Simulation of Stomatal Conductance and Water Use Efficiency of Tomato Leaves Exposed to Different Irrigation Regimes and Air CO2 Concentrations by a Modified “Ball-Berry” Model

Stomatal conductance (gs) and water use efficiency (WUE) of tomato leaves exposed to different irrigation regimes and at ambient CO2 (a[CO2], 400 ppm) and elevated CO2 (e[CO2], 800 ppm) environments were simulated using the “Ball-Berry” model (BB-model). Data obtained from a preliminary experiment (Exp. I) was used for model parameterization, where measurements of leaf gas exchange of potted tomatoes were done during progressive soil drying for 5 days. The measured photosynthetic rate (Pn) was used as an input for the model. Considering the effect of soil water deficits on gs, an equation modifying the slope (m) based on the mean soil water potential (Ψs) in the whole root zone was introduced. Compared to the original BB-model, the modified model showed greater predictability for both gs and WUE of tomato leaves at each [CO2] growth environment. The models were further validated with data obtained from an independent experiment (Exp. II) where plants were subjected to three irrigation regimes: full irrigation (FI), deficit irrigation (DI), and alternative partial root-zone irrigation (PRI) for 40 days at both a[CO2] and e[CO2] environment. The simulation results indicated that gs was independently acclimated to e[CO2] from Pn. The modified BB-model performed better in estimating gs and WUE, especially for PRI strategy at both [CO2] environments. A greater WUE could be seen in plants grown under e[CO2] associated with PRI regime. Conclusively, the modified BB-model was capable of predicting gs and WUE of tomato leaves in various irrigation regimes at both a[CO2] and e[CO2] environments. This study could provide valuable information for better predicting plant WUE adapted to the future water-limited and CO2 enriched environment.


INTRODUCTION
Plant stomata aperture play a predominant role in modulating the diffusion of CO 2 and H 2 O vapor between leaf and atmosphere (Buckley and Mott, 2013), and optimizing photosynthetic and transpiration rates, hereby the water use efficiency (WUE) at leaf scale (Liu et al., 2009). It is wellestablished that both reduced irrigation regimes, especially alternate partial root-zone irrigation (PRI) and elevated atmospheric CO 2 environment (e[CO 2 ]) could induce partial stomatal closure and synergistically enhance WUE (Pazzagli et al., 2016). Therefore, a better understanding of how to model stomatal conductance for water vapor (g s ) is essential for the accurate prediction of leaf transpiration and improvement of plant WUE in response to the future water limited and CO 2 enriched environments.
A number of approaches has been tested for modeling g s under well-watered conditions (Gutschick and Simonneau, 2002). Among those, the Ball-Berry model (BB-model) describing the linear coupling relation of g s to photosynthetic rate (P n ), relative humidity (h s ), and CO 2 concentration (C s ) on the leaf surface (Ball et al., 1987) has been broadly adopted and utilized from leaf to plant scale due to its apparent accuracy and simplicity (Miner et al., 2017). However, the P n -g s relationship of BB-model would be changed under water stress (Damour et al., 2010), hence modified model is needed for simulating g s , for instance, by incorporating empirical functions coupling with abscisic acid (ABA), leaf water potential or soil water potential (Sala and Tenhunen, 1996;Gutschick and Simonneau, 2002;Bauerle, et al. et al., 2004;Damour et al., 2010).
For drought-prone areas, more efficient irrigation techniques need to be developed and implemented in order to achieve optimal crop yield and quality (Du et al., 2015). PRI strategy has been demonstrated to save considerable amount of irrigation FIGURE 1 | Water retention curves of the peat substrate used in experiment one and of the sandy soil used in experiment two.
water without significantly reducing yield as compared to full irrigation (FI) (Kang and Zhang, 2004;Wei et al., 2016). It is well-known that reduced plant water consumption under PRI is resulted from a decreased g s , which is primarily regulated by the root-to-shoot ABA signaling triggered in the roots exposed to drying soil Liu et al., 2006). A modified BB-model based on the temporal and spatial change of soil water potential ( s ) in the soil columns has been reported to be capable of predicting g s and WUE of PRI treated potato leaves, indicating an enhancement of WUE for PRI in relation to FI plants (Liu et al., 2009).
Plants grown at e[CO 2 ] generally possess an increased P n but decreased g s , resulting in a greater WUE as compared to those growth at a[CO 2 ] (Pazzagli et al., 2016). An better understanding of the coordination between g s and P n in response to e[CO 2 ] is crucial for simulating the WUE at leaf level. The magnitude of e[CO 2 ] effect on g s is modulated substantially together with other environmental variables, if g s is independently acclimated to e[CO 2 ] from P n , this would alter the sensitivity of g s to [CO 2 ], P n , and/or h s , and thereby requiring re-parameterization of the BB-model for plants grown under different CO 2 environment (Ainsworth and Rogers, 2007). As lower g s and higher WUE are anticipated under both e[CO 2 ] and reduced irrigation strategies, and a synergic interaction of those two factors would further decrease g s and enhance WUE (da Silva et al., 2017). However, this will complicate the influence of e[CO 2 ] associated with PRI regime on the prediction of g s and WUE using the BB-model.
To date, the change of g s and WUE of tomato leaves in response to PRI strategy at e[CO 2 ] have not been depicted by any model. Therefore, the objective of this study was to examine whether g s is independently acclimated to e[CO 2 ] and if the BB-model is capable of predicting leaf g s and WUE of tomato leaves exposed to different irrigation regimes in combination with two CO 2 growth conditions. Such modeled g s might provide FIGURE 2 | Relationship between the observed stomatal conductance (g s ) and Ball-index (P n h s /C s ) of a tomato leaf during progressive soil drying at [CO 2 ] concentration of 400 and 800 ppm, respectively in experiment one. an effective way for estimating transpiration rate at canopy scale and optimizing WUE of tomato plant in the future drier and CO 2 enriched environment.

The BB-Model and Its Modification
The BB-model (Ball et al., 1987) describes the relationship between leaf stomatal conductance (g s ) and photosynthetic rate (P n ), relative humidity (h s ) and CO 2 concentration (C s ) on the leaf surface: where g 0 is the residual stomatal conductance if P n is zero, m is the slope of relation between g s and P n h s /C s (the Ballindex), also called the stomatal sensitivity factor. Under wellwatered condition, the m is a constant and this model is a simple linear correlation between g s and Ball-index. However, under soil water deficits, m could be varied largely, and the relationship between g s and P n h s /C s becomes curvilinear (Sala and Tenhunen, 1996). Accounting for the effect of soil water deficits on leaf g s , numerous approaches have been used to adjust the m. The modified m could be based on an exponential function related to the ABA concentration in the xylem sap (Gutschick and Simonneau, 2002). Our earlier studies found that plant ABA could be empirically expressed as a linear function of the mean soil water potential ( s ) in the root zone (Liu et al., 2008). Hereby, in this study, the xylem sap ABA was replaced by s : where m i is the initial slope of the BB-model without soil water deficits, β is a constant. In Equation (1), C s was calculated as: where C a is the atmospheric CO 2 concentration (i.e., 400 or 800 ppm in this study); g b is the boundary layer conductance and shown to be 9.29 mol m −2 s −1 in the leaf chamber according to the manufacture's directions. The h s is computed as the ratio of two partial pressures of water vapor at the leaf surface and in the leaf internal space of stomata, e s /e i . The e s was obtained by Equation (4) as shown by Gutschick and Simonneau (2002): By rearranging Equation (4), h s was calculated as: where e a is the partial pressure of water vapor in the air and is obtained during gas exchange measurement. e i could be computed by Equation (5) It is necessary to notice that g s is used as an input variable for computing h s (Equation 5). The g s could be obtained by rearranging the BB-model as: Equation (7) could then be elaborated as a quadratic equation of g s : (C s /gb)g 2 s +(C s −g 0 C s /g b −mP n /g b )g s +(−(g 0 C s +mP n e a /e i )) = 0 (8) g s was solved as: where A = (C s /g b ), B = (C s -g 0 C s /g b -mP n /g b ) and C = −(g 0 C s +mP n e a /e i ). By applying PROC NLIN (SAS 9.4 Ins. Inc.) of g s on the remaining variables, the parameters g 0 and m were derived. Here, m could be replaced by Equation (2) to consider the effect of soil water deficits on g s . After estimating g s by Equation (9), leaf transpiration rate (T r ) could be calculated as: where P a is the air pressure (1013 hPa). WUE of tomato leaves was then computed as: The aim of this study was to examine the capability of the BBmodel in predicting g s and WUE for tomato leaves at e[CO 2 ] in combination with different irrigation regimes as this has not been done up to date; therefore, we have taken the observed P n as an input for the BB-model rather than developing a coupled model for P n . Moreover, we compared the performance of the original BB-model (without m modification) and the modified BB-model (with m modification by Equation 2) in simulating g s and WUE in order to evaluate the importance of soil water deficits on the model performance under different irrigation regimes and CO 2 enriched environment.

Data
The data for this study are from two pot experiments conducted in a climate-controlled greenhouse at the experimental farm of the Faculty of Science, University of Copenhagen, Taastrup, Denmark. The experimental setups have been detailed elsewhere (Yan et al., 2017;Wei et al., 2018) and are only summarized here.
In both experiments, the tomato seeds (Solanum lycopersicum L., cv. Elin) were sown on 26th Sept. 2016. Half of the plants were grown in a greenhouse cell with ambient CO 2 concentration of 400 ppm (a[CO 2 ]), and another half were grown in a cell with elevated CO 2 concentration of 800 ppm (e[CO 2 ]). In the first experiment (Exp. I), plants were grown in 1.5 L pots filled with peat substance. Since 31st Oct., plants were subjected to progressive soil drying by withholding irrigation from the pots for 5 days when the g s decreased to ca. 10% of that on 31st Oct. (i.e., when pot weight ca. 320 g). Gas exchange measurements (P n and g s ) were made at midday around 10:00 h with a portable photosynthetic system (LiCor-6400XT, LI-Cor, NE, USA). Measurements were performed on one leaf per plant at 20 • C chamber temperature and 1200 µmol m −2 s −1 photon flux density, and at a [CO 2 ] of 400 ppm for a[CO 2 ] and 800 ppm for e[CO 2 ] treatment, respectively. The mean volumetric soil water content in the pots was monitored by weighing the pots daily. The mean s was then obtained based on the water retention curve for the peat substance (Figure 1). Other environmental variables such as e a and T were also obtained during gas exchange measurements. The data obtained from this experiment was used for parameterization. The models were then validated by data obtained from another experiment (Exp. II). In this experiment, tomato plants were grown in pots with roots divided into two equal compartments. The pots (10 L) were filled with a sandy soil. The water retention curve of the sandy soil is shown in Figure 1. Three weeks after transplanting, plants were subjected to three irrigation regimes: (1) full irrigation (FI), in which both soil columns were irrigated daily to 18% (vol.); (2) alternative partial root-zone drying (PRI), in which only one soil column was watered daily to 70% irrigation amount of FI while the other was allowed to dry until the soil water content had decreased to ca. 6%; then the irrigation was shifted; (3) deficit irrigation (DI), in which the same amount of water for PRI was irrigated evenly to the two soil columns. The irrigation treatments lasted 40 days and each soil compartment of the PRI plants had experienced five drying/wetting cycles. The mean soil water content of each soil column was determined by TDR (Time Domain R ctometry; TRASE, Soil Moisture Equipment Corp., USA). The P n and g s and other environmental variables were obtained in the same way as for experiment I.

Statistics
The performance of the original and the modified BB-model was compared by evaluating the coefficient of determination (r 2 ), the mean absolute error (MAE) and the root mean square of error (RMSE) of the linear regressions between the measured and the observed values of g s and WUE. Analysis of covariance (ANCOVA) (SAS 9.4 Ins. Inc.) was performed to reveal the regression lines between vapor pressure deficit (VPD) in the atmosphere and leaf WUE.

Model Parameterization
The data from experiment I was used for model parameterization. In brief, during the 5 days of soil drying s decreased from −0.01 to −0.53 MPa and −0.01 to −0.61 MPa, g s decreased from 0.61 to 0.03 mol m −2 s −1 and 0.39 to 0.03 mol m −2 s −1 , and P n decreased from 15.3 to 1.54 mmol m −2 s −1 and 18.3 to 3.77 mmol m −2 s −1 , for the plants grown at [CO 2 ] concentration of 400 and 800 ppm, respectively.
To calculate the slope (m) of the BB-model, we first calculated h s by using the observed g s values. The Ball-index, viz. P n h s /C s was then computed. By plotting the observed g s against the Ballindex, an exponential, rather than a linear relationship between the two variables was found in both a[CO 2 ] and e[CO 2 ] (400 and 800 ppm) environment (Figure 2). Thus, m was not a constant indicating that g s is not linearly correlated with P n and h s . The original BB-model has a constant slope, m; while the modified BB-model has an initial slope, m i and which is reduced exponentially by soil water deficits (Equation 2). Values in the parentheses are 95% confidence intervals of the parameters (n = 25).
Frontiers in Plant Science | www.frontiersin.org Accordingly, the effects of soil water deficits on m (i.e., Equation 2) must be taken into account. This was done by incorporating Equation (2) (Figure 3;

Model Validation
The BB-models (with or without m modifications) were validated by the data obtained from the experiment II (Wei et al., unpublished).  (Figures 6, 7).
The original model showed poor predictability for WUE (0.21 < r 2 < 0.64) of tomato leaves (Figures 8, 9). The modified BB-model performed better in predicting leaf WUE than did the original BB-model for any FI, DI or PRI tomato plant owing to its greater r 2 (0.46 < r 2 < 0.75), lower MAE and RMSE values in both a[CO 2 ] and e[CO 2 ] conditions. Figure 10 shows there was a negative relationship between VPD and WUE. The correlation between predicted VPD and WUE was not different among the FI, DI, and PRI treatments under the same BB-model and [CO 2 ] environment ( Table 2).

DISCUSSION
In the two pot experiments, the simulation of the BB-model in predicting g s and WUE of tomato leaves grown in a[CO 2 ]  and e[CO 2 ] environment under different irrigation regimes was performed. For both a[CO 2 ] and e[CO 2 ] plants, the original and modified BB-model could explain more than 71% of the observed variation in g s for all irrigation regimes; the modified BB-model was notably superior to the original model in predicting g s and WUE, although the two models showed relatively poor simulation in WUE of tomato leaves (r 2 < 0.75) under different irrigation treatments.
It is well-known that partial stomatal closure leading to decrease in leaf g s during progressive soil drying was mainly induced by the increased root-to-shoot chemical signaling (ABA) in moderate soil moisture deficit (Liu et al., 2005). Moreover, changes in environmental conditions especially h s or VPD might mediate ABA action and influence leaf gas exchange . A high h s or lowered VPD may decrease the delivery and concentration of ABA in the guard cells, resulting in minimal increase in g s and carbon assimilation without greater increase in transpiration (Speirs et al., 2013). In the current study, mostly the parameters e i and e a in the BB-model could well characterize the alteration in h s or VPD and modulation on ABA catabolism in relation to the growth environment of the plants. In addition, the modulation of CO 2 concentration in combination with VPD and soil water deficits on the ABA signaling and its regulation on g s has not been well illustrated and needs further investigations (Yan et al., 2017).
A better understanding of whether there is physiological acclimation of g s to water stress and e[CO 2 ] is crucial for describing plant responses using the BB-model (Miner et al., 2017). The influence of soil water stress on the slope (m) has been contradictory. Misson et al. (2004) measured 350% variation in m of ponderosa pine during the developing season; similarly, Héroult et al. (2013) found that two eucalyptus species had significant reductions in m under drought. However, Xu and Baldocchi (2003) showed that m remains relatively constant for blue oak even under severe water stress. Hence, disparate proposals have been suggested to account for the effect of water deficit on m of the BB-model (Buckley and Mott, 2013). To emphasize the significance of the root-to-shoot ABA signaling in regulating g s during mild soil drying, an ABAbased module was incorporated into the original BB-model to modify the m (Gutschick and Simonneau, 2002;Bauerle, et al. et al., 2004). The common point of those proposals for m modification has been that the initial slope of the BB-model, i.e., m i is scaled downward in response to progressive soil drying (Liu et al., 2009). In the present study, the modified m is related to soil water potential ( s ) and it was found that such modification remarkably improved the predictability of model especially for the drought stressed tomato leaves grown at either a[CO 2 ] and e[CO 2 ] environment (Figures 4,  5). Besides, the m obtained from the model parameterization concentration of 800 ppm. The predictability of the models was compared by the coefficient of determination (r 2 ), the mean absolute error (MAE), and the root mean square of error (RMSE) of the linear regressions between the observed and predicted g s . The dashed lines indicate the 1:1 relationship.
is higher (Table 1) than that observed mean value in C3 crops (Miner et al., 2017).
However, no change of m was noticed in five tree species (Medlyn et al., 2001) and soybean grown in a long-term free-air CO 2 enrichment (FACE) experiment (Leakey et al., 2006), suggesting that acclimation of g s is mostly along with the photosynthetic acclimation, resulting in an unchanged m in e[CO 2 ] condition (Ainsworth and Rogers, 2007;Gimeno et al., 2016). In contrast to those, here the m in e[CO 2 ] was significantly greater than that in a[CO 2 ] environment (Table 1), implying g s was independently acclimated to e[CO 2 ] from P n .
A greater m indicates a higher sensitivity of stomata to the plant growth environment (such as radiation, humidity, soil water availability, and CO 2 concentration). In this study, the intercept of the modified BB-model, i.e., g 0 for tomato leaves was relatively lower than that obtained in an earlier potato study by concentration of 800 ppm. The predictability of the models was compared by the coefficient of determination (r 2 ), the mean absolute error (MAE), and the root mean square of error (RMSE) of the linear regressions between the observed and predicted g s . The dashed lines indicate the 1:1 relationship. Liu et al. (2009), and was not statistically significant from zero. A decreased g 0 indicates a low stomatal conductance in the dark. It has been reported that at e[CO 2 ] the simulated g 0 could be lower, greater or no change as compared with that at a[CO 2 ] (Medlyn et al., 2001;Bunce, 2004;Leakey et al., 2006). The BB-model not only requires environmental variables, also need the plant photosynthesis. Earlier studies have adopted the Farquhar's photosynthesis model (Farquhar et al., 1980) as an integrated  Table 2. module in the BB-model for simulating leaf P n (Gutschick and Simonneau, 2002;Bauerle, et al. et al., 2004;Misson et al., 2004). However, such model is largely apt to a model of the correlation between P n and g s resulted from the interdependence of those two factors, and it is useless for interpreting the causality (Liu et al., 2009). Numerous reports have confirmed that, applying the observed P n as an input variable, the BB-model could performed successfully in predicting g s over different species (Misson et al., 2004;Liu et al., 2009). In addition, accompanied with decreased g s , leaf P n would maintained or increased under PRI or e[CO 2 ] (Pazzagli et al., 2016), which could complicate the simulation of P n for plants grown under PRI strategy combined with e[CO 2 ] environment. Thus, in this study, the measured leaf P n was used to improve the model's performance in predicting g s across a wide range of soil water status. Figures 6, 7 illustrated that the modified BB-model performed better than the original BB-model in predicting leaf g s of the three irrigation treatments at each atmospheric [CO 2 ] concentration, especially for the PRI tomato plant with highest high r 2 , lowest MAE and RMSE values. It is evident that a decreased g s of leaf grown at e[CO 2 ] would affect plant water use and further contribute to the varied s (Kaminski et al., 2014). Likewise, the soil water heterogeneity induced by the PRI treatment could markedly altered the root-to-shoot ABA signaling involved in regulating g s , and the physiological responses, including leaf g s become more sensitive to the reduction of s in the PRI than that in DI plant Liu et al., 2008). Besides, previous study has revealed that xylem ABA concentration of PRI-treated potatoes is determined by s and not by s -dry (Liu et al., 2008), consistent with the results that the model performed much better when using mean s in the whole root zone than using s -dry to account for the effect of soil drying on g s (Liu et al., 2009). Therefore, a modification of m using s incorporating into the BB-model is desired and necessary for achieving a better prediction of PRI tomato g s in the e[CO 2 ] environment.
In the present study, similar to that for g s simulation, the modified BB-model also showed a better predictability for tomato WUE at leaf scale of all irrigation regimes in both [CO 2 ] concentration as compared to the original model, particularly for PRI-treated plant, although the improvement of the prediction was less significant (Figures 8, 9). This could be ascribed to the accumulated errors in several variables for calculating WUE from the modeled g s . Nonetheless, regardless of the s effect, the original and modified BB-model normally could not distinguish the influence of irrigation regimes on WUE of tomato leaves in either [CO 2 ] condition ( Figure 10; Table 2), indicating a robust relationship between WUE and VPD among different soil moisture condition and atmospheric [CO 2 ] concentration. It is well-known that PRI combined with e[CO 2 ] plants could lead to an increase of P n , and a decrease of stomatal aperture, hence synergistically enhancing leaf WUE (Pazzagli et al., 2016). Thus, a greater WUE is expected for the PRI plant grown in e[CO 2 ] environment as simulated by the modified g s and BB-model.
Collectively, tomato g s was independent acclimation to e[CO 2 ] environment from P n . Introducing s of whole root zone into the BB-model could improve the model predictability of the model on g s and WUE of tomato leaves under combinations of different irrigation regimes and CO 2 environments. This information is useful for better predicting WUE of tomato plant in a future drier and CO 2 enriched environment.

AUTHOR CONTRIBUTIONS
ZW: Done the experiment and finished the first manuscript; TD: The corresponding author, the work was supervised by him; XL: Help to do the experiment; LF: Help to do the experiment; FL: The corresponding author, the work was supervised by him.