leaf stomatal responses to vapour pressure deficit under current … · 2-enriched atmosphere...

Leaf stomatal responses to vapour pressure deficit undercurrent and CO2-enriched atmosphere explained by theeconomics of gas exchange

GABRIEL G. KATUL1,2, SARI PALMROTH1 & RAM OREN1

1Nicholas School of the Environment and 2Department of Civil and Environmental Engineering, Duke University, Durham,North Carolina, USA

ABSTRACT

Using the economics of gas exchange, early studies derivedan expression of stomatal conductance (g) assuming thatwater cost per unit carbon is constant as the daily loss ofwater in transpiration (fe) is minimized for a given gain inphotosynthesis (fc). Other studies reached identical results,yet assumed different forms for the underlying functionsand defined the daily cost parameter as carbon cost per unitwater. We demonstrated that the solution can be recoveredwhen optimization is formulated at time scales commensu-rate with the response time of g to environmental stimuli.The optimization theory produced three emergent gasexchange responses that are consistent with observedbehaviour: (1) the sensitivity of g to vapour pressure deficit(D) is similar to that obtained from a previous synthesis ofmore than 40 species showing g to scale as 1 - m log(D),where m ∈ [0.5,0.6], (2) the theory is consistent with theonset of an apparent ‘feed-forward’ mechanism in g, and (3)the emergent non-linear relationship between the ratio ofintercellular to atmospheric [CO2] (ci/ca) and D agrees withthe results available on this response. We extended thetheory to diagnosing experimental results on the sensitivityof g to D under varying ca.

Key-words: gas exchange; optimal stomatal control; photo-synthesis; stomatal conductance; transpiration.

INTRODUCTION

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 alsoinvestigated water uptake by a willow tree in 1648, in effectperforming one of the earliest recorded experiments onstomatal conductance (g) to gas transfer. Centuries later,both of van Helmont’s activities converged in a modern-daystory:Atmospheric CO2 is rising largely because of the com-bustion of fossil fuel, and the ability of terrestrial plants touptake CO2 is currently a leading mitigation strategy tooffset this rise. Because the role of stomata in regulating the

exchange of CO2 for water is central to many plant andecosystem processes, services and products, variations in gand in their responses to environmental variables havebeen subjected to intense research for decades. And yet,despite numerous experiments and several modellingapproaches, the precise mechanisms responsible for sto-matal responses to certain environmental stimuli remainvague (see e.g. review by Buckley 2005).

Several empirical and semi-empirical models describingstomatal responses to environmental stimuli exist (e.g.Jarvis 1976; Collatz et al. 1991; Leuning 1995).These models,advanced primarily after the publication of the seminalwork by Jarvis (1976), are based on an electrical circuitanalogy – stomata are viewed as a resistor (or a conductor)with a maximum species-specific value of g attained whenstomatal pores are fully open.The maximum g is reduced bynon-linear functions that account for the effects of externalenvironmental factors via increases in the concentration ofCO2 in the leaf’s air space, or the capacity of the soil–planthydraulics to supply water to the leaf relative to the poten-tial rate of vapour loss rate from fully open stomata. Thesefunctions reflect decreasing light levels, increasing CO2

concentration and vapour pressure deficit, departure fromoptimum leaf temperature, and also decreasing leaf waterpotential representing the hydration state of the stomatalsystem. Such a parsimonious representation of stomatalconductance, combined with the increased availability ofportable equipment for measuring gas exchange, greatlycontributed to quantifying differential species sensitivitiesto environmental stimuli and stresses.

The wealth of data on g led to certain generalities onstomatal responses to the environment. For example, Mott& Parkhurst (1991) demonstrated that stomatal closure is aresponse to leaf transpiration rate rather than to varyingvapour pressure deficit (D). Nevertheless, a synthesis ofstomatal responses to varying D, obtained from studies onover 40 species from grasses to deciduous and evergreentrees, revealed a general functional form that can bedescribed as g = gref(1 - m log(D)), where m ª 0.5-0.6 andgref is g at D = 1 kPa (Oren et al. 1999). Further support forthis value of m was provided in Mackay et al. (2003). It wasalso demonstrated that the value of m ª 0.6 is consistentwith a hydraulic model in which plants control transpiration

Correspondence: G. Katul. Fax: +1 919 684 8741; e-mail: [email protected]

Plant, Cell and Environment (2009) 32, 968–979 doi: 10.1111/j.1365-3040.2009.01977.x

© 2009 Blackwell Publishing Ltd968

rate to protect the transport system from excessive loss ofhydraulic function (Sperry et al. 1998, 2002; Oren et al. 1999;Lai et al. 2002). Yet stomatal control cannot be only toregulate the rate of water loss. Indeed, it has long beensuggested that, at the leaf scale, natural selection may haveoperated to provide increasingly efficient means of control-ling the trade offs between carbon gain and the accompa-nying water vapour loss (e.g. Cowan 1977, 1982, 2002; Ball,Cowan & Farquhar 1988). If so, can such an optimizationprinciple be used to constrain certain parameters insemi-empirical models, perhaps even replacing them byfunctional responses that naturally emerge from such opti-mization? Emergent functions are more general, unlikeimposed functions that empirically describe data, limitingtheir application to the conditions represented by theexperiment.

First presented by Cowan (1977) and Cowan & Farquhar(1977), and reformulated by Hari et al. (1986) andBerninger & Hari (1993), the cost (= daily water loss intranspiration) to benefit (= daily carbon gain in photosyn-thesis) analysis was framed as an ‘economic’ optimization.While the assumptions on the form of the underlying func-tions differ between Cowan & Farquhar (1977) and Hariet al. (1986), their optimal solutions are, in fact, identical.Moreover, while both studies implicitly assumed an integra-tion time scale, their solution appeared independent of thetime scale of flux integration. The stomatal control overgas exchange is described through a concept of invariant‘carbon cost of water’ or ‘water cost of carbon’, withouta priori specification of stomatal response to D oratmospheric CO2. The predicted expressions of stomatalresponses to D or atmospheric CO2 are ‘emergent proper-ties’ of the optimization theory. We compare these emer-gent responses with data from studies from a wide range ofconditions. We demonstrate that the optimization theorypermits predictions of stomatal response to environmentalstimuli, especially with respect to D in both current andCO2-enriched atmosphere. The analysis limits ‘optimality’to bulk leaf gas-exchange; it may not be used to explain suchinter-related questions as how guard cells operate toachieve optimality, or why leaves are oriented in a specificway within ecosystems.

THEORY

The basic equations for the leaf-level CO2 and water vapourfluxes across stomata are given by:

f g c cc a i= −( ) (1)

f ag e ee i a= −( ), (2)

where fc is the CO2 flux, fe is the water vapour flux, g is thestomatal conductance, ca is ambient and ci intercellular CO2

concentration, a = 1.6 is the relative diffusivity of water withrespect to carbon, and ei is the intercellular and ea theambient water vapour concentration. Photosynthesis (p) isrelated to ci via the Farquhar model (Farquhar, Caemmerer& Berry 1980a):

pc

c= −

+α

α12

i

i

Γ, (3)

where a1 and a2 depend on whether the photosynthetic rateis light – or Rubisco-limited. For analytical tractability,assume that G/ci << 1 and the expression a1ci/(a2 + ci) ª a1ci/(a2 + sca), where s relates ci to ca:

p c≈ α i (4)

where a = g1Vc,max for temperature limited photosynthesisand a = g2 PAR for light-limited photosynthesis. Here, Vc,max

stands for maximum carboxylation capacity, g1 and g2 arephysiological parameters, and PAR is photosyntheticallyactive radiation. Equation 4 demonstrates the correspon-dence between the assumed p - ci curve in Hari et al. (1986)and the parameters in the Farquhar model, although theprecise value of a is irrelevant to the following optimizationdiscussion. Assuming steady-state conditions,

f p rc − + = 0 (5)

where r is the leaf respiration rate. Combining Eqns 1, 2, 4and 5 results in the following formulations:

cgc rg

ia= ++ α

(6)

pgc rg

= +( )+

αα

a . (7)

The two basic equations (Eqns 6 and 7) include threeunknown state variables (ci, g and p), generating a problemnot closed mathematically. Standard approaches to ‘close’this problem assume an empirical relationship between g, pand some environmental stimuli such as air relative humid-ity (RH) or D (Baldocchi & Meyers 1998; Lai et al. 2000).Two well-known formulations that fit a wide range of fielddata are given by the so-called ‘Ball-Berry model’ (Ball,Woodrow & Berry 1987; Collatz et al. 1991):

gmc

pRH b11

1= +a

, (8)

and the ‘Leuning model’ (Leuning 1995):

gmc

pDD

b22

1

11= +⎛⎝⎜

⎞⎠⎟ +

−

a o

, (9)

where b1 sets a minimum g, Do is the sensitivity of g tovapour pressure deficit, and m1 and m2 are empirical param-eters 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 workby Hari et al. (1986) and Berninger & Hari (1993). In theirformulation, g is expressed as g = gou where go is themaximum conductance and u is the degree of stomatal

Economics of gas exchange and optimal stomatal control 969

© 2009 Blackwell Publishing Ltd, Plant, Cell and Environment, 32, 968–979

opening (0 < u � 1). The carbon cost of a unit of transpiredwater, l, is formulated as the inverse of (∂fe/∂g)/(∂fc/∂g).Like in Cowan & Farquhar (1977), l is assumed to be con-stant at time scale of 1 d, and the fluxes of CO2 and watervapour are integrated over the same period.

Interestingly, in the following discussion, we show that asolution to an analogous optimization problem can bedeveloped without time-integration (given the steady-stateassumption in Eqn 5) and Lagrange multipliers. Resultsfrom such a solution are identical to those from Hari et al.(1986) and Cowan & Farquhar (1977), and actually estab-lish some constraints on how constant the cost parameterneeds to be for the solution to be accurate.

However, before presenting this optimum solution, wenote that when aca >> r, Eqns 2 and 7 can be combinedto arrive at an explicit relationship between fc and fe

given as fc f

f e ec

a e

e i a

≈+ −( )

αα

. This expression has a nega-

tive convexity for any positive fe and ei - ea because∂∂

2

2

2

3 0f

fc e e

f e ec

e

a i a

e i a

= − ( ) −( )+ −( )( )

<α

α.

Hence, despite the linearization in the p - ci curveadopted in Eqn 4, the fc (dependent) versus the fe (indepen-dent) expression proposed here maintains a negative con-vexity and thus our formulation admits an optimal solutionas discussed in Cowan & Farquhar (1977). To find thisoptimum for the linearized p - ci curve, the maximization ofthe carbon gain function f(u) with respect to stomatal aper-ture control (u) can be expressed as:

max

;

ue

e

ee

f u p ff u

upu

fu

pu

fu

fu

( ) = −( ) ⇒( )

= −

= − − =

λ λ

λ λ

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

0

if e e1 1λ λ( )( ) << ( )( ) ⇒∂ ∂ ∂ ∂u f f u

(10)

∂∂u

ug c rug

aug e eα

αλo a

oo i a

+( )+

− −( )⎛⎝⎜

⎞⎠⎟

= 0.

In comparison with Hari et al. (1986), it is clear thatthe optimization problem in Eqn 10 simplifies to a uni-

variate maximization problem provided∂∂

∂∂

λ λfu

fu

e e≈ .

This simplification was the basis of the definitionof the Cowan & Farquhar (1977) marginal water costper unit carbon, expressed as (∂fe/∂g)/(∂fc/∂g) [or∂∂

∂∂

∂ ∂ ∂ ∂pu

fu

p u f u− = ⇒ = ( ) ( )λ λee0 ]. Maximization in

Eqn 10 is achieved when

ua e e g a e e g c r

a e e g=

− −( ) + −( ) −( )−( )

( )α λ αλλ

i a o i a o a

i a o

2 1 2

2. (11)

The condition ∂f(u)/∂u = 0 in Eqn 10 alone does not rule outthat the result in Eqn 11 is a local minimum rather than amaximum for f(u).To ensure that f(u) is maximum for the ugiven by Eqn 11, f(u) must be concave (or negatively

convex). Upon twice differentiating f(u) with respect to u,

we obtain∂

∂

2

2

2

3

2f uu

g c r

g u

( )= − −( )

+( )α α

αo a

o

, which is monotonically

negative (u > 0) provided aca > r. Hence, this negative con-vexity in f(u) guarantees that u in Eqn 11 is a maximum andnot a minimum. For r/a << ca

ug

ca e e

≈ − +−( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

αλo

a

i a

11 2

. (12)

Hence,

g g uc

a e e= ≈ − +

−( )⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟o

a

i a

αλ

11 2

. (13)

This expression states that g decreases with increasingei - ea, and is sensitive to the slope of the p -ci curve.Because u is bounded between zero and unity, theoreticalbounds on the cost parameter l can be readily establishedand are given as:

λ

λ λ

α

max

minmax

, . .

, . . .

=−( )

=( )

=+( )

=( )

ca e e

u

gu

a

s a

o

i e

i e

0

112

(14)

The variations in l (as lmin/lmax) are entirely dictated by go/a– the maximum conductance and the basic physiologicalparameter of the linear p - ci curve. If go/a ~ 1, then lmin/lmax ~ 1/4 (or fourfold variation). Using a similar maximiza-tion approach, we also derived the optimum conductanceusing the non-linear p - ci curve (Eqn 3). However, theresulting formulation does not reveal primitive scaling rulesbetween environmental stimuli and stomatal conductanceowing to the larger number of parameters.

Because of its ‘non-physical’ nature, l cannot be inde-pendently inferred and the success of testing the optimi-zation hypothesis at the leaf-level has been variable(Farquhar, Schultze & Küppers 1980b; Hall & Schulze1980; Berninger, Mäkelä & Hari 1996; Guehl & Aussenac1987; Fites & Teskey 1988; Berninger & Hari 1993; Hariet al. 1999; Thomas, Eamus & Bell 1999; Hari, Mäkelä &Pohja 2000; Aalto, Hari & Vesala 2002). Instead of testingthe theory by searching experimental data for a constantl, we focus on searching data for the emergent propertiesof the optimal solution, namely the sensitivity of g to D.This is somewhat more pertinent for two reasons: (1) opti-mality may still persist with some variations in l pro-vided these variations abide by the condition |(1/l)(∂l/∂g)| << |(1/fe)(∂fe/∂g)| or simply |(∂l/l)| << |(∂fe/fe)|, and (2)experimentally, it is difficult to estimate l based on gasexchange data because uncertainties in measuring ∂fe/∂gand ∂fc/∂g may be large under certain combinations ofenvironmental conditions.

Recall that no a priori specification of the stomatalresponse to D is imposed and that the shape of the

970 G. G. Katul et al.


dependency of g on D emerges from the optimization. Hariet al. (2000) presented a convincing field test on the depen-dence of g on D-1/2, suggesting that such a dependency is a‘validation’ of the optimization hypothesis (assuming thatatmospheric vapour pressure deficit, D well approximatesleaf-to-air vapour pressure difference, ei - ea). However, thedependence of g on D-1/2 clearly conflicts with the func-tional 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 toconstrain certain parameters in semi-empirical models of g,we firstly consider the sensitivity of g to D, and then evaluatehow the sensitivity of g to D is impacted by changes in ca.

Sensitivity of g and fe to D

Consistency with observed response of g to D(Oren et al. 1999)Assuming that D is a surrogate for ei - ea, g is expressed inHari et al. (1986) as:

g g u Dca

= = ( )− +⎛⎝⎜

⎞⎠⎟

−o

aαλ

11 2

1 2 . (15)

The emerging conductance sensitivity is reflected in thedependence on D-1/2, which may be expressed via a Taylorseries expansion as:

D DD D

D

− = − ( ) + − ( )( )+ − ( )( )

+

+ −

1 22 3

112

1 22

1 23

1 2

loglog

!log

!

. . . .log(( )( )n

n!,

(16)

where the leading term is of the form 1 - (1/2)log(D).Figure 1 shows the variations of D-1/2 and

11 2

1

1

+ − ( )( )=

∑ log!

Dn

n

n

n

for n1 = 1, 2, 3 and for typical D

varying from a low of 0.5 to an extreme of 6. The seriesconverges rapidly, with n1 = 3 indistinguishable from D-1/2.Moreover, Fig. 1 shows that the Taylor series expansionD-1/2 ª 1 - 1/2 log(D) is accurate for D ∈ [0.5,2] (i.e. towithin 5% relative error), but for D > 2, higher order effects(n1 > 1) become large.

From a broad survey of plant species representing manyfunctional types, a functional form g in response to D wasderived as g = gref(1 - m log(D)), where gref is the so-calledreference conductance determined for similar light and soilmoisture conditions at D = 1 kPa (Oren et al. 1999). Thesynthesis showed that m ª 0.5 - 0.6.Since then,other studiesadded many additional species showing similar response(Mackay et al. 2003).To compare this well-supported empiri-cal finding with the optimization prediction, the empiricalfunction can be expressed as g = gref(1 - m log(D)). The

optimization result for g/gref is g/gref = (-1 + F/D1/2)/

(-1+ F), with Φ = ( )ca

a

λ

1 2

. Upon replacing D-1/2 ª 1 - 1/

2 log(D),g

gD

Dref

= − + − ( )( )− +

= −−

( )1 1 1 21

112 1

ΦΦ

ΦΦ

loglog .

For the case when F >> 1,g

gD

ref

= − ( )112

log , close to the

lower limit of the reported m values in Oren et al. (1999).

Even for large F,Φ

Φ −>

11 the optimization theory should

yield an m > 0.5, also consistent with the empirical findingsand the theory of water transport to leaves in Oren et al.(1999).

To further illustrate the similarity between the emergentbehaviour from the optimization model and the generalvalue of m [from 1 - m log(D)], we used the data of Faguscrenata Blume (Fig. 2; Iio et al. 2004) not included in Orenet al. (1999). The regression analysis on the data in Fig. 2results in m ª 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.

Consistency with the sensitivity of fe to D(Monteith 1995)A joint reduction in g and fe with increasing D can be seenas evidence of a feed-forward mechanism of stomatalresponse (Schulze et al. 1972). Such behaviour, under someconditions, can also be predicted based on optimal stomatalcontrol (Buckley 2005). Here, we demonstrate that theemergent g ~ D-1/2 from the optimization theory is qualita-tively consistent with Monteith’s (1995) view of the sensi-tivity of fe to D and the apparent feed-forward mechanism.

The function f a Dca

Dea= − + ( )⎛

⎝⎜⎞⎠⎟

αλ

1 21 2 suggests that fe is

dominated by two opposing terms when D is increased.

D (kPa)

0 2 4 6

f (D

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5D–1/2

1st

2nd

3rd

4th

Figure 1. Variations of the square root of water vapourpressure deficit (D-1/2) and its Taylor series expansion:

11 2

1

1

+ − ( )( )=

∑ log!

Dn

n

n

n

for n1 = 1,2,3 as a function of D.



These terms are represented by the sum − + ( )Dca

Da

λ

1 21 2

and imply that at low D, fe increases rapidly with increasingD because the first term is small compared to the secondterm. However, at very high D, the first term may dominateand fe begins to decline with increasing D, consistent withempirical findings (Monteith 1995; Pataki et al. 1998). Thisoutcome is identical to a derivation from the g -D responseperformed in Oren et al. (1999). The optimization modelcan 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, theapparent feed-forward mechanism occurs only when∂∂

fD

e ≤ 0 . This Dcirt can be readily computed as

∂∂

fD

aca

D Dca

e acrit

aor= − + ( )⎛⎝⎜

⎞⎠⎟

= =−αλ λ

112

014

1 21 2 ; , (17)

and depends only on ca and l.This critical limit can be assessed based on data from

four-step measurements on three leaves of Abutilon theo-phrasti, two of which display an apparent feed-forwardmechanism and one displaying a plateau [i.e. transpirationalmost independent of D; Bunce (1997)]. In this framework,a plateau indicates that Dcirt is not yet exceeded. A regres-sion model g = s1 + s2D-1/2 was fitted to the four-point con-ductance data, and then fe was computed from the modelledg (along with Dcrit determined from s1 and s2).The expectedtranspiration is similar to the data, as is the D at which theapparent feed-forward is observable (Fig. 3).

Finally, it should be noted that for a given l, the optimi-zation approach predicts that leaf transpiration becomes

negligible when f Dca

Dea= = − + ( )0

1 21 2

λ , which results in

D = 0 (a trivial solution) and Dca

D= =acritλ

4 (a non-trivial

solution). The second solution suggests that transpirationbecomes 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 l inhumid climates, which is a reasonable behaviour.

D (kPa)

0.5 1.0 1.5 2.0 2.5 3.0 3.5

g/g r

ef

0.2

0.4

0.6

0.8

1.0

1.2

1.4

y = 1s + s2D1/2; r

2 = 0.75

y = s1 + s2log(D) ; r

2 = 0.75

Figure 2. Comparison between measured (circles) and modelledg normalized by a reference g (gref), defined at D = 1 kPa, forFagus crenata Blume (Iio et al. 2004). The lines representpredictions based on the optimization theory (solid line) andthe functional relationship in Oren et al. (1999) (dashed line).

g (m

mol

m–2

s–1

)

0

300

600

900

1200

0 10 20 30 40 10 20 30 40 10 20 30 40

f e (

mm

ol m

–2 s

–1)

0

3

6

9

12

D (mPa Pa–1)

Figure 3. Comparison between predicted stomatal conductance (g) and transpiration rate (fe) based on optimization theory, and datafrom Abutilon theophrasti (Bunce 1997) for the onset of a feed-forward mechanism in three leaves (represented by different symbols).



Dependence of ci/ca on D

An important yet unexplored consequence of this optimi-zation model is that ci/ca varies in a predictable manner withD, and is given by

cc

gD

D

ca

ac

i

a

a

a

=+

≈+

( ) −

= − ⎛⎝⎜

⎞⎠⎟

−

1

1

1

11

1

1

1 21 2

1 21 2

α

λ

λ. (18)

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 pre-served by the optimization model (Fig. 4), suggesting thatthe linear p - ci approximation is perhaps appropriate for‘field’ conditions and is consistent with the arguments inLeuning (1995) and Katul et al. (2000). As discussed inKatul et al. (2000), Leuning’s semi-empirical conductancemodel leads to a linear decline of ci/ca with increasing D.In contrast, the optimization theory model predicts that asD decreases ci/ca increases asymptotically such thatci/ca → 1 when D → 0. Leuning (1995) used data on 9 ofthe 16 species from Turner, Schulze & Gollan (1984) tocalibrate the stomatal response to D. Although the datacan probably be represented equally well by a linear anda non-linear approximation, at low D, some degree ofnon-linearity in the dependence of ci/ca on D is apparentin the data of all species (see fig. 9 in Leuning 1995).Indeed, many field and laboratory studies have reported anon-linear decline of ci/ca with increasing D (Farquharet al. 1980b; Lloyd & Farquhar 1994), or present data thatappear to be a better fit with D1/2 than D, as shownin 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 ofci/ca on D can also be delineated within the context of theoptimization framework. By rewriting the gas-exchangeequations as

f fc c

a e ec e

a i

i a

= −( )−( )

,

a plausible condition can be derived by replacing theconstant marginal cost l = (∂fc/∂g)/(∂fe/∂g) with a constantflux-based water use efficiencylL = fc/fe, to yield

ff

ca e e

cc

c

e

a

s a

i

aL=

−( )−⎛

⎝⎜⎞⎠⎟ =1 λ .

A constant lL (i.e. independent of D) can only be achievedif

cc

ac

e ei

a

L

as a= − ⎛

⎝⎜⎞⎠⎟ −( )1

λ.

In other words, ci/ca may decline linearly with increasingD as suggested by Leuning (1995) if lL is a constant.By equating these two ci/ca formulations, it can beshown that lL can be related to l using lL = (lca/a)1/2D1/2.In typical field experiments, large variations in D areneeded to discern the dependence of lL on D (assumingconstant l), perhaps explaining why data cannot conclu-sively reject a linear ci/ca decline with increasing D (e.g.Fig. 5).

Two studies criticized the optimality hypothesis by dem-onstrating that l was not constant but varied with D (Fites& Teskey 1988; Thomas et al. 1999). As noted earlier, varia-tions in l alone do not disprove the optimality hypothesisand predictions of the response of g to D may still bereasonable, provided that |(1/l)(∂l/∂u)| << |(1/fe)(∂fe/∂u)|.Stated differently, as long as the relative variations in l aremuch smaller than the relative variations in fe, the scalingproperties emerging from the optimization theory arerobust. We analysed the data published in Fites & Teskey(1988; Fig. 6) and noted that (1) their g scales as D-1/2 andis consistent with the optimality hypothesis, (2) regressingtheir reported fc upon their fe results in a near-constantlL (in disagreement with the optimality hypothesis), (3)regressing ∂fe/∂g versus ∂fc/∂g directly estimated from thedata (digitized by us), suggests a constant l (consistentwith the optimality hypothesis), and (4) their ci/ca is notlinearly related D, especially at high D (consistent withthe optimality hypothesis but not the outcome in point 2).Notice that points (2) and (3) cannot be simultaneouslysatisfied given that lL = (lca/a)1/2D1/2. In estimating ∂fe/∂gand ∂fc/∂g, we used central differencing to approximatethese gradients from the data published by Fites & Teskey(1988; their Figs 1 and 2). Estimating ∂fe/∂g and ∂fc/∂g fromraw (and digitized) data increases the uncertainty whencomputing such derivatives because the random error isgenerally amplified by the differencing operator. We alsoconfirmed that the residuals from the regression in step(3) are not significantly dependent on D. While similaranalysis could not be repeated on the data in Thomas et al.

g (mmol m–2 s–1)

0 500 1000 1500

c i /

c a

0.2

0.4

0.6

0.8

1.0

Figure 4. The hyperbolic dependence of the ratio ofintercellular to atmospheric CO2 concentration (ci/ca) on stomatalconductance (g) among a wide range of species and experiments[0.46 � r2 � 0.99; data from fig. 2 in Katul et al. (2000)].



(1999) because their fluxes and ci/ca were not published,∂l/l calculated based on five different methods resultedin estimates varying within 20%, possibly reflecting|(1/l)(∂l/∂u)| << |(1/fe)(∂fe/∂u)|. Thus, with a less stringentcriterion for the constancy of l, the results from bothstudies cannot be used to undermine the optimizationtheory.

Effects of high atmospheric CO2 on theresponse of g to D

Because gca

D∼ ( )a

λ

1 21 2, the optimization approach can pro-

vide diagnostic results on how the g – D response senseschanges in ca, a topic that has received much attention forover a decade (Ellsworth et al. 1995; Bunce 1998; Heath1998; Medlyn, Barton & Broadmeadow 2001; Wullschlegeret al. 2002; Herrick, Maherali & Thomas 2004).

To facilitate comparisons with a number of publishedstudies, the effects of high ca on g at a reference environ-mental state (i.e. the sensitivity of g to D in the neigh-bourhood of D = 1 kPa) are firstly considered. Beginning

with gc

aD= − + ( )⎛

⎝⎜⎞⎠⎟

αλ

11 2

a , differentiating with respect to

D, setting D = 1 kPa, and re-arranging the terms, the sensi-tivity of g to D at D = 1 kPa can be expressed as

dgdD

gD=

= − −( )1

12kPa

α . (19)

This outcome suggests that within the optimization frame-work, the slope of dg/dD versus g is constant (= 1/2) and notimpacted by ca; however, the intercept a can vary apprecia-

bly with ca. Recall that α αα

≈+1

2 ci

, and an increase in ca

(and thus ci) will result in a decrease in a even if the basicphotosynthesis 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 forcomparison with these optimization results are presented aslinear relationships between g and D with no informationon the p - ci relationship (Heath 1998; Medlyn et al. 2001;Wullschleger et al. 2002; Herrick et al. 2004). However, itmay be possible to synthesize the results of such studies byevaluating dg/dD at D = 1 kPa as a ratio of the responsesobtained under high and current ca.

Beginning with dg dD g g= − −( )12

1 α and noting that

g = a[-1 + (ca/(al))1/2], the ratio of dg/dD (at D = 1 kPa)under high and current ca is given by:

0.0 0.5 1.0 1.5

c i /

c a

0.5

0.6

0.7

0.8

0.9

1.0

D (kPa)0.0 0.5 1.0 1.5

0.5 1.0 1.5

0.5

0.6

0.7

0.8

0.9

1.0

D1/2 (kPa1/2)

0.5 1.0 1.5

0 1 2 3

0.5 1.0 1.5

r

2 = 0.64

r

2 = 0.67r

2 = 0.97r

2 = 0.65

r

2 = 0.64r

2 = 0.95

Figure 5. The ratio of intercellular to atmospheric CO2 concentration (ci/ca) as a function of vapour pressure deficit (D, upper panels)and D1/2 (lower panels). The gas-exchanged data sets are from a 12-year-old forest dominated by Eucalyptus ssp. (circles; Wong & Dunin1987), a well-watered Pinus taeda seedlings (triangles up; Fites & Teskey 1988), and a 13-year-old P. taeda plantation at Duke ForestFACE facility (triangles down; Ellsworth 2000), where the measurements were collected 1996–1999 at multiple locations in the controlplots at air temperatures between 20 and 30 °C. Filled triangles are obtained on 13dC data reported in Mortazavi et al. (2005) for pine andhardwood foliage in the same stand.



ydg dD

dg dD

g

g c ca

=[ ]

[ ]

=[ ]

[ ]−

+⎛⎝

⎞⎠ −

⎛

+

+

c c

c

c c

c a a

e

a a

a

a a

a

δ

δ 11

11 2δ

λ⎝⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−⎛⎝

⎞⎠ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−

11

11 2

1

ca

a

a

or

λ

;

(20)

yc c

ac

a

= −+⎛

⎝⎞⎠ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−⎛⎝

⎞⎠ −

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

11

11

1

11 2 1 2

a a

e

a

a

δλ λ

−−1

x,

where dca is the atmospheric CO2 increment, andle and la are the equivalent water costs for highand current ca, respectively. Figure 7 shows thaty dg dD dg dD= [ ] [ ]+c c ca a aδ (dependent variable) as a

function of x g g= [ ] [ ]+c c ca a aδ (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-termexperiments (e.g. Free Air CO2 Enrichment facilities),where the CO2 enrichment, (dca + ca)/ca, ranged from 1.5 to2.0. Based on the optimization theory, the departure from

unity in this data is captured by the terms c ca

a a

e

+⎛⎝⎜

⎞⎠⎟

δλ

1 2

and ca

a

aλ⎛⎝⎜

⎞⎠⎟

1 2

. In the case of le = la, and for (dca + ca)/

ca > 1, the predicted y as a function of reasonable values ofl is close to unity though for large enough l, y becomesnegative. Accepting for the moment the linearity of theresponse of g to D, as reported by the authors, this analy-sis suggests that increased CO2 should affect le such thatle/la > 1 + dca/ca. Next, we analyse this ‘excess cost’ usingdata from two short-term exposure experiments (Bunce1998; Heath 1998).

The conductance at high ca relative to current ca may beexpressed as:

′( ) = = ⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

− + +

− +

−

R Dgg

D

D

c cac

a

e

a

e

a

e

a

a a

a

αα

λ

λ

δ1

1

1 21 2

1 2−−

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟1 2

. (21)

Hence, two effects must be simultaneously considered whenassessing the effects of high ca on the relationship between gand D: (1) the effect of ca on ae/aa, which reflects only thedegree to which the parameter of the linear p - ci curve shifts

under high ca (i.e. α αα

∼ 1

2 + sca

), and (2) the effect of the

variation in D emerging from the theory outside the p - ci

AHD (g m–3)7 9 11 13 15

g (m

mol

m–2

s–1

)

40

60

80

100

g = s1 – s2 D1/2; r 2= 0.89

fe (mmol m–2 s–1)

0.9 1.1 1.3 1.5

f c (mm

ol m

–2 s

–1)

4.2

4.4

4.6

4.8

Δfe/Δg

–0.04 –0.03 –0.02 –0.01 0.00

Δfc/

Δg

0.00

0.01

0.02

0.03

0.04

0.05

r 2 = 0.48

r 2 = 0.81

(a)

(b)

(c)

Figure 6. (a) Stomatal conductance (g) as a function ofabsolute humidity deficit (AHD), (b) the photosynthetic rate (fc)as a function of transpiration rate (fe), where the regression slopeis 0.0011 mol CO2 /mol H2O, and (c) the sensitivity of fc to achange in g (Dfc/Dg) as a function of sensitivity of fe to a changein g (Dfe/Dg) approximated using central differencing. Theregression slope is 0.0018 mol CO2/mol H2O. Data from Fites &Teskey (1988).

gca+dca/ gca

0.0 0.5 1.0 1.5

[dg/

d D] c a

+dc a

/ [dg

/dD

] c a

0.0

0.5

1.0

1.5FsQrPsQiPaPiaLsLs2Fs2CsQr2

Figure 7. The ratio of the sensitivities of conductance to vapourpressure deficit (D) under current (ca) and high CO2(dca + ca)conditions as a function of the corresponding ratio of g (atD = 1 kPa). Data were obtained from Medlyn et al. (2001; Fagussylvatica, Quercus robur, Pinus sylvestris, Quercus ilex, Philyreaangustifolia and Picea abies), Herrick et al. (2004; Liquidambarstyraciflua), Wullschleger et al. (2002; L. styraciflua2) and Heath(1998; Fagus sylvatica2, Castanea sativa, Quercus robur2). Symbolfill from open to black indicates the enrichment level:(dca + ca)/ca): ~1.5, ~1.7 and ~2.0, respectively. Dashed line is1:1 line.



physiology. Mathematically, if D → +•, R′(•) → ae/aa andbecomes independent of D; stated differently, at high D, thep - ci physiology alone does not introduce a dependence onD – it simply modifies it by a fraction <1. To facilitate aseparate analysis of the effects of high ca beyond the predict-able multiplier factor emerging from the p - ci response, wedefine R(D) = R′(D)/R(•). The problem considered next ishow R(D) changes with increasing D when |dca| > 0. Toaddress this problem, three cases are considered:Case (1): ∂R(D)/∂D > 0When |dca| > 0 and all other parameters, including∂l/∂ca = 0, are held constant, the optimization theory pre-dicts that R(D) must increase with increasing D. In fact,R(D) must increase with increasing D as long asλλ

δe

a

a a

a

< +⎛⎝⎜

⎞⎠⎟

c cc

. The predicted increase for this case is con-

sistent with data on three species (Heath 1998; Fig. 8).Case (2): ∂R(D)/∂D = 0

If λ δ λea a

aa= +⎛

⎝⎜⎞⎠⎟

c cc

, then R(D) is not affected by increasing

D. We have not encountered a data set where this caseemerges.

Case (3): ∂R(D)/∂D < 0

This case is possible only when λ δ λea a

aa> +⎛

⎝⎜⎞⎠⎟

c cc

. Bunce

(1998) presented experimental evidence suggesting thatR(D) can decrease with increasing D opposite to the pre-diction in case (1). Can le/la increase sufficiently under ca toresult in the emergence of responses of the type of case (3)?

The information provided in Bunce (1998) does notpermit us to address the question quantitatively. Thus weask: Can such increases in le/la be observed under high ca?Leaf-level gas exchange data collected on Pinus taedaat the Duke Forest Free Air CO2 Enrichment (FACE)facility were used to answer this question (data fromtrees experiencing current ca are shown in Fig. 5). The

value of acλa

⎛⎝⎜

⎞⎠⎟

1 2

for current and for elevated CO2 plots

(ca + 200 mmol mol-1) was first computed by regressing1 - ci/ca versus D1/2. No down-regulation in the p - ci curvewas documented for the period used in this analysis(Ellsworth 2000; Rogers & Ellsworth 2002). For D (in kPa),

these regression results suggest that acλa

⎛⎝⎜

⎞⎠⎟ =

1 2

0 096. for

current andacλa

⎛⎝⎜

⎞⎠⎟ =

1 2

0 230. for elevated ca. The scatter is

large (see Fig. 5), as expected when measurements aremade on different fascicles in different seasons, and soilmoisture conditions vary greatly among measurement cam-paigns. Nevertheless, the slopes of the regression were dif-ferent between the ca treatments (P < 0.05). Based on thisrough assessment, le = la(0.230/0.096)2(580/380) = 8.72,

satisfying the inequality λ δ λea a

aa> +⎛

⎝⎜⎞⎠⎟

c cc

, required for the

emergence of responses the like of case (3). Based on theFACE parameters, R was then modelled with Eqn 22 for arange of D and compared to data in Bunce (1998) – themodelled R agrees well with the data (Fig. 9).

The effects of elevated CO2 on stomatal conductanceresponse 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 atmo-spheric CO2. Thus, the results from the analysis above mustviewed with some caution. As we noted earlier, we derivedthe optimal stomatal conductance for the non-linear p - ci

curve and intend to use it in future analyses to furtherevaluate these responses.

CONCLUSIONS

Analytical results from the original optimization theorywere 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 conceptualframework of the ‘optimality hypothesis’ but assumed alinear response of p to ci. We showed that the expectedstomatal control (1) can be re-formulated as a univariatemaximization problem not needing Lagrange multipliers or

1.0

1.2

1.4

1.6

1.8

R

1.0

1.2

1.4

1.6

D (kPa)0.5 1.0 1.5 2.0 2.5 3.0

1.0

2.0

3.0

4.0

Fagus sylvatica

Castanea sativa

Quercus robur

Figure 8. The ratio (R) between stomatal conductance underhigh atmospheric CO2 conditions (current + 250 ppm) andcurrent conditions as a function of water vapour pressure deficit(D). Data are for three species from Heath (1998).



integration time scales, (2) is consistent with the onset of anapparent ‘feed-forward’ mechanism in g as discussed inMonteith (1995), (3) agrees with a synthesis survey suggest-ing that g scales as 1 - m log(D) where m ∈ [0.5,0.6] (Orenet al. 1999), and (4) agrees with experiments reporting anon-linear variation in ci/ca with D.We have also shown thatphysical constraints on the degree of stomatal opening (i.e.0 < u � 1, g � 0) provide logical limits to l that can beindependently derived from the p - ci curve, maximumtheoretical conductance, D and ca.

How g responds to D (or in some models RH) is ofcentral importance given that under future climate sce-narios, warming is expected not to affect air relative humid-ity but to increase D exponentially (Kumagai et al. 2004).Using the optimization theory, we analysed the conflictingexperimental results on the sensitivity of g to D undercurrent and high CO2 reported in Bunce (1998) and Heath(1998), among others. The approach provides a diagnostictool 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 timescale at which the optimization is operating is commensu-rate with the time scales of opening and closure of stomatalaperture u, which is too short to be interpreted as beingdriven by whole-plant resource optimization when fixing aresource constraint (as is often done in the economics of gasexchange). It is conceivable that ‘whole-plant’ scale carbongain may actually be achieved if stomatal aperture controlshave evolved to be ‘efficient’ at the finest possible timescales (i.e. scales at which max

uef u p f( ) = −( )λ ). This state-

ment follows from a variant of Pontryagin’s maximum prin-ciple, which informally implies that beginning from knowninitial conditions (say a certain amount of carbon in thewhole plant system), global optimality at the plant scale in

terms of maximizing its carbon gain (which is a linear sumacquired from all the leaves) is guaranteed if at each timestep, the ‘local’ maximum is always selected at the stomatallevel for the set of environmental conditions. In thiscontext, the framework proposed here assumes that thislocal maximum amounts to maximizing the leaf photosyn-thesis while minimizing water loss rate.

It should be emphasized, however, that the apparentagreements with data cannot be viewed as an endorsementof the validity of the optimality hypothesis on stomatalbehaviour. Nevertheless, such optimization formulationshave joined semi-empirical models, such as the Leuning(1995) and the Collatz et al. (1991), to facilitate couplingleaf-level gas exchange to canopy scale and predictingphotosynthesis, transpiration and tree growth under cur-rent and future climatic conditions (Mäkelä et al. 2006;Schymanski et al. 2007, 2008; Buckley 2008).

ACKNOWLEDGMENTS

This study was supported by the United States Departmentof Energy through the Office of Biological and Environ-mental Research Terrestrial Carbon Processes programand 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 theBi-National Agricultural Research Development fund(IS-3861-06). We thank Kim Novick and Stefano Manzonifor helpful comments on an earlier version of this work.

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R

0.4

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Received 1 December 2008; received in revised form 16 January2009; accepted for publication 20 January 2009



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