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Logo of annbotAboutAuthor GuidelinesEditorial BoardAnnals of Botany
Ann Bot. 2010 March; 105(3): 431–442.
Published online 2009 December 8. doi:  10.1093/aob/mcp292
PMCID: PMC2826246

A stomatal optimization theory to describe the effects of atmospheric CO2 on leaf photosynthesis and transpiration


Background and Aims

Global climate models predict decreases in leaf stomatal conductance and transpiration due to increases in atmospheric CO2. The consequences of these reductions are increases in soil moisture availability and continental scale run-off at decadal time-scales. Thus, a theory explaining the differential sensitivity of stomata to changing atmospheric CO2 and other environmental conditions must be identified. Here, these responses are investigated using optimality theory applied to stomatal conductance.


An analytical model for stomatal conductance is proposed based on: (a) Fickian mass transfer of CO2 and H2O through stomata; (b) a biochemical photosynthesis model that relates intercellular CO2 to net photosynthesis; and (c) a stomatal model based on optimization for maximizing carbon gains when water losses represent a cost. Comparisons between the optimization-based model and empirical relationships widely used in climate models were made using an extensive gas exchange dataset collected in a maturing pine (Pinus taeda) forest under ambient and enriched atmospheric CO2.

Key Results and Conclusion

In this interpretation, it is proposed that an individual leaf optimally and autonomously regulates stomatal opening on short-term (approx. 10-min time-scale) rather than on daily or longer time-scales. The derived equations are analytical with explicit expressions for conductance, photosynthesis and intercellular CO2, thereby making the approach useful for climate models. Using a gas exchange dataset collected in a pine forest, it is shown that (a) the cost of unit water loss λ (a measure of marginal water-use efficiency) increases with atmospheric CO2; (b) the new formulation correctly predicts the condition under which CO2-enriched atmosphere will cause increasing assimilation and decreasing stomatal conductance.

Keywords: Economics of gas exchange, free air CO2 enrichment, marginal water-use efficiency, photosynthesis, Pinus taeda, stomatal conductance, stomatal optimization


Two controls of stomatal behaviour are drawing attention because they are relevant to future climate and fresh-water resources. The first control relates stomatal response to variations in atmospheric CO2 (ca) and other environmental factors, which ‘regulate’ stomatal conductance (Ainsworth and Rogers, 2007) and the second deals with the ‘regulatory’ role stomata play in limiting water losses in the context of leaf economics (Scarth, 1927). As summarized by Scarth's review, the reductions in the partial pressure of atmospheric CO2 cause stomata to open while increasing ca leads stomata to close. Although this phenomenon was observed as early as the late 19th century by Sir Francis Darwin (Darwin, 1898), the mechanistic reasons for this first control on stomata and the variation in the intensity of the response are complex and not fully understood (Ainsworth and Rogers, 2007, and references therein). The second control reflects the link between CO2 absorbed in photosynthesis and water vapour loss through the stomata in transpiration. Studies on leaf transpiration have a long history – beginning perhaps with the seminal experiments of Edme Mariotte around 1660 (Meidner, 1987). However, still pertinent is a statement made in Scarth's review on the interaction between leaf transpiration and CO2 uptake (Scarth, 1927): ‘when stomata regulate one process they must regulate the other also but the question remains as to which of these actions represents the real role of the stomata in the economy of the plant’.

Today, the interaction between the processes of photosynthesis and transpiration, and stomatal response to CO2 and other environmental factors are recognized to affect global-scale phenomena. For example, global climate models predict future acceleration of continental scale run-off primarily because plant stomata open less as atmospheric CO2 concentrations increase, thereby reducing transpiration rates (Gedney et al., 2006; Betts et al., 2007). Reduced stomatal conductance is also predicted to reduce the potential uptake of CO2 by plants, contributing to increased atmospheric CO2 concentration (Cox et al., 2000). When less water is lost through transpiration in water-limited ecosystems, the CO2 assimilation period may be extended because of improved water balance of plants (Volk et al., 2000). Alternatively, canopy leaf area may increase (Woodward, 1990), leading to a more productive ecosystem (Oren et al., 1987). Either response would provide a feedback between vegetation and the climate system. Thus, a theory based on first-principles explaining the observed variation in stomatal response to CO2 must be urgently identified.

Two broad modelling approaches have been employed to describe stomatal conductance as a function of environmental stimuli. One approach focuses on environmental ‘regulation’ of stomatal opening and is based on semi-empirical formulations relating stomatal conductance to environmental parameters (Jarvis, 1976) and the rate of photosynthesis (e.g. Ball et al., 1987; Collatz et al., 1991; Leuning, 1995), and is routinely employed in ecological, hydrological and climate models (Sellers et al., 1995, 1996; Baldocchi and Meyers, 1998; Lai et al., 2000; Siqueira and Katul, 2002; Juang et al., 2008). The second approach is based instead on the economy of the plant and focuses on the ‘regulatory’ role of stomata. In these models, stomatal opening is optimized to maximize carbon gain for a unit water loss; thus water loss is considered a cost to the plant (Givnish and Vermeij, 1976; Cowan, 1977, 1982; Cowan and Farquhar, 1977; Hari et al., 1986; Berninger and Hari, 1993; Mäkelä et al., 1996). The appeal of this approach is that by virtue of its construction, it addresses the intrinsic interactions between assimilation and transpiration, and thus suggests an answer to Scarth's questions (Scarth, 1927). Despite the appeal and capability of models based on the stomatal optimization approach to simulate field conditions, this approach has not been employed in operational climate or ecological models (Hari et al., 1986, 1999, 2000; Berninger and Hari, 1993; Berninger et al., 1996; Mäkelä et al., 1996, 2004, 2006; Aalto et al., 2002; Thum et al., 2007).

Comparisons of the two approaches to modelling stomatal conductance (i.e. semi-empirical and based on optimization) using the same dataset have rarely been conducted and there has been no study of the effects of elevated ca on their parameterization. In particular, the effect of elevated ca on the optimization parameter representing the trade-off between photosynthetic carbon gain and transpirational water loss has not been assessed so far. The aim here is to address this omission and to compare the performance of the two modelling approaches. To achieve this aim, a simple analytical model of leaf gas exchange based on short-term (approx. 10 min) optimization of stomatal conductance is proposed, complemented by transport equations for CO2 and water vapour, and a non-linear photosynthesis model. Side-by-side model comparisons are conducted using an extensive gas exchange dataset collected in a pine forest growing under ambient and enriched CO2 atmosphere. The new short-term optimization theory and a unique dataset are used to assess how elevated ca might impact the trade-off parameter, the most pertinent to the economy of gas exchange.


Basic equations

Mass transfer of CO2 and water vapour between leaves and the bulk atmosphere can be described by Fickian diffusion through stomata when the boundary layer resistance is negligible:

equation image

where fc is the CO2 flux, fe is the water vapour flux, g is the stomatal conductance to CO2, ca is ambient and ci intercellular CO2 concentrations, respectively, a = 1·6 is the relative diffusivity of water vapour with respect to carbon dioxide, ei is the intercellular and ea the ambient water vapour concentration and D is the vapour pressure deficit representing eiea when the leaf is well coupled to the atmosphere, as is usually the case when gas exchange of narrow-leaved species is measured in a ventilated cuvette.

When respiration is small with respect to fc, the biochemical demand for CO2 is mathematically well described by the photosynthesis model of Farquhar et al. (1980):

equation image

where cp is the CO2 compensation point, a1 and a2 are selected depending on whether the photosynthetic rate is light or Rubisco limited. Under light-saturated conditions, as in all gas exchange measurements used in this study, a1 = Vcmax (maximum carboxylation capacity) and a2 = Kc(1 + Coa/Ko), where Kc and Ko are the Michaelis constants for CO2 fixation and oxygen inhibition and Coa is the oxygen concentration in air (see Table 1 for definitions, temperature dependencies and units). Hence, expressed in terms of conductance, eqns (1) and (2) can be combined to yield

equation image


equation image

when g > 0.

Table 1.
Biochemical model parameters for eqn (2) and their temperature adjustments (Katul et al., 2000)

The closure problem

Equations (3) and (4) are not mathematically ‘closed’ and require independent formulations of g so that fc and ci can be computed. Two conductance formulations, the so-called Ball–Berry (g1, see Ball et al., 1987) and the Leuning (g2, see Leuning, 1995) models are commonly used in climate (Sellers et al., 1995, 1996) or detailed biosphere–atmosphere models (Baldocchi and Meyers, 1998; Lai et al., 2000; Siqueira and Katul, 2002; Juang et al., 2008). The formulations, respectively, are:

equation image


equation image

where H is the mean air relative humidity, Do is a vapour pressure deficit constant and m1, m2, b1 and b2 are empirical fitting parameters. Equations (5a) and (5b) therefore provide the necessary mathematical closure (i.e. three equations with three unknowns: fc, ci and g). Equations (5a) and (5b) are semi-empirical, mainly determining a priori the effects of environmental stimuli (e.g. D or H) on conductance and thus fall into the first category of stomatal regulation models.

An optimality model

The second approach to stomatal conductance modelling employs an optimality principle originally proposed by Givnish and Vermeij (1976) and retained in the work of Cowan and Farquhar (1977), Hari et al. (1986), Berninger and Hari (1993) and more recently of Konrad et al. (2008). In this approach, an objective function (i.e. the net carbon gain for the leaf) is defined as

equation image

By maximizing the gross carbon gain fc while minimizing the water loss λfe (in units of carbon) for a species-specific cost parameter λ, an expression for conductance can be derived. Throughout, the notation of Hari et al. (1986) is used; the symbol λ in Cowan (1977) indicates the inverse of the cost parameter employed here. Such a maximization condition can be expressed as δf(g)/δg = 0 [f(g) is guaranteed to have a maximum given the expected concavity in eqn (6) of f(g)with respect to g]. The optimization problem framed in eqn (6) does not a priori assume a functional response of g to D or ca. Instead, it assumes (a priori) that excessive water losses do induce stomatal closure, which is consistent with the experimental findings of Mott and Parkhurst (1991) who, using a ‘helox’ gas medium, demonstrated that stomata respond to transpiration rates rather than measures of air humidity. The original formulae of Givnish and Vermeij (1976), Cowan (1977), Cowan and Farquhar (1977), Hari et al. (1986) and Berninger and Hari (1993) define eqn (6), or a variant of it, for an integration interval bounded by times t0 and t1. Since in this optimization problem there is no dynamic component (i.e. stomatal conductance is assumed to adapt instantaneously to changing environmental conditions), it can be shown that this arbitrary integration interval does not alter the mathematical expression for optimal conductance, provided that the integration limits are held constant. In other words, the maximization of the cumulative C gain over an arbitrary time interval is equivalent to maximizing eqn (6) at each instant in time (Pontryagin maximum principle applied to this disjoint optimization problem; see Denn, 1969).

Assuming that stomatal behaviour is optimal (and thus λ is a constant with respect to g) and differentiating, eqn (6) yields

equation image

When setting δf(g)/δg = 0 and solving for g, the following is obtained:

equation image

Similar to the two formulations of the first approach (eqns 5a and 5b), this formulation is explicit in relating conductance to ca and D. However, unlike these formulations, both relating conductance to the photosynthetic rate leaving two (g1 formulation) or three (g2 formulation) unknown parameters, eqn (8) relates conductance to the three parameters of the photosynthesis model leaving only one unknown physiological parameter (λ). Moreover, the model formulation in eqn (8) suggests that λ cannot be entirely ‘free’ and must be bounded between zero and (cacp)/(aD) to ensure real and positive conductance. This theoretical bound is in contrast to the less constrained choices of m1 and m2 in the conductance models of eqn (5).

It is worth emphasizing that for eqn (8) to be applicable, λ does not need to be exactly constant for a given leaf and under certain external conditions. Suboptimal conditions are still admissible in eqn (8) provided |δλ/λ| [double less-than sign]fe/fe|. Recalling that fe = agD (eqn 1), this necessary condition can be expressed as λ/g [dbl greater-than sign]|δλ/δg|. Future experimental studies might employ this constraint to test how close to optimality a single leaf is.

A linearized optimality model

Linearizing the biochemical demand function results in a much simpler and informative model for optimal g. Linearizing the model requires the assumptions that cp [double less-than sign] ci and that the variability of ci affects only marginally the denominator of eqn (2), leading to the approximation a2 + ci = a2 + (ci/ca)ca = a2 + sca. As a result,

equation image

where s is treated as a constant set equal to the long-term mean of ci/ca. We stress that s is treated as a model constant only in the denominator of eqn (9), while in eqn (1) ci/ca is allowed to vary. Note also that the geometric interpretation of the group of parameters a1/(a2 + sca) in eqn (9) is simply the slope of the fc(ci) curve. Combining this linearized photosynthesis model with eqn (1) results in

equation image


equation image

The objective function in eqn (6) now simplifies to

equation image

and upon differentiation with respect to g yields

equation image

By setting δf(g)/δg = 0 and solving for g results in

equation image

This expression is identical to that in Hari et al. (1986), though derived without the use of arbitrary time integration. Replacing eqn (14) into eqns (10) and (11) provides closed form expressions for ci and fc given by

equation image


equation image

Hereafter, the solution in eqn (8) is referred to as the ‘non-linear model’ and the result in eqn (14) as the ‘linear model’ (denoted by LI). The linear model was recently shown by Katul et al. (2009) to be consistent with (a) stomatal responses to changes in vapour pressure deficit and (b) the non-linearities in ci/ca variations with D.


The experiments, including details of the free-air CO2 enrichment (FACE) facility at Duke Forest, the specifics of the portable gas exchange system and the data analysis are presented elsewhere (Ellsworth, 1999, 2000; Hendrey et al., 1999; Katul et al., 2000; Maier et al., 2008). Gas exchange measurements were made at the Duke University FACE site, Orange County, North Carolina (35°98′N, 79°8′W, elevation = 163 m) at an even-aged loblolly pine (Pinus taeda) forest planted at 2·0 × 2·4 m spacing in 1983 following clear cutting and burning. The site index is between 20 m and 21 m at age 25 years. The FACE system consists of eight 30-m-diameter plots, four of which are fumigated with CO2 to maintain atmospheric CO2 concentration at ambient + 200 µmol mol−1 while the other four serve as controls and receive ambient air only. Two datasets, described below, were employed in this study.

Dataset 1 (1996–1999)

Steady-state gas exchange of leaves was measured in 1996–1999 in plots 1–6, at two growth CO2 concentrations (ambient and ambient + 200 µmol mol−1 CO2 concentration) and near ambient temperature using a portable infrared gas analyser system for CO2 and water vapour (CIRAS-1, PP-Systems, Amesbury, MA, USA). The system was operated in open flow mode with a 5·5-cm-long leaf chamber and an integrated CO2 supply system. The light-saturated photosynthetic rate and stomatal conductance were measured on upper canopy foliage, at 11–12 m above ground. For further details see Ellsworth (1999, 2000).

Dataset 2 (2005)

Light-saturated photosynthetic rates and leaf conductance were measured in 2005 in plots 1–8 on detached foliage using a LI-6400 (Li-Cor Inc., Lincoln, NE, USA). Gas exchange measurements at growth CO2 concentrations were completed within 30 min of removing the fascicles. Within this time period, the light-saturated leaf gas exchange rates remained stable with little change in stomatal conductance (see figure 1 in Maier et al., 2008). Chamber temperature was maintained at 20 °C (spring) and 25 °C (summer and autumn). Details are given in Maier et al. (2008).

Fig. 1.
Left panels: estimation of the effective parameters via regression analysis for the stomatal conductance models in eqns (5a) and (5b), the Ball–Berry model g1 = m1 fcRH/ca + b1 (A) and the Leuning model g2 = m2 fc[(1 + D/D0)ca]−1 + b2 ...

The combined dataset, resulting in 193 gas exchange measurements, 73 of which were collected under elevated ca, is used in the following analyses. All measured gas exchange rates are reported per unit leaf area, where the leaf area is expressed as half of the total needle surface area (i.e. one-sided leaf area).


Before comparing the four stomatal conductance model formulations and discussing the effects of elevated ca on their parameters, measurement errors and how these errors might impact parameter estimation are briefly discussed.

Measurement errors

Measurement errors during leaf gas exchange experiments may be due to (a) uncertainties in the measured CO2 and H2O concentrations in the gas exchange chamber, (b) unaccounted leaks from the chamber and (c) uncertainties in the calculated needle leaf area when results are scaled to a unit leaf area. Gas concentration measurement errors are in the order of 0·001 % for CO2, 0·1 % for H2O and 0·1 % for temperature. Diffusive leaks are not expected to be an issue in studies where gas exchange measurements are performed at external CO2 and water vapour concentrations. Hence, these two sources of error should not significantly contribute to the variability in the observations.

Leaf area estimates may be an important source of uncertainty. These estimates are calculated from needle width, which is measured to the nearest 0·1 mm and therefore can introduce an error on the order of 10 %. However, the error is the same for photosynthesis, leaf transpiration and stomatal conductance estimates, and thus propagates to some but not all parameters, as described later.

Ball–Berry and Leuning models

The effective model parameters for g1 and g2 (eqns 5a and 5b) and their frequency distributions were computed from the two gas exchange datasets. The values for m1, m2, b1 and b2 were computed using an ordinary least squares (OLS) approach (Fig. 1, left panels, and Table 2). Hereafter, these best-fit values from OLS are referred to as effective parameter values. Because the interest here is in the responses to environmental drivers, the focus is on the sensitivity parameters m1 and m2. Variations in m1 and m2 among leaves and trees were also computed by inverting eqns (5a) and (5b) for given values of the intercepts b1 and b2. To account for uncertainty in the intercept estimates (due to both natural variability and measurement errors induced by leaf area measurements), a Monte-Carlo approach was employed by selecting b1 and b2 from normal distributions having mean values identical to the effective parameter values in Table 2 and standard deviations commensurate with the standard error of estimate determined from OLS (see Table 2). Hence, for each measurement point in Fig. 1, 100 b1 and b2 realizations were generated and for each of these m1 and m2 were calculated. These estimated m1 and m2 parameters are summarized as frequency distributions, grouped by ambient and elevated ca (Fig. 1, right panels). Increasing the number of realizations for the intercepts did not change the resulting frequency distributions. It is noted that estimates of m1 and m2 are not affected by leaf area measurement errors affecting conductance and photosynthesis identically and thus cancelling out in the ratio.

Table 2.
Effects of elevated ca on the statistics of the stomatal model parameters, estimated using ordinary least-square regression (OLS) or by inverting the stomatal conductance equations (see Figs 1 and and33)

Optimal conductance models

The optimal conductance models in eqns (8) and (14) require the parameters a1, a2, cp and λ. To determine these parameters, the gas exchange data were analysed as follows. Parameters a2 and cp were determined using the standard temperature formulations summarized in Table 1 (i.e. assuming no variability across leaves). This assumption seems justified given the consistency of these two parameters in a wide range of species. Parameter a1 was determined by re-arranging eqn (2) as

equation image

and then using each of the 193 measured fc, ci and leaf temperature (T) values. It is emphasized that a1 was not determined from typical fc(ci) curves. Hence, the uncertainty in a1 originates from errors incurred when measuring fc and fe (from which g, and subsequently ci, are estimated) and/or from natural variability among leaves and individual trees. Because ci is computed from the ratio of photosynthesis to conductance, A/g, it is unaffected by uncertainties in leaf area; however, some of the variability in a1 might be traced to errors in fc caused by uncertainties in leaf area estimates.

Values of a1 computed from eqn (17) were then compared in Fig. 2 to the standard temperature formulation for the maximum carboxylation capacity (Table 1) using a value of Vcmax,25 = 59 µmol m−2 s−1 obtained from independent data (Katul et al., 2000). The agreement in leaf temperature responses for a1 is reasonable considering that the gas-exchange was measured under ambient and enriched ca using two different systems, and over a wide range of conditions, as described earlier. Such an agreement suggests that, regardless of some uncertainty introduced by measurement errors, the estimation method (eqn 17) is sufficiently reliable for the purposes of this work.

Fig. 2.
Estimated maximum carboxylation capacity (or a1 for Rubisco-limited photosynthesis) from 193 gas exchange measurements as a function of leaf temperature (eqn 17) for needles exposed to ambient and elevated atmospheric CO2 (sampled from 1996 to 1999). ...

When it is assumed that stomata are optimally controlled, the parameter λ can be estimated using the definition of marginal water-use efficiency found by differentiation of eqn (6) (see also Hari et al., 1986),

equation image

The definition in eqn (18) can be used to compute λ from a1, a2, cp (previously described) and the measured D and g,

equation image


equation image

was defined.

In the linear model, λ can be directly determined from measured ca, D, fc and g (or ci),

equation image

Note that measurement errors due to the leaf area estimate (the most relevant in the present dataset) do not affect λLI and λ, which depend on the ratios A/g or a1/g.

The effective (or best-fit) values of λ and λLI along with standard error of estimates, determined from the OLS using eqn (18) (Fig. 3A and 3C) are reported in Table 2. For each gas exchange measurement, the uncertainty in the estimated λ for the linear model originates from measurement errors in fc, g and D. Likewise, for the non-linear model, additional uncertainty originates from the same variables used to compute a1. However, much of the variability in λ stems from needle-to-needle variations in measured fc/g, rather than measurement errors, as discussed above.

Fig. 3.
Left panels: linear regression-based estimates of the parameter λ from eqn (18) for the non-linear (A) and the linear optimization models (C), using gas exchange measurements collected under ambient (light circles) and elevated (heavy circles) ...

In the OLS estimates of λ, the abscissa was aD and the ordinate was computed analytically by differentiating with respect to g in eqn (4) (in the non-linear model) and eqn (9) (in the linear model), using the gas exchange data (including D) as input in the evaluation. Figures 3B and 3D also show the frequency distributions of λ and λLI of individual leaves based on eqns (19) and (20). Figure 4 compares λ with λLI of all 193 data points. The correlation coefficient (R) between these two λ estimates is large (R = 0·96), yet the linear model overestimated λ by about 20 %. This comparison suggests that λLI can provide an approximation of λ obtained from the more realistic, non-linear biochemical model.

Fig. 4.
Comparison between λ (non-linear model, denoted by NL; eqn 19) and λLI (linearized model, LI; eqn 20) for needles exposed to ambient and elevated ca. To estimate λ, a1 was estimated from data points in Fig. Fig.2.2. The ...


Effects of elevated CO2 on the parameters of Ball–Berry and Leuning models

When using the effective values of the OLS, both Ball–Berry and Leuning models described the gas exchange data well, although they were collected from plots with a wide range of soil nitrogen over years with a wide range of environmental conditions (McCarthy et al., 2007) and at different times during the growing season (Ellsworth, 2000), all of which affect leaf nitrogen concentration and physiological state. The coefficient of variation (CV = standard deviation/mean), was fairly large for m1 and m2 among individual needles (approx. 35 % and approx. 10 %, respectively; see frequency distributions in Fig. 1). The mean of m1 in the Ball–Berry model did not differ between ambient and elevated ca (Table 2), as was also found by Medlyn et al. (2001). In contrast, in the Leuning model, the effective m2 was significantly different when comparing ambient and enriched CO2 conditions. In both models, the standard deviations of m1 and m2 was higher at elevated ca. A Kolmogorov–Smirnoff test has also been conducted on differences between the frequency distributions of m1 and m2 shown in Fig. 1, and again, there was a significant CO2-induced effect on the parameters in the Leuning model at the 95 % confidence level but not in the Ball–Berry model.

Effects of elevated CO2 on λ

Having determined λ, the effect of elevated ca on λ and λLI was assessed. Elevated CO2 increased the effective values for λ in both the linear and the non-linear model (Table 2). Frequency distributions of λ estimates based on eqns (19) and (20) at the leaf scale are compared in Fig. 3B and 3D (analogous to the histogram comparisons conducted for m1 and m2 in Fig. 1). Elevated ca increased the mean λ and λLI, but due to simultaneous increase in the standard deviations, their CV remained unaffected (see Table 2). The CV values of λ and λLI are comparable to those of m1 and m2.

Model inter-comparison

Given the large variation in a1 among leaves (e.g. Fig. 2), the two optimization models and the two semi-empirical models are compared in an ‘ensemble-averaged’ sense to assess how well each reproduced expected effects of elevated atmospheric ca on photosynthesis, g and ci/ca.

For this comparison, eqns (1) and (2) were combined with eqns (5a) and (5b) (Ball–Berry and Leuning stomatal models) and numerically solved to provide estimates for ci, g and fc. Equations (5a) and (5b) were parameterized with the coefficients in Table 1. The parameter a1 in eqn (2) was parameterized using the continuous line in Fig. 2 (i.e. the standard model with Vcmax,25 = 59 µmol m−2 s−1) assuming that this formulation represents the ensemble fc(ci) behaviour in all cases and is independent of elevated ca (meaning that there is no down-regulation). The constant Vcmax,25 from the 1996–1998 dataset was used because later studies of down-regulation at the site were inconclusive (Rogers and Ellsworth, 2002; Crous et al., 2004, 2008; Springer et al., 2005; Maier et al., 2008). If the magnitude of the down-regulation is precisely known for each foliage element, it can be incorporated into parameter a1. However, this information is not available, and attempting to incorporate down-regulation would introduce additional and uneven uncertainty to the results from the four models.

Two calculations were made, each based on the non-linear and linear optimization models. The first assumed that λ does not vary with changing ca and was set equal to the mean value in Table 2 for ambient ca conditions. The other assumed that λ ≈ 0·089ca/Dref, where Dref = 1 kPa. The coefficient 0·089 was intended to reproduce the estimated variation of λ with elevated ca, commensurate with the change in mean values reported in Table 2 for λLI, and is assumed to be an upper-envelope of plausible λ variations.

The inputs to all models were leaf temperature, vapour pressure deficit or relative humidity and ca = 360 µmol m−2 s−1 for ambient and ca = 560 µmol m−2 s−1 for elevated conditions. The ensemble model calculations and comparison to data are reported in Figs 5 and and6.6. When setting λ to a constant independent of ca, increasing ca led to an increasing conductance and ci/ca (Fig. 5). The increase in g contrasts with evidence suggesting unchanging or slightly decreasing conductance with increasing ca. The mathematical structure of the optimization models suggest that when ca → +∞, the linear model predicts that g scales as ca−1/2, while the non-linear predicts g scales as ca−1. Hence, both models eventually predict a decline in g with increasing ca, as long as ca is high enough (3-fold current concentrations), even if λ is held constant. This indicates that to correctly predict changes in conductance within a realistic range of ca, the dependence of λ on atmospheric CO2 cannot be neglected.

Fig. 5.
The effects of elevated ca on conductance g (top), photosynthesis An (middle) and ci/ca (bottom): a comparison between the non-linear (NL) and linearized (LI) optimality results for a constant (set at ambient) and variable λ forced by ensemble-averaged ...
Fig. 6.
The effects of elevated ca on conductance g (top), photosynthesis An (middle) and ci/ca (bottom) as modelled by the non-linear approach with variable λ, the Ball–Berry model and the Leuning model. For reference, the ensemble-averaged gas ...

When λ was increased with elevated ca, the non-linear model calculations agreed well with the gas exchange data for the 1996–1998 period (Fig. 5), correctly predicting a slight decrease in g and ci/ca and markedly increased assimilation rate under elevated CO2 in the Duke Forest dataset. Also, the linearized optimization approach provides reasonable estimates of photosynthesis, g and ci/ca when λ varies with ca, similar to the Ball–Berry and Leuning model predictions (Fig. 6). Independent datasets from other experiments on loblolly pine under elevated ca (ranging from chamber to FACE) were also included for reference in Fig. 6 and some do show g hardly changing or even exhibiting an increase with ca. Changes in stomatal conductance, however, were not statistically significant in these experiments and thus cannot be reconciled with the 25 % increase predicted by the optimization model when λ was assumed independent of ca (Fig. 5). Finally, it is noted that the increase in λ with ca is consistent with an independent analysis based on the sensitivity of g to D for ambient and enriched ca by Katul et al. (2009) using published experiments reported by Heath (1998), Bunce (1998) and Medlyn et al. (2001).


The stomatal conductance optimization hypothesis assumes that the regulatory role of stomata is to simultaneously maximize the carbon gain rate while minimizing the rate of water losses. Accepting this as the stomatal role, it was possible to predict how certain environmental stimuli control stomatal conductance thereby unifying the two controls discussed in Scarth's review (Scarth, 1927), i.e. ‘regulation’ by environmental conditions (atmospheric CO2 concentration and vapour pressure deficit) and ‘regulatory’ in the economics of leaf gas exchange. The mathematical equivalence between the maximization of daily and short-term assimilation leads us to interpret the optimization problem differently from the earlier theory. We argue that optimization operates on time-scales commensurate with opening and closure of stomatal aperture (approx. 10 min, below which the dynamic, delayed responses of stomata to external stimuli should be accounted for) and each leaf optimally and autonomously regulates stomatal conductance. Perhaps two well-investigated examples in support of this argument are the responses of leaves to transient sunlight due to canopy shading (Pearcy, 1990; Naumburg et al., 2001; Naumburg and Ellsworth, 2002) and to variable cloud cover in dry ecosystems (Knapp and Smith, 1989; Knapp et al., 1989; Knapp, 1993).

In the first case, light penetration through the canopy illuminates a small area of a leaf for a brief period resulting in rapid opening of the stomata in the illuminated region while those in shaded regions remain relatively closed. This action would allow the entire plant to use light in photosynthesis as it becomes available without unnecessarily losing water (Hetherington and Woodward, 2003). Hence, while this ‘adaptive strategy’ may have evolved to satisfy the carbon and water economies of the entire plant, it is achieved by the rapid response of stomata of individual leaves (or areas of a leaf) to light availability. Leaf scale optimization thus leads to an optimal strategy for managing the carbon–water economy of the whole plant.

In the second example, plants subjected to drying soil and highly variable light during the growing season face a similar (although more complex) optimization problem. Their ‘objective’ is to conserve water when uptake of CO2 is light limited (Knapp et al., 1989), rather than maximizing assimilation during the sun fleck periods as described above. Striking differences among leaf responses occur, depending on the plant growth form (Knapp and Smith, 1989) and photosynthetic pathway (Knapp, 1993). Shallow-rooted grasses, for example, are more prone to water stress than trees, resulting in greater ability to adjust stomatal conductance in response to rapid changes in meteorological and light conditions (Knapp and Smith, 1989). This example again shows that evolution has lead to short-term response strategies that are consistent with the hypothesis of short-term optimization. Mathematically, this hypothesis translates to maximization of the objective function f in eqn (6), without needing temporal integration (although it is noted again that the two formulations are mathematically equivalent due to Pontryagin maximum principle).

For this short-term, leaf-level optimization, the interpretation of λ also becomes highly localized and thus spatially variable within the canopy and across the individual plants in a tree stand. Nevertheless, λ for a given ca is no more variable within a stand than the parameters of models currently employed in climate systems (e.g. m1 and m2, as shown in Fig. 1). Clearly, the carbon–water economy of plants is also affected by processes occurring at longer time-scales and at the whole-plant level. For example, at the weekly time-scale, stomatal conductance and photosynthetic activity are reduced under water shortage through complex plant-scale hormonal signalling (Chapin, 1991). It is speculated that such responses may help the plant prevent severe stress events (imbalanced metabolite concentrations or high growth costs) and thus maximize long-term fitness. At even longer time-scales, carbon is partitioned to optimize water or nutrient absorption by roots versus carbon uptake by leaves (Givnish et al., 1984; Palmroth et al., 2006; Franklin, 2007). Overall, the short-term, leaf-level optimization can be interpreted as the end member of a hierarchy of responses to environmental conditions that span temporal scales ranging from minutes to years and spatial scales from a patch on a leaf to trees and stands.

The proposed optimization models require one physiological parameter – the marginal water-use efficiency, λ. Based on the relationship between the linearized and non-linear models (Fig. 4), λ for loblolly pine needles can be estimated from λLI, which in turn is inferred from measured assimilation rate, stomatal conductance and vapour pressure difference (eqn 20). Alternatively, λLI can be estimated from measured ci/ca, which is becoming increasingly available from a stable isotope network [BASIN (Biosphere–Atmosphere Stable Isotope Network);] bypassing the need to estimate λ from direct gas exchange measurements (see Lloyd and Farquhar, 1994; Palmroth et al., 1999). Data on additional species and conditions would be necessary to assess the generality of the relationship between λ and λLI.

The analysis showed that, unlike the sensitivity parameter of the Ball–Berry model (m1), both the sensitivity parameter of the Leuning model (m2) and λ change with ca (Figs 1 and and33 and Table 2). This might limit the practical use of stomatal optimization theories. However, the present results suggest that λ tends to vary predictably. The data and model results in Fig. 6 suggest that ci/ca is largely invariant with elevated ca, which (a) implies that the linearization in eqn (9) has some empirical support and (b) explicitly links λ and ca based on eqn (20). For example, noting that ci/ca in C3 plants is bounded between 0·65 and 0·80 (e.g. Jones, 1992), results in λLI ≈ 0·03ca – 0·08ca at a reference D = 1 kPa. Hence, based on the analytical results and the limited sensitivity of the effects of ci/ca to increasing ca, it is not surprising that λ increases with increasing ca. For C4 species, ci/ca is usually bounded between 0·4 and 0·6 (e.g. Jones, 1992), resulting in λLI ≈ 0·1ca – 0·22ca at a reference D = 1 kPa, suggesting a larger enhancement in the marginal water-use efficiency λ with increasing ca of C4 than C3 plants. This is consistent with a large number of studies of relationships of photosynthesis versus conductance (Hetherington and Woodward, 2003). The interpretation of this increase is also physiologically meaningful – the marginal water-use efficiency is known to increase with increasing ca for both C3 and C4 species, but at different rates. In short, the predictability of λ with ca removes the potential limitation of the optimality approach presented here.

In addition to the explicit link dictating an increase in λ with ca, λ should also increase with drought severity (Cowan and Farquhar, 1977; Cowan, 1982; Berninger and Hari, 1993; Mäkelä et al., 1996). Drought-induced changes in λ may be interpreted as plant responses to altered availability of either carbon or water. When water availability decreases at constant ca, the marginal cost of water losses (interpreted as risk of water stress damage; Berninger and Hari, 1993) becomes more important. However, in the present study the link between λ and leaf water status was not explored.

In conclusion, the set of equations derived from the optimization condition stated here has the advantage of having closed and explicit form for conductance, photosynthesis and intercellular CO2 vis-à-vis the analogous set of equations for the Ball–Berry or Leuning models that require numerical solution. In our approach, the numerical solution of the coupled eqns (1), (2) and (5) is not necessary, and the numerical instabilities that often arise at low assimilation rates (Zhan et al., 2003; Vico and Porporato, 2008) can be avoided. Thus, this approach is suitable for climate models (Sellers et al., 1995), or large eddy simulation models employed to represent the details of complex turbulent flow regimes in biosphere–atmosphere mass and energy exchange studies (Albertson et al., 2001). The proposed formulation will benefit from a theoretical explanation of the changes in λ due to CO2 concentration and soil moisture availability. These theoretical issues are currently being investigated and will be the subject of a future contribution.


This study was supported by the United States Department of Energy (DOE) through the Office of Biological and Environmental Research (BER) Terrestrial Carbon Processes (TCP) program (FACE and NICCR grants: DE-FG02-95ER62083, DE-FC02-06ER64156), by the National Science Foundation (NSF-EAR 0628342, NSF-EAR 0635787), by Bi-National Agricultural Research Development (BARD) fund (IS-3861-06) and by the US Department of Agriculture (USDA grant 58-6206-7-029). We also thank three anonymous reviewers and Giulia Vico for helpful comments.


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