Development of Modified Farquhar–von Caemmerer–Berry Model Describing Photodamage of Photosynthetic Electron Transport in C3 Plants under Different Temperatures

Photodamage of photosynthetic electron transport is a key mechanism of disruption of photosynthesis in plants under action of stressors. This means that investigation of photodamage is an important task for basic and applied investigations. However, its complex mechanisms restrict using experimental methods of investigation for this process; the development of mathematical models of photodamage and model-based analysis can be used for overcoming these restrictions. In the current work, we developed the modified Farquhar–von Caemmerer–Berry model which describes photodamage of photosynthetic electron transport in C3 plants. This model was parameterized on the basis of experimental results (using an example of pea plants). Analysis of the model showed that combined inactivation of linear electron flow and Rubisco could induce both increasing and decreasing photodamage at different magnitudes of inactivation of these processes. Simulation of photodamage under different temperatures and light intensities showed that simulated temperature dependences could be multi-phase; particularly, paradoxical increases in the thermal tolerance of photosynthetic electron transport could be observed under high temperatures (37–42 °C). Finally, it was shown that changes in temperature optimums of linear electron flow and Rubisco could modify temperature dependences of the final activity of photosynthetic electron transport under photodamage induction; however, these changes mainly stimulated its photodamage. Thus, our work provides a new theoretical tool for investigation of photodamage of photosynthetic processes in C3 plants and shows that this photodamage can be intricately dependent on parameters of changes in activities of linear electron flow and Rubisco including changes induced by temperature.


Introduction
Photosynthesis supports life on Earth through the fixation of solar energy [1,2], the synthesis of organic substances and the production of oxygen [3].All photosynthetic reactions can be divided into two groups: light reactions and dark reactions.During light reactions, a complex system of membrane-associated pigments absorb light [4]; the absorbed light energy is used to separate charges in photosynthetic reaction centers and create a transmembrane electrochemical potential across thylakoid membranes [5].An electron is transferred from water to the high-potential electron end-acceptor NADP + [1].This transfer is accompanied by the production of oxygen, which is released into the environment [6].At the same time, ATP synthesis is carried out due to the transmembrane potential [7].Thus, the products of light reactions are energy equivalents (ATP and NADFH [8]) and oxygen.Energy equivalents are used for the synthesis of carbohydrates from carbon dioxide (CO 2 ) and water during the Calvin cycle [1].CO 2 used by photosynthetic dark reactions is transported into leaves from the atmosphere through the stomata [3] and then diffuses to Plants 2023, 12, 3211 2 of 18 chloroplasts.CO 2 stomatal conductivity and the conductivity of the mesophyll of leaves affect the concentration of CO 2 in chloroplasts [9].
Photosynthesis can be subject to the negative influence of environmental stressors (excessive light, non-optimal temperatures, drought, salinization, etc.) [10,11].Stressors suppress photosynthetic light and dark reactions; this effect leads to a slowdown in the rate of biomass production [12].The suppression can be based on changes in the ultrastructure of chloroplasts, in the number of pigments and metabolites, in the stomatal opening and in other parameters [6].Particularly, photodamage of photosynthetic machinery including photosystem II [13,14] is one of key mechanisms of photosynthetic suppression because this damage can be induced by both the high activity of photosynthetic light reactions (e.g., under action of excess light [15]) and low activity of photosynthetic dark reactions (e.g., under temperatures that are not optimal for the Calvin-Benson cycle [16] or under drought, which closes stomata and decreases the CO 2 flux into leaves and chloroplasts [17]).
The central role of photodamage in the disruption of photosynthesis highlights the importance of analyzing this process.However, the efficiency of experimental investigations is restricted by the complexity of photodamage mechanisms, which includes the simultaneous participation of photosynthetic light reactions, photosynthetic dark reactions, and CO 2 fluxes through stomata, plasma membranes, and chloroplast envelopes in forming photodamage.These restrictions can be overcome by using mathematical modeling [18][19][20], which can simultaneously describe all processes influencing photodamage and their interactions.
Mathematical models can describe different levels of photosynthetic processes: from the reaction centers to plant canopies and ecosystems [21].Models of stationary energy flows in reaction centers are the basis of interpreting the results of chlorophyll fluorescence measurement which is a key method used to investigate photosynthetic light reactions [8,22].Models of the electron transport chain (ETC) in thylakoid membranes [19,[23][24][25] can additionally describe the regulation of photosynthetic light reactions by some environmental factors (e.g., light).There are also models describing the assimilation of CO 2 with a minimal description of the ETC, e.g., the Farquhar-von Caemmerer-Berry (FvCB) model which describes photosynthesis in C 3 plants (the basic FvCB model) [26][27][28] or the model by Laisk et al. [20] which is based on a detailed description of the Calvin-Benson cycle and the simulation of relations between photosynthesis and other physiological processes (e.g., amino acid synthesis).Further development of photosynthetic models can be based on the description of spatially heterogenous processes (e.g., in leaves [29][30][31][32] or plant canopies [33]), simulation of productivity [34], modelling interactions of plants with each other and with other living organisms [18] and others.
Considering the relation of photodamage to both light and dark photosynthetic reactions, the FvCB model [26][27][28], which minimally describes both groups of reactions, seems to be a prospective tool for the theoretical investigation of light-induced damages of photosynthetic machinery under adverse environmental conditions.This model is based on a description of limitations of photosynthetic CO 2 assimilation using the "slowest" processes including the rate of CO 2 carboxylation/oxygenation by Rubisco (RuBP), rate of ETC activity and rate of triose transport in the basic variant of the FvCB model [35].There are numerous modifications to the FvCB model; particularly, it can include descriptions of cyclic and pseudocyclic electron flows, CO 2 fluxes from air to the stroma of chloroplasts [28] including spatially heterogenous fluxes [31,32], temperature dependences of photosynthetic processes [35,36], etc.The latter descriptions can have especial importance regarding photodamage under action of non-optimal temperatures because low or high temperatures can stimulate light-induced damage of photosynthetic machinery [37]; however, the FvCB model and its modern modifications are not often focused on descriptions of photodamage including theoretical analysis of temperature dependences of this damage.
Thus, the aim of the current work was the development of a modified FvCB model describing photodamage of photosynthetic electron transport in C 3 plants under different temperatures.Pea plants are used as the research object for parameterization of the model to minimize the variability of its parameters; however, results of the model analysis should be Plants 2023, 12, 3211 3 of 18 qualitatively similar to the results of simulation concerning parameterization on the basis of other C 3 plants.It is expected that this model can be used as a relatively simple theoretical tool for understanding the characteristics of light-induced damages of photosynthetic machinery under action of adverse temperatures.[26,35,38].The steady-state rate of photosynthetic CO 2 assimilation was described as being equal to the slowest of the two main processes.The first process is carboxylation/oxygenation, related to RuBP activity.This process was limiting under the low CO 2 concentration in the modeled chloroplast.The second process is the regeneration of RuBP, associated with the production of NADPH from NADP + through photosynthetic light reactions.This regeneration was taken as a function of the potential rate of the linear electron flow through the ETC under the specific intensity of actinic light (J, µmol•m −2 •s −1 ).In accordance with a series of works [31,32,36], we did not include the limitation of photosynthetic CO 2 assimilation by the rate of triose transport because it was difficultly distinguished from the RuBP carboxylation/oxygenation limitation [36].

Modified Farquhar-von
Thus, the aim of the current work was the development of a modified FvCB model describing photodamage of photosynthetic electron transport in C3 plants under different temperatures.Pea plants are used as the research object for parameterization of the model to minimize the variability of its parameters; however, results of the model analysis should be qualitatively similar to the results of simulation concerning parameterization on the basis of other C3 plants.It is expected that this model can be used as a relatively simple theoretical tool for understanding the characteristics of light-induced damages of photosynthetic machinery under action of adverse temperatures.

General Description of the Model and Equations
Figure 1 shows the general scheme of the developed mathematical model of photodamage of photosynthetic electron transport in C3 plants.Our model was based on the Farquhar-von Caemmerer-Berry model [26,35,38].The steady-state rate of photosynthetic CO2 assimilation was described as being equal to the slowest of the two main processes.The first process is carboxylation/oxygenation, related to RuBP activity.This process was limiting under the low CO2 concentration in the modeled chloroplast.The second process is the regeneration of RuBP, associated with the production of NADPH from NADP + through photosynthetic light reactions.This regeneration was taken as a function of the potential rate of the linear electron flow through the ETC under the specific intensity of actinic light (J, µmol•m −2 •s −1 ).In accordance with a series of works [31,32,36], we did not include the limitation of photosynthetic CO2 assimilation by the rate of triose transport because it was difficultly distinguished from the RuBP carboxylation/oxygenation limitation [36].Thus, Equations ( 1) and (2) were used for the description of photosynthetic CO2 assimilation (Ahv): Thus, Equations (1) and (2) were used for the description of photosynthetic CO 2 assimilation (A hv ): W = min W j , W c (1) where W is the real CO 2 carboxylation rate (µmol•m −2 •s −1 ), W j is the potential CO 2 carboxylation rate under the limitation caused by ETC activity, W c is the potential CO 2 carboxylation rate under the limitation caused by RuBP carboxylation/oxygenation, C c is the concentration of CO 2 in the chloroplast (ppm) and Γ* is the photosynthetic CO 2 compensation point (ppm).Equation (3) describes W j which is dependent on the potential rate of linear electron flow through the ETC: In accordance with the FvCB model, J was approximated by a function which depended on the absorbed energy of the actinic light (Equation ( 4)) [35,39]: where θ is an empirical curvature factor, J max is a maximum electron transport rate through the ETC (µmol•m −2 •s −1 ) and I 2 is the absorbed light radiation by photosystem II (PSII) (µmol•m −2 •s −1 ) [35].I 2 was described as a constant fraction of the actinic light [40] (Equation ( 5)): where PAR is the actinic light intensity and β' is a coefficient calculated as the multiplication of the fraction of light absorbed by the modeled leaf and the fraction of light energy directed to the PSII.W c was calculated on basis of the Equation ( 6): where V cmax is the maximum rate of RuBP activity (µmol•m −2 •s −1 ), K c and K o are the Michaelis-Menten constants of carboxylation and oxygenation (ppm −1 ) and O is the O 2 concentration (ppm) [35].
The rate of carboxylation depended on the concentration of CO 2 inside the chloroplast.CO 2 was transported into the chloroplast from the air through the leaf tissue.Also, CO 2 entered from the mitochondria, where it was produced by respiration (R d , µmol•m −2 •s −1 ).The change in CO 2 concentrations in the intercellular space was described as: where C o and C i are concentrations of CO 2 in the air and in intercellular space, respectively; gm (mol•m −2 •s −1 ) and gs (mol•m −2 •s −1 ) are the mesophyll conductance and stomatal conductance to CO 2 , respectively.
After that, we calculated the changes in the concentration of CO 2 under both conditions: (i) the limitation caused by the rate of CO 2 carboxylation by RuBP (Equation ( 8)) and (ii) the limitation caused by the rate of electron flow through the ETC (Equation ( 9)): In our model, we calculated stationary C c c on the basis of Equations ( 7) and ( 8) and stationary C j c on the basis of Equations ( 7) and (9).Final C c was calculated as max C c c , C j c ; this C c was used for the calculation of the final C i using Equation (7).It is known that light (especially blue light) can strongly activate H + -ATPase in the plasma membrane [41] and activation of this transporter can increase the mesophyll conductance to CO 2 [42].To describe the dependence of mesophyll conductance on PAR, we used Equation (10): where ∆g max is the difference between the maximum value of the mesophyll CO 2 conductance under light and its dark level, n is the number of quanta of blue light per receptor required for increasing g m , K is the intensity of light inducing 50% increase of g m and g d is the mesophyll conductance to CO 2 under dark conditions.Equation ( 10) is based on the classical Hill equation, which describes the interaction of an enzyme with a ligand; in our case, PAR was used as an analog of the concentration of substrate in the Hill equation.
Photodamage plays an important role in the negative action of excess light and other adverse factors affecting photosynthetic processes [8,43].As a result, we included a description of photodamage of photosynthetic electron transport in our model.This damage can be associated both with the direct action of light and with the production of reactive oxygen species (ROS) [44], which destroy the enzymes of the photosynthetic apparatus [12]; however, these specific reasons were not explicitly described in our model.To describe the damage, the coefficient A j was included in the model that was multiplied electron flow (J).This coefficient was the variable and was described by Equation (11): where k 0 d is the damage rate constant and α is the ratio of the damage rate constant at W j = W to this rate constant at W j > W; α, should be less than 1.It was assumed that photodamage was relatively low at W j = W (W j < W c in accordance with Equation ( 1)) because all electrons transported by linear electron flow should be consumed by NADP + .In contrast, the damage was high at W j > W c because electrons could not leave the ETC and should induce overreduction of the chain in this case.To simplify model parametrization, we did not describe recovery of the damaged ETC in the model.We analyzed relatively short-term time intervals in our simulation (1 h); recovery processes require rather larger time intervals.
The model also included a block that characterized the temperature dependence of the modeled photosynthetic processes.We considered the temperature dependence of two processes: (i) the carboxylation/oxygenation by RuBP (the temperature influenced W c ) and (ii) the linear electron flow (the temperature influenced W j ).Both processes were described by the same empirical function (Equation ( 12)) with different parameters (based on [16,35,45]).
, t > t o (12) where t is the current temperature, t o is the optimal temperature for the activity of the enzyme complex (Act = 1) and σ 1 and σ 2 are the differences between the optimal temperature and the temperature at which the activity dropped to e −1 .For symmetric temperature dependence, σ 1 = σ 2 .Different coefficients Act were used for W c and W j ; W c and W j were multiplied by these coefficients.
Equations of the model were numerically calculated using the Euler method.

Experimental Procedure
Two-week-old pea seedlings (Pisum sativum L., cultivar "Albumen") were used for our investigation.The plants were hydroponically cultivated under a 23 • C temperature and a 16 h (light)/8 h (dark) regime of illumination in a Binder KBW 240 (BINDER GmbH, Tuttlingen, Germany).Humidity during cultivation was not controlled.The second mature pea leaf was used for measurements.
The parametrization of the model was conducted on the basis of experimental data.Parameters of photosynthetic light reactions were measured by a Dual-PAM-100 pulseamplitude modulation (PAM) fluorometer (Heinz Walz GmbH, Effeltrich, Germany).The main measured parameters were the quantum yields of photosystems I (Y(I)) and II (Y(II)) and the coefficient of non-photochemical quenching of the chlorophyll fluorescence (qN), which were automatically calculated using standard equations [46][47][48].Quantum yields and light fluxes were used to experimentally calculate linear electron flows (J) (see the next section).Blue actinic light (460 nm) with different intensities was used; the 239 µmol•m −2 •s −1 intensity was used if the light intensity is not separately specified.
Parameters of photosynthetic dark reactions were measured by a GFS-3000 infrared gas analyzer and portable gas exchange measuring system (Heinz Walz GmbH, Effeltrich, Germany).The main analyzed parameter was the photosynthetic assimilation of CO 2 (A hv ).A hv was calculated as the difference between the CO 2 assimilation under light and dark conditions.The content of CO 2 in the intercellular space (C i ), stomatal conductance to CO 2 (g s ) and respiration rate (R d ) were also measured.Different external CO 2 concentrations were used; 360 ppm was used if the CO 2 concentration is not separately specified.
The GFS-3000 system also provided controlled temperatures and their measurements.The 23 • C temperature was used as the basic setting; to analyze temperature's influence on photosynthetic parameters, action of other temperatures was used (10 A humidity level of 2000 ppm was used for measurements.Dual-PAM-100 and GFS-3000 apparatus were used in combination.

Model Parameterization
The parametrization of our model was performed in several stages.In the first stage, we calculated the photosynthetic linear electron flow (J) using Equation ( 13) [49] for each light intensity used in the experiment where Y(II) is the quantum yield of PSII.Absorbed light radiation by PSII (I2) was calculated in accordance with Equation (5).β was calculated in accordance with Equation ( 14): where β is the average value of the coefficient in Equation ( 5) under a high concentration of CO 2 (2000 ppm), dII is the fraction of the absorbed light directed to PSII under a 2000 ppm concentration of CO 2 and dII is this fraction under the other CO 2 concentrations.Equation (15), which was derived from Equations ( 2) and ( 3) at C c →∞ in accordance with a previously published work [50], was used for calculation of β on the basis of A hv , Y(II) and PAR: To estimate dII (or dII ), we used Equation ( 16) [34,42,50,51]: where Y(I) and Y(II) are the quantum yields of PSI and PSII measured under a low intensity of actinic light in accordance with [50].
Plants 2023, 12, 3211 7 of 18 Thus, the experimental linear electron flow was calculated on the basis of Equation ( 17): Figure 2 shows light dependences of the experimental J calculated using Equation ( 17) and its approximation by Equation (4) at J max = 55 µmol•m −2 •s −1 and θ = 0.25.Despite small differences between experimental and model J value, which were probably related to the simplified description of the linear electron flow in the FvCB model, the modeled J value imitated the experimental dependence well (R 2 was 0.9862).
To estimate dII (or dII′), we used Equation ( 16) [34,42,50,51]: where Y(I) and Y(II) are the quantum yields of PSI and PSII measured under a low intensity of actinic light in accordance with [50].Thus, the experimental linear electron flow was calculated on the basis of Equation ( 17): Figure 2 shows light dependences of the experimental J calculated using Equation ( 17) and its approximation by Equation ( 4) at Jmax = 55 µmol•m −2• s −1 and θ = 0.25.Despite small differences between experimental and model J value, which were probably related to the simplified description of the linear electron flow in the FvCB model, the modeled J value imitated the experimental dependence well (R 2 was 0.9862).In the second stage of parametrization, we estimated the maximum rate of RuBP activity (Vcmax).Based on Equations ( 2) and (3), C (for the ETR limitation) was calculated for all actinic light intensities (Table 1).In accordance with our previous work [31], the 239 µmol•m −2 •s −1 blue actinic light provides the limitation caused by the ETC activity in pea leaves.As a result, we used Equation (18), which was based on Equation ( 7), for calculation of gm under the 239 µmol•m −2• s −1 light intensity: In the second stage of parametrization, we estimated the maximum rate of RuBP activity (V cmax ).Based on Equations ( 2) and (3), C j c (for the ETR limitation) was calculated for all actinic light intensities (Table 1).In accordance with our previous work [31], the 239 µmol•m −2 •s −1 blue actinic light provides the limitation caused by the ETC activity in pea leaves.As a result, we used Equation (18), which was based on Equation (7), for calculation of g m under the 239 µmol•m −2 •s −1 light intensity: It was calculated that g m was 0.03 mol•m −2 •s −1 .We used Equation (19) for the calculation of C c at C o = 100 ppm and PAR = 239 µmol•m −2 •s −1 (the RuBP limitation) on the basis of the measured C i and calculated g m : It was calculated that C c was 90 ppm under these experimental conditions.Using Equations ( 2) and ( 6), we calculated C c and standard parameters of the FvCB model (Γ* = 38.6 ppm, Kc = 260 ppm, Ko = 179 ppm and O = 200,000 ppm [35]) and showed that V cmax was 18.2 µmol•m −2 •s −1 .Using Equations ( 2) and ( 6 intensities of actinic light (Table 1).The minimal rate of carboxylation corresponded to the maximum C c ; thus, we calculated the final C c as max C c c , C j c in accordance with basic equations of the FvCB model (Equations ( 1) and ( 2)).
Table 1.The concentrations of CO 2 in chloroplasts (Cc), which were calculated for the J (C Further, we calculated experimental g m under different light intensities using A hv , C c and C i in accordance with Equation (18). Figure 3 shows the experimental dependence of g m on the intensity of the blue actinic light.g m was increased with increases in the blue actinic light intensity which was in good accordance with the activation of H + -ATPase by blue light [41] and a positive influence of transporter activity on the mesophyll conductance to CO 2 [42].We used Equation (10) to approximate this dependence.Extrapolation of experimental g m to zero showed that g d was 0.018 mol•m −2 •s −1 .In accordance with [52,53], we used ∆g max equal to 0.25 mol•m −2 •s −1 .n = 2 and K = 1000 µmol•m −2 •s −1 were assumed.The determination coefficient between experimental and model dependences (R 2 ) was 0.992.It was calculated that gm was 0.03 mol•m −2 •s −1 .We used Equation ( 19) for the calculation of Cc at Co = 100 ppm and PAR = 239 µmol•m −2 •s −1 (the RuBP limitation) on the basis of the measured Ci and calculated gm: It was calculated that Cc was 90 ppm under these experimental conditions.Using Equations ( 2) and ( 6), we calculated Cc and standard parameters of the FvCB model (Г* = 38.6 ppm, Kc = 260 ppm, Ko = 179 ppm and O = 200 000 ppm [35]) and showed that Vcmax was 18.2 µmol•m −2 •s −1 .Using Equations ( 2) and ( 6), Ahv, calculated Vcmax and noted parameters of the FvCB model, we calculated C (for the RuBP limitation) under different intensities of actinic light (Table 1).The minimal rate of carboxylation corresponded to the maximum Cc; thus, we calculated the final Cc as max C , C in accordance with basic equations of the FvCB model (Equations ( 1) and ( 2)).
Further, we calculated experimental gm under different light intensities using Ahv, Cc and Ci in accordance with Equation (18). Figure 3 shows the experimental dependence of gm on the intensity of the blue actinic light.gm was increased with increases in the blue actinic light intensity which was in good accordance with the activation of H + -ATPase by blue light [41] and a positive influence of transporter activity on the mesophyll conductance to CO2 [42].We used Equation (10) to approximate this dependence.Extrapolation of experimental gm to zero showed that gd was 0.018 mol•m −2 •s −1 .In accordance with [52,53]   The determination coefficient between these dependences (R 2 ) was 0.992.
In the next stage, we parameterized the model considering the damage.For this, we used pulses of blue actinic light (70 s) with increasing intensity (PAR values were 108, 239, 425 and 758 µmol•m −2 •s −1 ).Between these illumination pulses were dark intervals contributing full relaxation of the energy-dependent component of the non-photochemical Plants 2023, 12, 3211 9 of 18 quenching of the chlorophyll fluorescence; as a result, qN values before light pulses were only dependent on photodamage because participation of the state transition in nonphotochemical quenching in pea plants is weak [50].It should also be noted that in this case, only the open reaction centers of photosystem II were investigated (after the dark interval).We used Equation (20) to estimate the damage rate: where qN i+1 and qNi are qN before initiation of light pulses i + 1 and i, respectively (after the corresponding dark intervals), and ∆t is the duration of the light pulses (70 s).Equation ( 21) was used for estimation of the experimental damage rate constant (kd): In accordance with Equation ( 11), 4 shows the dependence of the experimental k d on the blue actinic light intensity.It was shown that experimental dependence was well fitted by the model at k 0 d = 0.00008 s −1 and α = 0.09.The determination coefficient between these dependences (R 2 ) was 0.856.
In the next stage, we parameterized the model considering the damage.For this, we used pulses of blue actinic light (70 s) with increasing intensity (PAR values were 108, 239, 425 and 758 µmol•m −2 •s −1 ).Between these illumination pulses were dark intervals contributing full relaxation of the energy-dependent component of the non-photochemical quenching of the chlorophyll fluorescence; as a result, qN values before light pulses were only dependent on photodamage because participation of the state transition in non-photochemical quenching in pea plants is weak [50].It should also be noted that in this case, only the open reaction centers of photosystem II were investigated (after the dark interval).We used Equation (20) to estimate the damage rate: where qNi+1 and qNi are qN before initiation of light pulses i + 1 and i, respectively (after the corresponding dark intervals), and Δt is the duration of the light pulses (70 s).Equation ( 21) was used for estimation of the experimental damage rate constant (kd): In accordance with Equation ( 11) Figure 4 shows the dependence of the experimental kd on the blue actinic light intensity.It was shown that experimental dependence was well fitted by the model at k = 0.00008 s −1 and α = 0.09.The determination coefficient between these dependences (R 2 ) was 0.856.11) at k = 0.00008 s −1 and α = 0.09; Wj and W were calculated using Equations ( 1) and ( 3)-( 6), standard parameters of the FvCB model [35] and results from Table 1.The determination coefficient between these dependences (R 2 ) was 0.856.
Finally, we used experimental results to simulate temperature's influence on both activities of RuBP (Act(RuBP) and J (Act(RuBP)) (Figure 5).We assumed in the current  11) at k 0 d = 0.00008 s −1 and α = 0.09; W j and W were calculated using Equations ( 1) and ( 3)-( 6), standard parameters of the FvCB model [35] and results from Table 1.The determination coefficient between these dependences (R 2 ) was 0.856.
Finally, we used experimental results to simulate temperature's influence on both activities of RuBP (Act(RuBP) and J (Act(RuBP)) (Figure 5).We assumed in the current version of the model that non-optimal temperatures directly induced only reversible suppression of CO 2 carboxylation and ETC activity (see Equation (12), which could not show irreversible damages); in contrast, photodamage was irreversible in our model.
model-based analysis of photodamage under non-optimal temperatures was the next task of our work.
The standard procedure of the model analysis included the imitation of favorable conditions (actinic light) for 20 min, imitation of stressful conditions (actinic light + transient temperature changes or transient direct decreases in RuBP activity and J) and, finally, imitation of favorable conditions (actinic light).The final damage (Aj) was analyzed after a 60 min simulation.

Analysis of Photodamage under Combined Decreasing Activity of Linear Electron Flow and RuBP
The analysis of the developed and parameterized model began with an assessment of the influence of a combined decrease in the activity of linear electron flow and RuBP (Figure 6).This combined decrease is probable through the influence of stressors on both processes.However, magnitudes of their decreases can differ because the sensitivity of linear electron flow and RuBP to specific stressors is different (e.g., different temperature The standard procedure of the model analysis included the imitation of favorable conditions (actinic light) for 20 min, imitation of stressful conditions (actinic light + transient temperature changes or transient direct decreases in RuBP activity and J) and, finally, imitation of favorable conditions (actinic light).The final damage (Aj) was analyzed after a 60 min simulation.

Analysis of Photodamage under Combined Decreasing Activity of Linear Electron Flow and RuBP
The analysis of the developed and parameterized model began with an assessment of the influence of a combined decrease in the activity of linear electron flow and RuBP (Figure 6).This combined decrease is probable through the influence of stressors on both processes.However, magnitudes of their decreases can differ because the sensitivity of linear electron flow and RuBP to specific stressors is different (e.g., different temperature optimums for their activity [16]).Additionally, the activity of linear electron flow can be modified by factors that do not influence RuBP (e.g., non-photochemical quenching [54] or cyclic electron flow around PSI [54]) and vice versa.
Transient decreasing Act(J) and Act(RuBP) (from 20th to 40th min of simulation) strongly influenced the final Aj which showed photodamage (Figure 6).It was shown that the greatest damage to photosynthetic electron transport (the lowest final Aj) was imitated under the maximum activity of the ETC (Act(J) = 1) and the minimum activity of RuBP(Act(RuBP) = 0); this effect was observed under both light intensities.In contrast, decreasing Act(J) without decreasing Act(RuBP) lowered the magnitude of photodamage imitated by the model (increased the final Aj).  11) and can range from 0 to 1, are shown by pseudo-colors.This Aj shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.
These results can be explained through suppression of the linear electron flow in this case because the high linear electron flow without sink of electrons induces overreduction of the ETC and production of ROS [25,44,55,56] that contribute to damage of both the ETC and enzymes of the Calvin-Benson cycle [57].In contrast, the large electron sink (the active RuBP) and weak linear electron flow should minimize overreduction of the ETC and production of ROS, i.e., this combination can protect photosynthetic electron transport from photodamage.
It should be additionally noted that the simulated photodamage had thresholds in its dependences on decreasing Act(J) and Act(RuBP) under low-intensity light (Figure 6a).11) and can range from 0 to 1, are shown by pseudo-colors.This A j shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.
Analysis of the model showed that photodamage was simulated by both low (Figure 6a) and high (Figure 6b) intensities of actinic light under favorable conditions (Act(RuBP) = Act(J) = 1).The photodamage induction under low-intensity light was in good accordance with our previous results [25] which showed that 108 µmol•m −2 •s −1 blue actinic light decreased qN in pea plants without action of additional stressors.This photodamage was strongly stimulated under 758 µmol•m −2 •s −1 actinic light; it also corresponded to damage of part of the ETC by excess light [8].
Transient decreasing Act(J) and Act(RuBP) (from 20th to 40th min of simulation) strongly influenced the final A j which showed photodamage (Figure 6).It was shown that the greatest damage to photosynthetic electron transport (the lowest final A j ) was imitated under the maximum activity of the ETC (Act(J) = 1) and the minimum activity of RuBP(Act(RuBP) = 0); this effect was observed under both light intensities.In contrast, decreasing Act(J) without decreasing Act(RuBP) lowered the magnitude of photodamage imitated by the model (increased the final A j ).
These results can be explained through suppression of the linear electron flow in this case because the high linear electron flow without sink of electrons induces overreduction of the ETC and production of ROS [25,44,55,56] that contribute to damage of both the ETC and enzymes of the Calvin-Benson cycle [57].In contrast, the large electron sink (the active RuBP) and weak linear electron flow should minimize overreduction of the ETC and production of ROS, i.e., this combination can protect photosynthetic electron transport from photodamage.
It should be additionally noted that the simulated photodamage had thresholds in its dependences on decreasing Act(J) and Act(RuBP) under low-intensity light (Figure 6a).Particularly, under Act(J) = 1, decreasing the Act(RuBP) from 1 to 0.5 weakly influenced photodamage; in contrast, decreasing it from 0.5 to 0 gradually stimulated this damage.
Thus, results of simulation show that the transient decrease in activities of linear electron flow and RuBP strongly influences photodamage.Moreover, decreasing J can compensate for the negative influence of decreasing RuBP activity; this result is in good accordance with the positive influence of decreasing the linear electron flow on the tolerance of photosynthetic machinery (e.g., stimulation of non-photochemical quenching [54,58,59], increases in the cyclic electron flow around PSI [55] and other protective mechanisms are based on this effect).Considering the results of this stage regarding model analysis and different temperature dependences of Act(J) and Act(RuBP) (e.g., [16] or Figure 5 in the current work), it can be expected that some combinations of light conditions and values of non-optimal temperatures can decrease photodamage.Further analysis was focused on checking this hypothesis.

Theoretical Analysis of Temperature Dependences of Photodamage for the Different Model Parameters
The model-based analysis of the temperature dependence of photodamage of photosynthetic electron transport was first carried out using values of parameters which were calculated during parameterization.It was shown (Figure 7) that photodamage (decreasing the final A j ) was induced by all simulated light intensities; magnitudes of photodamage were increased with increasing light intensity.The maximum analyzed light intensity (758 µmol•m −2 •s −1 ) induced maximum photodamage (the final A j was about 0.72-0.76 at the different temperatures).
pensate for the negative influence of decreasing RuBP activity; this result is in good accordance with the positive influence of decreasing the linear electron flow on the tolerance of photosynthetic machinery (e.g., stimulation of non-photochemical quenching [54,58,59], increases in the cyclic electron flow around PSI [55] and other protective mechanisms are based on this effect).Considering the results of this stage regarding model analysis and different temperature dependences of Act(J) and Act(RuBP) (e.g., [16] or Figure 5 in the current work), it can be expected that some combinations of light conditions and values of non-optimal temperatures can decrease photodamage.Further analysis was focused on checking this hypothesis.

Theoretical Analysis of Temperature Dependences of Photodamage for the Different Model Parameters
The model-based analysis of the temperature dependence of photodamage of photosynthetic electron transport was first carried out using values of parameters which were calculated during parameterization.It was shown (Figure 7) that photodamage (decreasing the final Aj) was induced by all simulated light intensities; magnitudes of photodamage were increased with increasing light intensity.The maximum analyzed light intensity (758 µmol•m −2 •s −1 ) induced maximum photodamage (the final Aj was about 0.72-0.76 at the different temperatures).
Increasing temperature induced multi-phase changes which were dependent on light intensity (Figure 7).The 108 µmol•m −2 •s −1 light induced only weak changes in the final Aj including decreases in this value under temperatures lower than 30 °C and increases under higher temperatures.Under 239, 425 and 758 µmol•m −2 •s −1 light intensities, there were three phases of changes with increasing temperature: a weak increasing in final Aj (decreasing photodamage), a decrease in this value (increasing photodamage) and a second increase in the final Aj (the second decreasing photodamage).Increasing temperature induced multi-phase changes which were dependent on light intensity (Figure 7).The 108 µmol•m −2 •s −1 light induced only weak changes in the final A j including decreases in this value under temperatures lower than 30 • C and increases under higher temperatures.Under 239, 425 and 758 µmol•m −2 •s −1 light intensities, there were three phases of changes with increasing temperature: a weak increasing in final A j (decreasing photodamage), a decrease in this value (increasing photodamage) and a second increase in the final A j (the second decreasing photodamage).
Plants 2023, 12, 3211 13 of 18 These multi-phase dependences of photodamage can be explained by different temperature dependences of activities of linear electron flow and RuBP (Figure 5).Under low temperatures, both Act(J) and Act(RuPB) are decreased; therefore, W j and W c are also decreased.In accordance with Equations ( 1) and (11), decreasing W j and W c can induce opposite effects: decreasing W j should decrease photodamage and increasing W j -W c should stimulate this damage.Both processes are related to overreduction of ETC, which is the main reason behind photodamage [25,44,56], because decreasing W j is related to quantity of electrons transported into the ETC, and W j -W c is related to the balance between electrons entering the ETC and electrons leaving the ETC.Thus, weak temperature-dependent changes in the final photodamage within the 10-23 ºC temperature range are probably caused by opposite changes in the final A j in this case.
The second phase of temperature dependences of photodamage (its stimulation with the temperature increasing), which is mainly observed between 23 and 37 • C, is caused by the combination of decreasing Act(RuBP) and increasing Act(J) in this temperature range, because both W j and W j -W c should be strongly increased with increasing temperature in this case.It should be noted that data from the literature [16] support different temperature optimums of activities of linear electron flow and RuBP and show that the revealed effect can be observed in other plant species.
The last phase, which is observed under increased temperatures (37-42 • C), seems to be paradoxical because increasing temperature contributes to decreasing photodamage.This result can be explained by the strong decreases in both Act(RuBP) and Act(J) with increasing temperature; it decreases both W j and W j -W c and, therefore, lowers overreduction of the ETC to protect photosynthetic machinery [56,58,59].It should be noted that our model predicts further increases in the final Aj with strong increasing temperature (more than 42 • C; data not shown), i.e., further decreasing photodamage.However, this effect cannot correspond to real photosynthetic changes because it is known that heating to high temperatures (more than 42 • C) induces irreversible damage of photosynthetic machinery in pea plants [37].We do not describe this direct thermal damage in the current version of our model; thus, results of analysis of this model version at high temperatures (more than 42 • C) cannot be used for prediction of photosynthetic damage in plants.
After that, we investigated the similar temperature dependences under decreasing the stomatal conductance to CO 2 (g s ), which differed by 10% compared to the basic g s .It is known [60,61] that stomatal closing is a typical plant response to action of environmental stressors including drought.This closing can disrupt CO 2 flux from air to the chloroplast stroma and suppress photosynthetic dark reactions [31,32] stimulating damage of PSII that can protect PSI [56,62].
It was shown (Figure 8) that decreasing g s stimulated photodamage of photosynthetic electron transport and additionally lowered the magnitude of weak changes in the final A j within the 10-23 • C temperature interval.In contrast, the second and third phases of changes (the increase and following decrease in photodamage) were not qualitatively modified with decreased g s .Thus, our model-based results show that changes in stomatal opening have a weak influence on the temperature dependence of photodamage of the photosynthetic electron flow.In contrast, the absolute values of photodamage increase with decreasing g S especially, under high light intensity.This effect can be explained by a reduction in the CO 2 flux into chloroplasts, the suppression of CO 2 carboxylation by RuBP, and the overreduction of ETC under low g s .
to 20 °C strongly stimulated photodamage of photosynthetic electron transport with in-creasing temperature.The magnitude of increase in the final Aj at 37-42 °C was small.Increasing the RuBP temperature optimum from 25 °C to 30 °C induced the opposite changes: the final Aj was increased with increasing temperature within the investigated range.However, this effect could not be interpretated as unequivocally positive for photosynthetic tolerance because the values of final Aj in this case were mainly lower than ones at the basic temperature optimums (see Figure 7).Thus, results of the model-based analysis of influence of temperature optimums on photodamage of photosynthetic electron transport show that changes in these optimums Changes in the thermal tolerance of photosynthetic machinery and enzymes are considered to a way of supporting productivity of agricultural plants [63][64][65], e.g., through specific regimes of fertilizer application, plant selection or methods of genetic engineering.Particularly, these approaches increase the thermal tolerance of photosynthetic machinery (see, e.g., work [64] which focused on estimation of the thermal tolerance of plant productivity in accordance with tolerance of photosystem II to heating).
However, results of our model-based analysis (see, e.g., Figure 6) showed that the active linear electron flow in combination with suppressed RuBP activity can be more dangerous for photosynthetic electron transport than suppression of both J and RuBP.These results were in good accordance with data from the literature showing positive influence of suppression of linear electron flow on tolerance to stressors (see, e.g., works [56,62]).Thus, we analyzed the influence of changes in temperature optimums for linear electron flow and RuBP on photodamage of photosynthetic electron transport in plants in the final stage of the current investigation (Figure 9).
It was shown (Figure 9a) that decreasing the RuBP temperature optimum from 25 • C to 20 • C strongly stimulated photodamage of photosynthetic electron transport with increasing temperature.The magnitude of increase in the final A j at 37-42 • C was small.Increasing the RuBP temperature optimum from 25 • C to 30 • C induced the opposite changes: the final A j was increased with increasing temperature within the investigated range.However, this effect could not be interpretated as unequivocally positive for photosynthetic tolerance because the values of final A j in this case were mainly lower than ones at the basic temperature optimums (see Figure 7).

Conclusions
Photodamage of photosynthetic electron transport is the key mechanism of disruption of photosynthesis in agricultural plants under action of stressors which decrease their crop yield.This means that studying photodamage is an important task for basic and applied investigations.However, complex mechanisms of photodamage restrict the use of experimental methods for investigations of this process; the development of mathematical models of photodamage and model-based analysis can be used for overcoming these restrictions.
In the current work, we developed a modified FvCB model which described photodamage of photosynthetic electron transport in higher plants.This model was parameterized on the basis of experimental results (using an example of pea plants).Analysis of the model showed that combined inactivation of linear electron flow and RuBP could result in both increases and decreases in of the level of photodamage, depending on the extent of inactivation of linear electron flow and RuBP.Simulation of photodamage under different temperatures and light intensities showed that temperature dependences could be multi-phase; particularly, paradoxical increases in the thermal tolerance of photosynthetic electron transport could be observed under high temperatures (37-42 °C).Finally, it was Thus, results of the model-based analysis of influence of temperature optimums on photodamage of photosynthetic electron transport show that changes in these optimums contribute to photodamage and decrease plant tolerance to the light-induced damage of photosynthetic machinery.This means that the development of new methods of crop protection based solely on changes in temperature optimums can have limited efficiency.

Conclusions
Photodamage of photosynthetic electron transport is the key mechanism of disruption of photosynthesis in agricultural plants under action of stressors which decrease their crop yield.This means that studying photodamage is an important task for basic and applied investigations.However, complex mechanisms of photodamage restrict the use of experimental methods for investigations of this process; the development of mathematical models of photodamage and model-based analysis can be used for overcoming these restrictions.
In the current work, we developed a modified FvCB model which described photodamage of photosynthetic electron transport in higher plants.This model was parameterized on the basis of experimental results (using an example of pea plants).Analysis of the model showed that combined inactivation of linear electron flow and RuBP could result in both increases and decreases in of the level of photodamage, depending on the extent of inactivation of linear electron flow and RuBP.Simulation of photodamage under different temperatures and light intensities showed that temperature dependences could be multi-phase; particularly, paradoxical increases in the thermal tolerance of photosynthetic electron transport could be observed under high temperatures (37-42 • C).Finally, it was shown that changes in the temperature optimums of linear electron flow and Rubisco could modify temperature dependences of the final photosynthetic electron transport under photodamage induction; however, these changes mainly stimulated this photodamage.
Caemmerer-Berry Model with Description of Photodamage 2.1.General Description of the Model and Equations

Figure 1
Figure1shows the general scheme of the developed mathematical model of photodamage of photosynthetic electron transport in C 3 plants.Our model was based on the Farquhar-von Caemmerer-Berry model[26,35,38].The steady-state rate of photosynthetic CO 2 assimilation was described as being equal to the slowest of the two main processes.The first process is carboxylation/oxygenation, related to RuBP activity.This process was limiting under the low CO 2 concentration in the modeled chloroplast.The second process is the regeneration of RuBP, associated with the production of NADPH from NADP + through photosynthetic light reactions.This regeneration was taken as a function of the potential rate of the linear electron flow through the ETC under the specific intensity of actinic light (J, µmol•m −2 •s −1 ).In accordance with a series of works[31,32,36], we did not include the limitation of photosynthetic CO 2 assimilation by the rate of triose transport because it was difficultly distinguished from the RuBP carboxylation/oxygenation limitation[36].

Figure 1 .
Figure 1.The general scheme of the modified FvCB model describing photodamage of photosynthetic electron transport (see text for details).

Figure 1 .
Figure 1.The general scheme of the modified FvCB model describing photodamage of photosynthetic electron transport (see text for details).

Figure 2 .
Figure 2. Experimental (blue markers) and model (black line) dependences of the linear electron flow (J) through the ETC on the intensity of the blue actinic light.Experimental J values were calculated on basis of Equation (17) (n = 7).The model dependence was approximated by Equation (4) at Jmax = 55 µmol•m −2 •s −1 and θ = 0.25.The determination coefficient between these dependences (R 2 ) was 0.9862.

Figure 2 .
Figure 2. Experimental (blue markers) and model (black line) dependences of the linear electron flow (J) through the ETC on the intensity of the blue actinic light.Experimental J values were calculated on basis of Equation (17) (n = 7).The model dependence was approximated by Equation (4) at J max = 55 µmol•m −2 •s −1 and θ = 0.25.The determination coefficient between these dependences (R 2 ) was 0.9862.
), A hv , calculated V cmax and noted parameters of the FvCB model, we calculated C c c (for the RuBP limitation) under different Plants 2023, 12, 3211 8 of 18 j c ) and RuBP (C c c ) limitations, and the final C c , which was calculated as max C c c , C j c , under different intensities of the blue actinic light.PAR, µmol•m −2 •s −1 RuBP (C ) limitations, and the final Cc, which was calculated as max C , C , under different intensities of the blue actinic light.

Figure 3 .
Figure 3. Experimental (blue markers) and model (black line) dependences of the mesophyll conductance to CO2 (gm) on the intensity of the blue actinic light.The model dependence was

Figure 4 .
Figure 4. Experimental (blue markers) and model (black line) dependences of the damage rate constant (kd) on the intensity of the blue actinic light (n = 6-7).The model dependence was calculated on basis of Equation (11) at k = 0.00008 s −1 and α = 0.09; Wj and W were calculated using Equations (1) and (3)-(6), standard parameters of the FvCB model[35] and results from Table1.The determination coefficient between these dependences (R 2 ) was 0.856.

Figure 4 .
Figure 4. Experimental (blue markers) and model (black line) dependences of the damage rate constant (k d ) on the intensity of the blue actinic light (n = 6-7).The model dependence was calculated on basis of Equation (11) at k 0 d = 0.00008 s −1 and α = 0.09; W j and W were calculated using Equations (1) and (3)-(6), standard parameters of the FvCB model[35] and results from Table1.The determination coefficient between these dependences (R 2 ) was 0.856.

Figure 5 .
Figure 5. Experimental (blue markers) and model (black line) dependences of activity of the CO 2 carboxylation by RuBP (Act(RuBP) (a) and linear electron flow (Act(J)) (b) on temperature.Experimental parameters (A hv and J) were normalized on maximal values.Equation (12) was used for approximation of experimental dependences.Act(RuBP) was approximated at t o = 25 • C and σ 1 = σ 2 = 22 • C; R 2 = 0.942.Act(J) was approximated at t • = 31 • C and σ 1 = 36 and σ 2 = 15.5 • C; R 2 = 0.988.It was shown that Equation (12) accurately described normalized temperature dependences of CO 2 carboxylation by RuBP (Figure 5a) and linear electron flow (Figure 5b); parameters of these approximation and their determination coefficients are shown in the figure.It was important that the determined optimal temperatures (25 and 31 • C for Act(RuBP) and Act(J), respectively) were in good accordance with data from the literature [16].Thus, we developed and parameterized the mathematical model of photodamage of photosynthetic electron transport which was based on the classical FvCB model.The model-based analysis of photodamage under non-optimal temperatures was the next task of our work.The standard procedure of the model analysis included the imitation of favorable conditions (actinic light) for 20 min, imitation of stressful conditions (actinic light + transient temperature changes or transient direct decreases in RuBP activity and J) and, finally, imitation of favorable conditions (actinic light).The final damage (Aj) was analyzed after a 60 min simulation.

Figure 6 .
Figure 6.Heatmaps of the final damage of photosynthetic electron transport (the final Aj) induced by decreasing activity of J (Act(J)) and RuBP (Act(RuBP)) under the actinic light with low (108 µmol•m −2 •s −1 ) (a) and high (758 µmol•m −2 •s −1 ) (b) intensities.Act(J) and Act(RuBP) in heatmaps show their values under imitation of stressful conditions.The final Aj, which is calculated on basis of Equation (11) and can range from 0 to 1, are shown by pseudo-colors.This Aj shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.

Figure 6 .
Figure 6.Heatmaps of the final damage of photosynthetic electron transport (the final A j ) induced by decreasing activity of J (Act(J)) and RuBP (Act(RuBP)) under the actinic light with low (108 µmol•m −2 •s −1 ) (a) and high (758 µmol•m −2 •s −1 ) (b) intensities.Act(J) and Act(RuBP) in heatmaps show their values under imitation of stressful conditions.The final A j , which is calculated on basis of Equation (11) and can range from 0 to 1, are shown by pseudo-colors.This A j shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.

Figure 7 .
Figure 7. Simulated temperature dependences of the final rate of photosynthetic electron transport (A j ) under different intensities of light for the basic model parameters.The final A j , which can range from 0 to 1, shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.

Figure 8 .
Figure 8. Simulated temperature dependences of the final rate of photosynthetic electron transport (Aj) under different intensities of light at the decreased gs (10% from the basic value).The final Aj, which can range from 0 to 1, shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.Changes in the temperature optimum of the linear electron flow induced the opposite effects: decreasing this optimum from 31 °C to 26 °C was accompanied by increases in the final Aj (lowering photodamage) with increasing temperature and increasing the optimum from 31 °C to 36 °C was accompanied by decreases in the final Aj (stimulation of photodamage) with increasing temperature.Thus, results of the model-based analysis of influence of temperature optimums on photodamage of photosynthetic electron transport show that changes in these optimums

Figure 8 .
Figure 8. Simulated temperature dependences of the final rate of photosynthetic electron transport (A j ) under different intensities of light at the decreased g s (10% from the basic value).The final A j , which can range from 0 to 1, shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.
contribute to photodamage and decrease plant tolerance to the light-induced damage of photosynthetic machinery.This means that the development of new methods of crop protection based solely on changes in temperature optimums can have limited efficiency.

Figure 9 .
Figure 9. Simulated temperature dependences of the final rate of photosynthetic electron transport (Aj) under modifications of temperature optimums of RuBP or J under the 425 µmol•m −2 •s −1 light.(a) t = 31 °C and t = 20 °C .(b) t = 31 °C and t = 30 °C .(c) t = 26 °C and t = 25 °C .(d) t = 36 °C and t = 25 °C.The final Aj, which can range from 0 to 1, shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.

Figure 9 .
Figure 9. Simulated temperature dependences of the final rate of photosynthetic electron transport (A j ) under modifications of temperature optimums of RuBP or J under the 425 µmol•m −2 •s −1 light.(a) t J o = 31 • C and t C o = 20 • C. (b) t J o = 31 • C and t C o = 30 • C. (c) t J o = 26 • C and t C o = 25 • C. (d) t J o = 36 • C and t C o = 25 • C. The final A j , which can range from 0 to 1, shows photodamage of the electron transport because its increase corresponded to decreasing photodamage.Changes in the temperature optimum of the linear electron flow induced the opposite effects: decreasing this optimum from 31 • C to 26 • C was accompanied by increases in the final A j (lowering photodamage) with increasing temperature and increasing the optimum from 31 • C to 36 • C was accompanied by decreases in the final A j (stimulation of photodamage) with increasing temperature.Thus, results of the model-based analysis of influence of temperature optimums on photodamage of photosynthetic electron transport show that changes in these optimums contribute to photodamage and decrease plant tolerance to the light-induced damage of photosynthetic machinery.This means that the development of new methods of crop protection based solely on changes in temperature optimums can have limited efficiency.