In ebimodeling/biocro: Modular Crop Growth Simulations
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Overview
The Farquhar-von-Caemmerer-Berry (FvCB) model for C~3~ photosynthesis was first
presented in @farquhar_biochemical_1980, and since that initial publication, it
has become one of the most important models for understanding the photosynthetic
response of plants to key environmental variables such as incident light, CO~2~,
and temperature. Many researchers are familiar with the model in the contex of
fitting it to experimentally measured CO~2~ response curves, where certain
simplifying assumptions can be reasonably applied to the model. However, these
assumptions are not reasonable when using the model in a wider range of
conditions than those usually encountered during gas exchange measurements. The
goal of this document is to identify several possible pitfalls associated with
these assumptions and explain how to avoid them when using the FvCB model.
The Central FvCB Equation {#sec:centraleq}
The FvCB model is rooted in a biochemical understanding of the photosynthetic
process, especially the carboxylation (or oxygenation) of RuBP by the enzyme
Rubisco. At its core, it predicts the net CO~2~ assimilation rate ($A_n$) from
the values of several key variables. An initial equation for $A_n$ can be
developed from the following considerations:
-
One carboxylation of RuBP assimilates one molecule of CO~2~.
-
Two oxygenations of RuBP produces hazardous byproducts that require the
release of one molecule of CO~2~ to neutralize. (Oxygenation of RuBP is often
referred to as photorespiration.)
-
Mitochondrial respiration also releases CO~2~ molecules.
Putting this all together, we can write down an initial equation for the net
assimilation rate:
\begin{equation}
A_n = V_c - 0.5 \cdot V_o - R_d, (#eq:an)
\end{equation}
where $V_c$ is the RuBP carboxylation rate, $V_o$ is the RuBP oxygenation rate,
and $R_d$ is the mitochondrial respiration rate. As written, $V_c$, $V_o$, and
$R_d$ are all non-negative rates, typically with units of $\mu$mol m$^{-2}$
s$^{-1}$. The first two terms in the equation ($V_c - 0.5 \cdot V_o$)
represent the contribution of photosynthesis (or Rubisco) to the net CO~2~
assimilation, while the remaining term ($-R_d$) is the result of mitochondrial
respiration, a separate process.
In its current form, this equation is not very easy to interpret or use. There
are a few modifications we can use to improve it. One change is to introduce
$\phi$, the ratio of RuBP oxygenation to carboxylation: $\phi = V_o / V_c$. Now
we can rewrite Equation \@ref(eq:an) as follows:
\begin{equation}
A_n = V_c \cdot \left( 1 - 0.5 \cdot \phi \right) - R_d. (#eq:anphi)
\end{equation}
This version of the equation makes it clear that $\phi$ plays a key role in
determining whether RuBP carboxylation is able to outweigh oxygenation; in other
words, it determines whether the photosynthetic contribution to the net CO~2~
assimilation rate is positive or negative. If $1 - 0.5 \cdot \phi < 0$, then
oxygenation consumes more carbon than carboxylation produces; if
$1 - 0.5 \cdot \phi > 0$, then the opposite is true. The crossover point where
carboxylation and oxygenation exactly cancel each other out occurs when
$1 - 0.5 \cdot \phi = 0$. It can be shown that $\phi$ depends on the relative
specificity of Rubisco for CO~2~ versus O~2~ ($S_{c/o}$), the partial pressure
of CO~2~ in the vicinity of Rubisco ($C$), and the partial pressure of O~2~ in
the vicinity of Rubisco ($O$):
\begin{equation}
\phi = \frac{1}{S_{c/o}} \cdot \frac{O}{C}. (#eq:s)
\end{equation}
(See, for example, Equation 2.16 from @caemmerer_biochemical_2000 and the
surrounding discussion.)
Equation \@ref(eq:s) tells us that when the amount of CO~2~ near Rubisco
increases, or when Rubisco's specificity for CO~2~ increases, $\phi$ decreases
and hence the oxygenation rate also decreases. This makes sense at an intuitive
level. It also helps to clarify the criteria for photosynthesis to become a net
producer of assimilated carbon. One immediate conclusion is that because
Rubisco specificity and oxygen levels are generally constant for a particular
chloroplast, the only reasonable way to change $\phi$ is to change $C$. But we
can still learn more. We have already seen that the crossover occurs when
$1 - 0.5 \cdot \phi = 0$ or, equivalently, when $\phi = 2$. Now, we can rewrite
this condition using \@ref(eq:s):
\begin{equation}
\phi = 2 \
\rightarrow \frac{1}{S_{c/o}} \cdot \frac{O}{C} = 2 \
\rightarrow C = \frac{O}{2 \cdot S_{c/o}}.
\end{equation}
In other words, photosynthesis is a net producer of carbon whenever $C$ exceeds
a special value called $\Gamma^$ defined by:
\begin{equation}
\Gamma^ = \frac{O}{2 \cdot S_{c/o}}. (#eq:gstar)
\end{equation}
$\Gamma^$ is often referred to as the "CO~2~ compensation point in the absence
of mitochondrial respiration." (In general, the term "CO~2~ compensation point"
refers to the value of $C$ where $A_n = 0$.) Note that since atmospheric oxygen
levels are fairly constant, $O$ rarely changes under most conditions, so
$\Gamma^$ is largely a property of Rubisco; more specifically, it depends
primarily on $S_{c/o}$. (The major exception to this is when $O$ is artificially
decreased during gas exchange measurements in order to reduce $V_o$.)
With this new concept, we can use Equations \@ref(eq:s) and \@ref(eq:gstar) to
rewrite Equation \@ref(eq:anphi) as follows:
\begin{equation}
A_n = V_c \cdot (1 - \Gamma^* / C) - R_d (#eq:anfinal)
\end{equation}
This equation contains a great deal of useful information, and we will refer to
it as the central equation of the FvCB model. (This is our terminology, and not
a standard name for the equation.) It can be found in many notable descriptions
of the FvCB model, such as:
-
Equation 16.57 in @farquhar_modelling_1982.
-
Equation 1 in @kirschbaum_temperature_1984.
-
Equation 2.19 in @caemmerer_biochemical_2000.
-
Equation A6 in @von_caemmerer_steady-state_2013.
(A big exception is the original FvCB paper [@farquhar_biochemical_1980], which
does not contain this equation!)
A key piece of information that can be learned from this equation is that while
the RuBP carboxylation rate $V_c$ can never be negative, the contribution of
photosynthesis to the net assimilation rate ($V_c \cdot(1 - \Gamma^ / C)$)
can be negative; this will occur whenever $V_c > 0$ and $C < \Gamma^$.
The rest of the FvCB model consists of a method for calculating $V_c$; this will
be discussed in the following sections.
Choosing a Carboxylation Rate
In the original FvCB model, two scenarios are considered when determining the
RuBP carboxylation rate $V_c$:
-
When there is plenty of RuBP available for Rubisco to act upon, the
carboxylation rate is said to be RuBP-saturated (or Rubisco-limited). In this
case, it is common to denote the carboxylation rate by $W_c$. The equation for
$W_c$ and its derivation can be found in many places, such as Equation 1.9 in
@caemmerer_biochemical_2000 and the surrounding discussion.
-
Following a carboxylation or an oxygenation, each RuBP molecule must be
"regenerated," that is, returned to its original state and ready for another
carboxylation or oxygenation. If the RuBP regeneration rate is too slow, it
may impose a new maximum on the carboxylation rate, which is said to be
RuBP-regeneration-limited. In this case, it is common to denote the
carboxylation rate by $W_j$. The equation for $W_j$ and its derivation can be
found in many places, such as Equations 2.21 and 2.22 in
@caemmerer_biochemical_2000 and the surrounding discussion. Sometimes this
rate is referred to as the electron-transport-limited rate, but this
terminology is becoming less common since it has been realized that other
factors besides electron transport can limit RuBP regeneration.
For brevity, we will simply reproduce the equations here without their
derivations:
\begin{equation}
W_c = \frac{V_{cmax} \cdot C}{C + K_c \cdot \left( 1 + O / K_o \right)} (#eq:wc)
\end{equation}
and
\begin{equation}
W_j = \frac{J \cdot C}{4 \cdot C + 8 \cdot \Gamma^*}, (#eq:wj)
\end{equation}
where $V_{cmax}$ is the maximum rate of Rubisco carboxylation activity, $K_c$
and $K_o$ are Michaelis-Menten constants for CO~2~ and O~2~, and $J$ is the
RuBP regeneration rate, which may depend on several external factors such as the
incident irradiance. $V_{cmax}$, $K_c$, and $K_o$ are all positive, while $J$ is
non-negative.
Note that several different versions of Equation \@ref(eq:wj) can be found in
the literature, where the only changes are to the coefficients in the
denominator ($4$ and $8$ in our version; see Equation 16.60 in
@farquhar_modelling_1982 or Section 2.4.2 from @caemmerer_biochemical_2000 for
more information). These coefficients are related to energy requirements
involved with the electron transport chain, and different estimates exist for
these requirements. Because of these differences, the coefficients can sometimes
be considered as variables in the equation instead of taking fixed values.
One significant feature of Equations \@ref(eq:wc) and \@ref(eq:wj) is that
neither one can ever return a negative value, as expected (and required) for any
equation that calculates a carboxylation rate. Another important thing to notice
is that $V_{cmax}$, $K_c$, $K_o$, $O$, and $\Gamma^*$ are generally constant
within a chloroplast, so the main environmental influences on $W_c$ and $W_j$
are $C$ and $J$. As $C$ and $J$ vary, one or the other of $W_c$ and $W_j$ may
become the larger of the two.
At this point, a key question needs to be addressed: which carboxylation rate
($W_c$ or $W_j$) should be used in Equation \@ref(eq:anfinal) to determine the
net CO~2~ assimilation rate? The FvCB authors provide an answer: we should
choose whichever rate is smaller for the current set of conditions (as specified
by $C$, $J$, etc). Intuitively, this makes sense. It doesn't matter how many
RuBP molecules could be carboxylated by Rubisco if the RuBP is not being
regenerated fast enough; alternatively, it doesn't matter how many RuBP
molecules are being regenerated if Rubisco is unable to carboxylate them.
Converting this idea to mathematical form, we arrive at another essential
equation:
\begin{equation}
V_c = \min { W_c, W_j } (#eq:mincj)
\end{equation}
This is the most common way to choose between the rates, but it does produce
very sudden transitions between RuBP-saturated and RuBP-regeneration-limited
carboxylation. Sometimes alternatives to a simple minimum are used, such as a
quadratic mixing method, that produce smoother transitions. For example, see
@kirschbaum_temperature_1984.
Pitfall: Limitations at Low CO~2~ Concentration {#sec:pitfallci}
People who are familiar with analyzing experimentally measured A-Ci curves may
be under the impression that at low $C$, carboxylation (and hence net CO~2~
assimilation) is always RuBP-saturated (or Rubisco-limited). Here, "low $C$" may
mean below $\Gamma^*$, below the CO~2~ compensation point, or possibly below
some other threshold. For example, a paper with an excellent discussion of C~3~
A-Ci curve fitting [@sharkey_fitting_2007] includes the following discussion of
limitations to assimilation:
The routine described here requires identifying whether a data point is
limited by Rubisco, RuBP regeneration or TPU. A good starting point is to
assign points above 30 Pa as RuBP-regeneration-limited and points below 20 Pa
as Rubisco-limited; points between 20 and 30 Pa might be either.
While this may be good advice for interpreting an A-Ci curve, it is not
generally true, and instead relies on an unstated assumption that $J$ is quite
high (an assumption often met while measuring an A-Ci curve).
There is nothing in the FvCB model equations (Equations \@ref(eq:anfinal),
\@ref(eq:wc), \@ref(eq:wj), and \@ref(eq:mincj)) that enforces $W_c < W_j$ below
some threshold value of $C$. In fact, there is a simple counterexample
demonstrating a situation where $W_j < W_c$ at low $C$: darkness. Darkness
ensures that $J = 0$ and hence $W_j \leq W_c$ for all values of $C$.
It is possible to find the point where the process limiting the carboxylation
rate changes; this can be done by setting $W_c = W_j$ and solving for $C$. The
result is given by Equation 2.52 in @caemmerer_biochemical_2000, where this
special value of $C$ is called $C_c$:
\begin{equation}
C_c = \frac{K_c \cdot \left( 1 + O / K_o \right) \cdot J / \left( 4 \cdot V_{cmax} \right) - 2 \cdot \Gamma^}{1 - J / \left( 4 \cdot V_{cmax} \right)} (#eq:cc)
\end{equation}
From a physical point of view, $C_c$ cannot be negative; however, Equation
\@ref(eq:cc) can mathematically produce negative values for $C_c$. When this
happens, it simply means that there is no crossover, and one or the other
process limits carboxylation for all realistic values of $C$. This occurs
whenever the denominator and numerator have opposite signs. The denominator
becomes negative when $4 \cdot V_{cmax} < J$. This is a situation where
$V_{cmax}$ is small and carboxylation is always RuBP-saturated. On the other
hand, the numerator becomes negative when
$J < 8 \cdot V_{cmax} \cdot \Gamma^ / \left[ K_c \cdot \left( 1 + O / K_o \right) \right]$.
This is a situation where $J$ is small and carboxylation is always
RuBP-regeneration-limited. (Of course, $C_c$ will be positive if both the
numerator and denominator are negative, but this is unlikely to happen for
realistic values of $K_c$, $K_o$, $O$, and $\Gamma^*$.) Even when $C_c$ is
positive, it may not be realistically obtainable; for example, if $J$ is just a
tiny bit smaller than $4 \cdot Vcmax$, $C_c$ can take very large values that
would only be achievable in a gas evironment of nearly pure CO~2~. Essentially,
this analysis of Equation \@ref(eq:cc) tells us that there is no simple rule for
determining which process limits carboxylation in the FvCB model, and it is even
possible that a crossover between limiting processes may not occur (see Figure
\@ref(fig:cc)).
#| fig.cap = "Solution space of Equation \\@ref(eq:cc) indicating where a
#| crossover is impossible because `Cc` is negative (pink), unlikely because
#| `Cc` is too large (dark blue), or otherwise possible (white). The upper
#| pink region is bounded from below by
#| `Vcmax = J * [ Kc * (1 + O / Ko)] / (8 * GammaStar)` and the lower pink
#| region is bounded from above by `Vcmax = J / 4`. In this calculation,
#| `Kc` = 259 microbar, `O` = 200 mbar, `Ko` = 179 mbar, `GammaStar` =
#| 38.6 microbar, and `Cc` is considered to be too large when it exceeds
#| 2000 microbar."
# Define a function that calculates the crossover point and returns an
# indication of whether it can or cannot occur. Here, -1 means Cc is negative
# and a crossover is impossible; 1 means Cc exceeds a threshold value (2000
# microbar) and is unlikely to be realistically achievable; 0 means that a
# crossover occurs at a reasonable value.
crossover <- function(
J, # micromol / m^2 / s
Vcmax # micromol / m^2 / s
)
{
Kc <- 259 # microbar
O <- 200 # mbar
Ko <- 179 # mbar
Gstar <- 38.6 # microbar
jv <- J / (4 * Vcmax) # dimensionless
Cc <- (Kc * (1.0 + O / Ko) * jv - 2.0 * Gstar) / (1.0 - jv) # microbar
if (Cc < 0) {
-1
} else if (Cc > 2000) {
1
} else {
0
}
}
# Set up the input values and the output matrix
npts <- 1001
j_seq <- seq(0, 100, length.out = npts)
v_seq <- seq(1, 100, length.out = npts)
z_cc <- matrix(nrow = length(j_seq), ncol = length(v_seq))
# Fill in the matrix
for (i in seq_along(j_seq)) {
for (j in seq_along(v_seq)) {
z_cc[i,j] <- crossover(j_seq[i], v_seq[j])
}
}
# Make the image
cc_image <- list(x = j_seq, y = v_seq, z = z_cc)
# Make a custom image plotting function
plot_image <- function(F_image, ...) {
cols <- c('#F2CEE6', '#FFFFFF', '#7798AB')
breaks <- c(-10, -0.5, 0.5, 10)
image(
F_image,
breaks = breaks,
col = cols,
useRaster = TRUE,
...
)
}
# Plot the image
plot_image(
cc_image,
xlab = 'J (micromol / m^2 / s)',
ylab = 'Vcmax (micromol / m^2 / s)'
)
Pitfall: Choosing Minimal Assimilation Rates {#sec:pitfallassim}
When carboxylation is RuBP-saturated, the corresponding net CO~2~ assimilation
rate (called $A_c$ rather than $A_n$ to indicate the type of limitation) can be
found by combining Equations \@ref(eq:anfinal) and \@ref(eq:wc) as follows:
\begin{equation}
A_c = W_c \cdot (1 - \Gamma^ / C) - R_d = \frac{V_{cmax} \cdot \left( C - \Gamma^ \right)}{C + K_c \cdot \left( 1 + O / K_o \right)} - R_d. (#eq:ac)
\end{equation}
Likewise, when carboxylation is RuBP-regeneration-limited, the corresponding net
CO~2~ assimilation rate (called $A_j$) can be found by combining Equations
\@ref(eq:anfinal) and \@ref(eq:wj) as follows:
\begin{equation}
A_j = W_j \cdot (1 - \Gamma^ / C) - R_d = \frac{J \cdot \left( C - \Gamma^ \right)}{4 \cdot C + 8 \cdot \Gamma^*} - R_d. (#eq:aj)
\end{equation}
With this in mind, the FvCB model is sometimes expressed in the following form,
which is a bit simpler than using the version above because it bypasses the
separate calculation of carboxylation rates and only requires three equations
instead of four:
\begin{equation}
A_n = \min { A_c, A_j }. (#eq:analt)
\end{equation}
This equation can be found in many publications, and the earliest instance may
be Equation A1 from @collatz_physiological_1991. In that paper, the authors
state
Equation (A1) is equivalent in form to that proposed by Kirschbaum and
Farquhar (1984), but our definitions of $J_C$, $J_E$, and $J_S$ differ
somewhat from theirs.
(Note that @kirschbaum_temperature_1984 uses $W_c$ and $W_j$, as mentioned
above.) Unfortunately, the claim from this paper is simply untrue because
Equations \@ref(eq:anfinal) and \@ref(eq:mincj) produce different results from
Equation \@ref(eq:analt).
To see why, suppose that $W_c < W_j$ and $C < \Gamma^$. In this case,
$\min { W_c, W_j } = W_c$ and so $A_n = A_c$ according to the FvCB model.
However, $1 - \Gamma^ / C$ is negative, so
$A_c = W_c \cdot \left( 1 - \Gamma^ / C \right) - R_d$ is also negative.
Likewise, $A_j$ is negative, but its magnitude is larger than $A_c$ because
$W_c < W_j$. So, in this case, $\min { A_c, A_j } = A_j$. Thus, Equation
\@ref(eq:analt) has chosen the incorrect limiting factor. In fact, Equation
\@ref(eq:analt) always chooses the wrong limiting factor when $C < \Gamma^$.
When Equation \@ref(eq:analt) is implemented in computer code to aid with
fitting A-Ci curves or simulating photosynthesis, it is often paired with an
additional condition like if (Aj < 0) {Aj = 0}. This is an attempt to fix the
problem of Equation \@ref(eq:analt) choosing the wrong limiting factor at low
$C$. Unfortunately, this solution is based on another misunderstanding where it
is incorrectly assumed that carboxylation is always RuBP-saturated at low $C$
(see Section \@ref(sec:pitfallci)). Although this "partial solution" might not
cause any problems when fitting A-Ci curves, it will not produce the FvCB
model's true output when $J$ is low.
Because of this subtle issue for $C < \Gamma^*$, it is better to avoid Equation
\@ref(eq:analt) in favor of Equations \@ref(eq:anfinal) and \@ref(eq:mincj).
Although Equation \@ref(eq:analt) may seem like it is simply an alternative way
to specify the FvCB model, it is actually not, and it would be more accurate to
consider it as a separate "pseudo-FvCB" model, a term we will continue to use in
the rest of this document. Examples comparing the FvCB model against the
pseudo-FvCB model can be found in Section \@ref(sec:examples).
Including Triose Phosphate Utilization
The sum of the chemical reactions in the Calvin–Benson–Bassham (CBB) cycle can
be written as
\begin{multline}
\qquad 3 \, \mathrm{CO_2} + 6 \, \mathrm{NADPH} + 6 \, \mathrm{H^+} + 9 \, \mathrm{ATP} + 5 \, \mathrm{H_2O} \rightarrow \
\mathrm{(G3P)} + 6 \, \mathrm{NADP^+} + 9 \, \mathrm{ADP} + 8 \, \mathrm{P_i}, \qquad (#eq:cbb)
\end{multline}
where G3P refers to glyceraldehyde 3-phosphate (also known as triose phosphate)
and P~i~ refers to inorganic phosphate (an ion). Each molecule of ATP is
synthesized from three inorganic phosphate ions; thus, the CBB reaction requires
nine P~i~ ions, eight of which remain free at the end. The missing inorganic
phosphate is contained in the triose phosphate molecule, which is exported from
the chloroplast and eventually used to form monosaccharide sugars. Since the CBB
reaction requires three CO~2~ molecules, each of which originate from a RuBP
carboxylation, we can see that each carboxylation effectively consumes one third
of an inorganic phosphate. By separately considering the impact of RuBP
oxygenation (not included here), it can be shown that the overall rate of
inorganic phosphate consumption during photosynthesis ($R_{pc}$) can be given by
\begin{equation}
R_{pc} = \frac{V_c}{3} - \frac{V_o}{6} - \frac{\alpha \cdot V_o}{2},
\end{equation}
where $0 \leq \alpha \leq 1$ is the fraction of glycolate carbon not returned to
the chloroplast (see Section 2.4.3 in @caemmerer_biochemical_2000). As in
Section \@ref(sec:centraleq), we can use $V_o = \phi \cdot V_c$ and
$\phi / 2 = \Gamma^ / C$ to rewrite this equation in terms of $\Gamma^$ and
$C$:
\begin{equation}
R_{pc} = \frac{V_c}{3 \cdot C} \cdot \left[ C - \Gamma^* \left( 1 + 3 \cdot \alpha \right) \right]. (#eq:rpc)
\end{equation}
The inorganic phosphate supply is replenished when triose phosphate is used to
form sugars; although it happens elsewhere, the inorganic phosphate liberated
during this reaction is returned to the chloroplast. Under some circumstances,
the rate of triose phosphate utilization (TPU) is slow enough to limit the
inorganic phosphate available for use in the choloroplast for photosynthesis; in
this case, the carboxylation rate is said to be TPU-limited and $R_{pc} = T_p$,
where $T_p$ is the rate of triose phosphate utilization. The expression for
$R_{pc}$ from Equation \@ref(eq:rpc) can be substuted into $R_{pc} = T_p$ and
solved for the corresponding carboxylation rate, which is referred to as $W_p$
in this case:
\begin{equation}
W_p = \frac{3 \cdot T_p \cdot C}{C - \Gamma^* \cdot \left( 1 + 3 \cdot \alpha \right)}. (#eq:wp)
\end{equation}
This represents a third possible limitation to the carboxylation rate in
addition to the RuBP-saturated and RuBP-regeneration-limited rates. While this
limitation was not included in the original FvCB model, its importance was
realized soon afterwards [@sharkey_photosynthesis_1985; @harley_improved_1991]
and it has since become a standard part of the model. Since we are now
considering a new process in the model, we must update Equation \@ref(eq:mincj)
to include $W_p$:
\begin{equation}
V_c = \min { W_c, W_j, W_p }. (#eq:mincjp)
\end{equation}
Pitfall: Impossible Limitations
Inspecting Equation \@ref(eq:wp), we can see that when
$C < \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, the denominator of the
equation is negative, and hence $W_p < 0$. However, $W_p$ represents a
carboxylation rate, which cannot be negative. So what's going on here? One way
to understand the situation is to evaluate Equation \@ref(eq:rpc) with
$C$ set to $\Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$. Doing so, we find
that the rate of inorganic phosphate consumption in this case is exactly 0. In
other words, when $C$ is low enough, photosynthesis is not consuming any
inorganic phosphate, and it is therefore impossible for carboxylation to be
limited by TPU. One way to indicate this restriction is to modify Equation
\@ref(eq:wp) to the following:
\begin{equation}
W_p =
\begin{cases}
\infty \;, & 0 \leq C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right) \[1ex]
\frac{3 \cdot T_p \cdot C}{C - \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)} \;, & \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right) < C
\end{cases} (#eq:wpnew)
\end{equation}
With this modification, $W_p$ is never negative, and it can never be the
smallest of $W_c$, $W_j$, and $W_p$ when
$C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, so we can safely use
it in conjunction with Equation \@ref(eq:mincjp).
When carboxylation is TPU-limited, the corresponding net CO~2~ assimilation
rate (called $A_p$ rather than $A_n$ to indicate the type of limitation) can be
found by combining Equations \@ref(eq:anfinal) and \@ref(eq:wp) as follows:
\begin{equation}
A_p = W_p \cdot (1 - \Gamma^ / C) - R_d = \frac{3 \cdot T_p \cdot \left( C - \Gamma^ \right)}{C - \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)} - R_d. (#eq:ap)
\end{equation}
This equation is often incorporated into the pseudo-FvCB model discussed in
Section \@ref(sec:pitfallassim); in this case, Equation \@ref(eq:analt) is
updated to the following:
\begin{equation}
A_n = \min { A_c, A_j, A_p }. (#eq:analtnew)
\end{equation}
See, for example, Equations 2.26 and 2.27 in @caemmerer_biochemical_2000
(but be aware that Equation 2.26 has a typo in the denominator). Using Equation
\@ref(eq:analtnew) without considering the domain over which Equation
\@ref(eq:ap) is valid can cause some additional issues beyond the ones discussed
in Section \@ref(sec:pitfallassim); in particular, the net assimilation rate may
become TPU-limited when $C < \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$,
although this should not be possible. This is just one more reason to choose a
minimal carboxylation rate (Equation \@ref(eq:mincjp)) rather than a minimal net
assimilation rate (Equation \@ref(eq:analtnew)) when using or describing the
FvCB model. Examples comparing the FvCB model against the pseudo-FvCB model can
be found in Section \@ref(sec:examples).
Handling C = 0
One issue with Equation \@ref(eq:anfinal) is that its value becomes undefined
when $C = 0$. This is not a significant issue because $C > 0$ in any realistic
situation. However, it is possible to extend the model to $C = 0$ by considering
the limiting value as $C$ approaches 0 from the right.
First, because carboxylation cannot be TPU-limited when
$C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, it cannot be TPU-limited
when $C = 0$, and there is no need to consider it here. Next, note that although
$1 - \Gamma^ / C$ is undefined when $C = 0$, $A_c$ and $A_j$ (Equations
\@ref(eq:ac) and \@ref(eq:aj)) are not. In fact, we have the following when
$C = 0$:
\begin{equation}
A_c \big|{C = 0} = - \frac{\Gamma^ \cdot V_{cmax}}{K_c \cdot \left( 1 + O / K_o\right)} - R_d
\end{equation}
and
\begin{equation}
A_j \big|{C = 0} = - \frac{J}{8} - R_d.
\end{equation}
Remembering that Equation \@ref(eq:analt) _always makes the wrong choice for
$C < \Gamma^$, we can reverse its behavior to find $A_n$ when $C = 0$:
\begin{equation}
A_n \big|{C = 0} = \max \bigl{ A_c \big|{C = 0}, A_j \big|{C = 0} \bigr}.
\end{equation}
Examples Comparing the FvCB and Pseudo-FvCB Models {#sec:examples}
Here we will use R to demonstrate some of the differences between the FvCB model
and the pseudo-FcVB model. First, we will define a function that implements the
FvCB model:
fvcb_model <- function(
C, # microbar
Vcmax, # micromol / m^2 / s
Kc, # microbar
O, # mbar
Ko, # mbar
J, # micromol / m^2 / s
Gstar, # microbar
Tp, # micromol / m^2 / s
alpha, # dimensionless
Rd # micromol / m^2 / s
)
{
Wc <- Vcmax * C / (C + Kc * (1 + O / Ko))
Wj <- J * C / (4 * C + 8 * Gstar)
Wp <- 3 * Tp * C / (C - Gstar * (1 + 3 * alpha))
Wp[C <= Gstar * (1 + 3 * alpha)] <- Inf
Vc <- pmin(Wc, Wj, Wp)
An <- (1 - Gstar / C) * Vc - Rd
list(Wc = Wc, Wj = Wj, Wp = Wp, Vc = Vc, An = An)
}
Now, we will define a function that implements the pseudo-FvCB model; here we
will intentionally avoid restricting $A_p$ to its proper $C$ domain, allowing it
to become limiting at low $C$:
pseudo_fvcb_model <- function(
C, # microbar
Vcmax, # micromol / m^2 / s
Kc, # microbar
O, # mbar
Ko, # mbar
J, # micromol / m^2 / s
Gstar, # microbar
Tp, # micromol / m^2 / s
alpha, # dimensionless
Rd # micromol / m^2 / s
)
{
Ac <- Vcmax * (C - Gstar) / (C + Kc * (1 + O / Ko)) - Rd
Aj <- J * (C - Gstar) / (4 * C + 8 * Gstar) - Rd
Ap <- 3 * Tp * (C - Gstar) / (C - Gstar * (1 + 3 * alpha)) - Rd
An <- pmin(Ac, Aj, Ap)
list(Ac = Ac, Aj = Aj, Ap = Ap, An = An)
}
We will also define a function that runs each model across a range of $C$ values
for a set of input parameter values and compares the results by creating
figures. The code for this function is not included in this article, but can be
found in its source file. It will be used to generate the figures in the
following sections.
# Load the lattice package for creating graphs
library(lattice)
# Make a function that runs both versions of the model across a range of C
# values and then plots the results
fvcb_compare <- function(
Vcmax, # micromol / m^2 / s
Kc, # microbar
O, # mbar
Ko, # mbar
J, # micromol / m^2 / s
Gstar, # microbar
Tp, # micromol / m^2 / s
alpha, # dimensionless
Rd, # micromol / m^2 / s
plot_type
)
{
C_seq <- seq(1, 1000, by = 2)
fvcb_model_res <- fvcb_model(
C_seq,
Vcmax,
Kc,
O,
Ko,
J,
Gstar,
Tp,
alpha,
Rd
)
pseudo_fvcb_model_res <- pseudo_fvcb_model(
C_seq,
Vcmax,
Kc,
O,
Ko,
J,
Gstar,
Tp,
alpha,
Rd
)
clim <- c(0, 1000)
alim <- c(-20, 50)
colors <- c(
"#000000",
"#66C2A5",
"#FC8D62",
"#8DA0CB",
"#E78AC3"
)
if (plot_type == 1) {
xyplot(
An + Vc + Wc + Wj + Wp ~ C_seq,
data = fvcb_model_res,
type = 'l',
lty = c(1, 1, 2, 2, 2),
lwd = c(3, 3, 2, 2, 2),
par.settings = list(
superpose.line = list(col = colors),
superpose.symbol = list(col = colors)
),
grid = TRUE,
xlim = clim,
ylim = alim,
auto.key = list(
lines = TRUE,
points = FALSE
),
xlab = 'C (microbar)',
ylab = 'Rates (micromol / m^2 / s)',
main = 'FvCB Model'
)
} else if (plot_type == 2) {
xyplot(
An + Ac + Aj + Ap ~ C_seq,
data = pseudo_fvcb_model_res,
type = 'l',
lty = c(1, 2, 2, 2),
lwd = c(3, 2, 2, 2),
par.settings = list(
superpose.line = list(col = colors),
superpose.symbol = list(col = colors)
),
grid = TRUE,
xlim = clim,
ylim = alim,
auto.key = list(
lines = TRUE,
points = FALSE
),
xlab = 'C (microbar)',
ylab = 'Rates (micromol / m^2 / s)',
main = 'Pseudo-FvCB Model'
)
} else if (plot_type == 3) {
xyplot(
fvcb_model_res$An + pseudo_fvcb_model_res$An ~ C_seq,
type = 'l',
lty = c(1, 2),
lwd = c(3, 2),
par.settings = list(
superpose.line = list(col = colors),
superpose.symbol = list(col = colors)
),
grid = TRUE,
xlim = clim,
ylim = alim,
xlab = 'C (microbar)',
ylab = 'An (micromol / m^2 / s)',
main = 'FvCB vs. Pseudo-FvCB Models',
auto.key = list(
text = c('FvCB model', 'Pseudo-FvCB model'),
lines = TRUE,
points = FALSE
)
)
}
}
Finally, we will also define some default parameter values to use for our
simulations; these correspond to bright conditions with $\alpha = 0$. All values
are from the caption of Figure 2.6 in @caemmerer_biochemical_2000 with the
exception of $O$, which was not specified in the caption. The value for $O$
chosen here is approximately the atmospheric partial pressure of O~2~.
defaults <- list(
Vcmax = 100, # micromol / m^2 / s
Kc = 259, # microbar
O = 200, # mbar
Ko = 179, # mbar
J = 170, # micromol / m^2 / s
Gstar = 38.6, # microbar
Tp = 11.8, # micromol / m^2 / s
alpha = 0, # dimensionless
Rd = 1 # micromol / m^2 / s
)
Each of the following sections will include three separate figures representing
the model outputs at a certain set of conditions:
-
A figure showing $W_c$, $W_j$, $W_p$, $V_c$, and $A_n$ (calculated using
Equations \@ref(eq:anfinal) and \@ref(eq:mincjp)). This figure shows how the
net CO~2~ assimilation rate is determined in the FvCB model.
-
A figure showing $A_c$, $A_j$, $A_p$ and $A_n$ (calculated using Equation
\@ref(eq:analtnew)). This figure shows how the net CO~2~ assimilation rate is
determined in the pseudo-FvCB model.
-
A figure showing $A_n$ as calculated by the two different models,
highlighting the differences between them.
Bright Conditions, $\alpha = 0$
In this case, the only difference between the models occurs for $C < \Gamma^*$,
where carboxylation is RuBP-saturated in the FvCB model but assimilation is
RuBP-regeneration-limited in the pseudo-FvCB model.
do.call(fvcb_compare, within(defaults, {plot_type = 1}))
do.call(fvcb_compare, within(defaults, {plot_type = 2}))
do.call(fvcb_compare, within(defaults, {plot_type = 3}))
Bright Conditions, $\alpha = 0.5$
Here we set $\alpha$ to 0.5 and reduce $T_p$ to 10 (to make TPU limitations
easier to spot.) As before, there is a difference between the models when
$C < \Gamma^$, where carboxylation is RuBP-saturated in the FvCB model but
assimilation is RuBP-regeneration-limited in the pseudo-FvCB model. There is
also another difference because $Ap$ becomes negative for
$C < \Gamma^ \cdot \left(1 + 3 \cdot \alpha \right)$, leading to an unrealistic
and sudden dip in net assimilation in the pseudo-FvCB model near $C$ = 100
$\mu$bar.
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 1}))
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 2}))
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 3}))
Dim Conditions, $\alpha = 0$
Here we set $J$ to 10, simulating dim light. Again, there is a difference
between the models for $C < \Gamma^*$, but now the opposite scenario occurs with
respect to the bright conditions: carboxylation is RuBP-regeneration-limited in
the FvCB model but assimilation is RuBP-saturated in the pseudo-FvCB model.
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 1}))
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 2}))
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 3}))
Final Remarks
In this document, we have shown that although the pseudo-FvCB model equations
are frequently used by plant biologists, they are not exactly equivalent to the
original FvCB model. Differences occur at low CO~2~ concentrations where
$C < \Gamma^*$; in this situation, the two models always disagree as to which
process limits carboxylation and determines the overall net CO~2~ assimilation
rate. While these differences may not always be consequential (for example,
CO~2~ concentrations are rarely low enough in the context of a typical crop
growth simulation), it is nevertheless important to ensure that any mathematical
model is described accurately, and that any code implementation of a model
faithfully represents it. If this is not the case, then evaluating the ability
of a model to represent reality---an essential goal of computational
modeling---becomes an impossible task. For this reason, we recommend that the
original formulation of the model be preferred to the pseudo-FvCB version when
using the FvCB model.
References
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Overview
The Farquhar-von-Caemmerer-Berry (FvCB) model for C~3~ photosynthesis was first presented in @farquhar_biochemical_1980, and since that initial publication, it has become one of the most important models for understanding the photosynthetic response of plants to key environmental variables such as incident light, CO~2~, and temperature. Many researchers are familiar with the model in the contex of fitting it to experimentally measured CO~2~ response curves, where certain simplifying assumptions can be reasonably applied to the model. However, these assumptions are not reasonable when using the model in a wider range of conditions than those usually encountered during gas exchange measurements. The goal of this document is to identify several possible pitfalls associated with these assumptions and explain how to avoid them when using the FvCB model.
The Central FvCB Equation {#sec:centraleq}
The FvCB model is rooted in a biochemical understanding of the photosynthetic process, especially the carboxylation (or oxygenation) of RuBP by the enzyme Rubisco. At its core, it predicts the net CO~2~ assimilation rate ($A_n$) from the values of several key variables. An initial equation for $A_n$ can be developed from the following considerations:
-
One carboxylation of RuBP assimilates one molecule of CO~2~.
-
Two oxygenations of RuBP produces hazardous byproducts that require the release of one molecule of CO~2~ to neutralize. (Oxygenation of RuBP is often referred to as photorespiration.)
-
Mitochondrial respiration also releases CO~2~ molecules.
Putting this all together, we can write down an initial equation for the net assimilation rate: \begin{equation} A_n = V_c - 0.5 \cdot V_o - R_d, (#eq:an) \end{equation} where $V_c$ is the RuBP carboxylation rate, $V_o$ is the RuBP oxygenation rate, and $R_d$ is the mitochondrial respiration rate. As written, $V_c$, $V_o$, and $R_d$ are all non-negative rates, typically with units of $\mu$mol m$^{-2}$ s$^{-1}$. The first two terms in the equation ($V_c - 0.5 \cdot V_o$) represent the contribution of photosynthesis (or Rubisco) to the net CO~2~ assimilation, while the remaining term ($-R_d$) is the result of mitochondrial respiration, a separate process.
In its current form, this equation is not very easy to interpret or use. There are a few modifications we can use to improve it. One change is to introduce $\phi$, the ratio of RuBP oxygenation to carboxylation: $\phi = V_o / V_c$. Now we can rewrite Equation \@ref(eq:an) as follows: \begin{equation} A_n = V_c \cdot \left( 1 - 0.5 \cdot \phi \right) - R_d. (#eq:anphi) \end{equation}
This version of the equation makes it clear that $\phi$ plays a key role in determining whether RuBP carboxylation is able to outweigh oxygenation; in other words, it determines whether the photosynthetic contribution to the net CO~2~ assimilation rate is positive or negative. If $1 - 0.5 \cdot \phi < 0$, then oxygenation consumes more carbon than carboxylation produces; if $1 - 0.5 \cdot \phi > 0$, then the opposite is true. The crossover point where carboxylation and oxygenation exactly cancel each other out occurs when $1 - 0.5 \cdot \phi = 0$. It can be shown that $\phi$ depends on the relative specificity of Rubisco for CO~2~ versus O~2~ ($S_{c/o}$), the partial pressure of CO~2~ in the vicinity of Rubisco ($C$), and the partial pressure of O~2~ in the vicinity of Rubisco ($O$): \begin{equation} \phi = \frac{1}{S_{c/o}} \cdot \frac{O}{C}. (#eq:s) \end{equation} (See, for example, Equation 2.16 from @caemmerer_biochemical_2000 and the surrounding discussion.)
Equation \@ref(eq:s) tells us that when the amount of CO~2~ near Rubisco increases, or when Rubisco's specificity for CO~2~ increases, $\phi$ decreases and hence the oxygenation rate also decreases. This makes sense at an intuitive level. It also helps to clarify the criteria for photosynthesis to become a net producer of assimilated carbon. One immediate conclusion is that because Rubisco specificity and oxygen levels are generally constant for a particular chloroplast, the only reasonable way to change $\phi$ is to change $C$. But we can still learn more. We have already seen that the crossover occurs when $1 - 0.5 \cdot \phi = 0$ or, equivalently, when $\phi = 2$. Now, we can rewrite this condition using \@ref(eq:s): \begin{equation} \phi = 2 \ \rightarrow \frac{1}{S_{c/o}} \cdot \frac{O}{C} = 2 \ \rightarrow C = \frac{O}{2 \cdot S_{c/o}}. \end{equation}
In other words, photosynthesis is a net producer of carbon whenever $C$ exceeds a special value called $\Gamma^$ defined by: \begin{equation} \Gamma^ = \frac{O}{2 \cdot S_{c/o}}. (#eq:gstar) \end{equation} $\Gamma^$ is often referred to as the "CO~2~ compensation point in the absence of mitochondrial respiration." (In general, the term "CO~2~ compensation point" refers to the value of $C$ where $A_n = 0$.) Note that since atmospheric oxygen levels are fairly constant, $O$ rarely changes under most conditions, so $\Gamma^$ is largely a property of Rubisco; more specifically, it depends primarily on $S_{c/o}$. (The major exception to this is when $O$ is artificially decreased during gas exchange measurements in order to reduce $V_o$.)
With this new concept, we can use Equations \@ref(eq:s) and \@ref(eq:gstar) to rewrite Equation \@ref(eq:anphi) as follows: \begin{equation} A_n = V_c \cdot (1 - \Gamma^* / C) - R_d (#eq:anfinal) \end{equation} This equation contains a great deal of useful information, and we will refer to it as the central equation of the FvCB model. (This is our terminology, and not a standard name for the equation.) It can be found in many notable descriptions of the FvCB model, such as:
-
Equation 16.57 in @farquhar_modelling_1982.
-
Equation 1 in @kirschbaum_temperature_1984.
-
Equation 2.19 in @caemmerer_biochemical_2000.
-
Equation A6 in @von_caemmerer_steady-state_2013.
(A big exception is the original FvCB paper [@farquhar_biochemical_1980], which does not contain this equation!)
A key piece of information that can be learned from this equation is that while the RuBP carboxylation rate $V_c$ can never be negative, the contribution of photosynthesis to the net assimilation rate ($V_c \cdot(1 - \Gamma^ / C)$) can be negative; this will occur whenever $V_c > 0$ and $C < \Gamma^$.
The rest of the FvCB model consists of a method for calculating $V_c$; this will be discussed in the following sections.
Choosing a Carboxylation Rate
In the original FvCB model, two scenarios are considered when determining the RuBP carboxylation rate $V_c$:
-
When there is plenty of RuBP available for Rubisco to act upon, the carboxylation rate is said to be RuBP-saturated (or Rubisco-limited). In this case, it is common to denote the carboxylation rate by $W_c$. The equation for $W_c$ and its derivation can be found in many places, such as Equation 1.9 in @caemmerer_biochemical_2000 and the surrounding discussion.
-
Following a carboxylation or an oxygenation, each RuBP molecule must be "regenerated," that is, returned to its original state and ready for another carboxylation or oxygenation. If the RuBP regeneration rate is too slow, it may impose a new maximum on the carboxylation rate, which is said to be RuBP-regeneration-limited. In this case, it is common to denote the carboxylation rate by $W_j$. The equation for $W_j$ and its derivation can be found in many places, such as Equations 2.21 and 2.22 in @caemmerer_biochemical_2000 and the surrounding discussion. Sometimes this rate is referred to as the electron-transport-limited rate, but this terminology is becoming less common since it has been realized that other factors besides electron transport can limit RuBP regeneration.
For brevity, we will simply reproduce the equations here without their derivations: \begin{equation} W_c = \frac{V_{cmax} \cdot C}{C + K_c \cdot \left( 1 + O / K_o \right)} (#eq:wc) \end{equation} and \begin{equation} W_j = \frac{J \cdot C}{4 \cdot C + 8 \cdot \Gamma^*}, (#eq:wj) \end{equation} where $V_{cmax}$ is the maximum rate of Rubisco carboxylation activity, $K_c$ and $K_o$ are Michaelis-Menten constants for CO~2~ and O~2~, and $J$ is the RuBP regeneration rate, which may depend on several external factors such as the incident irradiance. $V_{cmax}$, $K_c$, and $K_o$ are all positive, while $J$ is non-negative.
Note that several different versions of Equation \@ref(eq:wj) can be found in the literature, where the only changes are to the coefficients in the denominator ($4$ and $8$ in our version; see Equation 16.60 in @farquhar_modelling_1982 or Section 2.4.2 from @caemmerer_biochemical_2000 for more information). These coefficients are related to energy requirements involved with the electron transport chain, and different estimates exist for these requirements. Because of these differences, the coefficients can sometimes be considered as variables in the equation instead of taking fixed values.
One significant feature of Equations \@ref(eq:wc) and \@ref(eq:wj) is that neither one can ever return a negative value, as expected (and required) for any equation that calculates a carboxylation rate. Another important thing to notice is that $V_{cmax}$, $K_c$, $K_o$, $O$, and $\Gamma^*$ are generally constant within a chloroplast, so the main environmental influences on $W_c$ and $W_j$ are $C$ and $J$. As $C$ and $J$ vary, one or the other of $W_c$ and $W_j$ may become the larger of the two.
At this point, a key question needs to be addressed: which carboxylation rate ($W_c$ or $W_j$) should be used in Equation \@ref(eq:anfinal) to determine the net CO~2~ assimilation rate? The FvCB authors provide an answer: we should choose whichever rate is smaller for the current set of conditions (as specified by $C$, $J$, etc). Intuitively, this makes sense. It doesn't matter how many RuBP molecules could be carboxylated by Rubisco if the RuBP is not being regenerated fast enough; alternatively, it doesn't matter how many RuBP molecules are being regenerated if Rubisco is unable to carboxylate them. Converting this idea to mathematical form, we arrive at another essential equation: \begin{equation} V_c = \min { W_c, W_j } (#eq:mincj) \end{equation}
This is the most common way to choose between the rates, but it does produce very sudden transitions between RuBP-saturated and RuBP-regeneration-limited carboxylation. Sometimes alternatives to a simple minimum are used, such as a quadratic mixing method, that produce smoother transitions. For example, see @kirschbaum_temperature_1984.
Pitfall: Limitations at Low CO~2~ Concentration {#sec:pitfallci}
People who are familiar with analyzing experimentally measured A-Ci curves may be under the impression that at low $C$, carboxylation (and hence net CO~2~ assimilation) is always RuBP-saturated (or Rubisco-limited). Here, "low $C$" may mean below $\Gamma^*$, below the CO~2~ compensation point, or possibly below some other threshold. For example, a paper with an excellent discussion of C~3~ A-Ci curve fitting [@sharkey_fitting_2007] includes the following discussion of limitations to assimilation:
The routine described here requires identifying whether a data point is limited by Rubisco, RuBP regeneration or TPU. A good starting point is to assign points above 30 Pa as RuBP-regeneration-limited and points below 20 Pa as Rubisco-limited; points between 20 and 30 Pa might be either.
While this may be good advice for interpreting an A-Ci curve, it is not generally true, and instead relies on an unstated assumption that $J$ is quite high (an assumption often met while measuring an A-Ci curve).
There is nothing in the FvCB model equations (Equations \@ref(eq:anfinal), \@ref(eq:wc), \@ref(eq:wj), and \@ref(eq:mincj)) that enforces $W_c < W_j$ below some threshold value of $C$. In fact, there is a simple counterexample demonstrating a situation where $W_j < W_c$ at low $C$: darkness. Darkness ensures that $J = 0$ and hence $W_j \leq W_c$ for all values of $C$.
It is possible to find the point where the process limiting the carboxylation rate changes; this can be done by setting $W_c = W_j$ and solving for $C$. The result is given by Equation 2.52 in @caemmerer_biochemical_2000, where this special value of $C$ is called $C_c$: \begin{equation} C_c = \frac{K_c \cdot \left( 1 + O / K_o \right) \cdot J / \left( 4 \cdot V_{cmax} \right) - 2 \cdot \Gamma^}{1 - J / \left( 4 \cdot V_{cmax} \right)} (#eq:cc) \end{equation} From a physical point of view, $C_c$ cannot be negative; however, Equation \@ref(eq:cc) can mathematically produce negative values for $C_c$. When this happens, it simply means that there is no crossover, and one or the other process limits carboxylation for all realistic values of $C$. This occurs whenever the denominator and numerator have opposite signs. The denominator becomes negative when $4 \cdot V_{cmax} < J$. This is a situation where $V_{cmax}$ is small and carboxylation is always RuBP-saturated. On the other hand, the numerator becomes negative when $J < 8 \cdot V_{cmax} \cdot \Gamma^ / \left[ K_c \cdot \left( 1 + O / K_o \right) \right]$. This is a situation where $J$ is small and carboxylation is always RuBP-regeneration-limited. (Of course, $C_c$ will be positive if both the numerator and denominator are negative, but this is unlikely to happen for realistic values of $K_c$, $K_o$, $O$, and $\Gamma^*$.) Even when $C_c$ is positive, it may not be realistically obtainable; for example, if $J$ is just a tiny bit smaller than $4 \cdot Vcmax$, $C_c$ can take very large values that would only be achievable in a gas evironment of nearly pure CO~2~. Essentially, this analysis of Equation \@ref(eq:cc) tells us that there is no simple rule for determining which process limits carboxylation in the FvCB model, and it is even possible that a crossover between limiting processes may not occur (see Figure \@ref(fig:cc)).
#| fig.cap = "Solution space of Equation \\@ref(eq:cc) indicating where a #| crossover is impossible because `Cc` is negative (pink), unlikely because #| `Cc` is too large (dark blue), or otherwise possible (white). The upper #| pink region is bounded from below by #| `Vcmax = J * [ Kc * (1 + O / Ko)] / (8 * GammaStar)` and the lower pink #| region is bounded from above by `Vcmax = J / 4`. In this calculation, #| `Kc` = 259 microbar, `O` = 200 mbar, `Ko` = 179 mbar, `GammaStar` = #| 38.6 microbar, and `Cc` is considered to be too large when it exceeds #| 2000 microbar." # Define a function that calculates the crossover point and returns an # indication of whether it can or cannot occur. Here, -1 means Cc is negative # and a crossover is impossible; 1 means Cc exceeds a threshold value (2000 # microbar) and is unlikely to be realistically achievable; 0 means that a # crossover occurs at a reasonable value. crossover <- function( J, # micromol / m^2 / s Vcmax # micromol / m^2 / s ) { Kc <- 259 # microbar O <- 200 # mbar Ko <- 179 # mbar Gstar <- 38.6 # microbar jv <- J / (4 * Vcmax) # dimensionless Cc <- (Kc * (1.0 + O / Ko) * jv - 2.0 * Gstar) / (1.0 - jv) # microbar if (Cc < 0) { -1 } else if (Cc > 2000) { 1 } else { 0 } } # Set up the input values and the output matrix npts <- 1001 j_seq <- seq(0, 100, length.out = npts) v_seq <- seq(1, 100, length.out = npts) z_cc <- matrix(nrow = length(j_seq), ncol = length(v_seq)) # Fill in the matrix for (i in seq_along(j_seq)) { for (j in seq_along(v_seq)) { z_cc[i,j] <- crossover(j_seq[i], v_seq[j]) } } # Make the image cc_image <- list(x = j_seq, y = v_seq, z = z_cc) # Make a custom image plotting function plot_image <- function(F_image, ...) { cols <- c('#F2CEE6', '#FFFFFF', '#7798AB') breaks <- c(-10, -0.5, 0.5, 10) image( F_image, breaks = breaks, col = cols, useRaster = TRUE, ... ) } # Plot the image plot_image( cc_image, xlab = 'J (micromol / m^2 / s)', ylab = 'Vcmax (micromol / m^2 / s)' )
Pitfall: Choosing Minimal Assimilation Rates {#sec:pitfallassim}
When carboxylation is RuBP-saturated, the corresponding net CO~2~ assimilation rate (called $A_c$ rather than $A_n$ to indicate the type of limitation) can be found by combining Equations \@ref(eq:anfinal) and \@ref(eq:wc) as follows: \begin{equation} A_c = W_c \cdot (1 - \Gamma^ / C) - R_d = \frac{V_{cmax} \cdot \left( C - \Gamma^ \right)}{C + K_c \cdot \left( 1 + O / K_o \right)} - R_d. (#eq:ac) \end{equation} Likewise, when carboxylation is RuBP-regeneration-limited, the corresponding net CO~2~ assimilation rate (called $A_j$) can be found by combining Equations \@ref(eq:anfinal) and \@ref(eq:wj) as follows: \begin{equation} A_j = W_j \cdot (1 - \Gamma^ / C) - R_d = \frac{J \cdot \left( C - \Gamma^ \right)}{4 \cdot C + 8 \cdot \Gamma^*} - R_d. (#eq:aj) \end{equation}
With this in mind, the FvCB model is sometimes expressed in the following form, which is a bit simpler than using the version above because it bypasses the separate calculation of carboxylation rates and only requires three equations instead of four: \begin{equation} A_n = \min { A_c, A_j }. (#eq:analt) \end{equation} This equation can be found in many publications, and the earliest instance may be Equation A1 from @collatz_physiological_1991. In that paper, the authors state
Equation (A1) is equivalent in form to that proposed by Kirschbaum and Farquhar (1984), but our definitions of $J_C$, $J_E$, and $J_S$ differ somewhat from theirs.
(Note that @kirschbaum_temperature_1984 uses $W_c$ and $W_j$, as mentioned above.) Unfortunately, the claim from this paper is simply untrue because Equations \@ref(eq:anfinal) and \@ref(eq:mincj) produce different results from Equation \@ref(eq:analt).
To see why, suppose that $W_c < W_j$ and $C < \Gamma^$. In this case, $\min { W_c, W_j } = W_c$ and so $A_n = A_c$ according to the FvCB model. However, $1 - \Gamma^ / C$ is negative, so $A_c = W_c \cdot \left( 1 - \Gamma^ / C \right) - R_d$ is also negative. Likewise, $A_j$ is negative, but its magnitude is larger than $A_c$ because $W_c < W_j$. So, in this case, $\min { A_c, A_j } = A_j$. Thus, Equation \@ref(eq:analt) has chosen the incorrect limiting factor. In fact, Equation \@ref(eq:analt) always chooses the wrong limiting factor when $C < \Gamma^$.
When Equation \@ref(eq:analt) is implemented in computer code to aid with
fitting A-Ci curves or simulating photosynthesis, it is often paired with an
additional condition like if (Aj < 0) {Aj = 0}. This is an attempt to fix the
problem of Equation \@ref(eq:analt) choosing the wrong limiting factor at low
$C$. Unfortunately, this solution is based on another misunderstanding where it
is incorrectly assumed that carboxylation is always RuBP-saturated at low $C$
(see Section \@ref(sec:pitfallci)). Although this "partial solution" might not
cause any problems when fitting A-Ci curves, it will not produce the FvCB
model's true output when $J$ is low.
Because of this subtle issue for $C < \Gamma^*$, it is better to avoid Equation \@ref(eq:analt) in favor of Equations \@ref(eq:anfinal) and \@ref(eq:mincj). Although Equation \@ref(eq:analt) may seem like it is simply an alternative way to specify the FvCB model, it is actually not, and it would be more accurate to consider it as a separate "pseudo-FvCB" model, a term we will continue to use in the rest of this document. Examples comparing the FvCB model against the pseudo-FvCB model can be found in Section \@ref(sec:examples).
Including Triose Phosphate Utilization
The sum of the chemical reactions in the Calvin–Benson–Bassham (CBB) cycle can be written as \begin{multline} \qquad 3 \, \mathrm{CO_2} + 6 \, \mathrm{NADPH} + 6 \, \mathrm{H^+} + 9 \, \mathrm{ATP} + 5 \, \mathrm{H_2O} \rightarrow \ \mathrm{(G3P)} + 6 \, \mathrm{NADP^+} + 9 \, \mathrm{ADP} + 8 \, \mathrm{P_i}, \qquad (#eq:cbb) \end{multline} where G3P refers to glyceraldehyde 3-phosphate (also known as triose phosphate) and P~i~ refers to inorganic phosphate (an ion). Each molecule of ATP is synthesized from three inorganic phosphate ions; thus, the CBB reaction requires nine P~i~ ions, eight of which remain free at the end. The missing inorganic phosphate is contained in the triose phosphate molecule, which is exported from the chloroplast and eventually used to form monosaccharide sugars. Since the CBB reaction requires three CO~2~ molecules, each of which originate from a RuBP carboxylation, we can see that each carboxylation effectively consumes one third of an inorganic phosphate. By separately considering the impact of RuBP oxygenation (not included here), it can be shown that the overall rate of inorganic phosphate consumption during photosynthesis ($R_{pc}$) can be given by \begin{equation} R_{pc} = \frac{V_c}{3} - \frac{V_o}{6} - \frac{\alpha \cdot V_o}{2}, \end{equation} where $0 \leq \alpha \leq 1$ is the fraction of glycolate carbon not returned to the chloroplast (see Section 2.4.3 in @caemmerer_biochemical_2000). As in Section \@ref(sec:centraleq), we can use $V_o = \phi \cdot V_c$ and $\phi / 2 = \Gamma^ / C$ to rewrite this equation in terms of $\Gamma^$ and $C$: \begin{equation} R_{pc} = \frac{V_c}{3 \cdot C} \cdot \left[ C - \Gamma^* \left( 1 + 3 \cdot \alpha \right) \right]. (#eq:rpc) \end{equation}
The inorganic phosphate supply is replenished when triose phosphate is used to form sugars; although it happens elsewhere, the inorganic phosphate liberated during this reaction is returned to the chloroplast. Under some circumstances, the rate of triose phosphate utilization (TPU) is slow enough to limit the inorganic phosphate available for use in the choloroplast for photosynthesis; in this case, the carboxylation rate is said to be TPU-limited and $R_{pc} = T_p$, where $T_p$ is the rate of triose phosphate utilization. The expression for $R_{pc}$ from Equation \@ref(eq:rpc) can be substuted into $R_{pc} = T_p$ and solved for the corresponding carboxylation rate, which is referred to as $W_p$ in this case: \begin{equation} W_p = \frac{3 \cdot T_p \cdot C}{C - \Gamma^* \cdot \left( 1 + 3 \cdot \alpha \right)}. (#eq:wp) \end{equation}
This represents a third possible limitation to the carboxylation rate in addition to the RuBP-saturated and RuBP-regeneration-limited rates. While this limitation was not included in the original FvCB model, its importance was realized soon afterwards [@sharkey_photosynthesis_1985; @harley_improved_1991] and it has since become a standard part of the model. Since we are now considering a new process in the model, we must update Equation \@ref(eq:mincj) to include $W_p$: \begin{equation} V_c = \min { W_c, W_j, W_p }. (#eq:mincjp) \end{equation}
Pitfall: Impossible Limitations
Inspecting Equation \@ref(eq:wp), we can see that when $C < \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, the denominator of the equation is negative, and hence $W_p < 0$. However, $W_p$ represents a carboxylation rate, which cannot be negative. So what's going on here? One way to understand the situation is to evaluate Equation \@ref(eq:rpc) with $C$ set to $\Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$. Doing so, we find that the rate of inorganic phosphate consumption in this case is exactly 0. In other words, when $C$ is low enough, photosynthesis is not consuming any inorganic phosphate, and it is therefore impossible for carboxylation to be limited by TPU. One way to indicate this restriction is to modify Equation \@ref(eq:wp) to the following: \begin{equation} W_p = \begin{cases} \infty \;, & 0 \leq C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right) \[1ex] \frac{3 \cdot T_p \cdot C}{C - \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)} \;, & \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right) < C \end{cases} (#eq:wpnew) \end{equation} With this modification, $W_p$ is never negative, and it can never be the smallest of $W_c$, $W_j$, and $W_p$ when $C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, so we can safely use it in conjunction with Equation \@ref(eq:mincjp).
When carboxylation is TPU-limited, the corresponding net CO~2~ assimilation rate (called $A_p$ rather than $A_n$ to indicate the type of limitation) can be found by combining Equations \@ref(eq:anfinal) and \@ref(eq:wp) as follows: \begin{equation} A_p = W_p \cdot (1 - \Gamma^ / C) - R_d = \frac{3 \cdot T_p \cdot \left( C - \Gamma^ \right)}{C - \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)} - R_d. (#eq:ap) \end{equation} This equation is often incorporated into the pseudo-FvCB model discussed in Section \@ref(sec:pitfallassim); in this case, Equation \@ref(eq:analt) is updated to the following: \begin{equation} A_n = \min { A_c, A_j, A_p }. (#eq:analtnew) \end{equation} See, for example, Equations 2.26 and 2.27 in @caemmerer_biochemical_2000 (but be aware that Equation 2.26 has a typo in the denominator). Using Equation \@ref(eq:analtnew) without considering the domain over which Equation \@ref(eq:ap) is valid can cause some additional issues beyond the ones discussed in Section \@ref(sec:pitfallassim); in particular, the net assimilation rate may become TPU-limited when $C < \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, although this should not be possible. This is just one more reason to choose a minimal carboxylation rate (Equation \@ref(eq:mincjp)) rather than a minimal net assimilation rate (Equation \@ref(eq:analtnew)) when using or describing the FvCB model. Examples comparing the FvCB model against the pseudo-FvCB model can be found in Section \@ref(sec:examples).
Handling C = 0
One issue with Equation \@ref(eq:anfinal) is that its value becomes undefined when $C = 0$. This is not a significant issue because $C > 0$ in any realistic situation. However, it is possible to extend the model to $C = 0$ by considering the limiting value as $C$ approaches 0 from the right.
First, because carboxylation cannot be TPU-limited when $C \leq \Gamma^ \cdot \left( 1 + 3 \cdot \alpha \right)$, it cannot be TPU-limited when $C = 0$, and there is no need to consider it here. Next, note that although $1 - \Gamma^ / C$ is undefined when $C = 0$, $A_c$ and $A_j$ (Equations \@ref(eq:ac) and \@ref(eq:aj)) are not. In fact, we have the following when $C = 0$: \begin{equation} A_c \big|{C = 0} = - \frac{\Gamma^ \cdot V_{cmax}}{K_c \cdot \left( 1 + O / K_o\right)} - R_d \end{equation} and \begin{equation} A_j \big|{C = 0} = - \frac{J}{8} - R_d. \end{equation} Remembering that Equation \@ref(eq:analt) _always makes the wrong choice for $C < \Gamma^$, we can reverse its behavior to find $A_n$ when $C = 0$: \begin{equation} A_n \big|{C = 0} = \max \bigl{ A_c \big|{C = 0}, A_j \big|{C = 0} \bigr}. \end{equation}
Examples Comparing the FvCB and Pseudo-FvCB Models {#sec:examples}
Here we will use R to demonstrate some of the differences between the FvCB model and the pseudo-FcVB model. First, we will define a function that implements the FvCB model:
fvcb_model <- function( C, # microbar Vcmax, # micromol / m^2 / s Kc, # microbar O, # mbar Ko, # mbar J, # micromol / m^2 / s Gstar, # microbar Tp, # micromol / m^2 / s alpha, # dimensionless Rd # micromol / m^2 / s ) { Wc <- Vcmax * C / (C + Kc * (1 + O / Ko)) Wj <- J * C / (4 * C + 8 * Gstar) Wp <- 3 * Tp * C / (C - Gstar * (1 + 3 * alpha)) Wp[C <= Gstar * (1 + 3 * alpha)] <- Inf Vc <- pmin(Wc, Wj, Wp) An <- (1 - Gstar / C) * Vc - Rd list(Wc = Wc, Wj = Wj, Wp = Wp, Vc = Vc, An = An) }
Now, we will define a function that implements the pseudo-FvCB model; here we will intentionally avoid restricting $A_p$ to its proper $C$ domain, allowing it to become limiting at low $C$:
pseudo_fvcb_model <- function( C, # microbar Vcmax, # micromol / m^2 / s Kc, # microbar O, # mbar Ko, # mbar J, # micromol / m^2 / s Gstar, # microbar Tp, # micromol / m^2 / s alpha, # dimensionless Rd # micromol / m^2 / s ) { Ac <- Vcmax * (C - Gstar) / (C + Kc * (1 + O / Ko)) - Rd Aj <- J * (C - Gstar) / (4 * C + 8 * Gstar) - Rd Ap <- 3 * Tp * (C - Gstar) / (C - Gstar * (1 + 3 * alpha)) - Rd An <- pmin(Ac, Aj, Ap) list(Ac = Ac, Aj = Aj, Ap = Ap, An = An) }
We will also define a function that runs each model across a range of $C$ values for a set of input parameter values and compares the results by creating figures. The code for this function is not included in this article, but can be found in its source file. It will be used to generate the figures in the following sections.
# Load the lattice package for creating graphs library(lattice) # Make a function that runs both versions of the model across a range of C # values and then plots the results fvcb_compare <- function( Vcmax, # micromol / m^2 / s Kc, # microbar O, # mbar Ko, # mbar J, # micromol / m^2 / s Gstar, # microbar Tp, # micromol / m^2 / s alpha, # dimensionless Rd, # micromol / m^2 / s plot_type ) { C_seq <- seq(1, 1000, by = 2) fvcb_model_res <- fvcb_model( C_seq, Vcmax, Kc, O, Ko, J, Gstar, Tp, alpha, Rd ) pseudo_fvcb_model_res <- pseudo_fvcb_model( C_seq, Vcmax, Kc, O, Ko, J, Gstar, Tp, alpha, Rd ) clim <- c(0, 1000) alim <- c(-20, 50) colors <- c( "#000000", "#66C2A5", "#FC8D62", "#8DA0CB", "#E78AC3" ) if (plot_type == 1) { xyplot( An + Vc + Wc + Wj + Wp ~ C_seq, data = fvcb_model_res, type = 'l', lty = c(1, 1, 2, 2, 2), lwd = c(3, 3, 2, 2, 2), par.settings = list( superpose.line = list(col = colors), superpose.symbol = list(col = colors) ), grid = TRUE, xlim = clim, ylim = alim, auto.key = list( lines = TRUE, points = FALSE ), xlab = 'C (microbar)', ylab = 'Rates (micromol / m^2 / s)', main = 'FvCB Model' ) } else if (plot_type == 2) { xyplot( An + Ac + Aj + Ap ~ C_seq, data = pseudo_fvcb_model_res, type = 'l', lty = c(1, 2, 2, 2), lwd = c(3, 2, 2, 2), par.settings = list( superpose.line = list(col = colors), superpose.symbol = list(col = colors) ), grid = TRUE, xlim = clim, ylim = alim, auto.key = list( lines = TRUE, points = FALSE ), xlab = 'C (microbar)', ylab = 'Rates (micromol / m^2 / s)', main = 'Pseudo-FvCB Model' ) } else if (plot_type == 3) { xyplot( fvcb_model_res$An + pseudo_fvcb_model_res$An ~ C_seq, type = 'l', lty = c(1, 2), lwd = c(3, 2), par.settings = list( superpose.line = list(col = colors), superpose.symbol = list(col = colors) ), grid = TRUE, xlim = clim, ylim = alim, xlab = 'C (microbar)', ylab = 'An (micromol / m^2 / s)', main = 'FvCB vs. Pseudo-FvCB Models', auto.key = list( text = c('FvCB model', 'Pseudo-FvCB model'), lines = TRUE, points = FALSE ) ) } }
Finally, we will also define some default parameter values to use for our simulations; these correspond to bright conditions with $\alpha = 0$. All values are from the caption of Figure 2.6 in @caemmerer_biochemical_2000 with the exception of $O$, which was not specified in the caption. The value for $O$ chosen here is approximately the atmospheric partial pressure of O~2~.
defaults <- list( Vcmax = 100, # micromol / m^2 / s Kc = 259, # microbar O = 200, # mbar Ko = 179, # mbar J = 170, # micromol / m^2 / s Gstar = 38.6, # microbar Tp = 11.8, # micromol / m^2 / s alpha = 0, # dimensionless Rd = 1 # micromol / m^2 / s )
Each of the following sections will include three separate figures representing the model outputs at a certain set of conditions:
-
A figure showing $W_c$, $W_j$, $W_p$, $V_c$, and $A_n$ (calculated using Equations \@ref(eq:anfinal) and \@ref(eq:mincjp)). This figure shows how the net CO~2~ assimilation rate is determined in the FvCB model.
-
A figure showing $A_c$, $A_j$, $A_p$ and $A_n$ (calculated using Equation \@ref(eq:analtnew)). This figure shows how the net CO~2~ assimilation rate is determined in the pseudo-FvCB model.
-
A figure showing $A_n$ as calculated by the two different models, highlighting the differences between them.
Bright Conditions, $\alpha = 0$
In this case, the only difference between the models occurs for $C < \Gamma^*$, where carboxylation is RuBP-saturated in the FvCB model but assimilation is RuBP-regeneration-limited in the pseudo-FvCB model.
do.call(fvcb_compare, within(defaults, {plot_type = 1}))
do.call(fvcb_compare, within(defaults, {plot_type = 2}))
do.call(fvcb_compare, within(defaults, {plot_type = 3}))
Bright Conditions, $\alpha = 0.5$
Here we set $\alpha$ to 0.5 and reduce $T_p$ to 10 (to make TPU limitations easier to spot.) As before, there is a difference between the models when $C < \Gamma^$, where carboxylation is RuBP-saturated in the FvCB model but assimilation is RuBP-regeneration-limited in the pseudo-FvCB model. There is also another difference because $Ap$ becomes negative for $C < \Gamma^ \cdot \left(1 + 3 \cdot \alpha \right)$, leading to an unrealistic and sudden dip in net assimilation in the pseudo-FvCB model near $C$ = 100 $\mu$bar.
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 1}))
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 2}))
do.call(fvcb_compare, within(defaults, {alpha = 0.5; Tp = 10; plot_type = 3}))
Dim Conditions, $\alpha = 0$
Here we set $J$ to 10, simulating dim light. Again, there is a difference between the models for $C < \Gamma^*$, but now the opposite scenario occurs with respect to the bright conditions: carboxylation is RuBP-regeneration-limited in the FvCB model but assimilation is RuBP-saturated in the pseudo-FvCB model.
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 1}))
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 2}))
do.call(fvcb_compare, within(defaults, {J = 10; plot_type = 3}))
Final Remarks
In this document, we have shown that although the pseudo-FvCB model equations are frequently used by plant biologists, they are not exactly equivalent to the original FvCB model. Differences occur at low CO~2~ concentrations where $C < \Gamma^*$; in this situation, the two models always disagree as to which process limits carboxylation and determines the overall net CO~2~ assimilation rate. While these differences may not always be consequential (for example, CO~2~ concentrations are rarely low enough in the context of a typical crop growth simulation), it is nevertheless important to ensure that any mathematical model is described accurately, and that any code implementation of a model faithfully represents it. If this is not the case, then evaluating the ability of a model to represent reality---an essential goal of computational modeling---becomes an impossible task. For this reason, we recommend that the original formulation of the model be preferred to the pseudo-FvCB version when using the FvCB model.
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