Modeling light response of effective quantum efficiency of photosystem II for C3 and C4 crops

Effective quantum efficiency of photosystem II (Φ_PSII) represents the proportion of photons of incident light that are actually used for photochemical processes, which is a key determinant of crop photosynthetic efficiency and productivity. A robust model that can accurately reproduce the nonlinear light response of Φ_PSII (Φ_PSII–I) over the I range from zero to high irradiance levels is lacking. In this study, we tested a Φ_PSII–I model based on the fundamental properties of light absorption and transfer of energy to the reaction centers via photosynthetic pigment molecules. Using a modeling-observation intercomparison approach, the performance of our model versus three widely used empirical Φ_PSII–I models were compared against observations for two C₃ crops (peanut and cotton) and two cultivars of a C₄ crop (sweet sorghum). The results highlighted the significance of our model in (1) its accurate and simultaneous reproduction of light response of both Φ_PSII and the photosynthetic electron transport rate (ETR) over a wide I range from light limited to photoinhibition I levels and (2) accurately returning key parameters defining the light response curves.

Introduction

Light intensity (I; see Table 1 for list of abbreviations and definitions) exhibits dynamic fluctuations across various temporal scales, ranging from seconds to months due to wind-induced leaf movements and diurnal solar variations (Liu et al., 2021). Under low I conditions, plants efficiently channel the majority of absorbed light to reaction centers for photochemical processes. Under high I conditions, plants dissipate approximately 80% of absorbed light as heat through non-photochemical quenching (NPQ) to prevent light damage (Niyogi and Truong, 2013). The generation of NPQ primarily originates from the de-excitation of light-harvesting pigment molecules in excited state (N_k). Consequently, there exists a significant correlation between the quantity of N_k and the magnitude of NPQ. Additionally, the light environment within leaves is highly heterogeneous due to the focusing effect of epidermal cells and the light-guiding properties of vascular bundle sheath structures (Xiao et al., 2016; Song et al., 2017). This heterogeneity results in significant I variations among different cells within the leaf and even among chloroplasts within the same mesophyll cell (Xiao et al., 2016). Photosystems within chloroplasts constantly operate in a highly dynamic light environment, and accurate modeling of photosynthetic responses to rapid change of I is important for us to understand the adaptive responses of plants to the changing light environments.

Table 1

Table 1. List of major model parameters defining the light response curves of effective quantum efficiency of photosystem II (Φ_PSII) and electron transport rate (ETR).

Photosystem II (PSII) is pivotal in the light-dependent reactions of photosynthesis, driving the initial steps of energy conversion. Its activity can be conveniently assayed using bio-optical techniques (Buckley and Farquhar, 2004; Robakowski, 2005; Baker, 2008; Pavlovič et al., 2011; Shevela et al., 2023), with chlorophyll a fluorescence being the most widely adopted method. This technique facilitates the research into several key photosynthetic properties, including the maximum quantum efficiency of PSII (F_v/F_m), the effective quantum efficiency of PSII [Φ_PSII = (F_m′ − F′)/F_m′, where F_m′ is the maximum fluorescence in the light and F′ is steady-state fluorescence], the photosynthetic electron transport rate (ETR) and NPQ. In addition, given that the generation of NPQ mainly results from the de-excitation of N_k, the concurrent change in N_k and NPQ with the increasing I should be able to be characterized with a robust model.

Φ_PSII and ETR are the most widely used photochemical parameters to assess the efficiency of plant photochemistry in different environments (Genty et al., 1989; Moin et al., 2016). Φ_PSII represents the proportion of photons of incident light that are actually used to drive photochemistry, and it is closely linked with the closure and opening of PSII in photosynthetic primary reactions and chlorophyll fluorescence emission (Baker, 2008). Meanwhile, ETR is closely related to Φ_PSII and I (Genty et al., 1989; Krall and Edwards, 1992). Many studies using fluorescence techniques to determine Φ_PSII found it decreasing nonlinearly with the increasing I (Robakowski, 2005; Pavlovič et al., 2011; van der Tol et al., 2014; Córdoba et al., 2016). With the increasing I, ETR initially increased, and then, it reached a platform or there occurred photoinhibition or dynamic downregulation of PSII at high light intensities. This nonlinear relationship reflects the complex interplay between light absorption, energy transfer, and dissipation mechanisms in the photosynthetic apparatus, highlighting the adaptive responses of plants to varying environments.

No model has yet been reported to simultaneously accurately reproduce Φ_PSII–I and ETR–I curves over a wide I range from zero to photoinhibitory I levels. Among the limited studies characterizing the Φ_PSII–I curve, the negative exponential function and the exponential function are the most widely used models (Webb et al., 1974; Smyth et al., 2004; Ritchie, 2008; Ritchie and Bunthawin, 2010; Silsbe and Kromkamp, 2012; Robakowski et al., 2018). However, it has been reported that the values of I_sat estimated by the negative exponential functions are significantly higher than the measured values (Robakowski, 2005; Ritchie, 2008; Ritchie and Bunthawin, 2010). In addition, the non-rectangular hyperbolic model (NRH model) is the most widely used to characterize the ETR–I curve and returning ETR_max (von Caemmerer, 2000; Long and Bernacchi, 2003; Miao et al., 2009; Yin et al., 2009; Gu et al., 2010; Bernacchi et al., 2013; von Caemmerer, 2013; Buckley and Diaz-Espejo, 2015; Cai et al., 2018; Yin et al., 2021). However, the NRH model significantly have been reported to overestimate ETR_max, and it cannot return a realistic I_sat due to its asymptotic function (Buckley and Diaz-Espejo, 2015; Yang et al., 2024). Experimentally, the value of Φ_PSIImax was measured when I was 0 μmol photons m⁻² s⁻¹.

In this study, we aimed to develop and test a Φ_PSII–I model based on the fundamental properties of light absorption of photosynthetic pigment molecules (Ye et al., 2013a, 2013b). We evaluated the performance of the model using a modeling-observation intercomparison approach against observations conducted on two C₃ corps (peanut and cotton) and two genotypes of a C₄ crop (sweet sorghum cultivars KFJT-1 and KFJT-4). We also compared the robustness of this model against three widely used empirical ETR–I and Φ_PSII–I models (i.e., the negative exponential function, the exponential function, and the NRH model) in their performances of (1) reproducing the observed light response curves and (2) returning key parameters defining the curves (i.e., Φ_PSIImax, ETR_max and the corresponding I_sat).

Materials and methods

Plant materials

Seeds of peanut (Arachis hypogaea L.) and cotton (Gossypium hirsutum L.) were surface disinfected with 70% ethanol and 20% bleach, then planted in trays and placed in an RDN-1000E-4 growth chamber (Ningbo Dongnan Instrument Co., China) under conditions of 23°C and 28°C (16 h/8 h light/dark cycle) for cultivation. When the seedlings developed two cotyledons, they were transplanted into the fields of the Botanical Garden at Nantong University. Field management was carried out according to the previously described methods (Wang et al., 2017). Seedling cultivation of sweet sorghum (Sorghum bicolor L. Moench, KFJT-1 and KFJT-4) followed the protocols established in our previous research (Yang et al., 2024). These two strains were developed by the Institute of Modern Physics, Chinese Academy of Sciences, through heavy ion irradiation of the parental line KFJT-CK. The cultivated seedlings were transferred to plastic pots and placed in a climate-controlled chamber, where they were grown under 25,000 lx light intensity, at 25°C, and with a 16 h/8 h light/dark cycle. Healthy plants bearing eight leaves were selected for data measurements for each species.

Analytical models

Model 1 (Ye model)

The Ye model provides a mechanistic framework for describing the light response of ETR in PSII based on the biophysical properties of light-harvesting pigment molecules using Equation 1 (Ye et al., 2013a, 2013b):

$\begin{array}{ll}ETR=\frac{\alpha \beta N_0{\sigma }_{\text{ik}}\phi }{S}\times \frac{1-\frac{(1-\frac{g_i}{g_k}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }I}{1+\frac{(1-\frac{g_i}{g_k}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }I}I& (1)\end{array}$

where φ is the exciton-use efficiency in PSII, N₀ is total photosynthetic pigment molecules of the measured leaf, S is the leaf area (m²), and g_i and g_k are the degeneration of energy levels of photosynthetic pigments in the ground state (i) and excited state (k), respectively. k_P and k_D are rates of the photochemical reaction and heat loss, respectively (Baker, 2008). ξ₁, ξ₂, and ξ₃ are the occupation probability of photochemistry, heat loss, and fluorescence emission, respectively. σ_ik is the eigen-absorption cross-section of photosynthetic pigments from the ground state i to the excited state k via light exposure, and τ is the average lifetime of the photosynthetic pigments in the lowest excited state k.

To simplify Equation 1, three aggregate parameters encapsulating biophysical dynamics are introduced: ${\alpha }_{\text{e}}=\frac{\alpha \beta N_0{\sigma }_{\text{ik}}\phi }{S}$ [μmol electrons (μmol photons)⁻¹] represents the initial slope of the light response curve of electron transport rate (ETR–I curve), ${\beta }_{\text{e}}=\frac{(1-\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }$ [m² s (μmol photons)⁻¹] reflects the dynamic downregulation term of PSII, and ${\gamma }_{\text{e}}=\frac{(1+\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }$ [m² s (μmol photons)⁻¹] represents the saturation term of photosynthesis.

With these parameters, Equation 1 simplifies to:

$\begin{array}{ll}ETR={\alpha }_{\text{e}}\frac{1-{\beta }_{\text{e}}I}{1+{\gamma }_{\text{e}}I}I& (2)\end{array}$

Equations 1 and 2 describe ETR–I function and depict the interdependence between ETR and biophysical parameters.

Since Equation 1 is a non-asymptotic function, it has the first derivative. When the first derivative of Equation 1 equals to zero, I_sat is calculated as follows:

$\begin{array}{ll}{I}_{\text{sat}}=\frac{\sqrt{\frac{({\beta }_{\text{e}}+{\gamma }_{\text{e}})}{{\beta }_{\text{e}}}}-1}{{\gamma }_{\text{e}}}& (3)\end{array}$

Substituting Equation 3 into Equation 2, the maximum ETR (ETR_max) can be determined using Equation 4:

$\begin{array}{ll}ET{R}_{\text{max}}={\alpha }_{\text{e}}{\left(\frac{\sqrt{{\beta }_{\text{e}}+{\gamma }_{\text{e}}}-\sqrt{{\beta }_{\text{e}}}}{{\gamma }_{\text{e}}}\right)}^2& (4)\end{array}$

Moreover, combing Equation 1 with ETR =α×β×Φ_PSII×I (Krall and Edwards, 1992), the relationship between Φ_PSII and I can be described as follows:

$\begin{array}{ll}{\Phi }_{\text{PSII}}=\frac{N_0{\sigma }_{\text{ik}}\phi }{S}\times \frac{1-\frac{(1-\frac{g_i}{g_k}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }I}{1+\frac{(1-\frac{g_i}{g_k}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }I}I& (5)\end{array}$

Simplified, this becomes:

$\begin{array}{ll}{\text{Φ}}_{\text{PSII}}={\text{Φ}}_{\text{PSIImax}}\frac{1-{\beta }_{\text{e}}I}{1+{\gamma }_{\text{e}}I}& (6)\end{array}$

where ${\text{Φ}}_{\text{PSIImax}}=\frac{{\alpha }_{\text{e}}}{\alpha \beta }$.

The effective absorption cross-section of light-harvesting pigment molecules (σ′_ik), which represents its ability to absorb light energy with I, can also be expressed as a function of I (Ye et al., 2013b). Namely,

$\begin{array}{ll}{\text{σ}}_{\text{ik}}^{\text{'}}=\frac{{\sigma }_{\text{ik}}}{1+\frac{(1-\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau I}{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }} \times [1-\frac{(1-\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau I}{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }]& (7)\end{array}$

Equation 7 shows that σ′_ik increases with k_P, k_D, ξ₁, ξ₂, ξ₃, and 1/τ but decreases with I. σ′_ik = σ_ik when I = 0 μmol photons m⁻² s⁻¹. As such, the light absorption cross-section is not a constant under any given I (excluding I = 0 μmol photons m⁻² s⁻¹). By introducing β_e and γ_e, Equation 7 can be simplified to:

$\begin{array}{ll}{\text{σ}}_{\text{ik}}^{\text{'}}=\frac{1-{\beta }_{\text{e}}I}{1+{\gamma }_{\text{e}}I}{\sigma }_{\text{ik}}& (8)\end{array}$

Comparing Equation 5 with Equation 7, the relationship between Φ_PSII and σ′_ik is described by Equation 9:

$\begin{array}{ll}{\text{Φ}}_{\text{PSII}}={\text{Φ}}_{\text{PSIImax}}\frac{{\text{σ}}_{\text{ik}}^{\text{'}}}{{\sigma }_{\text{ik}}}& (9)\end{array}$

For a given species under given environmental conditions, the values of Φ_PSIImax and σ_ik are constants. Equation 9 demonstrates that Φ_PSII is directly proportional to σ′_ik, and it changes as a function of σ′_ik.

The number of photosynthetic pigment molecules in the excited state k (N_k) can be expressed as:

$\begin{array}{ll}{N}_{\text{k}}=\frac{{\sigma }_{ik}\tau I}{(1-\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau I+({\xi }_3+{\xi }_1{k}_{\text{P}}\tau +{\xi }_2{k}_{\text{D}}\tau )}{N}_0& (10)\end{array}$

Equation 10 demonstrates that N_k is a dynamic variable, exhibiting continuous fluctuations rather than maintaining constant values. N_k decreases with k_P, k_D, ξ₁, ξ₂, and ξ₃ but increases with σ_ik, τ, and I. Under dark conditions (I = 0 μmol photons m⁻² s⁻¹), it must be that N_k equals 0.

By applying β_e and γ_e to Equation 10, it can be simplified to:

$\begin{array}{ll}{N}_{\text{k}}= \frac{1}{1-\frac{g_{\text{i}}}{g_{\text{k}}}}\times \frac{{\beta }_{\text{e}}I}{1+{\gamma }_{\text{e}}I}{N}_0& (11)\end{array}$

Equation 11 shows that N_k increases with the increasing I.

Considering that chlorophyll fluorescence primarily stems from light-harvesting pigment molecules in the excited state, it is logical to infer that NPQ response to I would closely mirror the response of N_k to I. Building on Equation 11, the light response expression for NPQ can be deduced from the light response model of N_k, the expression for NPQ in response to light can be derived as:

$\begin{array}{ll}NPQ=NP{Q}_{\text{max}}\frac{aI}{1+bI}+NP{Q}_0& (12)\end{array}$

where $a=\frac{{\beta }_{\text{e}}}{1-\frac{g_{\text{i}}}{g_{\text{k}}}}$ is initial slope of the light response curve of NPQ, $b=\frac{(1-\frac{g_{\text{i}}}{g_{\text{k}}}){\sigma }_{\text{ik}}\tau }{{\xi }_3+({\xi }_1{k}_{\text{P}}+{\xi }_2{k}_{\text{D}})\tau }$ is equivalent to γ_e, the saturation term of photosynthesis, and NPQ₀ is NPQ at I = 0 μmol photons m⁻² s⁻¹.

Equations 2, 5 (and 6), 8, 11, and 12 represent how Model 1 (Ye model) describes Φ_PSII–I, ETR–I, σ′_ik–I N_k–I, and NPQ–I curves, respectively.

Model 2 (negative exponential function)

It has been found experimentally that Φ_PSII, ranging from 0 to 1, usually follows a simple negative exponential function as follows (Ritchie, 2008; Ritchie and Bunthawin, 2010; Buckley and Diaz-Espejo, 2015):

$\begin{array}{ll}{\text{Φ}}_{\text{PSII}}={\text{Φ}}_{\text{PSIImax}}\times e^{-k_{\text{w}}I}& (13)\end{array}$

where Φ_PSIImax is defined as the maximum effective quantum efficiency when I = 0 μmol photons m⁻² s⁻¹, k_w is a scaling constant, and I is the light intensity. The values of Φ_PSIImax and k_w can be obtained when Φ_PSII–I curves are simulated by Equation 13.

Substituting Equation 13 into ETR = α×β×Φ_PSII×I (Krall and Edwards, 1992), we get the following expression for ETR:

$\begin{array}{ll}ETR=\alpha \times \beta \times I\times {\text{Φ}}_{\text{PSIImax}}\times e^{-k_{\text{w}}I}& (14)\end{array}$

When the first derivative of Equation 14 equals to zero, we can calculate saturation I (I_sat=1/k_w), then substitute k_w =1/I_sat into Equation 14 to determine the maximum electron transport rate (ETR_max = α×β×I_sat×Φ_PSIImax×e⁻¹).

Equations 13 and 14 represent how the negative exponential function (Model 2) describes Φ_PSII–I and ETR–I curves. It should be noted that there are two values of Φ_PSIImax to be returned when both Φ_PSII–I and ETR–I curves are simulated by Equations 13 and 14, respectively.

Model 3 (exponential function)

The exponential function was introduced to simulate Φ_PSII–I curves as follows (Smyth et al., 2004; Silsbe and Kromkamp, 2012):

$\begin{array}{ll}{\text{Φ}}_{\text{PSII}}=\frac{F_{\text{v}}}{F_{\text{m}}}\times \frac{I_{\text{sat}}}{I}[1-\text{exp}(-\frac{I}{I_{\text{sat}}}\left)\right]& (15)\end{array}$

where F_v/F_m is the “dark-adapted” maximum operating efficiency of PSII, and I_sat is the saturation I (Smyth et al., 2004). However, it should be noted that Φ_PSIImax cannot be estimated by Equation 15, but I_sat and F_v/F_m can be estimated when Φ_PSII–I curve is fitted by Equation 15.

Similarly, substituting Equation 15 into ETR = α×β×Φ_PSII×I (Krall and Edwards, 1992), we get the following expression for ETR:

$\begin{array}{ll}ETR=\alpha \times \beta \times \frac{F_{\text{v}}}{F_{\text{m}}}\times I_{\text{sat}}[1-\text{exp}(-\frac{I}{I_{\text{sat}}})]& (16)\end{array}$

The values of I_sat and F_v/F_m can be returned when ETR–I curves are simulated by Equation 16.

The maximum ETR can be calculated by Equation 17:

$\begin{array}{ll}ET{R}_{\text{max}}=\alpha \times \beta \times \frac{F_{\text{v}}}{F_{\text{m}}}\times I_{\text{sat}}[1-\text{exp}(-1)]& (17)\end{array}$

Equations 15 and 16 represent how the exponential function (Model 3) describes Φ_PSII–I and ETR–I curves. Similarly, it should be noted that there are two values of I_sat and F_v/F_m to be returned when both Φ_PSII–I and ETR–I curves are simulated by Equations 15 and 16, respectively.

Model 4 (non-rectangular hyperbolic model)

The non-rectangular hyperbolic (NRH) model has been mainly used to fit the ETR–I curves of plants (von Caemmerer, 2000; Long and Bernacchi, 2003; Miao et al., 2009; Yin et al., 2009; Gu et al., 2010; Bernacchi et al., 2013; von Caemmerer, 2013; Buckley and Diaz-Espejo, 2015; Cai et al., 2018; Yin et al., 2021), and it has been a sub-model in the FvCB model when irradiance is below the saturation level (Farquhar et al., 1980; von Caemmerer, 2000, 2013; Park et al., 2016; Yin et al., 2021). In the NRH model, the dependence of ETR on I can be expressed as follows:

$\begin{array}{ll}ETR=\frac{{\alpha }^{\prime }\times I+ET{R}_{\text{max}}-\sqrt{{({\alpha }^{\prime }\times I+ET{R}_{\text{max}})}^2-4\theta \times {\alpha }^{\prime }\times I\times ET{R}_{\text{max}}}}{2\theta }& (18)\end{array}$

where α′ is defined as the initial slope of the ETR–I curve, θ is a degree of curvature, and ETR_max is the maximum ETR. Because the first derivative of Equation 18 is always greater than zero, we cannot use Equation 18 to estimate I_sat.

Similarly, combing Equation 18 with ETR = α×β×Φ_PSII×I (Krall and Edwards, 1992), we get the following expression for Φ_PSII:

$\begin{array}{ll}{\text{Φ}}_{\text{PSII}}=\frac{{\alpha }^{\prime }\times I+ET{R}_{\text{max}}-\sqrt{{({\alpha }^{\prime }\times I+ET{R}_{\text{max}})}^2-4\theta \times {\alpha }^{\prime }\times I\times ET{R}_{\text{max}}}}{2\theta \times \alpha \times \beta \times I}& (19)\end{array}$

Equations 18 and 19 represent how the NRH model (Model 4) describes ETR–I and Φ_PSII–I curves. However, it should be noted that Φ_PSIImax and its corresponding I_sat cannot be estimated by Equation 19. In addition, there are two values of ETR_max to be returned when both ETR–I and Φ_PSII–I curves are simulated by Equations 18 and 19, respectively.

Chlorophyll a fluorescence measurement

Measurements were performed on plant leaves using a LI-6800 portable photosynthesis system (LI-COR Inc., USA) equipped with a LI-6800-01A leaf chamber fluorometer (LI-COR Inc., USA). A fully unfolded, dark green, and healthy leaf was used for each measurement. The initial fluorescence (F₀) was recorded after 25 min of dark adaptation in the cuvette. The maximal fluorescence level of the dark-adapted leaves (F_m) and light-adapted leaves (F_m′) were determined by applying saturating flashes (15,000 μmol photons m⁻² s⁻¹) lasting 1 s, to promote the closure of the PSII reaction centers (Maxwell and Johnson, 2000). F_v/F_m and NPQ were calculated as (F_m−F₀)/F_m and (F_m−F_m′)/F_m′, respectively (van Kooten and Snel, 1990).

Light response measurements were conducted on sunny days from 8:30–11:30 a.m. to 2:00–5:00 p.m. using the automatic measurement program of the LI-6800 system. Leaves were flatly clamped into the leaf chamber and gradually exposed to light intensities of 0 μmol photons m⁻² s⁻¹, 25 μmol photons m⁻² s⁻¹, 50 μmol photons m⁻² s⁻¹, 100 μmol photons m⁻² s⁻¹, 200 μmol photons m⁻² s⁻¹, 300 μmol photons m⁻² s⁻¹, 400 μmol photons m⁻² s⁻¹, 600 μmol photons m⁻² s⁻¹, 800 μmol photons m⁻² s⁻¹, 1,000 μmol photons m⁻² s⁻¹, 1,200 μmol photons m⁻² s⁻¹, 1,400 μmol photons m⁻² s⁻¹, 1,600 μmol photons m⁻² s⁻¹, 1,800 μmol photons m⁻² s⁻¹, 1,900 μmol photons m⁻² s⁻¹, to 2,000 μmol photons m⁻² s⁻¹. For each light intensity, a minimum waiting time of 2 min and a maximum waiting time of 3 min were set before recording data. The instrument automatically matched the reference and sample chambers before data recording to ensure accuracy. The ambient CO₂ concentration in the leaf chamber was maintained at 410 μmol mol⁻¹, supplied via an external CO₂ gas cylinder connected to the instrument’s CO₂ injection system, with a flowrate of 500 μmol s⁻¹. Air temperature in the leaf chamber was set at 30°C. Before measurements, leaves were exposed to sunlight or a light intensity of 1,800 μmol photons m⁻² s⁻¹ for 40 min to ensure activation. ETR was calculated as ETR = α×β×Φ_PSII×I, where α is the distribution coefficient of absorption light energy by PSII and PSI, assumed to be 0.5 (Krall and Edwards, 1992; Maxwell and Johnson, 2000; Evans, 2009), and β is leaf absorptance, assumed to be 0.84 (Ehleringer, 1981). In this study, the values of Φ_PSII at I = 0 μmol photons m⁻² s⁻¹ were taken as the maximum Φ_PSII (Φ_PSIImax). Key parameters (e.g., Φ_PSIImax, ETR_max, F_v/F_m, and I_sat) from ETR–I curves and Φ_PSII–I curves were fitted using Model 1–4, respectively, with SPSS 24.0 statistical software (SPSS, Chicago, IL).

Chlorophyll content

Leaf disks were removed from the labeled leaves, followed by rapidly clipping of leaf area of 1 cm diameter for each leaf, to be cut into fine shreds and placed into glass test tube containing 5 mL of 80% (v/v) acetone. The airtight tubes were placed in the dark overnight or until the leaf was blanched at 25°C. All treatments were performed in triplicate. The extracts were centrifuged at 4,000 rpm for 10 min. Absorbances at 663 nm and 645 nm were measured using a spectrophotometer (UVICON-930, Kontron Instruments, Zürich, Switzerland) to determine the contents of chlorophyll (Chl) a and Chl b according to previous reported method by Wellburn (1994) (Wellburn, 1994). In addition, we may use the measured chlorophyll content to estimate N₀ and then use Equation 1 to simulate the ETR–I curves of leaves to obtain α_e, β_e, and γ_e, respectively. The values of σ_ik can be estimated by ${\alpha }_{\text{e}}=\frac{\alpha \beta N_0{\sigma }_{\text{ik}}\phi }{S}$ (in this study, α is 0.5, β is 0.84, φ is 0.95, and S is 6×10⁻⁴ m²), the values of σ′_ik can be estimated by Equation 7 when the values of σ_ik, β_e, and γ_e were determined. ETR–I curves were fitted with the Photosynthesis Model Simulation Software (PMSS) at http://photosynthetic.sinaapp.com/index.html, in both Chinese and English, using Simulated Annealing and the Metropolis Algorithm to extract key parameters (e.g., α_e, β_e, γ_e, σ_ik, σ′_ik, ETR_max, and I_sat).

Statistical analyses

All variables are expressed as mean values ± SE from five samples for each species. Data were analyzed with one-way analysis of variance (ANOVA), and then, the values of ETR_max and I_sat estimated by four models were compared using a paired-sample t-test at p < 0.05 (p-significance level) using the SPSS 24.0 statistical software (SPSS, Chicago, IL). In addition, to compare the advantages and disadvantages of the study models, we took the Akaike’s information criterion (AIC), mean absolute error (MAE), and determination coefficient (R²) as indicators to assess the fitting results of the three models. AIC was calculated by reference to Akaike’s method (Akaike, 1974), which equals 2k+n×ln(SSR/n) (here, k is the number of parameters, n is the sample size, and SSR is the sum square of residuals) and R² was given directly by SPSS 24.0 after fitting the data.

Results

Light response curves of ETR

All plant species exhibited a characteristic rapid initial increase in ETR with rising I, followed by a saturation phase (Figure 1). Both C₃ crops (A. hypogaea and G. hirsutum) displayed a slight decline in ETR beyond the saturation I, indicating a dynamic downregulation of PSII or photoinhibition (Figures 1A, B). The observed values of I_sat were approximately 1,600 μmol photons m⁻² s⁻¹and 1,820 μmol photons m⁻² s⁻¹ for A. hypogaea and G. hirsutum, respectively, with the corresponding ETR_max values as approximately 195.49 µmol electrons m^–2 s^–1 and 228.83 µmol electrons m^–2 s^–1 (Figures 1A, B; Table 2). In contrast, the two cultivars of the C₄ crop S. bicolor showed less notable reduction in ETR after I surpassed I_sat (Figures 1C, D). The observed values of ETR_max for KFJT-1 and KFJT-4 were approximately 133.84 µmol electrons m^–2 s^–1 and 170.15 µmol electrons m^–2 s^–1, respectively, with the corresponding I_sat values approximately 1,600 μmol photons m⁻² s⁻¹ (Figures 1C, D; Table 3).

Figure 1

Figure 1. Light response curves of the electron transport rate (ETR–I) for various crops—Arachis hypogaea (A), Gossypium hirsutum (B), Sorghum bicolor cultivar KFJT-1 (C), and S. bicolor cultivar KFJT-4 (D). The curves were simulated by Models 1–4, respectively. Values were presented as means ± SE (n = 5). A horizontal dashed line represents the the fitted value of ETR_max from the model (μmol electrons m⁻² s⁻¹), while a vertical dashed line represents the fitted value of I_sat from the model (μmol photons m⁻² s⁻¹).

Table 2

Table 2. Fitted (Equations 2, 13, 15, 19) and measured (Obs.) values of parameters defining ETR–I curves of A. hypogaea and G. hirsutum.

Table 3

Table 3. Fitted (Equations 2, 13, 15, 19) and measured (Obs.) values of parameters defining ETR–I curves of two S. bicolor cultivars.

Compared to Models 1, 2, and 4, Model 3 largely failed to represent the observed ETR–I curves (Figure 1). Models 1, 2, and 4 demonstrated high goodness-of-fit, based on the coefficient of determination (R²) and mean absolute error (MAE) (Tables 2, 3). Although Model 4 showed higher R² values than Models 1 and 2 (Tables 2, 3), it overestimated ETR_max and cannot return I_sat. Models 1 and 2 provided ETR_max and I_sat values closely aligned with observed data across all species, but Model 1 demonstrated the lowest AIC, indicating an optimal balance between predictive accuracy and model parsimony. Model 3 underestimated both ETR_max and I_sat, with significant differences between fitted and measured data across all crops (p<0.05) (Figure 1C; Tables 2, 3). Additionally, despite Model 3 being able to produce F_v/F_m values by fitting ETR–I curves, these values significantly deviated from the observed F_v/F_m across all crops.

Light response curves of Φ_PSII

The four models varied significantly in characterizing Φ_PSII–I curves (Figure 2; Tables 4, 5). The Φ_PSII–I curves, fitted using Models 1 (Equation 6), 2 (Equation 14), 3 (Equation 16), and 4 (Equation 18), exhibited a characteristic decrease in Φ_PSII with the increasing I for all crops (Figure 2). Among the models, Model 1 most accurately simulated the nonlinear relationship between Φ_PSII and I, obtaining the highest R² and the lowest MAE compared to Models 2 and 3, which exhibited notable deviation from observations, particularly for A. hypogaea and G. hirsutum (Figures 2A, B; Tables 4, 5). Despite being able to estimate F_v/F_m, Model 3 produced F_v/F_m values that were significantly different from the measured values. Additionally, Model 3 generated an I_sat value with an unknown or unclear meaning (Tables 4, 5). While Model 4 exhibited the highest fitting degree for G. hirsutum and S. bicolor KFJT-4, the fitted curves fluctuated in the low light intensity range (below 600 μmol photons m⁻² s⁻¹) (Figure 2). Furthermore, Model 4 generated a significantly higher ETR_max than the measured value. Another key difference among the models is their ability to return the Φ_PSIImax. Models 1 and 2 can return Φ_PSIImax, while Models 3 and 4 cannot. Compared to Model 2, Model 1 returned Φ_PSIImax values closer to the observed values.

Figure 2

Figure 2. Light response curves of effective quantum efficiency (Φ_PSII–I) for various crops—A. hypogaea (A), G. hirsutum (B), S. bicolor cultivar KFJT-1 (C), and S. bicolor cultivar KFJT-4 (D). The curves were simulated by Model 1–4, respectively, and the fitted absolute error is shown. Values were presented as means ± SE (n = 5).

Table 4

Table 4. Fitted (Equations 6, 14, 16, 18) and measured (Obs.) values of parameters defining Φ_PSII–I curves of A. hypogaea and G. hirsutum..

Table 5

Table 5. Fitted (Equations 6, 14, 16, 18) and measured (Obs.) values of parameters defining Φ_PSII–I curves of two S. bicolor cultivars.

Light response curves of NPQ, N_k and σ′_ik

NPQ increased nonlinearly with I across all plant species, with distinct patterns observed between C₃ and C₄ plants (Figures 3A–D). When I was below 600 μmol photons m⁻² s⁻¹, the NPQ of A. hypogaea and G. hirsutum increased very slowly with the increasing I. When I was more than 600 μmol photons m⁻² s⁻¹, NPQ increased rapidly (Figures 3A–D). In contrast, the NPQ of the two S. bicolor cultivars showed nearly linear increase when I was below 1,800 μmol photons m⁻² s⁻¹. Under high I condition, the NPQ of A. hypogaea and G. hirsutum was significantly higher than that of the two S. bicolor cultivars. For example, at 1,800 μmol photons m⁻² s⁻¹, the NPQ values of A. hypogaea and G. hirsutum were 1.54 and 1.35, respectively, whereas the NPQ values of the two S. bicolor cultivars were only 0.77 and 0.81 (Figures 3A–D). N_k increased nonlinearly with the increasing I for all crops (Figure 3E–H), more rapidly in S. bicolor compared to that in A. hypogaea and G. hirsutum. At I = 1,800 μmol photons m⁻² s⁻¹, the N_k values for S. bicolor cultivars KFJT-1 and KFJT-4 were 0.61 and 0.64, respectively, and A. hypogaea and G. hirsutum demonstrated lower N_k values (0.48 and 0.45, respectively).

Figure 3

Figure 3. Non-photochemical quenching (NPQ), total light-harvesting pigment molecules in excited state (N_k), and effective light energy absorption cross-section (σ′_ik, m²) for A. hypogaea (A, E, I), G. hirsutum (B, F, J), S. bicolor cultivar KFJT-1 (C, G, K), and S. bicolor cultivar KFJT-4 (D, H, K). Values were presented as means ± SE (n = 5). The intersection of the black horizontal and vertical dashed lines in each graph represents their measured value at I = 1,800 μmol photons m⁻² s⁻¹.

During photosynthesis, light-harvesting pigment molecules absorb light energy and transition to different excited states. The ability of these molecules to absorb light energy is represented by their effective absorption cross-section (σ′_ik). As illustrated in Figures 3I–L, σ′_ik nonlinearly decreased with the increasing I for all crops, whose σ′_ik–I curves exhibited similar trends to their Φ_PSII–I curves (Figure 2), indicating a strong correlation between light absorption and photosynthetic efficiency. Within the tested range of I, A. hypogaea and G. hirsutum exhibited significantly higher σ′_ik values compared to the two cultivars of S. bicolor. When I was 1,800 μmol m^-2 s^-1, the σ′_ik of A. hypogaea and G. hirsutum were 1.19 × 10^-21 m² and 1.40 × 10^-21 m², respectively (Figures 3I, J), and the σ′_ik values of S. bicolor cultivars KFJT-1 and KFJT-4 were 0.68 × 10^-21 m² and 0.81 × 10^-21 m² (Figures 3K, L). σ_ik showed similar inter-specific difference as that of σ′_ik. The σ_ik values estimated by Model 1 (Equation 1) for A. hypogaea and G. hirsutum were (3.60 ± 0.10) × 10⁻²¹ m² and (3.74 ± 0.23) × 10⁻²¹ m², respectively. The σ_ik values of S. bicolor cultivars KFJT-1 and KFJT-4 were (2.42 ± 0.12) × 10⁻²¹ m² and (2.58 ± 0.54) × 10⁻²¹ m², respectively.

Discussion

Empirical models and biological integration

Classic empirical models typically rely on mathematical analyses of measured data to establish quantitative functions, often lacking explicit incorporation of biological processes. Through the intercomparison among the four models and observations, this study demonstrates that Model 1 (Ye model) can accurately and simultaneously simulate ETR–I and Φ_PSII–I curves. Model 1 demonstrated its consistent robustness and accuracy for the studied crops in (1) reproducing the ETR–I and Φ_PSII–I curves and (2) returning key quantitative traits defining the light response functions.

By employing an explicit and transparent analytical framework with consistent definitions, Ye model incorporates the fundamental processes of light energy absorption, conversion, and transfer to the reaction centers of PSII via photosynthetic pigments. These processes include light harvesting, exciton resonance transfer, quantum level transitions, and de-excitation (Ye et al., 2013a, 2013b; Shevela et al., 2023). Equation 5 incorporated the quantitative relationship between Φ_PSII and the intrinsic characteristics of light-harvesting pigment molecules (i.e., N₀, σ_ik, τ, φ, k_P, k_D, g_i, g_k, ξ₁, ξ₂, and ξ₃. Our results highlight that the consistent decrease in Φ_PSII and σ′_ik with the increasing I (Figures 2, 3I–L), a finding consistent with previous studies (Suggett et al., 2004, 2007; Ye et al., 2013a).

Validation of model predictions

The observed decrease in σ′_ik with increasing I (Figures 3I–L) support previous studies (Suggett et al., 2004, 2007; Ye et al., 2013a). For instance, Suggett et al. (2004) reported the increase in effective absorption cross-sections for PSII of Emiliania huxleyi with the decrease in I in the plant growth environment. These results demonstrated that plants could adjust their light absorption properties to optimize photosynthetic efficiency and minimize photodamage under the changing light environment.

Moreover, the values of ETR_max and I_sat fitted by Ye model (Equation 2) were in close agreement with the observed data (Figure 1; Tables 2, 3), supporting previous reports (Ye et al., 2013a, 2013b; Robakowski et al., 2018). In contrast, Model 3 underestimated ETR_max and the corresponding I_sat (Figure 1; Tables 2, 3). While Model 4 has been widely used to estimate ETR_max and is a sub-model of FvCB model (Farquhar et al., 1980; von Caemmerer, 2000; Long and Bernacchi, 2003; Sharkey et al., 2007; Miao et al., 2009; Yin et al., 2009; Gu et al., 2010; Bernacchi et al., 2013; von Caemmerer, 2013; Park et al., 2016; Cai et al., 2018; Yin et al., 2021), it overestimated ETR_max and cannot return the corresponding I_sat (Figure 1; Tables 2, 3). These results support the previous studies reporting the limitations of these empirical models in accurately characterizing ETR–I curves (Smyth et al., 2004; Silsbe and Kromkamp, 2012; Buckley and Diaz‐Espejo, 2015; Yang et al., 2024). Meanwhile, Ye model (Equation 6) can also accurately characterize the Φ_PSII–I curves (Figure 2; Tables 4, 5). The negative exponential function (Model 2) overestimated Φ_PSIImax (Figure 2; Tables 4). The exponential function (Model 3) and the NRH model (Model 4) cannot return Φ_PSIImax (Tables 4, 5).

Photosynthetic differences between C₃ and C₄ plants

Our study also highlights distinct photosynthetic responses between C₃ and C₄ plants. For instance, the Φ_PSII in A. hypogaea and G. hirsutum was significantly higher than that in two cultivars of S. bicolor, suggesting that C₃ plants may possess a greater capacity for light energy utilization compared to C₄ plants. The results demonstrate that Ye model (Model 1) can be used to estimate the absorption cross-section of pigment molecules for both C₃ and C₄ plants, supporting previous studies on other photosynthetic organisms (e.g., algae and cyanobacteria) (Yang et al., 2023; Ye et al., 2024). Ley and Mauzerall (1982) (Ley and Mauzerall, 1982) reported that the absolute absorption cross-section for oxygen production for chlorophyll in Chlorella vulgaris at 596 nm in vivo was 2.9×10^–21 m², which was independent of total cell chlorophyll content. In this study, σ_ik estimated by Ye model (Equation 1) for A. hypogaea, G. hirsutum, S. bicolor KFJT-1, and S. bicolor KFJT-4 were 3.60 × 10⁻²¹ m², 3.74 × 10⁻²¹ m², 2.42 × 10⁻²¹ m² and 2.58 × 10⁻²¹ m², respectively.

C₃ and C₄ plants represent distinct evolutionary adaptations to different environmental conditions. The differences in their photosynthetic machinery and carbon fixation mechanisms lead to distinct responses in parameters such as Φ_PSII, ETR, and NPQ across light intensities (Stefanov et al., 2022). The different σ_ik values between C₃ and C₄ plants observed in this study reflect their differential adaptations to light environments. C₃ plants, which evolved in more moderate light environments, typically have a higher light absorption capacity (i.e., higher σ_ik). It allows them to efficiently capture light in potentially light-limited conditions. However, this higher absorption capacity also makes them more susceptible to photoinhibition at high light intensities, as observed in the ETR–I curves where A. hypogaea and G. hirsutum showed more notable decline in ETR beyond I_sat (Figure 1). In contrast, the C₄ plants (S. bicolor) in this study showed lower σ_ik values, contributing to their ability to maintain relatively constant ETR even when I surpasses I_sat (Figure 1). The carbon-concentrating mechanism allows C₄ plants to maintain high photosynthetic rates under high light conditions without the need for excessive light absorption, thereby reducing the risk of photodamage (Yang et al., 2024).

Model limitations and implications

While the Ye model provides a robust representation of light response mechanisms in PSII, it has limitations that warrant further research. The model assumes steady-state photosynthetic conditions, which may not fully capture dynamic processes such as stomatal closure, changes in chloroplast morphology, or fluctuating light intensities. Additionally, factors such as nutrient availability, water stress, or temperature effects are not explicitly included in the current framework. These variables can significantly impact photosynthetic efficiency and could be integrated into future iterations of the model. This study validated the model on a limited set of crops (peanut, cotton, and sweet sorghum) under controlled conditions. Therefore, future studies should expand validation to a broader range of species (e.g., maize, rice, and wheat), and environmental contexts (e.g., fluctuating light, drought, and extreme temperatures) would strengthen the generalizability of model. Incorporating dynamic elements into the model to simulate transient responses to light fluctuations would be valuable. Furthermore, investigating the integration of abiotic stress factors into the model framework could improve predictions of photosynthetic efficiency under stress conditions such as drought or extreme heat.

The accurate assessment of key photosynthetic parameters, such as ETR_max, Φ_PSIImax, I_sat, σ_ik, σ′_ik, N_k, and NPQ, positions the Ye model as a transformative tool for advancing photosynthetic research. These parameters are critical in understanding how plants respond to fluctuating light environments, optimize photochemical efficiency, and manage photoprotection. For instance, the ability of Ye model to quantify N_k and σ′_ik enables detailed exploration of how light-harvesting complexes dynamically adjust to varying light intensities. Similarly, the coupling of Φ_PSII and NPQ predictions provides insight into the balance between photochemical utilization and dissipation of excess light energy, a critical factor under high-stress conditions such as drought or extreme light fluctuations.

This capability has direct implications for plant breeding and crop management in the context of climate change. By leveraging the outputs of the model, researchers can identify genotypes with optimized light absorption and photoprotective traits for high light variability scenarios, such as those experienced in marginal or degraded agricultural lands. Moreover, the model can support breeding programs aimed at developing cultivars with enhanced yield stability by selecting for traits that mitigate photoinhibition or excessive NPQ under fluctuating light. Additionally, incorporating these parameter assessments into ecosystem and agricultural productivity models can improve predictions of carbon assimilation and crop yield under diverse environmental conditions.

To the best of our knowledge, this is the first study reporting a robust model (Equations 2, 6) that can simultaneously and accurately simulate ETR–I and Φ_PSII–I curves and returning values of key physical and biochemical parameters of photosynthetic pigments (i.e., intrinsic absorption cross-section and the effective absorption cross-section of light-harvesting pigment molecules). The findings could also help quantify key light-harvesting properties associated with photoacclimation (Fiebig et al., 2023), photoprotection (Niyogi and Truong, 2013), dynamic downregulation of PSII (Ralph and Gademann, 2005), and/or photoinhibition (Govindjee, 2002) in response to environmental change. This study is useful for (1) plant experimentalists quantifying intra- and/or inter-specific variation in Φ_PSII–I responses and (2) modelers working on better model representation of photosynthetic processes under dynamic light environment.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding authors.

Author contributions

X-LY: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing. TA: Data curation, Formal analysis, Investigation, Writing – review & editing. Z-W-YY: Formal analysis, Software, Writing – review & editing. H-JK: Data curation, Formal analysis, Methodology, Writing – review & editing. PR: Methodology, Software, Writing – review & editing. F-BW: Data curation, Formal analysis, Software, Writing – review & editing. Z-PY: Conceptualization, Funding acquisition, Methodology, Software, Supervision, Writing – original draft, Writing – review & editing. S-XZ: Methodology, Writing – original draft, Writing – review & editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was supported by the National Natural Science Foundation of China (grant numbers 31960054, 41961005, and 31860045) and the Program for Construction of the Advantage Science and Technology Innovation Group of Jiangxi Province (grant number 20142BCB24010).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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