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Lecture 7 on Leaf Energy Balance, Photosynthesis and Respiration, Concepts and Models March 8, 2016 Instructor: Dennis Baldocchi Professor of Biometeorology Ecosystem Science Division Department of Environmental Science, Policy and Management 345 Hilgard Hall University of California, Berkeley Berkeley, CA 94720 Leaf Energy Balance A need for information on surface temperature is ubiquitous throughout environmental sciences. Enzyme kinetics of photosynthesis, leaf respiration, saturated vapor pressure, rates of transpiration, stomatal conductance and emissions of volatile organic carbon compounds are all tied to functions of surface (leaf temperature). Evolution and ecological success is also tied to leaf temperatures (Helliker and Richter, 2008; McElwain et al., 1999). Plants die if their temperature is too high because of its size, position and orientation cause the surface energy balance to be unfavorable. In this section we derive equations for computing leaf temperature and evaporation rates. Theories have been developed for linear and non-linear forms. Quadratic Solutions for Leaf Temperature and Latent Heat Exchange

Consider the ideal case of energy exchange on one side of a leaf. To compute the evaporation rate of leaves we start with two fundamental equations. One is the leaf energy balance is defined by how net radiation (Rn) is consumed by sensible (H) and latent energy (λE) exchange:

nR H Eλ= +

The other is the equation for the incoming flows (flux densities) of shortwave solar (R) and longwave terrestrial energy (L) that drive the sum of sensible and latent heat exchange and long wave energy emission, that is a function of leaf temperature to the fourth power.

4(1 ) lQ R L T H Eρ ε εσ λ= − ↓ + ↓= + +

We define these in terms of Q by moving the longwave emission to the right hand side of the equation so we can solve for surface temperature.

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We make the fundamental assumption that the leaf is exposed to steady steady-state conditions. In the following equations we will refer to the following physical constants and variable definitions: ε is emissivity, σ is the Stefan-Boltzman constant, ρa is air density, λ is the latent heat of vaporization, Tk is absolute temperature (K), gs is the stomatal conductance (m s-1), gh is the aerodynamic conductance for sensible heat transfer (m s-1), Cp is the specific heat of air, Q is absorbed energy (incoming short and long wave radiation, minus reflected shortwave radiation; W m-2), mv and ma are the molecular weights of vapor and dry air (g mol-1), P is pressure (kPa), es is saturated vapor pressure (kPa) and ea is the ambient vapor pressure (kPa). And m mv a/ is 0.622. The resistance network for water vapor transfer (rw) consists of a route through the leaf boundary layer (ra) and the stomata (rs). These two resistances are in series, so their conductances are in parallel:

1 1 1a sa s w

a s a s w

g gr r rg g g g g

++ = = + = = :

( / ) ( ( ) )

( / ) ( ( ) )

( / ) ( ( ) )( )

v a a w s l a

v a a s l a s a

a s

v a a s l a

s a

m m g e T eEP

m m e T e g gP g g

m m e T eP r r

λρλ

λρ

λρ

−= =

−=

+

−

+

To solve this set of equations we apply a relation that linearizes the Saturation Vapor Pressure relation and the Stefan Boltzmann Equation for the emission of long wave radiation. By apply Taylor’s Expansion Theory we estimate the saturation vapor pressure at leaf temperature as:

2"( ) ( ) '( ) ( )2s

s l x a s l a l aee T e T e T T T T= + − + −

In the equation above the single prime is the first derivative of es with respect to T,

( )sde TdT

, and the double prime is the second derivative, 2

2

( )sd e TdT

With algebraic manipulation the leaf energy balance can expressed a quadratic equation

for latent heat exchange (λE; W m-2):

3

a LE2 + b LE +c =0

We can solve directly for latent heat flux density

0.622 ( ( ) ) 0.622 ( ( ) )( )

a s l a s a a s l a

a s s a

e T e g g e T eEP g g P r r

λρ λρλ

− −= =

+ +

20.622 ( ) ''(( ( ) ) ( )( ) ( ) )2

a w s as a a a l a l a

g e TE e T e s T T T T TPλρ

λ = − + − + −

Substituting

4

34a

l aa p h a

Q E TT TC g Tλ σε

ρ εσ− −

− =+

One derives a quadratic equation for λE (aλE2+bλE+c=0). The coefficients for λE are:

2

3 2

( )8 ( 4 )

a s v s

a a p h k

g m d e Tam P C g T dT

ρ λρ σε

=+

3

24

3 2

( )42

( )2 ( 4 ) 2

s v sk p h

a

a s v sk

a a p h k

g m de Tb T C gm P dT

g m d e T Q Tm P C g T dT

ρλε σ ρ

ρ λσε

ρ σε

= − ⋅ ⋅ ⋅ − − +

−⎡ ⎤+⎢ ⎥+ ⎣ ⎦

3 4

2 24 2 4

3 2

( ( ) ) ( )( 4 ) ( )2

( ) ( )2 ( 4 ) 4

a s v s a a s v sa p h k k

a a

a s v sk k

a a p h k

g m e T e g m de T Qc C g T Tm P m P dT

g m d e T Q T Q Tm P C g T dT

ρ λ ρ λρ ε σ ε σ

ρ λσε σε

ρ σε

−= + ⋅ ⋅ + − ⋅ ⋅ +

⎡ ⎤+ − ⋅⎢ ⎥+ ⎣ ⎦

A quadratic equation, defining the difference between leaf and air temperature (ΔT),

was derived from the leaf energy balance relationship so an analytical solution could be used

to compute leaf temperature (Paw U and Gao, 1988):

a ΔT2 + b ΔT + c =0 (A18)

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The coefficients are defined as:

22

2

( )122a s v s

ka

g m d e Ta Tm P dT

ρ λε σ= ⋅ ⋅ ⋅ +

3 ( )8 2a s v sk p h

a

g m de Tb T C gm P dT

ρ λε σ ρ= ⋅ ⋅ ⋅ + +

4 ( ( ) )2 2a s v s ak p s

a

g m e T ec T C g Qm P

ρ λε σ ρ

−= ⋅ ⋅ ⋅ + + −

Partitioning of Sensible and Latent Heat Exchange. Given an amount of energy how do we know how much will be converted into sensible or latent heat exchange?? We can examine this question be examining the Bowen ratio, H/LE.

( )a p l a aH C T T gρ= −

( '( ))a p w s l aC g D e T TE

ρλ

γ

+ −=

( )( '( ))

a p l a a

a p w s l a

C T T gHC g D e T TEρ

βρλ

γ

−= =

+ −

( )( '( ))

l a a

w s l a

T T gHE g D e T T

γβ

λ−

= =+ −

where

4

3

0.622 /4 0.622 ( ) ' /

a wl a

a p h a s a a w

Q T g D PT TC g T e T g P

εσ λρ εσ λ ρ

− −− =

+ +

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Derivations need to be revisited if one is considering energy exchange on one or two sides of a leaf, or if the leaf is amphi or hypo stomatous. In each case factors of two abound on different terms. It is left to the student to perform the derivation for the case of interest. If a leaf is hypostomatous, energy exchange will occur on both sides of the leaf, but evaporation will occur only on one side, for instance.

7.1 Leaf Photosynthesis Models As biometeorologists, our goal is to develop a canopy-scale model that couples photosynthesis and stomatal conductance to compute leaf transpiration and leaf temperature. To do so we must evaluate the type of photosynthesis models that are available in the literature and understand how to implement them and parameterize them. Today it is common to use either the Ball-Berry-Collatz (Collatz et al., 1991), Leuning (Leuning, 1990) or carbon-water optimization models (Katul et al., 2010; Medlyn et al., 2011) to evaluate stomatal conductance, gs. All estimate stomatal conductance are a function of leaf photosynthesis (A) and some measure of humidity deficit of the leaf, either in terms of relative humidity (rh) or vapor pressure deficit (D). In terms of parsimony, the Ball-Berry-Colltaz is simplest and contains one key unknown, the slope m; the other equations often yield similar results and require information on more constants.

02[ ]s

Arhg g mCO

= +

While the following discussion may seem abstract to those schooled whole in fluid mechanics, the adoption of coupled energy balance photosynthesis models has become standard in state of art climate models, like the Climate-Land Model, CLM, (Bonan et al., 2011; Bonan et al., 2014) Over the years numerous mathematical models have been developed to simulate leaf photosynthesis, A. The basis of the models have been using ideas relating to diffusion of CO2 to the chloroplast and enzyme kinetics (Farquhar, 1989; Farquhar et al., 2001; Harley and Tenhunen 1991); for an overview the more serious student is referred to the book of Suzanne von Caemmerer (von Caemmerer, 2000). Gaastra’s (1959) model for photosynthesis is a diffusion-limited model, and set the intellectual stage for using the Ohm’s law analogy to study leaf fluxes; on sabbatical I was able to get a pdf copy of this paper from the Wageningen University Library.

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Gaastra’s model computes photosynthesis as the ratio of the potential difference in CO2 between the free atmosphere and that in the chloroplast and the sum of resistances to this transfer:

( )a c

leaf mesophyll chloroplast

C CAr r r

−=

+ +

The leaf resistance is a series of resistances between the leaf boundary layer, the stomata, the sub stomatal cavity. The mesophyll resistance involves the cell wall, plasmalemma and cytoplast. The Chloroplast resistance involves the chloroplast membrane and the stroma. Note the stomatal diffusion resistance, normally determined for water, must be modified for the diffusion of CO2. The rate velocity of reactions, v, associated with enzyme kinetics of the concentration of the substrate, C, can be defined using the Michaelis-Menten equation:

max max

1

v C vv K C KC

⋅= =

++

This equation is often used to define how photosynthesis varies with the substrate, CO2

max max

1

P C PP K C KC

⋅= =

++

Here the rate of reaction is proportional to a maximum value (Pmax), the amount of substrate (C) and a reaction rate, K, or Michaelis-Menten constant. This relation forms a diminishing returns function. Note that K represents the concentration C when P equals 0.5 Pmax. Others have developed empirical models that relate photosynthesis to the amount of available light. One example is the empirical model of Thornley and Johnson (1990)

max

max

I PPI P

αα⋅ ⋅

=⋅ +

Our first task is to develop a simple model that reflects the trade-offs between sink and diffusion limitations. If we combine: 1) a substrate dependent model and 2) a diffusion-limited model then we can derive a polynomial solution for photosynthesis:

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For example, consider the balance between the Michaelis-Menten Equation and the Fickian diffusion model for leaf photosynthesis:

max av C C CvK C r

−= = −

+

This equation can be solved for Ca by eliminating for C (C C rva= − ). Substituting this identity into the upper equations produces a quadratic equation on v.

2 maxmax( ) 0a aK C v Cv v v

r r+

+ + − =

This equation can be solved with the quadratic equation. This equation provides an important concept for it illustrates the non-linearity that arises by balancing diffusion and demand for a substrate such as CO2. Here we assume that the uptake by the leaf has a positive sign. Thisbalanceequationcanserveasafoundationtowardsunderstandinghowconsumptionprocessesassociatedwithtracegasexchangemayvarywithplanttraitsandenvironmentalconditions.Themodelparameter,vmax,givesinformationonthecapacityofthesetofreaction.Innaturalsystemsthisparameterwillvarywithageofleaf,itsnitrogencontentandleafthickness(Reichetal.,1997).Theparameter,K,implieshowefficientlyCisusedbythereactions.

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Theresistance,r,isatotalresistancethatrepresentstheeffectsofdiffusionacrosstheleafboundarylayer,thestomatalporeandthroughthemesophylltothechloroplast.Thisresistancewillvarywithleafsize,windspeed,humiditydeficits,soilmoistureandleafthickness.

TovisualizetheneteffectofinteractionsweshowthefamilyofcurvesthatsolveforvforarangeofCandrvalues.Forthiscasewithconstant,vmax,weseethathigherresistanceimpedesthediffusionofCtotheinterioroftheleaf,causingadeficitinC,whichisfollowedbyareductioninthedemand,viav(C),untilanewbalanceismet.Astheresistance,r,getssmallesttheresponsecurvebetweenVandCapproachthelimitsetbytheMichaelis-Mentenfunction.

Others, such as Marshall and Biscoe (1980) have used polynomial equations (2nd order) to model the response of photosynthesis, then use the quadratic equation to extract values of P.

{ }1/22max max max

1 ( ) 42

P I P I P I Pα α θαθ

⎡ ⎤= ⋅ + − ⋅ + − ⋅ ⋅⎣ ⎦

2( ) ( )( ) 0m mA R I A A R IAθ φ φ+ − + + + =

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Farquhar-von Caemmerer-Berry Model Modern models couple equations for leaf photosynthesis, stomatal conductance and dark and photorespiration to form a mechanistic tool for examining environmental control and feedbacks. The biochemical equations for the carbon exchange processes are taken from Farquhar et al., (1980) Leaf photosynthesis (A) is a function of the rates of carboxylation (Vc), oxygenation (Vo, photorespiration) and dark respiration (Rd) (all have units of µmol m-

2 s-1).

*

(1 )dc o c dc

A= -0.5 - V RV V R CΓ

= − −

The term, c o-0.5V V , is evaluated as the minimum value between Wc, the rate of carboxylation when ribulose bisphosphate (RuBP) is saturated, and Wj, the carboxylation rate when RuBP regeneration is limited by electron transport. The term: Vc - 0.5 Vo is expressed by Farquhar et al. (1980) as:

*

minc o c jc

-0.5 = [ , ](1- )V V W WCΓ

The variable, *Γ , is the photosynthetic compensation point in the absence of dark respiration (mol mol-1) . It occurs when 2c oV V= The photosynthetic compensation point is distinct from the CO2 compensation point which occurs when carboxylation is in balance with photo- and dark respiration, 2c o dV V R= +

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Technically, the equations are derived in terms of the chloroplast CO2 concentation, Cc, because they were developed for photosynthetic material in vitro. In practice, it is common to evaluate these equations in terms of Ci , the intercellular CO2 concentration (mol mol-1). Estimates of Ci, implicitly assume that the mesophyll conductance, gm, is infinite

c im

AC Cg

= − ; this topic will be explored more below.

If Wc is minimal, then: maxC cc

2cc

o

V CW = [ ]O+ (1+ )C KK

* *max0.5 C c

c o c2c

cco

( - )V CV V W (1- )= [ ]OC + (1+ )C KK

Γ Γ− =

In this case VCmax is the maximum carboxylation rate when RuBP is saturated and Ko and Kc are the Michaelis-Menten coefficients for O2 and CO2.. In principle, the M-M coefficients are the concentration at one-half the maximum rate velocity. And leaf photosynthesis, A, is defined as:

[CO2] (ppm)

0 200 400 600 800 1000

Car

boxy

latio

n ra

te (µ

mol

m-2

s-1

)

0

10

20

30

40

50

60

Ci

supply~stomatal conductance (gs)

Wc : demand limited by RUBISCO saturation

Wj : demand limited by RuBP regenerationby electron transport

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*

maxc

C d2

cco

( - )CA RV [ ]O+ (1+ )C KK

Γ= −

If Wj is minimal, then:

cj

c

JCW =4 +8C Γ

* *

*0.5 cc o j

c c

J( - )CV V W (1- )=4 +8C C

Γ Γ− =

Γ

And photosynthesis is computed as:

*

*4 8c

dc

CA J RC−Γ

= −+ Γ

Here J is the potential rate of electron transport. J is evaluated as a function of incident. photosynthetic photon flux density (I). Users must be careful because some equations evaluate J in terms of moles of electrons (µmol e- m-2 s-1) and others evaluate J in terms of moles of CO2. In principle 4 electrons must be absorbed for each CO2 molecule. Consequently, the maximum quantum yield in terms of electrons is equal to 4 times the maximum quantum yield in terms of CO2 (Harley and Tenhunen 1991; Long and Bernacchi, 2003). We can prove that 4 electrons are needed to fix one CO2 molecule by the coupled redox reaction Oxidation: 2 H

2O -> O

2 + 4 e- + 4 H+

Electron Donor: -78 kJ/e- x 4 e- = -312 kJ/mole Reduction: CO

2 + 4H+ + 4 e- -> CH

2O + H

2O

Electron Acceptor: 47 kJ/e- x 4 e- = 188 kJ/mole Net: CO

2 + H

2O -> CH

2O + O

2

One function for computing J is (Harley and Tenhunen 1991):

12

2 2

2max

1

IJI

J

α

α=

+

The variable, α, is the quantum yield (mol e- mol-1 quanta) and Jmax is the maximum rate of electron transport. Other investigators use a rectangular hyperbola to model J (Collatz et al., 1991; Voncaemmerer and Farquhar, 1981).

22 max 2 max 2 max( ) 4

2psII

psII

Q J Q J Q JJ

θ

θ

+ − + −=

psIIθ is a curvature coefficient for PSII

2 ,psII maxQ Iαφ β=

I is the photon flux density of visible light, α is absorptance, ,psII maxφ is the quantum yield for PSII and β is the fraction of light reaching PSII (0.5) Analytical Model for Leaf Photosynthesis To derive an analytical equation for photosynthesis we must couple the biochemical equations for A with the diffusion equations (Baldocchi, 1994). A simple conductance relation is employed to express Ci:

i ss

A= -C Cg

where Cs is the surface CO2 concentration and gs is stomatal conductance (mol m-2 s-1). Stomatal conductance can be computed with the algorithm of Collatz et al. (1991) which couples gs to leaf photosynthesis and relative humidity.

0ss

m A rh= + ggC

The coefficient m is a dimensionless slope, rh is relative humidity at the leaf surface, g0 is the zero intercept, and A (µmol m-2 s-1) is leaf photosynthesis. Finally, the system of equations and unknowns, for computing leaf photosynthesis, is closed by expressing the CO2 concentration at the leaf's surface (Cs) in terms of the atmosphere's CO2

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concentration (Ca) and the conductance across the laminar boundary layer of a leaf (gb):

s ab

A= -C Cg

The variables, Ca and gb, are external inputs to the leaf biochemistry model and are determined from the micrometeorology of the canopy. Either numerical or analytical solutions (Baldocchi, 1994) for the coupled leaf photosynthesis-stomatal conductance model can be used to compute these fluxes.

Table 1 List of equations and unknowns for solving coupled photosynthesis-stomatal conductance model (after Baldocchi, 1994)

Unknowns Equation

Cs a

b

A-Cg

Ci s

s

A-Cg

gs 0

s

m A rh + gC

A+Rd *

minc o c ji

-0.5 = [ , ](1- )V V W WCΓ

Wj,Wc i

i

a - adCe +bC

Algebraic manipulations of five equations and unknowns yield a cubic equation for photosynthesis (A):

3 2 ( ) ( )

0

d d da

dd

a a

be A A e b a e R A e a ad eR R bC

R bada eRC C

γα β θ α α γ β θ β θ

γγγ γ

+ + − + + + − + + +

− + + + =

The undefined coefficients are:

1b

b m rhg

α = + − ⋅

( 2 )a b bC g m rh b gβ = ⋅ ⋅ − −

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and

( )bg m rh bθ = ⋅ ⋅ −

A numerical solution to the cubic equation was used (Press et al, Numerical Recipes in C). In class I share the matlab code that couples leaf photosynthesis, stomatal conductance and leaf energy balance.

SubModels and Their Parameters CO2 Compensation The CO2 compensation point in the absence of mitochondrial respiration is a function of the oxygen content of the atmosphere, kinetic coefficients for oxygen and CO2 and the maximum oxygenation velocity, associated with photorespiration. It can be defined from

*maxC c

c o c2

cco

( - )V C-0.5 =W =V V [ ]O+ (1+ )C KK

Γ

when Cc and Γ∗ are equal, forcing 2 c oV V=

* max 2 2

max

0.5 [ ] 0.5o c

c O

V K O OV K S⋅ ⋅

Γ = =

where S is the specificity factor:

max

max

c o

o c

V KSV K

=.

Typical values for the photosynthetic compensation point range between 35 and 40 ppm. The specificity factor, S, arises from the ratio of the rates of carboxylation and oxygenation, Vc/Vo.

2

2

[ ][ ]

c

o

V COSV O

=

15

The ratio between the activities of oxygenation and carboxylation are defined as:

,max 2

,max 2

[ ][ ]

oo c

c c o

VV K OV V K CO

=

which is derived by manipulating Vo and Vc and dividing one by one another

,max 2 ,max 2

2 2 2 2

[ ] [ ] /[ ] (1 [ ] / ) [ ] / 1 [ ] / )

o o oo

o c o c

V O V O KV

O K CO K O K CO K= =

+ + + +

,max 2 ,max 2

2 2 2 2

[ ] [ ] /[ ] (1 [ ] / ) [ ] / 1 [ ] /

c c cc

c o o c

V CO V CO KV

CO K O K O K CO K= =

+ + + +

In vitro data by Brooks, Farquhar and colleagues indicate that

,max

,max

0.21o

c

VV

=

The Specificity Factor controls the partitioning of carbon between the carboxylase and oxygenase functions of Rubisco. The specificity factor is a function of temperature and the solubilities of oxygen and carbon dioxide. It ranges between about 80 and 110 at 25C. For Quercus robur, its value is 102.33 at 25C. From (Balaguer et al., 1996) species S Spinacia oleracea 82-94 Nicotiana tabacum 51-104 Castanea sativa 50 Fagus sylvatica 49 Syringa vlugaris 83-88 Quercus robur 102 Quercus ilex 93 The concentrations of CO2 and O2 refer to the liquid phase. If we are to use ambient air concentrations we must account for the solubility of CO2 and O2 in the chloroplast, which requires us to know something about the Henry’s Law coefficients. The Henry’s constant for CO2 (KH,CO2) at 22C is about 0.03636 mol l-1 bar-1. Its value for O2 is 0.00125 mol l-1 bar-1 at 22.5C. The temperature dependents of the Henry’s law coefficients are, according to Roupsard et al. (Roupsard et al., 1996)

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3 3 6 2 6 3

, 2 ( ) 78.5 10 2.89 10 54.7 10 0.417 10H COK T T T T− − − −= ⋅ − ⋅ + ⋅ − ⋅

3 6 6 2 9 3, 2 ( ) 2.1 10 57.1 10 1.0247 10 7.503 10H OK T T T T− − − −= ⋅ − ⋅ + ⋅ − ⋅

The following figure produces information on the temperature dependency of the Henry’s Law partitioning coefficients. When deducing parameter values from the literature, one has to be careful and assess whether or not appropriate corrections have been made. For example, Harley et al report a Specificity coefficient of τ equal to 2904, which is much larger than the specificity factors already reported, that are on the order of 100. In practice τ is equal to:

, 2

, 2

28.38 102.33H CO

H O

KS

Kτ = ≈ ⋅

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Figure 1 Temperature dependence of rate coefficients for photosynthesis

Temperature dependencies One of the premises of biometeorology is the need to estimate the temperature of the organ because so many constituent processes have highly non-linear responses to temperature. The dependency of many key photosynthesis parameters is the topic of this section. The parameters Jmax, Vcmax, Ko, Kc, Rd are all non-linear functions of temperature, separate from the Henry’s law solubility effects discussed above. Many follow the exponential function according to the activation energy of the enzymes. But others become rate limited due to deactivation. In recent years Bernacchi and colleagues (Bernacchi et al., 2003; Bernacchi et al., 2001) revisited the temperature functions

Temperature (C)

0 10 20 30 40 50

Part

ition

ing

Coe

ffici

ent (

mol

l-1 b

ar-1

)

0.0001

0.001

0.01

0.1

Temperature (C)

0 10 20 30 40 50

K C

O2/

K O

2

22242628303234363840

CO2

O2

18

because they were originally derived for in vitro plant material and for a narrow range of temperature. The newer functions are for in vivo plant material and for temperature ranges between 10 to 40 C. The general functional form of the temperature response function for photosynthetic processes is given as:

( ) exp( )aHf T cRTΔ

= −

c is a dimensionless coefficient and ΔHa is the change in activation energy. R is the universal gas constant and T is absolute temperature. Coefficients are listed in Table 2 for all the parameters listed above. Table 2 coefficients for temperature functions of Bernacchi, for tobacco

f(T298) c Ha kJ mol

-1

Jmax

1 17.7 43.9

Vcmax

1 26.35 65.33

Vomax

1 22.98 60.11

*Γ 42.75 19.02 37.83

Kc 404.9 38.05 79.43

Ko

278.4 20.3 36.3

19

Rd

1 18.72 46.39

If the temperature response function decreases with temperature an alternate equation is ues:

298

exp( )( ) ( )

1 exp( )

a

K

dK

K

HcRTf T f T HsTRT

Δ−

=Δ

+ Δ −

In the past, the coefficients for Jmax, VCmax and Γ, KO2, KC and Rd were assessed with functions that derive from Arrhenius equation

298( 298)( ) exp298a k

kK

E Tf T kR T

⎡ ⎤−= ⎢ ⎥⋅ ⋅⎣ ⎦

This equation predicts how the rate of enzyme kinetics increases with temperature⎯either increase exponentially or its rate peaks at a given temperature. One commonly used temperature function used for Jmax and VCmax peaks at an optimal temperature, Topt , and follows a Boltzmann distribution (Harley and Tenhunen 1991):

( )exp ( ) / ( )( )

)1 exp

a l opt l opt

l d

l

E T T R T Tf T

S T HR T

⋅ − ⋅ ⋅=

⎛ ⎞Δ ⋅ −Δ+ ⎜ ⎟⋅⎝ ⎠

Ea is the activation energy, R is the universal gas constant, Tl is leaf temperature and Topt is the optimum temperature. The terms ΔHd and ΔS represent changes in enthalpy and entropy. ΔHd is also considered as term for deactivation of energy. These equations and a set of model parameters have been reviewed recently by Medlyn et al. (Medlyn et al., 2002).

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Table 3 Model coefficients for the Arrhenius equation as applied to Vcmax; surveyed by Medlyn et al. 2000

species K298 Ea (kJ mol-1) Glycine max 97.76 54.08 Gossypium hirustum 91.48 93.56 Acer pseudoplantanus 72.96 33.92 Fraxinus regia 77.97 50.61 Juglans regia 62.1 43.98 Prunus persica 65.05 73.74 Quercus petraea 79.5 56.28 Abies alba 41.64 35.16 Pinus pinaster 89.98 62.22 Pinus sylvestris 53.99 35.53 Pinus taeda 57.05 60.88 The functional behavior of the Bolzmann curve is shown below. While H and S seem conservative, from our literature search, the relation between s, Topt and H is quite important and cannot be chosen arbitrarily if one desired to attain a prescribed functional shape. A curve with an extreme peak occurs when S Topt exceeds H and is reduced as S Topt is less than H

21

Figure 2 Functional relation between the Boltzmann distribution for photosynthesis and temperature

Carboxylation Velocity

Vcmax is the maximum capacity for RuBP carboxylation and is a function of leaf nitrogen content. Jeff Amthor models this variable as a function of leaf nitrogen content:

Vcmax = 0.00125 kT k2 frubisco Nleaf kt is a temperature coefficient, k2 is the catalytic constant for RuBP carboxylation, f is the fraction of N that is in RuBISCO; typical values range between 0.087 and 0.23 . Vcmax can be determined from gas exchange as the initial slope of the A-C curve

max* (1 / )

c

c o

VdAdC K O K

=Γ + +

In 1993, Stan Wullschleger (Wullschleger, 1993) produced one of the first global syntheses of leaf photosynthesis and found a strong relation between Vcmax and Jmax. Information of this type have been a godsend and have led to a more universal

f(T)=exp(c - Hd/(R T))/(1+exp(S T- Ha/(R T)))

Tk

260 270 280 290 300 310 320 330

f(T)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

S = 650Ha=201,000

S = 650Hd = 200,000

S = 650Hd = 202000

22

application of leaf photosynthesis models. It showed us that we don’t have to worry so much about particularly parameters for each and every species, but that model parameters are self-constrained.

The survey by Wullschleger showed that model parameters could be grouped and ranked according to different functional types.

plant Vcmax range Jmax Range crops dicots 90 29-194 171 87-329 monocots 68 35-108 157 87-229 hort crops fruit trees 37 11-69 82 29-148 vegetables 59 15-97 137 40-290 temperate forest hardwood 47 11-119 104 29-237 conifer 25 6-46 40 17-121 tropical forest 51 9-126 107 30-222 understory herb 66 11-148 149 31-269 desert annual 153 91-186 306 264-372 sclerophyollous shrub

53 35-71 122 94-167

Recently Kattge (Kattge et al., 2009) produced a newer summary of model coefficients by examining the TRY database http://www.try-db.org/index.php?n=Site.Database

23

Vcmax@25C Na

mmolm-2s-1 gm

-2

TropicalTrees(oxisols) 29+/-7.7 2.17+/-0.8

TropicalTrees(nonoxisols) 41+/-15.1 1.41+/-0.56

TemperateBroadleavedEvergreen 61.4+/-27.7 1.87+/-0.93

TemperateBroadleavedDeciduous 57.7+/-21.2 1.74+/-0.71

Evergreenconiferous 62.5+/-24.7 3.10+/-1.35

Evergreendeciduous 39.1+/-11.7 1.81+/-0.64

Evergreenshrubs 61.7+/-24.6 2.03+/-1.05

Deciduousshurbs 54.0+/-14.5 1.69+/-0.62

C3herbaceous 78.2+/-31.1 1.75+/-0.76

C3crops 100.7+/-36.6 1.62+/-0.61

Nitrogen is a major component of RUBP. Many scientists have reported tight correlations between leaf N and Vcmax. We have also learned that resources are optimized in a canopy so leaf N and Vcmax vary with depth in a canopy. Quantifying these relationships have been the subject of much work by ecophysiologists over the years.

24

Figure 3 Relation between Vcmax and leaf N on a mass per unit area basis. The data are from Wilson et al (2000) and represent data from oak, maple and sweet gum.

Na (g m-2)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Vcm

ax (µ

mol

m-2

s-1

)

0

20

40

60

80

25

Figure 4 Field data showing the relation between Jmax and Vcmax by vertically sampling leaves in a forest, as N changes with depth

In practice it is difficult to measure many A-C over the course of a day, hence statistical representativeness is limited, as is the ability to sample leaves from an ecosystem with a large number of species. Several investigators have shown a reasonable relationship between Vcmax and maximum photosynthesis for multi-specied forest and grassland ecosystems (Wilson et al., 2000; Wohlfahrt et al., 1999).

J max

µm

ol e

- m-2

s-1

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

Acer rubrum

albaQuercus

Vc max µmol m-2 s-1

(data from Harley and Baldocchi, 1995and Wullschleger, 1993)

Walker Branch Watershed

26

Figure 5 Amax vs Jmax. Data of Kell Wilson, NOAA

As data sets grow, we are learning that photosynthetic model parameters are not static, but experience seasonality. One of the first data sets to show this was taken by my former postdoc Kell Wilson (Wilson et al., 2000).

Figure 6 Seasonal variation in Vcmax, (Wilson et al., 2000)

Data of KB Wilson

Amax

0 3 6 9 12 15 18 21 24

Vcm

ax

0

10

20

30

40

50

60

70

80

90

100

b[0] -2.6111018578b[1] 4.7224835061r ≤ 0.8172445191

Day of year

100 125 150 175 200 225 250 275 300 325

V cmax

(µm

ol m

-2 s

-1)

0

10

20

30

40

50

60

70

White oak

27

Our recent data shows how Vcmax varies with soil moisture, as measured with pre-dawn water potential (Baldocchi and Xu, 2007). These data lead to our conclusion that the Ball-Berry coefficient of the stomatal conductance model is constant and it is Vcmax that varies with soil water deficits. These data are consistent with theoretical developments by Katul et al, who relate Vcmax and soil water potential (Katul et al., 2003).

Quercus douglasii

Pre-dawn Water Potential (MPa)

-7 -6 -5 -4 -3 -2 -1 0

Vcm

ax

0

20

40

60

80

100

120

140

b[0]: 111.3b[1]: 22.69b[2]: 1.734r ≤ 0.7532

Mesophyll Conductance and Chloroplast CO2 Commercial gas exchange cuvette systems evaluate the internal CO2 concentration, Ci, not the Chloroplast CO2, the parameter on which the original Farquhar-von Cammerer model is based; remember this equation was generated on the basis of photosynthesis of in vitro chloroplasts (von Caemmerer, 2000). Out of convenience it is often assumed that the mesophyll conductance is infinite and that Cc and Ci are nearly equivalent. Workers such as Loreto and Evans (Evans and Loreto, 2000; Evans et al., 1986) and Ethier and Livingston (Ethier and Livingston, 2004) have challenged conventional wisdom and have evaluated the values of gm, Cc and examined their effect on computing Vcmax. The mesophyll conductance involves CO2 diffusion through the leaf mesophyll. This path includes diffusion through intercellular air spaces, through the cell wall and the intracellular liquid pathway (Flexas et al., 2007b) and is equivalent to the term used in some papers, internal conductance, gi.

28

There are two general methods commonly used to estimate mesophyll conductance and chloroplast CO2 concentration. One method is based on measuring stable isotope concentration in the leaf and the other is based on fluorescence measurements of the electron transport rate (Evans and Loreto, 2000; Evans et al., 1986; Flexas et al., 2007a; Harley et al., 1992; Loreto et al., 1992; Warren, 2006). There have also been efforts to estimate mesophyll conductance via curve-fitting of the A-C curve (Ethier and Livingston, 2004). First question is how to assess gm and Cc. This is a difficult and complex task that needs additional information to close a system of equations. To recap, let’s start with the set of Fickian equations for the diffusion of CO2 from the free atmosphere, through the leaf boundary layer, the stomata and the mesophyll to the chloroplast: A = gb(Ca - Cs) = gs(Cs - Ci) = gm(Ci – Cc) = min(Wj(Cc), Wc(Cc)). By measuring leaf photosynthesis and estimating the CO2 concentration in the intercellular space and at the chloroplast one can estimate mesophyll conductance, gm:

mi c

AgC C

=−

Problem is we need an independent measure of Cc, which often depends on knowing gm. To get around this closure problem, more unknowns than equations, we can invoke and utilize the standard equation for electron transport, as is typically measured with fluorescence methods:

*

*

( )(4 8 )c

c

A R CJC

+ + Γ=

−Γ

Returning to Fick’s Law where we express mesophyll conductance as a function of photosynthesis, A, and the internal and chloroplast CO2 concentrations

mi c

AgC C

=−

we can also solve for the chloroplast concentration Cc:

c im

AC Cg

= −

And substitute it into the equation for J, producing:

29

*

*

4( )(4 ) 8 )im

im

AA R CgJ AC

g

+ − + Γ=

− −Γ

Next we solve for mesophyll conductance in terms of Ci, rather than Cc, through a number of algebraic steps.

* *4( ) ( )(4 ) 8 )i im m

A AJ C A R Cg g

− −Γ = + − + Γ

* *4( )4 ( ) ( )8 )i im m

A AJC J J A R C A R A Rg g

− − Γ = + − + + + Γ

Re-arrange the equation so gm is on one side, together:

* * 4( )8 ) ( )4 ( )i im m

A AJC J A R A R C A R Jg g

− Γ − + Γ − + = − + +

* * * ( 4 4 ))( ( )4) 8 8i

m

A J A RC J A R J A Rg

− −− + − Γ − Γ − Γ =

* * *

( 4 4 ))( 4 4 ) 8 8mi

A J A RgC J A R J A R

− −=

− − − Γ − Γ − Γ

Factor out 4 4J A R− −

*( 8 8 )( 4 4 )

m

i

AgJ A RCJ A R

=Γ + +

−− −

Here the extra unknown is J, the rate of whole chain electron transport. The fluorescence method provides a direct measurement of J, whereas A can be assessed from gas exchange measurements using a cuvette and Ci can be deduced measuring stomatal conductance, via leaf transpiration, with the same cuvette system. Based on the method of Genty et al. 1989, the quantum yield of photosystem II can be assessed from fluorescence as:

30

'' '

mpsII

m m

F F FF F

φ− Δ

= =

Where Fm’ is the maximum rate of fluorescence and F is the steady state rate. From these measurements, we can assess the rate of whole chain electron transport as f psII pJ Qφ α β= ⋅ ⋅ ⋅

Qp is photon flux density, α is absorptance and β is the fraction of photons that reach PSII (0.5). Measurements of the stable carbon isotope 13C enables us to deduce mesophyll conductance using diffusion theory, information on the relationship between Ci/Ca and isotope discrimination due to photosynthesis (see Evans and Loreto, 2000). Isotope discrimination is described according to a series of fractionations as air diffuses through the boundary layer, the stomata, internal mesophyll and reacts with enzymes (Warren, 2006):

*/

a s s i i cb i

a a a

c

a a

p p p p p pa a ap p p

p eR k fbp p

− − −Δ = + +

+ Γ+ −

The isotope content is described in delta notation as 1a

p

RR

Δ = − , where Ra and Rp are the

molar ratios of 13 122 2/CO CO in the air and plant. We can also substitute terms with

i cm

Ap pg

− =

*/

a s s ib i

a a m a

c

a a

p p p p Aa a ap p g p

p eR k fbp p

− −Δ = + +

+ Γ+ −

ab=2.9‰

31

a = 4.4‰ ai=1.8‰ b=27-30‰ Ignoring isotope fractionation through the laminar boundary layer produces

*/

a ii

a m a

c

a a

p p Aa ap g p

p eR k fbp p

−Δ = +

+ Γ+ −

and We can also estimate an isotope fractionation for the conditions that mesophyll conductance is infinite:

( ) ( )a i i ii

a a a

a p p p pb a b ap p p−

Δ = + = + −

Next we take the difference iΔ−Δ and develop a function that relates gm and A.

* ( )/

a ii i

a m a

c a i i

a a a a

p p Aa ap g p

p a p p peR k fb bp p p p

−Δ −Δ = +

−+ Γ+ − − −

Simplifying further

*( ) /

i im a

c i

a a

Aag p

p p eR k fbp p

Δ −Δ =

− + Γ+ −

and

*/

i im a

m a a

Aag peR k fAb

g p p

Δ −Δ =

+ Γ− −

and

32

*/( )i im a a

eR k fAa bg p p

+ ΓΔ −Δ = − −

The slope of the relation between iΔ −Δ and a

Ap

is proportional to 1/gm.

Figure 7 von Caemmerer and Evans 1992

A third way to assess JA is to measure photosynthesis under low O2, when little RUBP is consumed by photosynthetic carbon oxidation, PCO, or photorespiration (Loreto et al., 1992). How important is gm? The following figure shows how mesophyll conductance scales with Ci-Cc (Warren, 2006).

33

Warren and Abrams (2006 PCE)

Ci - Cc

0 20 40 60 80 100 120 140 160

g m (m

ol m

-2 s

-1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

b[0] 0.6888b[1] -8.274e-3b[2] 2.5305e-5r ≤ 0.650

And a recent survey by Flexas et al (Flexas et al., 2007b) clearly indicates that mesophyll conductance is finite.

34

Why must we be concerned about a finite gm? Ethier and Livingston (2004) investigated the impact of Cc and gm on the assessment of Vcmax. They report that assuming gm is infinite underestimate Vcmax. First they demonstrate that gm is not infinite by surveying the literature and reporting a range of values between 0.02 and 0.5 mol m-2 s-1

. We can also deduce from the equation defining Vc max how differences in Cc or Ci will affect its final version. Remember we estimate Vcmax from:

max

(1 / )c

c o

VdAdC K O K

=Γ + +

If Cc < Ci for a given A, the initial slope of dA/dCc will be greater than dA/dCi. For example let’s assume Cc is 0.95 Ci, Vcmax will be 5% greater if computed with Cc rather than Ci. One Dimensional Models, the Need to Assess Cc vs Ci One dimensional Diffusion model of Terrashima et al.

2

2

( ) ( )c xD A xx

ρτ

∂=

∂

is a function of tortuosity (τ), volumetric fraction of intercellular space (ρ) or mesophyll porosity. D is the binary diffusivity.

2

( ) ( ( ) ( ))[ ]( ( ) )

w i c

v i c d

A x g s C x C xRubp k C x k R

φ φ

φ

= −

= − −

Analytical solution to Ci

3 1 2( ) exp( ) exp( )iC x k C x C xα α= + + − Phi is defined as the Bunsen coefficient (?? Henry’s Law coefficient?? 0.829 at 25 C) DARK RESPIRATION.

35

Respiration provides energy for metabolism and synthesis. At the leaf level, Collatz et al. (1991) and Amthor (1994) model dark respiration as a function of VCmax.─a typical value being Rd equals 0.015 times VCmax. Their assumption implies that Rd is a function of leaf nitrogen. Conceptual the rate of autotrophic respiration can be viewed as a function of the sum of constituent processes (Amthor ): a b I N T MR r B r I r N r T r M= + + + +

where Bis the rate of biosynthesis of new tissue, I is the rate of active ion uptake by roots, N is the rate of nitrogen assimilation, T is the rate of translocation of carbohydrates, amino acids and T is translocation due to phoelem loading. The r factors involve the cost Gifford (Gifford, 2003)) reports on new data by Loreto using 13C to examine Kok effect. Finds that autotrophic respiration of mitochondria are same day and night, different from old results of Laisk and Kok, showing day respiration is less than night respiration. Inside respired respiration was refixed in mesophyll. Dark respiration is divided into growth and maintenance components, which are functions of ongoing photosynthesis and total biomass (W). • Rdark = Rgrowth + R main = a Ps + bW (McCree, 1970) Collatz et al. (1990) models it as a function of Vcmax, which implicitly means it (Rd) is a function of leaf nitrogen, as explicitly modelled by Amthor (1994). • Rmain=k1 k2(Tleaf) Nleaf Bibliography

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39

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