Investigation on absorption cross-section of photosynthetic pigment molecules based on a mechanistic model of the photosynthetic electron flow-light response in C3, C4 species and cyanobacteria grown under various conditions

Investigation on intrinsic properties of photosynthetic pigment molecules participating in solar energy absorption and excitation, especially their eigen-absorption cross-section (σ ik) and effective absorption cross-section (σ ′ ik), is important to understand photosynthesis. Here, we present the development and application of a new method to determine these parameters, based on a mechanistic model of the photosynthetic electron flow-light response. The analysis with our method of a series of previously collected chlorophyll a fluorescence data shows that the absorption cross-section of photosynthetic pigment molecules has different values of approximately 10−21 m2, for several photosynthetic organisms grown under various conditions: (1) the conifer Abies alba Mill., grown under high light or low light; (2) Taxus baccata L., grown under fertilization or non-fertilization conditions; (3) Glycine max L. (Merr.), grown under a CO2 concentration of 400 or 600 μmol CO2 mol−1 in a leaf chamber under shaded conditions; (4) Zea mays L., at temperatures of 30°C or 35°C in a leaf chamber; (5) Osmanthus fragrans Loureiro, with shaded-leaf or sun-leaf; and (6) the cyanobacterium Microcystis aeruginosa FACHB905, grown under two different nitrogen supplies. Our results show that σ ik has the same order of magnitude (approximately 10−21 m2), and σ ′ ik for these species decreases with increasing light intensity, demonstrating the operation of a key regulatory mechanism to reduce solar absorption and avoid high light damage. Moreover, compared with other approaches, both σ ik and σ ′ ik can be more easily estimated by our method, even under various growth conditions (e.g., different light environment; different CO2, NO2, O2, and O3 concentrations; air temperatures; or water stress), regardless of the type of the sample (e.g., dilute or concentrated cell suspensions or leaves). Our results also show that CO2 concentration and temperature have little effect on σ ik values for G. max and Z. mays. Consequently, our approach provides a powerful tool to investigate light energy absorption of photosynthetic pigment molecules and gives us new information on how plants and cyanobacteria modify their light-harvesting properties under different stress conditions.

Investigation on intrinsic properties of photosynthetic pigment molecules participating in solar energy absorption and excitation, especially their eigenabsorption cross-section (s ik ) and effective absorption cross-section (s ′ ik ), is important to understand photosynthesis.Here, we present the development and application of a new method to determine these parameters, based on a mechanistic model of the photosynthetic electron flow-light response.The analysis with our method of a series of previously collected chlorophyll a fluorescence data shows that the absorption cross-section of photosynthetic pigment molecules has different values of approximately 10 −21 m 2 , for several photosynthetic organisms grown under various conditions: (1) the conifer Abies alba Mill., grown under high light or low light; (2) Taxus baccata L., grown under fertilization or non-fertilization conditions; (3) Glycine max L. (Merr.),grown under a CO 2 concentration of 400 or 600 mmol CO 2 mol −1 in a leaf chamber under shaded conditions; (4) Zea mays L., at temperatures of 30°C or 35°C in a leaf chamber; (5) Osmanthus fragrans Loureiro, with shaded-leaf or sun-leaf; and (6) the cyanobacterium Microcystis aeruginosa FACHB905, grown under two different nitrogen supplies.Our results show that s ik has the same order of magnitude (approximately 10 −21 m 2 ), and s ′ ik for these species decreases with increasing light intensity, demonstrating the operation of a key regulatory mechanism to reduce solar absorption and avoid high light damage.Moreover, compared with other approaches, both s ik and s ′ ik can be more easily estimated by our method, even under various growth conditions (e.g., different light environment; different CO 2 , NO 2 , O 2 , and O 3 concentrations; air temperatures;
Previous studies have mainly focused on investigating the relationship between the chlorophyll content and the plant's light absorption and excitation capacity, without considering the impact of parameters such as the intrinsic and effective absorption crosssections of photosynthetic pigment molecules on this process (Baker, 2008;Ahammed et al., 2018;Robakowski et al., 2018).However, the intrinsic properties of photosynthetic pigment molecules in light absorption and excitation are essential to understand the light-harvesting mechanism in photosynthetic organisms.Between different parameters studied, the following are essential to understand this process: (1) the effective absorption crosssections of photosystem II (PSII) (s PSII ) (Suggett et al., 2007;Klughammer and Schreiber, 2015; see Table 1 for the list of abbreviations); (2) the absorption cross-section for oxygen production (s O 2 ) (Ley and Mauzerall, 1982); (3) the absorption cross-section of chlorophyll molecules (s chl ) (Ley and Mauzerall, 1982;Suggett et al., 2007); (4) the eigen-absorption cross-section (s ik , from the ground state to the first excited state) (Ye et al., 2013a;Ye et al., 2013b;Zuo et al., 2019;Hashemi and Leppert, 2021;Hu et al., 2021;He et al., 2022); (5) the effective absorption cross-section of photosynthetic pigment molecules (s ′ ik ) (Ye et al., 2013a;Ye et al., 2013b;Hu et al., 2021;He et al., 2022); and (6) the optical crosssection (a L *(l), where l is the wavelength) (Celebi-Ergin et al., 2022).These absorption cross-sections measured in cyanobacteria, algae, and plants vary among taxa and/or with various growth conditions, as shown in the studies mentioned above.
Currently, there are several methods or techniques for determining the values of these types of parameters.For example, Ley and Mauzerall (1982) estimated s chl in Chlorella vulgaris by measuring the rates of O 2 production and found that s O 2 decreased with increasing growth light intensity.Surprisingly, there were no significant differences in s chl between cells grown under low-light or high-light conditions.De Boni et al. (2007) used the white light continuum (WLC) Z-scan technique to measure the resonant nonlinear spectrum of chlorophyll a extracted from Spinacia oleracea, to study s 01 (from the ground state to the first excited state), as well as s 1n (from the first excited state to higher excited states), and showed that s 01 is dependent on wavelength.Based on the fast repetition rate (FRR) fluorescence technique, Suggett et al. (2007) found that as the growth light intensity increased, the s PSII and s chl in Emiliania huxleyi (Lohmann) decreased.Klughammer and Schreiber (2015) used a kinetic multi-color PAM chlorophyll fluorometer (referred to as MC-PAM method), which allows measurements of the sub-millisecond fluorescence rise kinetics upon the onset of strong variously colored actinic light and found that s PSII was a function of wavelength.Moreover, Ye et al. (2013a) proposed the YRS method, which can be used to determine s ik and s ′ ik when the light-response curve of electron transport rate (ETR-I curve) and chlorophyll content are measured.In the results mentioned above, the dependence of s PSII , s chl , and a L *(l) on wavelength has been confirmed (De Boni et al., 2007;Klughammer and Schreiber, 2015;Celebi-Ergin et al., 2022).Furthermore, previous studies have demonstrated that the s PSII and s chl values can vary during growth under different light intensities (Ley and Mauzerall, 1982;Suggett et al., 2007;Yang et al., 2018;Hu et al., 2021).In addition, Celebi-Ergin et al. (2022) found that the decrease in total chlorophyll resulted in an increased a L *(l) when growth [CO 2 ] increases.However, it is interesting to note that, despite variations among different photosynthetic organisms and environmental conditions, the magnitude of s chl [s ik , s 01 , and a L *(l)] remains consistent at approximately 10 −21 m 2 (Ley and Mauzerall, 1982;De Boni et al., 2007;Suggett et al., 2007;Ye et al., 2016;Yang et al., 2018;Zuo et al., 2019;Hu et al., 2021;He et al., 2022).Only cell suspensions were used for the measurements mentioned above, except for the YRS method (2013a), which also measured leaves.The YRS is the only method capable of investigating s ik and s ′ ik in the leaves of C 3 , C 4 , and CAM species under various conditions because it requires only a commercial portable gas-exchange and fluorescence systems and measured chlorophyll content (Ye et al., 2018;Zuo et al., 2019;Hu et al., 2021;He et al., 2022;Yang et al., 2023).
In order to address uncertainties about the relationship between light intensity, environmental factors (such as temperature, CO 2 , O 2 The mechanistic model of the ETR-I curve of Chl a fluorescence ETR via PSII can be calculated with Eq. 1 (Ye et al., 2013a;Ye et al., 2013b) as: where a ' is fraction of light absorbed by PSII (dimensionless), b ' is leaf absorptance (dimensionless), j is the exciton-use efficiency of PSII (dimensionless), N 0 is total photosynthetic pigment molecules of the measured leaf sample, S is the measured area of leaf sample (m 2 ), and g i and g k are the degeneracy of energy level of photosynthetic pigment molecules in the ground state (i) and excited state (k) (dimensionless), respectively.k P and k D are rates of the photochemical reaction and heat loss (s −1 ), respectively (Baker, 2008).x 1 , x 2 , and x 3 were referred to as "statistical weight of photochemistry, heat loss, and fluorescence emission" (dimensionless), respectively, in our previous work (Ye, 2012;Ye et al., 2013a, Ye et al., 2013b).Furthermore, s ik (m 2 ) is the eigen-absorption cross-section of photosynthetic pigment molecules from the ground state i to the excited state k (after light exposure), t (s) is the average lifetime of the photosynthetic pigment molecules in the lowest excited state k, and I is the light intensity (mmol photons m −2 s −1 ).For the details of the derivation of the mechanistic model, see Ye et al. (2013a, b).
According to Ye et al. (2013a, b), s ik , t, x 1 , x 2 , x 3 , g i , g k , k P , and k D (Eq. 1) are inherently specific but have different values, depending on the species and the environmental conditions (e.g., light, temperature, CO 2 concentration, and the nutrient used).Therefore, for a given species and at a specific environmental condition, all parameters such as s ik , t, x 1 , x 2 , x 3 , g i , g k , k P , and k D should be constant.In this case, we may assume that a = a 0 b 0 ], referred to as the "dynamic downregulation term of PSII/photoinhibition", and g = (1+g i =g k )s ik t [(mmol photons) −1 m 2 s] referred to as "the saturation term of photosynthesis" in our previous work (Ye, 2012;Ye et al., 2013a, Ye et al., 2013b).Then, Eq. 1 can be simplified as: Equations 1 and 2 describe the ETR-I function and characterize the interdependence between ETR and light intensity.Since Eq. 1 is a non-asymptotic function, it has the first derivative.When the first derivative of Eq. 1 equals zero, the saturation light intensity (I sat ) corresponding to the maximum ETR can be calculated as follows: Substituting Eq. 3 into Eq.2, the maximum ETR (ETR max ) can be calculated as: (4) Moreover, when chlorophyll content (unit: mg m −2 ) is determined, s ik can be calculated as: where n 0 is the number of photosynthetic pigment molecules of the measured leaf sample per unit volume (m −3 ), and d is the thickness of leaf sample (m).The biological significance of the other parameters in Eq. 5 is the same as that in Eq. 1.
In this work, we neither fit the values of the initial parameters of the model (x i , k P , k D , t, and others) nor used their values assessed by other authors.We realize that the values of some of these parameters may differ significantly from the values known from the literature, and this is the weakness of the YRS model.Instead, we used Eq. 2 to fit the parameters a, b, and g composed of the initial ones.We realize that this approach means using the YRS model actually as a phenomenological model.To some extent, this is due to the fact that the YRS model does not explicitly take into account the reactions of electron transport.Taking into account the reactions of electron transport and finding an interpretation of results of the fitting in terms of the values of the initial parameters of the model is the task for future research.
Equation 1 has been widely used in the literature to simulate the ETR-I curves of photosynthetic organisms and to extract several parameters from those curves (Serodio et al., 2013;Morfopoulos et al., 2014;Sun et al., 2015;Ahammed et al., 2018;Robakowski et al., 2018;Yang et al., 2018;Zuo et al., 2019;Hu et al., 2021;He et al., 2022;Robakowski et al., 2022;Yang et al., 2023).In addition, when the chlorophyll content (mg m −2 ) is measured, besides a, ETR max , and I sat , some other key parameters characterizing photosynthetic pigment molecules (e.g., s ′ ik ; the minimum average life time of the light-harvesting pigment molecules in the lowest excited state, t min ; the light-harvesting pigment molecules in the excited state, N k ) can also be obtained (Ye et al., 2013a;Ye et al., 2013b).

Effective absorption cross-section of photosynthetic pigment molecules
The s ′ ik , based on uniform light absorption in leaves, can be described as follows (Ye et al., 2013a;Ye et al., 2013b): The biological significance of the parameters in Eq. 6 is the same as that in Eq. 1.We note, from Eq. 6, that s ′ ik will decrease with increasing I when photosynthetic organisms are under certain environmental conditions; from Eq. 6, s ′ ik = s ik when I = 0 mmol photons m −2 s −1 .As such, the s ′ ik is not a constant under any given light intensity (except when I = 0 mmol photons m −2 s −1 ).

Data processing and statistical analysis
The mechanistic model presented above was fitted to obtain key parameters (e.g., ETR max , I sat , s ik , and s ′ ik ) defining the ETR-I curves, using the Photosynthesis Model Simulation Software (PMSS) (http://photosynthetic.sinaapp.com,Jinggangshan University, Ji'an, China).When fitting the ETR-I curves using the YRS model, the initial values are set as follows: a e = 0.06, b e = 0.002, g e = 0.01.
All statistical tests were performed using the statistical package SPSS 18.5 statistical software (SPSS, Chicago, IL).Student's t-test was conducted to test whether there were significant differences between the fitted and the measured values of quantitative traits (such as ETR max and I sat ).Goodness of the fit of experimental observations with results obtained with the mathematical model was assessed using the coefficient of determination (R 2 = 1 -SSE/ SST, where SST is the total sum of squares and SSE is the error sum of squares).

Examples of model application
Details of the growth conditions, the species examined, and the procedures for generating the ETR-I curves to be examined with our model are described in the Supporting Information [for A. alba, see Robakowski et al., 2022; for T. baccata, see Robakowski et al., 2018; for G. max at 400 and 600 mmol CO 2 mol −1 , see Ye et al., 2018; for Z. mays at 30°C and 35°C temperatures, see Wang et al., 2022; for O. fragrans (both for sun-leaf and shaded-leaf), see Hu et al., 2021; and for M. aeruginosa, under two different nitrogen supplies, see Yang et al., 2023].In the present study, representative ETR-I curves (fitting the model of Eq. 1) for four C 3 species (i.e., A. alba, T. baccata, G. max, and O. fragrans), one C 4 species (Z.mays), and one cyanobacterium (M.aeruginosa) are shown in Figure 1.For A. alba grown under HL (Figure 1A), O. fragrans (shaded-leaf) (Figure 1E) and M. aeruginosa grown under two nitrogen supplies (Figure 1F), the value of ETR initially increased (almost linearly) with the light intensity towards saturation, and subsequently, beyond the highest irradiance (i.e., I sat ), it showed a significant degree of decline, likely reflecting an onset of dynamic downregulation of PSII, especially in  M. aeruginosa.On the other hand, A. alba grown under LL (Figure 1A), T. baccata (fertilization) (Figure 1B), G. max at 400 mmol CO 2 mol −1 under shaded conditions (Figure 1C), Z. mays (Figure 1D), and O. fragrans (sun-leaf) (Figure 1E) exhibited only a small decline of ETR with increasing light intensity beyond the I sat .Data for T. baccata (non-fertilization) (Figure 1B) and G. max at 600 mmol CO 2 mol −1 under shaded conditions (Figure 1C) showed that ETR hardly increases with increasing light intensity beyond the I sat .Moreover, the fitted curves show that Eq. 1 reproduces quite well the ETR-I curves of all the six species, regardless of whether there is dynamic downregulation, and this with extremely good fits (R 2 ≥ 0.995) (Figure 1; Table 2).
Table 2 lists the photosynthetic parameters (such as a, ETR max , and I sat ) estimated by Eq. 1 and their corresponding observed values for the four C 3 species, a C 4 species, and a cyanobacterium grown under different conditions, described earlier.In this study, we did not observe significant differences between the estimated values of ETR max (and I sat ) and their corresponding observed values (p > 0.05).Furthermore, Table 2 shows that the values of s ik , calculated by Eq. 5, have different values ranging from 1.25 × 10 −21 m 2 to 5.68 ×10 −21 m 2 for the six species used in this study.For A. alba, there was a significant difference between s ik at HL and at LL (p< 0.05), with the values of s ik at HL being higher than that at LL; for O. fragrans, there was a significant difference between s ik evaluated for the shaded-leaf and for the sun-leaf (p< 0.05), with the values of s ik of the sun-leaf being larger than that of the shaded-leaf; for M. aeruginosa, there was a significant difference between s ik under NO 3 − -N supply and under NH 4 + -N supply (p< 0.05), with the values of s ik under NO 3 − -N supply being smaller than that under NH 4 + -N supply.In addition, no significant difference was found between s ik evaluated in G. max at 400 mmol CO 2 mol −1 and 600 mmol CO 2 mol −1 ; furthermore, no significant difference was found for s ik between Z. mays at 30°C and 35°C.Also there were no significant differences in s ik between T. baccata seedlings grown in fertilization and in non-fertilization conditions.

Effective absorption cross-section of photosynthetic pigment molecules
The values of s ′ ik nonlinearly decrease with I in A. alba at HL and LL (Figure 2A), in T. baccata grown under fertilized and nonfertilized conditions (Figure 2B), in G. max at 400 mmol CO 2 mol −1 and 600 mmol CO 2 mol −1 (Figure 2C), in O. fragrans of sun-leaf and shaded-leaf (Figure 2E), and in M. aeruginosa grown under NO 3 − -N supply and NH 4 + -N supply (Figure 2F).For Z. mays at 30°C versus that at 35°C (Figure 2D), we observed that s ′ ik decreases almost nonlinearly with increasing I.In addition, for A. alba, there is a significant difference between s ′ ik at HL and at LL for any I (p< 0.05) (Figure 2A), and the values of s ′ ik at HL are higher than those of s ′ ik at LL.For example, at 783 mmol photons m −2 s −1 for A. alba, the value of s ′ ik at HL is 0.697 (± 0.060) × 10 −21 m 2 , while the value of s ′ ik at LL is 0.263 (± 0.032) × 10 −21 m 2 .For Z. mays, there is a significant difference between s ′ ik at 30°C and at 35°C (p< 0.05) when the light intensity ranges from 800 to 1,800 mmol photons m −2 s −1 , with the s ′ ik values at 35°C higher than those at 30°C.For instance, at 1,800 mmol photons m −2 s −1 for Z. mays, the value of s ′ ik at 35°C is 0.898 (± 0.049) × 10 −21 m 2 , while the value of s ′ ik at 30°C is 0.717 (± 0.030) × 10 −21 m 2 .For O. fragrans, there is a significant difference between s ′ ik for shaded-leaf and sun-leaf (p< 0.05), as the values of s ′ ik of sun-leaf are higher than those of shaded-leaf at any light intensity.For instance, at 606 mmol photons m −2 s −1 for O. fragrans, the value of s ′ ik for the sun-leaf is 0.588 (± 0.044) × 10 −21 m 2 , while the value of s ′ ik for the shaded-leaf is 0.266 (± 0.043) × 10 −21 m 2 .For M. aeruginosa, there is a significant difference between s ′ ik under NO 3 -N supply.In contrast, for G. max, no significant difference between s ′ ik at 400 mmol CO 2 mol −1 and at 600 mmol CO 2 mol −1 is found (Figure 2C).

Discussion
We have simulated the ETR−I curves of six different species under different conditions using the mechanistic model developed by Ye et al. (2013a, b).Fitting this model to previously collected data (including HL and LL conditions, fertilized and non-fertilized conditions, two CO 2 concentrations, two temperatures, and different species, i.e., four C 3 , one C 4 , and one cyanobacterium) showed extremely good fits (R 2 ≥ 0.995), and the fitted values for ETR max and I sat were consistent with all the observations (Table 2), and with no significant difference between the fitted values for ETR max (and I sat ) and their corresponding observations (p< 0.05; Table 2).This shows that Eq. 1 is highly adequate for fitting ETR−I curves regardless of dynamic downregulation of PSII/ photoinhibition in the photosynthetic organisms used in our current study.Furthermore, our results are consistent with previous studies (Ye et al., 2013a;Ye et al., 2013b;Ye et al., 2016;Yang et al., 2018;Zuo et al., 2019;Hu et al., 2021;He et al., 2022).
There are several methods that can be used to determine s chl .For example, Ley and Mauzerall (1982) estimated s PSII of C. vulgaris by measuring the rates of O 2 production and found that the estimated value of s chl was 2.90 (± 0.10) × 10 −21 m 2 .At the same time, these authors found no significant differences in s chl measured on LL-grown and HL-grown cells.After 25 years of Ley and Mauzerall's work, De Boni et al. (2007) used WLC Z-scan technique to study the values of s 01 and s 1n of Chl a and found that the value of s 01 was ~4.7 × 10 −21 m 2 at 680 nm, while the value of s 1n was ~0 m 2 .Furthermore, De Boni et al. (2007) found that s 01 is wavelength-dependent, while Klughammer and Schreiber (2015) using a MC-PAM method determined that s chl is a function of wavelength and can be determined in combination with chlorophyll content.Moreover, Suggett et al. (2007) using the fast repetition rate fluorescence technique studied that both s PSII and s chl in E. huxleyi decrease as the growth light intensity increases.For instance, s chl ]; I sat , saturation irradiance (mmol photons m −2 s −1 ); J max , the maximum electron transport rate (mmol electrons m −2 s −1 ); s ik , eigen-absorption cross-section of photosynthetic pigment molecules (m 2 ); R 2 , determination coefficient.The different superscript letters followed by the values are significantly different between fitted values and observation values within the same species using Student's t-test at p< 0.05 (n = 3-12).For s ik and chlorophyll content, the different superscript capital letters followed by the values are significantly different between fitted values and observation values for the same species with different treatments using Student's t-test at p< 0.05 (n = 3-12).).These results are consistent with our results given in Table 2.However, s chl , s 01 , a L *(l), and s ik have different definitions and different biological significance.For example, if i = 0 and k = 1, s ik represents the chlorophyll molecules that are excited from the ground state (i = 0) to the first excited state (k = 1), i.e., s 01 .In addition, when k ≥ 1, s ik represents the average light absorption cross-section of a photosynthetic pigment since the accessory pigments transfer the absorbed solar energy to chlorophyll molecules, which will make a transition from the ground state (i = 0) to different excited states (k ≥ 1), depending on the wavelength of light.It, thus, follows that s ik has the same biological significance as s chl and a L *(l).Consequently, both s chl and a L *(l) indicate the average light absorption crosssection of the photosynthetic pigment under light illumination.
From an operational perspective of the YRS method (Ye et al., 2013a, Ye et al., 2013b), the value of s ik can be easily estimated by using Eq. 5, for any species and under different environmental conditions.For instance, the value of s ik is 1.12 × 10 −21 m 2 for T. baccata (fertilization), while it is 5.68 × 10 −21 m 2 for O. fragrans.In this study, the values of s ik for other photosynthetic organisms were found to be between these two values (Table 2).Thus, our work demonstrates that although the experimental materials and methods used have been different (Ley and Mauzerall, 1982;De Boni et al., 2007;Suggett et al., 2007;Ye et al., 2013a, Ye et al., 2013b;Klughammer and Schreiber, 2015;Celebi-Ergin et al., 2022), the values of s ik or s chl are approximately of the same order of magnitude (approximately 10 −21 m 2 ).Previous studies have demonstrated that the relationship between s chl and light adaptation is crucial for understanding how plants and algae adapt to different light environments (Ley and Mauzerall, 1982;Suggett et al., 2007;Neale and Melis, 2010).Plant species and algae grown under different light conditions may have different evolutionary strategies in optimizing light absorption and utilization (Neale and Melis, 2010;Buckley and Diaz-Espejo, 2015).The magnitude of s chl can reflect the light-harvesting capability and light adaptation ability of plants and algae.In this study, the value of s ik for A. alba grown under LL conditions is smaller than that of s ik for A. alba grown under HL (Table 2), and there is a significant difference between them.This result is different from the results obtained by Ley and Mauzerall (1982) and Suggett et al. (2007).Ley and Mauzerall (1982) found that there was no significant difference between s chl of C. vulgaris in LL and in HL, while Suggett et al. (2007) found that s chl of E. huxleyi (B11) at LL growth was higher than those under HL growth.This indicates that the shade-tolerant conifer A. alba has a different strategy from the alga E. huxleyi grown under different light environments.This inconsistency between our results and those of others may reflect evolutionary differences between the terrestrial plants and algae, which are adapted to totally different light environments.In addition, in our present study, we found that there was no significant difference between s ik values at 30°C and 35°C for Z. mays, or at 400 mmol CO 2 mol −1 and 600 mmol CO 2 mol −1 for G. max.These results are inconsistent with those obtained by Celebi-Ergin et al. (2022) on eelgrass (Zostera marina L.), as they found that the decrease in total chlorophyll at greater [CO 2 ] values resulted in an increased a L *(L).It is uncertain whether this difference between our results and those of the other authors result from the different experimental conditions or the speciesspecific, genetic features, and thus further research is needed.Suggett et al. (2007) found that s chl of E. huxleyi (B11) decreased as the growth light intensity increased.Their results showed that E. huxleyi (B11) at the lowest growing irradiance (25 mmol photons m −2 s −1 ) had the highest values of s chl (i.e., 3.25 × 10 −21 m 2 ), while at the highest light intensity (600 mmol photons m −2 s −1 ), it had the smallest values of s chl (i.e., 2.95 × 10 −21 m 2 ).For the intermediate growth irradiances (i.e., 150 and 300 mmol photons m −2 s −1 ), the values of s chl were between 2.95 × 10 −21 m 2 and 3.25 × 10 −21 m 2 .These results suggest that the value of s chl is a function of light intensity, and that it decreases with light intensity.Therefore, the results of Suggett et al. (2007) are consistent with our conclusion on s ′ ik , which decreases nonlinearly with increasing I.However, Ley and Mauzerall (1982) found no significant differences in s chl measured on LL-grown and HL-grown cells.Moreover, Klughammer and Schreiber (2015) found that s PSII (625 nm) nonlinearly increases with increasing light, and then it decreases nonlinearly at high light intensities, but the decrease in s PSII (625 nm) with light intensity depends on the value of J p (where J p is a parameter characterizing the energetic connectivity between PSII units; see Stirbet, 2013).For instance, Klughammer and Schreiber (2015) reported that s PSII (625 nm) nonlinearly increases with I beyond 2,000 mmol photons m −2 s −1 for J p = 2.This obviously contradicts our common sense expectation, since photoinhibition would occur in most dilute suspensions of Chlorella exposed to high light intensities of over 2,000 mmol photons m −2 s −1 .
We emphasize that our results have been derived from instantaneous measurement of ETR-I curves of plants and one cyanobacterium; thus, some differences between s chl obtained by Ley and Mauzerall (1982) and by Suggett et al. (2007) and s ′ ik may result from the different growth irradiance conditions.Compared to the complex methods and techniques required for obtaining parameters such as s PSII , s chl , and a L *(l) in Ley and Mauzerall (1982); Suggett et al. (2007); De Boni et al. (2007); Klughammer and Schreiber (2015), and Celebi-Ergin et al. (2022), we can easily determine the values of s ′ ik by Eq. 6 for any given species under any environmental condition.In addition, it should be noted that due to Eq. 1 being a non-asymptotic function, it possesses a first derivative.According to the principles of calculus, when the first derivative of Eq. 1 equals zero, I sat and its corresponding ETR max can be calculated using Eqs.( 4) and (3), respectively.However, it is only when the selected model perfectly fits the ETR-I curve that the fitted values will match the observed values.Empirical models such as the single exponential model, double exponential model, nonrectangular hyperbolic model, and Suggett et al.'s semi-mechanistic model are asymptotic and cannot accurately fit the ETR-I curve when plants or algae experience dynamic downregulation of PSII/ photoinhibition (Platt et al., 1980;Suggett et al., 2007;Brading et al., 2011;Yang et al., 2023).As a result, obtaining ETR max and I sat becomes challenging, let alone other parameters reflecting the intrinsic characteristics of photosynthetic pigment molecules (such as s ik , s ′ ik , t min , and N k ).Although the double exponential model can be used to fit the ETR-I curve of algae, it needs to be coupled with modified exponential model to derive ETR max and I sat , and the obtained ETR max and I sat values are significantly higher than the measured values (Maxwell and Johnson, 2000;Brading et al., 2011;Buckley and Diaz-Espejo, 2015).On the other hand, the YRS method not only effectively fits the ETR response curves of plants and algae to light but also shows no significant difference between the fitted ETR max and I sat values and the observation values (Liang et al., 2018;Yang et al., 2018;Furutani et al., 2022).In this study, the results obtained using the YRS model for fitting also exhibit no significant difference from the observation values, further demonstrating the applicability and reliability of this model.
Furthermore, the reason why s ′ ik decreases with light intensity in our study, in addition to the characteristics of harvesting pigment molecule itself (e.g., change in s ik , x 1 , x 2 , x 3 , g i , and g k ), may be due to the photo-relocation movement of chloroplasts to anticlinal cell walls (i.e., negative blue/UV-A light-dependent chloroplast phototaxis) where light absorption is reduced (Kasahara et al., 2002;Lazaŕ et al., 2022), presumably to minimize the saturation of photosynthesis by decreasing the amount of excess excitation energy (Murchie and Niyogi, 2011).Such observations may help to investigate the underlying nature of plants' photoprotection mechanism under high light intensity (Murchie and Niyogi, 2011), since decreases in s ′ ik with light intensity would prevent the pigment molecules from absorbing light energy further, and could thus provide us a means to quantitatively investigate the photoinhibition and photoprotection in photosynthetic organisms.
Although different fluorescence and O 2 evolution methods, such as the specific O 2 evolution technique (Ley and Mauzerall, 1982;Celebi-Ergin et al., 2022), WLCZ-scan technique (De Boni et al., 2007), and FRR technique (Suggett et al., 2007), may estimate the light absorption of chlorophyll pigments, these methods have the disadvantage to be technically challenging.Moreover, they use only dilute suspensions of cells, which is why these methods have not been widely applied for the s chl evaluation.On the other hand, while the MC-PAM technique (Klughammer and Schreiber, 2015) can be used not only for dense suspensions of unicellular algae, but also for optically more complex samples (like leaves and corals), there still remains a difficulty in determining s chl , due to the complexity of their methods used for data processing.In contrast with the above approaches, combined commercial portable gasexchange and fluorescence systems (e.g., LI-6400, Li-Cor Inc., Lincoln, Nebraska, USA; CIRAS-II, PP Systems, Hitchin, UK; LCA4 ADC-Biosciences, Hoddesdon, UK; Dual-PAM-100, Walz Inc., Germany), the values of s ik can be obtained easily by using the YPS method presented here for any plant species, algae, and cyanobacteria, under diverse environmental conditions, once their chlorophyll content is measured.Therefore, our method has a greater advantage over the methods and techniques in determining the values of s ik (see Eq. 5).
In conclusion, (1) the mechanistic model, used in this research, has universality in fitting ETR−I curves and in estimating photosynthetic parameters such as ETR max , I sat , s ik , and s ′ ik ; (2) compared with other methods, estimating s chl , s PSII (l), or a L *(l), our method is much more convenient in determining s ik and s ′ ik than those used by other authors (Ley and Mauzerall, 1982;De Boni et al., 2007;Suggett et al., 2007;Klughammer and Schreiber, 2015;Celebi-Ergin et al., 2022); (3) Eqs. 5 and 6 can be used to investigate the effects of different environmental factors (e.g., CO 2 , O 2 , O 3 , NO 2 , temperature, and nutrition) on s ik and s ′ ik , respectively.We conclude that Eq. 1 (see above for the description of the YRS method) provides a powerful means to predict and simulate the ETR−I curves, and to obtain several important parameters such as ETR max , I sat , s ik , and s ′ ik .Furthermore, the latter can be used to investigate light energy absorption properties of photosynthetic pigment molecules and to understand how plants, algae, and cyanobacteria modify their lightharvesting properties unde rlying photo-acclimation, photoprotection, and abiotic stress.Dynamic changes of s ′ ik in response to light indicate that these pigment-protein complexes are used by the photosynthetic organisms, not only as antennae capturing light energy, but also to serve an important role in light energy partitioning and dissipation.Thus, we recommend the use of the YRS method to calculate the values of s ik and s ′ ik to investigate light energy absorption properties of photosynthetic pigment molecules for all plants, algae, and cyanobacteria under any environmental condition.We are planning further studies (i) for evaluating the performance of Eq. 1 when applied to observations on different photosynthesizing organisms under extreme environmental conditions; (ii) for estimating s ik and s ′ ik when the samples, under investigation, are concurrently stressed by various environmental factors, including, e.g., low or high temperatures, different CO 2 , O 2 , O 3 , and NO 2 concentrations, water stress, or salt stress; and (iii) for comparison of s ik and s ′ ik among different functional groups of plants, cyanobacteria, and algae, to better understand evolutionary mechanisms of adaptation in photosynthetic organisms to different environmental conditions.
In order to facilitate others to utilize our developed mechanistic model of ETR−I, we have developed and exploited PMSS (http:// photosynthetic.sinaapp.com).Using measured ETR−I data and combining it with measurements on chlorophyll content (mg m −2 ), many characteristics, associated with light energy absorption of chlorophyll molecules, can be obtained with this software.
FIGURE 1Light-response curves of photosynthetic electron flow and curves fitted by Eq.I for six species.(A) for A. alba; (B) for T. baccata; (C) for G. max; (D) for Z. mays; (E) for O. fragrans; and (F) for M aeruginosa.An asterisk indicates that mean values of ETR at the given / value are significantly different using Student's t-test at p<0.05 (n-5-12).Data are mean±SE (n-3-12).The results of the statistical analyses, seeTable 2. I, lightintensity.
(p< 0.05) when the light intensity is below 559 mmol photons m −2 s −1 , and in this case, the values of s ′ ik under NO 3 − -N supply are less than those under NH 4 + -N supply.For instance, at 559 mmol photons m −2 s −1 , for M. aeruginosa, the value of s ′ ik is 1.666 (± 0.037) × 10 −21 m 2 under NO 3 − -N supply, and the value of s ′ ik is 2.230 (± 0.162) × 10 −21 m 2 under NH 4 + was 3.25 × 10 −21 m 2 in cells grown under low light (25 mmol photons m −2 s −1 ), but decreased to 2.95 × 10 −21 m 2 under high light (600 mmol photons m −2 s −1 ).Their results revealed that growth light intensity affects the values of s chl , and indirectly demonstrated that s chl is a function of light intensity, and seems to decrease with light intensities.Thus, although the values of s chl obtained with various methods are different, the estimated values are in the same order of magnitude (approximately 10 −21 m 2 FIGURE 2Light-response curves of the effective absorption cross-section of photosynthetic pigment molecules calculated by Eq. 6 for six species.(A) for A. alba; (B) for T. baccata; (C) for G. max; (D) for Z. mays; (E) for O. fragrans; and (F) for M aeruginosa.An asterisk indicates that mean values of s'ik at the given I value are significantly different using Student's t-test at p< 0.05 (n = 3-12).Data are mean ± SE (n=3-12).I, light intensity.

TABLE 1
Definitions of the abbreviations., and the values of s chl or s PSII , we have used here the YRS method to investigate s ik and s ′ ik in plant leaves and a cyanobacterium in both dilute and dense suspensions.The objective of this study is to estimate both s ik and s ′

TABLE 2
Results fitted by Eq. 1 and observation values of photosynthetic parameters for six species under various conditions (mean ± SE, n = 3-12).