Novel Model for Stomatal Conductance: Enhanced Accuracy Under Variable Irradiance and CO₂ in C₃ Plant Species

Simple Summary

This study introduces a new model for predicting stomatal conductance (g_sc)—the rate at which plants exchange gas such as CO₂ and water vapor with the atmosphere through stomata—in three common C₃ plant species. The researchers compared a new model proposed by Ye et al. with two widely used models (Ball–Woodrow–Berry and Medlyn models) under varying light intensities and CO₂ conditions. The study found that the new model more accurately describes how g_sc responds to changes in the environment, especially under high light or fluctuating CO₂ levels. The results highlight the limitations of existing models and demonstrate the improved predictive power of the new approach. This work is important for better understanding plant water use efficiency, improving crop productivity, and predicting how plants may respond to climate change.

Abstract

This study analyzes stomatal conductance (g_sc) in Trifolium repens L., Lolium perenne L., and Triticum aestivum L. under varying environmental conditions. Light-response curves for photosynthesis (A_n–I) at 420 μmol mol⁻¹ CO₂ were used to determine saturating irradiance (I_sat) using a light-response model for photosynthesis, and CO₂-response curves for photosynthesis (A_n–C_i) were measured at I_sat and half I_sat for these C₃ plant species. The Ball–Woodrow–Berry (BWB) model, Medlyn model, and a new model were compared for their ability to describe the net photosynthetic rate (A_n) relative to g_sc under changing irradiance or CO₂. The BWB model overestimated g_sc response, simplifying stomatal behavior, while the Medlyn model deviated at high A_n values, indicating limitations in dynamic responses. The new model showed a better empirical fit under the tested conditions, achieving high R² values and low AIC values across all three species, and demonstrated a strong alignment with empirical data. Our findings highlight the complexity of g_sc regulation and the need for improved models to better represent stomatal dynamics under different environmental conditions. This research is vital for optimizing water use efficiency, enhancing crop productivity, and understanding plant resilience to climate change.

1. Introduction

Stomatal conductance (g_s; abbreviations listed in Supplementary Table S1) is a critical physiological parameter in plant biology, playing a central role in regulating gas exchange between the plant and the atmosphere [1,2]. Specifically, stomatal conductance refers to the rate at which water vapor and carbon dioxide move through the stomatal pores, which are microscopic openings on the surfaces of plant leaves. This regulation is essential for maintaining plant health and productivity, as it directly influences key processes such as photosynthesis, transpiration, and the plant’s overall ability to cope with environmental stressors [3,4,5].

The dynamics of stomatal opening and closing are influenced by a range of environmental factors, including light intensity, atmospheric carbon dioxide concentration, humidity, and soil water availability [6,7,8,9,10]. These responses are part of a complex feedback mechanism that enables plants to optimize gas exchange. Through this process, plants balance the need to absorb carbon dioxide for photosynthesis with the risk of water loss due to transpiration. This delicate balance is crucial for plant survival, particularly in the face of changing environmental conditions such as drought or fluctuating temperatures [5,11,12].

Historically, stomatal conductance has been modeled using a variety of approaches, ranging from simple empirical relationships to more mechanistic models that account for the underlying physiological processes [1,2,13,14,15,16,17]. One of the most widely used models is the Ball–Woodrow–Berry (BWB) model, which provides a framework for understanding how stomatal conductance is influenced by atmospheric demand for water vapor and the plant’s internal water status [18]. This model has been instrumental in understanding the coupling between stomatal conductance and transpiration, but it does not fully account for the complex interactions between stomatal behavior and other environmental variables [19]. Based on the BWB model, other models of stomatal conductance were developed [1,11,19,20,21,22].

As research in plant physiology has advanced, there is an increasing recognition that more sophisticated, process-based models are necessary to accurately predict stomatal behavior across a wide range of environmental conditions. These models seek to integrate multiple physiological processes, including the regulation of stomatal aperture by both biochemical and biophysical factors, and can help predict plant responses to climate change and water stress [3,8]. For instance, models that incorporate mechanistic descriptions of stomatal signaling pathways, such as those proposed by Ainsworth and Rogers [23], provide a more comprehensive understanding of how plants adjust stomatal conductance in response to both internal and external cues.

In recent years, advancements in computational plant modeling, along with the development of high-throughput phenotyping techniques, have significantly improved our ability to predict stomatal responses in dynamic and complex environments [17,24,25]. These cutting-edge models are not only essential for advancing fundamental research in plant physiology but also have important practical implications, especially for agriculture. Optimizing water use efficiency, enhancing crop productivity, and understanding plant resilience to climate change are key areas where accurate predictions of stomatal conductance can have a direct impact [26]. Currently, while the stomatal conductance model developed by Medlyn et al. [26] enjoys widespread adoption, it is not without its recognized challenges and constraints. Research findings underscore the model’s difficulty in precisely forecasting stomatal responses to fluctuations in atmospheric CO₂ levels (C_a). Specifically, under conditions where carboxylation is the limiting factor—characterized by intense light and scant CO₂—the model erroneously anticipates a stomatal opening, contrary to the observed stomatal closure [27]. Concurrently, the model falls short in comprehensively capturing the structural acclimation of plants to enduring CO₂ variations, potentially leading to inaccuracies in parameters such as maximum hydraulic conductance and photosynthetic capacity adjusted for temperature [27]. Moreover, the Medlyn model [26], which is grounded in the optimization theory of water-use efficiency, transitions to an empirically sound yet suboptimal framework under conditions of light saturation, potentially failing to accurately mirror the physiological dynamics of stomata [27]. These limitations underscore an imperative for ongoing research and model enhancement to deepen our understanding and predictive capabilities regarding stomatal behavior across diverse environmental scenarios. Furthermore, in various stomatal conductance models [18,20,21,22,26], the term g₀ refers to the residual stomatal conductance, which is generally assumed to approach zero [2,28,29]. However, the extent to which this assumption holds true in actual conditions remains uncertain, warranting further investigation into this matter.

The objectives of this study are: (i) to plot A_n–C_a and A_n–g_sc response curves for Trifolium repens L., Lolium perenne L., and Triticum aestivum L under both saturating irradiance (I_sat) and half I_sat, thereby elucidating the response patterns of A_n to g_sc under these light conditions; (ii) to compare the ability of the novel model proposed by Ye et al. [2,30] with the BWB and Medlyn models in describing the relationship between A_n and g_sc under conditions of varying irradiance or CO₂ concentration; and (iii) to analyze the rationality of considering the g₀ values obtained from different model’s fittings as the conventionally defined residual stomatal conductance, or whether they are small enough to be negligible, by comparing the g₀ values derived from the different models. This comprehensive approach will not only allow us to evaluate the effectiveness of these models but also deepen our understanding of the physiological mechanisms governing stomatal conductance in response to environmental variations.

2. Materials and Methods

2.1. Stomatal Conductance Models Description

2.1.1. Stomatal Conductance of Ball–Woodrow–Berry Model

The Ball–Woodrow–Berry (BWB) model is an empirical construct that captures the dynamic relationship between stomatal conductance and the net photosynthetic rate (A_n). Originally developed by Ball et al. [18], this model characterizes stomatal conductance as a function of A_n, leaf surface CO₂ concentration (C_s), and relative humidity (h_r). It posits that under steady-state conditions, stomatal conductance exhibits a linear correlation with A_n, provided that C_s and h_r remain constant. This relationship is predicated on the stomata’s role in modulating intercellular CO₂ concentration to foster a stable environment for photosynthesis. The stomatal conductance to CO₂ (g_sc) is determined using the Ball–Woodrow–Berry model [18]:

$$g_{\mathrm{sc}}=g_1{\displaystyle \frac{A_{\mathrm{n}}{h}_{\mathrm{r}}}{C_{\mathrm{s}}}}+g_0$$

(1)

where g₀ (uint: mol·m⁻²·s⁻¹) and g₁ (dimensionless) are the fitting parameters [2,18,31], A_n is net photosynthetic rate (μmol·m⁻²·s⁻¹), h_r is relative humidity at the leaf surface (dimensionless), and C_s is atmospheric CO₂ concentration at the leaf surface. The parameter g₁ represents the slope of the relationship between g_sc and A_nh_r/C_s [26] or is referred to as the empirical slope [32].

However, in mathematical contexts, the ratio of two large numbers is often highly sensitive to minor changes in the denominator. To mitigate this sensitivity, Equation (1) can be reformulated as:

$$A_{\mathrm{n}}={\displaystyle \frac{1}{g_1{h}_{\mathrm{r}}}}\left(g_{\mathrm{sc}}-g_0\right)C_{\mathrm{s}}$$

(2)

Therefore, the parameters g₁ and g₀ can be determined by applying Equation (2). Furthermore, in Equation (2), when A_n is equal to zero, it follows that g_sc must equal g₀.

2.1.2. Stomatal Conductance Model of Medlyn et al. [26]

The optimal stomatal conductance model presented by Medlyn et al. [26] is defined by the following equations:

$$g_{\mathrm{sc}}=g_0+\left(1+{\displaystyle \frac{g_1}{\sqrt{D}}}\right){\displaystyle \frac{A_{\mathrm{n}}}{C_{\mathrm{s}}}}$$

(3)

$$g_1\propto \sqrt{{\Gamma }_{*}\lambda }$$

(4)

where g₀ signifies the stomatal conductance when the net photosynthetic rate is zero (0 μmol⋅m⁻²⋅s⁻¹), g₁ represents the slope of the stomatal conductance response, λ is the marginal water cost of carbon gain, and Γ_* denotes the CO₂ compensation point concentration in the absence of day respiratory rate (R_day). The stomatal conductance slope g₁ is proportional to the combination of terms $\sqrt{{\Gamma }_{*}\lambda }$ [26], suggesting that g₁ rises with increasing λ values. Similarly to Equation (2), the third equation can be rewritten as:

$$A_{\mathrm{n}}={\displaystyle \frac{1}{\left(1+\frac{g_1}{\sqrt{D}}\right)}}\left(g_{\mathrm{sc}}-g_0\right)C_{\mathrm{s}}$$

(5)

Hence, the parameters g₁ and g₀ can be determined by applying Equation (5). Consistent with Equation (2), in Equation (5), when A_n is zero, it is necessary that g_sc equals g₀.

2.1.3. Stomatal Conductance Model by Ye et al. [2]

The stomatal conductance model of Ye et al. [2] is expressed as:

$$g_{\mathrm{sc}}=g_1{\displaystyle \frac{A_{\mathrm{n}}}{C_{\mathrm{a}}-C_{\mathrm{i}}}}+g_0$$

(6)

where C_i is the intercellular CO₂ concentration, g₁ is a coefficient (dimensionless).

However, to address the sensitivity of large number ratios and to align more closely with the mathematical expression of Fick’s first law of diffusion, Equation (6) can be reformulated as follows:

$$A_{\mathrm{n}}={\displaystyle \frac{1}{g_1}}\left(g_{\mathrm{sc}}-g_0\right)\left(C_{\mathrm{a}}-C_{\mathrm{i}}\right)$$

(7)

Equation (7) elucidates that A_n is directly proportional to the difference between C_a and C_i, and it exhibits a linear relationship with g_sc under specific environmental conditions. The parameters g₁ and g₀ can be determined by employing Equation (7) to model the response of A_n to variations in C_a, C_i and g_sc. Moreover, in Equation (7), when A_n equals zero, it follows that either g_sc equals g₀ or C_a equals C_i.

Additionally, in Equation (7), when g₁ is set to 1 and g₀ to 0, the equation reduces to the form of Fick’s first law of diffusion. Following the principles of Fick’s first law, under steady-state conditions, the A_n can be articulated as detailed below [9,33,34,35]:

$$A_{\mathrm{n}}=g_{\mathrm{sc}}\left(C_{\mathrm{a}}-C_{\mathrm{i}}\right)$$

(8)

2.2. Study Site and Plants

T. repens, L. perenne (Zhongxin 830) and T. aestivum (Jimai 22) were used as test materials. The experimental site was located at Yucheng Station in the southwest of Yucheng County, Chinese Academy of Sciences, Shandong Province. The plants were sown on 15 October 2022, and maintained under standard field conditions. Data collection was performed on sunny days from 28 April to 10 May 2023. T. repens was in a period of the BBCH 14–16 (Biologische Bundesanstalt, Bundessortenamt und CHemical Industry, Julius Kühn-Institut (JKI), Quedlinburg, Germany) growth stage (vigorous vegetative stage with 4–6 leaves unfolded), with an approximate plant height of 15 cm, and mature leaves at the top were selected for testing. L. perenne was in the BBCH 45–49 growth stage (booting stage), with an approximate plant height of 1.3 m. The new mature, fully unfolded leaves were selected for testing. T. aestivum was in the BBCH 45–55 growth stage (booting to early flowering stage), with an approximate plant height of 60–70 cm, and the flag leaf was selected for testing.

2.3. Gas Exchange and Chl Fluorescence Measurement

From 28 April to 10 May 2023, A_n–I curves were measured on each study plant with an open-path gas exchange system (LI-6400; Li-Cor, Lincoln, NE, USA), equipped with a leaf chamber fluorometer (6400-40; Li-Cor). Measurements were conducted between 9:30–11:30 and 14:30–17:00 on sunny days. The ambient [CO₂] concentration was maintained at 420 μmol·mol⁻¹ for 15 or 13 light levels in the following order (first to last): 2000, 1800, 1600, 1400, 1200, 1000, 800, 600, 400, 200, 150, 100, 80, 50, 0 μmol·m⁻²·s⁻¹ for L. perenne and T. aestivum; and 1600, 1400, 1200, 1000, 800, 600, 400, 200, 150, 100, 80, 50, 0 μmol·m⁻²·s⁻¹ for T. repens. The plants were allowed to acclimate to changes in light intensity for approximately 2–3 min before measurements were logged; it took about 50 min to complete an entire A_n–I curve. After data collection, a mechanistic model of A_n–I in “Photosynthesis Model Simulation Software (PMSS version 2.0)” platform was used to simulate the A_n–I curves [36], This simulation determined the saturating irradiance (I_sat) as 1200 μmol·m⁻²·s⁻¹ for T. repens, 900 μmol·m⁻²·s⁻¹ for L. perenne, and 2000 μmol·m⁻²·s⁻¹ for T. aestivum, respectively. Then A_n–C_i and J–C_i curves were simultaneously recorded at saturating irradiance for 12 CO₂ concentrations in the following order (first to last measurement): 1600 (except for T. aestivum), 1400, 1200, 1000, 800, 600, 420, 300, 200, 100, 60, and 0 μmol·mol⁻¹. The measurements were conducted using an open-path gas exchange system. A stable injected gas supply was provided by CO₂ cartridges (iSi Gesellschaft mit beschränkter Haftung (GmbH), AUSTRIA U2614, Vienna, Austria) compatible with the LI-6400, coupled with the instrument’s built-in CO₂ mixer. The flow rate entering the leaf chamber was set and maintained at 400 μmol s⁻¹. The temperature within the leaf chamber was set at 30 ± 1.3 °C, and the relative humidity was controlled between 45% and 75%. To ensure steady-state conditions, the plants were given approximately 5 min to acclimate to ambient CO₂ (420 μmol·mol⁻¹) in the gas exchange chamber before beginning each A_n–C_i and J–C_i curve, and then logged. It took approximately 45 min to complete a single A_n–C_i and J–C_i curve. The gas exchange properties including photosynthesis rate, C_i, leaf temperature and stomatal conductance were logged at each C_a once the system had reached a predetermined stability point (coefficient of variation ≤ 1%). If this criterion could not be met within a reasonable waiting period (typically 3–5 min), manual logging was performed once the readings were confirmed to have stabilized. Gas exchange measurements were conducted on three independent biological replicates for each plant species (n = 3). The three models of stomatal conductance (BWB model, Medlyn model and a new model proposed by Ye et al.) were integrated into the Photosynthesis Model Simulation Software (PMSS) platform (http://www.zipiao.tech, accessed on 13 November 2024) (Zipiao software development Co., Ltd., Ji’an China), which was then used to simulate these data to obtain g₀ and g₁ for the three C₃ plants. The parameters g₀ and g₁ were estimated using nonlinear least squares regression implemented in the “Photosynthesis Model Simulation Software (PMSS)” platform.

2.4. Statistical Analysis

All variables are presented as mean values (±SE) based on three replicate samples for each species. The data were subjected to one-way analysis of variance (ANOVA). To determine if there were significant differences between A_n values calculated using Equations (2), (5) and (7), and the corresponding observed values, we employed a paired-sample t-test at the 5% level of significance (p < 0.05), utilizing the statistical software package SPSS 18.0 (SPSS, Chicago, IL, USA). The concordance between the mathematical models and the experimental data was evaluated using the adjusted coefficient of determination (R²) and Akaike Information Criterion (AIC) value.

3. Results

3.1. Photosynthesis and Stomatal Conductance Response Under Variable Irradiance

The light-response curves of photosynthesis (A_n–I) for the three species in question followed the anticipated pattern. At high irradiance levels, T. repens showed a slight decline in A_n beyond 1 200 μmol·m⁻²·s⁻¹, suggesting the occurrence of dynamic down-regulation or photoinhibition (Figure 1A). In the case of L. perenne and T. aestivum, the A_n tends to reach saturation as light intensity increases under high light intensity conditions (Figure 1B,C).

In the present study, Equations (2), (5) and (7) are capable of describing the tendency of the stomatal conductance-response curves of photosynthesis (A_n–g_sc) for the three species at a CO₂ concentration of 420 μmol·mol⁻¹ (Figure 2). It can be clearly observed that A_n increases with rising g_sc, yet the relationship between A_n and g_sc is nonlinear, particularly at high A_n values. Upon examination of Figure 2, it is apparent that the fitted curves from Equations (2) and (5) diverge from the observed data points, with the most significant deviations occurring at high A_n values. In contrast, Equation (7) provides a robust characterization of the A_n–g_sc relationship for L. perenne and T. aestivum, yielding extremely high determination coefficients (R²) and low AIC values (

Table 1. The estimates by Equations (2), (5) and (7) for three C₃ species when light-response curves of photosynthesis were performed under a CO₂ concentration of 420 μmol·mol⁻¹ (mean ± SE, n = 3), respectively. Estimated values within one plant which are statistically significantly different (p < 0.05) are annotated with different superscript letters, with the same superscript letter indicating no difference for Equations (2), (5) and (7). The unit of g₀ is mol·m⁻²·s⁻¹.

	T. repens			L. perenne			T. aestivum
	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)
g0	0.087 ± 0.008 a	0.085 ± 0.008 a	−0.002 ± 0.005 b	0.045 ± 0.009 a	0.042 ± 0.009 a	−0.002 ± 0.000 b	0.040 ± 0.012 a	0.037 ± 0.010 a	−0.002 ± 0.000 b
g1	3.446 ± 0.294	0.209 ± 0.209	0.756 ± 0.039	4.515 ± 0.065	1.499 ± 0.089	0.727 ± 0.004	4.574 ± 0.455	1.513 ± 0.223	0.724 ± 0.006
R2	0.914	0.912	0.992	0.857	0.882	1.000	0.936	0.950	1.000
AIC	12.08	12.07	10.18	21.01	18.57	4.32	26.75	25.53	3.91

), indicating a strong fit to the observed data points.

The comparative analysis of the g₀ and g₁ parameter values estimated by Equations (2), (5) and (7) shows that for the same species, there is no significant difference in g₀ values derived from Equations (2) and (5) (p > 0.05), while the g₀ value derived from Equation (7) is significantly lower than those obtained from Equations (2) and (5) (p < 0.05, Table 1). Given that the dimension of g₁ varies between Equations (2), (5) and (7), Equation (2) (BWB model) provides the numerically highest g₁ values for the three species. Furthermore, the fitting values derived from Equation (7) indicate that the most accurate fit is achieved for L. perenne and T. aestivum.

3.2. Photosynthesis and Stomatal Conductance Response Under Variable CO₂ Concentration at I_sat

The CO₂-response curves of photosynthesis (A_n–C_a) for the three species under investigation adhered to the expected trends (Figure 3). T. repens demonstrated a marked increase in A_n in response to elevated CO₂ levels (Figure 3A). For L. perenne and T. aestivum, the A_n approached saturation as CO₂ concentration increased under high CO₂ conditions, signifying a plateau in photosynthetic capacity and the onset of triose phosphate utilization (TPU) limitation (Figure 3B,C).

Under saturating irradiance conditions, the relationship between A_n and g_sc for the three species is shown in Figure 4. It is clear that A_n decreases as g_sc increases except for T. aestivum, but the relationship between A_n and g_sc is nonlinear, especially at high A_n values. In addition, although Equations (2), (5) and (7) can all describe the tendency of the A_n–g_sc curves for the three species under conditions of their respective saturating irradiance, the concordance between the mathematical models and the observed data shows obvious differences. A close look at Figure 4 reveals that the fitted curves from Equations (2) and (5) deviate from the observed data points, with the most notable discrepancies at low A_n values for T. repens (Figure 4A), at both low and high A_n values for L. perenne (Figure 4B), and at high A_n values for T. aestivum (Figure 4C). In contrast, Equation (7) offers a reliable representation of the A_n–g_sc relationship for all three species, resulting in extremely high R² and low AIC values (

Table 2. The estimates by Equations (2), (5) and (7) for three C₃ species when CO₂-response curves of photosynthesis were performed at their corresponding saturating irradiance (mean ± SE, n = 3), respectively. Estimated values within one plant which are statistically significantly different (p < 0.05) are annotated with different superscript letters, with the same superscript letter indicating no difference. The unit of g₀ is mol·m⁻²·s⁻¹.

	T. repens			L. perenne			T. aestivum
	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)
g0	−0.162 ± 0.051 a	−0.105 ± 0.005 a	−0.002 ± 0.000 b	0.000 ± 0.039 a	0.018 ± 0.028 a	−0.001 ± 0.003 a	−0.280 ± 0.220 a	0.035 ± 0.079 a	−0.264 ± 0.157 a
g1	34.391 ± 8.301	6.701 ± 1.599	0.742 ± 0.015	8.633 ± 1.613	2.214 ± 0.564	0.740 ± 0.018	17.406 ± 7.652	4.572 ± 2.820	1.952 ± 0.695
R2	0.964	0.843	0.999	0.925	0.924	0.999	0.939	0.964	1.000
AIC	22.39	19.21	6.06	26.27	22.87	9.36	23.68	23.42	8.69

), which signifies an excellent fit to the empirical data.

The comparative analysis of the g₀ and g₁ parameter values estimated by Equations (2), (5) and (7) shows that, there is no significant difference in g₀ values derived from Equations (2), (5) and (7) for L. perenne and T. aestivum (p > 0.05), while the g₀ value derived from Equation (7) is significantly higher than those obtained from Equations (2) and (5) for T. repens (p < 0.05, Table 2). Notably, the dimension of g₁ differs between Equations (2), (5) and (7); hence, Equation (2) (BWB model) yields the highest numerical g₁ values among the three species. Moreover, the fitting values derived from Equation (7) suggest that the optimal fit is obtained for all three species.

3.3. Photosynthesis and Stomatal Conductance Response Under Variable CO₂ Concentration at Half of I_sat

The A_n–C_a relationship for the species under investigation followed the anticipated trends (Figure 5). T. repens exhibited a significant increase in A_n in response to elevated CO₂ levels; however, A_n declined at a concentration of 1600 μmol·mol⁻¹ (Figure 5A). In the case of L. perenne and T. aestivum, an approached saturation as the CO₂ concentration increased under high CO₂ conditions, indicating a plateau in photosynthetic capacity and the initiation of triose phosphate utilization (TPU) limitation (Figure 5B,C).

The relationship between A_n and g_sc for the three species under conditions of half their respective saturating irradiance can be observed in Figure 6. It is evident that the relationship between A_n and g_sc is complex and nonlinear. Although Equations (2), (5) and (7) can all describe the tendency of the A_n–g_sc curves for the three species under conditions of half their respective saturating irradiance, the concordance between the mathematical models and the observed data shows obvious differences (Figure 6). Upon close examination of Figure 6, it becomes apparent that the fitted curves from Equations (2) and (5) diverge from the observed data points, particularly at low A_n values for T. repens (Figure 6A), and across a range of A_n values for L. perenne (Figure 6B) and T. aestivum (Figure 6C). Conversely, Equation (7) provides a reliable representation of the A_n–g_sc relationship for all three species, yielding extremely high R² values and low AIC values (

Table 3. The estimates by Equations (2), (5) and (7) for the three C₃ species when CO₂-response curves of photosynthesis were performed at half of their corresponding saturating irradiance (mean ± SE, n = 3), respectively. Within a single plant species, estimated values that are statistically distinct (p < 0.05) are marked with unique superscript letters, while identical superscript letters signify no significant difference. The unit of g₀ is mol·m⁻²·s⁻¹.

	T. repens			L. perenne			T. aestivum
	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)	Equation (2)	Equation (5)	Equation (7)
g0	−0.082 ± 0.038 a	−0.033 ± 0.015 a	−0.002 ± 0.000 a	−0.324 ± 0.176 a	−0.312 ± 0.175 a	−0.005 ± 0.003 a	−0.269 ± 0.150 a	−0.586 ± 0.309 a	−0.013 ± 0.008 a
g1	25.502 ± 5.474	4.853 ± 1.891	0.760 ± 0.046	41.855 ± 15.809	18.901 ± 8.429	0.782 ± 0.025	22.981 ± 10.228	21.427 ± 10.123	0.811 ± 0.067
R2	0.882	0.777	0.989	0.741	0.720	0.996	0.744	0.849	0.999
AIC	21.98	19.61	12.18	25.74	24.79	12.02	27.94	27.81	10.05

), which indicate an excellent fit to the observed data for L. perenne (Figure 6B) and T. aestivum (Figure 6C).

The comparative analysis of the g₀ and g₁ parameter values estimated by Equations (2), (5) and (7) shows that, no significant difference is observed in g₀ values derived from Equations (2), (5) and (7) for all three species under conditions of half their respective saturating irradiance (p > 0.05, Table 3). Notably, the dimension of g₁ differs between Equations (2), (5) and (7); hence, Equation (2) (BWB model) yields the highest numerical g₁ values among the three species. Moreover, the fitting values derived from Equation (7) suggest that the optimal fit is obtained for L. perenne and T. aestivum.

4. Discussions

The present study offers a detailed analysis of stomatal conductance (g_sc) in three species—T. repens, L. perenne, and T. aestivum—across a range of environmental conditions. Our findings underscore the intricacy of g_sc regulation and its response to variations in light intensity and CO₂ concentration, aligning with the broader understanding of stomatal behavior as a key integrator of plant responses to the environmental factors [2,37].

4.1. Theoretical Framework and Innovation of the New Model by Ye et al. [2]

The new model proposed by Ye et al. (Equation (7)) inherits its mathematical form from Fick’s first law, which ensures a solid physical foundation [33,34,35]. However, its core innovation lies in extending this physical law—which describes ideal gas diffusion—into a parameterized modeling framework capable of capturing complex plant physiological regulation. The key distinction resides in the introduction and interpretation of the parameters g₀ and g₁. In Fick’s law, the proportional relationship between A_n and g_sc (A_n–g_sc) is fixed at one, assuming stomata are the sole limiting factor for photosynthesis and that the plant operates in a theoretically optimal state [2,33]. However, real-world plant physiological processes often cause systematic deviations in photosynthetic behavior. By introducing the fitting parameters g₁ and g₀, Equation (7) effectively captures these deviations (Figure 2, Figure 4 and Figure 6). In contrast, the BWB model, being highly empirical, lacks a clear physiological interpretation for its parameter g₁, while the Medlyn model, grounded in optimization theory, shows limitations in dynamic responses and structural acclimation—as evidenced in this study by its clear deviations under high light or low CO₂ conditions (Figure 2, Figure 4 and Figure 6). Therefore, the novelty of the model proposed by Ye et al. is not merely an algebraic rearrangement of Fick’s first law; rather, it enables us to effectively quantify the regulatory effects of non-stomatal limitations—such as g_m or biochemical processes—on the photosynthesis-stomatal conductance relationship within a unified framework.

4.2. Model Comparison and Fit

The comparative analysis of the BWB, Medlyn, and the new models has revealed significant discrepancies in their ability to capture the relationship between the net photosynthetic rate (A_n) and stomatal conductance to CO₂ (g_sc). The BWB model, a cornerstone in the field, yielded the highest g₁ values, suggesting an overly steep slope in the g_sc response. This characteristic may not accurately encapsulate the nuanced physiological mechanisms that govern stomatal behavior, as recent research has indicated [5,11,27,38]. The curves generated by the Medlyn et al. [26] model diverged from the observed data points at high A_n for the three species, suggesting that the model may not fully capture the dynamic stomatal responses under diverse environmental conditions.

In contrast, Equation (7) emerged as a more robust descriptor of the A_n–g_sc relationship, delivering high R² values and low AIC values across all plant species under variable irradiance and CO₂ concentrations. This outcome indicates a superior alignment with empirical data, underscoring the equation’s potential as a more accurate predictive tool. This aligns with the call for models that better represent stomatal dynamics and account for the complex interactions between environmental stressors and plant physiology.

Equation (7)’s success in capturing the A_n–g_sc relationship may be attributed to its ability to incorporate a broader range of environmental variables and plant physiological responses, providing a more comprehensive framework for understanding stomatal behavior. This is particularly important in the context of global change, where plants are subjected to increasing atmospheric CO₂ concentrations, temperature variability, and drought stress. The model’s robustness across different plant species suggests its versatility and potential applicability in diverse ecosystems and under varying environmental conditions.

However, it is important to note that while Equation (7) shows promise, there is still a need for further research to refine and validate these models under a wider range of conditions. This includes understanding how plants acclimate structurally and physiologically to persistent changes in atmospheric CO₂, and how these acclimations interact with other environmental variables such as light and temperature [23]. Additionally, this model must be tested against a broader dataset, including different plant species and environmental scenarios, to ensure their predictive accuracy and applicability in actual growth environment conditions.

4.3. Photosynthetic Response and Stomatal Conductance

The photosynthetic response to fluctuating irradiance and CO₂ concentrations was consistent with anticipated patterns, particularly for T. repens, which showed a significant enhancement in net photosynthetic rate (A_n) in response to increased CO₂ levels (Figure 1, Figure 3 and Figure 5). Notably, at a CO₂ concentration of 420 μmol·mol⁻¹, T. repens exhibited a decline in A_n at a light intensity of 1200 μmol·m⁻²·s⁻¹ (Figure 1A), suggesting the potential engagement of photoinhibition or dynamic down-regulation mechanisms [30]. These responses may act as protective measures against excessive irradiance exposure, as previously detailed by Murchie and Niyogi [39]. In the case of L. perenne and T. aestivum, the A_n under high irradiance levels indicated a plateau in photosynthetic capacity, signifying the occurrence of photosynthetic saturation (Figure 1B,C). Moreover, both species displayed a saturation trend in A_n under high CO₂ conditions at their respective saturating irradiance (I_sat) and half of I_sat (Figure 2B,C and Figure 3B,C), indicating a plateau in photosynthetic capacity and the initiation of triose phosphate utilization (TPU) limitation. This TPU limitation is a pivotal factor influencing the photosynthetic performance of plants in environments with elevated CO₂ levels [12,26,29,34,40].

Furthermore, this study numerically disclosed substantial variability in the g₀ and g₁ parameters as estimated by Equations (2), (5) and (7) among the three species in question. Notably, Equation (2) yielded g₀ values that extended from −0.280 to 0.087 mol·m⁻²·s⁻¹, with corresponding g₁ values spanning a broad range of 3.446 to 34.391 (Table 1, Table 2 and Table 3). When applying Equation (5), the g₀ values were found to range between −0.586 to 0.085 mol·m⁻²·s⁻¹, and g₁ values fluctuated between 0.209 and 21.427 (Table 1, Table 2 and Table 3). Equation (7) provided g₀ estimates that fell within a narrower band, from −0.264 to 0.000 mol·m⁻²·s⁻¹, and g₁ values that were comparatively stable, extending from 0.724 to 1.952 (Table 1, Table 2 and Table 3). Strikingly, Equation (7), with the exception of T. aestivum at I_sat, produced g₀ values tightly clustered between −0.013 and 0.000 mol·m⁻²·s⁻¹, and g₁ values that were remarkably consistent, centering around 0.8 (Table 1, Table 2 and Table 3). The remarkable stability of g₁ strongly suggests that it may represent a conserved parameter characterizing the intrinsic coupling of the stomatal-photosynthetic system in C₃ plants. When g₁ ≈ 0.8, it implies that under measured conditions, the net photosynthetic rate is approximately 80% of the value predicted by the ideal Fick’s law of diffusion. This systematic “discount” can reasonably be attributed to the combined effects of limitations in g_m and other biochemical processes. Therefore, g₁ can be interpreted as a “comprehensive physiological regulation factor,” whose magnitude directly reflects the degree to which internal leaf physiological resistances attenuate ideal gas exchange. This interpretation endows g₁ with clear biological significance, surpassing the purely empirical nature of the parameter g₁ in the BWB model while also circumventing the dimensional issues associated with g₁ in the Medlyn model. This consistency points to a reliable and predictable stomatal response across the species under scrutiny, particularly when subjected to varying irradiance levels and CO₂ concentrations. These observations highlight the diversity in stomatal conductance parameters and emphasize the importance of model-specific analysis when interpreting plant responses to environmental scenarios.

Conversely, g₀ is recognized as the residual stomatal conductance in several models [2,18,22,26]. Typically, g₀ is considered to be minuscule and is often assumed to be zero in many models [2,28,29]. However, our study reveals that for both the BWB and Medlyn models, g₀ exceeds zero significantly for the three plant species under varying light irradiance and CO₂ concentrations, as detailed in Table 1, Table 2 and Table 3, indicating that its impact cannot be overlooked. In the context of Equation (7), g₀ is relatively smaller and can generally be considered negligible, except for T. aestivum under saturating light intensities (Table 1, Table 2 and Table 3). Consequently, our findings suggest that it may be more accurate to view g₀ as a fitting constant rather than residual stomatal conductance. Otherwise, it would be difficult to explain why the estimated values of g₀ for T. aestivum under saturating light conditions, as calculated by Equations (2) and (7), are below −0.2 mol·m⁻²·s⁻¹ (Table 2), and for T. aestivum under half-saturating light conditions, as calculated by Equations (2) and (5), are also below −0.2 mol·m⁻²·s⁻¹ (Table 3). Regarding the occurrence of negative g₀ values obtained from model fittings, particularly the frequent instances observed with the BWB and Medlyn model fits (Table 1, Table 2 and Table 3), these are likely mathematical artifacts arising from the fitting processes of the respective formulas. The functional forms of these models may force the fitting procedure to produce a negative intercept to achieve the best possible fit to the nonlinear data, thereby resulting in g₀ values that contradict biological reality. Consequently, the g₀ parameters derived from these model fits may not reflect the true physiological reality of plant stomata, as stomatal conductance cannot be negative [2,32]. Such negative g₀ values may occur when the model attempts to minimize residuals under conditions where stomatal conductance is very low or approaching zero [18,26]. Furthermore, we note that compared to Equation (7), lower negative g₀ values are more commonly obtained when fitting the A_n–C_a response curves for the three C₃ species under half-saturating irradiance using Equations (2) and (5). This observation further supports the need for a more physiologically consistent model.

4.4. Limitations and Future Perspectives

While this study presents a promising model for stomatal conductance, it is important to acknowledge its limitations, which also outline clear pathways for future research. Firstly, the evaluation and parameterization of the models are conducted on three C₃ species (T. repens, L. perenne, and T. aestivum). Although this provides a robust comparison within a specific plant species, the generalizability of our findings to other species, particularly woody plants, C₄ species, or plants from extreme habitats, remains to be tested. Future work should include a broader phylogenetic range of species to validate and potentially refine the model across diverse plant kingdoms. Secondly, our measurements were performed under controlled, steady-state conditions in a gas exchange chamber. While this allows for precise isolation of the effects of irradiance and CO₂, it does not fully capture the dynamic and fluctuating nature of the natural environment, such as rapid changes in light (e.g., sunflecks), wind, and vapor pressure deficit. Testing the model’s performance under field conditions with real-world environmental variability is a crucial next step. Finally, the model was parameterized and validated using the same dataset. To unequivocally establish its predictive power and avoid overfitting, an independent validation with a completely separate dataset is essential. Future research should prioritize applying the model to an independent dataset collected from different growth conditions or by independent research groups. Addressing these limitations will be critical for advancing the model from a promising theoretical framework to a robust tool for predicting plant-water-carbon interactions in real-world ecosystems under global change.

5. Conclusions

This study systematically evaluated the predictive performance of three stomatal conductance models—BWB, Medlyn, and a new model proposed by Ye et al. (Equation (7))—under variable light intensity and CO₂ conditions for three C₃ plant species. The results indicated that the traditional BWB and Medlyn models exhibited significant deviations in describing the relationship between A_n and g_sc, particularly under conditions of high A_n or extreme environments. In contrast, the new model demonstrated exceptional robustness and accuracy, achieving high R² values and low AIC values between simulated and measured data across all three species and conditions. Furthermore, the analysis of key parameters revealed that the fitted parameters (g₀ and g₁) in the new model showed superior consistency and stability. The g₀ values obtained from the new model were notably more stable, supporting their interpretation as a fitting constant rather than a fixed physiological parameter. This characteristic stands in clear contrast to the often non-negligible or negative g₀ estimates from the BWB and Medlyn models, clearly challenging the conventional concept of “residual conductance”. Therefore, the new model provides a more reliable tool for simulating stomatal behavior. It offers significant value for improving predictions of crop water use efficiency and understanding plant responses to global change. However, while the new model performs well across the three C₃ species tested, further validation across a wider range of species and environmental conditions is necessary before broader generalizations can be made.