Stomata regulate the rate of water vapour escaping leaves (transpiration, E) by adjusting stomatal aperture, inevitably restricting the rate of CO2 uptake, called assimilation (A). This results in a difference in CO2 concentration from higher atmospheric (Ca), to lower internal concentration (Ci). The ease with which water vapour permeates stomata is referred to as stomatal conductance to water (gSW), which is higher than that for CO2 due to the smaller size of the H2O molecule (gSW≈1.6gSC). The theoretical basis for identifying the extent to which stomata restrict assimilation was laid by Gaastra (1959) and further developed by Jones (1985). Stomatal restriction was associated to sensitivity: how much A varies in response to a change in gSC. Mathematically, the latter is achieved by expressing A as a function of multiple variables and calculating the change in A resulting from a small alteration of gSC, while keeping all other variables unchanged. In formal notation, this is the partial derivative, \(\partial A/\partial {g}_{\mathrm{SC}}\).

The trade-off between assimilation and the rate of water loss can be represented by a similar metric, the marginal carbon revenue of water, \(\partial \mathrm{A }/ \partial E\) (Buckley et al. 2017), or, alternatively, by water use efficiency. The notion of water use efficiency had its origins in agronomy, and measured the production (grain harvested, above ground biomass, total plant or biomass) per unit of water used (water supplied through irrigation, rain, water transpired, or evapotranspiration) (Hatfield and Dold 2019). At the leaf level, two formulations of water use efficiency are commonly used. The ratio between assimilation and gSW is referred to as intrinsic water use efficiency (iWUE), while the ratio between assimilation and transpiration is often termed instantaneous water use efficiency. However, this terminology can lead to confusion. To mitigate this ambiguity, I encourage using the term ‘instantaneous transpiration efficiency’ (ITE) for the ratio between A and E, as suggested by Duursma et al. (2013). While iWUE is dependent solely on leaf properties, ITE increases without bound as air humidity rises, irrespective of the plant. This is because transpiration is the product of gSW and the air-to-leaf water mole fraction gradient, DS \(( {\text{ITE}}= {}_{\mathrm{i}}{\text{WUE}}/{D}_{\mathrm{S}})\). Consequently, iWUE is typically favoured over ITE when comparing plants (Table 1).

Table 1 Acronyms, definitions, variables, and units used
Full size table

Regression analysis, a widely employed statistical technique, offers a easily accessible approach for exploring relationships between variables. The output of regression analysis is the slope and the intercept of the line that best fits this relationship, and, optionally, the associated statistics. Unfortunately, confusion between the partial derivative, the slope of the regression, and the average value of the datapoints used for the regression has led to the incorrect utilization of regression analysis as a substitute for sensitivity analysis or water use efficiency. I will present two instances of these concepts being confounded.

The slope of the regression of A on g SC does not measure stomatal control over assimilation

The simple argument that if plants with a steeper slope in the regression of A against gSC were to increase gSC, they would enhance A more than plants with a lower slope has been frequently put forth (Yan et al. 2016; Kawamitsu et al. 1993; Carriquí et al. 2015), but is actually incorrect. Assimilation typically responds to multiple factors, therefore \(\partial A/\partial {g}_{\mathrm{SC}}\) captures the incremental change in CO2 assimilation resulting from a slight alteration in stomatal conductance to CO2, while maintaining other factors constant. However, in experimental conditions, any change observed in gSC is typically not independent of other variables affecting A. Instead, gSC may respond to environmental drivers that typically also affect other processes involved in photosynthesis. For instance, the imposition of drought leads to alterations in the plant water relations, stomatal closure, reduced biochemical activities, and decreased efficiency in energy conversion processes (Bellasio et al. 2023). These changes can impact intercellular and intracellular CO2 and bicarbonate diffusion, light interception, structural integrity, and nutrient uptake, all of which have a depressive effect on assimilation (Lawlor and Cornic 2002). When a population of experimental datapoints of A are plotted over gSC, the response of A will therefore aggregate both the impact of gS on A and the influence of other drivers on A, hence include both stomatal and non-stomatal effects. As a result, the slope of a regression fitted through experimental data points of A against gSC is typically not indicative of \(\partial A/\partial {g}_{\mathrm{SC}}\). As Wong (1979) wrote, “linear relationships between A and gS do not necessarily indicate that stomata control the rate of assimilation”. Postulating a link between stomatal control and the slope of the regression fitted to a set of empirical A and gSC data pairs becomes particularly problematic when comparing C3 and C4 plants. This is due to the fact that regression lines of A to gSC are typically steeper in C4 plants than in C3 plants (Quirk et al. 2019); however, C4 plants usually have less advantage than C3 plants when they open their stomata (Farquhar and Sharkey 1982). I will now illustrate this numerically.

To calculate operational values of A and gS starting from input values of stomatal and non–stomatal limitations (LS and LNS), I start from describing the dependence of A upon Ci with a function after Prioul and Chartier (1977) in the formulation of Bellasio et al. (2016b) as:

$${A}=\frac{{\text{CE}} \left({C}_{\mathrm{i}}-\varGamma \right) + {A}_{\mathrm{SAT}}-\sqrt{{({\text{CE}}[{C}_{\mathrm{i}}-\varGamma ]+{A}_{\mathrm{SAT}})}^{2}-(4 \omega {A}_{\mathrm{SAT}} {\text{CE}} [{C}_{\mathrm{i}}-\Gamma ]})}{2\omega },$$
(1)

where ASAT is the CO2-saturated rate, CE is the initial slope, Γ is the x-intercept, ω is defining curvature. Typical A/Ci curves (Fig. 1A) were obtained using a generic C3 and C4 parameterisation (inset in Fig. 1B).

Fig. 1
figure 1

Simulation of iWUE, \(\partial A/\partial {g}_{\mathrm{SC}}\), and regression of A on gS. Panel A shows the modelled dependence of C3 (thick dashed red line) and C4 (thick solid blue line) assimilation (A) upon CO2 concentration in the substomatal cavity (Ci, Eq. 1), parameterised to generically represent the photosynthetic optimum for young leaves under moderate light intensity (inset in Panel B). The rationale of the simulations is exemplified for C3 assimilation. Point 1 (Ca, APot), lying on the uppermost A/Ci curve, shows the potential assimilation attained if intercellular spaces were directly exposed to external CO2 concentration, Ca (Eq. 1 calculated for Ci = Ca). In Point 2 (Ca, APotCa) assimilation is only limited by non-stomatal limitations (LNS, Eq. 2); the thin dashed line is the C3 A/Ci curve passing through APotCa, derived by reducing saturated rate (ASAT) and initial slope (CE) proportionally (Eq. 3). Point 3 (Ciop, Aop) is a generic operational point where stomatal limitation (LS) may occur (Eq. 4); the slope of the supply function s (Eq. 6) is gSC. Panel B shows the response of assimilation to gSC obtained under a Ca of 420 μmol mol−1 for plants operating at their optimum, or with non-stomatal limitation (LNS = 0.2). The inset shows model inputs, modified from Bellasio et al. (2016a, b) to generically describe C3 and C4 assimilation. Panel C shows the plot of Aop and gSCop obtained using 400 random combinations of LNS and LS between a minimum and maximum shown in the inset. Outputs in the inset are the mean iWUE (Aop/gSWop) calculated for 400 (gSWop, Aop) points; the mean of the \(\partial A/\partial {g}_{\mathrm{SC}}\) of the A/gSC curves passing through each of the 400 (gSCop, Aop) points, and the slope of the regression fitted to the same points after adding normally distributed noise with a standard deviation of 10% of the simulated Aop and gS. Panel D shows the plot of Aop and gSCop obtained using the LNS and LS limits shown in the inset, output as in Panel C. Units of slopes and iWUE are \(\frac{\mu \,\mathrm{mol }\,{\mathrm{CO}}_{2}}{\mathrm{mol Air}}\)

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The potential assimilation that would occur if intercellular spaces were directly exposed to Ca is calculated by substituting Ca for Ci in Eq. 1 (APot Fig. 1A, point 1). Ordinarily, non-stomatal limitation (LNS) may reduce assimilation to a point (Fig. 1A, point 2) called APotCa, which, from the definition of non-stomatal limitation (Bellasio et al. 2018) is:

$${A}_{\mathrm{PotCa}}={A}_{\mathrm{Pot }}\left(1-{L}_{\mathrm{NS}}\right).$$
(2)

To find the A/Ci curve passing through APotCa one can assume that CE and ASAT scale linearly to the values of CEʹ and ASATʹ. If the ratio \(\mu ={A}_{\mathrm{SAT}}/{\text{CE}}\) is invariant, then the standard quadratic form of Eq. 1, \(\left\{\omega {A}^{2}- \left({\text{CE}}\left[{C}_{\mathrm{i}}-\Gamma \right]+{A}_{\mathrm{SAT}}\right)A+ {A}_{\mathrm{SAT}} {\text{CE}} \left[{C}_{\mathrm{i}}-\Gamma \right]=0\right\},\) can be solved for CEʹ as:

$${\text{CE}}^{\prime} = \frac{{A(\mu + C - \Gamma ) + \sqrt {[A(\Gamma - C - \mu )]^{2} - [4\omega A^{2} \mu (C - \Gamma )]} }}{{2\mu (C - \Gamma )}}$$
(3)

CEʹ, the estimated value of CE of the leaf in the presence of LNS, is calculated by inputting APotCa (Eq. 2) and Ca in Eq. 3; ASATʹ, the estimated value of ASAT in the presence of LNS, is, by assumption, μ CEʹ.

Stomatal limitation (LS) would further reduce assimilation to the ‘operational’ Aop (Fig. 1A point 3), which, from the definition of stomatal limitation (Farquhar and Sharkey 1982) is:

$${A}_{\mathrm{op}}={A}_{\mathrm{PotCa}}-{A}_{\mathrm{Pot}} {L}_{\mathrm{S}}.$$
(4)

The new corresponding Ciop is found by solving Eq. 1 as:

$$C_{\text{iop}} = \frac{{\omega A_{\text{op}} ^{2} + {\text{CE}}^{\prime}\Gamma A_{\text{op}} - A_{\text{SAT}} ^{{\prime 2}} A_{\text{op}} - A_{\text{SAT}} ^{\prime } {\text{CE}}^{\prime}\Gamma }}{{{\text{CE}}^{\prime}A_{{{\text{op}}}} - A_{\text{SAT}} ^{\prime } {\text{CE}}^{\prime}}}.$$
(5)

Finally, gSC is:

$${g}_{\mathrm{SC}}=\frac{{A}_{\mathrm{op}}}{{C}_{\mathrm{a}}-{C}_{\mathrm{iop}}}.$$
(6)

The general behaviour of the model is shown if Fig. 1B. When there is only LS, at any given gSC the points (Ciop, Aop) satisfying Eq. 1 lie on the upper A/Ci curves. Introducing LNS lowers Aop mainly when gSC is high.

Four hundred random combinations of LNS and LS were generated in a manner that they fall within the minimum and the maximum intervals specified in the inset of Fig. 1C, and their sum would be within the range of 0.05 to 0.5. The proportion of LS was intentionally set lower for C4 assimilation than for C3, replicating physiological operational conditions (Ghannoum et al. 2003; Bellasio et al. 2018). Equations 4 and 6 were employed to simulate A and gSC, respectively, 10% normally distributed-error was added, and a regression line was fitted to the resulting (Ciop, Aop) pairs (Fig. 1C). Each (Ciop, Aop) pair corresponds to an A/Ci curve (Eq. 1) parameterised with CEʹ, ASATʹ (Eq. 3), ω and Γ (inset in Fig. 1B). For each of these A/Ci curves, \(\partial A/\partial {g}_{\mathrm{SC}}\) was calculated numerically at each (Ciop, Aop) point for small increments of LS in Eq. 3.

Despite imposing a physiologically lower proportion of LS in C4 assimilation, the slope of the regression line fitted to the simulated data points was higher for C4 than C3 assimilation. The average \(\partial A/\partial {g}_{\mathrm{SC}}\) of the individual A/gSC curves passing through the (gSCop, Aop) pairs differed from the slope of the regression line fitted through the same (gSCop, Aop) pairs (insets of Fig. 1C and 1D). This means that the slope of the regression line fitted through operational Aop and Ciop does not serve as a measure of LS or of \(\partial A/\partial {g}_{\mathrm{SC}}\).

The slope of the regression of A on g SW is not water use efficiency

The slope of the regression for a set of gas exchange points measured in operational conditions (gSWop, Aop) has been erroneously employed to measure the average iWUE of a set of data points, as seen in studies like Vogan and Sage (2011) and Killi et al. (2017). However, the slope of a regression fitted to a set of (gSWop, Aop) pairs would equal their average iWUE only if the unconstrained regression line passed through the origin, which is typically not the case.

To illustrate this numerically, four hundred random combinations of LNS and LS were generated in a way that they fall within the minimum and the maximum intervals specified in the inset of Fig. 1D. Subsequently, assimilation and gSC were simulated as described above, and a regression line was fitted to the resulting (gSCop, Aop) pairs (Fig. 1D) and (gSWop, Aop) pairs. The slope of the regression fitted through the (gSWop, Aop) pairs (inset in Fig. 1D) was lower in C4 than in C3, but the ranking of iWUE was in the opposite order (inset in Fig. 1D). This shows that the slope of the regression of Aop on gSW may not even scale to iWUE, and therefore should not be employed as a measure of iWUE.

Instead of by fitting a regression line through the (gSWop, Aop) pairs, water use efficiency should be calculated for each individual data point as \({}_{\mathrm{i}}{\text{WUE}}=A/{g}_{\mathrm{SW}}\). The implementation is entirely straightforward, because gas exchange measurements automatically provide the necessary information (A and gSW).

Conclusion

I have provided numerical evidence that inferring water use efficiency or stomatal control over assimilation from the slope of the empirical regression of operational CO2 assimilation on stomatal conductance is inappropriate. I recommend calculating both metrics: ‘intrinsic water use efficiency’ (representing the assimilation-to-stomatal conductance ratio that is independent of air humidity and therefore preferable for plant comparisons) and ‘instantaneous transpiration efficiency’ (the term I suggest reserving for the assimilation-to-transpiration ratio), for each individual gas exchange data point.