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**|**Ann Bot**|**v.105(3); 2010 March**|**PMC2826246

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Ann Bot. 2010 March; 105(3): 431–442.

Published online 2009 December 8. doi: 10.1093/aob/mcp292

PMCID: PMC2826246

Received 2009 August 4; Revised 2009 October 21; Accepted 2009 November 10.

Copyright © The Author 2009. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article has been cited by other articles in PMC.

Global climate models predict decreases in leaf stomatal conductance and transpiration due to increases in atmospheric CO_{2}. 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 CO_{2} 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 CO_{2} and H_{2}O through stomata; (*b*) a biochemical photosynthesis model that relates intercellular CO_{2} 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 CO_{2}.

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 CO_{2}, 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 CO_{2}; (*b*) the new formulation correctly predicts the condition under which CO_{2}-enriched atmosphere will cause increasing assimilation and decreasing stomatal conductance.

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 CO_{2} (*c*_{a}) 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 CO_{2} cause stomata to open while increasing *c*_{a} 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 CO_{2} 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 CO_{2} 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 CO_{2} 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 CO_{2} 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 CO_{2} by plants, contributing to increased atmospheric CO_{2} concentration (Cox *et al.*, 2000). When less water is lost through transpiration in water-limited ecosystems, the CO_{2} 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 CO_{2} 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 *c*_{a} on their parameterization. In particular, the effect of elevated *c*_{a} 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 CO_{2} 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 CO_{2} atmosphere. The new short-term optimization theory and a unique dataset are used to assess how elevated *c*_{a} might impact the trade-off parameter, the most pertinent to the economy of gas exchange.

Mass transfer of CO_{2} and water vapour between leaves and the bulk atmosphere can be described by Fickian diffusion through stomata when the boundary layer resistance is negligible:

1

where *f*_{c} is the CO_{2} flux, *f*_{e} is the water vapour flux, *g* is the stomatal conductance to CO_{2}, *c*_{a} is ambient and *c*_{i} intercellular CO_{2} concentrations, respectively, *a* = 1·6 is the relative diffusivity of water vapour with respect to carbon dioxide, *e*_{i} is the intercellular and *e*_{a} the ambient water vapour concentration and *D* is the vapour pressure deficit representing *e*_{i} – *e*_{a} 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 *f*_{c}, the biochemical demand for CO_{2} is mathematically well described by the photosynthesis model of Farquhar *et al.* (1980):

2

where *c*_{p} is the CO_{2} compensation point, *a*_{1} and *a*_{2} 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, *a*_{1} = *V*_{cmax} (maximum carboxylation capacity) and *a*_{2} = *K*_{c}(1 + *C*_{oa}/*K*_{o}), where *K*_{c} and *K*_{o} are the Michaelis constants for CO_{2} fixation and oxygen inhibition and *C*_{oa} 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

3

and

4

when *g* > 0.

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

Equations (3) and (4) are not mathematically ‘closed’ and require independent formulations of *g* so that *f*_{c} and *c*_{i} can be computed. Two conductance formulations, the so-called Ball–Berry (*g*_{1}, see Ball *et al.*, 1987) and the Leuning (*g*_{2}, 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:

5a

and

5b

where *H* is the mean air relative humidity, *D*_{o} is a vapour pressure deficit constant and *m*_{1}, *m*_{2}, *b*_{1} and *b*_{2} are empirical fitting parameters. Equations (5a) and (5b) therefore provide the necessary mathematical closure (i.e. three equations with three unknowns: *f*_{c}, *c*_{i} 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.

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

6

By maximizing the gross carbon gain *f*_{c} while minimizing the water loss λ*f*_{e} (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 δ*f*(*g*)/δ*g* = 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 *c*_{a}. 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 *t*_{0} and *t*_{1}. 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

7

When setting δ*f*(*g*)/δ*g* = 0 and solving for *g*, the following is obtained:

8

Similar to the two formulations of the first approach (eqns 5a and 5b), this formulation is explicit in relating conductance to *c*_{a} and *D*. However, unlike these formulations, both relating conductance to the photosynthetic rate leaving two (*g*_{1} formulation) or three (*g*_{2} 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 (*c*_{a} – *c*_{p})/(*aD*) to ensure real and positive conductance. This theoretical bound is in contrast to the less constrained choices of *m*_{1} and *m*_{2} 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| |δ*f*_{e}/*f*_{e}|. Recalling that *f*_{e} = *agD* (eqn 1), this necessary condition can be expressed as λ/*g* |δλ/δ*g*|. Future experimental studies might employ this constraint to test how close to optimality a single leaf is.

Linearizing the biochemical demand function results in a much simpler and informative model for optimal *g*. Linearizing the model requires the assumptions that *c*_{p} *c*_{i} and that the variability of *c*_{i} affects only marginally the denominator of eqn (2), leading to the approximation *a*_{2} + *c*_{i} = *a*_{2} + (*c*_{i}/*c*_{a})*c*_{a} = *a*_{2} + *sc*_{a}. As a result,

9

where *s* is treated as a constant set equal to the long-term mean of *c*_{i}/*c*_{a}. We stress that *s* is treated as a model constant only in the denominator of eqn (9), while in eqn (1) *c*_{i}/*c*_{a} is allowed to vary. Note also that the geometric interpretation of the group of parameters *a*_{1}/(*a*_{2} + *sc*_{a}) in eqn (9) is simply the slope of the *f*_{c}(*c*_{i}) curve. Combining this linearized photosynthesis model with eqn (1) results in

10

and

11

The objective function in eqn (6) now simplifies to

12

and upon differentiation with respect to *g* yields

13

By setting δ*f(g*)/δ*g* = 0 and solving for *g* results in

14

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 *c*_{i} and *f*_{c} given by

15

and

16

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 *c*_{i}/*c*_{a} variations with *D*.

The experiments, including details of the free-air CO_{2} 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 CO_{2} to maintain atmospheric CO_{2} 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.

Steady-state gas exchange of leaves was measured in 1996–1999 in plots 1–6, at two growth CO_{2} concentrations (ambient and ambient + 200 µmol mol^{−1} CO_{2} concentration) and near ambient temperature using a portable infrared gas analyser system for CO_{2} 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 CO_{2} 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).

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 CO_{2} 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).

The combined dataset, resulting in 193 gas exchange measurements, 73 of which were collected under elevated *c*_{a}, 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 *c*_{a} on their parameters, measurement errors and how these errors might impact parameter estimation are briefly discussed.

Measurement errors during leaf gas exchange experiments may be due to (*a*) uncertainties in the measured CO_{2} and H_{2}O 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 CO_{2}, 0·1 % for H_{2}O and 0·1 % for temperature. Diffusive leaks are not expected to be an issue in studies where gas exchange measurements are performed at external CO_{2} 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.

The effective model parameters for *g*_{1} and *g*_{2} (eqns 5a and 5b) and their frequency distributions were computed from the two gas exchange datasets. The values for *m*_{1}, *m*_{2}, *b*_{1} and *b*2 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 *m*_{1} and *m*_{2}. Variations in *m*_{1} and *m*_{2} among leaves and trees were also computed by inverting eqns (5a) and (5b) for given values of the intercepts *b*_{1} and *b*_{2}. 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 *b*_{1} and *b*_{2} 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 *b*_{1} and *b*_{2} realizations were generated and for each of these *m*_{1} and *m*_{2} were calculated. These estimated *m*_{1} and *m*_{2} parameters are summarized as frequency distributions, grouped by ambient and elevated *c*_{a} (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 *m*_{1} and *m*_{2} are not affected by leaf area measurement errors affecting conductance and photosynthesis identically and thus cancelling out in the ratio.

The optimal conductance models in eqns (8) and (14) require the parameters *a*_{1}, *a*_{2}, *c*_{p} and λ. To determine these parameters, the gas exchange data were analysed as follows. Parameters *a*_{2} and *c*_{p} 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 *a*_{1} was determined by re-arranging eqn (2) as

17

and then using each of the 193 measured *f*_{c}, *c*_{i} and leaf temperature (*T*) values. It is emphasized that *a*_{1} was not determined from typical *f*_{c}(*c*_{i}) curves. Hence, the uncertainty in *a*_{1} originates from errors incurred when measuring *f*_{c} and *f*_{e} (from which *g*, and subsequently *c*_{i}, are estimated) and/or from natural variability among leaves and individual trees. Because *c*_{i} 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 *a*_{1} might be traced to errors in *f*_{c} caused by uncertainties in leaf area estimates.

Values of *a*_{1} 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 *V*_{cmax,25} = 59 µmol m^{−2} s^{−1} obtained from independent data (Katul *et al.*, 2000). The agreement in leaf temperature responses for *a*_{1} is reasonable considering that the gas-exchange was measured under ambient and enriched *c*_{a} 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.

Estimated maximum carboxylation capacity (or *a*_{1} 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 CO_{2} (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),

18

The definition in eqn (18) can be used to compute λ from *a*_{1}, *a*_{2}, *c*_{p} (previously described) and the measured *D* and *g*,

19

where

19a

was defined.

In the linear model, λ can be directly determined from measured *c*_{a}, *D*, *f*_{c} and *g* (or *c*_{i}),

20

Note that measurement errors due to the leaf area estimate (the most relevant in the present dataset) do not affect λ_{LI} and λ, which depend on the ratios *A*/*g* or *a*_{1}/*g*.

The effective (or best-fit) values of λ and λ_{LI} 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 *f*_{c}, *g* and *D*. Likewise, for the non-linear model, additional uncertainty originates from the same variables used to compute *a*_{1}. However, much of the variability in λ stems from needle-to-needle variations in measured *f*_{c}/*g*, rather than measurement errors, as discussed above.

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 λ_{LI} of individual leaves based on eqns (19) and (20). Figure 4 compares λ with λ_{LI} 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 λ_{LI} can provide an approximation of λ obtained from the more realistic, non-linear biochemical model.

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 *m*_{1} and *m*_{2} among individual needles (approx. 35 % and approx. 10 %, respectively; see frequency distributions in Fig. 1). The mean of *m*_{1} in the Ball–Berry model did not differ between ambient and elevated *c*_{a} (Table 2), as was also found by Medlyn *et al.* (2001). In contrast, in the Leuning model, the effective *m*_{2} was significantly different when comparing ambient and enriched CO_{2} conditions. In both models, the standard deviations of *m*_{1} and *m*_{2} was higher at elevated *c*_{a}. A Kolmogorov–Smirnoff test has also been conducted on differences between the frequency distributions of *m*_{1} and *m*_{2} shown in Fig. 1, and again, there was a significant CO_{2}-induced effect on the parameters in the Leuning model at the 95 % confidence level but not in the Ball–Berry model.

Having determined λ, the effect of elevated *c*_{a} on λ and λ_{LI} was assessed. Elevated CO_{2} 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 *m*_{1} and *m*_{2} in Fig. 1). Elevated *c*_{a} increased the mean λ and λ_{LI}, but due to simultaneous increase in the standard deviations, their CV remained unaffected (see Table 2). The CV values of λ and λ_{LI} are comparable to those of *m*_{1} and *m*_{2}.

Given the large variation in *a*_{1} 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 *c*_{a} on photosynthesis, *g* and *c*_{i}/*c*_{a}.

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 *c*_{i}, *g* and *f*_{c}. Equations (5a) and (5b) were parameterized with the coefficients in Table 1. The parameter *a*_{1} in eqn (2) was parameterized using the continuous line in Fig. 2 (i.e. the standard model with *V*_{cmax,25} = 59 µmol m^{−2} s^{−1}) assuming that this formulation represents the ensemble *f*_{c}(*c*_{i}) behaviour in all cases and is independent of elevated *c*_{a} (meaning that there is no down-regulation). The constant *V*_{cmax,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 *a*_{1}. 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 *c*_{a} and was set equal to the mean value in Table 2 for ambient *c*_{a} conditions. The other assumed that λ ≈ 0·089*c*_{a}/*D*_{ref}, where *D*_{ref} = 1 kPa. The coefficient 0·089 was intended to reproduce the estimated variation of λ with elevated *c*_{a}, commensurate with the change in mean values reported in Table 2 for λ_{LI}, 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 *c*_{a} = 360 µmol m^{−2} s^{−1} for ambient and *c*_{a} = 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 *c*_{a}, increasing *c*_{a} led to an increasing conductance and *c*_{i}/*c*_{a} (Fig. 5). The increase in *g* contrasts with evidence suggesting unchanging or slightly decreasing conductance with increasing *c*_{a}. The mathematical structure of the optimization models suggest that when *c*_{a} → +∞, the linear model predicts that *g* scales as *c*_{a}^{−1/2}, while the non-linear predicts *g* scales as *c*_{a}^{−1}. Hence, both models eventually predict a decline in *g* with increasing *c*_{a}, as long as *c*_{a} 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 *c*_{a}, the dependence of λ on atmospheric CO_{2} cannot be neglected.

The effects of elevated *c*_{a} on conductance *g* (top), photosynthesis *A*_{n} (middle) and *c*_{i}/*c*_{a} (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 **...**

The effects of elevated *c*_{a} on conductance *g* (top), photosynthesis *A*_{n} (middle) and *c*_{i}/*c*_{a} (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 *c*_{a}, 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 *c*_{i}/*c*_{a} and markedly increased assimilation rate under elevated CO_{2} in the Duke Forest dataset. Also, the linearized optimization approach provides reasonable estimates of photosynthesis, *g* and *c*_{i}/*c*_{a} when λ varies with *c*_{a}, similar to the Ball–Berry and Leuning model predictions (Fig. 6). Independent datasets from other experiments on loblolly pine under elevated *c*_{a} (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 *c*_{a}. 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 *c*_{a} (Fig. 5). Finally, it is noted that the increase in λ with *c*_{a} is consistent with an independent analysis based on the sensitivity of *g* to *D* for ambient and enriched *c*_{a} 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 CO_{2} 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 CO_{2} 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 *c*_{a} is no more variable within a stand than the parameters of models currently employed in climate systems (e.g. *m*_{1} and *m*_{2}, 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 λ_{LI}, which in turn is inferred from measured assimilation rate, stomatal conductance and vapour pressure difference (eqn 20). Alternatively, λ_{LI} can be estimated from measured *c*_{i}/*c*_{a}, which is becoming increasingly available from a stable isotope network [BASIN (Biosphere–Atmosphere Stable Isotope Network); http://basinisotopes.org/] 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 λ_{LI}.

The analysis showed that, unlike the sensitivity parameter of the Ball–Berry model (*m*_{1}), both the sensitivity parameter of the Leuning model (*m*_{2}) and λ change with *c*_{a} (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 *c*_{i}/*c*_{a} is largely invariant with elevated *c*_{a}, which (*a*) implies that the linearization in eqn (9) has some empirical support and (*b*) explicitly links λ and *c*_{a} based on eqn (20). For example, noting that *c*_{i}/*c*_{a} in C_{3} plants is bounded between 0·65 and 0·80 (e.g. Jones, 1992), results in λ_{LI} ≈ 0·03*c*_{a} – 0·08*c*_{a} at a reference *D* = 1 kPa. Hence, based on the analytical results and the limited sensitivity of the effects of *c*_{i}/*c*_{a} to increasing *c*_{a}, it is not surprising that λ increases with increasing *c*_{a}. For C_{4} species, *c*_{i}/*c*_{a} is usually bounded between 0·4 and 0·6 (e.g. Jones, 1992), resulting in λ_{LI} ≈ 0·1*c*_{a} – 0·22*c*_{a} at a reference *D* = 1 kPa, suggesting a larger enhancement in the marginal water-use efficiency λ with increasing *c*_{a} of C_{4} than C_{3} 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 *c*_{a} for both C_{3} and C_{4} species, but at different rates. In short, the predictability of λ with *c*_{a} removes the potential limitation of the optimality approach presented here.

In addition to the explicit link dictating an increase in λ with *c*_{a}, 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 *c*_{a}, 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 CO_{2} 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 CO_{2} 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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