Estimation of intrinsic water-use efficiency from δ13C signature of C3 leaves: Assumptions and uncertainty

Carbon isotope composition (δ13C) has been widely used to estimate the intrinsic water-use efficiency (iWUE) of plants in ecosystems around the world, providing an ultimate record of the functional response of plants to climate change. This approach relies on established relationships between leaf gas exchange and isotopic discrimination, which are reflected in different formulations of 13C-based iWUE models. In the current literature, most studies have utilized the simple, linear equation of photosynthetic discrimination to estimate iWUE. However, recent studies demonstrated that using this linear model for quantitative studies of iWUE could be problematic. Despite these advances, there is a scarcity of review papers that have comprehensively reviewed the theoretical basis, assumptions, and uncertainty of 13C-based iWUE models. Here, we 1) present the theoretical basis of 13C-based iWUE models: the classical model (iWUEsim), the comprehensive model (iWUEcom), and the model incorporating mesophyll conductance (iWUEmes); 2) discuss the limitations of the widely used iWUEsim model; 3) and make suggestions on the application of the iWUEmes model. Finally, we suggest that a mechanistic understanding of mesophyll conductance associated effects and post-photosynthetic fractionation are the bottlenecks for improving the 13C-based estimation of iWUE.

Carbon isotope composition (d 13 C) has been widely used to estimate the intrinsic water-use efficiency (iWUE) of plants in ecosystems around the world, providing an ultimate record of the functional response of plants to climate change. This approach relies on established relationships between leaf gas exchange and isotopic discrimination, which are reflected in different formulations of 13 C-based iWUE models. In the current literature, most studies have utilized the simple, linear equation of photosynthetic discrimination to estimate iWUE. However, recent studies demonstrated that using this linear model for quantitative studies of iWUE could be problematic. Despite these advances, there is a scarcity of review papers that have comprehensively reviewed the theoretical basis, assumptions, and uncertainty of 13 C-based iWUE models. Here, we 1) present the theoretical basis of 13 C-based iWUE models: the classical model (iWUE sim ), the comprehensive model (iWUE com ), and the model incorporating mesophyll conductance (iWUE mes ); 2) discuss the limitations of the widely used iWUE sim model; 3) and make suggestions on the application of the iWUE mes model. Finally, we suggest that a mechanistic understanding of mesophyll conductance associated effects and post-photosynthetic fractionation are the bottlenecks for improving the 13 C-based estimation of iWUE. KEYWORDS water-use efficiency, carbon isotope discrimination, mesophyll conductance, postphotosynthetic fractionation, climate change, photosynthesis Introduction During photosynthesis, plant stomata act as a control valve for the diffusion of CO 2 and water vapor, regulating the rates of water and carbon exchange between the biosphere and the atmosphere (de Boer et al., 2011;Adams et al., 2020;Walker et al., 2021). Intrinsic water-use efficiency (iWUE), defined as the ratio of net photosynthetic rate (A n ) to stomatal conductance for water vapor (g sw ), plays a key role in quantifying carbon uptake and water loss at leaf to continental scales (Seibt et al., 2008;Keenan et al., 2013). The response of iWUE is fundamental to climate change research since small changes in iWUE can have profound impacts on global carbon and water cycles. Furthermore, iWUE can provide insights into the mechanisms of plant physiological responses to climate change and support the screening and breeding of climate-resilient crops (Farquhar and Richards, 1984;von Caemmerer et al., 2014;Gresset et al., 2014). Central to these research domains is the quantification of iWUE.
Stable carbon isotope discrimination (D) can be used as an integrated measure of iWUE in C 3 plants (Farquhar et al., 1989). Plants discriminate against 13 C in favour of 12 C during photosynthetic CO 2 assimilation in C 3 leaves, and the variation in carbon isotope composition (d 13 C) from source CO 2 to photosynthetic products (e.g., bulk leaf organic carbon or sugars) is termed as D, following Farquhar et al. (1982b);Farquhar et al. (1989): where atmospheric d 13 C a is approximately -7~-8‰ during the 20th century. D can also be estimated from d 13 C of CO 2 entering (d in and C in ) and leaving (d out and C out ) the cuvette during gas exchange, termed as online 12 C/ 13 C discrimination (Evans et al., 1986): where x= C in /(C in -C out ). In this way, D can be measured nondestructively to probe real-time responses of photosynthesis at high temporal resolution. Changes in photosynthetic parameters (A n and g s ) are captured in D online and the isotopic signatures are further imprinted on plant tissues during biosynthesis. As such, biomass-based D reflects physiological status of plants throughout the growth period of plant tissues (Cernusak et al., 2013;Soh et al., 2019). Different from classical approaches such as gas exchange or growth analysis, biomass-based D can be applied retrospectively, providing a useful record of iWUE at large spatial and temporal scales (Frank et al., 2015;Adams et al., 2020;Gong et al., 2022).
Inferring iWUE from isotopic records relies on theoretical models. In the current literature, most studies have utilized the simple, linear equation of photosynthetic discrimination to estimate iWUE. However, it can be problematic to interpret iWUE using this linear model which ignores effects other than diffusion through stomata and carboxylation. For instance, Seibt et al. (2008) suggested that the uncertainty in iWUE-13 C models was related to the simplification of mesophyll conductance (g m ). g m represents the conductance to CO 2 diffusion from the intercellular space to the carboxylation site in chloroplasts, a key limiting factor of photosynthesis in addition to stomatal conductance and biochemical capacity (Tholen et al., 2012;Stangl et al., 2019). However, recent advances in d 13 C-based iWUE estimation have not been systematically reviewed. The main objective of this mini review is to concisely summarize the theoretical basis and uncertainties of d 13 C-based iWUE models. We (i) present different formulations of D and the associated assumptions, (ii) present D-based iWUE models derived from those formulations: the classical model (iWUE sim ), the comprehensive model (iWUE com ), the model incorporating g m (iWUE mes ), (iii) discuss the limitations of the widely used iWUE sim model; and make suggestions on the application of the iWUE mes model.

Comprehensive model of photosynthetic 13 C discrimination and simplifications
A comprehensive description of 13 C discrimination (D com ) during C 3 photosynthesis was given by Farquhar et al. (1982b) and extended to include ternary effects of transpiration on CO 2 assimilation by Farquhar and Cernusak (2012): and where a b (2.9‰) and a s (4.4‰) are fractionations associated with the diffusion of CO 2 through leaf boundary layer and in the air, respectively. a m (1.8‰) is the fractionation associated with the dissolution and diffusion of CO 2 in mesophyll (see Table S1 for the list of parameters). C a , C s , C i and C c represent the mole fraction of CO 2 in air, at leaf surface, in the intercellular spaces and chloroplast, respectively ( Figure 1). D c o m can be separated into a series of fractionation components of leaf boundary layer conductance (D gbc ), stomatal conductance (D gsc ), mesophyll conductance (D gm ), Rubisco (ribulose-1,5-bisphosphate carboxylase/ oxygenase) carboxylation (D b ), day respiration (D e ), and photorespiration (D f ). Note that t is included to account for the ternary effects of transpiration rate (E) on photosynthetic discrimination (Farquhar and Cernusak, 2012). Usually, the effect of t is small and can be omitted under low or moderate vapor pressure deficit (VPD) (Farquhar and Cernusak, 2012;Evans and Caemmerer, 2013). If D gbc is also omitted (C a =C s and g ac =g sc ), the D com model is simplified as: Diagram of the CO 2 diffusion pathway in C 3 leaves and different formulations of iWUE (iWUE com , iWUE mes R , iWUE mes , and iWUE sim ) derived from the Farquhar et al. model for photosynthetic 13 where the subscript "mes R" indicates that the expression takes mesophyll conductance, day respiration and photorespiration into account.
D e , the respiratory contribution to discrimination is mainly determined by respiratory fractionation (e), and R d /(An+ R d ). D e has rarely been accurately quantified largely due to the difficulty of estimating R d (Tcherkez et al., 2017;Gong et al., 2018). Moreover, fractionation of day respiration has rarely been reported, and e estimated from respiration in the dark varies between 0 and -6‰ Tcherkez et al., 2010). Under natural conditions, D e is usually small and negligible (Seibt et al., 2008;Ubierna and Farquhar, 2014). Notably, a significant apparent respiratory fractionation may occur when the CO 2 source used for combined gas exchange and isotopic measurements has a d 13 C differed from that of the ambient air (Gillon and Griffiths, 1997;Gong et al., 2015). Under such conditions, e should be corrected to account for the isotopic disequilibria between photosynthetic and respiratory fluxes (Wingate et al., 2007;Gong et al., 2015). Assuming D e =0, Equation 6 can be simplified as: C c is usually unknown since its calculation requires g m which cannot be directly measured. g m is assumed to be infinite in early studies (for a review see Flexas et al., 2012); that is, CO 2 mole fraction in the chloroplast is equal to that in the intercellular space. Assuming C i =C c and D f =0, Equation 7 is simplified as: Comprehensive model of iWUE and simplifications The comprehensive model of iWUE which includes all fractionation components of Equation 3 was first derived by Ma et al. (2021): where e'=ea b /a e , f'=fa b /a f and є=(1/k-W/2)/(1/k+W/2). This formulation is particularly useful for assessing the contribution of each fractionation component to iWUE estimates. Ma et al. (2021) performed sensitivity tests using theoretical data of the standard photosynthetic scenarios. Their results indicated that ternary correction and D gbc had little influence on iWUE com estimates (error< 2 mmol mol -1 ), which is in agreement with Seibt et al. (2008). Neglecting the contribution of t and D gbc , Equation 9 can be simplified as:

Equation 10
D e in the iWUE mes R model could be ignored as it caused an error of less than 2 mmol mol -1 in typical photosynthetic scenarios (Ma et al., 2021). Excluding the contribution of day respiratory, Equation 10 can be simplified as: The iWUE mes model provided iWUE estimates that are numerically very similar with iWUE com (error< 3 mmol mol -1 ) (Ma et al., 2021). Neglecting the contribution of g m and photorespiration, the simplified equation for iWUE is given as: This linear relationship between iWUE and photosynthetic 13 C discrimination is the most used to estimate iWUE, however, the limitations of this formulation have been raised (Seibt et al., 2008;Ubierna and Farquhar, 2014;Ma et al., 2021).

Uncertainty in iWUE sim estimation associated with mesophyll conductance
Experimental evidence shows that g m exerts a significant limitation on CO 2 diffusion and leads to a significant drawdown from C i to C c (Loreto et al., 1992;Flexas et al., 2008;Cano et al., 2014). It is apparent from Equation 11 that, assuming an infinite g m will lead to overestimation of iWUE, this is supported by experimental observations (Barbour et al., 2010;Stangl et al., 2019;Adams et al., 2020). Experimental results showed that the relationship between D and water-use efficiency is at least partly a function of g m (Warren and Adams, 2006). Therefore, it is important to incorporate g m in the parameterization of the iWUE model. Ma et al. (2021) showed that iWUE sim overestimated iWUE by c. 65%. Importantly, the magnitude of overestimation is dependent on D, making correction using empirical relations difficult. These results raise concerns regarding the accuracy of iWUE sim estimations.
The overestimation of the iWUE sim model has also been observed in recent studies using 13 C series of environmental archives (Baca Cabrera et al., 2021;Bing et al., 2022). More importantly, iWUE sim model could provide biased estimations of historical iWUE trend. Gong et al. (2022) analyzed tree ring 13 C series across the globe using the iWUE mes model, and reported that iWUE sim model significantly overestimated iWUE (by c. 100%) and the rate of iWUE gain with time or C a (by c. 70%) during the 20th century. This finding has been confirmed by studies carried out in distinct ecosystems (Bing et al., 2022;Mathias and Hudiburg, 2022). Failure to consider g m must lead to an overestimated historical trend as implied by the partial derivative of iWUE mes (Equation 11) : Given that 13 C series of environmental archives (e.g. tree rings) provide a unique proxy for benchmarking the output of land surface models (Frank et al., 2015;Wang et al., 2017;Lavergne et al., 2022), cautions should be paid when iWUE sim model is used to predict historical trend of iWUE.

Uncertainty in iWUE sim estimation associated with b'
The value and physiological meaning of b' in the equation of D sim or iWUE sim remain subjects of debate. Initially, Farquhar et al. (1982a) proposed that b' (27‰) could be derived from early in vitro estimations of Rubisco carboxylation. As it agreed well with the relationship between measured biomass-based D and C i /C a , it was interpreted as a fitted value. However, when measured D from online instantaneous measurements was used to fit Equation 8, the fitted b' appears to be lower than 27‰ (Caemmerer and Evans, 1991;Ma et al., 2021).
b' was also explained as the net fractionation caused by Rubisco and PEPC (phosphoenolpyruvate carboxylase). Farquhar and Richards (1984) described b' as a function of relative contribution of Rubisco and PEPC carboxylation: where b (29-30‰) and b 4 (usually taken as -5.7‰ at 25°C) are fractionation factors of Rubisco and PEPC carboxylation, respectively. b is the proportion of carbon fixation through PEPC carboxylation. PEPC uses HCO À 3 produced by CO 2 hydration as the substrate for the synthesis of aspartate or malate, which is important for the control of cellular pH (Davies, 1979). Generally, carboxylation by Rubisco contributes a greater fraction of carbon in plants and respiratory substrates. But it is also suggested that the PEPC carboxylation could be important under the conditions of low stomatal conductance or carboxylation in darkness (Ikeda and Yamada, 1981;Gupta et al., 1994;Hibberd and Quick, 2002). Furthermore, several studies have revealed that N source and concentration were potential factors affecting carbon fixation by PEPC, which indicates that b' could vary with nitrogen metabolism (Raven and Farquhar, 1990;Douthe et al., 2012;Lian et al., 2021). That is, Equation 14 is not particularly useful for iWUE estimation because b is variable and difficult to quantify.
b' has also been described by Ubierna and Farquhar (2014) as a parameter that included carboxylation, mesophyll conductance, and photorespiration: According to Equation 15, b' is largely dependent on C c /C i which is modulated by g m . It should be noted that the most used b'=27‰ is consistent with the C c /C i value of 0.9, higher than the common values of 0.7-0.8 (Caemmerer and Evans, 1991;Warren et al., 2003). So far, Equations 14 and 15 have only been used to discuss the potential origin of variation in b', but have not been incorporated in the model of iWUE estimation. In short, there is still no consensus concerning the interpretation of b', and current discussion on b' (Equations 14, 15) illustrated that it should not be treated as a constant value of 27‰.

Uncertainty in iWUE associated with post-photosynthetic fractionation
Post-photosynthetic fractionation (D post ) includes the discrimination processes that follow photosynthetic carbon fixation, altering d 13 C signals in plant organs and leaves at different development stages (Badeck et al., 2005;Vogado et al., 2020). In general, heterotrophic organs (branches, stems and roots) are 13 C-enriched compared with autotrophic organs (leaves) (Badeck et al., 2005;Bowling et al., 2008;Cernusak et al., 2009;Lamade et al., 2016), and the immature leaves (heterotrophic phase) are 13 C-enriched (by c. 2‰) compared to mature leaves (autotrophic phase) in both deciduous and evergreen species (Lamade et al., 2009;Vogado et al., 2020). However, the contribution of D post to d 13 C of plant tissues and its influence on iWUE estimation are poorly understood.
Several studies has accounted for D post to estimate iWUE (Table S2). Gimeno et al. (2021) found an improvement in correlation between iWUE estimated from gas exchange and that from D when g m and D post were accounted for. In that study, D post was taken as -2.5‰ estimated from the d 13 C difference between phloem contents and whole-tree photosynthesis. Similarly, d 13 C of wood and cellulose have been corrected by -3.2‰ and -1.3‰, respectively, to account for the offset from leaf d 13 C ( Thomas et al., 2013;Brownlee et al., 2016). In other studies, iWUE was calculated from tree-ring with a d 13 C offset of -2‰ to account for D post (Michelot et al., 2011;Frank et al., 2015). Without correcting a D post of about 2.5‰, iWUE estimated from the d 13 C of tree-ring could be overestimated by 20% (Gessler et al., 2009).
The likely mechanisms underlying D p o s t include fractionation associated with respiration, transport and mixing of assimilates Brüggemann et al., 2011;Bögelein et al., 2019). Respiratory fractionation ranges from -6 to 0‰ in various species (Duranceau et al., 1999;Ghashghaie et al., 2001;Tcherkez et al., 2003). Also, there are some variations in the apparent fractionation during transport processes (e.g., daynight differences in d 13 C of leaf-export organic matter and different leaf-to-phloem d 13 C signatures along vertical canopy gradients) and very few direct measurements of isotopic differences between components at molecule/atom scale (Gessler et al., 2008;Mauve et al., 2009;Gilbert et al., 2011;Gilbert et al., 2012;Bögelein et al., 2019). In addition, postphotosynthetic fractionation is complicated by ontogenic effects (e.g., size, height, and age of individuals) that can confound the relationship between iWUE and environmental factors (Vadeboncoeur et al., 2020). That is, the influence of D post could accumulate over time and lead to age-dependent patterns (Cernusak et al., 2009). Therefore, using a constant, empirical value of D post could be unreliable. A mechanistic description of D post should be very useful to improve the iWUE estimates, which requires further study on the fractionations associated with respiration, transport and allocation of assimilates. iWUE mes , a useful, simplified model for estimating iWUE iWUE mes takes the influence of mesophyll conductance and photorespiratory fractionation into account. We propose to use the iWUE mes model (Equation 11) since it considers the components that have a significant influence on the iWUE estimation. In particular, it includes g m effect which is known to affect iWUE prediction.
Parameterizing the iWUE mes model requires g sc /g m rather than g m (Ma et al., 2021). Some studies use a constant g m in the equation to estimate iWUE (Keeling et al., 2017), which makes more sense than disregarding g m . However, the assumption of constant g m is not supported by experimental evidence. The positive relationships between g sc and g m have been reported in many studies (Flexas et al., 2013;Gong et al., 2018), and this relationship is rather conserved across levels of CO 2 , irradiance, and drought stress and functionally distinct species (Flexas et al., 2008;Ma et al., 2021;Gong et al., 2022). Incorporating the g sc /g m ratio improves the predictive accuracy of the iWUE model, as demonstrated in gas exchange experiments (Ma et al., 2021). Without knowing g m , it is preferable to use the average g sc /g m of 0.79 ( ± 0.07) derived from a global synthesis to parameterize the iWUE mes model rather than using the iWUE sim model.
We acknowledge that using a constant g sc /g m is not always adequate. As more g m data become available, interspecific differences in g sc /g m can be identified and should be accounted for in the estimated iWUE. Theoretically, species-specific g sc /g m is more appropriate to be used in the iWUE mes estimation. It is also noteworthy that short-term responses of g m are still not well defined, implying that neglecting short-term variation in g sc /g m might lead to errors in estimating iWUE at nonsteady-states, thus should be addressed in further work. Moreover, the iWUE mes model does not account for the post-photosynthetic fractionation, due to a lack of knowledge on post-photosynthetic fractionation. Therefore, we recommend for biomass-based analyses to distinguish the age of organs to minimize the influence of post-photosynthetic fractionation.

Conclusion remarks
The comprehensive model of photosynthetic 13 C discrimination of Farquhar and Cernusak (2012) is a synthesis of current understanding, and provides the theoretical basis for estimating iWUE from the 13 C composition of plant materials. The classical iWUE sim model has been shown to strongly overestimate iWUE and its historical trends due to the neglect of g m associated effect, limiting its use in quantitative studies. iWUE mes is suggested as a useful, simplified model for quantitative estimation of iWUE, which has been included in a standardized, open-source tool (R package) for calculation of iWUE from stable isotope signatures (Mathias and Hudiburg, 2022). Nonetheless, the formulation of iWUE mes could still be further improved. For example, a fixed, empirical g sc /g m value could be replaced by species-specific values or mechanistic relationships derived from experimental results. 13 C discrimination of plant material, combining with appropriate iWUE models, is also an ultimate tool for screening genetic resources to enhance the iWUE of crops under climate change scenarios. One of the primary questions is how g sc and g m of plants will respond to changes in temperature, water availability, and carbon dioxide concentration. Furthermore, the response of postphotosynthetic fractionation to climate change factors remains unknown. We conclude that mechanistic descriptions of g m associated effect and post-photosynthetic fractionation are the bottlenecks for improving the 13 C-based estimation of iWUE.

Author contributions
XG conceptualized the topic of this review, WM and YY wrote the first draft, and all authors contributed to the writing and revision of the manuscript. All authors contributed to the article and approved the submitted version.

Funding
This work was supported by the National Natural Science Foundation of China (NSFC 31870377, 32120103005, 32201277).

Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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