2 Theory
2.1 Photosynthesis
2.1.1 Instantaneous Photosynthesis
Photosynthesis is the process by which plants use the energy from sunlight to turn CO\(_2\) and H\(_2\)O into biomass and O\(_2\). A commonly used photosynthesis model, the Farquhar-vonCaemmerer-Berry Model (Berry & Bjorkman, 1980) describes gross photosynthesis (\(A_{gross}\)) as the smaller of two reactions:
\[ A_{gross} = \text{min}(A_c,\; A_j) \tag{2.1}\]
\(A_c\) is the carboxylation-limited photosynthesis rate. Carboxylation is the reaction of CO\(_2\) with the enzyme key photosynthesis enzyme Rubisco. Therefore, the maximum rate of carboxylation (\(V_{cmax}\)) directly controls \(A_c\). Additionally, \(A_c\) is constrained depends on the amount of CO\(_2\) that is available in the leaf (\(c_i\)), and is constrained by additional enzyme-dynamics that we skip for now (given by the Michaelis-Menten Constant \(K\) and the photo respiration \(\Gamma^*\)). \(A_c\) is defined as:
\[ A_c = V_{cmax}\frac{c_i - \Gamma^*}{c_i-K}\ \tag{2.2}\]
\(A_j\) is the RuBP-photosynthesis rate. RuBP is another crucial enzyme in the photosynthetic cycle and its regeneration depends on the electron transport coming from harvesting the incoming light. Thus, this photosynthesis rate is also often called “electron transport-limited rate”. \(A_j\) is therefore directly dependent on \(J\), the electron transport rate, as shown below:
\[ A_j = \frac{J}{4}\frac{c_i-\Gamma^*}{c_i+2\Gamma^*} \tag{2.3}\]
The electron transport rate is further defined by a saturating function, whereby \(J\) increases with the absorbed light (\(I_{abs\)) up to an maximum electron transport rate \(J_{max}\):
\[ J=\frac{4 \varphi I_{abs}}{\sqrt{1+\left(\dfrac{4 \varphi I_{abs}}{J_{max}}\right)^{2}}} \tag{2.4}\]
Here, \(\varphi\) is the quantum yield efficiency of photosynthesis and describes how many moles of carbon are bound per mole of electrons. Technically, eight electrons are necessary to bind one CO\(_2\), so the quantum yield efficiency is often set to 0.125 (but may vary in reality due to abiotic and biotic conditions).
For now, this is all we need to know about how to model the instantaneous response of photosynthesis, meaning how photosynthesis responds to immediate (think of milliseconds and faster) changes in temperature, light, and so on. Key to take away from this short primer is that a photosynthesis model requires the three variables: \(c_i\), \(V_{cmax}\), \(J_{max}\). We know from the literature these leaf traits are not fixed in time and that plants adjust them over time. This means that plants acclimate to their environment by changing their physiology based on the environmental condition they experience over weeks (Smith & Dukes, 2013). As described below, this knowledge helps us to predict long-term average leaf traits from which we can then derive the instantaneous response of photosynthesis.
Before we get to the acclimated part, we must also understand how CO\(_2\) actually enters a leaf. All gaseous exchange with the atmosphere goes through small openings in a leaf, called the stomata, which are actively opened and closed by the leaf itself. As stomata open CO\(_2\) enters and H\(_2\)O leaves the leaf. This is a classic diffusion, which is controlled by the diffusivity of the compound in question and the gradient in compound concentration in the leaf and in the atmosphere. Using Fick’s law, the transpiration of H\(_2\)O is:
\[ E = g_s 1.6D \tag{2.5}\]
Here, \(E\) is the transpiration of water, \(D\) is the vapor pressure deficit (the difference between actual and saturated water vapor pressure), \(g_s\) is the stomatal conductance of CO\(_2\) (the \(1.6\) are to transform the H\(_2\)O flux into units of CO\(_2\)).
2.1.2 Acclimated Photosynthesis
2.1.2.1 The Original P-Model
To predict the three key variables mentioned above, we use the “P-Model” (the “P” stands for “Productivity”) formulation that was originally published by Prentice et al. (2014). The P-Model makes assumption that the plants optimize their physiology towards average long-term climatic conditions. This means that over a timescale of days to weeks, plants acclimate to their environment. Making this assumption allows to predict acclimated values of \(c_i\), \(V_{cmax}\), \(J_{max}\) across ecosystems and species, knowing nothing else but the climatic condition of that ecosystem.
The Coordination Hypothesis states that a leave coordinates its physiology so that during midday conditions (high light, high temperature) neither photosynthesis rate is limiting (Maire et al., 2012):
\[ A_{gross} = A_c = A_j \tag{2.6}\]
The Least-Cost Hypothesis applies the economic principle of resource optimization to a plant. To maintain its product (biomass, here \(A_{gross}\)), a plant needs to balance its carbon resource investments for building its “infrastructure” to bind CO\(_2\) and to maintain a matching supply of H\(_2\)O (Prentice et al., 2014). In other words, there must exists an optimal ratio of internal to atmospheric CO\(_2\) concentrations (\(\chi\)), so that the marginal costs for maintaining electron transport and carboxylation are minimal:
\[ a \frac{\partial(E / A_{gross})}{\partial \chi}+b \frac{\partial\left(V_{cmax} / A_{gross}\right)}{\partial \chi} = 0 \tag{2.7}\]
Here, \(a\) is the marginal cost to maintain a given water transport rate, \(b\) is the marginal cost to maintain a given carboxylation transport rate. Accounting for the temperature dependency of water (\(\eta^*\)), we can express the marginal costs as a ratio - the ratio between the marginal costs of maintaining carboxylation over maintaining water transport:
\[ \beta = b/a \,\eta^* \tag{2.8}\]
Following the rationale of the cost minimization, the P-Model further uses the criteria that a leaf also minimizes the marginal carbon cost (\(c\)) for maintaining its electron transport capacity:
\[ \frac{\partial A}{\partial J_{\max }}=c \tag{2.9}\]
Applying these equations predicts \(V_{cmax}\), \(g_{s}\), and \(J_{max}\)jointly and how they vary with climatic conditions for absorbed sunlight (\(I_{abs}\)), ambient CO\(_{2}\) (\(c_a\)), vapor pressure deficit (\(D\)), air temperature (\(T\)), and atmospheric pressure (\(P_{atm}\)). Note that a detailed explanation and derivation of the P-Model is given in B. D. Stocker et al. (2020) and available as code in the {rpmodel} package (B. Stocker & Hufkens, 2021).
2.1.2.2 A Numerical Approach
The short-coming of the analytic approach to the P-Model describe above is that it takes air temperatures to predict the acclimated traits and cannot account for any de-coupling of leaf and air temperatures. Also, since \(g_s\) is coupled jointly with the other traits, it cannot vary independently to potentially modify the leaf temperature so that the total carbon costs are minimized. To resolve this issue, we can create a numerical minimization problem where each leaf trait is optimized separately - and through the liberation of \(g_s\), we can couple an energy balance to additionally optimize a leaf’s temperature. Following the rationale of the Least-Cost Hypothesis, we use here the following optimization problem:
\[ \frac{\beta \;V_{cmax} + 1.6\;D\;g_s + c \;J_{max} }{A_{gross}} = min. \tag{2.10}\]
Note that we are now lacking a formulation for \(A_{gross}\). However, we can use Fick’s Law again and set the carbon flux equal to assimilation:
\[ A_c \text { or } A_j=g s\left(c_a-c_i\right) \tag{2.11}\]
Inserting \(A_c\) or \(A_c\) creates a quadratic equation that can be solved for in \(c_i\) with the general form:
\[ 0=a c_i^2+b c_i+c \tag{2.12}\]
For \(A_c\), this gives:
\[ A_c=g_s\left(c_a-c_i\right)=V_{c \max } \frac{c_i-\Gamma^*}{c_i+K} \tag{2.13}\] with the parameters:
\[ a=-g_s \quad b=g_s c_a-g_s K-V_{c \max } \quad c=\Gamma^* V_{c \max }+g_s c_a K \tag{2.14}\] For \(A_j\), this gives: \[ A_j=g_s\left(c_a-c_i\right)=\frac{J}{4} \frac{c_i+\Gamma^*}{c_i+2 \Gamma^*} \tag{2.15}\] with the parameters: \[ a=-g_s \quad b=g_s c_a-2 \Gamma^* g_s-J / 4 \quad c=2 \Gamma^* g_s c_a+\Gamma^* J / 4 \tag{2.16}\]
2.2 Energy Balance
A leaf’s temperature is controlled by its gain and loss of energy. The total incoming energy of a leaf is the sum of absorbed solar short-wave and absorbed thermal long-wave radiation. The outgoing energy consists of the long-wave thermal radiation by the leaf, the sensible heat flux (i.e., thermal convection) and the latent heat flux (i.e., energy consumed through vaporization or released from dew). Under steady-state conditions, a constant leaf temperature exists at which incoming and outgoing energy are equal.
For simplicity, we do not delve into the physics of the energy balance. The aspect to understand is that the calculation of the leaf temperature has a re-cursive component to it. For example, to calculate the leaf temperature, one must know the stomatal boundary layer for water. However, this boundary layer itself depends on the leaf temperature. Such recursion complicates the calculation of the leaf temperature.
This tutorial follows a numerical approach to calculate leaf temperatures. The energy balance model implemented here is adapted from the {plantecophys} package (Duursma, 2015). It is based on the model published by Leuning et al. (1995) and Wang & Leuning (1998), where more in-depth descriptions can be found.