Despite the critical role that stomata play in terrestrial carbon and water flux (Hetherington & Woodward, 2003), there remains no consensus theoretical model that can explain and predict variations in stomatal conductance to water vapour (gsw) in relation to short- and long-term variations in environmental forcing (Berry et al., 2010; Damour et al., 2010; Buckley & Mott, 2013). One appealing prospect to fill this modelling gap is the hypothesis that plants regulate gsw optimally, that is, they vary gsw so that carbon gain is maximized for a given water loss (Cowan & Farquhar, 1977). Challenges arise when using this hypothesis to predict stomatal behaviour, particularly in relation to elevated atmospheric CO2 concentration (ca). One challenge is that the optimal short-term response of gsw to ca is widely perceived to differ from the observed response (Katul et al., 2010; Medlyn et al., 2011, 2013). Another is the apparent lack of a theoretical framework to extend the theory to long-term changes in ca, such as those associated with climate change. Recent work by Katul et al. (2009, 2010) and Medlyn et al. (2011, 2013) has revitalized the optimization approach by beginning to address these challenges. Our objective in this Letter is to clarify several issues raised by that work, and to offer an alternative perspective based on integration of leaf and whole plant function.
Overview of stomatal optimisation
The aim of ‘stomatal optimization theory’ is to find the pattern of gsw that optimizes the tradeoff between carbon gain (net CO2 assimilation rate, A) and water loss (transpiration rate, E). There are several ways to formalize this question. One approach that may seem obvious at first glance – maximizing the instantaneous ratio of A/E – actually leads to a trivial solution, because under most conditions A/E is greatest when gsw is zero. Another approach, pioneered by Cowan & Farquhar (1977), uses the method of Lagrange multipliers to maximize the integral of A over some period, ∫A, subject to the constraint that total water use, ∫E, is the same among all candidate patterns of gsw. That is,
where μ is an arbitrary constant. The general solution is simply:
Thus, the ratio of the marginal sensitivities of A and E to gsw should be invariant over the space/time interval in question. This result is only valid if the relationship between A and E created by varying gsw is convex (i.e. (∂A/∂gsw)/(∂E/∂gsw) decreases as gs increases).
What happens at elevated CO2?
At right in Eqn 1, the expression is simplified to ∂A/∂E and written as a function of time (t) and space (s) to remind us that ∂A/∂E, the marginal carbon product of water, is a biological variable, whereas μ is an undetermined constant with no a priori biological meaning. Eqn 2 does not tell us how to estimate the numerical value of μ from ca or other biophysical or environmental data. Its meaning only crystallizes when we take account of processes on longer timescales than that on which the dynamic stomatal response to ca operates – that is, scales that encompass physiological and developmental acclimation and evolutionary adaptation, which combine to produce changes in μ. The question of how to compute or predict changes in μ therefore requires that we refocus our attention on processes that occur at higher organizational scales (Cowan, 2002).
We can gain some insight by computing ∂A/∂E from the equations of gas exchange, and asking how the parameters in the resulting expression may be affected by acclimatory responses on longer timescales. It is easily shown (Buckley et al., 2002) that, provided leaf temperature is invariant with gsw,
where gtc and gtw are total conductances to CO2 and H2O, respectively, cc is chloroplastic CO2 concentration, Δw is the leaf to air water vapour mole fraction gradient and k is the slope of the photosynthetic CO2 demand curve (the slope ∂A/∂cc of the A vs cc relationship obtained by varying ca while keeping photosynthetic capacity, irradiance, gtc and temperature constant) (see Supporting Information Notes S1 for details). Eqn 2 cannot be solved for gsw without adopting simplifications, such as linearizing the demand curve (assuming k is constant) or ignoring boundary layer and mesophyll resistances (Table 1). Numerical solution predicts a positive response of gsw to ca at low ca and a negative response at higher ca (e.g. Fig. 1). Linearized solutions predict a positive response at all ca (e.g. Lloyd & Farquhar, 1994). By contrast, most experimental observations show that the short-term dynamic response of gsw to ca is negative at all ca (e.g. Morison, 1998).
Table 1. Parameters used to compute optimal responses of stomatal conductance to atmospheric CO2
Carboxylation capacity at 25°C
μmol m−2 s−1
Ratio of electron transport capacity at 25°C to Vm25
Leaf to air water vapour mole fraction gradient
μmol m−2 s−1
Marginal carbon product of water
Maximum quantum yield of electrons from incident PPFD
Curvature parameter for response of potential electron transport rate (J) to PPFD and Jm25
Curvature parameter for response of net assimilation rate to carboxylation- and regeneration-limited rates
Boundary layer conductance to H2O
mol m−2 s−1
Mesophyll conductance to CO2
mol m−2 s−1
This apparent discrepancy between observed and optimal short-term dynamic responses of gsw to ca has spawned two recent theoretical developments in an attempt to reconcile the theory with the data. One of these developments, by Medlyn et al. (2011), focused on the short-term dynamical response itself. Those authors noted that, because Eqn 2 generally predicts a positive response when photosynthesis is limited by RuBP carboxylation but a negative response when regeneration is limiting (Fig. 1a), it would appear that stomata behave as if regeneration were always limiting. Although such behaviour diverges from the optimal solution under carboxylation-limited conditions, Medlyn et al. (2011) suggested that this may reflect a physiological constraint on stomatal function (they noted evidence that stomatal guard cells lack the machinery needed to distinguish these limitations), and that the discrepancy may not be particularly important in practice, because photosynthesis is more often limited by regeneration.
Katul et al. (2009, 2010) proposed an alternative resolution. They redefined the optimization problem by positing that stomata maximize the instantaneous difference between A and the ‘water loss in units of carbon,’ which they assumed was proportional to E by a parameter, ξ (Eqn 9 in Katul et al., 2009 or Eqn 5 in Katul et al., 2010, adapted to our notation):
This leads to the solution (modified from Eqn 9 in Katul et al., 2009; see Notes S1 for details):
Katul et al. (2009, 2010) then assumed that ξ is far less sensitive than E to gsw, implying that
and reducing Eqn. 5 to
If ξ is constant, this solution is identical to Eqn 1 However, Katul et al. (2010) proposed that ξ is not constant, but instead is proportional to ca, that is,
where ζ is constant. This assumption transforms the solution to:
When Eqn 8 is applied to Eqn 2, it eliminates the positive response of gsw to ca, thus resolving the discrepancy. It also appears to tell us how ca affects μ. Eqn 8 has already been adopted in numerous modeling studies (Launiainen et al., 2011; Manzoni et al., 2011a,b; Volpe et al., 2011; Way et al., 2011; Palmroth et al., 2013).
This resolution has several fundamental flaws, however. First, it is unclear why it should be ecologically advantageous to maximize A – ζcaE rather than ∫A – μ∫E. In fact it is not: the latter solution leads to greater carbon gain under varying ca when controlling for water loss (Fig. 2). Second, this resolution is premised on two assumptions (Eqns 6 and 8) about the mathematical and biological properties of the parameter ξ. Yet because ξ is never rigorously defined in biophysical terms, Eqns 6 and 8 cannot be justified on biophysical grounds. Katul et al. (2009) thus attempted to justify these assertions empirically: they showed that Eqn 8 fitted the data better than assuming that ξ is invariant with ca, and they justified Eqn 6 by attempting to show empirically that ∂A/∂E is far less sensitive than E to gsw. However, the latter empirical comparisons were flawed and circular. The flaw was that Katul et al. (2009) estimated ∂A/∂E from the slope of a relationship between A and E created by changing evaporative demand (Δw) rather than stomatal conductance (gsw). The slope thus computed is actually the ratio of total derivatives of A and E with respect to Δw (dA/dΔw)/(dE/dΔw), which is negative (cf. fig. 6 in Katul et al., 2009), whereas the correct quantity is the ratio of partial derivatives with respect to gsw ((∂A/∂gsw)/(∂E/∂gsw) = ∂A/∂E), which is positive. Thus, Eqn 6 has not in fact been validated empirically. The circularity arose from the attempt to justify Eqn 6 by estimating the sensitivity of ∂A/∂E to gsw. In order to use empirical estimates of (∂ln(∂A/∂E)/∂gsw)/(∂lnE/∂gsw) to validate Eqn 6, Katul et al. (2009) had to assume that ξ = ∂A/∂E (Eqn 7) – that is, they had to adopt Eqn 7 in order to derive Eqn 7, which is circular.
The a priori identification of ξ as ∂A/∂E may have been motivated by the perception that ∂A/∂E and μ are the same quantity in the original Cowan–Farquhar problem. This is not correct, despite the impression given by Eqn 2. Eqn 2 is not a definition of the Lagrange multiplier μ, nor of ∂A/∂E; rather, it is the solution to maximizing ∫A – μ∫E. It says that stomatal conductance should vary so that the marginal carbon product of water (∂A/∂E) remains equal to some undefined constant μ. The link between the multiplier μ and the derivative ∂A/∂E only arises after one has solved the constrained optimization problem (Eqn 1). An alternative perspective is that Katul et al. (2009) implicitly adopted ξ = ∂A/∂E in the problem statement itself. This avoids the circularity but entails a fundamentally different goal function, in which E is multiplied not by an undetermined constant, but by ∂A/∂E itself. This transforms Eqn 5 and its solution (Eqn 6) to
However, the latter solution resolves to
This merely states that ∂A/∂E should be invariant as gsw changes, which is identical to the original Cowan–Farquhar solution. Unlike the latter solution, however, the instantaneous approach does not specify a timescale at which ∂A/∂E should be invariant. Furthermore, the goal function in Eqn 10 is of dubious merit. If boundary layer and mesophyll resistance are small, this goal function is equivalent to
(see Notes S1). It is unclear what ecological advantage a leaf would gain by maximising this quantity.
We argue that, although the instantaneous approach per se is not inherently flawed, the multiplier for E in the goal function (ξ) cannot simply be ∂A/∂E. What, then, does ξ represent, if not ∂A/∂E? If we are to accept A - ξE as an ecologically meaningful instantaneous goal function, then the product ξE must represent a carbon cost of water loss (indeed, Katul et al. (2010) defined ξE as the ‘water loss in units of carbon’). Thus, ξ is the carbon cost of water loss per unit of transpiration, or the unit carbon cost of water loss. This is a very different quantity from ∂A/∂E. Thus, ∂A/∂E represents how much carbon the plant gains for every additional unit of water that it transpires, whereas ξ represents how much carbon the plant loses for every unit of water it transpires. The plant loses carbon by transpiring because, to replace evaporative losses, it must invest carbon in roots and xylem to capture and transport water (Givnish, 1986). These two quantities are both derivatives, but in very different domains: ∂A/∂E is the derivative of leaf carbon gain with respect to transpiration rate, whereas ξ is the inverse of the derivative of water supply rate with respect to plant carbon investment in water supply.
A different perspective
To understand how and why ξ or μ should change with ca, we must consider the question of timescale. Part of the motivation for assuming ξ ∝ ca (Eqn 9) was the need to reconcile optimal and observed short-term dynamic responses of gsw to ca (Medlyn et al., 2013; Vico et al., 2013). However, if ξ involves whole-plant carbon costs of water loss, as we argue earlier, then it is clear that the timescale for variations in ξ is much longer than for the short-term dynamic response: namely, it is the timescale at which the carbon costs of water loss change due to changes in soil moisture and root and xylem growth. That timescale ranges from days to decades or even longer. Notably, this is the same as the timescale for changes in total water supply, and thus for the Lagrange multiplier μ in the constrained optimization problem. The question, then, is how to predict effects of ca on ξ and μ over these longer timescales. Katul et al. (2009) argued that μ is nonphysical and therefore cannot be independently inferred. However, μ has long been understood to involve water supply (Cowan & Farquhar, 1977; Cowan, 1982; Hari et al., 1986; Mäkelä et al., 1996; Schymanski et al., 2008), and Givnish (1986) showed that μ should also be affected by functional balance between root and shoot function. We argue that μ is in fact highly constrained by biology and physics, and that, with certain assumptions, its numerical value and response to long-term changes in ca can be directly computed.
For example, if stomata open too widely, then water loss will reduce leaf water potential below either the turgor loss point or the threshold causing runaway xylem cavitation. This places an upper limit on transpiration rate, Emax, which can be calculated from biophysical properties, including soil water potential, cavitation threshold water potential, leaf osmotic pressure and plant hydraulic conductance. One possible strategy would be to choose μ so that E reaches but does not exceed Emax during the course of a day (Buckley, 2005). This could be modelled by setting μ to the largest value that ∂A/∂E reaches (calculated at Emax) each day (as illustrated in Fig. 3). Like Emax, that value of ∂A/∂E can be computed on a biophysical basis (e.g. Fig. 3b), as shown by Konrad et al. (2008). Although other factors might require this strategy to be modified – for example, effects of progressive soil drought (Mäkelä et al., 1996), competition for water (Cowan, 1982), trunk water storage (Scholz et al., 2011; Pfautsch & Adams, 2012) and osmotic adjustment (Bartlett et al., 2012; Sanders & Arndt, 2012) – our point is that for any strategy, the appropriate value of ∂A/∂E is largely determined by measurable biophysical properties that may be affected by CO2 enrichment. These include leaf and xylem properties and aspects of plant structure that affect water supply, but they also include carbon investments in roots and leaves, which influence photosynthetic function and thus the return from investing carbon to deliver water to leaves. Enrichment may also affect stomatal size and density, which influence how individual guard cell responses translate into changes in gsw (Franks & Beerling, 2009; Lammertsma et al., 2011; Doheny-Adams et al., 2012). It is unlikely that a simple proportionality will emerge between ca and μ when effects of enrichment on all these properties are accounted for.
To demonstrate how one might begin to assess these integrated effects of ca on μ, we used two approaches to estimate changes in μ following CO2 enrichment. First, we applied to Eqn 3 the common finding from FACE experiments that the ratio of ci/ca is unaffected by enrichment (Ainsworth & Long, 2005), which suggests that enrichment increases μ by c. 17–41% (Fig. 4) (the range reflects differences between regeneration- and carboxylation-limited conditions, and uncertainty about enhancement of dark respiration; details in Supporting Information Notes S1). Second, we present simulations of structural and photosynthetic acclimation of mature trees following a step increase in ca from 370 to 570 ppm (previously published by Buckley, 2008), based on a tree growth model in which carbon allocation is optimized with respect to whole-plant carbon gain (DESPOT, Buckley & Roberts, 2006). The model operates on an annual time step and assumes the value of ψleaf prevailing during active photosynthesis is constant from year-to-year. Because DESPOT itself is based on optimization, this model is especially suited for the task of predicting long-term structural acclimation in response to CO2 enrichment in relation to the multipliers in stomatal optimization. These simulations predicted an immediate 19% enhancement in μ (relative to a control simulation at constant ca), followed by fluctuations between 36% and 11% over the ensuing years as a result of continuing structural adjustments (Buckley, 2008) (Fig. 4). By contrast, the hypothesis that ∂A/∂E is simply proportional to ca (Eqn 8) predicts a much larger increase of 54% in μ following CO2 enrichment (Fig. 4). These results complement the widespread finding that enrichment causes rapid direct responses at the leaf scale, but that these effects are often damped by more gradual changes at the plant scale and above (Saxe et al., 1998; Ainsworth & Rogers, 2007; Kirschbaum, 2011; Wang et al., 2012).
Embracing complexity to move optimisation forward
Medlyn et al. (2013) commented that the effects of ca on μ are unlikely to be understood without considering optimization on a longer timescale. We strongly agree, and we suggest that the appropriate timescale is that at which whole plant photosynthetic resource balance, and therefore carbon allocation, are modulated. The brief analysis earlier shows that it is possible to consider effects of CO2 enrichment on the mysterious Lagrange multiplier, μ, at the heart of stomatal optimization, on a biophysical basis – we simply need to expand our perspective from the leaf to the whole plant. This brings additional complexity and uncertainty, but the benefits of rigorously extending stomatal optimization theory to future climates certainly outweigh the costs.
The authors thank Graham Farquhar for helpful discussions, and three anonymous reviewers for useful comments on an earlier draft of this paper. T.N.B. was supported by the US National Science Foundation (Award no. 1146514) and by the Grains Research and Development Corporation (GRDC).