ 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 CO₂ and water vapor fluxes; see “Materials and Methods”).

However, because the residence time of CO₂ 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 CO₂ 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 CO₂-H₂O oxygen isotope exchange). The latter is usually derived using an approach similar to that described for ¹³C 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 C₄ 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 CO₂ within the leaf mesophyll and the CA-catalyzed CO₂-H₂O 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 ¹⁸O/¹⁶O ratio of CO₂ 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 CO₂-H₂O 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 CO₂-H₂O isotopic exchange rate is much greater than any C¹⁸OO 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 CO₂, 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 ∆¹⁸O 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 CO₂ hydration rate k_h and the residence time τ_res is taken as the ratio of the total amount of CO₂ inside the leaf to the one-way flux of CO₂ 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 CO₂ 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 C₄ species.

The CO₂ gas-exchange rate A is the net result of CO₂ hydration, carboxylation, and respiration rates (Fig. 1, biochemical view). At steady state, isotopic equilibrium may not be reached, even at the CA site, if CO₂ 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 CO₂ flux (see Appendix C for a derivation) is determined as follows:

where R_eq = R_esα_wc, ϕ_r is the fraction of respired CO₂ not recycled by the chloroplast stroma (C₃ plant) or not produced in the bundle sheath (C₄ plant), and F_r is the ratio of the respiratory flux to the net flux: F_r = (V_r + 0.5V_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 CO₂ 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 C₃ species and equal to the CO₂ isotope ratio at the CA site (R_CA). In C₄ species, we argue (see Appendix C) that R_c (the ¹⁸O/¹⁶O ratio of CO₂ 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 CO₂ 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_CAp_i), k_CA is the measured leaf CA activity rate [expressed in µmol(CO₂) m⁻² s⁻¹ Pa⁻¹], a_cb = α_cb − 1, and ∆_eq = (R_es/R_x − 1)α_cwR_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 CO₂ 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 CO₂ compensation point in the absence of day respiration; see Eq. C17), this requires solving a quadratic (for C₃ plants) or a cubic (for C₄ 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 C₃ 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 CO₂ respired by C₃ 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 CO₂ 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 C₃ and C₄ 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 C₃ and C₄ 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 C₃ and C₄ plants exposed to different light levels (see Table II in Cousins et al., 2006b).

We estimated the effect of assuming or not assuming full CO₂-H₂O 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 CO₂ 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).

Figure 2.

Light response of gas-exchange parameters in F. bidentis leaves exposed to increasing levels of photosynthetic photon flux density (PPFD). A, CO₂ partial pressure ratio p_CA/p_a. B, ∆¹⁸O-derived mesophyll conductance (g_m). C, ∆¹⁸O-derived degree of isotopic equilibrium (θ). D, Ratio ρ = A/(k_CAp_CA). Data were taken from Cousins et al. (2006b). Original and revised values, with three different values of the respiratory fraction V_r/A, or assuming full equilibration, are shown. The CO₂ partial pressure ratio p_i/p_a and the stomatal conductance to CO₂ (g_sc) also are shown, in A and B, respectively.

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 C¹⁸OO 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 C₃ 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 C¹⁸OO discrimination model for C₄ species (Eq. 7).

Figure 3.

Light response of gas-exchange parameters in tobacco leaves exposed to increasing levels of photosynthetic photon flux density (PPFD). A, CO₂ partial pressure ratio p_CA/p_a. B, ∆¹⁸O-derived mesophyll conductance (g_m). C, ∆¹⁸O-derived degree of isotopic equilibrium (θ). D, Ratio ρ = A/(k_CAp_CA). Data were taken from Cousins et al. (2006b). Original and revised values, with three different values of the respiratory fraction V_r/A, or assuming full equilibration, are shown. The CO₂ partial pressure ratio p_i/p_a and the stomatal conductance to CO₂ (g_sc) also are shown, in A and B, respectively.

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 CO₂ hydration and carboxylation, and the contribution of respiratory fluxes (Eq. 7), it is possible to reconcile the in vitro CA assays and the Δ¹⁸O 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 CO₂ (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 C₃ species and marginally small in C₄ species (Fig. 4).

Figure 4.

Degree of isotopic equilibrium (θ) and mesophyll conductance (g_m) for three C₃ (left) and three C₄ (right) plants studied by Barbour et al. (2016). Original and revised values, with three different values of the respiratory fraction V_r/A, or assuming full equilibration, are shown. CA activity for Gossypium hirsutum (cotton) and Triticum aestivum (wheat) was assumed to be equal to that of tobacco, and CA activity for Z. mays (corn) was taken from Cousins et al. (2006b). For cotton, wheat, and corn, only data for mature leaves are shown here. For corn, one individual data point did not lead to a plausible solution to Equation C23 (i.e. negative ε_ci and g_m) and, thus, was discarded when computing the mean value. For consistency, we also discarded this individual data point when computing the mean g_m value corresponding to full isotopic equilibrium. Had it not been discarded, we would have obtained the same g_m value obtained by Barbour et al. (2016), as is the case for the other species.

These results show that the degree of equilibration is expected to be near unity in all species (Fig. 4), and especially in C₃ plants, partially justifying a posteriori the assumption made by Barbour et al. (2016). However, accounting for incomplete equilibration between CO₂ and leaf water led to ∆¹⁸O-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 C₃ plants, the ∆¹⁸O-derived g_m assuming full equilibration were sometimes of a magnitude similar to that of the g_m estimated from ∆¹³C 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 ∆¹³C- and ∆¹⁸O-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 ∆¹³C- and ∆¹⁸O-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 C₄ plants agreed well with Δ¹⁸O-derived g_m assuming θ = 1. Their reported PEPC-derived g_m values for Z. mays and Setaria viridis agree well also with the Δ¹⁸O-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 CO₂ hydration and carboxylation would reconcile the two approaches even more, by leading to ∆¹⁸O-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 Δ¹⁸O-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 ∆¹⁸O-derived g_m estimates are slightly higher (up to +20%), than those estimated using full equilibration (Fig. 5).

Figure 5.

Effects of PEPC activity (k_PEPC) on gas-exchange parameters in wild-type (WT) and heterozygous (Pp) and homozygous (pp) PEPC-deficient A. edulis plants grown in elevated (0.98 kPa) CO₂. A, CO₂ partial pressure ratio p_CA/p_a. B, ∆¹⁸O-derived mesophyll conductance (g_m). C, ∆¹⁸O-derived degree of isotopic equilibrium (θ). D, Ratio ρ = A/(k_CAp_CA). Data were taken from Cousins et al. (2007). Original and revised values, with three different values of the respiratory fraction V_r/A, or assuming full equilibration, are shown. In C, numbers in parentheses indicate isotopic discrimination (∆¹⁸O).

The results shown in Figure 5 also may help explain, at least qualitatively, the data from Stimler et al. (2011), who reported differences in Δ¹⁸O-derived θ between C₃ and C₄ 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 Δ¹⁸O-derived θ values, Stimler et al. (2011) suggested that Equation 5 should be revisited. To explain the lower Δ¹⁸O values of C₄ plants relative to those of C₃ 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 C₄ species to a point where it would affect CA activity (towards CO₂ but not COS). Indeed, a depletion of bicarbonate would deplete p_CA because the ratio of CO₂ to bicarbonate is fixed by pH, and this should lead to a decrease in the residence time of CO₂ 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 Δ¹⁸O (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 CO₂. 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 C₃ or C₄ 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).

Figure 6.

Effects of leaf CA activity (k_CA) on gas-exchange parameters in different wild-type and CA-deficient F. bidentis leaves grown and measured at ambient CO₂ concentrations. A, CO₂ partial pressure ratio p_CA/p_a. B, ∆¹⁸O-derived mesophyll conductance (g_m). C, ∆¹⁸O-derived degree of isotopic equilibrium (θ). D, Ratio ρ = A/(k_CAp_CA). Data were taken from Cousins et al. (2006b). Values, with three different values of the respiratory fraction V_r/A, or assuming full equilibration, are shown. Dashed lines indicate second-order polynomial fits to the individual data points.

Figure 7.

Degree of equilibration (θ) as a function of the ratio of net photosynthesis to CA hydration rate for the different experiments revisited in this study (Figs. 2 and 3, light response; Fig. 4, interspecies differences; Fig. 5, PEPC response; and Fig. 6, CA response) and the remaining data set of Barbour et al. (2016) testing the effect of leaf age on mesophyll conductance. Only values for V_r/A = 0 are shown for clarity. Increasing V_r/A tends to lift the curve up, with all θ values above 0.6 for V_r/A = 0.6.

CONCLUSION

All the results presented here indicate that ∆¹⁸O-derived g_m values can be estimated robustly at steady state by considering the competition between CO₂ hydration and carboxylation, which determines the incomplete CO₂-H₂O equilibration inside the leaf. Even though CO₂ 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 Δ¹⁸O-derived g_m, and its significance for CO₂ assimilation, are still difficult to grasp, particularly for C₃ plants that exhibit CA activity in different mesophyll compartments. For both C₃ and C₄ plants, the contribution of the respiratory fluxes to the overall net C¹⁸OO 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 ∆¹³C discrimination data (Tholen et al., 2012). The new model formulation presented in this study, by accounting for the compartmentalization of leaf water and CO₂ 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 CO₂ and C¹⁸OO 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 Δ¹⁸O (and Δ¹³C) 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 ¹⁸O-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 CO₂ (A) fluxes, and stomatal (g_sc) and boundary layer (g_bc) conductances for CO₂.

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_esα_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 CO₂ 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 ¹⁸O loss of a subsaturating, labeled C¹⁸O₂-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(CO₂) 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 CO₂ hydration rate under the conditions in vivo (i.e. at physiological pH), K_H (mol m⁻³ Pa⁻¹) is the solubility of CO₂ in water, V_assay (m³) is the volume of the assay solution, and S_leaf (m²) 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 CO₂ is less sensitive to the buffer solution used (Hatch and Burnell, 1990). More importantly, because the CO₂-H₂O 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 CO₂ 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 CO₂ 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 – pH} (von Caemmerer et al., 2004). A modification of Equation 9 was then applied here:

where k_CA,orig is the original (reported) CA activity. For C₃ 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 C₄ 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 C₄ 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′ = 0.5E/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 CO₂ 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 CO₂ 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 (C₃ plants) or Equation C23 (C₄ plants), from which we could compute p_CA = p_iε_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 CO₂ in equilibrium with the water at the evaporation site and in the intercellular air space.

• Supplemental Figure S4. Differences between the intercellular CO₂ in equilibrium with the water at the evaporation site and in the intercellular air space.

APPENDIX A: TERNARY EFFECTS AND C¹⁸OO PHOTOSYNTHETIC DISCRIMINATION

Derivation of Equation 12

Neglecting ternary effects, the net CO₂ 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 C¹⁸OO flux, the ratio R_A = ¹⁸A′/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 ¹⁸O EXCHANGE DURING CO₂ HYDRATION AND BICARBONATE DEHYDRATION IN CLOSED AND OPEN SYSTEMS

Rationale

The dynamics of ¹⁸O exchange between CO₂ 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 ¹⁸O labelling techniques, kinetic and equilibrium isotopic effects were ignored as a first approximation (e.g. no distinction was made between hydration rate constants for CO₂ and C¹⁸OO). In this situation, the isotope ratio of dissolved CO₂ 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 CO₂ and water (Beck et al., 2005; Zeebe, 2007). A brief description of the kinetic isotope effects during CO₂ hydration and bicarbonate dehydration is presented below.

Kinetic Isotope Effects during CO₂ Hydration and Bicarbonate Dehydration

Using the same notation as in previous studies (Mills and Urey, 1940; Uchikawa and Zeebe, 2012), that is, [66] for C¹⁶O₂ concentration, [68] for C¹⁶O¹⁸O concentration, [6] for H₂¹⁶O concentration, [668] for H₂C¹⁶O₂¹⁸O 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 ¹⁸O 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 H₂CO₃ and HCO₃⁻ (and H₂O and OH⁻) are isotopically inseparable, Equation B5 remains valid even when CA activity is low and CO₂ hydroxylation dominates ¹⁶k_h (and ¹⁸k_h…).

Analysis of the Dynamics of the System

Equation B5 describes the rate of change of C¹⁸O¹⁶O and H₂C¹⁸O¹⁶O₂ 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 ¹⁸O in either the dissolved CO₂ 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 = [H₂CO₃] + [HCO₃⁻] + [CO₃²⁻] and α_cb is provided by Equation B6. To a good approximation, we can assume that the ratio [CO₂]/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 [CO₂]/S becomes large and the square root term in Equation B8 can be approximated as α_cb[CO₂]/S + 1/3, so that τ⁻¹ (=k_iso) approaches ¹⁸k_h[H₂O]/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 0.028¹⁸k_h[H₂O] (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 ¹⁸k_h and, thus, CA activity in vivo using ¹⁸O 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 (τ₁) and after (τ₂) the addition of leaf extracts in the reactor, and CA activity then is computed as (τ₁/τ₂)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 CO₂ 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 CO₂ 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 C¹⁸O₂ fraction and the ¹⁸O in the CO₂ atom fraction are (Gerster, 1971a):

where V_G and V_L are the volumes of the gas and liquid phases, B is the dimensionless CO₂ 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 = ¹⁸k_h[H₂O]). 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 ¹⁸O 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 ¹⁸O in CO₂ and bicarbonate in folio. However, in this case, other CO₂ fluxes (carboxylation, respiration, and atmosphere) competing with CA activity will maintain the CO₂ 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 C¹⁸OO PHOTOSYNTHETIC DISCRIMINATION IN C₃ AND C₄ SPECIES

Rationale

As mentioned already, the ¹⁸O photosynthetic discrimination model proposed by Farquhar and Lloyd (1993; denoted FL93 hereafter) was developed for C₃ 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 C¹⁸OO discrimination during photosynthesis that is applicable to both C₃ and C₄ 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 CO₂ in the cytoplasm of mesophyll cells, so that the CO₂ flux A entering the cytosol must balance the CO₂ flux out:

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

and no accumulation of CO₂ in the chloroplasts of C₃ plants or the bundle sheath cells of C₄ plants:

where V_p is assumed to represent well the rate of CO₂ that is released from C₄ acids. The mass balance of bicarbonate in the chloroplasts of C₃ plants leads to V_hc = V_dc, so that, for both C₃ and C₄ plants, the sum of Equations C1 to C3 gives A = V_c − 0.5V_o − V_r.

Similar mass balances can be written for the ¹⁸O in CO₂ and bicarbonate, where the net leaf photosynthetic uptake of C¹⁸OO, ¹⁸A, must balance all the C¹⁸OO flux out of the cytoplasm of the mesophyll cells. Using Equation B5 above to explicate the hydration-dehydration terms, and writing V_hm = k_hmC_m and V_dm = k_dmB_m, this gives:

where ¹⁸k_hm is the hydration rate for C¹⁸OO (Zeebe, 2014), a_ch and a_sh are fractionation factors during CO₂ diffusion across the chloroplast and bundle sheath cells, respectively, is the equilibrium oxygen isotope fractionation factor between CO₂ and carbonic acid, and R_m and are the oxygen isotope composition of dissolved CO₂ 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, [C¹⁸OO] = 2[CO₂]R.

Similarly, the mass balance of HCO₂¹⁸O⁻ 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 C₃ plants, a similar equation can be derived for the mass balance of HCO₂¹⁸O⁻ 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 HCO₂¹⁸O⁻ in the chloroplasts of mesophyll cells gives: .

Finally, the budget of ¹⁸O in CO₂ in the chloroplasts of C₃ plants or in the bundle sheath cells of C₄ plants leads to:

where b is the isotope fractionation during Rubisco-catalyzed carboxylation and R_p corresponds to the isotope ratio of the CO₂ released from C₄ acids in the bundle sheath of C₄ plants.

Model Simplifications for C₃ and C₄ Plants

The value of b is unknown, but noting that the oxygen atoms of CO₂ 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 HCO₃⁻ + PEP → CO₂ + 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 CO₂ 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 = 0.5¹⁸A/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 C₃ plants (Fabre et al., 2007) but also algae (Giordano et al., 2003). To our knowledge, the existence of mitochondrial CA in C₄ plants has never been shown, but given the relatively small efflux of CO₂ from the mitochondria, we will assume that respired CO₂ 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 ¹⁸O discrimination measurements on Ricinius communis performed in the dark (Cernusak et al., 2004) and concluded that respired CO₂ 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 CO₂ out of the leaf.

Model Simplifications Specific to C₃ Plants

A last simplification may arise regarding the distinction made between R_m and R_c. Could these isotope ratios be equal? In C₄ plants, it seems quite unlikely, because they represent CO₂ 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 CO₂ diffusion (a_w ≈ 0.8‰), we could expect in C₃ plants the CO₂ 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 C₃ plants, then Equation C7 simplifies even further and leads to:

which can be rearranged:

where V_r + 0.5V_o = V_cΓ/C_c and ρ_c = V_c/(¹⁸k_hmC_m + ¹⁸k_hcC_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_cΓ/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 CO₂ hydration in the cytosol. An important implication is that ρ_c is not simply related to a single CO₂ partial pressure anymore. If we define p_CA such that ¹⁸k_hmC_m + ¹⁸k_hcC_c = k_CAp_CA, then Equation C13 can be used to retrieve p_CA from online CO¹⁸O 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 (¹⁸k_hm + ¹⁸k_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 ¹⁸k_hc >> ¹⁸k_hm, k_CA ≈ ¹⁸k_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_cΓ/C_c). The latter also can be expressed as ρ/(1 + 3ρF_r), where ρ = ρ_c(1 − Γ/C_c) = A/(k_CAp_CA) and F_r is the ratio of the respiratory flux to the net flux: F_r = (V_r + 0.5V_o)/A. In other words, the respiratory terms tend to reduce ρ*, bringing the CO₂ closer to full equilibration. With these notations and now assuming b = 0, Equation C11 becomes:

Noting that ρ = A/(k_CAp_CA) also can be reexpressed as ρ_i(1 + ε_ci)/ε_ci, where ρ_i = A/(k_CAp_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 CO₂ 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_iε_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 C₃ Model Simplifications

To summarize, for C₃ plants, (1) if the respiratory terms are known (F_r or at least V_r/A), (2) if the respired CO₂ has an isotope ratio in equilibrium with leaf water at the evaporation site (R_mi ≈ R_eq), and (3) if the CO₂ in the cytosol and the chloroplasts have similar isotope ratios (R_m ≈ R_c), then we can compute a g_m from ¹⁸O discrimination measurements that will correspond to the CO₂ 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 C₃ plants (Eq. C14) is such that it should correspond more closely to the CO₂ partial pressure in the chloroplast (p_CA ≈ PC_c).

Model Simplifications Specific to C₄ plants (a Correction to this section has been posted, see at the end of Appendix C)

For C₄ 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 C₄ plants, then Equation C7 can be rewritten:

The main CO₂ source in the bundle sheath comes from the decarboxylation of the C₄ 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 CO₂ and bicarbonate in this compartment to be close to isotopic equilibrium: , where α_cb is the isotopic fractionation between CO₂ 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 C₃ and C₄ 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 C₃ 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 C¹⁸OO 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 C₄ 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 δ¹⁸O of CO₂ in the intercellular air space did not lie between the δ¹⁸O of CO₂ 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 CO₂ 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 CO₂ 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 C₄ Model Simplifications

To summarize, for C₄ plants, (1) if the respiratory terms (ϕ_r and V_r/A) are known, (2) if respired CO₂ 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 CO₂ 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 ¹⁸O discrimination measurements that will correspond to the CO₂ transfer resistance between the intercellular air space (partial pressure p_i) and the CO₂ 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_i0p_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 ¹⁸O discrimination measurements (i.e. in the presence of CA activity). This is problematic, especially in C₄ 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 C₄ 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

Table D1. List of symbols.

Symbol	Definition (and First Appearance in the Text)	Unit
	Isotope fractionation factor during CO2 diffusion through the leaf boundary layer and the stomata (Eq. 12)	‰
ach, ash	Isotope fractionation factor during CO2 diffusion across the chloroplast (ach; C3 plant) or the bundle sheath cells (ash; C4 plant) (Eq. C4)	‰
acb	Isotope fractionation factor between CO2 and bicarbonate at equilibrium (i.e. acb = αcb − 1) (Eq. 7)	‰
aw	Isotope fractionation factor during CO2 dissolution and diffusion from the substomatal cavity to the CA site (Eq. 1)	‰
A, 18A	Net total CO2 and C18OO flux (Eq. 1)	mol m−2 s−1
b, bPEPC	Isotope fractionation factor during Rubisco and PEPC carboxylation (Appendix C)	‰
B	Dimensionless solubility of CO2 in water (i.e. 8.314KHT, where T is temperature) (Eq. B11)	m3 m−3
Cc, Bc	CO2 and bicarbonate concentrations at the Rubisco site (chloroplast stroma in C3 plants, bundle sheath in C4 plants) (Fig. 1)	mol mol−1
Cm, Bm	CO2 and bicarbonate concentrations in the cytoplasm of mesophyll cells (Fig. 1)	mol mol−1
E	Transpiration rate (Eq. 11)	mol m−2 s−1
Fr	Ratio of respiratory to net CO2 flux [i.e. (Vr + 0.5Vo)/A] (Eq. 6)	Unitless
gbc, gsc, gtc	Boundary layer, stomatal, and total conductance to CO2 (Eq. 11)	mol m−2 s−1
gch, gsh	Conductance to CO2 from the CA site to the Rubisco site (i.e. across the chloroplast [gch; C3 plant] or the bundle sheath cells [gsh; C4 plant])	mol m−2 s−1
gm	Mesophyll conductance to CO2 from the intercellular air space to the CA site (Fig. 1)	mol m−2 s−1
kCA	Leaf CA activity rate (Eqs. 7 and 9)	mol m−2 s−1 Pa−1
kCA,orig	Leaf CA activity rate, before applying the pH correction (Eq. 10)	mol m−2 s−1 Pa−1
kcat,assay, kuncat,assay	CA-catalyzed and uncatalyzed CO2 hydration rates under CA assay conditions (Eq. 9)	s−1
18kd, 16kd	H2C18OO2 and H2C16O3 dehydration rates (Eq. B1)	s−1
KH	Solubility of CO2 in water (Eq. 9)	mol m−3 Pa−1
kh, 16kh, 18kh, 18k′h	CO2 hydration rate (introduction, after Eq. 5) and CO2-H2O, C18OO-H2O, and CO2-H218O reaction rates (Eq. B1)	s−1
18khm, 18khc	C18OO-H2O reaction rate in the mesophyll and the chloroplast, respectively (Eqs. C4 and C6)	s−1
kiso	CO2-H2O isotopic exchange rate (Eq. 4)	s−1
kuncat,invivo	Uncatalyzed CO2 hydration rate under in vivo conditions (Eq. 9)	s−1
P	Atmospheric pressure (Fig. 1)	Pa
pa, pi, pCA	CO2 partial pressure in outside air, intercellular air space, and at the CA site (Fig. 1)	Pa
pHassay, pHinvivo	pH of the assay solution and of the leaf CA-containing compartment (Eq. 10)	Unitless
RA	Isotope ratio of net CO2 flux (=0.518A/A)	Unitless
R′i	Isotope ratio of CO2 in intercellular air space when ternary effects are neglected (Eq. A1)	Unitless
Ra, Ri, RCA	Isotope ratio of CO2 in outside air, intercellular air space, and at the CA site (Fig. 1)	Unitless
Rb	Isotope ratio of bicarbonate (Eq. B5)	Unitless
Rb,eq	Isotope ratio of bicarbonate in equilibrium with water (Eq. B5)	Unitless
Rc, R′c	Isotope ratio of CO2 and bicarbonate at the Rubisco site (Fig. 1)	Unitless
Rc	Isotope ratio of CO2 (Eq. B5)	Unitless
Rc,eq	Isotope ratio of CO2 in equilibrium with water (i.e. Rwαwc) (Eq. B4)	Unitless
Req	Isotope ratio of CO2 in equilibrium with water at the evaporation site (i.e. Resαwc) (Eq. 6)	Unitless
R′eq	Isotope ratio of CO2 in equilibrium with water in bundle sheath cells (i.e. Rxαwc) (Eq. 6)	Unitless
Res	Isotope ratio of water at the evaporation site (Eq. 2)	Unitless
Ri0, RCA0, Rc0	Ri, RCA, and Rc in the absence of CA activity (Eqs. 8 and C26)	Unitless
Rm, R′m	Isotope ratio of CO2 and bicarbonate in the cytoplasm of mesophyll cells (Fig. 1)	Unitless
Rmi, R′mi	Isotope ratio of CO2 produced in mitochondria of mesophyll and bundle sheath cells (Fig. 1)	Unitless
Rp	Isotope ratio of CO2 released from C4 acids in bundle sheath cells (Eq. C6)	Unitless
Rtrans	Isotope ratio of transpired water vapor (Appendix C)	Unitless
Rx	Isotope ratio of water in bundle sheath cells (Eq. 6)	Unitless
Rw	Isotope ratio of water (Eq. B4)	Unitless
S	Total bicarbonate species (i.e. [H2CO3] + [HCO3−] + [CO32−]) (Eq. B8)	mol m−3
SG	Surface area of the gas-liquid interface (Eq. B11)	m2
Sleaf	Leaf area used for the CA assay (Eq. 9)	m2
t	Time (Eq. 4)	s
t′	Ternary correction factor (Eq. 11)	Unitless
ta	Ternary correction factor for CO2 isotopes (Eq. 13)	Unitless
Vassay	Volume of the CA assay solution (Eq. 9)	m3
Vc	Rubisco carboxylase activity rate in the chloroplast stroma of C3 plants or bundle sheath cells of C4 plants (Fig. 1)	mol m−2 s−1
VG, VL	Volumes of the gas and liquid phases (Eq. B11)	m3
Vhc, Vdc	CO2 hydration and dehydration rates in the chloroplast stroma of C3 plants (Fig. 1)	mol m−2 s−1
Vhm, Vdm	CO2 hydration and dehydration rates in the cytoplasm of mesophyll cells (Fig. 1)	mol m−2 s−1
Vo, Vr	Photorespiration and mitochondrial respiration rates (Fig. 1)	mol m−2 s−1
Vp	PEPC activity rates in the cytoplasm of C4 mesophyll cells (and CO2 release rate from C4 acid in bundle sheath cells) (Fig. 1)	mol m−2 s−1
,αcw	Isotope fractionation between CO2 and water at equilibrium (i.e. Rc,eq/Rw) (Eqs. 2 and B4)	Unitless
, αcb	Isotope fractionation between CO2 and bicarbonate at equilibrium (i.e. Rc,eq/Rb,eq) (Eq. B6)	Unitless
Γ*	CO2 compensation point in the absence of mitochondrial respiration (Appendix C)	mol mol−1
∆A	Photosynthetic C18OO discrimination (i.e. Ra/RA − 1) (Eq. 12)	‰
∆ci	Isotope ratio of CO2 at the CA site, expressed relative to that in the intercellular air space (i.e. RCA/Ri − 1) (Eq. 1)	‰
∆ci0	∆ci in the absence of CA activity (Eq. 3)	‰
∆ea	Isotope ratio of CO2 in equilibrium with the evaporation site, expressed relative to that in air (i.e. Req/Ra − 1) (Eq. 11)	‰
∆ei	Isotope ratio of CO2 in equilibrium with the evaporation site, expressed relative to that in the intercellular air space (i.e. Req/Ri − 1) (Eq. 2)	‰
∆eq	Isotope ratio of the evaporation site, expressed relative to that of bundle sheath cell water (i.e. Res/Rx − 1) (Eq. 7)	‰
∆i	Photosynthetic C18OO discrimination, expressed relative to the isotope ratio of CO2 in the intercellular air space (i.e. Ri/RA − 1) (Eq. 1)	‰
∆ia	Isotope ratio of CO2 in the intercellular air space, expressed relative to that in the outside air (i.e. Ri/Ra − 1) (Eq. A4)	‰
∆′ia	∆ia without ternary corrections (Eq. 11)	‰
∆mc	Isotope fractionation factor between respired CO2 and CO2 at the Rubisco carboxylation site (i.e. Rmi/Rc − 1) (Appendix C)	‰
εci	pCA/(pi – pCA) (Eq. 1)	Unitless
εia	pi/(pa – pi) (Eq. 11)	Unitless
θ	Degree of isotopic equilibration (Eq. 3)	Unitless
πt	Transfer coefficient between the gas and liquid phases (Eq. B11)	m s−1
ρ	Ratio of net CO2 flux to CA activity [i.e. A/(kCApCA)] (Eq. 7)	Unitless
ρi	Ratio of net CO2 flux to kCApi (Eq. 7)	Unitless
τ	Time scale of CO2-H2O isotopic exchange dynamics (Eq. B8)	s
τ′	Decay rate of 18O in the CO2 atom fraction (Eq. B11)	s
τ1, τ2	C18O2 decay rates before and after leaf extract addition during the CA assay (Appendix B)	s
τres	Residence time of CO2 inside the leaf mesophyll (Eq. 5)	s
ϕr	Fraction of respired CO2 not recycled by the chloroplast stroma (C3 plant) or not produced in the bundle sheath (C4 plant) (Fig. 1)	Unitless

Table D2. List of acronyms.

Acronym	Definition
CA	Carbonic anhydrase
PEPC	Phosphoenolpyruvate carboxylase
PPFD	Photosynthetic photon flux density