- © 2019 American Society of Plant Biologists. All Rights Reserved.

## Abstract

Theoretical models of photosynthetic isotopic discrimination of CO_{2} (^{13}C and ^{18}O) are commonly used to estimate mesophyll conductance (*g*_{m}). This requires making simplifying assumptions and assigning parameter values so that *g*_{m} can be solved for as the residual term. Uncertainties in *g*_{m} estimation occur due to measurement noise and assumptions not holding, including parameter uncertainty and model parametrization. Uncertainties in the ^{13}C model have been explored previously, but there has been little testing undertaken to determine the reliability of *g*_{m} estimates from the ^{18}O model (*g*_{m18}). In this study, we exploited the action of carbonic anhydrase in equilibrating CO_{2} with leaf water and manipulated the observed photosynthetic discrimination (Δ^{18}O) by changing the oxygen isotopic composition of the source gas CO_{2} and water vapor. We developed a two-source δ^{18}O method, whereby two measurements of Δ^{18}O were obtained for a leaf with its gas-exchange characteristics otherwise unchanged. Measurements were performed in broad bean (*Vicia faba*) and Algerian oak (*Quercus canariensis*) in response to light and vapor pressure deficit. Despite manipulating the Δ^{18}O by over 100‰, in most cases we observed consistency in the calculated *g*_{m18}, providing confidence in the measurements and model theory. Where there were differences in *g*_{m18} estimates between source-gas measurements, we explored uncertainty associated with two model assumptions (the isotopic composition of water at the sites of CO_{2}-water exchange, and the humidity of the leaf internal airspace) and found evidence for both. Finally, we provide experimental guidelines to minimize the sensitivity of *g*_{m18} estimates to measurement errors. The two-source δ^{18}O method offers a flexible tool for model parameterization and provides an opportunity to refine our understanding of leaf water and CO_{2} fluxes.

Investigation of mesophyll conductance (*g*_{m}) by the isotope method has rapidly increased in use over the past 20 years with advances in laser technology and the capacity to measure isotope fluxes in real time. In C_{3} species, *g*_{m} can be estimated by combining gas-exchange data with measurements of online ^{13}C-photosynthetic discrimination (Δ^{13}C; Evans et al., 1986) and ^{18}O-photosynthetic discrimination (Δ^{18}O; Gillon and Yakir, 2000; Barbour et al., 2016) of CO_{2}. While the ^{13}C-isotope method is only precise for C_{3} photosynthesis, the ^{18}O-isotope method can be applied to both C_{3} and C_{4} photosynthetic types. Additionally, for C_{3} species, the ^{18}O-isotope method has the potential, when combined with measurements of Δ^{13}C, to validate model assumptions between the two approaches if the diffusion pathways are considered congruent, or if not, to provide more information on the distribution of conductances in the liquid phase between wall and chloroplast membranes.

Despite wide application of the isotope theory, as *g*_{m} cannot be directly measured, model validation is nearly impossible, and this has raised concerns of artifacts in the interpretation of isotopic estimates of *g*_{m}. Unlike variable stomatal conductance, which we can attribute to changes in pore aperture, mechanisms to explain observations of variable *g*_{m} in response to short-term environmental perturbation remain largely unknown. Instead, the field has relied on reproducibility and comparative approaches to confirm patterns of *g*_{m}, including, for example, the variable J method (Fabre et al., 2007; Flexas et al., 2007), physically based anatomical models (Tosens et al., 2012; Tomás et al., 2013), and systems of equations to constrain solutions (Hanson et al., 2016; Ubierna et al., 2017). Discrepancies between model predictions and observations, for example, led Wingate et al. (2007) to reformulate fractionation associated with respiration in the ^{13}C model to account for disequilibrium. The ^{18}O-isotope method has been used much less than the ^{13}C-isotope method for *g*_{m} estimation and, importantly, there has been no testing undertaken to determine the robustness of estimates by this method. Therefore, we sought to develop a technique to test the robustness of *g*_{m} estimation by the ^{18}O-isotope method (*g*_{m18}).

Theory describing how photosynthesis modifies the δ^{18}O of CO_{2} in air passing over a leaf was developed by Farquhar and Lloyd (1993). In comparison with carbon, there is thought to be no or very little enzymatic fractionation associated with fixation of ^{18}O-CO_{2} by rubisco. Instead, the apparent ^{18}O discrimination during photosynthesis arises due to equilibrium fractionation when oxygen isotopes of CO_{2} exchange with leaf water during the interconversion with bicarbonate. This reaction is fast when facilitated by the enzyme carbonic anhydrase (CA [turnover rate per subunit in the order of 10^{5} to 10^{6} s^{−1}]; Rowlett et al., 1994). Therefore, complete or almost complete isotopic equilibrium between CO_{2} and water is expected, although deviations can occur (Ogée et al., 2018). It is uncertain where this reaction occurs because, in C_{3} plants, CAs have been observed in the chloroplast, cytosol, mitochondria, and plasma membrane (Fabre et al., 2007). Initially, Gillon and Yakir (2000) suggested that the final sites of equilibration were the cytosol/chloroplast interfaces. However, a recent δ^{18}O model by Ogée et al. (2018) recognizes that CA sites are likely to be a mean point of CA activity within the mesophyll cells. Therefore, the [CO_{2}] relevant for Δ^{18}O is denoted *C*_{m}, with the understanding that, in C_{3} plants, this can occur at any place from the plasma membrane to within the chloroplast, and in C_{4} plants, anywhere within the mesophyll. The apparent ^{18}O discrimination during photosynthesis is largely dependent on the δ^{18}O of the CO_{2} supplying the leaf and the δ^{18}O of the leaf water. As such, there is no unique Δ^{18}O for a given *C*_{m}/*C*_{a}. Thus, unlike in the photosynthetic discrimination of ^{13}C, there is capacity to manipulate the observed Δ^{18}O experimentally without impacting leaf physiological processes. In what follows, we introduce our use of this feature to validate *g*_{m18} estimates.

Using the ^{18}O photosynthetic discrimination model, *g*_{m18} (mol m^{−2} s^{−1} bar^{−1}) can be calculated as (see Supplemental Material S1 for derivation and “Appendix” for a list of symbols):

where *A* (μmol m^{−2} s^{−1}) is the assimilation rate, *P* (bar) is the total atmospheric pressure, and *C*_{i} (μmol mol^{−1}) is the CO_{2} concentration in the intercellular spaces, and where a subscripted δ (‰) refers to the ^{18}O isotope composition of the CO_{2} in equilibrium (δ_{ce}) with leaf water at the sites of evaporation (δ_{w_e}), CO_{2} that is being assimilated (δ_{A}), and CO_{2} in the intercellular leaf spaces (δ_{i}). The oxygen isotope effect during CO_{2} diffusion (*α*_{w}), is expressed as (1 + *a*_{w}), where *a*_{w} is the ^{18}O fractionation during dissolution and diffusion of CO_{2} in liquid.

It is usual to obtain only one measurement of Δ^{18}O for a given condition and hence only one *g*_{m18} estimate (Barbour et al., 2016; Ubierna et al., 2017; Kolbe and Cousins, 2018), but if we switch the δ^{18}O of the CO_{2} and/or water vapor supplied to the cuvette during measurements we can obtain two different Δ^{18}O values for a given leaf physiological state. This is the basis of what we will refer to as the two-source δ^{18}O method (Fig. 1). With this approach, two sets of numerator and denominator pairs for Equation 1 are obtained. The influence of the cuvette air CO_{2} isotopic composition (δ_{a}) on Δ^{18}O can be seen from the definition of , where *R*_{a} is the isotope ratio of the CO_{2} in the ambient air and *R*_{A} is the isotope ratio of assimilated CO_{2}, whereas the influence of the water vapor isotopic composition (δ_{w_v}) is via its influence on the evaporative site leaf water isotopic composition (δ_{w_e}) and hence the δ^{18}O of the CO_{2} in equilibrium with this water (δ_{ce}). In theory, the ratio of each pair of measurements should return the same *g*_{m18} estimate because the leaf physiological state is unchanged. The linear regression of the values for the numerator against the denominator of Equation 1 should have a slope equal to *g*_{m18} and a zero intercept. Any deviation of the intercept from zero results from differences in individual *g*_{m18} estimates (Supplemental Fig. S1) and highlights inaccuracies in the measurements or in the underlying assumptions for Δ^{18}O.

Offsets in the intercept may arise due to (1) measurement inaccuracies, (2) uncertainties in parameter values, and/or (3) flaws in the theory. We used simulation modeling to assess the sensitivity of *g*_{m18} estimates to measurement inaccuracies under different CO_{2} δ^{18}O conditions and explored the utility of the two-source δ^{18}O method for identifying uncertainties in model assumptions with two experimental examples: Algerian oak (*Quercus canariensis*) and broad bean (*Vicia faba*) leaves measured at two different light levels, and *V. faba* leaves measured at different leaf-to-air vapor pressure differences (VPD).

In the first experiment, we explored model assumptions in the expected composition of the water at the sites of CO_{2}-water exchange (δ_{w_ex}). It is assumed that this water can be estimated using a modified Craig-Gordon equation applied to the evaporative sites of leaves (Farquhar and Lloyd, 1993). Measurements of the δ^{18}O of respired CO_{2} support this assumption (Cernusak et al., 2004). In contrast, the isotopic composition of Suc, which is enriched by about 27‰ with respect to substrate water, partly reflects cytosolic water, which in turn tends to lie somewhere between the Craig-Gordon prediction (δ_{w_e}) and xylem water isotopic composition (Cernusak et al., 2003). Spatial variation in the δ^{18}O of leaf water is also well documented (Cernusak et al., 2016). As such, differences in where CO_{2} exchanges with water and where net evaporation occurs may lead to spatial disparity in the isotopic calculations. We used two different light levels to manipulate the evaporative conditions and contrasting leaf anatomies of amphistomatous (*V. faba*) and hypostomatous (*Q. canariensis*) leaves. If δ_{w_ex} did not equal the modeled assumption, δ_{w_e}, the two-source δ^{18}O method would reveal this inconsistency by returning different *g*_{m18} estimates between isotopic conditions.

In the second experiment, we examined whether, as the VPD increased, the intercellular relative humidity (ratio of the vapor pressure to the saturated vapor pressure; *e*_{i}/*e*_{s} = *h*) could decrease, if there was insufficient compensation in transpiration rate resulting from stomatal closure. The assumption that *h* = 1 has recently been challenged in coniferous trees (Cernusak et al., 2018) and in an abscisic acid-insensitive poplar (*Populus* × *canescens*; Cernusak et al., 2019).

In both of our experiments, we proceeded to a two-source solution whenever *g*_{m18} estimates obtained for leaves with similar physiological conditions, but with different Δ^{18}O values, differed. This calculation resulted in one unique *g*_{m18} by forcing the intercept to be equal to zero by adjusting a second uncertain model parameter (Fig. 1; Supplemental Fig. S1), which was either δ_{w_ex} in the first experiment or *h* in the second. In both experiments, *g*_{m13} estimates were also obtained and compared with *g*_{m18} to provide insights into the locations of CO_{2}-water exchange.

## RESULTS

### Evaluating Sources of Error

#### Measurement Noise: Monte Carlo Simulation

When *g*_{m18} is written in the form of Equation 1, it becomes apparent that the sensitivity of *g*_{m18} estimates to experimental error will be greatest as the difference between the δ^{18}O of the CO_{2} in the intercellular spaces and that in equilibrium with water, , becomes small. To investigate this further, we used simulated gas-exchange data (*A* = 31 μmol m^{−2} s^{−1}; stomatal conductance [*g*_{s}] = 0.4 mol m^{−2} s^{−1}; expected *g*_{m18} = 1 mol m^{−2} s^{−1} bar^{−1}) with three different starting isotopic scenarios of and propagated noise in the simulated measurement of δ^{18}O of CO_{2} (sd 0.3‰) and water vapor (sd 0.35‰) using Monte Carlo simulation. Cuvette inlet and outlet δ^{18}O values of CO_{2} and water vapor were generated randomly 10,000 times using a normal distribution, based on the average of reported machine precisions from two similar studies utilizing δ^{18}O information (Loucos et al., 2017; Sauze et al., 2018). For each isotopic scenario, the mean and sd of the *g*_{m18} estimate was obtained. Figure 2 demonstrates that given measurement imprecision, for the same gas-exchange properties there is a greater spread in the proportional errors of *g*_{m18} estimates when the isotopic difference is small compared with when it is large.

We found similar patterns of *g*_{m18} sensitivity in our own data using the same Monte Carlo approach. In this case, our measured machine precision was on average 0.04‰ for δ^{18}O of CO_{2} and 0.2‰ for δ^{18}O of water vapor. Again, uncertainty around a *g*_{m18} estimate (represented as the sd) was greatest when was small (Fig. 3).

#### Uncertainty in Model Parameter Values: Sensitivity Analysis

We now move away from Monte Carlo simulation of measurement noise and consider the impact of systematic errors. We take, as examples, errors in δ_{w_e}, *h*, and leaf temperature (*T*_{leaf}) and examine their effect on the calculated *g*_{m18}. An overestimation of δ_{w_ex} by 30% of the difference between δ_{w_e} and the δ^{18}O of the transpired vapor (δ_{w_E}) resulted in the greatest proportional error in the estimated *g*_{m18} when was small. Furthermore, the proportional error in the estimated *g*_{m18} was the same whether the expected *g*_{m18} was 0.5 or 2 mol m^{−2} s^{−1} bar^{−1} (Fig. 4A). Conversely, systematic errors in *h* (e.g. assuming unity when the true value is 0.9; Fig. 4B) or *T*_{leaf} (±0.5°C; Fig. 4, C and D) resulted in an error pattern with that was dependent on the magnitude of *g*_{m18}. Importantly, larger did not necessarily reduce uncertainty in the *g*_{m18} estimate with errors in *T*_{leaf} or *h*.

#### Observing Errors Using the Two-Source δ^{18}O Method

Using the sensitivity analysis produced for δ_{w_e} as an example, the numerator versus denominator values from Equation 1 were plotted for each simulated *g*_{m18} estimate (Fig. 5B). The linear relationship indicates that the slope between two points and the extrapolated intercept remains the same despite each point having a different and associated proportional error in the calculated *g*_{m18} (Fig. 5A). This means that no matter what two points are compared, the absolute error in the overestimation of δ_{w_e} remains the same. This is the basis of using the two-source δ^{18}O method to estimate model parameter error, where two estimates of *g*_{m18} that are different are assumed to carry the same magnitude of error. All errors present themselves as offsets in the intercept from zero. If the true δ_{w_e} were used, each simulated point would return the expected *g*_{m18} value of 1 mol m^{−2} s^{−1} bar^{−1} and the intercept would equal zero. Thus, when two *g*_{m18} estimates are experimentally obtained, the true *g*_{m18} estimate can be retrieved by adjusting the parameter considered to be uncertain until the intercept is equal to zero. This was done using the SOLVER function in Excel (see “Materials and Methods” for more detail).

When measurement noise can be minimized or statistically evaluated, the remaining variation in parameter values estimated with the two-source δ^{18}O method may carry information of biological interest. To proceed to a two-source solution, we considered two *g*_{m18} estimates from the same leaf to be different when either (1) the pair of *g*_{m18} estimates were statistically different at the level of *P* < 0.05 (we performed this with a mixed-effects model, with plant replicate as a random factor) or (2) the simulated Monte Carlo sd values for the pair of *g*_{m18} estimates did not overlap.

### Experimental Manipulations

#### Light Experiment

To manipulate the observed photosynthetic discrimination , we switched the supply of CO_{2} and water vapor to the cuvette between CO_{2} tanks and water vapor sources of different oxygen isotopic compositions (Fig. 1). The resulting range in was over 100‰ (Fig. 6, inset). Despite this, estimates of *g*_{m18} were generally well conserved across light levels and isotopic conditions (Fig. 6), even though assimilation rate doubled with light in *V. faba* (16.9 versus 34.5 µmol m^{−2} s^{−1}) but not in *Q. canariensis* (16.1 versus 19.5 µmol m^{−2} s^{−1}). The *Q. canariensis g*_{m} estimates were significantly different between light levels for both *g*_{m13} and *g*_{m18}: under high versus low light, mean *g*_{m18} was 0.32 ± 0.03 versus 0.39 ± 0.03 mol m^{−2} s^{−1} bar^{−1} and mean *g*_{m13} was 0.27 ± 0.03 versus 0.29 ± 0.03 mol m^{−2} s^{−1} bar^{−1}. In *V. faba*, there was no significant difference between light levels, where on average *g*_{m18} was 0.77 ± 0.15 mol m^{−2} s^{−1} bar^{−1} and *g*_{m13} was 0.68 ± 0.16 mol m^{−2} s^{−1} bar^{−1}. Combining all measurements, for a given light level and species, *g*_{m18} was always significantly higher than *g*_{m13}.

Between the CO_{2} and water vapor switches, significant differences between *g*_{m18} estimates were found for *V. faba* independent of light level, but not for *Q. canariensis*. In both the light data sets of *V. faba* and *Q. canariensis*, there were enough *g*_{m18} pairs where the sd values slightly overlapped that exclusion of data pairs would have compromised the two-source analysis. This was despite the precision of the individual *g*_{m18} estimates being robust, with the sd being on average only 2.4% of the mean simulated value for *Q. canariensis* and 7.1% for *V. faba*. The larger sd of *V. faba* was driven by eight out of 58 data points, and when these were excluded, the simulated sd was 4% of the mean (Fig. 3).

There were no statistical grounds by which to proceed to a two-source solution for *Q. canariensis*; however, for *V. faba*, evidence was supported by one criterion (statistical differences) but not the other (overlapping simulated sd). We proceeded to a two-source solution with the *V. faba* data set but interpret the output with caution.

Numerator and denominator values for Equation 1 were regressed for *V. faba*, and the slopes and intercepts were determined (Fig. 7). The intercept offsets from zero were 0.16 ± 0.07 (sd) under high light and 0.08 ± 0.02 (sd) under low light. Using the two-source δ^{18}O method, we solved for the composition of water at the sites of exchange (δ_{w_ex}) that resulted in the same *g*_{m18} estimate between CO_{2} source gases (the value when the intercept = 0). Calculated δ_{w_ex} deviated from the assumed δ_{w_e} by −0.14‰ to −1.37‰. There was a significant difference in the magnitude of the deviation (δ_{w_ex} – δ_{w_e}) between light levels under each water vapor isotopic condition: −0.58‰ ± 0.12‰ versus −0.22‰ ± 0.13‰, high and low light, respectively (depleted water vapor), and −1.04‰ ± 0.12‰ versus −0.51‰ ± 0.13‰, high and low light, respectively (enriched water vapor).

In *V. faba*, the correlation between *g*_{m18} and *g*_{m13} of the original data had an *R*^{2} = 0.27 (*P* < 0.001), with a reduced major axis slope of 0.93; however, after correcting δ_{w_ex} using the two-source δ^{18}O method, the overall relationship was no longer significant (*R*^{2} = 0.04; Fig. 8, A versus B). In *Q. canariensis*, the *R*^{2} was 0.78 (*P* < 0.001) and the slope was 1.5 (Fig. 8E).

#### VPD Experiment

In general, we observed significant declines in *g*_{s}, *g*_{m13}, and *g*_{m18} with increasing VPD in *V. faba* leaves (Fig. 9; Supplemental Fig. S2). While the response curves were constructed using three plants, obtaining biological replicates at a given VPD is challenging. This is especially pertinent at high VPDs, where it is difficult to maintain a set VPD due to the feedback of stomatal closure and reduced transpiration on the cuvette VPD. Thus, to determine whether to proceed with a two-source solution, we relied on Monte Carlo simulated uncertainty (criterion *b*) rather than on statistical differences between biological replicates (criterion *a*). In all except two cases, the sd values did not overlap. These data points remain in the analyses but are identified in Figure 10. Using each pair of Δ^{18}O observations to solve simultaneously for *g*_{m18} and *h*, we were able to detect both a decline in *g*_{m18} (Fig. 9) and *h* (Fig. 10) with increasing VPD. Interestingly, at the highest VPDs there was a tendency for *h* to recover (Fig. 10). In contrast, we were unable to detect subsaturation of the intercellular airspace using the method of Cernusak et al. (2018), which assumed *g*_{m18} to be constant across VPDs. Comparing the regression coefficients between raw and two-source corrected conductances in relation to VPD, we found no significant differences between slopes, but the *h*-corrected intercepts were all significantly different from the uncorrected values: 0.06 ± 0.02 higher for *g*_{sc}, 0.39 ± 0.12 lower for *g*_{m18}, and 0.41 ± 0.12 lower for *g*_{m13} (Supplemental Fig. S2).

In correcting the internal humidity with the two-source δ^{18}O method, the recalculated *C*_{i} was between 2 and 21 μmol mol^{−1} higher, and for *g*_{sw}, it was between 0 and 0.1 mol m^{−2} s^{−1} higher than the uncorrected values (Supplemental Fig. S3, B and C). We also considered inconsistency in the assumption that δ_{w_ex} = δ_{w_e} by simultaneously solving for *h* and δ_{w_ex}, and the leaf internal air remained vapor subsaturated at moderate VPDs, albeit less so (Supplemental Fig. S4).

Unlike the *V. faba* results from the light experiment, the relationship between *g*_{m18} and *g*_{m13} was much stronger when VPD was the source of variation. Both raw and *h*-corrected slopes of *g*_{m18} versus *g*_{m13} were significant (*P* < 0.001), with values of 0.92 and 0.97, respectively. After correcting for the internal relative humidity, the *R*^{2} improved by 0.2 (Fig. 8, C versus D).

## DISCUSSION

### Interpretational Philosophy of Theoretical Isotope Models and Their Uncertainties

The theoretical model of ^{18}O photosynthetic discrimination has been used, among other things, to calculate *g*_{m18} as a residual term when measurements of isotope fluxes are combined with gas exchange. Simplifying assumptions and presumed parameter values are required to undertake this approach, so that any uncertainty in these assumptions is then passed onto the estimate of *g*_{m18}. As there is no way to directly measure *g*_{m}, model validation is nearly impossible, and so rarely attempted. However, we were able to exploit the action of CA, labeling different source gas CO_{2} compositions with leaf water to achieve different photosynthetic discriminations. In doing so, we tested the robustness of *g*_{m18} estimates within the same experimental framework, making this a direct and flexible approach.

Differences in *g*_{m18} estimates between isotopically distinct conditions could be interpreted in three ways: (1) measurement inaccuracies, (2) uncertainties in parameter values, and/or (3) flaws in the theory. While measurement inaccuracies owing to instrument imprecision and methodological issues (e.g. poor calibration) can be minimized, they will always present a source of uncertainty. Importantly, through our model simulations, we showed that a major factor in the sensitivity of *g*_{m18} estimates to experimental error was the absolute value of , with sensitivity increasing as the difference became smaller. From an experimental perspective of retrieving robust estimates of *g*_{m18}, we provide a number of ways to achieve a large in order to reduce the sensitivity of *g*_{m18} to experimental uncertainty (Supplemental Table S1; Supplemental Material S2).

The two assumptions in the ^{18}O model that we consider to be in doubt and that we tested in this study include the composition of water at the site of exchange and the humidity of the internal leaf airspaces. The assumption of isotopic equilibrium associated with CO_{2}-water exchange was recently raised as an issue particularly in C_{4} species and CA-deficient mutants (Ogée et al., 2018). As we measure C_{3} species, we consider it unlikely that there is systematic deviation from disequilibrium associated with light intensity or VPD. Until more of these measurements are undertaken, we also cannot rule out the third possibility that the ^{18}O photosynthetic discrimination model requires theoretical reconsideration.

To prevent overinterpretation of the data, we used statistical approaches to establish whether *g*_{m18} estimates were different between isotopic conditions, before proceeding to a two-source solution to retrieve one unique estimate for *g*_{m18} and the second parameter of interest.

### Can We Assume Craig-Gordon Evaporative Site Isotopic Composition at the Sites of CO_{2}-Water Exchange?

In the first experiment, we switched between CO_{2} tanks of contrasting δ^{18}O and also switched incoming water vapor isotopic compositions. We expected that discrepancies between *g*_{m18} estimates for different isotopic conditions would be minimal based on previous rationalizations that the close proximity of the chloroplasts to the cell wall in a C_{3} leaf should ensure that the chloroplastic water is equal to the Craig-Gordon-predicted evaporative site isotopic composition (Gillon and Yakir, 2000; Cernusak et al., 2004). Regressing numerator values against denominator values of Equation 1, we found that the offsets in the intercepts were indeed small (Fig. 7). Furthermore, when δ_{w_ex} was adjusted to retrieve one *g*_{m18} estimate (intercept = 0), the errors in the estimated δ^{18}O at the sites of exchange were no more than 1.4‰. To put this into context, leaf water δ^{18}O can vary in the horizontal plane for a dicot by ∼17‰ (Gan et al., 2002) and even more along grass blades (Cernusak et al., 2016). However, the small deviations from predicted δ_{w_e} in the present experiments were sufficient to cause bias in *g*_{m18} estimates for *V. faba*, where *g*_{m18} was large and |δ_{i} − δ_{ce}| was small (Fig. 6B). Such differences may yield valuable information about the supply of water to evaporative sites.

Uncertainties in the ^{18}O isotope composition of the water at the sites of exchange may arise from differences in where CO_{2} exchanges with water and where net evaporation occurs. This could lead to spatial disparity in the calculations of δ_{A} and δ_{i} from observations of the δ^{18}O of CO_{2} and in the calculation of δ_{ce} from observations of the δ^{18}O of water vapor. As xylem water is usually less enriched than water at the evaporative sites, small-scale water isotopic gradients can develop along the water pathway from xylem vessels through files of mesophyll cells to evaporative sites (Péclet effect; Farquhar and Lloyd, 1993). Recently, it was proposed that the extent of a Péclet effect may depend on the location of the sites of net evaporation and condensation (Barbour et al., 2017), which, according to physical models (Rockwell et al., 2014; Buckley et al., 2017), vary with environmental conditions and leaf anatomy. It is noteworthy that the intercept of the relationship between numerator and denominator in the two-source analysis was offset to a larger extent in the high-light condition in amphistomatous *V. faba* (Fig. 7), but there was insufficient evidence to justify the need for a two-source solution in *Q. canariensis*. It will be interesting to see whether general patterns in the deviation of δ_{w_ex} from δ_{w_e} emerge with a greater comparison of leaf anatomies and evaporative conditions.

### Can We Assume That the Leaf Intercellular Airspace Is Always Vapor Saturated?

The calculation of *g*_{s} and *C*_{i} from measurements of gas exchange rely on the assumption that the leaf internal air is saturated (*h* = 1) at the measured leaf temperature. As water moves from a wet apoplastic surface to a drier atmosphere, there is no debating that there will be a gradient in vapor pressure. What the results of Cernusak et al. (2018) call into question is whether this gradient extends sufficiently into the substomatal cavity to invalidate the assumption that the saturated plane is close to the stomatal pore so that changes in transpiration rate may be attributed to changes in stomatal aperture. In the case of *h* < 1, *g*_{s} (and hence *C*_{i}) is underestimated, because its estimation will partly reflect the increased pathlength and internal resistance to the saturated plane.

Buckley and Sack (2019) acknowledge, as we do, the challenge unsaturation presents to our current understanding of leaf water transport, in particular, the fact that humidities lower than unity imply very negative water potentials, at least at evaporation points associated with the net flux out of the leaf. In this study, we observed a minimum internal humidity of 95%, which suggests that water is being held within the cell apoplast at approximately −7 MPa (at 25°C; Fig. 10). A bulk water potential this low for *V. faba*, and most plants, would be lethal, with the disruption of membrane integrity and cell lysis occurring at water potentials well greater than this (Fellows and Boyer, 1978). Thus, the decline in hydraulic conductance must be localized to certain internal points within the cell apoplast. Reconciling these point estimates of water potential with our understanding of how the bulk leaf water potential limits stomatal conductance and water transport is clearly difficult. However, researchers have alluded to this disparity for a long time. For example, as reported by Farquhar and Raschke (1978), when the epidermis of *Commelina communis* was removed, the transpiration rate declined to 19% after 50 min with exposure to air of ∼73% relative humidity, yet the bulk water potential had only declined to −3.5 MPa.

The method we utilize to calculate the internal humidity improves on that of Cernusak et al. (2018), because with the two-source δ^{18}O method no a priori determination of *g*_{m18} is required, removing one uncertainty. The method of Cernusak et al. (2018) first estimated *g*_{m18} under low VPD conditions where *h* = 1 could be assumed. This *g*_{m18} was then used to calculate δ_{c}, the ^{18}O isotopic composition of CO_{2} at the sites of CO_{2}-water exchange (with Supplemental Material S1, Eq. S1) for other VPDs. Finally, a solution for *e*_{i} was iteratively determined with the constraint δ_{c} = δ_{ce}, where δ_{ce} was independently obtained with Equation 4. When we applied this approach to *V. faba* leaves measured at different VPDs, we were unable to detect subsaturation of the internal airspace (*h* < 1). This might have originated by holding *g*_{m18} constant across VPDs. Using our *V. faba* data, we modeled the response of δ_{i} and δ_{ce} to different values of *h* for two case scenarios (Fig. 11): δ_{in} lower or higher than δ_{ce}. When δ_{in} is less than δ_{ce} (Fig. 11A), δ_{i} will be less than δ_{ce}, and vice versa (Fig. 11B). Figure 11 illustrates that large |δ_{i} − δ_{ce}| occurs when *g*_{m18} is low (Eq. 1) and/or *h* < 1. Thus, observations of large |δ_{i} − δ_{ce}| cannot be solely attributed to decreased *g*_{m18} in situations when unsaturation could be suspected, such as when VPD is high. Similarly, in using the larger *g*_{m18} value derived at low VPD for higher VPD situations, the extent of unsaturation can be hidden and even lead to values of *h* > 1. This was evident in a more recent investigation by Cernusak et al. (2019) comparing the estimated internal humidity patterns between wild-type and transgenic poplar with functionally impaired stomatal responses to abscisic acid: for the wild-type plants, fixing *g*_{m18} at a single value for estimating *h* resulted in *h* increasing above unity at the highest VPDs, suggesting that at some point *g*_{m18} most likely declined. Indeed, in using the two-source δ^{18}O method in this study, the solution indicated a decline in both *g*_{m18} and *h* with increasing VPD (Fig. 9).

The decline in *g*_{s}, *g*_{m13}, and *g*_{m18} with increasing VPD, with (Fig. 9) or without (Supplemental Fig. S2) correcting for internal vapor pressure, agrees with the findings of Loucos et al. (2017) in cotton (*Gossypium hirsutum*) leaves, who similarly used online isotopic discrimination methods to estimate *g*_{m}. The two other studies that have investigated VPD responses used chlorophyll fluorescence methods leading to contrasting conclusions (Bongi and Loreto, 1989; Warren, 2008). Further investigations are required to determine whether this reflects species-specific responses or differences associated with methods; however, it is noteworthy that in our study, two essentially different isotopic approaches gave similar *g*_{m} trends (Fig. 9), and the relationship between *g*_{m18} and *g*_{m13} was improved when *h* was corrected (Fig. 8).

In both the light and VPD experiments, we also solved the intercept for errors in leaf temperature. According to Mott and Peak (2011), there is a tendency for leaf thermocouples to measure partly air temperature. When the leaf cuvette air is cooler, the reported *T*_{leaf} will tend to be underestimated and vice versa, if other influences are held constant. However, the error in *T*_{leaf} was not always consistent with the direction of the gradient in temperature from air to leaf (Supplemental Fig. S5). Furthermore, within a given leaf in the VPD experiment, the proportional error in *T*_{air} − *T*_{leaf} varied with VPD, whereas if the error resulted from heat conduction of the thermocouple, the proportion should be constant.

### Do *g*_{m13} and *g*_{m18} Estimates Represent Different Diffusion Pathways?

Given the different assumptions involved in Δ^{13}C and Δ^{18}O models, it is attractive to use both isotopes to constrain *g*_{m} estimation and interpretation. To do so requires the assumption that the diffusion pathways are similar. Gillon and Yakir (2000) first proposed that the sites of ^{18}O exchange were located at the outer chloroplast surface so that the total conductance from *C*_{i} to *C*_{c} could be proportioned into wall [*g*_{w} = *A*/*P*(*C*_{i} − *C*_{m})] and chloroplast [*g*_{ch} = *A/P*(*C*_{m} − *C*_{c})] components. However, Ogée et al. (2018) recently argued that the apparent locations of CA represent weighted mean activities from the cytosol to the sites of carboxylation, with more weighting toward the chloroplast because of the more basic pH and higher CA concentration. In this scenario, the diffusional pathways represented by the two isotopic methods will be more aligned. It is therefore interesting that published ratios of *g*_{m18} to *g*_{m13} tend to be large and as high as 4 (as observed in *Glycine max*; Gillon and Yakir, 2000).

One possibility to explain large differences between *g*_{m18} and *g*_{m13} estimates is that part of the respiratory flux diffuses out of the leaf via the cytosol, passing through gaps between the chloroplasts rather than through them. The original Δ^{13}C model assumes that the respiratory fluxes must first enter the chloroplast before diffusing out of the leaf. In this form, *g*_{m} estimates can be interpreted as physical diffusional properties in series. However, if the respiratory gases partly diffuse out of the leaf directly via the cytosol, estimates of *g*_{m13} must instead be considered as a flux-weighted variable sensitive to the ratio of photorespiration to CO_{2} assimilation (Tholen et al., 2012; and expanded on in von Caemmerer, 2013; Yin and Struik, 2017; Ubierna et al., 2019). In this case, *g*_{m18} may be larger and constant, whereas apparent *g*_{m13} may vary. Recently, we built on earlier work that considers this scenario (Evans and von Caemmerer, 2013) and developed updated equations for the ^{13}C model (Ubierna et al., 2019). As we measured here under 2% (v/v) O_{2} and under ambient CO_{2} concentrations, the impact on our findings of uncertainty in the spatial direction of photo(respiratory) fluxes can be considered negligible. However, the implication of these different interpretations is that the ratio of *g*_{m18} to *g*_{m13} will not necessarily be constant (i.e. linearly related to each other), although we would expect some ordered pattern.

In this study, the strength of the relationship (*R*^{2}) between *g*_{m18} and *g*_{m13} varied between 0.04 and 0.78, suggesting that traits driving differences between genotypes or responses to environmental conditions may not always similarly influence both conductances. In general, *g*_{m18} was larger than *g*_{m13}; however, the magnitude of the slope between these variables varied across species and measurement conditions. In *Q. canariensis*, the slope of *g*_{m18} versus *g*_{m13} was 1.5, which is similar to the 1.3 value observed by Gillon and Yakir (2000). Under the tested conditions, this suggests that CA and carboxylation sites are not the same, although the difference between *g*_{m} estimates is not large and could be removed by changing parameter values in the two isotope models. Across the two experiments for *V. faba*, including original and corrected estimates, the individual ratios of *g*_{m18} to *g*_{m13} varied between 0.5 and 2.3. Data points under the 1:1 line (ratio < 1) suggest an inconsistency in the model or in the assumed parameter values because, across all flux scenarios, *g*_{m13} represents the lower bound on *g*_{m} estimates (Fig. 8). For example, increasing the presumed ^{13}C isotopic fractionation for rubisco from the 29‰ value used here shifts the data to the left of the 1:1 line. Given the uncertainties in both isotopic models, further research is required before differences in *g*_{m18} and *g*_{m13} can be interpreted in terms of wall and chloroplast components, respectively (Barbour et al., 2016).

## CONCLUSION

In capitalizing on the capacity of CA to equilibrate CO_{2} with leaf water, we were able to use a two-source δ^{18}O approach to test the reliability of the ^{18}O-isotope method to estimate the CO_{2} diffusional path from the intercellular airspaces to the sites of CA activity. Under the experimental conditions tested, *g*_{m18} estimates were mostly robust, providing confidence in the theoretical model. Furthermore, in manipulating the difference between the δ^{18}O of the CO_{2} in the intercellular spaces and that in equilibrium with water, we demonstrated that the impact of measurement errors and uncertainty in model assumptions on *g*_{m18} could be reduced when was large. Importantly, the two-source method has the potential to identify situations where model discrepancies occur even when the difference between *g*_{m18} estimates is small, offering a flexible tool for model parameterization. Here, we used the method to explore inconsistencies in the assumptions of the δ^{18}O of water at the site of exchange and of the humidity inside the leaf, providing an opportunity to refine our understanding of leaf water and CO_{2} fluxes.

## MATERIALS AND METHODS

### Plant Material

Broad bean (*Vicia faba*) was grown from seed in 4.5-L pots with potting mix and Osmocote. Seeds for the light experiment were sown in May 2017, and paired leaves (one for each light level) from four plants (replicates) were measured on 5- to 6-week-old plants. Each light condition was measured on a separate day, as it took 6 to 7 h to obtain the seven measurements for one light condition. Seeds sown in June 2017 were used for the VPD experiment, and measurements were conducted on 8-week-old plants. Five leaves from three plants were used to construct the VPD response curves of *g*_{s}, *g*_{m13}, and *g*_{m18}. Four year-old Algerian oak (*Quercus canariensis*) seedlings grown in 11-L pots were measured for the light experiment in January 2018. Measurements were similarly made on paired leaves for each of three plant replicates. For all experiments, plants were grown in controlled-temperature glasshouses maintained at 25°C during the day and 17°C at night, with natural illumination. Plants were watered daily, and additional liquid fertilizer was applied as needed.

### Coupled Gas-Exchange and Online Isotope Discrimination Measurements

An LI-6400XT open gas-exchange system (Li-Cor) was coupled with a cavity-ring down absorption spectroscope (L-2130; Picarro) and a dual quantum cascade laser (QCL) absorption spectroscope (Aerodyne Research). The L-2130 measured the H_{2}^{16}O and H_{2}^{18}O isotopologues of water vapor and the QCL measured isotopologues of CO_{2}, including ^{12}CO_{2}, ^{13}CO_{2}, C^{16}O^{17}O, and C^{16}O^{18}O.

The L-2130 was calibrated using working standards spanning −13.55‰ to 54.99‰ (referenced against IAEA standards VSMOW, VSLAP, and GISP). These were injected (30 μL) into a sealed vessel and allowed to completely evaporate before being drawn out by the L-2130. The volume of gas in the vessel was maintained by a flow of dry synthetic air (2% [v/v] O_{2}) into the vessel with an open split to prevent overpressuring. The vessel was insulated to prevent temperature-induced isotopic gradients. The L-2130 showed no discernible water vapor concentration dependence of the δ^{18}O above 5,000 μmol mol^{-1}. One to two standards were run at the end of each measurement day to establish a common calibration curve of expected versus observed δ^{18}O values (*n* = 60), which was applied across all the data.

The LI-6400XT was assembled with the 6400-07 needle cuvette (12 cm^{2}) fitted with an LI-6400-18 RGB light source (set to red:blue, 90:10). The inlet gas line to the LI-6400XT console (reference gas) and the outflow of the matching valve of the leaf cuvette (sample gas) were split so that part of this flow could be diverted to the L-2130 and another part to the QCL after being dried by a Nafion dryer and dry-ice trap. Synthetic air of oxygen and N_{2} was mixed using two mass-flow controllers to achieve the 2% (v/v) O_{2} used in both experiments. Part of this mix flowed through a temperature-controlled bubbler for humidification, which provided the reference gas for the cuvette. The other part was mixed with another stream carrying CO_{2} of known isotopic composition from a capillary system (calibration gas). This high-pressure CO_{2} tank supplied both the calibration capillary system and the LI-6400XT console, so that when the source gas for the cuvette was switched, the data were corrected to the calibration gas of the same isotopic composition. The flow of the CO_{2} stream was controlled in order to regulate the CO_{2} concentration of the calibration gas (see Supplemental Fig. S6 for a diagram of the experimental setup). Two concentrations of the calibration gas spanning the leaf cuvette reference and sample gas concentrations were measured at the beginning and end of a measurement cycle, which included four to five alternating reference/sample pairs. This accounted for both the concentration dependence and drift of the machine between each measurement series. The observed deltas were linear in the concentration range measured (350–550 μmol mol^{−1}), allowing us to linearly interpolate between the (observed − expected) δ^{13}C or δ^{18}O offset with [^{12}CO_{2}]. This regression was then used to apply an offset to the raw delta values: . For each isotopologue, the delta values were calculated as: and , where is the ratio of the Vienna Pee Dee Belemnite standard. Delta values were scaled to per mil (‰) by multiplying by 1,000 (see Supplemental Material S3 for further details on the gas-sampling protocol and data corrections). After correcting the data, the last three reference/sample pairs were averaged with mean precision of 0.04‰. All raw measurements and corrections were made with respect to the VPDB scale, but when δ^{18}O of CO_{2} values were combined with δ^{18}O of water vapor values, the δ^{18}O of the CO_{2} was converted to the VSMOW scale using the conversion formula .

### Light Experiment

Young fully expanded leaves were used for all measurements, and the cuvette gaskets were sealed with Terostat putty (Terostat IX; Henkel Technologies). The cuvette conditions were set to a sample [CO_{2}] of 400 μmol mol^{−1}, 2% (v/v) O_{2}, and 25°C ± 0.1°C leaf temperature. Photosynthetically active radiation (μmol m^{−2} s^{−1}) was 1,300 or 300 for the high- and low-light conditions, respectively. For the two-source analysis to be valid, gas-exchange rates (CO_{2} and water vapor) need to be stable between switches. Thus, leaves were allowed to stabilize 1 to 2 h before measurements commenced. It took at least 20 min to sufficiently turn over the CO_{2} gas and for the leaf to reach isotopic stability during a CO_{2} switch. This was because one of the source gases was highly enriched in ^{18}O, which shifted the leaf water isotopic composition during CO_{2}-water exchange. We confirmed this through observations of the δ^{18}O of the sample water vapor, which changed during a CO_{2} switch despite gas exchange remaining stable (Supplemental Fig. S7). For the water vapor switch, it took longer for isotopic stability to be achieved due in part to the sorption properties of water on tubing and the cuvette interior (despite being above dew point temperature). The isotopic composition of the reference vapor was stable ∼15 min post switch, but the sample vapor took ∼45 min to sufficiently turn over, at which point a measurement was taken. Subsequent measurements were made over the next 2 to 3 h during the slower transition phase of the leaf to the new isotopic steady state.

### VPD Experiment

Young fully expanded leaves were placed in a leaf cuvette sealed with Terostat. The cuvette conditions were set to a sample [CO_{2}] of 400 μmol mol^{−1}, 2% (v/v) O_{2}, 1,300 μmol m^{−2} s^{−1} light intensity, and leaf temperature of 30°C ± 0.4°C. Cuvette flow rate and the humidity of the reference air were modified by adjusting the temperature of the water bath housing the bubbler to achieve different VPDs. At the highest VPDs, the bubbler was bypassed and dry air was supplied to the cuvette. Achieving stable gas conditions was difficult, especially between 1.5 and 2 kPa, when stomata were closing. When there was a significant drift in gas exchange during the CO_{2} switch, the measurement was discarded.

The composition of the CO_{2} tanks and water vapor composition for each experiment are given in Table 1. The isotope composition for the CO_{2} tanks was measured in our stable isotope laboratory at the Australian National University on either a dual-inlet mass spectrometer (Micromass Isoprime) or by running pulses of a known gas through one inlet of a dual-gas reference box on the same machine, interspersed with pulses of the unknown tank CO_{2}.

### Calculation of Mesophyll Conductance

Equation 18 of Farquhar and Cernusak (2012) can be rearranged to calculate the mesophyll conductance to CO_{2} from the intercellular airspaces to the sites of CO_{2}-water exchange by substituting *C*_{m} with using Fick’s law (for full derivation, see Supplemental Material S1), to give Equation 1.

is estimated from direct measurements of δ^{18}O and dry concentrations of CO_{2} in air entering and exiting the cuvette:

From which, δ_{i} can be calculated as:

where *t* is the ternary correction, is the CO_{2} concentration in the ambient air, *C*_{i} is the concentration of CO_{2} in the leaf intercellular airspaces, and is the weighted diffusional fractionation across the boundary layer and stomata.

, the ^{18}O isotopic composition CO_{2} at the sites of CO_{2}-water exchange, cannot be directly measured. But if it is assumed that the CO_{2} is in isotopic equilibrium with leaf water and the isotopic composition of this water is equal to the Craig-Gordon prediction , can be taken as :

where

and where is the ^{18}O fractionation between CO_{2} and water at equilibrium at leaf temperature, is the composition of the water vapor surrounding the leaf, is the composition of the transpired water vapor (Eq. 6 below), is the ratio of ambient to intercellular vapor pressure (*e*_{s} is the saturated vapor pressure and *h* is the internal relative humidity), is the equilibrium fractionation for the phase change from liquid to vapor, and is the kinetic fractionation for diffusion through the stomata and boundary layer (“Appendix”).

The composition of the transpired water vapor can be calculated as:

where and are water vapor mole fractions of air entering and exiting the cuvette and and are δ^{18}O of water vapor entering and exiting the cuvette.

The *g*_{m13} was calculated with Equation 43 from Ubierna et al. (2018), assuming fractionation values *b* = 29‰, *f* = 11‰, and *e* = −3‰. Dark respiration, R_{d,} was measured for each leaf at the end of the day in the dark.

### Statistical Analysis

To calculate the two-source solution for *g*_{m18}, the slope and intercept of numerator and denominator values (Eq. 1) from paired Δ^{18}O measurements were obtained. Reduced major axis for line fitting was selected owing to measurement errors associated with both numerator and denominator values. The Excel SOLVER function was used to find the value of the model parameter that resulted in the intercept = 0. When this constraint was met, the slope and individual *g*_{m18} values were equal. The model parameters solved for included in the light experiment and *h* (*e*_{i}/*e*_{s}) in the VPD experiment. Note that is dependent on *h* (Eq. 5); however, measurements for the light experiment were undertaken at low VPD, where subsaturation was assumed to be negligible. For the VPD experiment, we compared the two-source solution of the internal humidity with the method of Cernusak et al. (2018). To recalculate *g*_{s} and *C*_{i} using the estimated internal humidity, we used the gas-exchange equations described by Cernusak et al. (2018). Briefly, gas-exchange equations reported from the LI-6400XT assume that the vapor pressure inside the leaf (*e*_{i}) is equal to the saturation vapor pressure at that leaf temperature (*e*_{s}). Thus, we recalculated *e*_{i} according to .

The SAS procedure PROC MIXED with leaf replicate as a random effect was used to determine if differences in the mean *g*_{m18} estimates between light levels, δ^{18}O of CO_{2} source switches, and δ^{18}O of water vapor inlet conditions were significantly different at *P* < 0.05 (Fig. 6). Simple effects were investigated for significant interactions. PROC REG was used to determine if the linear relationship between *g*_{m13} and *g*_{m18} estimates (Fig. 8), and the linear relationship between conductance parameters and VPD (Fig. 9), were significant. In the former, model coefficients were obtained from reduced major axis analysis, and in the latter, ordinary least-squares analysis. PROC GLM was used to test for significant differences between slopes and intercepts of raw and two-source-corrected conductances in relation to VPD (Supplemental Fig. S2).

### Supplemental Data

The following supplemental materials are available.

**Supplemental Figure S1.**Example of the two-source δ^{18}O method for detection of errors in measurements or uncertainties in model assumptions according to the offset in the intercept when plotting the numerator against the denominator of Equation 1 (Supplemental Material S1, Eq. S4).**Supplemental Figure S2.**Comparison of uncorrected and two-source-corrected conductance estimates as a function of the VPD between leaf and air in*V. faba*.**Supplemental Figure S3.**Errors in gas-exchange estimates when*h*is less than unity.**Supplemental Figure S4.**Impact of simultaneously solving for the composition of water at the sites of CO_{2}-water exchange (δ_{w_ex}) and*h*.**Supplemental Figure S5.**Absolute error in the measured*T*_{leaf}as a function of the difference between measured air and leaf temperatures.**Supplemental Figure S6.**Schematic showing how the LI-6400XT open gas-exchange system (Li-Cor) was coupled to the cavity-ring down absorption spectroscope (L-2130; Picarro) and dual QCL absorption spectroscope (Aerodyne Research).**Supplemental Figure S7.**Example of the effects of the enriched CO_{2}source tank on leaf water in cotton, as observed via the δ^{18}O of the sample water vapor.**Supplemental Table S1.**Experimental approaches to manipulating the δ^{18}O of the intercellular airspace ( compared with the δ^{18}O of CO_{2}in equilibrium with leaf water .**Supplemental Material S1.**Derivation of*g*_{m}from Δ^{18}O theoretical model.**Supplemental Material S2.**Experimental approaches to manipulating the isotopic difference (*δ*_{i}-*δ*_{ce}).**Supplemental Material S3.**Calibration of raw QCL data.

## ACKNOWLEDGMENTS

We thank Barry McManus from Aerodyne for his time and patience in ensuring that our machine was set up and optimized for our requirements as well as Lisa Wingate and Jérôme Ogée and their lab group for sharing their ideas on how they were using a two-source δ^{18}O isotope approach for soil applications.

## Appendix

List of Symbols

Symbol | Description | Unit/Calculation |
---|---|---|

Oxygen isotope effects | ||

Weighted fractionation across the boundary layer, stomata, and mesophyll in series | ‰ | |

Weighted fractionation across the boundary layer and stomata | ; ‰ | |

| ^{18}O fractionation across the boundary layer | 5.8‰ |

| ^{18}O fractionation across the stomata | 8.8‰ |

| ^{18}O fractionation during dissolution and diffusion in liquid | 0.8‰ |

Oxygen isotope effect during CO_{2} diffusion | ; unitless | |

Oxygen isotope effect during dissolution and diffusion in liquid | a_{w} ; unitless | |

^{18}O fractionation between liquid water and vapor at equilibrium | ; ‰ | |

^{18}O fractionation during diffusion of water from the leaf intercellular airspaces to the atmosphere | ; ‰ | |

^{18}O fractionation between CO_{2} and water at equilibrium | ; ‰ | |

Gas exchange | ||

A | Rate of CO_{2} assimilation | LI-6400XT output; μmol m^{−2} s^{−1} |

C_{a} | Mole fraction of CO_{2} in the ambient air | Measured; μmol mol^{−1} |

C_{c} | Equivalent mole fraction of CO_{2} at the site of carboxylation in the chloroplast | ; μmol mol^{−1} |

C_{i} | Mole fraction of CO_{2} in the intercellular spaces | LI-6400XT output; μmol mol^{−1} |

C_{m} | Equivalent mole fraction of CO_{2} at the site of CO_{2}-water exchange | ; μmol mol^{−1} |

C_{s} | Mole fraction of CO_{2} at the leaf surface | LI-6400XT output; μmol mol^{−1} |

E | Transpiration rate | LI-6400XT output; mol m^{−2} s^{−1} |

e_{a} | Atmospheric vapor pressure | Measured; mbar |

e_{i} | Vapor pressure of leaf internal airspaces | Estimated from two-source method; mbar |

e_{s} | Saturated vapor pressure of leaf internal airspaces at leaf temperature | Estimated from T_{leaf}: ; mbar |

g_{b} | Boundary layer conductance | LI-6400XT output; mol m^{−2} s^{−1} |

g_{m13} | Mesophyll conductance to diffusion of CO_{2} as calculated from Δ^{13}C model | Equation 43 (Ubierna et al., 2018); mol m^{−2} s^{−1} bar^{−1} |

g_{m18} | Mesophyll conductance to diffusion of CO_{2} as calculated from Δ^{18}O model | Equation 1; mol m^{−2} s^{−1} bar^{−1} |

g_{sc} | Stomatal conductance to diffusion of CO_{2} | LI-6400XT output; mol m^{−2} s^{−1} |

g_{sw} | Stomatal conductance to diffusion of water vapor | LI-6400XT output; mol m^{−2} s^{−1} |

g_{tc} | Total conductance to diffusion of CO_{2} | ; mol m^{−2} s^{−1} |

h = e_{i}/e_{s} | Relative humidity of leaf internal airspaces | Estimated from the two-source method or the approach of Cernusak et al. (2018) |

P | Atmospheric pressure | Measured; bar |

t | Ternary correction coefficient | ; unitless |

T_{K} | Leaf temperature in Kelvin | Measured; K |

T_{leaf} | Leaf temperature in Celsius | Measured; °C |

w_{a} | Mole fraction of water vapor in the ambient air | LI-6400XT output; mol water mol^{−1} air |

w_{in} | Mole fraction of water vapor entering the cuvette | LI-6400XT output; mol water mol^{−1} air |

Isotopic compositions | ||

R | Isotope ratio | |

δ^{18}O | ^{18}O isotope ratio of CO_{2} | ; where |

δ^{18}O | ^{18}O isotope ratio of water | ; where |

δ^{13}C | ^{13}C isotope ratio of CO_{2} | ; where |

is or | Definition of ^{18}O and ^{13}C photosynthetic discrimination | ; unitless or if multiplied by 1,000 becomes scaled by ‰. |

δ_{A} | Isotopic composition of CO_{2} that is being assimilated | ; ‰ |

δ_{a} | Isotopic composition of CO_{2} in the ambient air exiting the leaf cuvette | Measured; ‰ |

δ_{i} | ^{18}O isotopic composition of CO_{2} in the intercellular spaces | Equation 3; ‰ |

δ_{in} | Isotopic composition of CO_{2} entering the leaf cuvette | Measured; ‰ |

δ_{c} | ^{18}O isotopic composition of CO_{2} at the sites of CO_{2}-water exchange | Supplemental Material S1, Equation S1; ‰ |

δ_{ce} | ^{18}O isotopic composition of CO_{2} in equilibrium with leaf water at composition δ_{w_e} | Equation 4; ‰ |

δ_{w_v} | ^{18}O isotopic composition of water vapor exiting the leaf cuvette | Measured; ‰ |

δ_{w_v(in)} | ^{18}O isotopic composition of water vapor entering the leaf cuvette | Measured; ‰ |

δ_{w_E} | ^{18}O isotopic composition of the transpired water vapor | Equation 6; ‰ |

δ_{w_e} | ^{18}O isotopic composition of water at the sites of evaporation | Equation 5; ‰ |

δ_{w_ex} | ^{18}O isotopic composition of water at the sites of CO_{2}-water exchange | Assumed equal to δ_{w_e}, otherwise estimated from two-source method; ‰ |

## Footnotes

M.H.-P., L.A.C., and G.D.F. discussed the original idea; M.H.-P. and H.S.-W. developed the two-source online-isotope measurement system; M.H.-P. designed and conducted the experiments; N.U. reviewed the theory and mathematics; M.H.-P. analyzed and wrote the article with contributions from all authors.

↵1 This work was supported by Australian Research Council Discovery Grant DP150100588.

↵3 Senior author.

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- Received June 24, 2019.
- Accepted September 5, 2019.
- Published September 13, 2019.