Using the economics of gas exchange, early studies derived an expression of stomatal conductance (g) assuming that water cost per unit carbon is constant as the daily loss of water in transpiration (fe) is minimized for a given gain in photosynthesis (fc). Other studies reached identical results, yet assumed different forms for the underlying functions and defined the daily cost parameter as carbon cost per unit water. We demonstrated that the solution can be recovered when optimization is formulated at time scales commensurate with the response time of g to environmental stimuli. The optimization theory produced three emergent gas exchange responses that are consistent with observed behaviour: (1) the sensitivity of g to vapour pressure deficit (D) is similar to that obtained from a previous synthesis of more than 40 species showing g to scale as 1 − m log(D), where m? [0.5,0.6], (2) the theory is consistent with the onset of an apparent ‘feed-forward’ mechanism in g, and (3) the emergent non-linear relationship between the ratio of intercellular to atmospheric [CO2] (ci/ca) and D agrees with the results available on this response. We extended the theory to diagnosing experimental results on the sensitivity of g to D under varying ca.
Jan Baptist van Helmont is credited with coining the word ‘gas’ in the 17th century and noting that ‘gas sylvestre’ (carbon dioxide) is given off by burning charcoal. He also investigated water uptake by a willow tree in 1648, in effect performing one of the earliest recorded experiments on stomatal conductance (g) to gas transfer. Centuries later, both of van Helmont's activities converged in a modern-day story: Atmospheric CO2 is rising largely because of the combustion of fossil fuel, and the ability of terrestrial plants to uptake CO2 is currently a leading mitigation strategy to offset this rise. Because the role of stomata in regulating the exchange of CO2 for water is central to many plant and ecosystem processes, services and products, variations in g and in their responses to environmental variables have been subjected to intense research for decades. And yet, despite numerous experiments and several modelling approaches, the precise mechanisms responsible for stomatal responses to certain environmental stimuli remain vague (see e.g. review by Buckley 2005).
Several empirical and semi-empirical models describing stomatal responses to environmental stimuli exist (e.g. Jarvis 1976; Collatz et al. 1991; Leuning 1995). These models, advanced primarily after the publication of the seminal work by Jarvis (1976), are based on an electrical circuit analogy – stomata are viewed as a resistor (or a conductor) with a maximum species-specific value of g attained when stomatal pores are fully open. The maximum g is reduced by non-linear functions that account for the effects of external environmental factors via increases in the concentration of CO2 in the leaf's air space, or the capacity of the soil–plant hydraulics to supply water to the leaf relative to the potential rate of vapour loss rate from fully open stomata. These functions reflect decreasing light levels, increasing CO2 concentration and vapour pressure deficit, departure from optimum leaf temperature, and also decreasing leaf water potential representing the hydration state of the stomatal system. Such a parsimonious representation of stomatal conductance, combined with the increased availability of portable equipment for measuring gas exchange, greatly contributed to quantifying differential species sensitivities to environmental stimuli and stresses.
The wealth of data on g led to certain generalities on stomatal responses to the environment. For example, Mott & Parkhurst (1991) demonstrated that stomatal closure is a response to leaf transpiration rate rather than to varying vapour pressure deficit (D). Nevertheless, a synthesis of stomatal responses to varying D, obtained from studies on over 40 species from grasses to deciduous and evergreen trees, revealed a general functional form that can be described as g = gref(1 − m log(D)), where m ≈ 0.5−0.6 and gref is g at D = 1 kPa (Oren et al. 1999). Further support for this value of m was provided in Mackay et al. (2003). It was also demonstrated that the value of m ≈ 0.6 is consistent with a hydraulic model in which plants control transpiration rate to protect the transport system from excessive loss of hydraulic function (Sperry et al. 1998, 2002; Oren et al. 1999; Lai et al. 2002). Yet stomatal control cannot be only to regulate the rate of water loss. Indeed, it has long been suggested that, at the leaf scale, natural selection may have operated to provide increasingly efficient means of controlling the trade offs between carbon gain and the accompanying water vapour loss (e.g. Cowan 1977, 1982, 2002; Ball, Cowan & Farquhar 1988). If so, can such an optimization principle be used to constrain certain parameters in semi-empirical models, perhaps even replacing them by functional responses that naturally emerge from such optimization? Emergent functions are more general, unlike imposed functions that empirically describe data, limiting their application to the conditions represented by the experiment.
First presented by Cowan (1977) and Cowan & Farquhar (1977), and reformulated by Hari et al. (1986) and Berninger & Hari (1993), the cost (= daily water loss in transpiration) to benefit (= daily carbon gain in photosynthesis) analysis was framed as an ‘economic’ optimization. While the assumptions on the form of the underlying functions differ between Cowan & Farquhar (1977) and Hari et al. (1986), their optimal solutions are, in fact, identical. Moreover, while both studies implicitly assumed an integration time scale, their solution appeared independent of the time scale of flux integration. The stomatal control over gas exchange is described through a concept of invariant ‘carbon cost of water’ or ‘water cost of carbon’, without a priori specification of stomatal response to D or atmospheric CO2. The predicted expressions of stomatal responses to D or atmospheric CO2 are ‘emergent properties’ of the optimization theory. We compare these emergent responses with data from studies from a wide range of conditions. We demonstrate that the optimization theory permits predictions of stomatal response to environmental stimuli, especially with respect to D in both current and CO2-enriched atmosphere. The analysis limits ‘optimality’ to bulk leaf gas-exchange; it may not be used to explain such inter-related questions as how guard cells operate to achieve optimality, or why leaves are oriented in a specific way within ecosystems.
The basic equations for the leaf-level CO2 and water vapour fluxes across stomata are given by:
where fc is the CO2 flux, fe is the water vapour flux, g is the stomatal conductance, ca is ambient and ci intercellular CO2 concentration, a = 1.6 is the relative diffusivity of water with respect to carbon, and ei is the intercellular and ea the ambient water vapour concentration. Photosynthesis (p) is related to ci via the Farquhar model (Farquhar, Caemmerer & Berry 1980a):
where α1 and α2 depend on whether the photosynthetic rate is light – or Rubisco-limited. For analytical tractability, assume that Γ/ci << 1 and the expression α1ci/(α2 + ci) ≈ α1ci/(a2 + sca), where s relates ci to ca:
where α = γ1Vc,max for temperature limited photosynthesis and α = γ2PAR for light-limited photosynthesis. Here, Vc,max stands for maximum carboxylation capacity, γ1 and γ2 are physiological parameters, and PAR is photosynthetically active radiation. Equation 4 demonstrates the correspondence between the assumed p − ci curve in Hari et al. (1986) and the parameters in the Farquhar model, although the precise value of α is irrelevant to the following optimization discussion. Assuming steady-state conditions,
where r is the leaf respiration rate. Combining Eqns 1, 2, 4 and 5 results in the following formulations:
The two basic equations (Eqns 6 and 7) include three unknown state variables (ci, g and p), generating a problem not closed mathematically. Standard approaches to ‘close’ this problem assume an empirical relationship between g, p and some environmental stimuli such as air relative humidity (RH) or D (Baldocchi & Meyers 1998; Lai et al. 2000). Two well-known formulations that fit a wide range of field data are given by the so-called ‘Ball-Berry model’ (Ball, Woodrow & Berry 1987; Collatz et al. 1991):
where b1 sets a minimum g, Do is the sensitivity of g to vapour pressure deficit, and m1 and m2 are empirical parameters that vary among species. Another closure assumption, first proposed by Cowan (1977) and Cowan & Farquhar (1977), is a constant marginal water cost per unit carbon, (∂fe/∂g)/(∂fc/∂g). This basic premise is retained in the work by Hari et al. (1986) and Berninger & Hari (1993). In their formulation, g is expressed as g = gou where go is the maximum conductance and u is the degree of stomatal opening (0 < u ≤ 1). The carbon cost of a unit of transpired water, λ, is formulated as the inverse of (∂fe/∂g)/(∂fc/∂g). Like in Cowan & Farquhar (1977), λ is assumed to be constant at time scale of 1 d, and the fluxes of CO2 and water vapour are integrated over the same period.
Interestingly, in the following discussion, we show that a solution to an analogous optimization problem can be developed without time-integration (given the steady-state assumption in Eqn 5) and Lagrange multipliers. Results from such a solution are identical to those from Hari et al. (1986) and Cowan & Farquhar (1977), and actually establish some constraints on how constant the cost parameter needs to be for the solution to be accurate.
However, before presenting this optimum solution, we note that when aca >> r, Eqns 2 and 7 can be combined to arrive at an explicit relationship between fc and fe given as . This expression has a negative convexity for any positive fe and ei − ea because .
Hence, despite the linearization in the p − ci curve adopted in Eqn 4, the fc (dependent) versus the fe (independent) expression proposed here maintains a negative convexity and thus our formulation admits an optimal solution as discussed in Cowan & Farquhar (1977). To find this optimum for the linearized p − ci curve, the maximization of the carbon gain function f(u) with respect to stomatal aperture control (u) can be expressed as:
In comparison with Hari et al. (1986), it is clear that the optimization problem in Eqn 10 simplifies to a univariate maximization problem provided . This simplification was the basis of the definition of the Cowan & Farquhar (1977) marginal water cost per unit carbon, expressed as (∂fe/∂g)/(∂fc/∂g) [or ]. Maximization in Eqn 10 is achieved when
The condition ∂f(u)/∂u = 0 in Eqn 10 alone does not rule out that the result in Eqn 11 is a local minimum rather than a maximum for f(u). To ensure that f(u) is maximum for the u given by Eqn 11, f(u) must be concave (or negatively convex). Upon twice differentiating f(u) with respect to u, we obtain , which is monotonically negative (u > 0) provided αca > r. Hence, this negative convexity in f(u) guarantees that u in Eqn 11 is a maximum and not a minimum. For r/α << ca
This expression states that g decreases with increasing ei − ea, and is sensitive to the slope of the p −ci curve. Because u is bounded between zero and unity, theoretical bounds on the cost parameter λ can be readily established and are given as:
The variations in λ (as λmin/λmax) are entirely dictated by go/α– the maximum conductance and the basic physiological parameter of the linear p − ci curve. If go/α ∼ 1, then λmin/λmax ∼ 1/4 (or fourfold variation). Using a similar maximization approach, we also derived the optimum conductance using the non-linear p − ci curve (Eqn 3). However, the resulting formulation does not reveal primitive scaling rules between environmental stimuli and stomatal conductance owing to the larger number of parameters.
Recall that no a priori specification of the stomatal response to D is imposed and that the shape of the dependency of g on D emerges from the optimization. Hari et al. (2000) presented a convincing field test on the dependence of g on D−1/2, suggesting that such a dependency is a ‘validation’ of the optimization hypothesis (assuming that atmospheric vapour pressure deficit, D well approximates leaf-to-air vapour pressure difference, ei − ea). However, the dependence of g on D−1/2 clearly conflicts with the functional dependencies of g1 or g2 assumed by Ball et al. (1987) and Leuning (1995), respectively.
ANALYSES AND DISCUSSION
To assess whether the optimization principle can be used to constrain certain parameters in semi-empirical models of g, we firstly consider the sensitivity of g to D, and then evaluate how the sensitivity of g to D is impacted by changes in ca.
The emerging conductance sensitivity is reflected in the dependence on D−1/2, which may be expressed via a Taylor series expansion as:
where the leading term is of the form 1 − (1/2)log(D). Figure 1 shows the variations of D−1/2 and for n1 = 1, 2, 3 and for typical D varying from a low of 0.5 to an extreme of 6. The series converges rapidly, with n1 = 3 indistinguishable from D−1/2. Moreover, Fig. 1 shows that the Taylor series expansion D−1/2 ≈ 1 − 1/2 log(D) is accurate for D ? [0.5,2] (i.e. to within 5% relative error), but for D > 2, higher order effects (n1 > 1) become large.
From a broad survey of plant species representing many functional types, a functional form g in response to D was derived as g = gref(1 − m log(D)), where gref is the so-called reference conductance determined for similar light and soil moisture conditions at D = 1 kPa (Oren et al. 1999). The synthesis showed that m ≈ 0.5 − 0.6. Since then, other studies added many additional species showing similar response (Mackay et al. 2003). To compare this well-supported empirical finding with the optimization prediction, the empirical function can be expressed as g = gref(1 − m log(D)). The optimization result for g/gref is g/gref = (−1 + Φ/D1/2)/(−1+ Φ), with . Upon replacing D−1/2 ≈ 1 − 1/2 log(D), For the case when Φ >> 1, , close to the lower limit of the reported m values in Oren et al. (1999). Even for large Φ, the optimization theory should yield an m > 0.5, also consistent with the empirical findings and the theory of water transport to leaves in Oren et al. (1999).
To further illustrate the similarity between the emergent behaviour from the optimization model and the general value of m[from 1 − m log(D)], we used the data of Fagus crenata Blume (Fig. 2; Iio et al. 2004) not included in Oren et al. (1999). The regression analysis on the data in Fig. 2 results in m ≈ 0.45, similar to D−1/2 scaling (P > 0.05). Thus, the optimization theory appears consistent with the well-documented general behaviour of g with respect to D.
A joint reduction in g and fe with increasing D can be seen as evidence of a feed-forward mechanism of stomatal response (Schulze et al. 1972). Such behaviour, under some conditions, can also be predicted based on optimal stomatal control (Buckley 2005). Here, we demonstrate that the emergent g ∼ D−1/2 from the optimization theory is qualitatively consistent with Monteith's (1995) view of the sensitivity of fe to D and the apparent feed-forward mechanism.
The function suggests that fe is dominated by two opposing terms when D is increased. These terms are represented by the sum and imply that at low D, fe increases rapidly with increasing D because the first term is small compared to the second term. However, at very high D, the first term may dominate and fe begins to decline with increasing D, consistent with empirical findings (Monteith 1995; Pataki et al. 1998). This outcome is identical to a derivation from the g −D response performed in Oren et al. (1999). The optimization model can be used to predict the value of D at which the onset of such apparent feed-forward is likely to occur (critical D, Dcirt). Because ∂g/∂D ∼ −(1/2)D−3/2 < 0 for all D > 0, the apparent feed-forward mechanism occurs only when . This Dcirt can be readily computed as
and depends only on ca and λ.
This critical limit can be assessed based on data from four-step measurements on three leaves of Abutilon theophrasti, two of which display an apparent feed-forward mechanism and one displaying a plateau [i.e. transpiration almost independent of D; Bunce (1997)]. In this framework, a plateau indicates that Dcirt is not yet exceeded. A regression model g = σ1 + σ2D−1/2 was fitted to the four-point conductance data, and then fe was computed from the modelled g (along with Dcrit determined from σ1 and σ2). The expected transpiration is similar to the data, as is the D at which the apparent feed-forward is observable (Fig. 3).
Finally, it should be noted that for a given λ, the optimization approach predicts that leaf transpiration becomes negligible when , which results in D = 0 (a trivial solution) and (a non-trivial solution). The second solution suggests that transpiration becomes negligible when actual D exceeds 4Dcrit. Hence, leaves in dry climates, which routinely experience high D, must close stomata more often than leaves with similar λ in humid climates, which is a reasonable behaviour.
Dependence of ci/ca on D
An important yet unexplored consequence of this optimization model is that ci/ca varies in a predictable manner with D, and is given by
As reported in a large number of studies (see Katul, Ellsworth & Lai 2000), the hyperbolic dependence of ci/ca on g (i.e. first equality on the right-hand side) is well preserved by the optimization model (Fig. 4), suggesting that the linear p − ci approximation is perhaps appropriate for ‘field’ conditions and is consistent with the arguments in Leuning (1995) and Katul et al. (2000). As discussed in Katul et al. (2000), Leuning's semi-empirical conductance model leads to a linear decline of ci/ca with increasing D. In contrast, the optimization theory model predicts that as D decreases ci/ca increases asymptotically such that ci/ca → 1 when D → 0. Leuning (1995) used data on 9 of the 16 species from Turner, Schulze & Gollan (1984) to calibrate the stomatal response to D. Although the data can probably be represented equally well by a linear and a non-linear approximation, at low D, some degree of non-linearity in the dependence of ci/ca on D is apparent in the data of all species (see fig. 9 in Leuning 1995). Indeed, many field and laboratory studies have reported a non-linear decline of ci/ca with increasing D (Farquhar et al. 1980b; Lloyd & Farquhar 1994), or present data that appear to be a better fit with D1/2 than D, as shown in Fig. 5 (Wong & Dunin 1987; Fites & Teskey 1988; Mortazavi et al. 2005; and several discussed in Katul et al. 2000).
The conditions that lead to a near-linear dependence of ci/ca on D can also be delineated within the context of the optimization framework. By rewriting the gas-exchange equations as
a plausible condition can be derived by replacing the constant marginal cost λ = (∂fc/∂g)/(∂fe/∂g) with a constant flux-based water use efficiencyλL = fc/fe, to yield
A constant λL (i.e. independent of D) can only be achieved if
In other words, ci/ca may decline linearly with increasing D as suggested by Leuning (1995) if λL is a constant. By equating these two ci/ca formulations, it can be shown that λL can be related to λ using λL = (λca/a)1/2D1/2. In typical field experiments, large variations in D are needed to discern the dependence of λL on D (assuming constant λ), perhaps explaining why data cannot conclusively reject a linear ci/ca decline with increasing D (e.g. Fig. 5).
Two studies criticized the optimality hypothesis by demonstrating that λ was not constant but varied with D (Fites & Teskey 1988; Thomas et al. 1999). As noted earlier, variations in λ alone do not disprove the optimality hypothesis and predictions of the response of g to D may still be reasonable, provided that |(1/λ)(∂λ/∂u)| << |(1/fe)(∂fe/∂u)|. Stated differently, as long as the relative variations in λ are much smaller than the relative variations in fe, the scaling properties emerging from the optimization theory are robust. We analysed the data published in Fites & Teskey (1988; Fig. 6) and noted that (1) their g scales as D−1/2 and is consistent with the optimality hypothesis, (2) regressing their reported fc upon their fe results in a near-constant λL (in disagreement with the optimality hypothesis), (3) regressing ∂fe/∂g versus ∂fc/∂g directly estimated from the data (digitized by us), suggests a constant λ (consistent with the optimality hypothesis), and (4) their ci/ca is not linearly related D, especially at high D (consistent with the optimality hypothesis but not the outcome in point 2). Notice that points (2) and (3) cannot be simultaneously satisfied given that λL = (λca/a)1/2D1/2. In estimating ∂fe/∂g and ∂fc/∂g, we used central differencing to approximate these gradients from the data published by Fites & Teskey (1988; their Figs 1 and 2). Estimating ∂fe/∂g and ∂fc/∂g from raw (and digitized) data increases the uncertainty when computing such derivatives because the random error is generally amplified by the differencing operator. We also confirmed that the residuals from the regression in step (3) are not significantly dependent on D. While similar analysis could not be repeated on the data in Thomas et al. (1999) because their fluxes and ci/ca were not published, ∂λ/λ calculated based on five different methods resulted in estimates varying within 20%, possibly reflecting |(1/λ)(∂λ/∂u)| << |(1/fe)(∂fe/∂u)|. Thus, with a less stringent criterion for the constancy of λ, the results from both studies cannot be used to undermine the optimization theory.
Effects of high atmospheric CO2 on the response of g to D
To facilitate comparisons with a number of published studies, the effects of high ca on g at a reference environmental state (i.e. the sensitivity of g to D in the neighbourhood of D = 1 kPa) are firstly considered. Beginning with , differentiating with respect to D, setting D = 1 kPa, and re-arranging the terms, the sensitivity of g to D at D = 1 kPa can be expressed as
This outcome suggests that within the optimization framework, the slope of dg/dD versus g is constant (= 1/2) and not impacted by ca; however, the intercept α can vary appreciably with ca. Recall that , and an increase in ca (and thus ci) will result in a decrease in α even if the basic photosynthesis model parameters do not change with ca (e.g. there is no reduction in Vc,max because of down-regulation). In many published studies, data available for comparison with these optimization results are presented as linear relationships between g and D with no information on the p − ci relationship (Heath 1998; Medlyn et al. 2001; Wullschleger et al. 2002; Herrick et al. 2004). However, it may be possible to synthesize the results of such studies by evaluating dg/dD at D = 1 kPa as a ratio of the responses obtained under high and current ca.
Beginning with and noting that g = α[−1 + (ca/(aλ))1/2], the ratio of dg/dD (at D = 1 kPa) under high and current ca is given by:
where δca is the atmospheric CO2 increment, and λe and λa are the equivalent water costs for high and current ca, respectively. Figure 7 shows that (dependent variable) as a function of (independent variable) from a number of studies (Heath 1998; Medlyn et al. 2001; Wullschleger et al. 2002; Herrick et al. 2004). These studies include both short-term CO2 exposures and long-term experiments (e.g. Free Air CO2 Enrichment facilities), where the CO2 enrichment, (δca + ca)/ca, ranged from 1.5 to 2.0. Based on the optimization theory, the departure from unity in this data is captured by the terms and . In the case of λe = λa, and for (δca + ca)/ca > 1, the predicted y as a function of reasonable values of λ is close to unity though for large enough λ, y becomes negative. Accepting for the moment the linearity of the response of g to D, as reported by the authors, this analysis suggests that increased CO2 should affect λe such that λe/λa > 1 + δca/ca. Next, we analyse this ‘excess cost’ using data from two short-term exposure experiments (Bunce 1998; Heath 1998).
The conductance at high ca relative to current ca may be expressed as:
Hence, two effects must be simultaneously considered when assessing the effects of high ca on the relationship between g and D: (1) the effect of ca on αe/αa, which reflects only the degree to which the parameter of the linear p − ci curve shifts under high ca (i.e. ), and (2) the effect of the variation in D emerging from the theory outside the p − ci physiology. Mathematically, if D → +∞, R′(∞) → αe/αa and becomes independent of D; stated differently, at high D, the p − ci physiology alone does not introduce a dependence on D– it simply modifies it by a fraction <1. To facilitate a separate analysis of the effects of high ca beyond the predictable multiplier factor emerging from the p − ci response, we define R(D) = R′(D)/R(∞). The problem considered next is how R(D) changes with increasing D when |δca| > 0. To address this problem, three cases are considered:
Case (1): ∂R(D)/∂D > 0
When |δca| > 0 and all other parameters, including ∂λ/∂ca = 0, are held constant, the optimization theory predicts that R(D) must increase with increasing D. In fact, R(D) must increase with increasing D as long as . The predicted increase for this case is consistent with data on three species (Heath 1998; Fig. 8).
Case (2): ∂R(D)/∂D = 0
If , then R(D) is not affected by increasing D. We have not encountered a data set where this case emerges.
Case (3): ∂R(D)/∂D < 0
This case is possible only when . Bunce (1998) presented experimental evidence suggesting that R(D) can decrease with increasing D opposite to the prediction in case (1). Can λe/λa increase sufficiently under ca to result in the emergence of responses of the type of case (3)?
The information provided in Bunce (1998) does not permit us to address the question quantitatively. Thus we ask: Can such increases in λe/λa be observed under high ca? Leaf-level gas exchange data collected on Pinus taeda at the Duke Forest Free Air CO2 Enrichment (FACE) facility were used to answer this question (data from trees experiencing current ca are shown in Fig. 5). The value of for current and for elevated CO2 plots (ca + 200 µmol mol−1) was first computed by regressing 1 − ci/ca versus D1/2. No down-regulation in the p − ci curve was documented for the period used in this analysis (Ellsworth 2000; Rogers & Ellsworth 2002). For D (in kPa), these regression results suggest that for current and for elevated ca. The scatter is large (see Fig. 5), as expected when measurements are made on different fascicles in different seasons, and soil moisture conditions vary greatly among measurement campaigns. Nevertheless, the slopes of the regression were different between the ca treatments (P < 0.05). Based on this rough assessment, λe = λa(0.230/0.096)2(580/380) = 8.72, satisfying the inequality , required for the emergence of responses the like of case (3). Based on the FACE parameters, R was then modelled with Eqn 22 for a range of D and compared to data in Bunce (1998) – the modelled R agrees well with the data (Fig. 9).
The effects of elevated CO2 on stomatal conductance response to D were evaluated based on a linear p − ci curve. However, it is well known that non-linearities in the p − ci curve become more pronounced under elevated atmospheric CO2. Thus, the results from the analysis above must viewed with some caution. As we noted earlier, we derived the optimal stomatal conductance for the non-linear p − ci curve and intend to use it in future analyses to further evaluate these responses.
Analytical results from the original optimization theory were first derived by Cowan (1977) and Cowan & Farquhar (1977) assuming the daily water cost per unit carbon, (∂fe/∂g)/(∂fc/∂g) is strictly a constant. Hari et al. (1986) and Berninger & Hari (1993) retained the conceptual framework of the ‘optimality hypothesis’ but assumed a linear response of p to ci. We showed that the expected stomatal control (1) can be re-formulated as a univariate maximization problem not needing Lagrange multipliers or integration time scales, (2) is consistent with the onset of an apparent ‘feed-forward’ mechanism in g as discussed in Monteith (1995), (3) agrees with a synthesis survey suggesting that g scales as 1 − m log(D) where m ? [0.5,0.6] (Oren et al. 1999), and (4) agrees with experiments reporting a non-linear variation in ci/ca with D. We have also shown that physical constraints on the degree of stomatal opening (i.e. 0 < u ≤ 1, g ≥ 0) provide logical limits to λ that can be independently derived from the p − ci curve, maximum theoretical conductance, D and ca.
How g responds to D (or in some models RH) is of central importance given that under future climate scenarios, warming is expected not to affect air relative humidity but to increase D exponentially (Kumagai et al. 2004). Using the optimization theory, we analysed the conflicting experimental results on the sensitivity of g to D under current and high CO2 reported in Bunce (1998) and Heath (1998), among others. The approach provides a diagnostic tool and coherent predictions of changes in gas exchange (at least in the response of g to D and ca).
In the optimization framework proposed here, the time scale at which the optimization is operating is commensurate with the time scales of opening and closure of stomatal aperture u, which is too short to be interpreted as being driven by whole-plant resource optimization when fixing a resource constraint (as is often done in the economics of gas exchange). It is conceivable that ‘whole-plant’ scale carbon gain may actually be achieved if stomatal aperture controls have evolved to be ‘efficient’ at the finest possible time scales (i.e. scales at which ). This statement follows from a variant of Pontryagin's maximum principle, which informally implies that beginning from known initial conditions (say a certain amount of carbon in the whole plant system), global optimality at the plant scale in terms of maximizing its carbon gain (which is a linear sum acquired from all the leaves) is guaranteed if at each time step, the ‘local’ maximum is always selected at the stomatal level for the set of environmental conditions. In this context, the framework proposed here assumes that this local maximum amounts to maximizing the leaf photosynthesis while minimizing water loss rate.
It should be emphasized, however, that the apparent agreements with data cannot be viewed as an endorsement of the validity of the optimality hypothesis on stomatal behaviour. Nevertheless, such optimization formulations have joined semi-empirical models, such as the Leuning (1995) and the Collatz et al. (1991), to facilitate coupling leaf-level gas exchange to canopy scale and predicting photosynthesis, transpiration and tree growth under current and future climatic conditions (Mäkeläet al. 2006; Schymanski et al. 2007, 2008; Buckley 2008).
This study was supported by the United States Department of Energy through the Office of Biological and Environmental Research Terrestrial Carbon Processes program and National Institute for Climate Change Research (DE-FG02-00ER53015, DE-FG02-95ER62083 and DE-FC02-06ER64156), by the National Science Foundation (NSF-EAR 0628342, NSF-EAR 0635787), and by the Bi-National Agricultural Research Development fund (IS-3861-06). We thank Kim Novick and Stefano Manzoni for helpful comments on an earlier version of this work.