G. Katul1,2,*, S. Manzoni3, S. Palmroth1, A. Porporato2,1, D. Way1,4 & R. Oren1

1Nicholas School of the Environment & 2 Department of Civil and Environmental Engineering, Duke

University, Durham, North Carolina, USA; 3Department of Physical Geography, Stockholm University,

Sweden; 4Department of Biology, University of Western Ontario, Ontario, Canada

*Email: gaby@duke.edu

Alpine Summer School, Course XXIII

Valsavarenche, Valle D’Aosta, Italy, 22 June – 1 July, 2015

Special Lecture on the Economics of Leaf-Gas Exchange

(Thursday, June 25, 2015)

Jan Baptist van Helmont coined the word ‘gas’ in the 17th century and noted that ‘gas sylvestre’ (carbon dioxide) is given off by burning charcoal.

He also investigated water uptake by a willow

tree, thereby pioneering some of the earliest experiments on gas transfer (after the seminal work of Edme Mariotte around 1660).

Centuries later, van Helmont’s activities converged into a modern-day story:

Atmospheric CO2 (ca) is rising largely due to

fossil fuel combustion, and the ability of terrestrial plants to uptake CO2 is currently a leading mitigation strategy to offset this rise.

WATER CYCLE: Increase in continental scale

runoff because plant stomata open less as CO2 concentrations increase (Betts et al., 2007; Gedney et al., 2006).

H2O CO2

CARBON CYCLE: Reduced stomatal conductance - predicted to lead to saturation of CO2 uptake by plants, contributing to acceleration of global warming (Cox et al., 2000).

Land plants appeared in the fossil record some 500 Million years ago, presumably evolving from aquatic chlorophyte algae.

Compared to aquatic systems, the terrestrial environment posed major challenges to terrestrial plant survival.

From: sharonhs-taxa-2013-p3.wikispaces.com

To exchange gas with a desiccating atmosphere (a necessary requirement for carbon fixation via photosynthesis), plants needed to engineer systems to bring water to the exchanging tissue.

Connections between plant hydraulic structure (water delivery system) and function (carbon fixation) must exist.

Plants evolved a coordinated photosynthetic – hydraulic machinery, where the maximum capacity of the hydraulic machinery reflects tradeoffs between safety and efficiency.

Leaf photosynthesis to be minimum of 3 rates:

min

E

c c

s

J

f J

J

RuBP regeneration-limited

Rubisco-limited

Sucrose-limited

*

1

2

i

c

i

Cf

C

Mathematical Form:

M. Calvin

*

1

2

i

c

i

Cf

C

( )c s a if g C C

Farquhar model: Biochemical Demand

Mass Transfer (Fickian): Atmospheric Supply

2 equations,

3 unknowns: fc, gs, Ci

A. Fick

Francis Darwin already noted in 18981 that

“transpiration is stomatal rather than curticular, so that other things being equal, the yield of watery vapour depends on the degree to which stomata is open, and may be used as an index to their condition”.

So, what drives stomatal opening?

Darwin, F. (1898), Observations on Stomata. Proceedings of the Royal Society of

London (1854-1905). 1898-01-01. 63:413–417.

www.evolution.Berkeley.edu

)()()()( 4321

max,

al

s

s cffDfPARfg

g

No synergistic interactions

One of the earliest empirical

approaches (Jarvis, 1976)

Jarvis, P., 1976, The Interpretation of the Variations in Leaf Water Potential and

Stomatal Conductance Found in Canopies in the Field, Phil. Trans. R. Soc.

Lond. B vol. 273, 593-610

P. Jarvis

1

1 21 1 2 1; 1c c

a a o

m m Dg f RH b g f b

c c D

‘Ball-Berry’

(Collatz et al., 1991)

Leuning (1995)

Two well-known formulations that fit a wide range of data:

Ball-Berry - allowed interactions between the biosphere and

atmosphere when ‘greening’ climate models (Sellers et al.,

1996).

Note the linear relation between g and fc/ca

SUPPORTED BY A LARGE CORPUS OF LEAF-LEVEL DATA

Science, Vol. 275, 24 January, 1997

Climate projections for warming scenarios:

Usually predict invariant air relative humidity (RH). If so, then vapor pressure deficit (D) increases exponentially with increasing temperature (T) as discussed in Kumagai et al. (2004).

𝐷 = 𝑒∗(𝑇)(1 − 𝑅𝐻)

Kumagai, T., G. G.Katul, A. Porporato, T. M.Saitoh, M. Ohashi, T. Ichie, M.

Suzuki, 2004, Carbon and water cycling in a Bornean tropical rainforest under

current and future climate scenarios, Advances in Water Resources, 27, 135-150

1

1 21 1 2 1; 1c c

a a o

m m Dg f RH b g f b

c c D

Stomatal conductance as a “compromise between the need to provide a passage for assimilation and the prevention of excessive transpiration” (Cowan and Troughton, 1971; Givnish, 1976).

c

e s

Carbon Gain f

Water Loss f a g D

Define the instantaneous:

Objective: maximize C gain

Dalton’s Law

Lagrange

Multiplier

T

sc dtgf0

Subject to soil moisture availability constraint

sesc gfgfH Hamiltonian:

H has to be maximized `

0)()(

sesc

s

gfgfg

Problem mathematically CLOSED - Analytical solution

accounting for all non-linearities in the fc-Ci curve and

photosynthesis limitations was derived elsewhere (Katul et

al., 2010).

*

1

2

i

c

i

Cf

C

( )c s a if g C C

Biochemical Demand

Atmospheric Supply

Optimality Hypothesis

(Temporal changes in λ neglected at time scales faster than soil moisture changes)

Hamiltonian H

Analytical foresight: Photosynthesis model simplified (as Hari et al., 1986; Lloyd, 1991)

Linearized biochemical demand

function: *

2

1

i

i

c cc

f

aaai scccc 22 )/(

s= assumed constant

(e.g. determined from stable isotopes)

P. Hari

Maximize Hamiltonian given by:

Dag

scg

cggfgfgH s

as

assescs

21

1)()()(

)()(

21

1

as

assc

scg

cggf

Carbon gain Water loss

fc(gs)

gs

Downward

concavity!

fe(gs) Correct downward concavity & saturation

with increased CO2

Upon differentiating H(gs) with respect to gs and setting it to zero (Hari et al., 1986; Lloyd, 1991; Katul et al., 2009):

1/2

1/21a

i

a

a

c

cD

c

1/ 2

1

2

1 a

a

sgD

c

sc a

1

2

c a a

a

f c a Dcsc

Combine the formulation for conductance and photosynthesis (Katul et al., 2010)

2/1 Da

c

c

fg a

a

c

1

1 21 1 2 1; 1c c

a a o

m m Dg f RH b g f b

c c D

Linear Biochemical Demand Function

Leuning Ball-Berry

See Medlyn et al. (2011) for a variant on the light-limitation version.

Proportionality of gc and A

(see also Hari et al., 2000, Aus. J. Plant Phys.)

Pa

lmro

th e

t a

l., 1

99

9, O

eco

log

ia

2/1 Da

c

c

AEDg a

a

s

~40 species that include diffuse

porous, ring-porous, non-porous, and

desert shrubs.

Stomatal responses to vapor pressure deficit: Is the D-1/2 consistent with literature responses?

𝐺𝑠

𝐺𝑠𝑟𝑒𝑓= 1 − 𝑚 𝑙𝑜𝑔(𝐷)

Empirical findings of Oren et al (1999):

Analysis on more than 40 species found that

m=0.5-0.6 when fitting leaf and tree conductance data using:

1/2

1

2

1 a

a

ca

a sg

Dc a

RECALL: OPTIMAL SOLUTION

𝑔

𝑔𝑟𝑒𝑓= 1 − 𝑚 𝑙𝑜𝑔(𝐷)

𝑔𝑟𝑒𝑓 defined at D=1 kPa, high light levels and ample soil moisture

1/211 (1 1/ 2log( ))

1 log( ); 11 2 1

a

ref

g DD

g

c

a

1/ 2

1

2

1a

refaca

a sg

c a

1 log(m )ref

gD

g Compare to Oren et al. (1999):

Linear optimization model with D=1 gives reference conductance

m=0.5-0.6

1

1/ 2

1

( )

1/ 2log( )1

!

11 log( ) ..

2

n

n

n

f D D

D

n

D

RAPIDLY CONVERGING

Fagus crenata

Drop in

Transpiration

VPD

Transpiration

𝑓𝑒 = 𝑔 𝐷 𝐷

𝑓𝑒 ∝ 𝑓 𝐷 𝐷

Driving force for leaf transpiration:

leaf conductance: 𝑔 ∝ 𝑓 𝐷

∝ 𝐷

Leaf transpiration:

Optimization models and apparent feed-forward mechanism

Apparent feedforward mechanism is a shutdown in 𝑔 that is faster than the

increase in driving force 𝐷 so that with increased 𝐷, transpiration losses are

reduced to prevent leaf de-hydration.

All else being the same, this process should be reversible.

Apparent feed-forward mechanism:

Literature:

Rare (at best), reversibility rarely tested.

CAN NECESSARY CONDITIONS TO

OBSERVE IT USING BE ESTABLISHED

USING OPTIMAL STOMATAL

THEORIES?

Apparent feed-forward mechanism For Rubisco-limited Photosynthesis

𝑓𝑒 = 𝑎𝐷𝛼1

𝛼2 + 𝑠𝑐𝑎−1 +

𝑐𝑎

𝑎𝜆𝐷

12

= 𝑎𝛼1

𝛼2 + 𝑠𝑐𝑎−𝐷 + 𝐷1/2

𝑐𝑎

𝑎𝜆

1/2

𝜕𝑓𝑒

𝜕𝐷= −1 +

1

2

𝑐𝑎

𝑎𝜆

1/2

𝐷−1/2 = 0 𝐷𝑐𝑟𝑖𝑡 =1

4

𝑐𝑎

𝑎𝜆

𝐷 > 𝐷𝑐𝑟𝑖𝑡 =1

4

𝑐𝑎

𝑎𝜆

Hence, apparent feed-forward mechanism discussed in Monteith (1995) occurs when

Solve for D:

Recall that linearized solution yields for leaf transpiration:

Increasing D increases then decreases leaf transpiration (hence a maximum):

N.B. Some studies (e.g. Buckley, 2005) already speculated a linkage

between the apparent feed-forward mechanism and stomatal optimization.

Apparent feed-forward mechanism: Revisiting Bunce (1997) data

Comparison between

optimization model

calculations of stomatal

conductance (g)

and transpiration rate

(fe) and the data from

Bunce (1997) for the

onset of a feed-forward

mechanism in three

leaves of Abutilon

theophrasti plotted

using different symbols.

Conditions of experiments (i.e. how D is generated) affect to what degree the apparent feedforward mechanism is observed.

Fix Temperature but

vary vapor pressure

Fix vapor pressure but

vary temperature

Rubisco Rubisco RuBP

RuBP

From: Vico, G.,S. Manzoni, S. Palmroth, M. Weih, and G.G. Katul, 2013, A perspective on optimal leaf

stomatal conductance under CO2 and light co-limitations, Agricultural and Forest Meteorology, 182-

183, 191-199

Other optimization theories

Prentice et al. (2014):

Idea: Plants tend to operate at an optimal 𝑐𝑖

𝑐𝑎 and

maintain it constant.

1/2

1/21a

i

a

a

c

cD

c

Optimization models and ratio of intercellular to ambient CO2

This ratio only holds when plants maximize their carbon gain

for a given water loss. It is independent of stomatal operation

as the stomatal aperture here is already at optimum.

Nonlinear response of ci/ca with D

1/2

1/21a

i

a

a

c

cD

c

Nonlinear response

Linear response

Eucalyptus Pine Seedling Pine Canopy

Stable

Isotopes

Gas

exchange

CG=constant conductance

CC = constant ci/ca

BB = Ball-Berry model

LE = Leuning model

JO = Jarvis-Oren model

LO = Linear optimality

AT ET

Way, D., et al. 2011, Journal of Geophysical Research, 116, G04031, doi:10.1029/2011JG001808

Seasonal Carbon balance

MODELS

Extension to include Mesophyll conductance and salt stress

Revise theories to accommodate

water and salt stress under

ambient and elevated CO2.

Manzoni, S., G. Vico, G.G. Katul, P.A. Fay, H.

W. Polley, S. Palmroth, and A. Porporato,

2011, Optimizing stomatal conductance for

maximum carbon gain under water stress: a

meta-analysis across plant functional types

and climates, Functional Ecology, 25, 456-

467

Volpe, V., S. Manzoni, M. Marani, and G.G.

Katul, 2011, Leaf conductance and carbon

gain under salt-stressed conditions, Journal

of Geophysical Researach, 116, G04035,

doi:10.1029/2011JG001848

Blue-print on how to proceed sketched in:

Linearized formulation

𝑔𝑒𝑓𝑓 =𝑔𝑚𝑔𝑠

𝑔𝑚 +𝑔𝑠 CO2 pathway (Mesophyll & Stomatal)

H2O pathway (only stomatal)

𝜕𝑔𝑠

𝜕𝑔𝑚= 0 Assume

𝑔𝑠 =𝑎1𝑔𝑚

𝑎1 + 𝑔𝑚(𝑎2 + 𝑠𝑐𝑎)−1 +

𝑐𝑎 − 𝑐𝑝

𝑎𝜆𝐷

𝑔𝑠

Fresh

Salt

Fresh

Salt

Interestingly, 𝑔𝑠 − 𝑓𝑐 form is

maintained as with empirical models.

2/1

Da

cc

c

fg

pa

a

cs

𝑔𝑠 =𝑎1𝑔𝑚

𝑎1 + 𝑔𝑚(𝑎2 + 𝑠𝑐𝑎)−1 +

𝑐𝑎 − 𝑐𝑝

𝑎𝜆𝐷

PDF(s)

s

Water availability was not considered in the stomatal optimization

theories. Soil moisture content varies across a wide range of time

scales.

Process-based optimal control problem

1) Objective: maximize photosynthesis (A) over a

period T

2) Control: stomatal conductance to CO2 (gC)

3) Constraint: soil water is limited

Hypothesis:

Water use strategies are optimal in a given environment

(idea pioneered by Givnish, Cowan and Farquhar)

T

T

JdtttxtgAJ 0

,,

ttxLttgEtRdt

dxnZr ,,

ttxtgftttxtgAttxtgH ,,,,,,

,0

g

f

g

A

g

H .

x

H

dt

d

Hydrologic

Balance

Objective

Function

Hamiltonian

Derivation of

optimal g(t)

T

c dtgA0

Objective: maximize

Soil moisture changes slowly compared to light and VPD

Soil moisture is assumed constant

Optimal stomatal conductance

1

aD

ckg a

C

λ is constant, but undetermined! (classical solution by Cowan and Farquhar; Hari and Mäkelä)

0

E

At

T

c dtgA0

Objective: maximize

s

R E

QLgERdt

dsnZ cr

Subject to the

constraint

L

Zr

Q

Marginal water use efficiency

t

Optimal stomatal conductance

λ is defined by the boundary conditions of the optimization (Manzoni et al. 2013, AWR)

1

aD

ckg a

C with time

λ increases as drought progresses across

species, ecosystems, and climates

(Manzo

ni et al., 2011, F

unctional E

col)

Water stress

Wa

ter

use

eff

icie

ncy

Ψ

λ/λ

ww

eww

-ψ

λ

λww

-ψ

gc

to satisfy the carbon demand

1-D Richard’s equation with H2O sinks

solved by Finite Element Methods

Stem hydraulics – based on cavitation

curves from Sperry et al. (2000).

Maximum H2O loss from carbon demands

of the leaf.

Stomatal shut-down: From Sperry - Critical

pressure

To assess the maximum hydraulic transport capacity supporting photosynthesis, the focus is on maximum transpiration rate E in the xylem system under WELL-WATERED SOIL conditions.

Detailed representations available1 – but simpler ones needed to interpret the numerous data sources

1Bohrer, G., H. Mourad, T. A. Laursen, D. Drewry, R. Avissar, D. Poggi, R. Oren, and G. G. Katul,

2005, Finite-Element Tree Crown Hydrodynamics model (FETCH) using porous media flow within

branching elements - a new representation of tree hydrodynamics, Water Resources Research, 41,

W11404, doi:10.1029

~0S

L

g ( )P L

LLPgE

Simplifying assumptions:

1) Focus on maximum rates, attained

under well-watered conditions

→ ΨS-ΨL~-ΨL

2) Assume that cavitation occurs first

in the most distal parts of the plant (hydraulic segmentation, Zimmerman,

Can J Bot 1978)

→ ΨX~ΨL at the site of cavitation

3) Gravitational effects are negligible with

respect to frictional pressure losses

From: Manzoni, S., G. Vico, G. Katul, S. Palmroth, R.B. Jackson, and A. Porporato, 2013, Hydraulic limits on

maximum plant transpiration and the emergence of the safety-efficiency trade-off, New Phytologist, 198, 169-178

~0S

L

g ( )P L

1

50

, 1

a

LmaxPLP gg

Data

fo

r A

lnu

s c

ord

ata

(To

gn

ett

i a

nd

Bo

rgh

ett

i 1

99

7)

- (MPa)

Pg

( )

X

X

50

a

0 1 2 30

0.5

1

Vulnerability curve

Xylem conductivity declines with

decreasing water potential due

to cavitation and embolism

-ΨL

gP(Ψ

L)/

gP,m

ax

From: Manzoni, S., G. Vico, G. Katul, S. Palmroth, R.B. Jackson, and A. Porporato, 2013, Hydraulic limits on maximum plant transpiration

and the emergence of the safety-efficiency trade-off, New Phytologist, 198, 169-178

Maximize E with respect to ΨL

amaxL a

1

50, 1

L,maxL,maxPmax gE

maxPL,maxP gag ,

11 0 2 4 60

1

2

0 2 4 60

0 2 4 60

0.1

0.2

|L| (MPa)

L,max

Emax

High gP,max

, low |50

|

Low gP,max

, high |50

|

gP

(m

3 d-1 M

Pa

-1)

S-

L~-

L

E (

m3 d

-1)

4

2

(MP

a)

0 LLP

LL

gd

d

d

dE

Maximum E is computed as

3 parameters: gP,max, a, Ψ50

amax

1PLC

% loss of

conductivity

From: Manzoni, S., G. Vico, G. Katul, S. Palmroth, R.B. Jackson, and A. Porporato, 2013,

Hydraulic limits on maximum plant transpiration and the emergence of the safety-efficiency

trade-off, New Phytologist, 198, 169-178

LLPgE

Individual plant

transpiration Transpiration per unit

sapwood area

𝐸𝑚𝑎𝑥 = 𝑔𝑃,𝑚𝑎𝑥𝜓 50

𝑎 − 1 1−1/𝑎

𝑎

Varies between

0.5 and 0.7.

550 species

and cultivars

Correlations among plant traits suggest

evolutionary convergence towards a

balanced water-carbon economy.