A small dynamic leaf-level model predicting photosynthesis in greenhouse tomatoes

Dominique Joubert; Ningyi Zhang; Sarah.R. Berman; Elias Kaiser; Jaap Molenaar; J.D. Stigter

doi:10.1371/journal.pone.0275047

Abstract

The conversion of supplemental greenhouse light energy into biomass is not always optimal. Recent trends in global energy prices and discussions on climate change highlight the need to reduce our energy footprint associated with the use of supplemental light in greenhouse crop production. This can be achieved by implementing “smart” lighting regimens which in turn rely on a good understanding of how fluctuating light influences photosynthetic physiology. Here, a simple fit-for-purpose dynamic model is presented. It accurately predicts net leaf photosynthesis under natural fluctuating light. It comprises two ordinary differential equations predicting: 1) the total stomatal conductance to CO₂ diffusion and 2) the CO₂ concentration inside a leaf. It contains elements of the Farquhar-von Caemmerer-Berry model and the successful incorporation of this model suggests that for tomato (Solanum lycopersicum L.), it is sufficient to assume that Rubisco remains activated despite rapid fluctuations in irradiance. Furthermore, predictions of the net photosynthetic rate under both 400ppm and enriched 800ppm ambient CO₂ concentrations indicate a strong correlation between the dynamic rate of photosynthesis and the rate of electron transport. Finally, we are able to indicate whether dynamic photosynthesis is Rubisco or electron transport rate limited.

Citation: Joubert D, Zhang N, Berman S, Kaiser E, Molenaar J, Stigter J (2023) A small dynamic leaf-level model predicting photosynthesis in greenhouse tomatoes. PLoS ONE 18(3): e0275047. https://doi.org/10.1371/journal.pone.0275047

Editor: Arun Kumar Shanker, Central Research Institute for Dryland Agriculture, INDIA

Received: September 8, 2022; Accepted: March 1, 2023; Published: March 16, 2023

Copyright: © 2023 Joubert et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data files are available from the Kaggle database (https://www.kaggle.com/datasets/dominiquejoubert/photosynthesis).

Funding: This project (No. 17173) was funded by the Netherlands Organisation for Scientific Research (NWO), with contributions by Signify, Glastuinbouw Nederland, Ridder Growing Solutions, and Adviesbureau JFH Snel.

Competing interests: The authors have declared that no competing interests exist.

Abbreviations: Γ*, The Rubisco compensation point [μmolm⁻²s⁻¹]; A_n, Net photosynthetic rate [μmolm⁻²s⁻¹]; c_a, Ambient CO₂ concentration [μmolm⁻²s⁻¹]; c_c, CO₂ concentration in the chloroplast stroma [μmolm⁻²s⁻¹]; c_i, CO₂ concentration inside a leaf [μmolm⁻²s⁻¹]; d, Leaf diameter [m]; G, Steady-state total stomatal conductance to CO₂ diffusion [molm⁻²s⁻¹]; g_m, Mesophyllic conductance [molm⁻²s⁻¹]; g_tc, Total stomatal conductance to CO₂ diffusion [molm⁻²s⁻¹]; I, Irradiance [μmolm⁻²s⁻¹]; J, Steady-state electron transport rate [μmolm⁻²s⁻¹]; K_c, Michaelis-Menten constant for CO₂ [μmolm⁻²s⁻¹]; k_d, Time constant for decreases in g_tc [s]; K_o, Michaelis-Menten constant for O₂ [kPa]; k_u, Time constant for increases in g_tc [s]; O_i, Partial pressure of O₂ in air [kPa]; P_a, Atmospheric pressure [Pa]; R, Ideal gas constant [m³Pa¹mol⁻¹K⁻¹]; R_d, Mitochondrial respiration which includes CO₂ release in light other than photo-respiration [μmolm⁻²s⁻¹]; RH, Relative humidity [%]; T_l, Leaf temperature [K]; V_cmax, Maximum obtainable carboxylation rate [μmolm⁻²s⁻¹]; W_c, Rubisco limited carboxylation rate [μmolm⁻²s⁻¹]; W_j, Electron transport rate limited carboxylation rate [μmolm⁻²s⁻¹]

Introduction

The cultivation of greenhouse crops under optimised conditions will become increasingly important, with the need for year-round crop harvesting under changing environmental conditions as a driving factor. The widespread use of supplemental lighting to optimise growing conditions is a key tool at our disposal. The efficiency of converting light energy into plant photosynthesis is however not always optimal. This is particularly true when lights are first turned on, where time is required for the activation of key photosynthetic enzymes and for adjustments in stomatal pore aperture [1].

Given recent trends in global energy prices and continuous discussions on climate change, our energy footprint associated with the use of supplemental light needs to be reduced. Two potential avenues that may lead to the efficient conversion of light energy into photosynthesis are: 1) the use of light-emitting diodes (LEDs) in supplemental lighting applications. LEDs increase the efficiency with which electrical energy is converted to photosynthetically active radiation (PAR), the radiation with wavelength 400–700nm, which powers photosynthesis [2]. 2) A model-based implementation of “smart” lighting regimens. This approach necessitates a good understanding of how plants harness light energy under natural fluctuating irradiance (I).

Plant responses to fluctuating irradiance occur across numerous levels of complexity, ranging from the whole canopy (at crop level and governed by plant structure) to the biochemistry of a single reaction (at leaf level and governed by physiology) and across different orders of magnitude in time. At the top of the canopy light levels depend on factors such as the solar angle and amount of cloud cover. Lower parts of a crop canopy rely on light in the form of sun-flecks, and the amount of these in turn depends on the canopy’s structure [3].

Light intensity is the most dynamic condition to which greenhouse crops need to respond and can change at time scales ranging from a season (winter versus summer) to less than 1 second (passing clouds) [4]. The variation of incident light on leaves in the upper canopy can have a substantial effect on photosynthesis in this upper layer since it accounts for up to 75% of crop canopy carbon assimilation [5].

At leaf level, a plant’s ability to regulate photosynthesis in response to rapid variations in irradiance may be restricted by the following factors: 1) the opening/closing of stomata, 2) the activation/deactivation of Calvin cycle enzymes, 3) the up-regulation/down-regulation of photoprotective processes [3], 4) transiently changing mesophyllic conductance [6].

Previous leaf-level models have ranged both in complexity and in time scales of prediction. Depending on the research question, a model may for example include detailed mechanisms such as the regulation of enzymatic activity and both photochemical and non-photochemical quenching. The result is a model that comprises a multitude of differential and rate equations, which may include time constants in different scales (as an example consider the e-photosynthesis model published in 2013 [7] with 75 differential equations and approximately 120 rate equations).

These models often include the well-know Farquhar-von Caemmerer-Berry (FvCB) model, which mathematically describes key Calvin cycle processes and linear transport rates [8]. The result is an estimated net photosynthetic rate (A_n) which stems from competitive enzymatic processes involving CO₂ and O₂ binding under different environmental conditions [9]. A brief summary of selected small dynamic models is given in Table 1. We included a summary of the model presented here for comparison.

Download:

Table 1. Summary of selected small dynamic leaf-level photosynthesis models.

(Ordinary differential equations abbreviated as ODEs).

https://doi.org/10.1371/journal.pone.0275047.t001

The rest of this article is organised as follows. In section 1 we provide the theoretical background to the model and describe the plant physiology underpinning the well-known FvCB model. We also discuss three of the factors that may restrict A_n in greater detail. The model is defined in section 2, and the materials and methods used are discussed in section 3. Results and discussion are presented in sections 4 and 5, respectively, and conclusions are given in section 6.

Theory

We briefly introduce three of the factors that affect A_n, and which are included in the model: 1) stomatal conductance, 2) the Rubisco limited carboxylation rate, and 3) the electron transport limited carboxylation rate.

The opening and closing of stomata

Stomata are located on both the upper (u) and lower (l) surfaces of tomato leaves. They are tasked with regulating the flux of gaseous H₂O and CO₂ between a leaf and its surroundings. They do so by adjusting their pore aperture and this is achieved by changing the form of their 2 guard cells. These structures can, therefore, be thought of as conductors of CO₂ diffusion and, therefore, only allow a certain amount of CO₂ to enter a leaf.

Stomatal aperture depends on numerous environmental cues. For C3 plants (plants that allow for the direct carbon fixation of CO₂) under non-limiting growth conditions, pore sizes increase with increased irradiance and low CO₂ concentrations [13].

The metabolic regulation of these guard cells is highly complex and accordingly, an empirical model that predicts dynamic changes in the conductance of CO₂ related to perturbations in irradiance is used here [12]. Refer to S1 File for a discussion on how the total stomatal conductance to CO₂ diffusion (g_tc) is defined.

The FvCB model

This model describes the steady-state net photosynthetic (A_n) rate as [8, 14]: (1) (2) where 1 defines A_n as the difference between the gross photosynthetic rate (A_g) and R_d, the mitochondrial respiration which includes the release of CO₂ in light other than photo-respiration [14]. From 2 follows that A_n can be limited by the Rubisco limited carboxylation rate (W_c), the electron transport limited carboxylation rate (W_j) or the rate at which triose phosphates are utilised (W_p). Opting to keep our model structure concise and the number of unknown system parameters to a minimum, we omit the limiting factor W_p.

The Rubisco limited carboxylation rate (W_c).

Once CO₂ enters a leaf through the stomata, it diffuses through the inter-cellular spaces into the chloroplasts by means of a diffusion gradient between the chloroplast and the rest of the leaf. The numerous photosynthesis reactions that occur inside the chloroplast are summarised in the Calvin cycle, a process which comprises both light dependent and independent reactions.

During the first phase of this cycle, a single CO₂ molecule is fixed onto a Ribulose 1,5-bisphosphate (RuBP) molecule to form two 3-Phosphoglyceric acid (3-PGA) molecules. Important here in the context of modelling enzyme kinetics is that this process is catalysed by the enzyme Ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco), and its activation state is in turn increased by Rubisco activase. The structure of 3-PGA allows it to be combined and rearranged to form sugars which can be transported or stored for energy. The rate at which CO₂ fixation takes place is known as the carboxylation rate. However, Rubisco also catalyses RuBP oxygenation (binding RuBP to O₂). This reduces the efficiency of the Calvin cycle. The rate at which this takes place is called the oxygenation rate.

Mathematically, the Rubisco limited carboxylation rate is given as [8, 14, 15]: (3) where V_cmax is the maximum obtainable carboxylation rate, and c_c is the partial pressure of CO₂ in the chloroplast stroma whereas O_c is the partial pressure of O₂ in the chloroplast stroma. K_c is the Michealis-Menten constant for CO₂, K_o is the Michaelis-Menten constant for O₂, [R] is the concentration of unbound (available) RuBP, and K_r’ is the effective Michealis-Menten constant for RuBP. Assuming that RuBP is in excess, (3) reduces to [16], (4)

A relationship between W_c and the oxygenation rate is introduced in 2 by Γ*, the CO₂ concentration at which oxygenation proceeds at twice the rate of carboxylation causing the photosynthetic uptake of CO₂ to be compensated for by the photorespiratory release of CO₂ [17].

Notice that expression 4 is defined for CO₂ concentrations in the chloroplast stroma (c_c). This concentration can however not be measured directly and so is predicted if A_n, c_i, and g_m are known, by, (5) where g_m is the mesophyllic conductance encountered along the CO₂ diffusion pathway. By assuming that this conductance is infinitely large 5 simplifies to c_c = c_i.

Electron transport limited carboxylation rate (W_j).

The synthesis of RuBP also requires energy in the form of ATP and NADPH, and both ADP and NADP⁺ are continuously converted to these energy supplying molecules in light dependent reactions which are dependent on the rate of electron transport (J). Accordingly, the electron transport limited carboxylation rate (W_j) is given as [8, 14], (6) Eq 6 assumes 4 electrons per carboxylation and oxygenation and so, based on the number of electrons required for NADP+ reduction, the standard values used are 4 and 8. However, there are uncertainties in the relationship between electron transport and ATP synthesis. For example, 4.5 and 10.5 have also been used [17].

Given the above mentioned assumptions, expression 2 can be written as (7)

Temperature and light intensity effects on steady-state photosynthesis

The dependence of key FvCB model parameters on both leaf temperature (T_l) and irradiance (I) was already included in the 1980 publication by Farquhar et al. [8].

Changes in the values of parameters Γ*, K_o, and K_c (in 7) associated with leaf temperature changes are described using the Arrhenius equation [17], (8) (9) (10)

Here, the ideal gas constant R₁, is expressed in units kJmol⁻¹K⁻¹. Constants ΔH_ai and c_i, where i = 1, …, 3, are the respective energies of activation and scaling constants for parameters Γ*, K_o and K_c.

Electron transport becomes limited when insufficient quanta are absorbed [8]. Accordingly, J is modelled as a function of irradiance [18], (11) where parameter J_max is the upper limit to potential chloroplast electron transport determined by the components of the chloroplast electron transport chain [14]. Parameters θ and γ are unit-less (refer to S1 File for details). All parameter values are given in Table 3.

Dynamic model structure

The model we present in this paper comprises only two ordinary differential equations, the first predicting the total stomatal conductance to CO₂ diffusion (g_tc) and the second the CO₂ concentration inside the leaf (c_i). Here, g_tc is the sum of the boundary layer and stomatal conductances (see S1 File). An algebraic relation between the predicted states is used to approximate the net photosynthetic rate (A_n). An overview of the dynamic states, system parameters, model inputs, and the measured output is given by the standard state-space representation, (12) (13)

Function f is defined in Eqs 15–18. States g_tc and c_i, and the predicted A_n are measured model outputs. The three system parameters that need to be inferred from the measured data are k_u, k_d and c₃ in expressions 10, and 15 and 16, respectively. Three environmental conditions, I(t), c_a(t) and T_l(t), are directly measurable and modelled as inputs/disturbances to the system. Predictions for A_n are made using Fick’s law of diffusion. Also known as the net flux of CO₂ that enters a leaf, the dynamic A_n (achieved at a specific light intensity, as opposed to the attainable steady-state values predicted in 7) is calculated as [19, 20] (14) The asymmetric exponential response of stomata to increases and decreases in irradiance has often been reported [21]. This is modelled by introducing 2 time constants to the system, k_u describing the rate of increase in g_tc observed with an increase in irradiance, and k_d describing the rate of decrease in g_tc following a decrease in irradiance [10, 13, 21–25]. The resulting model structure is, (15) (16) G(I(t),c_a(t)) can be interpreted as the steady-state target function of g_tc for a particular combination of I(t) and c_a(t). A description of how G(I(t),c_c(t)) should be calibrated is given in S1 File.

The CO₂ concentration inside the leaf is modelled using a mass balance equation (refer to S1 File for a discussion), (17) The minimum function stems from the FvCB model in Eq 7. Substituting the functions W_c(t) and W_j(t), that have been adjusted to take T_l and I into account, into 17 gives, (18)

Materials and methods

Growing conditions of plants

Tomato plants were cultivated in a climate chamber (size: 16 m²) in Wageningen University &amp; Research, Wageningen, the Netherlands (52°N, 6°E). Seeds were germinated in rockwool plugs (diameter: 2 cm) and transferred to rockwool cubes (10 × 10 × 7 cm) one week after sowing.

Unless stated otherwise, day and night temperatures were set at 23°C and 20°C, respectively. Relative humidity was set at 70%. The CO₂ concentration was kept at ambient (450 ppm). Plants were irrigated automatically twice per day using an ebb & flow system (at 7 AM and 7 PM) with tomato nutrient solution (EC:2.2±0.1 mScm⁻¹, pH:5.5) (see S1 File).

Plants used to estimate V_cmax and R_d.

These plants were exposed to an irradiance of 250 μmolm⁻²s⁻¹ provided by two types of high-pressure sodium lamps (SON-T and HPI-T PLUS, Philips Lighting). The photoperiod in the chamber was 16 hours. The SON-T lamps were switched on one hour before the HPI-T PLUS lamps and were switched off one hour after the HPI-T PLUS lamps in an attempt to mimic the gradual increase and decrease of irradiance during sunrise and sunset.

Plants used to parameterise function J(I(t)).

The plants were exposed to an average irradiance of 200 μmolm⁻²s⁻¹, with irradiance fluctuating between 50 and 500 μmolm⁻²s⁻¹ every minute. The photoperiod in the chamber was 16 hours. The irradiance pattern was randomly changed on a daily basis to simulate natural fluctuations in irradiance. Key properties including the photoperiod, minimum and maximum irradiance, daily average irradiance, and the overall shape of the light pattern were kept the same. Dynamic irradiance was provided by GreenPower LED top lighting compact modules (Philips Lighting).

Plants used to measure photosynthesis under natural fluctuating irradiance.

The plants were brought to the greenhouse compartment to grow for another four weeks. Plants were grown on growth tables in the compartment of a Venlo-type glasshouse. One layer of cloth was put on the growth table and the greenhouse compartment had a photoperiod of 16 hours to allow for ample root growth. Only when global radiation outside the greenhouse dropped below 150 Wm⁻², were high-pressure sodium (HPS) lamps (600 W, Philips) used during the light period. These were switched off when outside global radiation increased to values above 250 Wm⁻². When the HPS lamps were on, the light intensity from these was approximately 150 μmolm⁻²s⁻¹ at plant level. The shading screen (HARMONY 4215 O FR, Ludvig Svensson) was closed when outside global radiation increased to values above 600 Wm⁻² and was opened when outside global radiation dropped below 500 Wm⁻². Set points of day and night temperature were 22°C and 18°C, respectively. Relative humidity was set at 65% and plants were irrigated four times per day with tomato nutrient solution (see S1 File).

Measurements conducted

Unless stated otherwise, gas exchange measurements were conducted on the fourth or fifth leaf of four-week-old plants (after transplanting) using a portable gas exchange system (LI-6400XT, Li-Cor Bioscience) equipped with a 6 cm² leaf chamber fluorometer. Airflow was set to 500 μmols⁻¹ during measurements and relative humidity was controlled at 75%. Irradiance was provided by a mixture of red (90%) and blue (10%) LEDs in the fluorometer.

Measurements to estimate V_cmax and R_d.

Leaf temperature was kept around 25°C. CO₂ response curves of photosynthesis (A_n/c_i curves) were measured by changing the atmospheric CO₂ concentration in the following order: 400, 300, 200, 100, 50, 400, 400, 500, 600, 800, 1000, 1200 ppm while keeping the light intensity at 1800 μmolm⁻²s⁻¹. Each CO₂ concentration step took about 2–5 minutes to finish. Measured photosynthetic rates during A_n/c_i curve constructions were corrected for diffusion leaks according to the Li-Cor manual [26]. In total, eight replicates were obtained (see S1 File for details).

Measurements to parameterise function J(I(t)).

Measurements were performed at an air temperature of 23°C, and at two atmospheric CO₂ concentrations: 400 ppm and 800 ppm. For each CO₂ concentration, the leaf was exposed to a respective irradiance of 0, 50, 100, 200, 400, 600, 800, 1000 and 1200 μmolm⁻²s⁻¹ for at least 45 minutes to allow both leaf net photosynthetic rate and stomatal conductance to reach steady-state. Six replicates were obtained for each CO₂ concentration (refer to S1 File for details).

Measurements to track photosynthesis under natural fluctuating irradiance.

Dynamic photosynthesis measurements were conducted between 3 and 24 September 2021 in a compartment (8×8 m) of a Venlo-type glasshouse located in Wageningen.

Measurements were conducted on leaves at the top of the plant that were fully exposed to sunlight and an air temperature of 23°C. Two sets of atmospheric CO₂ concentrations, i.e. 400 ppm and 800 ppm were used. The photosynthetically active radiation (PAR) in the leaf chamber fluorometer was continuously adjusted to match the readings from an adjacent PAR sensor placed next to the chamber.

Gas exchange data were logged every two seconds between approximately 9:00 to 16:00 hours. In total, five measured replicates were obtained at 400 ppm CO₂ and one measurement was taken at 800 ppm CO₂.

Photosynthesis under naturally fluctuating irradiance: Parameter estimation data set

Consider the set of observed greenhouse conditions shown in Fig 1. The dynamics of the three model inputs, I(t), c_a(t) and T_l(t), observed over a 6 hour period [09:37–15:33] were measured on 8 September 2021, with c_a(t) kept constant at 400 μmolm⁻²s⁻¹. Notice that irradiance (Fig 1a) peaks during 3 stages, 30–70 min, 105–170 min, and again after 335 min. It reaches a maximum of approximately 1193 μmolm⁻²s⁻¹ for a brief period. For the majority of the day light levels fluctuate between 200 and 400 μmolm⁻²s⁻¹.

Download:

Fig 1. Parameter estimation data set: Model inputs measured on 8 September 2021.

(a) Measured irradiance (I) μmolm⁻²s⁻¹ (b) Measured leaf temperature (T_l) K.

https://doi.org/10.1371/journal.pone.0275047.g001

Parameter estimation

Values for the 3 unknown system parameters, k_u, k_d and c₃, were estimated using Matlab’s global optimisation function, the genetic algorithm (ga). This method is well suited to the optimisation of highly nonlinear problems and problems with a discontinuous objective function [27]. The parameters were inferred by minimising the objective function, (19) where g_tc,m, c_i,m and A_n,m denote the measured outputs defined in 13 and N is the number of observed data points, recorded every 2 seconds. By attributing weights to the individual terms, we ensured that an accurate prediction of the important metric A_n was favoured. Measured g_tc and A_n values are shown in blue in Fig 5a and 5b.

Initial conditions.

Given that both g_tc and c_i are measured outputs, the initial conditions of the 2 state equations are known.

Results

Estimated parameter values

Figs 2–4 show the converged results of the genetic algorithm optimisation. Here, this global search method generated 300 random parameter combinations from the 3-dimensional space bounded by the intervals as indicated in Figs 2a–4a. These results reveal the frequency with which a particular parameter value was computed as optimal. In Fig 3a for example, the algorithm converged after a certain number of iterations, and computed roughly 200 of the 300 optimal points on the interval [800, 850] for parameter k_d.

Download:

Fig 2. Global optimisation results for the stomatal parameter k_u in expression 15.

(a) The range of potential k_u values considered is shown along with the optimised k_u distribution. The genetic algorithm converged to a parameter value on the interval [170, 180], with the optimised k_u = 179.4 s. (b) Objective function values: computed profile likelihood for a range of k_u values. The 95% confidence interval is shown in red as [157.5, 202.5] s.

https://doi.org/10.1371/journal.pone.0275047.g002

Download:

Fig 3. Global optimisation results for the stomatal parameter k_d in expression 16.

(a) The range of potential k_d values considered is shown along with the optimised k_d distribution. The genetic algorithm converged to a parameter on the interval [800, 850], with the optimised k_d = 830.3 s. (b) Objective function values: computed profile likelihood for a range of k_d values. The 95% confidence interval is shown in red as [735.6, 927.5] s.

https://doi.org/10.1371/journal.pone.0275047.g003

Download:

Fig 4. Global optimisation results for parameter c₃ in expression 10 used in 18.

(a) The range of potential c₃ values considered is shown along with the optimised c₃ distribution. The genetic algorithm converged to an optimal parameter value on the interval [37.9,38] as c₃ = 37.96. (b) Objective function values: computed profile likelihood for a narrow range of c₃ values. The 95% confidence interval is shown in red as [37.95, 37.98] s.

https://doi.org/10.1371/journal.pone.0275047.g004

Notice that the derived time constants k_u = 179.4 s and k_d = 830.3 s suggest that for a leaf under natural fluctuating irradiance, the overall stomatal dynamics related to an increase in irradiance is faster than those associated with a decrease in irradiance.

The optimised c₃ value 37.96 (see Fig 4) increases the value of W_c (defined in expression 4) compared to this function’s value computed using the commonly used c₃ = 38.28 [28]. All parameter values in 15 and 16 are given in Table 2. Parameter values in 18 given in Table 3.

Download:

Table 2. Values of the model parameters in expressions 15 and 16.

Parameters of the steady-state target function G(I(t),c_a(t)), computed under c_a = 400ppm and c_a = 800ppm, respectively, are also included. Estimated unknown system parameters are given with accompanying confidence intervals.

https://doi.org/10.1371/journal.pone.0275047.t002

Download:

Table 3. Values of the model parameters in expression 18.

A priori estimated parameters of function J(I(t)) and the respective steady-state parameters V_cmax and R_d are given. The unknown system parameter and its confidence interval is also shown. Finally, the parameters related to the temperature dependence of key FvCB model parameters are also given.

https://doi.org/10.1371/journal.pone.0275047.t003

Photosynthesis under naturally fluctuating irradiance: Parameter estimation

We parameterise the model using data measured on 8 September 2021 under natural fluctuating light as this contains rich information pertaining to fluctuations between W_c and W_j limitation (refer to Fig 5c).

Download:

Fig 5. Results obtained after parameter estimation.

(a) Total stomatal conductance to CO₂ diffusion molm⁻²s⁻¹. (b) Net photosynthetic rate μmolm⁻²s⁻¹ (c) min(W_c,W_j) μmolm⁻²s⁻¹, indicates whether photosynthesis is limited by the carboxylation or electron transport rate.

https://doi.org/10.1371/journal.pone.0275047.g005

Predictions for g_tc are shown in red in Fig 5a. A 9% increase in predicted accuracy (from the objective function in 19) was obtained by modelling g_tc with an asymmetric as opposed to a symmetric response to irradiance (see S1 File).

The dynamic relationship between the respective W_c and W_j limitations is shown in red in Fig 5c. Results indicate that Rubisco limited photosynthesis (W_c) coincides with elevated levels of natural irradiance seen Fig 1a. The model predicts that photosynthesis is limited by the electron transport rate (W_j) for the majority of the day, thus for irradiance levels that remain below 400 μmolm⁻²s⁻¹. The correlation between A_n and the rate of electron transport J(I(t)) is shown in Fig 6.

Download:

Fig 6. Results obtained after parameter estimation.

The correlation between A_n and the electron transport rate (J(I(t)) is shown for both measured and simulated data sets.

https://doi.org/10.1371/journal.pone.0275047.g006

Photosynthesis under natural fluctuating irradiance: Model validation

We proceed by using the parameter values obtained in Tables 2 and 3 during model validation. The outcome is summarised in Table 4.

Download:

Table 4. Greenhouse data measured under natural fluctuating irradiance.

Summary of RMSEs computed to assess the accuracy of A_n. Here , with the measured output denoted by A_n,m. Results comparing the use of the optimised parameter c₃ = 37.96 to the original c₃ = 38.28 [28] are shown. Experiments were conducted under 2 sets of ambient CO₂ concentrations, 400ppm and 800ppm, respectively. R² values are given in [].

https://doi.org/10.1371/journal.pone.0275047.t004

The results obtained for measurements taken on 6 Sept 2021 under an ambient CO₂ concentration of 400ppm are shown in Fig 7, whilst the results for measurements conducted on 7 Sept 2021 under a CO₂ concentration of 800ppm are shown in Fig 8. Both Figs 7b and 8b show good agreement between measured and modelled A_n. Simulations shown in Fig 8c suggest that when c_a is 800ppm, photosynthesis is solely limited by the electron transport rate. Furthermore, one observes similarities between the shapes of the modelled W_j and A_n. Given that W_jc is computed using J(I(t)) from expression 11, this suggests that for tomato under the conditions reported here, it may be sufficient to use electron transport rates, inferred from steady-state data, in a dynamic setting. The correlation between A_n and J(I(t)) is shown in Fig 9.

Download:

Fig 7. Model validation results obtained for measurements taken on 6 Sept 2021 with c_a = 400ppm.

(a) Measured irradiance μmolm⁻²s⁻¹. (b) Net photosynthetic rate μmolm⁻²s⁻¹ (c) min(W_c,W_j) μmolm⁻²s⁻¹, indicates whether photosynthesis is limited by the carboxylation or electron transport rate.

https://doi.org/10.1371/journal.pone.0275047.g007

Download:

Fig 8. Model validation results obtained for measurements taken on 7 Sept 2021 with c_a = 800ppm.

(a) Measured irradiance μmolm⁻²s⁻¹. (b) Net photosynthetic rate μmolm⁻²s⁻¹ (c) min(W_c,W_j) μmolm⁻²s⁻¹, indicates whether photosynthesis is limited by the carboxylation or electron transport rate.

https://doi.org/10.1371/journal.pone.0275047.g008

Download:

Fig 9. Results obtained for the validation data set measured on 7 Sept 2021 under elevated c_a = 800ppm.

The correlation between A_n and the electron transport rate (J(I(t)) is shown for both measured and simulated data sets.

https://doi.org/10.1371/journal.pone.0275047.g009

Discussion

It is highly unlikely that A_n is in steady-state under natural fluctuating irradiance conditions and so observing natural dynamic as opposed to step-change responses to light is useful in aiding our understanding of this key photosynthetic property. However, nonsteady-state photosynthesis is often overlooked, with kinetic measurements of A_n reported less due to the complexity associated with measuring and analysing them [29].

We set out to develop a fit-for-purpose dynamic photosynthesis model. The model is both calibrated and validated using measurements taken under naturally fluctuating greenhouse conditions. Sufficiently accurate A_n predictions in Table 4 suggest that the model (given in expressions 15–17) can potentially be used in greenhouse lighting control applications.

Model

Our model comprises 2 ODEs, predicting the total stomatal conductance to CO₂ (g_tc) diffusion and the CO₂ concentration inside a leaf (c_i). These predictions are required to compute the net photosynthetic rate (A_n) using Fick’s law of diffusion (expression 14). Our results show that satisfactory fit for purpose A_n values can be obtained by merely predicting the elements that comprise Fick’s law of diffusion.

The dynamic binding of CO₂ inside a leaf is modelled using the FvCB model that has been adapted to predict the steady-state J values at different light intensities. Here, we used this application in a dynamic setting, defining the dynamic electron transport rate function, J(I(t)). It does not include the process of how J gradually increases after light increase.

Given that parameter V_cmax is estimated from a priori A/c_i measurements (see S1 File for details), we chose to optimise a parameter used to describe the temperature dependence of the Michaelis-Menten constant for CO₂, c₃, to increase the accuracy of our A_n predictions (see Table 4).

The model is unique given its small size and the fact that it only comprises 3 unknown system parameters. We opt not to predict any detailed molecule concentrations such as RuBP (see [20] for example), and by assuming that the leaf is homogeneous, we do not include an additional equation that accounts for mesophyllic conductance (see [12] for example).

Furthermore, for V_cmax as defined in the FvCB model [8], we too assume that Rubisco is activated [22]. Predictions for A_n reported for tomato suggest that adequate A_n values can be obtained under this stringent model assumption.

Finally, it is important to realise that the implementation of a system which contains the FvCB model necessitates the a priori calculation of a multitude of species specific steady-state parameters. This requires repeated experimental measurements and some background knowledge on how to interpret and assimilate data.

Results

The first interesting point that emerges from using rapid fluctuating measurements to calibrate a model is the inferred time constants pertaining to the total stomatal conductance to CO₂ diffusion. Notice from Table 2 that k_u, associated with an increase in irradiance is faster than k_d. When inferring these parameter values from data measured after a single step change in irradiance, k_u is slower than k_d.

We highlighted the correlation between A_n and the electron transport rate (see Figs 6 and 9). We know that the formation of ATP and NADPH molecules are dependent on the rate of electron transport and that this is light dependent. Accordingly, we observe a strong linear relationship between between A_n and J under low irradiance levels. This indicates that under such conditions, photosynthesis is W_j limited. Accurate results shown in Figs 7 and 8b in particular, suggest that: 1) it is sufficient to use an electron transport rate function, at least for our greenhouse-grown tomato leaves, calibrated using steady-state values at different irradiance levels, in a dynamic setting, and 2) the calibration of this function i.e. the parameter values J_max, θ and γ, is critically important.

Conclusions

The importance of monitoring nonsteady-state responses to natural fluctuations in irradiance for improving crop photosynthesis has gained substantial support in recent years [30]. Our aim was to develop a small fit-for-purpose dynamic photosynthesis model that can be used in supplemental lighting control applications in greenhouses. We set out to build a model that accurately predicts the net photosynthetic rate (A_n) by taking plant physiology into account, and both calibrated and validated our model using nonsteady-state data measured under rapid fluctuating light conditions.

Four main points have emerged from our analysis:

We corroborated the added value of accounting for differences in stomatal responses to both increasing and decreasing fluctuations in irradiance. We observed a 9% increase in model accuracy when using 2 different time constants to describe the total stomatal conductance to CO₂ (refer to S1 File).
In contrast, we observed no significant increase in the predicted accuracy of A_n when modelling the steady-state target function of the total stomatal conductance to CO₂, denoted by G, as a function of both irradiance and ambient CO₂ concentration (refer to S1 File).
We showed that incorporating the FvCB equations into a dynamic model is sufficient for obtaining accurate fit for purpose A_n predictions under rapid natural light fluctuations. In particular, we found that the a priori parameterisation of the steady-state electron transport rate with respect to different irradiance levels (J(I(t)) is very effective in capturing the dynamics of photosynthesis in tomato leaves.
We showed the added value of optimising a parameter in one of the Michaelis-Menten constants in the FvCB model. Here, we opted to adjust parameter c₃ related to K_n. Its value is often used from literature, despite being calibrated for different plant species. This cautions us when we are simply using parameter values from literature derived for different plant species. Accordingly, our model can predict dynamic photosynthesis for other crops, provided that parameters are adjusted, a priori, using both steady-state and dynamic measurements.
We showed that satisfactory photosynthesis results can be obtained even when a model does not account for complex biological factors such as enzymatic inhibition, liquid-gas interactions or chloroplast movement.

Supporting information

S1 File. A small dynamic leaf-level model predicting photosynthesis in greenhouse tomatoes.

https://doi.org/10.1371/journal.pone.0275047.s001

(PDF)

References

1. Kaiser E, Kromdijk J, Harbinson J, Heuvelink E, Marcelis LF. Photosynthetic induction and its diffusional, carboxylation and electron transport processes as affected by CO₂ partial pressure, temperature, air humidity and blue irradiance. Ann Bot. 2017;119(1):191–205. pmid:28025286
- View Article
- PubMed/NCBI
- Google Scholar
2. Kuijpers WJP, Katzin D, van Mourik S, Antunes DJ, Hemming S, van de Molengraft MJG. Lighting systems and strategies compared in an optimally controlled greenhouse. Biosystems Engineering. 2021;202:195–216.
- View Article
- Google Scholar
3. Slattery RA, Walker BJ, Weber APM, Ort DR. The Impacts of Fluctuating Light on Crop Performance. Plant Physiology. 2017;176(2):990–1003. pmid:29192028
- View Article
- PubMed/NCBI
- Google Scholar
4. Assmann SM, Wang XQ. From milliseconds to millions of years: guard cells and environmental responses. Curr Opin Plant Biol. 2001;4(5):421–428. pmid:11597500
- View Article
- PubMed/NCBI
- Google Scholar
5. Long SP, Zhu X, Naidu SL, Ort DR. Can improvement in photosynthesis increase crop yields? Plant Cell Environ. 2006;29(3):315–330. pmid:17080588
- View Article
- PubMed/NCBI
- Google Scholar
6. Liu T, Barbour MM, Yu D, Rao S, Song X. Mesophyll conductance exerts a significant limitation on photosynthesis during light induction. New Phytol. 2022;233(1):360–372. pmid:34601732
- View Article
- PubMed/NCBI
- Google Scholar
7. Zhu X, Wang Y, Ort DR, Long SP. e-Photosynthesis: a comprehensive dynamic mechanistic model of C3 photosynthesis: from light capture to sucrose synthesis. Plant Cell Environ. 2013;36(9):1711–1727. pmid:23072293
- View Article
- PubMed/NCBI
- Google Scholar
8. Farquhar GD, von Caemmerer S, Berry JA. A biochemical model of photosynthetic CO2 assimilation in leaves of C₃ species. Planta. 1980;149:78–90. pmid:24306196
- View Article
- PubMed/NCBI
- Google Scholar
9. von Caemmerer S. Steady-state models of photosynthesis. Plant Cell Environ. 2013;36(9):1617–1630. pmid:23496792
- View Article
- PubMed/NCBI
- Google Scholar
10. Pearcy RW, Gross LJ, He D. An improved dynamic model of photosynthesis for estimation of carbon gain in sunfleck light regimes. Plant, Cell & Environment. 1997;20(4):411–424.
- View Article
- Google Scholar
11. Noe SM, Giersch C. A simple dynamic model of photosynthesis in oak leaves: coupling leaf conductance and photosynthetic carbon fixation by a variable intracellular CO₂ pool. Funct Plant Biol. 2004;31(12):1195–1204. pmid:32688986
- View Article
- PubMed/NCBI
- Google Scholar
12. Vialet-Chabrand S, Matthews JSA, Brendel O, Blatt MR, Wang Y, Hills A, et al. Modelling water use efficiency in a dynamic environment: An example using Arabidopsis thaliana. Plant Sci. 2016;251:65–74. pmid:27593464
- View Article
- PubMed/NCBI
- Google Scholar
13. Vialet-Chabrand SRM, Matthews JSA, McAusland L, Blatt MR, Griffiths H, Lawson T. Temporal Dynamics of Stomatal Behavior: Modeling and Implications for Photosynthesis and Water Use. Plant Physiol. 2017;174(2):603–613. pmid:28363993
- View Article
- PubMed/NCBI
- Google Scholar
14. von Caemmerer S, Farquhar G, Berry J. Biochemical Model of C3 Photosynthesis. In: Laisk A, Nedbal L, Govindjee (eds) Photosynthesis in silico. Advances in Photosynthesis and Respiration, vol 29. Springer, Dordrecht; 2009.
15. Walker AP, Beckerman AP, Gu L, Kattge J, Cernusak LA, Domingues TF, et al. The relationship of leaf photosynthetic traits—V_cmax and J_cmax—to leaf nitrogen, leaf phosphorus, and specific leaf area: a meta-analysis and modeling study. Ecol Evol. 2014;4(16):3218–3235. pmid:25473475
- View Article
- PubMed/NCBI
- Google Scholar
16. Farquhar GD. Models describing the kinetics of ribulose biphosphate carboxylase-oxygenase. Arch Biochem Biophys. 1979;193(2):456–468. pmid:464606
- View Article
- PubMed/NCBI
- Google Scholar
17. Sharkey TD, Bernacchi CJ, Farquhar GD, Singsaas EL. Fitting photosynthetic carbon dioxide response curves for C₃ leaves. Plant Cell Environ. 2007;30(9):1035–1040.
- View Article
- Google Scholar
18. Buckley TN, Diaz-Espejo A. Reporting estimates of maximum potential electron transport rate. New Phytol. 2015;205:14–17. pmid:25196056
- View Article
- PubMed/NCBI
- Google Scholar
19. Lommen PW, Smith SK, Yocum CS, Gates DM. Photosynthetic Model. In: Gates DM, Schmerl RB (eds) Perspectives of Biophysical Ecology. Ecological Studies, vol 12. Springer, Berlin; 1975.
20. Morales A, Kaiser E, Yin X, Harbinson J, Molenaar J, Driever SM, et al. Dynamic modelling of limitations on improving leaf CO2 assimilation under fluctuating irradiance. Plant Cell Environ. 2018;41(3):589–604. pmid:29243271
- View Article
- PubMed/NCBI
- Google Scholar
21. Ooba M, Takahashi H. Effect of asymmetric stomatal response on gas-exchange dynamics. Ecological Modelling. 2003;164(1):65–82.
- View Article
- Google Scholar
22. Bellasio C. A generalised dynamic model of leaf-level C3 photosynthesis combining light and dark reactions with stomatal behaviour. Photosynth Res. 2019;141(1):99–118. pmid:30471008
- View Article
- PubMed/NCBI
- Google Scholar
23. Kirchbaum MUF, Gross LJ, Pearcy RW. Observed and modelled stomatal responses to dynamic light environments in the shade plant Alocasia macrorrhiza. Plant, Cell & Environment. 1988;11(2):111–121.
- View Article
- Google Scholar
24. Knapp AK. Gas Exchange Dynamics in C₃ and C₄ Grasses: Consequence of Differences in Stomatal Conductance. Ecology. 1993;74(1):113–123.
- View Article
- Google Scholar
25. Vico G, Manzoni S, Palmroth S, Katul G. Effects of stomatal delays on the economics of leaf gas exchange under intermittent light regimes. New Phytologist. 2011;192(3):640–652. pmid:21851359
- View Article
- PubMed/NCBI
- Google Scholar
26. LI-COR, Inc. Support: LI-6400/XT Portable Photosynthesis System. In: [Internet] LI-COR, Inc; [cited 20 Apr.2022]. Available: https://www.licor.com/env/support/LI-6400/topics/leak-considerations.html?Highlight=diffusion.
27. Mathworks. Genetic Algorithm. In: [Internet] Mathworks; [cited 20 Apr.2022]. Available: https://nl.mathworks.com/help/gads/genetic-algorithm.html?s_tid=CRUX_lftnav.
28. Bernacchi CJ, Portis AR, Nakano H, von Caemmerer S, Long SP. Temperature response of mesophyll conductance. Implications for the determination of Rubisco enzyme kinetics and for limitations to photosynthesis in vivo. Plant Physiol. 2002;130(4):1992–1998. pmid:12481082
- View Article
- PubMed/NCBI
- Google Scholar
29. Kaiser E, Zhou D, Heuvelink E, Harbinson J, Morales A, Marcelis L. Elevated CO2 increases photosynthesis in fluctuating irradiance regardless of photosynthetic induction state. J Exp Bot. 2017;68(20):5629–5640. pmid:29045757
- View Article
- PubMed/NCBI
- Google Scholar
30. Long SP, Taylor SH, Burgess SJ, Carmo-Silva E, Lawson T, De Souza AP, et al. Into the Shadows and Back into Sunlight: Photosynthesis in Fluctuating Light. Annual Review of Plant Biology. 2022;73(1):617–648. pmid:35595290
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. Kaiser E, Kromdijk J, Harbinson J, Heuvelink E, Marcelis LF. Photosynthetic induction and its diffusional, carboxylation and electron transport processes as affected by CO₂ partial pressure, temperature, air humidity and blue irradiance. Ann Bot. 2017;119(1):191–205. pmid:28025286
View Article
PubMed/NCBI
Google Scholar

[2] View Article

[3] PubMed/NCBI

[4] Google Scholar

[ref2] 2. Kuijpers WJP, Katzin D, van Mourik S, Antunes DJ, Hemming S, van de Molengraft MJG. Lighting systems and strategies compared in an optimally controlled greenhouse. Biosystems Engineering. 2021;202:195–216.
View Article
Google Scholar

[6] View Article

[7] Google Scholar

[ref3] 3. Slattery RA, Walker BJ, Weber APM, Ort DR. The Impacts of Fluctuating Light on Crop Performance. Plant Physiology. 2017;176(2):990–1003. pmid:29192028
View Article
PubMed/NCBI
Google Scholar

[9] View Article

[10] PubMed/NCBI

[11] Google Scholar

[ref4] 4. Assmann SM, Wang XQ. From milliseconds to millions of years: guard cells and environmental responses. Curr Opin Plant Biol. 2001;4(5):421–428. pmid:11597500
View Article
PubMed/NCBI
Google Scholar

[13] View Article

[14] PubMed/NCBI

[15] Google Scholar

[ref5] 5. Long SP, Zhu X, Naidu SL, Ort DR. Can improvement in photosynthesis increase crop yields? Plant Cell Environ. 2006;29(3):315–330. pmid:17080588
View Article
PubMed/NCBI
Google Scholar

[17] View Article

[18] PubMed/NCBI

[19] Google Scholar

[ref6] 6. Liu T, Barbour MM, Yu D, Rao S, Song X. Mesophyll conductance exerts a significant limitation on photosynthesis during light induction. New Phytol. 2022;233(1):360–372. pmid:34601732
View Article
PubMed/NCBI
Google Scholar

[21] View Article

[22] PubMed/NCBI

[23] Google Scholar

[ref7] 7. Zhu X, Wang Y, Ort DR, Long SP. e-Photosynthesis: a comprehensive dynamic mechanistic model of C3 photosynthesis: from light capture to sucrose synthesis. Plant Cell Environ. 2013;36(9):1711–1727. pmid:23072293
View Article
PubMed/NCBI
Google Scholar

[25] View Article

[26] PubMed/NCBI

[27] Google Scholar

[ref8] 8. Farquhar GD, von Caemmerer S, Berry JA. A biochemical model of photosynthetic CO2 assimilation in leaves of C₃ species. Planta. 1980;149:78–90. pmid:24306196
View Article
PubMed/NCBI
Google Scholar

[29] View Article

[30] PubMed/NCBI

[31] Google Scholar

[ref9] 9. von Caemmerer S. Steady-state models of photosynthesis. Plant Cell Environ. 2013;36(9):1617–1630. pmid:23496792
View Article
PubMed/NCBI
Google Scholar

[33] View Article

[34] PubMed/NCBI

[35] Google Scholar

[ref10] 10. Pearcy RW, Gross LJ, He D. An improved dynamic model of photosynthesis for estimation of carbon gain in sunfleck light regimes. Plant, Cell & Environment. 1997;20(4):411–424.
View Article
Google Scholar

[37] View Article

[38] Google Scholar

[ref11] 11. Noe SM, Giersch C. A simple dynamic model of photosynthesis in oak leaves: coupling leaf conductance and photosynthetic carbon fixation by a variable intracellular CO₂ pool. Funct Plant Biol. 2004;31(12):1195–1204. pmid:32688986
View Article
PubMed/NCBI
Google Scholar

[40] View Article

[41] PubMed/NCBI

[42] Google Scholar

[ref12] 12. Vialet-Chabrand S, Matthews JSA, Brendel O, Blatt MR, Wang Y, Hills A, et al. Modelling water use efficiency in a dynamic environment: An example using Arabidopsis thaliana. Plant Sci. 2016;251:65–74. pmid:27593464
View Article
PubMed/NCBI
Google Scholar

[44] View Article

[45] PubMed/NCBI

[46] Google Scholar

[ref13] 13. Vialet-Chabrand SRM, Matthews JSA, McAusland L, Blatt MR, Griffiths H, Lawson T. Temporal Dynamics of Stomatal Behavior: Modeling and Implications for Photosynthesis and Water Use. Plant Physiol. 2017;174(2):603–613. pmid:28363993
View Article
PubMed/NCBI
Google Scholar

[48] View Article

[49] PubMed/NCBI

[50] Google Scholar

[ref14] 14. von Caemmerer S, Farquhar G, Berry J. Biochemical Model of C3 Photosynthesis. In: Laisk A, Nedbal L, Govindjee (eds) Photosynthesis in silico. Advances in Photosynthesis and Respiration, vol 29. Springer, Dordrecht; 2009.

[ref15] 15. Walker AP, Beckerman AP, Gu L, Kattge J, Cernusak LA, Domingues TF, et al. The relationship of leaf photosynthetic traits—V_cmax and J_cmax—to leaf nitrogen, leaf phosphorus, and specific leaf area: a meta-analysis and modeling study. Ecol Evol. 2014;4(16):3218–3235. pmid:25473475
View Article
PubMed/NCBI
Google Scholar

[53] View Article

[54] PubMed/NCBI

[55] Google Scholar

[ref16] 16. Farquhar GD. Models describing the kinetics of ribulose biphosphate carboxylase-oxygenase. Arch Biochem Biophys. 1979;193(2):456–468. pmid:464606
View Article
PubMed/NCBI
Google Scholar

[57] View Article

[58] PubMed/NCBI

[59] Google Scholar

[ref17] 17. Sharkey TD, Bernacchi CJ, Farquhar GD, Singsaas EL. Fitting photosynthetic carbon dioxide response curves for C₃ leaves. Plant Cell Environ. 2007;30(9):1035–1040.
View Article
Google Scholar

[61] View Article

[62] Google Scholar

[ref18] 18. Buckley TN, Diaz-Espejo A. Reporting estimates of maximum potential electron transport rate. New Phytol. 2015;205:14–17. pmid:25196056
View Article
PubMed/NCBI
Google Scholar

[64] View Article

[65] PubMed/NCBI

[66] Google Scholar

[ref19] 19. Lommen PW, Smith SK, Yocum CS, Gates DM. Photosynthetic Model. In: Gates DM, Schmerl RB (eds) Perspectives of Biophysical Ecology. Ecological Studies, vol 12. Springer, Berlin; 1975.

[ref20] 20. Morales A, Kaiser E, Yin X, Harbinson J, Molenaar J, Driever SM, et al. Dynamic modelling of limitations on improving leaf CO2 assimilation under fluctuating irradiance. Plant Cell Environ. 2018;41(3):589–604. pmid:29243271
View Article
PubMed/NCBI
Google Scholar

[69] View Article

[70] PubMed/NCBI

[71] Google Scholar

[ref21] 21. Ooba M, Takahashi H. Effect of asymmetric stomatal response on gas-exchange dynamics. Ecological Modelling. 2003;164(1):65–82.
View Article
Google Scholar

[73] View Article

[74] Google Scholar

[ref22] 22. Bellasio C. A generalised dynamic model of leaf-level C3 photosynthesis combining light and dark reactions with stomatal behaviour. Photosynth Res. 2019;141(1):99–118. pmid:30471008
View Article
PubMed/NCBI
Google Scholar

[76] View Article

[77] PubMed/NCBI

[78] Google Scholar

[ref23] 23. Kirchbaum MUF, Gross LJ, Pearcy RW. Observed and modelled stomatal responses to dynamic light environments in the shade plant Alocasia macrorrhiza. Plant, Cell & Environment. 1988;11(2):111–121.
View Article
Google Scholar

[80] View Article

[81] Google Scholar

[ref24] 24. Knapp AK. Gas Exchange Dynamics in C₃ and C₄ Grasses: Consequence of Differences in Stomatal Conductance. Ecology. 1993;74(1):113–123.
View Article
Google Scholar

[83] View Article

[84] Google Scholar

[ref25] 25. Vico G, Manzoni S, Palmroth S, Katul G. Effects of stomatal delays on the economics of leaf gas exchange under intermittent light regimes. New Phytologist. 2011;192(3):640–652. pmid:21851359
View Article
PubMed/NCBI
Google Scholar

[86] View Article

[87] PubMed/NCBI

[88] Google Scholar

[ref26] 26. LI-COR, Inc. Support: LI-6400/XT Portable Photosynthesis System. In: [Internet] LI-COR, Inc; [cited 20 Apr.2022]. Available: https://www.licor.com/env/support/LI-6400/topics/leak-considerations.html?Highlight=diffusion.

[ref27] 27. Mathworks. Genetic Algorithm. In: [Internet] Mathworks; [cited 20 Apr.2022]. Available: https://nl.mathworks.com/help/gads/genetic-algorithm.html?s_tid=CRUX_lftnav.

[ref28] 28. Bernacchi CJ, Portis AR, Nakano H, von Caemmerer S, Long SP. Temperature response of mesophyll conductance. Implications for the determination of Rubisco enzyme kinetics and for limitations to photosynthesis in vivo. Plant Physiol. 2002;130(4):1992–1998. pmid:12481082
View Article
PubMed/NCBI
Google Scholar

[92] View Article

[93] PubMed/NCBI

[94] Google Scholar

[ref29] 29. Kaiser E, Zhou D, Heuvelink E, Harbinson J, Morales A, Marcelis L. Elevated CO2 increases photosynthesis in fluctuating irradiance regardless of photosynthetic induction state. J Exp Bot. 2017;68(20):5629–5640. pmid:29045757
View Article
PubMed/NCBI
Google Scholar

[96] View Article

[97] PubMed/NCBI

[98] Google Scholar

[ref30] 30. Long SP, Taylor SH, Burgess SJ, Carmo-Silva E, Lawson T, De Souza AP, et al. Into the Shadows and Back into Sunlight: Photosynthesis in Fluctuating Light. Annual Review of Plant Biology. 2022;73(1):617–648. pmid:35595290
View Article
PubMed/NCBI
Google Scholar

[100] View Article

[101] PubMed/NCBI

[102] Google Scholar

Figures

Abstract

Introduction

Theory

The opening and closing of stomata

The FvCB model

The Rubisco limited carboxylation rate (Wc).

Electron transport limited carboxylation rate (Wj).

Temperature and light intensity effects on steady-state photosynthesis

Dynamic model structure

Materials and methods

Growing conditions of plants

Plants used to estimate Vcmax and Rd.

Plants used to parameterise function J(I(t)).

Plants used to measure photosynthesis under natural fluctuating irradiance.

Measurements conducted

Measurements to estimate Vcmax and Rd.

Measurements to parameterise function J(I(t)).

Measurements to track photosynthesis under natural fluctuating irradiance.

Photosynthesis under naturally fluctuating irradiance: Parameter estimation data set

Parameter estimation

Initial conditions.

Results

Estimated parameter values

Photosynthesis under naturally fluctuating irradiance: Parameter estimation

Photosynthesis under natural fluctuating irradiance: Model validation

Discussion

Model

Results

Conclusions

Supporting information

S1 File. A small dynamic leaf-level model predicting photosynthesis in greenhouse tomatoes.

References

The Rubisco limited carboxylation rate (W_c).

Electron transport limited carboxylation rate (W_j).

Plants used to estimate V_cmax and R_d.

Measurements to estimate V_cmax and R_d.