An integrated model of stomatal development and leaf physiology



  • Stomatal conductance (gs) is constrained by the size and number of stomata on the plant epidermis, and the potential maximum rate of gs can be calculated based on these stomatal traits (Anatomical gsmax). However, the relationship between Anatomical gsmax and operational gs under atmospheric conditions remains undefined.
  • Leaf-level gas-exchange measurements were performed for six Arabidopsis thaliana genotypes that have different Anatomical gsmax profiles resulting from mutations or transgene activity in stomatal development.
  • We found that Anatomical gsmax was an accurate prediction of gs under gas-exchange conditions that maximized stomatal opening, namely high-intensity light, low [CO2], and high relative humidity. Plants with different Anatomical gsmax had quantitatively similar responses to increasing [CO2] when gs was scaled to Anatomical gsmax. This latter relationship allowed us to produce and test an empirical model derived from the Ball–Woodrow–Berry equation that estimates gs as a function of Anatomical gsmax, relative humidity, and [CO2] at the leaf.
  • The capacity to predict operational gs via Anatomical gsmax and the pore-specific short-term response to [CO2] demonstrates a precise link between stomatal development and leaf physiology. This connection should be useful to quantify the gas flux of plants in past, present, and future CO2 regimes based upon the anatomical features of stomata.


Gaseous diffusion through the stomatal pores of terrestrial plants plays a fundamental role in primary productivity, ecosystem health, and global energy cycles (Berry et al., 2010). Stomata are formed by two symmetric epidermal guard cells that regulate this flow by changing the size of the pore in response to both external environmental and internal plant cues (Kim et al., 2010). This cellular activity is ultimately aimed at limiting water lost for carbon gained and enables plants to optimize gas exchange with their surrounding environment (Cowan & Farquhar, 1977). In addition to such physiological controls, the number of stomata on the epidermis also constrains the maximum possible diffusive capacity, which, in turn, is a consequence of the genetic and cellular machinery that controls stomatal development and patterning (Lau & Bergmann, 2012). This points to a fundamental connection between stomatal development and plant physiology that begs the question, what role does stomatal development play in the optimization of gas exchange and plant productivity?

In over 300 yr of stomatal research, we have yet to realize a comprehensive mechanistic model linking such modalities (Meidner, 1987). A mix of modeling approaches has been effective in reproducing components of the system and most of these have focused on how environmental or biochemical factors contribute to operational stomatal conductance (gs) (Farquhar et al., 1980; Leuning, 1995; Buckley et al., 2003; Konrad et al., 2008). Anatomical gsmax (previously termed gmax, gsmax, or gwmax) represents the maximum rate of gs to water vapor as determined by stomatal size and density in a diffusion-based equation (Eqn 1 in the 'Materials and Methods' section) (Brown & Escombe, 1900; Franks & Farquhar, 2001; Franks & Beerling, 2009b). The concept of Anatomical gsmax has been useful for assessing stomatal development with respect to a variety of phenomena, including: responses across temporal scales and changing climates, or among phylogenetic families (Franks & Beerling, 2009b; Taylor et al., 2012); quantifying differences in plant physiology and growth, or ecological adaptation (Franks et al., 2009; Lammertsma et al., 2011; Doheny-Adams et al., 2012); and modeling the influences of gas flux on climatic forcing (de Boer et al., 2011). These studies have made valuable contributions to the field despite the fact that operational gs is almost always less than Anatomical gsmax. Elucidating the precise relationship between these two gas-exchange parameters would provide an important insight into how plants balance stomatal development and physiology with environmental perception.

Anatomical gsmax itself is the product of the internal signaling elements and external environmental cues that define stomatal development, which determines the number and size of stomata on the epidermis of the leaf (Bergmann & Sack, 2007; Casson & Hetherington, 2010). A suite of genetic tools is now available that can alter stomatal patterning while maintaining stomatal function (Lau & Bergmann, 2012). We took advantage of this ability to manipulate stomatal production in a single species to examine the impacts of altering Anatomical gsmax without sensitizing the plant to different environmental growth conditions or comparing diverse plant species. These tools also allowed us to specifically modify stomatal characteristics without altering other components of plant function or form. Here we show that plants with different Anatomical gsmax had quantitatively similar responses to increasing [CO2] when operational gs was scaled to Anatomical gsmax. This relationship allowed us to produce and test an empirical model derived from the Ball–Woodrow–Berry (BWB) equation that estimates gs as a function of Anatomical gsmax, relative humidity (RH), and [CO2] at the leaf.

Materials and Methods

Plant material and growth conditions

All genotypes tested were in the Columbia (Col-0) ecotype of Arabidopsis thaliana and Col-0 was used as the control in all experiments. The following previously described genotypes were used: epf1-1 (Hara et al., 2007), epf1-1;epf2-1 (Hunt & Gray, 2009), SPCHpro::SPCH-YFP and SPCHpro::SPCH 2-4A-YFP (Lampard et al., 2008). In addition, the line SPCH SILENCE is a stably silenced (> four generations) line derived from SPCHpro::SPCH 2-4A-YFP. Of these, only epf1 and epf1epf2 have a noticeable stomatal clustering phenotype in cotyledons and that phenotype is greatly reduced in mature rosette leaves, with < 5% of stomata in clusters. Further details for each genotype can be found in Table 1 and Supporting Information, Table S1. Seeds were surface-sterilized and stratified at 4°C for 3–5 d in 0.15% agarose solution and then sown directly into pots of size 3.25 × 3.25 × 3 inches filled with Pro-Mix HP soil (Premier Horticulture, Quakerstown, PA, USA) and supplemented with Scott's Osmocote Classic 14-14-14 fertilizer (Scotts-Sierra, Marysville, OH, USA). At 10–14 d, seedlings were thinned so only one seedling per pot remained. Plants were grown to maturity in growth chambers under the following conditions: 16 : 8 h, day : night cycle, 22 : 20°C day :  night temperature, c. 100 μmol photon m−2 s−1, unless otherwise noted.

Table 1. Summary of stomatal characteristics and gas flux dynamics in Arabidopsis thaliana genotypes
Genotype (growth condition)Diffusive gsmax (mol m−2 s−1)Anatomical gsmax (mol m−2 s−1)Stomatal densitya (mm−2)Stomatal sizeb (μm2)Plants tested (n)Reference for line
  1. Values are mean ± SE.

  2. a

    Stomatal density is the combined number of adaxial and abaxial stomata.

  3. b

    Stomatal size is only of adaxial stomata (additional information in Table S1).

SPCH SILENCE0.299 ± 0.0290.452 ± 0.05084.8 ± 9.4234.5 ± 3.34This Study
Col-0 (Low Light)0.408 ± 0.0670.689 ± 0.105130.1 ± 19.4268.8 ± 5.24Ecotype control
Col-00.585 ± 0.0531.049 ± 0.072185.7 ± 12.7249.2 ± 4.58Ecotype control
SPCH-YFP0.788 ± 0.0481.106 ± 0.053182.6 ± 8.7269.1 ± 7.18Lampard et al. (2008)
epf1 0.936 ± 0.0940.923 ± 0.059209.3 ± 12.6280.1 ± 6.67Hara et al. (2007)
Col-0 (High Light)1.707 ± 0.1762.127 ± 0.088308.0 ± 12.5345.2 ± 6.84Ecotype Control
epf1epf2 1.715 ± 0.3241.928 ± 0.122441.6 ± 27.6244.5 ± 5.88Hunt & Gray (2009)
SPCH 2-4A1.995 ± 0.2691.722 ± 0.204359.3 ± 42.5250.9 ± 5.57Lampard et al. (2008)

Leaf-level gas-exchange measurements

Measurements were taken on the largest and most accessible mature rosette leaves at 5–7 wk postgermination using an LI-6400 Portable Photosynthesis System with the 6400-02B LED Light Source (LI-COR Biosciences Inc, Lincoln, NE, USA). Only one leaf was sampled per plant and between four and eight plants were sampled per genotype. To increase plant throughput, experiments were performed using a combination of the LI-6400 for gas-exchange measurements and a customized growth chamber to preincubate plants at different [CO2]. The chamber consisted of a glass aquarium of dimensions 20 × 10 × 12 inches with a sealed plexiglass lid that had a small hatch to insert and remove plants from the chamber. [CO2] in the chamber was monitored and adjusted by an infrared gas analyzer (Infrared Industries Inc., Santa Barbara, CA, USA) attached to a solenoid valve and a compressed CO2 gas tank. [CO2] below ambient was achieved by forced air circulation inside the chamber over a small container of soda lime (Alfa Aesar, Ward Hill, MA, USA). The [CO2] in the chamber was raised stepwise from 100 to 350, 500, 750, and finally 1000 ppm in advance of LI-6400 measurements at each of the same [CO2]. Other conditions in the chamber remained constant: RH  > 80% and temperature c. 22°C (Taylor Indoor Hygrometer/Thermometer, Oak Brook, IL, USA); light at 100–150 μmol photon m−2 s−1 was provided by blue and red LEDs mounted above the chamber (PAR38 Ultra Bright LED Light Bulbs; Plants were able to equilibrate at each [CO2] for at least 45 min before sampling with the LI-6400, with a longer incubation of 1–2 h taking place in the initial conditions at 100 ppm CO2. Conditions in the LI-6400 were as follows: flow, 500 μmol s−1; block temperature, 22°C; light, 1000 μmol photons m−2 s−1; RH, sample, 75-80%; [CO2], sample, 100–1000 μmol mol−1. Incoming RH was controlled by a custom-made dew point controller attached to the air input of the LI-6400 (Larry Giles; Carnegie Institution for Science, CA, USA). Once attached to the LI-6400, leaf gas-exchange was allowed to come to a steady state for 5–15 min at the [CO2] of the preincubation and then measurements were recorded by the LI-6400 for 5 min. Final parameter values were determined as the average value over those final 5 min. Leaves were excised from the plant after completing LI-6400 measurements and prepared for stomatal phenotype analysis (see Stomatal phenotype analysis for details). Leaf size was determined after cutting the leaf at the inside boundary of the LI-6400 sensor head gasket so that the LI-6400 file could be recomputed with proper leaf area (no leaves filled the entire 2 × 3 cm chamber) and ratio of abaxial to adaxial gsmax. The 6400-02B LED Light Source and 2 × 3 cm default-sized chamber were used in favor of the 6400-15 Arabidospsis Chamber because of the improved gas-exchange measurements we could attain with better temperature control and increased leaf area. The average Arabidopsis leaf area we achieved in the 6 cm2 chamber was 3.76 cm2 (min, 1.97; max, 5.71), compared with a maximum leaf area of 0.79 cm2 with the 6400-15 chamber. This increase in leaf area allowed for greatly improved sensitivity and stability in measuring conductance between plants with distinct, but sometimes very similar, stomatal density.

Stomatal phenotype analysis

Rosette leaves used in gas-exchange experiments were prepared for stomatal phenotype analysis in the following steps: cleared with 7 : 1 ethanol : acetic acid solution overnight or longer; softened for 30 min in 1 M potassium hydroxide; rinsed with water; mounted on slides with Hoyer's solution; and visualized by differential interference contrast microscopy on a Leica DM2500 microscope at ×20 magnification (0.320 mm−2 field of view). In the middle of the leaf, between the midvein and edge, four images on both the abaxial and adaxial sides of the leaf were taken. Stomatal density (D, mm−2) was manually counted for all pictures and all leaves using the Cell Counter in Image J ( Stomatal dimensions were counted on two to four leaves per genotype (75 > > 100 stomata) to calculate the average dimensions for each genotype. For this study, stoma size (S, μm2) was defined as an ellipse with its major axis equal to guard cell length and its minor axis equal to the width of the entire stoma; pore depth (l, μm) was equal to the guard cell width at the center of the stoma; pore length (μm) was equal to the length of the pore at the longest axis; and mean maximum stomatal pore area (amax, μm2) was defined as an ellipse with major axis equal to the pore length and minor axis equal to half the pore length.

Calculating Anatomical gsmax

Maximum stomatal conductance to water vapor as defined by stomatal anatomy (Anatomical gsmax, mol m−2s−1) was estimated for each leaf using a double end-correction version of the equation by Franks & Farquhar (2001):

display math(Eqn 1)

where d is the diffusivity of water in air (m2 s−1, at 22°C), v is the molar volume of air (m3 mol−1, at 22°C), and π is the mathematical constant, approximated to 3.142. Anatomical gsmax for each leaf was calculated as the sum of Anatomical gsmax abaxial (gab) and Anatomical gsmax adaxial (gad) using empirical values of D, l, and amax for stomata on each side of the leaf (Anatomical gsmax = gab + gad). D was determined independently for each leaf, while values of l and amax were genotype averages. Genotype averages for all stomatal dimensions used to calculate Anatomical gsmax can be found in Table S1.

Statistical analysis

All statistical analyses were performed using R ( Linear regression models were used to determine correlation coefficients (adjusted R2) and the statistical significance of covariation between parameters (< 0.05). Comparison of measured gs and modeled gAEM values were performed using a Wilcoxon signed-rank test for paired, nonparametric samples.


We measured gas-exchange parameters in a panel of developmental mutant lines, along with a wildtype control (Col-0), that varied in stomatal density but maintained proper stomatal spacing in mature rosette leaves (Tables 1, S1, Fig. S1). Proper stomatal spacing was defined as stomata that are separated by at least one epidermal cell (Sachs, 1991; Geisler et al., 2000). We intentionally selected lines with transgenes or mutations in a variety of the signaling components and transcriptional regulators acting in stomatal development to avoid any bias with respect to specific gene function. This broad sample allowed us to focus on the outcome of changing stomatal density rather than the processes occurring within the developmental pathway. We excluded from consideration any mutants that had obvious pleiotropic effects on plant morphology or physiological function in addition to altering stomatal development, such as erecta or yoda, for example (Bergmann et al., 2004; Masle et al., 2005). For these reasons, we often combined our lines into a single group for analysis, and specific genotypes are sparsely referenced in the text.

Our first objective was to determine if we could indeed measure maximum gs values (Diffusive gsmax) that were equivalent to the predicted values of Anatomical gsmax. We incubated Arabidopsis plants in a customized chamber under conditions that favored maximum stomatal opening – low [CO2], high RH, and blue-red light – and subsequently performed leaf-level gas-exchange measurements to determine Diffusive gsmax. Anatomical gsmax was calculated from the stomatal dimensions of the same leaves used in the gas-exchange measurements and a strong correlation between gsmax values was observed among all individuals (Fig. 1a, n = 50, R2 = 0.60, < 0.001). A reference line (1 : 1) that represents a perfect match between gsmax values had a slope and intercept within the 95% confidence intervals of the data regression. These results demonstrated that Anatomical gsmax provides an accurate prediction of maximum stomatal conductance under ideal gas-exchange conditions and across plants with wide ranges in stomatal density. At [CO2] concentrations that resemble our recent or potential future atmosphere (350 and 500 ppm CO2, respectively), the linear model between gs and Anatomical gsmax shifted away from the 1 : 1 reference line because gs declined with increasing [CO2] (Fig. 1b, 350 ppm CO2, = 50, R2 = 0.36, < 0.001; Fig. 1c, 500 ppm CO2, = 50, R2 = 0.14, < 0.01). However, Anatomical gsmax still maintained a quantifiable relationship with gs at each respective [CO2], indicating that Anatomical gsmax could be a useful tool for predicting operational gs under various CO2 regimes, including current atmospheric conditions.

Figure 1.

Anatomical gsmax provided an accurate prediction of Diffusive gsmax under conditions that maximize stomatal opening. (a) The diffusive maximum rate of stomatal conductance (Diffusive gsmax (dif gsmax)) was determined by leaf-level gas-exchange measurements of individual rosette leaves of Arabidopsis thaliana at 100 ppm CO2. The stomatal traits of each leaf were used to determine the anatomical maximum rate of stomatal conductance (Anatomical gsmax (anat gsmax)) as defined by Eqn 1 in the 'Materials and Methods' section. Linear regression of the data is significant (= 50, < 0.001), while the slope and intercept of a reference line (x, dashed line) are within the 95% confidence intervals of the regression (shaded gray). (b, c) The same leaves were used to determine the relationship between gs and Anatomical gsmax at increasing [CO2], 350 and 500 ppm, respectively (= 50, < 0.001, < 0.01).

In order to examine the relationship between Anatomical gsmax and operational gs rates at different [CO2], we compared the response curves of Arabidopsis lines with significantly different stomatal density (Fig. 2, n = 4 per genotype). Different rates of gs were most apparent at the lowest [CO2], but those absolute differences diminished as all lines responded in a similar fashion to increasing [CO2]. Notably, the total decrease in gs was proportional to the initial rate, Diffusive gsmax (Fig. 3a, n = 20, R2 = 0.96, < 0.001), and when individual gs responses were normalized to their respective values of Anatomical gsmax, there was no correlation between stomatal density and the relative decrease in gs (Fig 3b, n = 20, R2 = 0.000, = 0.954). Essentially, all lines responded equally to changes in [CO2] when scaled to Anatomical gsmax, regardless of large differences in stomatal density or initial stomatal conductance. This result highlights an underutilized approach for quantifying stomatal responses to the environment – stomatal conductance appears to respond to local conditions as a proportion of Anatomical gsmax. In fact, stomatal responses to increasing [CO2] from all lines can be normalized to a single response function with nominal variance that represents gs as a percentage of Anatomical gsmax (Fig. 3c, n = 20). Under our gas-exchange conditions, stomata operated at c. 50% capacity of Anatomical gsmax at 350 ppm CO2, 30% at 500 ppm, and 23% at 1000 ppm (Fig. 3c). Given this consistency in gs with respect to Anatomical gsmax, it should be possible to model gs rates provided with anatomical information and environmental conditions. Furthermore, our results indicate that the short-term gs response to CO2 (Fig. 3c) is actually a pore-specific property that can simply be scaled to stomatal density (Fig 2). This provides new evidence that the CO2 sensing mechanism is driven by guard cell activities that result in stomatal aperture modulation.

Figure 2.

Stomatal conductance (gs) was proportional to stomatal density in response to increasing [CO2]. Average gs responses to increasing [CO2] (= 4 per genotype, error bars ± SEM) of four stomata developmental mutant lines and a nonmutant control (Col-0) in Arabidopsis thaliana. All lines have distinct stomatal densities (inset bar graph, < 0.05).

Figure 3.

Absolute responses in stomatal conductance (gs) were independent of stomatal density after normalization with Anatomical gsmax. (a) The decrease in gs from the response curves of individual plants in Fig. 2 was proportional to values of the diffusive maximum rate of stomatal conductance (Diffusive gsmax) for each Arabidopsis thaliana plant (= 20, < 0.001). (b) Decreases in gs were normalized to the anatomical maximum rate of stomatal conductance (Anatomical gsmax) for each individual and there was no correlation between stomatal density and percentage change in gs (= 20, = 0.954). (c) All lines (= 20, mean ± SE) are pooled to show the general response in gs as a percentage of Anatomical gsmax.

Previous modeling efforts have used a variety of external or internal plant conditions, or both, as parameters to estimate rates of gs (Damour et al., 2010). Multiplicative models of environmental influence have been used to determine gs as a function of gsmax and external conditions, such as the Jarvis–Stewart model (Jarvis, 1976; Stewart, 1988). However, the Jarvis–Stewart model did not incorporate [CO2] as an independent variable, which limits its usefulness in projections that consider rising atmospheric CO2, and other components required extensive parameterization. One of the earliest and most utilized empirical models of gs, the BWB model (Ball et al., 1987), focused on the relationship between stomatal conductance and photosynthetic rates, and versions of the BWB are still incorporated into popular land-surface models that are used in current projections of climate change (CLM4, Oleson et al., 2010; JULES, Best et al., 2011). The BWB model incorporated the positive correlation between net carbon assimilation (A) and gs and the effects of RH and [CO2] on stomatal function at the leaf surface (hs and cs, respectively) to determine gs. These variables are combined in a straightforward equation:

display math(Eqn 2)

where g0 is the residual conductance when A approaches zero, g1 is an empirically defined constant, and the combined term (A · hs/cs) is called the index. g1 and g0 are determined by linear regression of the index vs gs using empirical data obtained from leaf-level gas-exchange measurements. Although modifications to the BWB model have been made (Aphalo & Jarvis, 1993; Leuning, 1995), the core model provides a validated framework for investigating the relationship between Anatomical gsmax and operational gs.

We modified the BWB equation by direct substitution of A with Anatomical gsmax to produce an integrative index that incorporates both anatomical and environmental factors to predict gs (anatomical-environmental maximum, AEM):

display math(Eqn 3)

The term maximum is included in the nomenclature because we collected our data using near-optimal conditions for stomatal opening, including saturating intensities of light, high RH, and ample soil moisture. While such benign environments are certainly observed in nature, there are many situations where the determinants of stomatal operation are less than favorable. This model does not, therefore, account for all conditions a plant experiences in nature, but it does capture the maximum diffusion rates (Diffusive gsmax) that can be achieved when considering the interaction between stomatal anatomy and typical environmental conditions.

Following the process established in the BWB model, gas-exchange measurements under various conditions (all five [CO2] points) were pooled and linear regression of the data produced a robust correlation between gs and gAEM (Fig. 4, n = 210, R2 = 0.65, < 0.001). The regression slope, g1 = 1.02, indicates that gAEM provides an accurate prediction of gs across a wide range of stomatal densities and [CO2], thus supporting the claim that Anatomical gsmax can be a useful tool for estimating gs. The best-fit model had a larger residual conductance than expected, g0 = 0.20, considering that the value of g0 established by the BWB model was not significantly different from the origin, even if some species did show a positive intercept (Ball et al., 1987). In support of this standard, our regression data nearest to the gs-axis were all below the g0 intercept, and clearly focused towards the origin, so we decided to use a constrained gAEM model (g1 = 1.02 and g0 = 0.00) to test the accuracy of a parameterized equation to predict gs.

Figure 4.

Stomatal conductance (gs) can be estimated as a function of Anatomical gsmax and environmental conditions at the leaf. An empirical model (gAEM) defined the linear relationship between gs and the product of stomatal anatomy (Anatomical gsmax, mol m−2 s−1) and external conditions at the leaf (relative humidity, RH, %; ambient [CO2], ppm). gs measurements from all [CO2] points and all Arabidopsis thaliana genotypes were pooled for the regression analysis (= 210, < 0.001, five points beyond 2.0 mol m−2 s−1 not shown; the x line is shown for reference, dashed).

Nonmutant Arabidopsis plants grown under low- or high-intensity light were used as a test group because they possessed different Anatomical gsmax (average Anatomical gsmax values of 0.69 and 2.13 mol m−2 s−1, respectively). Once again, gas-exchange measurements were taken at five [CO2] points, then cuvette conditions and Anatomical gsmax values were input to the constrained gAEM equation to produce modeled values of gs. Upon comparison, there was no significant difference between modeled gAEM and measured gs values for either group of plants (Fig. 5, n = 20, = 0.133 for high-light and = 0.058 for low-light plants) and the mean error rate was < 0.1 mol m−2 s−1 (0.056 mol m−2 s−1 for high-light and 0.117 mol m−2 s−1 for low-light plants). In general, the gAEM model was quite accurate in predicting gs values even though some rates were slightly underestimated, in part because we used a constrained version of the regression model. In most plant species, residual conductance (g0) should be near zero because there is an incentive to conserve water when the potential for carbon assimilation is very low (Cowan & Farquhar, 1977; Ball et al., 1987), although stomatal opening has been observed at night (Seibt et al., 2007). Making this assumption in our gAEM model did not hinder its ability to predict gs for Arabidopsis plants. These results demonstrate a reproducible and predictable interaction between Anatomical gsmax and operational gs, whereby the anatomical features of stomata set the limits of diffusion and local environmental conditions determine the actual rate.

Figure 5.

The gAEM model can be used to predict stomatal responses to increasing [CO2]. Two groups of Arabidopsis thaliana Col-0 plants grown under low- or high-light intensity (50 or 200 μmol photons m−2s−1, respectively) that have significantly different Anatomical gsmax (= 4, dotted lines, < 0.001) were used to test the accuracy of a constrained version of the gAEM model developed in Fig. 4 (gs = 1.02[gAEM] + 0). For both groups of plants, the modeled gAEM response (dashed line) was not significantly different from the measured stomatal conductance (gs) response (solid line) (= 20, Wilcoxon paired t-test, = 0.133 for high-light plants and = 0.058 for low-light plants). All values are means ± SE.


Our results strongly support the use of Anatomical gsmax as a proxy for maximum stomatal conductance (Fig. 1a). To our knowledge, this is the first empirical proof of the accuracy of Eqn 1 over wide ranges of stomatal density in a single species. While gs approached Anatomical gsmax under conditions that maximized stomatal pore opening (amax), Anatomical gsmax was no longer sufficient to predict gs under increasing CO2 because aperture size had decreased from amax (Fig. 1b,c). The Anatomical gsmax equation (Eqn 1) could be modified to predict gs under different conditions by the substitution of a new aperture size for amax, but it is difficult to know what size that would be. As a more convenient alternative, we produced a regression model using the gAEM index (Anatomical gsmax × RH/[CO2]) that could predict operational gs as a function of Anatomical gsmax and the environmental conditions at the leaf (Fig. 4). The gAEM model, which assumes light saturation and ample soil-water availability, can also be considered as a prediction of Diffusive gsmax, whereby the maximum operational conductance is constrained by the local environment:

display math(Eqn 4)

where g1 is an empirically determined constant (≈ 1) and g0 is the residual conductance when carbon assimilation approaches zero (≈ 0). Confidence in this model would obviously benefit from validation in other plant species, but its compatibility with the original terms of the BWB model is initially very supportive. For broad applications, the approach here would have to be adapted to predict conductance under light-limiting conditions, perhaps by scaling with A/Amax.

This novel utilization of Anatomical gsmax does, however, represent an order of magnitude improvement in estimating operational gs rates from stomatal dimensions. For example, at [CO2] concentrations reflecting recent and possible future atmospheric conditions (350 and 500 ppm), Anatomical gsmax was 290% greater than operational gs, on average, for low- and high-light plants, whereas modeled gAEM values were only 29% different (Fig. 5). From another perspective, the average difference in Anatomical gsmax between plants grown under low- or high-intensity light was 1.44 mol m−2 s−1, while the difference in operational gs between the same plants was actually just 0.21 mol m−2 s−1. Similar differences in modeled values were 0.27 mol m−2 s−1, a 20-fold improvement in relative terms (Fig. 5). These results also reinforced the observation that operational conductance is normally much lower than Anatomical gsmax, but why should that be the case? Under generalized atmospheric conditions (80% RH, 395 ppm CO2), our model predicts that operational gs would be c. 20% of Anatomical gsmax. Conductance rates c. 20% of total gas-exchange capacity might signify a kind of ‘sweet spot’, or highly sensitive region, of stomatal function (Franks et al., 2012). This degree of stomatal opening may define a position where changes in guard cell turgor pressure are particularly effective at controlling conductance rates in response to rapid changes in the local environment. At only 20% operating capacity, guard cells would be primed to increase pore size if favorable conditions persisted, but would be equally quick to close the pore if necessary. Interestingly, the long-term developmental response to increasing CO2, which is well documented in herbarium samples as a decrease in Anatomical gsmax as CO2 increased from the pre-industrial to its present concentration, might in fact be an attempt to keep stomatal function in this 20% sweet spot because operational gs has decreased in the short term below its optimal range (Woodward, 1987; Franks et al., 2013).

Our model was based on the observation that stomatal responses to CO2 were proportional to Anatomical gsmax, irrespective of stomatal density (Fig. 3). However, stomatal development is also influenced by [CO2], among other external cues, to produce an Anatomical gsmax profile that is aptly suited to the local environment (Casson & Hetherington, 2010; Lammertsma et al., 2011). The combination of these two things – normalized stomatal responses to the environment and environmentally sensitive development – provides the framework for a feedback-sensitive model that predicts how plants might interact with their abiotic surroundings over time and in the face of increasing CO2. For example, potential increases in atmospheric CO2 to 700 ppm by the end of this century (intermediate A1B projection (IPCC, 2001)) would result in decreases of gs close to 10% of Anatomical gsmax, a 50% reduction from current rates, as predicted by our model. This will likely be coupled with a developmental decrease in Anatomical gsmax itself, perhaps as much as 30% per 100 ppm CO2 (Lammertsma et al., 2011). Considered together, the effect of increasing atmospheric CO2 to 700 ppm has the potential to decrease rates of gs to 3.4% of today's Anatomical gsmax values, a sixfold decrease from current operational rates of gs. These estimated decreases in gas flux, both H2O and CO2, could have large effects at the canopy level and be used to assess decreases in transpiration, increases in temperature, changes in carbon sequestration, and continued feedbacks on plant development and productivity (Woodward et al., 1998; de Boer et al., 2011). This information could also be valuable for parameterization of the biosphere component in global circulation models, where biosphere–atmosphere feedbacks have already been linked to accelerating rates of global warming (Cox et al., 2000). Future work that explores how Anatomical gsmax relationships might be scaled to the canopy, for the aforementioned purposes, would thus be very meaningful.

On the other hand, both guard cell physiology and stomatal development demonstrate nonlinear responses to CO2, whereby stomata are more sensitive to changes below than above the ambient concentration because of the reciprocal relationship with CO2 (a function of 1/[CO2]) (Fig. 2; and de Boer et al., 2011). Understanding how plants detect and respond to changes in CO2 will be critical for determining how impactful the physiological effects on climate might be in the future. Although the exact mechanism for the CO2 sensor in plants has not yet been determined, recent work has implicated the carbonic anhydrases in the signaling response of guard cells to CO2 as well as stomatal development processes (Hu et al., 2010). In addition, our results suggest that the short-term response to CO2 is a pore-level property, independent of stomatal density, which should motivate further inquiry into guard cell-specific mechanisms of sensing CO2 concentration. Such approaches that utilize Arabidopsis mutants highlight the importance of plant model systems and genetic resources in defining the arc of climate change over the next century and beyond.

One of the emergent outcomes of using the BWB equation to develop our model was the apparent connection between net carbon assimilation (A) and stomatal development. The direct substitution of A for Anatomical gsmax was the only thing necessary to reproduce a robust index for gs, indicating that Anatomical gsmax could also be a relevant proxy for estimating A. While the correlation between A and gs has been widely acknowledged for some time (Wong et al., 1979), the generally accepted theory is that the resistance to CO2 flow across the epidermis has little effect on photosynthetic capacity (Farquhar & Sharkey, 1982). In fact, estimates of the stomatal limitation on photosynthesis are only 10–20% (Bjorkman et al., 1972; Nobel, 1999). If stomatal control of CO2 flux is not a main determinant of photosynthetic rates, what might be responsible for this apparent association? Recent evidence points toward coordinated developmental processes in the epidermis and mesophyll tissues of leaves: Anatomical gsmax has been correlated with leaf nitrogen content and the maximum rate of Rubisco carboxylation (Vcmax) (Franks & Beerling, 2009a; Franks et al., 2009). Additionally, leaf hydraulic capacity and vein density have also been correlated with photosynthetic capacity, which provides further evidence of tight developmental coordination between leaf tissues (Brodribb et al., 2005, 2007). Future exploration of the connection between Anatomical gsmax and photosynthetic capacity is certainly warranted and could be very inportant for understanding the relationship between gs and A.

Undoubtedly, the utility of Anatomical gsmax is linked to the simplicity of the measurements involved in gathering the data. The stomatal traits needed to calculate Anatomical gsmax might be easily and cost-effectively obtained from epidermal peels, impressions of leaves, directly from the epidermis itself, or from fossilized leaves (Casson et al., 2009; Franks & Beerling, 2009a,b). The gAEM model adds little complication to this process, but greatly increases the accuracy of estimating stomatal conductance. Indeed, climate sampling in the field, or from online databases that provide [CO2] and RH information, can be used in concert with stomatal measurements to produce gas flux estimates across multiple scales or vegetation types by application of the gAEM model. Of similar interest, the negative correlation between stomatal density and atmospheric CO2 and the availability of fossilized leaves has enabled proxy reconstruction of paleoclimatic CO2 conditions by quantifying changes in stomatal density over millions of years (Woodward, 1987; Van Der Burgh et al., 1993; McElwain & Chaloner, 1995). The strong influence of [CO2] in the gAEM model, and evidence that plants acquired active responses to CO2 early in their evolutionary history (Ruszala et al., 2011), suggests that it could be possible to reconstruct operational gs rates from these same fossilized plant materials. Measurements of paleobiological gas-exchange rates could help to quantify historical biosphere–atmosphere interactions and the extent of any physiological controls on paleoclimatic shifts in CO2 or climate (Beerling & Berner, 2005). The effect of plant growth conditions on stomatal sensitivity should also be considered when applying the gAEM model and we have not tested conditions beyond changes in light intensity (Fig. 5). While free-air CO2 enrichment (FACE) studies have shown little impact on the character of the CO2 response, the same might not be true for plants grown at low [CO2] in paleohistoric times (Medlyn et al., 2001).

In this study, we have integrated predictive models of gs that focused on developmental or physiological characteristics of stomata to produce a new method for estimating gs using the interaction of anatomical features and environmental conditions. We believe this compilation of modeling approaches will provide a framework for the inclusion of additional components to estimate gs across a broad spectrum of plant conditions, for example, those in shaded environments or under drought stress, akin to the Jarvis–Stewart model (Stewart, 1988). Our model also provides context for estimating operational rates of gs from the anatomical features of plants in historical CO2 climates and the response of plants to future increases in CO2. Finally, there is significant feedback that connects stomatal development with plant physiology and climate – Anatomical gsmax is developmentally sensitive to the very conditions that it influences via gas exchange (Hetherington & Woodward, 2003; Franks & Beerling, 2009b; Berry et al., 2010; Casson & Hetherington, 2010). The interdependency of this cycle reinforces the need to produce models that can incorporate multiple levels of this process. These integrative tools will be an invaluable approach for understanding how the Earth System works, and how it might change in the future.


This work was supported by a Stanford University Bio-X Interdisciplinary Fellowship awarded to G.J.D. and funding from the Carnegie Institution for Science and Gordon and Betty Moore Foundation. D.C.B. is an investigator of the Howard Hughes Medical Institute. We thank Larry Giles, Roland Pieruschka, and Jen Johnson for technical support with gas-exchange equipment and experiments, and Leila Moinpour for assistance in performing gas-exchange experiments and stomatal quantification.