Optimal Allocation of Leaf-Level Nitrogen: Implications ...ramirez/ce_old/projects/Optimal... ·...

Optimal Allocation of Leaf-Level Nitrogen:

Implications for Covariation of Vcmax and Jmax and

Photosynthetic Downregulation

J.A. Quebbeman,1 J.A. Ramirez1

Corresponding author: J.A. Quebbeman, Department of Civil and Environmental Engineer-

ing, Colorado State University, Campus Delivery 1372, Fort Collins, CO 80523-1372, USA

([email protected])

1Department of Civil and Environmental

Engineering, Colorado State University,

Fort Collins, Colorado, USA.

This article has been accepted for publication and undergone full peer review but has not been through

the copyediting, typesetting, pagination and proofreading process, which may lead to di↵erences

between this version and the Version of Record. Please cite this article as doi: 10.1002/2016JG003473

c�2016 American Geophysical Union. All Rights Reserved.

Abstract. The maximum rate of carboxylation, Vcmax, and the maximum

rate of electron transport, Jmax, describe leaf-level capacities of the photo-

synthetic system and are critical in determining the net fluxes of carbon diox-

ide and water vapor in the terrestrial biosphere. Although both Vcmax and

Jmax exhibit high spatial and temporal variability, most descriptions of pho-

tosynthesis in terrestrial biosphere models assume constant values for Vcmax

and Jmax at a reference temperature ignoring intra-seasonal, inter-annual,

and water stress-induced variations. Although general patterns of variation

of Vcmax and Jmax have been correlated across groups of species, climates,

and nitrogen concentrations, scant theoretical support has been provided to

explain these variations. We present a new approach to determine Vcmax and

Jmax based on the assumption that a limited amount of leaf nitrogen is al-

located optimally among the various components of the photosynthetic sys-

tem in such a way that expected carbon assimilation is maximized. The op-

timal allocation is constrained by available nitrogen, and responds dynam-

ically to the near-term environmental conditions of light and water supply

and to their variability. The resulting optimal allocations of a finite supply

of nitrogen replicate observed relationships in nature, including the ratio of

Jmax/Vcmax, the relationship of leaf nitrogen to Vcmax, and the changes in ni-

trogen allocation under varying water availability and light environments.

This optimal allocation approach provides a mechanism to describe the re-

sponse of leaf-level photosynthetic capacity to varying environmental and re-

source supply conditions that can be incorporated into terrestrial biosphere


models providing improved estimates of carbon and water fluxes in the soil-

plant-atmosphere-continuum.


1. Introduction

Photosynthesis uses atmospheric carbon dioxide (CO2), water, nitrogen, and other

molecules to transform solar energy (i.e., from photons) into chemical energy (i.e., glucose,

ATP, and other forms of chemical energy), a process also termed carbon assimilation.

Photosynthesis occurs in the leaf chloroplasts and involves light-dependent and light-

independent biochemical reactions. The former reactions produce O2, ATP and NADPH

and are constrained by the maximum rate of electron transport, Jmax (µmol e� m�2s�1),

whereas the latter reactions use the products of the light-dependent reactions to produce

glucose from CO2, and are constrained by the maximum rate of carboxylation, Vcmax

(µmol CO2 m�2s�1).

CO2 enters the leaf through stomata, small pores on the epidermis of leaves (most

abundant on the abaxial side) whose aperture varies depending on environmental and

biochemical factors. The flux of CO2 can be estimated as the product of the stomatal

conductance and the concentration gradient of CO2 between the leaf pore spaces and

the atmosphere, assuming well-coupled conditions between the leaf and the above-canopy

environment. However, the rate of carbon assimilation depends also on the biochemical

capacity of the photosynthetic system defined by Vcmax and Jmax.

Carboxylation includes the reaction of CO2 with Ribulose-1,5-Bisphosphate (RuBP),

which is catalyzed by the enzyme Ribulose-1,5-Bisphosphate Carboxylase-Oxygenase (Ru-

BisCO); Vcmax is the maximum rate of RuBP carboxylation. RuBP and RuBisCO are

essential in the light-independent reactions leading to the production of glucose (i.e., of

the Calvin-Benson cycle) [Nobel , 2009]. Jmax is the maximum rate of electron transport


in the light-dependent reactions that limits the supply of energy (i.e., ATP and NADPH)

for carboxylation and for regeneration of RuBP in the Calvin-Benson cycle. Therefore,

both Vcmax and Jmax are critical in determining the rate of carbon assimilation.

The biochemical model of photosynthesis of Farquhar, von Caemmerer and Berry [Far-

quhar et al., 1980], hereafter referred to as the FvCB model, is widely employed to describe

C3 and C4 photosynthesis over a range of climates and species (see Appendix A for de-

tails about the FvCB model). Typically, Vcmax and Jmax are estimated by fitting the

FvCB model to gas-exchange measurements [Sharkey et al., 2007]. This has resulted in

a plethora of published values varying between species, climates and environmental con-

ditions [Wullschleger , 1993; Walker et al., 2014; Medlyn et al., 1999; Verheijen et al.,

2013; Leuning , 1997]. Furthermore, Vcmax and Jmax also vary with depth in the canopy

[Niinemets et al., 2004; Niinemets and Tenhunen, 1997], and through the season [Xu and

Baldocchi , 2003; Wilson et al., 2000; Onoda et al., 2005; Broeckx et al., 2014]. However, a

general approach for describing variable Vcmax and Jmax dependent on environmental and

biochemical conditions has not been developed. As a result, many photosynthesis models

currently used in terrestrial biosphere models (TBMs) either assume constant values for

Vcmax and Jmax normalized to a reference temperature (usually 25 deg.C, referenced as

Vcmax25 and Jmax25), or use simplified regression approaches[Rogers , 2014], which can lead

to inappropriate estimates of carbon and water fluxes in the biosphere.

We propose a new optimal allocation approach for C3 photosynthesis such that Vcmax

and Jmax vary with mean environmental and biochemical conditions so as to make optimal

use of a finite supply of resources (i.e., light, nitrogen and water). Optimal use is that


which maximizes the expected value of carbon assimilation over appropriate timescales

(e.g., seasonal, inter-annual, climatic).

2. Approach

2.1. Optimality

Many optimality hypotheses describing photosynthetic responses to varying environ-

mental conditions have been proposed and studied for well over 50-years, including optimal

stomatal responses [Cowan and Farquhar , 1977; Lin, 2015], optimal nitrogen partitioning

in a canopy [Sands , 1995; Dewar et al., 2012; Chen et al., 1993], optimal partitioning of

leaf level nitrogen [Hikosaka and Hirose, 1997; Prentice et al., 2014], optimal leaf area

indices [Anten et al., 1995], and mutually dependent (i.e., coordinated) optimality of

plant hydraulic traits (i.e., water transport) and photosynthesis [Peltoniemi et al., 2012;

Manzoni et al., 2014].

We consider a novel optimal allocation approach for partitioning leaf-level nitrogen

between the two photosynthetic sub-systems associated with light-dependent and light-

independent reactions that maximizes the expected value of assimilation, and leads to

optimal values for carboxylation and electron transport capacities for a given variable light

environment and mean environmental conditions. As opposed to previous approaches that

either fix the magnitude of one of the systems at a reference temperature, or constrain

the ratio of Jmax25 to Vcmax25 , this approach allows both Vcmax and Jmax to vary in a

coordinated manner.

The photosynthetic system includes a series of inter-operating sub-systems where solar

energy is used for converting atmospheric carbon and water into active and stored forms of

energy [Farquhar et al., 1980; von Caemmerer , 2000]. This includes sub-systems for light


harvesting and electron transport, and for the development and maintenance of RuBisCO,

critical for carboxylation. Larger values of Vcmax or Jmax represent greater photosynthetic

capacities of each sub-system, although at a cost of increased requirements for limited

resources (primarily nitrogen) [Poorter et al., 2006].

Stomata adjust to environmental and biotic conditions on short time scales on the

order of minutes [Lange et al., 1971; Jarvis , 1976; Tardieu et al., 1996], whereas the leaf

photosynthetic components determining Vcmax and Jmax adjust seasonally and on longer

time scales [Onoda et al., 2005]. Approximately 2-5% of biomass proteins (e.g., RuBisCO,

etc.) are turned over and regenerated daily [Suzuki et al., 2001], which is approximately

100% turnover every 30 days. Therefore, a monthly time scale is more appropriate for

assessment of adaptation of Vcmax and Jmax. Using similar arguments about the half-life

time of RiBisCO, Maire et al., [2012] [Maire et al., 2012] also use a monthly time scale

for their analysis of photosynthetic coordination.

If most nitrogen resources were invested primarily in the development and maintenance

of RuBisCO, the leaf would have a relatively high maximum rate of carboxylation (Vcmax),

but limited resources remaining for investment in the electron transport system resulting

in a relatively low Jmax. Under this condition, the assimilation rate will remain limited

by Jmax regardless of irradiance level – clearly not an optimal allocation of nitrogen.

Conversely, the system could have a high value of Jmax, but no capacity for RuBP car-

boxylation with limited investment in RuBisCO. An optimal allocation is likely closer

to a state of co-limitation between both photosynthetic systems [von Caemmerer and

Farquhar , 1981]. We posit that there exists an optimal nitrogen allocation strategy that


maximizes expected assimilation over the indicated monthly time scale for variation of

Vcmax and Jmax.

For a given concentration of leaf-level organic nitrogen, Norg, we define the optimal

values of Vcmax and Jmax corresponding to the optimal allocation of organic nitrogen by

the following expression,

V ⇤cmax, J

⇤max = argmax

Z

t

An(Vcmax, Jmax, gs;⇥) dt | Norg (1)

where An is the net rate of assimilation (gross assimilation minus leaf respiration), gs

is the stomatal conductance, and ⇥ is a vector of exogenous time-dependent variables

including irradiance, temperature, soil moisture, and vapor pressure deficit. The period

of integration, t, corresponds to the time scale of variation of Vcmax and Jmax (e.g., a

month). In the above expression, Norg, Vcmax, and Jmax are constant for the given t, but

An, gs, and ⇥ vary continuously.

For a given period of integration, the values of Vcmax and Jmax that solve Eq.(1) also

maximize the expectation of An. Therefore, we can similarly restate the above as,

V ⇤cmax, J

⇤max = argmax

Z

⇥

f(⇥)

An(Vcmax, Jmax, gs;⇥) d⇥| Norg (2)

where f(⇥) is the joint probability density function (PDF) of variable(s) in ⇥ for the

given time period t.


In the solution presented below, only the irradiance is allowed to vary probabilistically,

while temperature, vapor pressure deficit, etc. vary deterministically. Therefore, Eq.(2)

becomes,

V ⇤cmax, J

⇤max = argmax

Z

I

f(I)

An(Vcmax, Jmax, gs; I, µT , µV PD) dI| Norg (3)

where f(I) is the PDF of irradiance, and µT and µV PD are the mean values of temperature

and vapor pressure deficit over the given time period t.

2.2. Nitrogen and Photosynthetic Capacity

Nitrogen is critical for development and maintenance of all components of the photosyn-

thetic system [Poorter et al., 1995] and is allocated to its various components as follows,

[Evans , 1989],

Norg = NP +NE +NR +NS +NO (4)

where, NP is the nitrogen invested in pigment proteins such as chlorophyll a and b, NE is

nitrogen allocated to the electron transport system including cytochrome f and coupling

factor, NR is the nitrogen allocated to RuBisCO, NS is nitrogen in soluble proteins other

than RuBisCO, and NO is additional organic leaf nitrogen not invested in photosynthetic

functions such as in cell walls (all terms in Eq.(4) are in units of mmol N m�2.)

In order to solve Eq.(1) through Eq.(3), constrained by leaf-level nitrogen, an ex-

plicit link relating leaf nitrogen concentration to Vcmax and Jmax is required. Several re-

searchers have determined these relationships for C3 species using observations [Hikosaka

and Terashima, 1995; Evans and Poorter , 2001].


Following Evans and Poorter, the non-soluble thylakoid nitrogen, equal to the sum

NP +NE, may be estimated as [Evans and Poorter , 2001; Zaehle and Friend , 2010],

NP +NE = 0.079Jmax + 0.0331� (5)

where Jmax is in (µmol e�) m�2 s�1, � is the concentration of chlorophyll per

unit area (µmol Chl m�2), 0.079 is in mmol N s (µmol e�)�1, and 0.0331 is in

mmol N (µmol Chl)�1.

The soluble nitrogen excluding that allocated to Rubisco is,

NS = ⌫Jmax (6)

where ⌫ ⇡ 0.3 (mmol N s (µmol e�)�1).

To make the connection between Vcmax and RuBisCO, we utilize relationships from

Niinemets et al. [Niinemets and Tenhunen, 1997; Niinemets et al., 1998],

NR = Vcmax/(6.25⇥ Vcr ⇥ ⇠) (7)

where Vcmax is in µmol CO2 m�2 s�1, Vcr is the specific activity of RuBisCO, that is, the

maximum rate of RuBP carboxylation per unit Rubisco

(⇡ 20.5 µmol CO2 (g RuBisCO)�1 s�1), 6.25 is grams RuBisCO per gram nitro-

gen in RuBisCO, and ⇠ is the mass in grams of one millimole of nitrogen equal to

0.014 gN (mmol N)�1.

Finally, the term NO is a constant of ’extra’ organic leaf nitrogen allocated to non-

photosynthetic resources, such as leaf cell walls.

Therefore, substituting Eqs.(5), (6), and (7) into Eq.(4), leads to,

Norg = 0.079Jmax + 0.0331� + Vcmax/(6.25Vcr⇠) + ⌫Jmax +NO (8)

which directly relates leaf organic nitrogen Norg to Vcmax and Jmax.c�2016 American Geophysical Union. All Rights Reserved.

2.3. Assimilation and Stomatal Control

We implement the FvCB model to determine the rate of net assimilation in Eqs.(1)

- (3) for a given Vcmax and Jmax, (see Appendix A for details about the FvCB model).

However, the FvCB model does not explicitly include stomatal function regulating the

internal concentration of CO2 and the supply of CO2 into the leaf. There are several mod-

els describing the relationship between assimilation and stomatal conductance, including

the Ball-Woodrow-Berry model Ball et al. [1987] and its variant the Leuning model [Le-

uning , 1995], as well as optimality based models including the seminal work of Farquhar

and Cowan [1977], and more recent adaptations of optimal stomatal control by Katul et

al.[2010] and Manzoni et al.[2011].

In order to determine the variations of stomatal conductance and the associated rate of

assimilation, we implement an optimality based approach similar to that of Manzoni et

al. [2011] (see Appendix B for the derivation of the optimal stomatal control model).

3. Results

3.1. Optimization

A closed form analytical solution of Eq.(3) is not feasible, therefore we implemented a

numerical procedure as described in detail in Appendix C. The maximum expected value

of assimilation is determined by using a gridded-search in the feasible decision space of

Vcmax and Jmax.

For a given Norg, distribution of irradiance, and set of mean environmental parameters,

the numerical procedure leads to a decision surface as shown in Figure 1. The figure

shows that there exists a unique pair of carboxylation and electron transport capacities


that maximize the expected assimilation; that is, there exists an optimal pair (V ⇤cmax,

J⇤max) that solves Eq.(3).

Suboptimal values occur for alternate allocations of nitrogen between NP , NR, NS and

NE. For example, holding the optimal value of Jmax, while increasing allocation of nitrogen

in chlorophyll (NP ) and concurrently decreasing nitrogen in RuBisCO (NR) to maintain a

constant Norg, would decrease the expected value of assimilation. Even as the absorptance

fraction (solid white lines) increases from higher levels of chlorophyll, the system becomes

increasingly carboxylation-limited due to lower investment of nitrogen in RuBisCO. This

condition is demonstrated in Figure 1 by moving along lines of constant Jmax in a direction

toward values of Vcmax lower than the optimal value.

3.2. Variable Leaf-Level Nitrogen

The process of finding optimal allocations was repeated for a number of leaf organic

nitrogen levels, holding all other parameters constant, resulting in an optimal response

function. The optimal Vcmax values over a typical range of observed nitrogen levels for C3

angiosperms were then compared to regressed relationships by Kattge et al.[2009], who

used a collection of observations from a number of di↵erent studies and plant functional

types. This comparison is shown in Figure 2.

For temperate broadleaf trees, shrubs and herbaceous groups, Figure 2 shows that

the optimal allocation approach leads to relationships between Norg and Vcmax that are

nearly identical to the relationships of Kattge et al.[2009], supporting the hypothesis

that nitrogen is allocated between various photosynthetic components to maximize net

expected assimilation.


3.3. Variable Light Environment

Variability in the characteristics of the light environment (i.e., changes in the distri-

bution of irradiance) may result from seasonal e↵ects [Bauerle et al., 2012], from canopy

closure [Wilson et al., 2000], or both.

For a given amount ofNorg, the optimal allocation approach for variable irradiance levels

leads to results as those shown in Figure 3. Reduction of irradiance leads to a reduction

in V ⇤cmax (and an increase in the ratio J⇤

max/V⇤cmax = !⇤), representing an optimal solution

with greater investment of organic nitrogen in chlorophyll a and b for light harvesting as

compared to that invested in carboxylation systems.

The predicted pattern of change in !⇤ shown in Figure 3 is similar to observations in

nature [Wilson et al., 2000], and is explained by the optimal allocation approach. Thus,

the optimal allocation of nitrogen between photosynthetic sub-systems also explains why

the ratio of J⇤max/V

⇤cmax, (i.e., !

⇤), is commonly observed to equal 2.1 +/- 0.6 across a

wide range of biomes and species [Wullschleger , 1993; Onoda et al., 2005; Leuning , 1997;

Kattge and Knorr , 2007; Bauerle et al., 2007; Medlyn et al., 1999].

3.4. Photosynthetic Down-Regulation during Prolonged Water Stress

During periods of low soil moisture supply (i.e., periods of moisture stress), stomata may

partially or fully close to prevent excessive negative leaf, shoot, or root water potentials

causing xylem embolism [Jones and Sutherland , 1991; Tyree and Sperry , 1989; Lopez

et al., 2013]. Closure of stomata reduces transpiration and the flux of CO2 into the leaf

resulting in inhibited assimilation – rates below the maximum potential for well-watered

conditions (i.e., no moisture stress). This available but unused photosynthetic capacity

incurs continued maintenance costs from respiration and protein re-synthesis.


Rather than maintaining this unused capacity during prolonged water stress, the pho-

tosynthetic system may be down-regulated (sometimes referred to as a non-stomatal re-

sponse), resulting in a reduction of Vcmax and Jmax from well-watered values [Flexas and

Medrano, 2002; Tezara et al., 1999]. This non-stomatal down-regulation process requires

modification of the photosynthetic system and redistribution of resources (e.g., nitrogen,

etc.). For a given Norg, light, and other environmental conditions, the corresponding re-

duction of Vcmax and Jmax from the optimum values should occur in a coordinated manner

that still maximizes the assimilation at reduced photosynthetic capacity.

In terms of Vcmax and Jmax, an optimal down-regulated state is where the reduction

in expected assimilation is minimum for a unit change in photosynthetic capacity (i.e.,

unit changes in Vcmax, Jmax), implying a reallocation of the maximum amount of Norg to

other biochemical uses. Mathematically, the optimal down-regulation path is the locus of

(Vcmax, Jmax), starting from the (V ⇤cmax, J

⇤max), satisfying the condition that the magnitude

of the gradient is minimum, that is, |rAn(Vcmax, Jmax)| = minimum.

Figure 4 shows a sample path of down-regulation on an assimilation decision surface.

The optimal down-regulation path represents a path of optimal reallocation of nitrogen –

this is also an allocation strategy that closely follows a state of co-limitation between the

photosynthetic systems. Chen et al.,[1993] proposed the Coordination Theory describ-

ing nitrogen allocation through a canopy where photosynthetic systems are co-limited

[Chen et al., 1993]. More recently, Maire et al., [2012] assessed 293 observations from 31

di↵erent species under a range of environmental conditions, and found that under mean

environmental conditions during the preceding month, RuBP carboxylation equaled RuBP


regeneration [Maire et al., 2012] further supporting this coordination. Our results further

support these ideas from an optimality perspective.

This path of down-regulation provides a direct link between the expected value of as-

similation, which is proportional to the expected value of stomatal conductance, and the

optimal values of V ⇤cmax and J⇤

max for a given set of environmental conditions. Therefore,

knowing the leaf-level Norg, the irradiance density distribution, the constrained expected

value of An (lower than the maximum well-watered capacity), and other mean environ-

mental conditions, the optimal allocation resulting in the stomatal-constrained values of

V ⇤cmax and J⇤

max can be found following the optimal down-regulation path.

However, a relationship between expected An (or equivalently gs) and soil moisture sta-

tus is required. Generally, these relationships can be estimated using linear-step functions

similar to maximum stomatal conductance responses [Tuzet et al., 2003; Farrior et al.,

2015], or any other available species-specific approach.

3.5. Optimal Allocation Down-Regulation against Observations

The optimal down-regulation procedure postulated here was tested against reported

seasonal values of Jmax and Vcmax in a summer semi-arid environment. We utilized data

reported by Xu and Baldocchi [2003], who studied an oak savanna system in California

exposed to extended summer dry periods. Their study provided detail of temperatures,

vapor pressure deficits, and soil moisture throughout the growing season. Additionally,

they measured stomatal conductance and carbon assimilation, which were used to deter-

mine the seasonally variable Vcmax and Jmax.


The study by Xu and Baldocchi represents a system whose moisture stress is slowly

increased through a season – a condition allowing photosynthetic system down-regulation

compared to solely regulating assimilation through stomatal control.

The path of down-regulation was determined by starting from the unstressed optimum

as postulated above and following the path of shallowest descent, resulting in optimal

down-regulated values of V ⇤cmax and J⇤

max. Along this path we then determine the asso-

ciated stomatal conductances and expected assimilation rates. Finally, using only the

stomatal conductance values reported over the season, the corresponding optimal values

of V ⇤cmax and J⇤

max were returned from the optimal path. Figure 5 compares the resulting

optimal values of V ⇤cmax and J⇤

max (without any calibration) to those reported by Xu and

Baldocchi [2003].

As desribed in Appendix C, estimates of leaf nitrogen concentration were obtained

from Xu and Baldocchi [2003], temperature adjustments of the Michaelis-Menten half-

rate constant, Km, were based on [Medlyn et al., 2002], and the density function of daily

irradiance was estimated from meteorologic records corresponding to this region in South-

ern California. As shown in Figure 5, the observed seasonal variation of Vcmax and Jmax

is well reproduced with the optimal down-regulated path from the maximum value of

assimilation.

The peak values of Vcmax and Jmax during leaf flushing (initial growth of leaves during

the spring) are not fully captured, and result most likely from increased leaf respiration

not included in the model during this period. Regardless, as assimilation and stomatal

conductance become limited during the summer due to soil moisture stress, the down-

regulation of the photosynthetic system is replicated. This result further supports the idea


of optimal allocation, and demonstrates how the optimal approach allows for a dynamic

seasonal response of photosynthetic capacity.

4. Summary and Concluding Remarks

We have introduced a new optimality based approach for considering leaf level parti-

tioning of nitrogen in various environments. Although many simplifications were made,

the optimal allocation of organic nitrogen between components of the photosynthetic sys-

tems as defined here results in photosynthetic system capacities as well as relationships

between those capacities and levels of organic nitrogen and irradiance that replicate those

observed in nature. This includes relationships between Norg and Vcmax, light environ-

ment and ! (i.e. Jmax/Vcmax), and the response of photosynthetic system capacity during

periods of down-regulation. Xu et.al. [Xu et al., 2012] also proposed a ”...mechanistic

nitrogen allocation model based on a trade-o↵ of nitrogen allocated between growth and

storage, and an optimization of nitrogen allocated among light capture, electron trans-

port, carboxylation, and respiration...”. Their approach assumed optimality conditions

under mean environmental conditions, but used a Metropolis-Hastings parameter fitting

approach. Here, we consider an optimality approach solely focused on the photosynthetic

system and using mechanistic relationships coupled with a general optimality condition

maximizing expected assimilation. In addition, our approach also allows for determina-

tion of optimality conditions during prolonged water stress (i.e., photosynthetic system

down-regulation), and does not require resorting to any fitting or calibration approach.

The optimal allocation approach presented here provides a new mechanism to incor-

porate dynamic responses of Vcmax and Jmax for C3 photosynthesis in TBMs. Accurate

representation of the photosynthetic system is paramount in coupled TBMs and global


circulation models with respect to carbon and water cycling processes, and critical to

these processes are the maximum capacities of electron transport and carboxylation. As-

suming constant values for these capacities, as is commonly done (e.g., at Jmax25 and

Vcmax25), may introduce significant accumulated errors in the carbon and water balances

over multi-decadal model projections [Rogers , 2014]. The new approach addresses these

issues by allowing optimal variation of Jmax and Vcmax. Although the new approach is

more demanding computationally, the added ability of the models to adjust optimally to

climate variability warrants that cost.

Finally, this new optimal allocation approach allows a unique perspective into seasonal

and climatic response of photosynthetic capacity, and opens many avenues of research

allowing an improved understanding of vegetation dynamics in our changing climate.

Appendix A: Farquhar, von Caemmerer and Berry Model

Following the Farquhar, von Caemmerer and Berry [Farquhar et al., 1980] model of pho-

tosynthesis (FvCB), the carboxylation-limited rate of assimilation for C3 photosynthesis

is,

Ac =Vcmax(Ci � �⇤)

Ci +Km

�Rd (A1)

where, Ac is the net rate of carboxylation-limited assimilation, Vcmax is the maximum rate

of carboxylation, Ci is the leaf internal CO2 concentration, �⇤ is the CO2 compensation

point in the absence of dark (or light-independent) respiration, Rd is the rate of mito-

chondrial respiration, and Km is the Michaelis-Menten half-rate constant accounting for

the competition for active sites of the enzyme RuBisCO between CO2 and O2 [Farquhar


et al., 1980]. Expanding the constant Km,

Km = Kc ⇥ (1 +Ko/O2) (A2)

where Kc is the rate constant for carboxylation reactions, and Ko is the rate constant for

oxygenation reactions. Km describes the inhibited carboxylation rate due to the presence

of oxygen.

The rate of mitochondrial respiration can be approximated as Rd ⇡ 0.015⇥ Vcmax [von

Caemmerer , 2000].

Further, the CO2 compensation point can be estimated as, �⇤ ⇡ (KcO2)/(8Ko), where

O2 is the partial pressure of oxygen. This assumes Vcmax ⇡ 4⇥ Vomax, where Vomax is the

maximum rate of RuBP oxygenation [von Caemmerer , 2000].

The RuBP limited rate of assimilation is defined as,

Aj =J(Ci � �⇤)

4(Ci + 2�⇤)�Rd (A3)

where, Aj is the RuBP limited rate of net assimilation, and J is the electron transport

rate. The rate J depends on absorbed irradiance for photosynthetic processing, and can

be approximated with a non-rectangular hyperbola [Ogren and Evans , 1993],

J =1

2✓

hI2 + Jmax �

p(I2 + Jmax)2 � 4✓I2Jmax

i(A4)

where ✓ is a curvature factor within [0 (for a rectangular hyperbola), 1 (for a Blackman-

type hyperbola)], Jmax is the maximum rate of electron transport, and I2 is the irradiance

absorbed by the chlorophyll such that,

I2 = ↵I(1� f)/2 (A5)

where I is the photosynthetically active radiation (PAR, wavelengths between 400-

700nm [Jones , 2014], µmol m�2s�1), ↵ is the absorptance fraction [Evans and Poorter ,c�2016 American Geophysical Union. All Rights Reserved.

2001], f adjusts for light quality (typically 0.15), and it is divided by 2 to approximate

the split between Photosystems I and II [Nobel , 2009].

The relationship between chlorophyll concentration and absorptance can be described

as [Evans and Poorter , 2001],

↵ =�

�+ 76(A6)

Eqs.(A1) and (A3) can be written as,

An =a1(Ci � �⇤)

Ci + a2�Rd (A7)

where, a1 is either Vcmax or J/4, and a2 is either Km or 2�⇤, for Ac and Aj, respectively.

The net assimilation rate of Eqs.(A1) and (A3) can be thought of as representing the

photosynthetic demand for CO2 flux. However, the actual supply flux of CO2, therefore,

the actual net assimilation rate, also depends on stomatal aperture. For equilibrium

conditions and assuming Fickian-like di↵usion across the stomata, the net assimilation

rate is,

An = gs(Ca � Ci) (A8)

where Ca is the ambient atmospheric CO2 concentration.

A challenge in utilizing this model is the determination of the internal CO2 concentration

Ci. One option is to explicitly solve Eq.(A8) for Ci, and substitute into Eq.(A7), resulting

in a non-linear relationship between An and Ci. However, it has been observed that

the ratio Ci/Ca remains relatively constant [Katul et al., 2010; Manzoni et al., 2011].

Therefore, assuming that Ci in the denominator of Eq.(A7) is a constant fraction (r) of

the atmospheric concentration [Yoshie, 1986; Prentice et al., 2014], where r = Ci/Ca and


solving for An leads to,

An =gs(kCa � �⇤k �Rd)

k + gs(A9)

where k = a1/(rCa + a2) and is referred to as the carboxylation e�ciency. This form

includes the explicit link between biochemical demand for CO2 and the supply of CO2

through stomatal limitations.

Appendix B: Optimal Stomatal Control

In this context, optimal stomatal control assumes that stomatal aperture is adjusted

so that the integrated amount of assimilation over a given period of time is maximum

for a constrained amount of available water for transpiration [Cowan and Farquhar , 1977;

Givnish, 1986; Buckley et al., 2002]. A robust proof of this approach is not included here,

but the canonical form using the calculus of variations is simply stated as a solution of,

@An/@gs � �@Et/@gs = 0 (B1)

where Et is the rate of transpiration, and � is a Lagrange multiplier that, as implied by

Eq.(B1), represents the marginal water use e�ciency (i.e., @An/@Et). A Fickian di↵usive

approach may be used to estimate Et,

Et = agsD (B2)

where a ⇡ 1.6 is assumed constant converting conductance of CO2 to water vapor, and D

is the vapor pressure deficit between the leaf internal pore spaces and the local atmosphere.

Taking the derivative of Eqs.(A9) and (B2) with respect to gs, substituting into Eq.(B1),

and solving for gs, results in,

gs =

rk(kCa � k�⇤ �Rd)

�aD� k (B3)


This optimal stomatal conductance (gs) may then be used to determine the optimal

assimilation rate in A9.

Appendix C: Numerical Procedure

The optimal resource allocation problem of Eq.(3) is not easily solved analytically.

However, because the assimilation rate for a given Norg, Vcmax and Jmax can be uniquely

determined, the optimal resource allocation problem can be solved numerically as follows:

1. Set an estimate of leaf nitrogen concentration Norg, and select values of Vcmax and

Jmax from a feasible set,

2. Estimate NO (mmol N m�2) as a function of Norg (gm�2), using NO = 17⇥Norg�8.

This regression developed from data in Appendix A of Evans[2001].

3. Solve Eq.(8) for the chlorophyl content as,

� =1

0.0331[Norg � 0.079Jmax

� Vcmax

6.25Vcr⇠� ⌫Jmax �NO] (C1)

4. Determine the absorptance, ↵, using Eq.(A6),

5. Adjust Km for temperature based on mean monthly values [Medlyn et al., 2002],

6. For a given discrete level of irradiance, determine the rate of assimilation using the

FvCB model and the optimal stomatal conductance of Eqs.(A9) and (B3), respectively,

7. Iterate for each level of irradiance from the density function in Eq.(3) and obtain

estimates of the expected rate of assimilation,

8. This process is then repeated for new values of Vcmax and Jmax (with Norg constant),

resulting in a di↵erent expected value of assimilation.


Spanning the feasible solution space of Vcmax and Jmax results in a surface of expected

assimilation values for a given concentration of leaf nitrogen and light distribution, referred

to as the decision surface. The maximum value of expected assimilation, and therefore

the corresponding values of V ⇤cmax and J⇤

max solving Eq.(3), may then be extracted from

the decision surface providing an estimate of the optimal solution. This process was

completed using an exhaustive gridded search; however, it may be more e�ciently found

using a simple Particle Swarm optimization approach Kennedy and Eberhart [1995].

The above process may be repeated for di↵erent light levels and nitrogen concentrations

in order to develop the relationships with V ⇤cmax and J⇤

max.

Acknowledgments. This work was funded by the U.S. Bureau of Reclamation

with substantial contributions from Colorado State University. Partial funding was

provided by the U.S. Forest Service. All correspondence should be directed to

[email protected]

References

Anten, N., F. Schieving, E. Medina, M. Werger, and P. Schu↵elen (1995), Optimal leaf area

indices in c3 and c4 mono- and dicotyledonous species at low and high nitrogen availabil-

ity, Physiologia Plantarum, 95 (4), 541–550, doi:10.1111/j.1399-3054.1995.tb05520.x.

Ball, J. T., I. E. Woodrow, and J. A. Berry (1987), A model predicting stomatal conduc-

tance and its contribution to the control of photosynthesis under di↵erent environmental

conditions, Progress in Photosynthesis Research, Vol. 4, 221–224.

Bauerle, W. L., J. D. Bowden, and G. G. Wang (2007), The influence of temperature on

within-canopy acclimation and variation in leaf photosynthesis: spatial acclimation to


microclimate gradients among climatically divergent acer rubrum l. genotypes, Journal

of Experimental Botany, 58 (12), 3285–3298, doi:10.1093/jxb/erm177.

Bauerle, W. L., R. Oren, D. A. Way, S. S. Qian, P. C. Stoy, P. E. Thornton, J. D. Bowden,

F. M. Ho↵man, and R. F. Reynolds (2012), Photoperiodic regulation of the seasonal pat-

tern of photosynthetic capacity and the implications for carbon cycling, Proceedings of

the National Academy of Sciences, 109 (22), 8612–8617, doi:10.1073/pnas.1119131109.

Broeckx, L., R. Fichot, M. Verlinden, and R. Ceulemans (2014), Seasonal variations

in photosynthesis, intrinsic water-use e�ciency and stable isotope composition of

poplar leaves in a short-rotation plantation, Tree Physiology, 34 (7), 701–715, doi:

10.1093/treephys/tpu057.

Buckley, T. N., G. D. Farquhar, and J. M. Miller (2002), The mathematics of linked

optimisation for water and nitrogen use in a canopy, Silva Fennica, 36 (3), 639–669.

Chen, J.-L., J. F. Reynolds, P. C. Harley, and J. D. Tenhunen (1993), Coordination theory

of leaf nitrogen distribution in a canopy, Oecologia, 93 (1), pp. 63–69.

Cowan, I., and G. Farquhar (1977), Stomatal function in relation to leaf metabolism and

environment, Experimental Biology Symposium, pp. 471–505.

Dewar, R. C., L. Tarvainen, K. Parker, G. Wallin, and R. E. McMurtrie (2012), Why does

leaf nitrogen decline within tree canopies less rapidly than light? an explanation from

optimization subject to a lower bound on leaf mass per area, Tree Physiology, 32 (5),

520–534, doi:10.1093/treephys/tps044.

Evans, J. R. (1989), Partitioning of nitrogen between and within leaves grown un-

der di↵erent irradiances, Australian Journal of Plant Physiology, 16, 533–548, doi:

http://dx.doi.org/10.1071/PP9890533.


Evans, J. R., and H. Poorter (2001), Photosynthetic acclimation of plants to growth irra-

diance: the relative importance of specific leaf area and nitrogen partitioning in max-

imizing carbon gain, Plant, Cell & Environment, 24 (8), 755–767, doi:10.1046/j.1365-

3040.2001.00724.x.

Farquhar, G., S. von Caemmerer, and J. Berry (1980), A biochemical model of photosyn-

thetic co2 assimilation in leaves of c3 species, Planta, Vol. 149, 78–90.

Farrior, C. E., I. Rodriguez-Iturbe, R. Dybzinski, S. A. Levin, and S. W. Pacala (2015),

Decreased water limitation under elevated co2 amplifies potential for forest carbon sinks,

Proceedings of the National Academy of Sciences, doi:10.1073/pnas.1506262112.

Field, C. (1983), Allocating leaf nitrogen for the maximization of carbon gain: Leaf

age as a control on the allocation program, Oecologia, 56 (2-3), 341–347, doi:

10.1007/BF00379710.

Flexas, J., and H. Medrano (2002), Drought-inhibition of photosynthesis in c3 plants-

stomatal and non-stomatal limitations revisited, Annals of Botany, Vol. 89, 183–189.

Givnish, T. J. (1986), On the economy of plant form and function, chap. Optimal stomatal

conductance, allocation of energy between leaves and roots, and the marginal cost of

transpiration, pp. 171–213, Cambridge University Press.

Hikosaka, K., and T. Hirose (1997), Leaf angle as a strategy for light competition: optimal

and evolutionarily stable light-extinction coe�cient within a leaf canopy, Ecoscience,

4 (4), 501–507.

Hikosaka, K., and I. Terashima (1995), A model of the acclimation of photosynthesis in

the leaves of c3 plants to sun and shade with respect to nitrogen use, Plant, Cell &

Environment, 18 (6), 605–618, doi:10.1111/j.1365-3040.1995.tb00562.x.


Jarvis, P. (1976), The interpretation of the variations in leaf water potential and stomatal

conductance found in canopies in the field, Philosophical Transactions of the Royal

Society of London. Series B, Biological Sciences, Vol. 273, 593–610.

Jones, H., and R. Sutherland (1991), Stomatal control of xylem embolism, Plant , Cell

and Environment, Vol. 14, 607–612.

Jones, H. G. (2014), Plants and Microclimate - A Quantitative Approach to Environmental

Plant Physiology, 3rd ed., Cambridge University Press.

Kattge, J., and W. Knorr (2007), Temperature acclimation in a biochemical model of

photosynthesis: a reanalysis of data from 36 species, Plant , Cell and Environment,

30 (9), 1176–1190.

Kattge, J., W. Knorr, T. Raddatz, and C. Wirth (2009), Quantifying photosynthetic

capacity and its relationship to leaf nitrogen content for global-scale terrestrial biosphere

models, Global Change Biology, 15 (4), 976–991, doi:10.1111/j.1365-2486.2008.01744.x.

Katul, G., S. Manzoni, S. Palmroth, and R. Oren (2010), A stomatal optimization theory

to describe the e↵ects of atmospheric co2 on leaf photosynthesis and transpiration,

Annals of Botany, Vol. 105 (No. 3), 431–442.

Kennedy, J. & Eberhart, R. Particle swarm optimization Neural Networks, 1995. Pro-

ceedings., IEEE International Conference on, 1995, 4, 1942-1948 Vol.4

Lange, O., R. Lsch, E.-D. Schulze, and L. Kappen (1971), Responses of stomata to changes

in humidity, Planta, 100 (1), 76–86, doi:10.1007/BF00386887.

Leuning, R. (1995), A critical appraisal of a combined stomatal-photosynthesis model for

c3 plants, Plant, Cell and Environment, Vol. 18, 339–355.


Leuning, R. (1997), Scaling to a common temperature improves the correlation between

the photosynthesis parameters Jmax and Vcmax, Journal of Experimental Biology,

48 (2), 345–347.

Lin, Y.-S. e. (2015), Optimal stomatal behaviour around the world, Nature Climate

Change, 5, 459–464.

Lopez, R., U. L. de Heredia, C. Collada, F. J. Cano, B. C. Emerson, H. Cochard, and

L. Gil (2013), Vulnerability to cavitation, hydraulic e�ciency, growth and survival in

an insular pine (pinus canariensis), Annals of Botany, 111 (6), 1167–1179.

Maire, V., P. Martre, J. Kattge, F. Gastal, and G. Esser (2012), The coordination of

leaf photosynthesis links C and N fluxes in C3 plant species, PLoS ONE, 7 (6), doi:

10.1371/journal.pone.0038345.

Manzoni, S., G. Vico, G. Katul, P. A. Fay, 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, Vol. 25,

456–467.

Manzoni, S., G. Vico, G. Katul, S. Palmroth, and A. Porporato (2014), Optimal plant

water-use strategies under stochastic rainfall, Water Resources Research, 50, 5379–5394,

doi:10.1002/2014WR015375.

Medlyn, B. E., F. W. Badeck, D. G. G. De Pury, C. V. M. Barton, M. Broadmeadow,

R. Ceulemans, P. De Angelis, M. Forstreuter, M. E. Jach, S. Kellomki, E. Laitat,

M. Marek, S. Philippot, A. Rey, J. Strassemeyer, K. Laitinen, R. Liozon, B. Portier,

P. Roberntz, K. Wang, and P. G. Jstbid (1999), E↵ects of elevated [co2] on photosyn-

thesis in european forest species: a meta-analysis of model parameters, Plant, Cell &


Environment, 22 (12), 1475–1495, doi:10.1046/j.1365-3040.1999.00523.x.

Medlyn, B. E., E. Dreyer, D. Ellsworth, M. Forstreuter, P. C. Harley, M. U. F.

Kirschbaum, X. Le Roux, P. Montpied, J. Strassemeyer, A. Walcroft, K. Wang, and

D. Loustau (2002), Temperature response of parameters of a biochemically based model

of photosynthesis. ii. a review fo experimental data, Plant, Cell & Environment, 25 (9),

1167–1179.

Niinemets, ., and J. D. Tenhunen (1997), A model separating leaf structural and physiolog-

ical e↵ects on carbon gain along light gradients for the shade-tolerant species acer sac-

charum, Plant, Cell & Environment, 20 (7), 845–866, doi:10.1046/j.1365-3040.1997.d01-

133.x.

Niinemets, l., O. Kull, and J. D. Tenhunen (1998), An analysis of light e↵ects

on foliar morphology, physiology, and light interception in temperate deciduous

woody species of contrasting shade tolerance, Tree Physiology, 18 (10), 681–696, doi:

10.1093/treephys/18.10.681.

Niinemets, U., O. Kull, and J. Tenhunen (2004), Within-canopy variation in the rate

of development of photosynthetic capacity is proportional to integrated quantum flux

density in temperate deciduous trees, Plant , Cell and Environment, 27, 293–313.

Nobel, P. S. (2009), Physicochemical and Environmental Plant Physiology, 4th ed., Else-

vier.

Ogren, E., and J. Evans (1993), Photosynthetic light-response curves- i. the influence of

co2 partial pressure and leaf inversion, Planta, 189 (2), 182–190.

Onoda, Y., K. Hikosaka, and T. Hirose (2005), Seasonal change in the balance between

capacities of rubp carboxylation and rubp regeneration a↵ects co2 response of photosyn-


thesis in polygonum cuspidatum, Journal of Experimental Botany, 56 (412), 755–763,

doi:10.1093/jxb/eri052.

Peltoniemi, M. S., R. A. Duursma, and B. E. Medlyn (2012), Co-optimal distribution

of leaf nitrogen and hydraulic conductance in plant canopies, Tree Physiology, 32 (5),

510–519, doi:10.1093/treephys/tps023.

Poorter, H., C. van de Vijver, R. Boot, and H. Lambers (1995), Growth and carbon

economy of a fast-growing and a slow-growing grass species as dependent on nitrate

supply, Plant and Soil, 171 (2), 217–227, doi:10.1007/BF00010275.

Poorter, H., S. Pepin, T. Rijkers, Y. de Jong, J. R. Evans, and C. Krner (2006), Construc-

tion costs, chemical composition and payback time of high- and low-irradiance leaves,

Journal of Experimental Botany, 57 (2), 355–371, doi:10.1093/jxb/erj002.

Prentice, I. C.; Dong, N.; Gleason, S. M.; Maire, V., and Wright, I. J. (2014), Balancing

the costs of carbon gain and water transport: testing a new theoretical framework for

plant functional ecology Ecology Letters, 17, 82–91

Rogers, A. (2014), The use and misuse of vc,max in earth system models, Photosynthesis

Research, 119, 15–29, doi:10.1007/s11120-013-9818-1.

Sands, P. (1995), Modelling canopy production. i. optimal distribution of photosynthetic

resources, Australian Journal of Plant Physiology, 22 (4), 593–601.

Sharkey, T. D., C. J. Bernacchi, G. D. Farquhar, and E. L. Singsaas (2007), Fitting

photosynthetic carbon dioxide response curves for c3 leaves, Plant, Cell & Environment,

30 (9), 1035–1040, doi:10.1111/j.1365-3040.2007.01710.x.

Suzuki, Y., A. Makino, and T. Mae (2001), Changes in the turnover of rubisco and levels

of mrnas of rbcl and rbcs in rice leaves from emergence to senescence, Plant, Cell &


Environment, 24 (12), 1353–1360, doi:10.1046/j.0016-8025.2001.00789.x.

Tardieu, F., T. Lafarge, and T. Simonneau (1996), Stomatal control by fed or endogenous

xylem aba in sunflower: interpretation of correlations between leaf water potential and

stomatal conductance in anisohydric species, Plant, Cell & Environment, 19 (1), 75–84,

doi:10.1111/j.1365-3040.1996.tb00228.x.

Tezara, W.; Mitchell, V. J.; Driscoll, S. D. and Lawlor, D. W. (1999) Water stress inhibits

plant photosynthesis by decreasing coupling factor and ATP Nature, 401, 914–917

Tuzet, A., A. Perrier, and R. Leuning (2003), A coupled model of stomatal conductance,

photosynthesis and transpiration, Plant , Cell and Environment, Vol. 26, 1097–1116.

Tyree, M. T., and J. S. Sperry (1989), Vulnerability of xylem to cavitation and embolism,

Annual Review of Plant Physiology and Plant Molecular Biology, 40 (1), 19–36, doi:

10.1146/annurev.pp.40.060189.000315.

Verheijen, L. M., V. Brovkin, R. Aerts, G. Bonisch, J. H. C. Cornelissen, J. Kattge,

P. B. Reich, I. J. Wright, and P. M. van Bodegom (2013), Impacts of trait variation

through observed traitclimate relationships on performance of an earth system model:

a conceptual analysis, Biogeosciences, 10 (8), 5497–5515, doi:10.5194/bg-10-5497-2013.

von Caemmerer, S. (2000), Biochemical Models of Leaf Photosynthesis, CSIRO Publishing.

von Caemmerer, S., and G. D. Farquhar (1981), Some relationships between the biochem-

istry of photosynthesis and the gas exchange of leaves, Planta, 153 (4), 376–387.

Walker, A. P., A. P. Beckerman, L. Gu, J. Kattge, L. A. Cernusak, T. F. Domingues, J. C.

Scales, G. Wohlfahrt, S. D. Wullschleger, and F. I. Woodward (2014), The relationship

of leaf photosynthetic traits Vcmax and Jmax to leaf nitrogen, leaf phosphorus, and

specific leaf area: a meta-analysis and modeling study, Ecology and Evolution, 4 (16),


3218–3235, doi:10.1002/ece3.1173.

Wilson, K. B., D. D. Baldocchi, and P. J. Hanson (2000), Spatial and seasonal variability

of photosynthetic parameters and their relationship to leaf nitrogen in a deciduous

forest, Tree Physiology, 20, 565–578.

Wullschleger, S. D. (1993), Biochemical limitations to carbon assimilation in c3 plants–

a retrospective analysis of the a/ci curves from 109 species, Journal of Experimental

Botany, Vol. 44, 907–920.

Xu, L., and D. D. Baldocchi (2003), Seasonal trends in photosynthetic parame-

ters and stomatal conductance of blue oak (quercus douglasii) under prolonged

summer drought and high temperature, Tree Physiology, 23 (13), 865–877, doi:

10.1093/treephys/23.13.865.

Xu C, Fisher R, Wullschleger SD, Wilson CJ, Cai M, et al. (2012) Toward a Mechanistic

Modeling of Nitrogen Limitation on Vegetation Dynamics. PLoS ONE 7(5): e37914.

doi:10.1371/journal.pone.0037914

Yoshie, F. (1986), Intercellular CO2 concentration and water-use e�ciency of temperate

plants with di↵erent life-forms and from di↵erent microhabitats Oecologia, Vol. 68 (3),

370–374.

Zaehle, S., and A. D. Friend (2010), Carbon and nitrogen cycle dynamics in the o-cn land

surface model: 1. model description, site-scale evaluation, and sensitivity to parameter

estimates, Global Biogeochemical Cycles, 24 (1), n/a–n/a, doi:10.1029/2009GB003521.


Figure 1. Photosynthetic Parameter Decision Space - Leaf level photosynthetic parameters and

the expected value of daily assimilation. White contours represent the irradiance absorptance

fraction resulting from increased investment in chlorophyll. The optimal allocation resulting in

maximum expected assimilation is marked with a small triangle (↵ ⇡ 82%). Increased investment

in Vcmax and/or Jmax requires less investment in chlorophyll, hence absorptance decreases. [The

graph shown corresponds to Norg = 2.7g m�2, Km = 450 µmol mol�1, and a scaled beta-

distributed irradiance with parameters: Irrmax = 1000W m�2, Irradiance density parameters:

�a = 1.2, �b = 1.5]


Figure 2. Comparison of optimal values of Vcmax with regressed relationships from Kattge

[2009] for di↵erent vegetation classes from a range of climates. Notice that the optimal V ⇤cmax

allocation matches the observed regressed mean values for a range of C3 classes. Also shown

in the figure are the optimal values of ! = Jmax/Vcmax predicted by the optimal allocation

approach. (Regressions from Kattge et al., 2009, Table 2, using just Vcmax regressions) [Km =

450 µmol mol�1, Irrmax = 1000W m�2, �a = 1.2, �b = 1.5].


Figure 3. Optimal values of Vcmax and ! with variable light environment [Norg = 2.7g m�2,

Km = 450 µmol mol�1, Irrmax = variable, �a = 1.2, �b = 1.5, Rd = 0.015⇥ Vcmax]


Figure 4. Most e�cient non-stomatal down-regulation pathway (dashed black line) for Vcmax

and Jmax. White contours display mean stomatal conductance (mol m�2s�1). For a given mean

stomatal conductance, the intersection with the black line is the optimal allocation of resources

between photosynthetic systems.


Figure 5. Comparison of seasonal co-limited down regulation to observed data. Blue line is

a trace of the Vcmax and Jmax values at a state of co-limitation using only observed maximum

assimilation rate data. Red points are observed data obtained from Figure 3 of a study by Xu

and Baldocchi, 2003 [Xu and Baldocchi , 2003]. Note: these results are uncalibrated.


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