Journal of Geophysical Research: Biogeosciences
RESEARCH ARTICLE
10.1002/2016JG003473
Key Points:
• Covariation between two common
photosynthetic parameters is
explained optimally
• Allocation of leaf-level nitrogen is
constrained, and optimal allocation
replicates observations
• Water stress leads to unused
photosynthetic capacity and optimal
capacity downregulation
Correspondence to:
J. A. Quebbeman,
[email protected]
Citation:
Quebbeman, J. A., and
J. A. Ramirez (2016), Optimal
allocation of leaf-level nitrogen:
Implications for covariation
of Vcmax and Jcmax and
photosynthetic downregulation,
J. Geophys. Res. Biogeosci.,
121, doi:10.1002/2016JG003473.
Received 5 MAY 2016
Accepted 27 AUG 2016
Accepted article online 6 SEP 2016
Optimal allocation of leaf-level nitrogen: Implications for
covariation of V cmax and Jmax and photosynthetic
downregulation
J. A. Quebbeman1 and J. A. Ramirez1
1 Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, Colorado, USA
Abstract The maximum rate of carboxylation, Vcmax , and the maximum rate of electron transport, Jmax ,
describe leaf-level capacities of the photosynthetic system and are critical in determining the net fluxes
of carbon dioxide and water vapor in the terrestrial biosphere. Although both Vcmax and Jmax exhibit high
spatial and temporal variability, most descriptions of photosynthesis in terrestrial biosphere models assume
constant values for Vcmax and Jmax at a reference temperature ignoring intraseasonal, interannual, 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 allocated optimally among the various components
of the photosynthetic system in such a way that expected carbon assimilation is maximized. The optimal
allocation is constrained by available nitrogen and responds dynamically 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 nitrogen allocation under varying water availability
and light environments. This optimal allocation approach provides a mechanism to describe the response
of leaf-level photosynthetic capacity to varying environmental and resource 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, adenosine triphosphate (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 nicotinamide adenine dinucleotide phosphate (NADPH) and are constrained by
the maximum rate of electron transport, Jmax (μmol e− m−2 s−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−2 s−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 .
©2016. American Geophysical Union.
All Rights Reserved.
QUEBBEMAN AND RAMIREZ
Carboxylation includes the reaction of CO2 with Ribulose-1,5-Bisphosphate (RuBP), which is catalyzed by the
enzyme Ribulose-1,5-Bisphosphate Carboxylase-Oxygenase (RuBisCO); 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
COVARIATION OF VCMAX AND JMAX
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
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 [Farquhar 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 details 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 conditions [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∘ 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, interannual, and climatic).
2. Approach
2.1. Optimality
Many optimality hypotheses describing photosynthetic responses to varying environmental conditions have
been proposed and studied for well over 50 years, including optimal stomatal responses [Cowan and Farquhar,
1977; Field, 1983; 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 subsystems 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 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 interoperating subsystems 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 subsystems 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 subsystem, 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., 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) 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] 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 carboxylation with
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
limited investment in RuBisCO. An optimal allocation is likely closer to a state of colimitation 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,
∗
∗
Vcmax
, Jmax
= argmax
∫t
An (Vcmax , Jmax , gs ; Θ) dt | Norg ,
(1)
where An is the net rate of assimilation (gross assimilation minus leaf respiration), gs is the stomatal conductance, and Θ 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 Θ vary continuously.
For a given period of integration, the values of Vcmax and Jmax that solve equation (1) also maximize the
expectation of An . Therefore, we can similarly restate the above as
∗
∗
Vcmax
, Jmax
= argmax
∫Θ
f (Θ)An (Vcmax , Jmax , gs ; Θ) dΘ| Norg ,
(2)
where f (Θ) is the joint probability density function (PDF) of variable(s) in Θ 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, equation (2) becomes
∗
∗
Vcmax
, Jmax
= argmax
∫I
f (I)An (Vcmax , Jmax , gs ; I, 𝜇T , 𝜇VPD ) dI| Norg ,
(3)
where f (I) is the PDF of irradiance and 𝜇T and 𝜇VPD 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 photosynthetic 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 equation (4) are in units of mmol N m−2 ).
In order to solve equation (1) through equation (3), constrained by leaf-level nitrogen, an explicit link relating leaf nitrogen concentration to Vcmax and Jmax is required. Several researchers have determined these
relationships for C3 species using observations [Hikosaka and Terashima, 1995; Evans and Poorter, 2001].
Following Evans and Poorter, the nonsoluble thylakoid nitrogen, equal to the sum NP + NE , may be estimated
as [Evans and Poorter, 2001; Zaehle and Friend, 2010]
(5)
NP + NE = 0.079Jmax + 0.0331𝜒,
where Jmax is in (μmol e ) m s , 𝜒 is the concentration of chlorophyll per unit area (μ mol Chl m ), 0.079 is
in mmol N s (μmol e− )−1 , and 0.0331 is in mmol N (μmol Chl)−1 .
−
−2 −1
−2
The soluble nitrogen excluding that allocated to Rubisco is
NS = 𝜈Jmax ,
(6)
where 𝜈 ≈ 0.3 (mmol N s (μ mol 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 × 𝜉),
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
(7)
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
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 nitrogen in
RuBisCO, and 𝜉 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 nonphotosynthetic resources, such as leaf cell
walls.
Therefore, substituting equations (5), (6),
and (7) into equation (4) leads to
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 (𝛼 ≈ 82%). Increased investment in Vcmax
and/or Jmax requires less investment in chlorophyll, hence, absorptance
decreases. (The graph shown corresponds to Norg = 2.7 g m−2 ,
Km = 450μmol mol−1 , and a scaled beta-distributed irradiance with
parameters: Irrmax = 1000 W m−2 and irradiance density parameters:
𝛽a = 1.2, 𝛽b = 1.5).
Norg = 0.079Jmax + 0.0331𝜒
+ Vcmax ∕(6.25Vcr 𝜉) + 𝜈Jmax + NO
(8)
which directly relates leaf organic nitrogen Norg to Vcmax and Jmax .
2.3. Assimilation and Stomatal
Control
We implement the FvCB model to determine the rate of net assimilation in
equations (1)–(3) for 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 models describing the relationship between assimilation and
stomatal conductance, including the Ball-Woodrow-Berry model Ball et al. [1987] and its variant the Leuning
model [Leuning, 1995], as well as optimality based models including the seminal work of Cowan and Farquhar
[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 equation (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 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
∗
∗
, Jmax
) that solves equation (3).
an optimal pair (Vcmax
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
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
demonstrated in Figure 1 by moving
along lines of constant Jmax in a direction toward values of Vcmax lower than
the optimal value.
Figure 2. Comparison of optimal values of Vcmax with regressed
relationships from Kattge et al. [2009] for different vegetation classes
∗
allocation
from a range of climates. Notice that the optimal Vcmax
matches the observed regressed mean mean values for a range
of C3 classes. Also shown in the figure are the optimal values of
𝜔 = Jmax /Vcmax predicted by the optimal allocation approach.
(Regressions from Kattge et al. [2009, Table 2] using just Vcmax
(Km = 450 μmol mol−1 , Irrmax = 1000 W m−2 , 𝛽a = 1.2, and 𝛽b = 1.5.
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 different
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 distribution of irradiance) may
result from seasonal effects [Bauerle et al., 2012], from canopy closure [Wilson et al., 2000], or both.
For a given amount of Norg , 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 Vcmax
(and an increase in the ratio
∗
∗
∗
Jmax /Vcmax = 𝜔 ), 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 𝜔∗ 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
∗
∗
∕Vcmax
, (i.e., 𝜔∗ ) is commonly observed to equal
photosynthetic subsystems also explains why the ratio of Jmax
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].
Figure 3. Optimal values of Vcmax and 𝜔 with variable light environment
(Norg = 2.7 g m−2 , Km = 450μmol mol−1 , Irrmax = variable, 𝛽a = 1.2,
𝛽b = 1.5, and Rd = 0.015 × Vcmax ).
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
3.4. Photosynthetic Downregulation
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; López
et al., 2013]. Closure of stomata reduces
transpiration and the flux of CO2
into the leaf resulting in inhibited
5
Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
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 resynthesis.
Rather than maintaining this unused
capacity during prolonged water
stress, the photosynthetic system
may be downregulated (sometimes
referred to as a nonstomatal response),
resulting in a reduction of Vcmax and
Jmax from well-watered values [Flexas
and Medrano, 2002; Tezara et al., 1999].
This nonstomatal downregulation
Figure 4. Most efficient nonstomatal downregulation pathway (dashed
process requires modification of the
black line) for Vcmax and Jmax . White contours display mean stomatal
photosynthetic system and redistribuconductance (mol m−2 s−1 ). For a given mean stomatal conductance, the
tion of resources (e.g., nitrogen). For
intersection with the black line is the optimal allocation of resources
given Norg , light, and other environbetween photosynthetic systems.
mental conditions, the corresponding
reduction 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 downregulated state is where the reduction in expected assimilation is
minimum for a unit change in photosynthetic capacity (i.e., unit changes in Vcmax and Jmax ), implying a reallocation of the maximum amount of Norg to other biochemical uses. Mathematically, the optimal downregulation
∗
∗
path is the locus of (Vcmax , Jmax ), starting from the (Vcmax
, Jmax
), satisfying the condition that the magnitude of
the gradient is minimum, that is, |∇An (Vcmax , Jmax )| = minimum.
Figure 4 shows a sample path of downregulation on an assimilation decision surface. The optimal downregulation path represents a path of optimal reallocation of nitrogen—this is also an allocation strategy that
closely follows a state of colimitation between the photosynthetic systems. Chen et al. [1993] proposed the
Coordination Theory describing nitrogen allocation through a canopy where photosynthetic systems are colimited [Chen et al., 1993]. More recently, Maire et al. [2012] assessed 293 observations from 31 different 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 downregulation provides a direct link between the expected value of assimilation, which is pro∗
∗
portional to the expected value of stomatal conductance, and the optimal values of Vcmax
and Jmax
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 envi∗
∗
ronmental conditions, the optimal allocation resulting in the stomatal-constrained values of Vcmax
and Jmax
can be found following the optimal downregulation path.
However, a relationship between expected An (or equivalently gs ) and soil moisture status 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 Downregulation Against Observations
The optimal downregulation procedure postulated here was tested against reported seasonal values of Jmax
and Vcmax in a summer semiarid 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 determine the seasonally
variable Vcmax and Jmax .
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
Figure 5. Comparison of seasonal colimited down regulation to observed data. Blue line is a trace of the Vcmax and Jmax values at a state of colimitation using
only observed maximum assimilation rate data. Red points are observed data obtained from Figure 3 of a study by Xu and Baldocchi [2003]. Note: these results
are uncalibrated.
The study by Xu and Baldocchi represents a system whose moisture stress is slowly increased through a
season—a condition allowing photosynthetic system downregulation compared to solely regulating assimilation through stomatal control.
The path of downregulation was determined by starting from the unstressed optimum as postulated above
∗
∗
and following the path of shallowest descent, resulting in optimal downregulated values of Vcmax
and Jmax
.
Along this path, we then determine the associated stomatal conductances and expected assimilation rates.
Finally, using only the stomatal conductance values reported over the season, the corresponding optimal
∗
∗
optimal values of Vcmax
and Jmax
were returned from the optimal path. Figure 5 compares the resulting optimal
∗
∗
values of Vcmax and Jmax (without any calibration) to those reported by Xu and Baldocchi [2003].
As described 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 Southern California. As shown in Figure 5, the observed seasonal variation of Vcmax and Jmax
is well reproduced with the optimal downregulated 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 downregulation 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 partitioning of nitrogen in
various environments. Although many simplifications were made, the optimal allocation of organic nitrogen between components of the photosynthetic systems 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 environment and
𝜔 (i.e., Jmax ∕Vcmax ), and the response of photosynthetic system capacity during periods of downregulation.
Xu et al. [2012] also proposed a “...mechanistic nitrogen allocation model based on a trade-off of nitrogen
allocated between growth and storage, and an optimization of nitrogen allocated among light capture, electron transport, carboxylation, and respiration...” Their approach assumed optimality conditions under mean
environmental conditions but used a Metropolis-Hastings parameter fitting approach. Here we consider an
QUEBBEMAN AND RAMIREZ
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Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
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 determination of optimality conditions during prolonged water stress (i.e., photosynthetic system
downregulation) and does not require resorting to any fitting or calibration approach.
The optimal allocation approach presented here provides a new mechanism to incorporate 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,
and critical to these processes are the maximum capacities of electron transport and carboxylation. Assuming
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 multidecadal 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 photosynthesis (FvCB), the
carboxylation-limited rate of assimilation for C3 photosynthesis is
Ac =
Vcmax (Ci − Γ∗ )
− Rd ,
Ci + Km
(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, Γ∗ is the CO2 compensation point in the absence of dark (or light independent) respiration, Rd is the rate of mitochondrial 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, Γ∗ ≈ (Kc O2 )∕(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 − Γ∗ )
− Rd ,
4(Ci + 2Γ∗ )
(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 nonrectangular
hyperbola [Ogren and Evans, 1993],
[
]
√
1
2
I + Jmax − (I2 + Jmax ) − 4𝜃I2 Jmax ,
J=
(A4)
2𝜃 2
where 𝜃 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 = 𝛼I(1 − f )∕2,
(A5)
where I is the photosynthetically active radiation (wavelengths between 400 and 700 nm [Jones, 2014],
μmol m−2 s−1 ), 𝛼 is the absorptance fraction [Evans and Poorter, 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].
QUEBBEMAN AND RAMIREZ
COVARIATION OF VCMAX AND JMAX
8
Journal of Geophysical Research: Biogeosciences
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The relationship between chlorophyll concentration and absorptance can be described as [Evans and
Poorter, 2001]
𝜒
.
𝛼=
(A6)
𝜒 + 76
Equations (A1) and (A3) can be written as
An =
a1 (Ci − Γ∗ )
− Rd ,
Ci + a2
(A7)
where a1 is either Vcmax or J∕4 and a2 is either Km or 2Γ∗ , for Ac and Aj , respectively.
The net assimilation rate of equations (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 diffusion 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 equation (A8) for Ci and substitute into equation (A7), resulting in a nonlinear 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 equations (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
g (kC − Γ∗ k − Rd )
An = s a
,
(A9)
k + gs
where k = a1 ∕(rCa + a2 ) and is referred to as the carboxylation efficiency. 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 − 𝜆𝜕Et ∕𝜕gs = 0,
(B1)
where Et is the rate of transpiration and 𝜆 is a Lagrange multiplier that, as implied by equations (B1), represents
the marginal water use efficiency (i.e., 𝜕An ∕𝜕Et ). A Fickian diffusive approach may be used to estimate Et
Et = ags D,
(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 equations (A9) and (B2) with respect to gs , substituting into equation (B1), and solving
for gs , results in
√
k(kCa − kΓ∗ − Rd )
− k.
gs =
(B3)
𝜆aD
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 equation (3) is not easily solved analytically. However, because the
assimilation rate for given Norg , Vcmax , and Jmax can be uniquely determined, the optimal resource allocation
problem can be solved numerically as follows:
QUEBBEMAN AND RAMIREZ
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9
Journal of Geophysical Research: Biogeosciences
10.1002/2016JG003473
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 (g m−2 ), using NO = 17 × Norg − 8. This regression developed
from data in Appendix A of Evans and Poorter [2001].
3. Solve equation (8) for the chlorophyl content as
[
]
Vcmax
1
Norg − 0.079Jmax −
− 𝜈Jmax − NO .
𝜒=
(C1)
0.0331
6.25Vcr 𝜉
4. Determine the absorptance, 𝛼 , using equation (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 equations (A9) and (B3), respectively.
7. Iterate for each level of irradiance from the density function in equation (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 different
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 Vcmax
and Jmax
solving equation (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 efficiently found using a simple
Particle Swarm optimization approach [Kennedy and Eberhart, 1995].
The above process may be repeated for different light levels and nitrogen concentrations in order to develop
∗
∗
the relationships with Vcmax
and Jmax
.
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.
QUEBBEMAN AND RAMIREZ
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