Extending the Farquhar–von Caemmerer–Berry photosynthesis model to account for various photorespiratory bypasses

Summary

• The Farquhar–von Caemmerer–Berry model is used as the standard to investigate leaf photosynthetic CO₂‐ and light‐responses in plant physiology. The model was developed for C₃ photosynthesis with the native photorespiratory cycle. However, it is also frequently used to parameterize photosynthesis of plants with synthetic photorespiratory bypasses.

• Here, we point out that this latter practice is not advised for two reasons. First, unlike the native photorespiratory cycle, synthetic bypasses alter amounts and location of photorespiratory CO₂ release. Second, requirements for reductants and ATP are altered by bypass pathways. We extend the model to account for these changes.

• Estimates of both V _cmax (Rubisco carboxylation capacity) and J _max (light‐saturated linear electron transport rate) differ between original and extended models, and the difference is greater for V _cmax than for J _max. Model analyses show that 17 bypasses reported so far perform contrastingly in terms of photosynthetic efficiency parameters (carboxylation efficiency and quantum yield) and photosynthetic capacity parameter (light‐saturated photosynthetic rate). The amount rather than the location of photorespiratory CO₂ release largely determines the benefit of bypasses.

• Full‐decarboxylating bypasses, which have been claimed to act as a CO₂‐concentrating mechanism around Rubisco, are least efficient, whereas a carbon‐fixing bypass is most promising for improving C₃ photosynthesis.

We extend a widely used C₃‐photosynthesis model to account for various photorespiratory bypasses, thereby, providing a useful resource and framework that enforces future novel bypass work and scrutinize if the results go counter to theory. Our model analyses show that of the 17 published bypass pathways, full‐decarboxylating bypasses (that have been claimed to act as a CO₂‐concentrating mechanism around Rubisco) are least efficient whereas a carbon‐fixing bypass is most promising for improving C3 photosynthesis. Thus, it is the amount rather than the location of photorespiratory CO₂ release that largely determines the benefit of bypasses.

Introduction

The model of Farquhar et al. (1980) for C₃ photosynthesis (the FvCB model) is the basic model for understanding C₃ leaf physiology from analysis of leaf gas exchange measurements (von Caemmerer & Farquhar, 1981). It has been used to estimate important biochemical parameters such as V _cmax (maximum carboxylation rate of Rubisco) and J _max (maximum rate of linear electron transport under saturating light) as well as CO₂‐diffusion parameter mesophyll conductance g _m (Harley et al., 1992) and day respiration R _d (Yin et al., 2011). It has also been used for model scaling‐up to predict responses of canopy photosynthesis to environmental changes. In short, it is widely accepted as a standard tool for understanding leaf physiology and predicting crop yield and ecosystem functioning.

The FvCB model was developed, based on understandings of Rubisco kinetics and energy requirements for the Calvin–Benson–Bassham (CBB) cycle and the native photorespiratory cycle (see Supporting Information Notes S1). The enzyme Rubisco catalyzes not only CO₂‐fixation or carboxylation of RuBP in the CBB cycle for assimilation but also the reaction of O₂‐binding of RuBP (Bowes & Ogren, 1972). The product of the latter oxygenation reaction is 2‐phosphoglycolate (PGly), which is toxic and must be first dephosphorylated to glycolate and further detoxified, thereby a series of reactions follow involving three organelles – chloroplast, peroxisome, and mitochondrion, forming the photorespiratory cycle (Figs 1 and 2). This cycle contains a decarboxylating step by glycine decarboxylase (GDC), releasing CO₂ as well as NH₄ ⁺ in the mitochondrion. The released NH₄ ⁺ needs to be re‐assimilated, which consumes reductants and ATP. In addition, part of PGly‐derived carbon returns to the chloroplast, and there are also energy costs to convert the salvaged carbon into a usable intermediate for joining the CBB cycle. Thus, the photorespiratory cycle not only leads to loss of photosynthetically fixed carbon but also requires reductants and ATP to operate. Under the ambient atmospheric conditions, photorespiration results in 25–50% loss of photosynthetic efficiency in C₃ plants (Walker et al., 2016). This loss can exacerbate with warming, as high temperature favours RuBP oxygenation over carboxylation reactions (Jordan & Ogren, 1984).

Fig. 1.

Full‐ and partial‐decarboxylating photorespiratory bypasses (bypasses 1 to 10). Arrows and integer numbers in colors correspond to bypass numbers in Table 1 and in the main text – red: bypass 1; orange: bypass 2; dark red: bypass 3; green: bypass 4; light blue: bypass 5; darker blue: bypass 6; purple: bypass 7; pink: bypasses 8 and 9 (i.e. pBHAC and cBHAC, and they are mainly in peroxisome and in chloroplast, respectively), and dark green: bypass 10. Black arrows represent native pathways. Full arrows indicate single‐step reactions, whereas dashed arrows indicate multi‐step reactions. Stoichiometric coefficients are given only for bypass pathways, whereas energy (ATP and NAD(P)H or electron (e⁻) equivalent) requirements are given basically only for bypass‐specific reactions (see second‐to‐the‐last column of Supporting Information Table S1). Enzymes for specific reactions are not shown, except those for scavenging H₂O₂ to distinguish relevant bypasses. Reactions for oxidation of glycolate by O₂ to glyoxylate and H₂O₂ and subsequent breaking‐down of H₂O₂ are given in detail only for the native photorespiratory pathway. Neither NH₄ ⁺ recycling in the native photorespiratory pathway nor the whole CBB cycle in detail is shown in this figure (but see Fig. 2). Grey boxes refer to carbon (C) sinks either from the CBB cycle or utilizing oxaloacetate for some (anabolic) reactions. 2‐OG, 2‐oxoglutarate; 3‐PGA, 3‐phosphoglycerate; APX, ascorbate peroxidase; CAT, catalase; CBB, Calvin–Benson–Bassham; CH₂‐THF, 5,10‐methylene‐tetrahydrofolate; Gln, glutamine; Glu, glutamate; RuBP, ribulose 1,5‐bisphosphate; THF, tetrahydrofolate; TP, triose phosphate.

Fig. 2.

Carbon‐neutral and carbon‐positive photorespiratory bypasses (bypasses 11–17). Arrows and integer numbers in colours correspond to bypass numbers in Table 1 and in the main text – red: bypass 11; orange: bypass 12; green: bypass 13; pink: bypass 14; blue: bypass 15; purple: bypass 16; and dark red: bypass 17. Black arrows represent native pathways. Full arrows indicate single‐step reactions, dashed arrows indicate multi‐step reactions, whereas green dotted arrows indicate uncertain steps for bypass 13 (see footnote of Supporting Information Table S1). Stoichiometric coefficients are given only for bypass pathways, while energy (ATP and NAD(P)H or electron (e⁻) equivalent) requirements are given basically only for bypass‐specific reactions (see the second‐to‐the‐last column of Table S1). Enzymes required for specific reactions are not shown, nor are reactions for oxidating glycolate to glyoxylate and H₂O₂ and subsequent breaking down of H₂O₂ for the native photorespiratory pathway (but see Fig. 1). Grey boxes refer to sinks for carbon (C) either from the CBB cycle or from end products in bypasses 12 and 13. Carbon saved from other bypasses, relative to the native photorespiratory pathway, is coupled with intermediates of the CBB cycle. 2‐OG, 2‐oxoglutarate; 3‐PGA, 3‐phosphoglycerate; CBB, Calvin–Benson–Bassham; CH₂‐THF, 5,10‐methylene‐tetrahydrofolate; DHAP, dihydroxyacetone phosphate; E4P, erythrose 4‐phosphate; F6P, fructose 6‐phosphate; FBP, fructose 1,6‐bisphosphate; GAP, glyceraldehyde 3‐phosphate; Gln, glutamine; Glu, glutamate; McG, malyl‐Co‐A‐glycerate cycle; PEP, phosphoenolpyruvate; PGly, phosphoglycolate; R5P, ribose‐5‐phosphate; Ru5P, ribulose 5‐phosphate; RuBP, ribulose 1,5‐bisphosphate; S7P, sedoheptulose 7‐phosphate; SBP, sedoheptulose 1,7‐bisphosphate; THF, tetrahydrofolate; Xu5P, xylulose 5‐phosphate.

An effective way to mitigate this photosynthetic inefficiency is to have a photorespiratory bypass, whereby CO₂ is relocated for being released in the chloroplast (instead of the mitochondrion, so that the released CO₂ can be re‐fixed by Rubisco) or/and energy required for the operation of the photorespiratory cycle can be lowered. Given that naturally occurred bypasses (e.g. Jiang et al., 2025) hardly relocate the CO₂ release or lower energy requirements, synthetic biology approaches have been explored to modify the photorespiratory pathway. Since the first synthetic bypass was installed in Arabidopsis thaliana (Kebeish et al., 2007), many bypass pathways have been reported for different species including crop plants (reviewed by Wang et al., 2022; Jin et al., 2023; Meacham‐Hensold et al., 2026). Most bypasses have been shown to increase leaf photosynthetic rate and its underlying parameters, and some of them even improve plant biomass accumulation and crop yield.

Of the underlying parameters, reported values of V _cmax and/or J _max are often estimated from fitting the FvCB model to gas exchange measurements on bypass plants (e.g. South et al., 2019; Meacham‐Hensold et al., 2024; Lin et al., 2025). However, using the FvCB model to estimate photosynthetic parameters of plants with synthetic photorespiratory bypasses is not advised. First, unlike the native photorespiratory cycle, synthetic photorespiratory bypasses alter amounts and location of photorespiratory CO₂ release. Second, the native photorespiratory cycle requires 2 mol NADPH‐equivalent reductants and 3.5 mol ATP per oxygenation (Notes S1), and these requirements for either reducing equivalents or ATP are altered by bypass pathways. It is conceivable that these changes have an impact on the estimation of both V _cmax and J _max.

Models have been reported for quantifying how photorespiratory bypasses work in improving photosynthesis (Xin et al., 2015; Basler et al., 2016; Trudeau et al., 2018; Smith et al., 2025). These models mostly are in a complex large‐scale format, based on individual reactions' kinetics or flux balance analyses, and therefore, not handy for synthetic biologists to use for estimating basic photosynthetic parameters like V _cmax and J _max. von Caemmerer (2013) described a revised FvCB model to quantify the bypass of Kebeish et al. (2007). Given that energy requirements depend on individual bypasses (e.g. Peterhansel et al., 2013; Smith et al., 2025), there is a need for a general model in an FvCB format, which can deal with diverse bypasses. The objective of this study is to develop such a general model and to show how this model could be used to assess the performance of diverse bypass pathways and to derive important photosynthetic parameters from conventional gas exchange data.

Description

Basic equations

The FvCB model (see Notes S1) describes the net CO₂‐assimilation rate (A) for plants with the native photorespiratory cycle as V _c – F – R _d = V _c – 0.5V _o – R _d = (1–0.5φ)V _c – R _d, where V _c is the rate of carboxylation, V _o is the rate of oxygenation, which ultimately leads to photorespiration (F), that is, 0.5 mol CO₂ released per oxygenation via glycine decarboxylation (the value that receives experimental support for the native, closed photorespiratory cycle, e.g. Abadie et al., 2016), and φ is the V _o : V _c ratio – a parameter characterising intrinsic properties of Rubisco. For bypass plants that are expected to mix both native and bypass pathways, F can be formulated as:

$$F=\left[0.5\left(1-x^{\prime }\right)\phi +g{x}^{\prime }\phi \right]V_{\mathrm{c}}$$ (Eqn 1)

where x′ and (1 − x′) are fractions of the glycolate‐carbon that follows bypass and native pathways, respectively, and g is the mol CO₂ released per oxygenation by the bypass pathway.

In the canonical FvCB model, V _c is limited either by Rubisco activity or by electron transport. The Rubisco‐limited V _c here is the same as in the FvCB model, whereas the electron transport‐limited V _c has to account for altered requirements for reductants and ATP.

In terms of the reductant requirement, per native photorespiratory cycle requires reductants equivalent to 2 mol NADPH, whereas this requirement varies depending on bypasses, for which we introduce parameter p to represent. Thus, electron‐transport limited V _c based on NADPH supply and demand balance can be defined as:

$$V_{\mathrm{c}}=\frac{\frac{1}{2}J}{2+\left(1-x^{\prime }\right)2\phi +x^{\prime }p\phi }=\frac{\frac{1}{4}J}{1+\left(1-x^{\prime }\right)\phi +x^{\prime }p\phi /2}$$ (Eqn 2(a))

where J is the rate of linear electron transport, ½ in the numerator refers to ½ mol NADPH formed per mol electron transferred, and the first 2 in the denominator refers to 2 mol NADPH required per turn of the CBB cycle.

In terms of ATP requirement, per native photorespiratory cycle consumes 3.5 mol ATP, whereas this requirement varies with the bypass. We introduce parameter q to represent this requirement by the bypass pathway. Thus, electron‐transport limited V _c based on ATP supply and demand balance can be defined as follows:

$$V_{\mathrm{c}}=\frac{\mathit{zJ}}{3+\left(1-x^{\prime }\right)3.5\phi +x^{\prime }q\phi }=\frac{\frac{z}{3}J}{1+\frac{\left(1-x^{\prime }\right)3.5}{3}\phi +\frac{x^{\prime }q}{3}\phi }$$ (Eqn 2(b))

where z is ATP production factor per electron transferred, and the first 3 in the denominator refers to 3 mol ATP required per turn of the CBB cycle.

As with the FvCB model, φ should be quantified as: φ = 2Γ _*/C _c (Supporting Information Notes S1), where C _c is chloroplast CO₂ level, and Γ _* is CO₂‐compensation point (i.e. the C _c level at which V _c = F) for native‐photorespiration plants. Combining this equation with Eqns 1, 2(a), 2(b) and Supporting Information Eqn S1.2(a) in Notes S1 for Rubisco‐limited carboxylation rate gives a general model for bypass plants:

$$A=V_{\mathrm{c}}-F-R_{\mathrm{d}}=\frac{\left\{C_{\mathrm{c}}-\left[\left(1-x^{\prime }\right)+2g{x}^{\prime }\right]{\Gamma }_{*}\right\}{x}_1}{C_{\mathrm{c}}+x_2}-R_{\mathrm{d}}$$ (Eqn 3)

where

$$x_1=\left\{\begin{array}{ll}{V}_{\text{cmax}}& \text{if Rubsico activity limits}\\ \frac{1}{4}J\text{or}\frac{z}{3}J& \text{if electron transport limits}\end{array}\right.$$

and

$$x_2=\left\{\begin{array}{ll}{K}_{\mathrm{mC}}\left(1+O/K_{\mathrm{mO}}\right)& \text{if Rubsico activity limits}\\ \left[\left(1-x^{\prime }\right)+\frac{x^{\prime }p}{2}\right]\left(2{\Gamma }_{*}\right)\text{or}& \text{if electron transport limits}\\ \left[\frac{\left(1-x^{\prime }\right)3.5}{3}+\frac{x^{\prime }q}{3}\right]\left(2{\Gamma }_{*}\right)& \end{array}\right.$$

where K _mC and K _mO are the Michaelis–Menten constants of Rubisco for CO₂ and O₂, respectively. The two options for electron transport‐limited rates stem from Eqns 2(a) and 2(b) that describe carboxylation rate in terms of NADPH and ATP balance, respectively, where the term J is described by Eqn S1.4(a) or Eqn S1.4(b) in Notes S1.

Bypasses and their energy budget

Seventeen photorespiratory bypasses are listed in Table 1, in an order according to whether they are implemented in plants and if so, then largely according to their publication year. They are illustrated in Fig. 1 for bypasses 1–10 and in Fig. 2 for bypasses 11–17. These bypasses differ in parameters g, p, and q as well as in the cellular location, where CO₂ is released. Bypass‐specific reductant and ATP requirements per oxygenation are outlined in Table S1. Note that regardless of the native pathway or any bypass pathway, there is always 1 mol ATP required for RuBP regeneration to enable a continued pathway (Table S1). Differences in energy requirements among bypasses lie not only in bypass‐specific reactions but also in whether NH₄ ⁺‐recycling is eliminated and whether parts of photorespiratory carbon return to the CBB cycle. If there is a return, it is all in the form of glycerate, which consumes ATP for being converted to 3‐phosphoglycerate (3‐PGA) and consumes both NADPH and ATP for the ensuing 3‐PGA reduction to triose phosphate (Table S1). The altered energy requirement by the bypasses occurs in other steps, as explained below.

1. Kebeish: This first bypass was achieved by introducing the Escherichia coli enzymes glycolate dehydrogenase (GDH), glyoxylate carboligase (GCL), and tartronic semialdehyde reductase (TSR) into the chloroplasts (Kebeish et al., 2007). The reaction of TSR requires 0.5 mol NADPH per oxygenation, but this is overcompensated for by the release of 1 mol NADPH‐equivalent by the GDH reaction although the identity of endogenous electron acceptors and donors associated with chloroplastic GDH is unknown. Energy required by 3‐PGA phosphorylation and reduction remains the same as the native pathway, as the GCL reaction releases 0.5 mol CO₂, and the subsequent TSR reaction generates 0.5 mol glycerate which is converted to 0.5 mol 3‐PGA for joining the CBB cycle. The bypass avoids NH₄ ⁺ recycling. Total energy required for this bypass is 1 mol NADPH and 3 mol ATP per oxygenation.

2. Carvallo: Like bypass 1, this bypass also introduces the bacterial GCL, and the end effect of this is to effectively avoid the NH₄ ⁺ recycling. However, the photorespiratory CO₂ release (0.5 mol) is relocated in the peroxisome, as it does not introduce GDH that enables the conversion of glycolate into glyoxylate in the chloroplast (Fig. 1). As a result, there is no NADPH benefit from the GDH reaction. Thus, the total energy required for this bypass is 2 mol NADPH and 3 mol ATP.

3. GMK: Maier et al. (2012) established this bypass by expressing enzymes in chloroplasts – (1) glycolate oxidase (GLO) for oxidizing glycolate into glyoxylate; (2) malate synthase for forming malate from glyoxylate; (3) malate enzyme (ME) to decarboxylate malate, forming pyruvate; and (4) pyruvate dehydrogenase (PDH) to decarboxylate pyruvate (Fig. 1). In addition, the katE gene encoding an E. coli plastidial catalase (CAT) was introduced to decompose the byproduct H₂O₂ of GLO to prevent oxidative damage. As a result of both ME and PDH activities, this full‐decarboxylation bypass releases 2 mol NADPH‐equivalents. The net energy requirement is −1 mol NADPH and 2 mol ATP per oxygenation.

4. AP3: This bypass was created in tobacco by South et al. (2019), where they also named the Kebeish bypass and the GMK bypass as AP1 and AP2, respectively. AP3 is similar to the GMK bypass, except that instead of using GLO, GDH was used to oxidize glycolate into glyoxylate (thereby, no CAT installation was needed). The net energy requirement is −2 mol NADPH and 2 mol ATP per oxygenation.

5. GOC: This is the first complete bypass implemented in crop plants, first in rice (Shen et al., 2019) and then in potato (Lin et al., 2025). Three enzymes, GLO, oxalate oxidase (OXO), and CAT, were overexpressed in chloroplasts. The net energy requirement is 1 mol NADPH and 2 mol ATP per oxygenation for this full‐decarboxylating bypass.

6. GCGT: This bypass was constructed by introducing rice GLO, E. coli CAT, GCL, and TSR into rice chloroplasts (Wang et al., 2020), thereby being an intermediate type of GOC and Kebeish bypasses. Total energy required for this partial decarboxylating bypass (releasing 0.5 mol CO₂) is 2 mol NADPH and 3 mol ATP.

7. GMA: This bypass (Xu et al., 2023) is similar to the GMK bypass, except that GLO‐produced H₂O₂, which is detoxified by CAT in GMK, is converted to H₂O by ascorbate peroxidase (APX) in GMA. Unlike CAT, APX requires an external electron donor. We assume that two electrons (1 NADPH‐equivalent) required for reducing H₂O₂ to H₂O by APX come from the photosynthetic electron transport chain as occurring in the ascorbate‐glutathione cycle. Thus, the energy requirement for this bypass is 0 mol NADPH and 2 mol ATP per oxygenation.

8. pBHAC: Inspired by the discovery on the β‐hydroxyaspartate (BHAC) cycle in marine proteobacteria (von Borzykkowski et al., 2019), Roell et al. (2021) introduced the cycle into A. thaliana to establish a peroxisomal BHAC (pBHAC) bypass. This BHAC cycle consists of four enzymes, and the reaction of iminosuccinate reductase (ISR) requires NADH. The cycle converts 1 mol glyoxylate to 0.5 mol oxaloacetate (OAA) without loss of carbon as CO₂ in the bypass pathway itself. However, the fate of OAA is uncertain. It could enter the tricarboxylic acid cycle or act as a substrate for amino‐acid biosynthesis or other anabolic reactions (von Borzykkowski et al., 2019). Alternatively, OAA is shuttled to the cytosol and decarboxylated by phosphoenolpyruvate carboxykinase to generate 3‐carbon acids that re‐enter the chloroplast and join the CBB cycle (Smith et al., 2025). Here we assume that OAA is half decarboxylated, and the energy demands are calculated accordingly. Thus, for this bypass, 0.25 mol CO₂ is released and 1.75 mol NADPH‐equivalents and 2.75 mol ATP are required per oxygenation.

9. cBHAC: This is similar to the pBHAC bypass, except that the cycle is expressed in the chloroplast (Chen et al., 2025a). For that to happen, GDH was co‐introduced to convert glycolate to glyoxylate in the chloroplast, whereby the BHAC cycle runs. This bypass was realized in rice. To quantify carbon loss and energy requirement, the same assumption was made here as for the pBHAC bypass. As the GDH reaction generates reductants, 0.75 mol NADPH‐equivalents, and 2.75 mol ATP are required per oxygenation for this bypass.

10. GCBG: This bypass, also reported by Chen et al. (2025a) for rice, is in line with the cBHAC bypass, where GDH oxidizes glycolate to glyoxylate, and the glyoxylate is then directed into the BHAC pathway through the coordinated action of β‐hydroxyaspartate aldolase and dehydrogenase (BhcC and BhcB). Instead of relying on the aspartate : glyoxylate aminotransferase as in the cBHAC bypass, the glutamate : glyoxylate aminotransferase (GGT) is introduced. This process converts 1 mol glycolate into 0.5 mol OAA and 0.5 mol NH₄ ⁺. The released NH₄ ⁺ is reassimilated in the chloroplast, via the GS‐GOGAT (glutamate synthase/glutamine:2‐oxoglutarate aminotransferase) cycle as occurring in the native photorespiratory pathway. The energy requirement for this bypass is 0.75 mol NADPH‐equivalent and 3.25 mol ATP.

11. TaCo: This tartronyl‐CoA (TaCo) pathway fixes, instead of losing, carbon (Fig. 2). It was designed initially by Trudeau et al. (2018) and demonstrated then in vitro (Scheffen et al., 2021) and implemented in rice (Chen et al., 2025b). This carbon‐positive bypass comprises enzymatic reactions that convert glycolate into glycerate. Glycolate is converted to glycolyl‐CoA by glycolyl‐CoA synthetase (GCS) as an initial step, requiring energy equivalent to 2 mol ATP. Glycolyl‐CoA undergoes a carboxylation reaction driven by glycolyl‐CoA carboxylase (GCC) to form tartronyl‐CoA, requiring 1 mol ATP. The tartronyl‐CoA is reduced to glycerate by tartronyl‐CoA reductase (TCR) and tartronate semialdehyde reductase (TSR) respectively, costing 2 mol NADPH. The glycerate is phosphorylated to 3‐PGA (costing 1 mol ATP), which joins the CBB cycle. As this bypass fixes 1 mol carbon, the cost for the ensuing 3‐PGA reduction is 2 mol NADPH and 2 mol ATP (instead of the usual 1.5 mol for each oxygenation as in the native photorespiratory pathway). The TaCo bypass costs a total of 4 mol NADPH‐equivalents and 7 mol ATP.

12. McG: The design of the McG cycle consists of two functional parts: the Mc part − comprising phosphoenolpyruvate carboxylase (PEPc), malate thiokinase (MTK), and malyl‐CoA lyase (MCL), and the G part – comprising GDH, GCL, and TSR (Yu et al., 2018). This McG pathway was used as part of the dual‐cycle CO₂ fixation (together with CBB), whereby the product of McG, acetyl‐CoA, is further used for lipid synthesis in A. thaliana (Lu et al., 2025). To compare it with other bypasses, the energy budget beyond acetyl‐CoA and the interconnection in the dual‐cycle design for an irreversible path from glycerate of the McG cycle to 3‐PGA of the CBB cycle are not considered here (Fig. 2). As 0.5 mol CO₂ released from decarboxylative condensation of glyoxylate to tartronic semialdehyde by GCL is equal to carbon required in HCO₃ ⁻ for PEP carboxylation into OAA in the McG cycle, the net CO₂ production is zero (‘carbon‐neutral’). Based on the energy release and consumption of the cycle, the net energy requirement for this bypass is 1 mol NADPH and 3 mol ATP per oxygenation.

13. imGS: This pathway was designed in rice to avoid the GDC reaction but convert glycine towards glycine betaine in the mitochondrion (Mo et al., 2025), thereby eliminating photorespiratory CO₂ release (‘carbon‐neutral’). Energy costs for synthesis of glycine betaine were not given by Mo et al. (2025). As glycine‐betaine synthesis involves a series of methylation reactions (not a reductive one), the process does not require external reductants but does so at high ATP costs. We estimate that the overall energy requirement for the imGS bypass is 2 mol NADPH and 6 mol ATP per oxygenation.

Table 1.

Various photorespiratory bypasses, and their mol CO₂ released (g), NADPH‐equivalent demand (p) and ATP demand (q) per RuBP oxygenation, and the fraction of total photorespiratory CO₂ released in the chloroplast (${\lambda }_{\mathrm{F}}$) (details for calculating parameters p and q are given in Supporting Information Table S1).

Bypass a	Species or in vitro implemented	Model parameters
		g b	p	q	${\lambda }_{\mathrm{F}}$ c	q/p
1. Kebeish	Arabidopsis thaliana (Kebeish et al., 2007), Camelina sativa (Dalal et al., 2015), potato (Ahmad et al., 2016), tobacco (as AP1 of South et al., 2019), cucumber (Chen et al., 2019)	0.5, Ch	1	3	x′	3.00
2. Carvalho	Tobacco (Carvalho et al., 2011)	0.5, P	2	3	0	1.50
3. GMK	A. thaliana (Maier et al., 2012), tobacco (as AP2 of South et al., 2019)	2, Ch	−1	2	x′	−2.00
4. AP3	Tobacco (South et al., 2019), potato (Cavanagh et al., 2022; Meadam‐Hensold et al., 2024), poplar (Tao et al., 2023), rice (Chen et al., 2026)	2, Ch	−2	2	x′	−1.00
5. GOC	Rice (Shen et al., 2019), potato (Lin et al., 2025)	2, Ch	1	2	x′	2.00
6. GCGT	Rice (Wang et al., 2020), rice (Li et al., 2024)	0.5, Ch	2	3	x′	1.50
7. GMA	Rice (Xu et al., 2023)	2, Ch	0 e	2	x′	∞
8. pBHAC	A. thaliana (Roell et al., 2021)	0.25, Cy d	1.75	2.75	0	1.57
9. cBHAC	Rice (Chen et al., 2025a)	0.25, Cy d	0.75	2.75	0	3.67
10. GCBG	Rice (Chen et al., 2025a)	0.25, Cy d	0.75	3.25	0	4.33
11. TaCo	in vitro validated (Trudeau et al., 2018; Scheffen et al., 2021), rice (Chen et al., 2025b)	−1, Ch f	4	7	x′	1.75
12. McG	Cyanobacteria (Yu et al., 2018), A. thaliana (Lu et al., 2025)	0 g	1	3	x′	3.00
13. imGS	Rice (Mo et al., 2025)	0	2	6 h	0	3.00
14. Ara5P	in vitro validated (Trudeau et al., 2018)	0	2	4	0	2.00
15. Ru1P	in vitro validated (Trudeau et al., 2018)	0	2	4	0	2.00
16. Eryth	in vitro validated (Trudeau et al., 2018)	0	2	5	0	2.50
17. Xyl	in vitro validated (Trudeau et al., 2018)	0	2	5	0	2.50
Native		0.5, M	2	3.5	0	1.75

^a Bypasses are ordered first according to whether they are implemented in plants and if so, then largely according to their publication year, and named as the acronym if indicated in their original publication, otherwise after the first‐author name.

^b Per‐oxygenation mol CO₂ release, followed by the organelle where CO₂ is released (Ch, chloroplast; Cy, cytosol; M, mitochondrion; P, peroxisome).

^c ${\lambda }_{\mathrm{F}}$ ≤ x′ (where x′, as defined in the text, is the fraction of the glycolate following the bypass pathway); ${\lambda }_{\mathrm{F}}$ = x′ occurs only if any release of CO₂ by the bypass pathway takes place in the chloroplast; otherwise, it is set to 0 (including for the carbon‐neutral bypasses where g = 0).

^d The end product of these bypasses is oxaloacetate (OAA); but the fate of OAA is uncertain. It could be diverted (without decarboxylation) to amino‐acid synthesis or other routes (Chen et al., 2025a; Meacham‐Hensold et al., 2026). OAA is also possibly decarboxylated by PEP‐CK (phosphoenolpyruvate carboxykinase) in the cytosol to generate 3‐carbon acids that re‐enter the chloroplast and join the CBB cycle (Smith et al., 2025). Here we assume that OAA is half decarboxylated, and the energy demands are calculated according to this assumption (see the main text and Table S1).

^e This GMA bypass is similar to the GMK bypass, except that hydrogen peroxide (H₂O₂) as a result of glycolate oxidase is detoxified to H₂O by catalase in GMK, whereas it is by ascorbate peroxidase (APX) in GMA. Unlike catalase, APX requires an external electron donor. We assume that two electrons (1 NADPH‐equivalent) required for reducing H₂O₂ to H₂O by APX come from the photosynthetic electron transport chain as occurring in the ascorbate‐glutathione cycle (see Table S1).

^f This bypass fixes 1 mol carbon during the carboxylation step of glycolyl‐CoA to tartronyl‐CoA; so, mol CO₂ released is marked as −1.

^g Assuming that 0.5 mol CO₂ released from decarboxylative condensation of glyoxylate to tartronic semialdehyde by glyoxylate carboligase is used for being converted by carbonic anhydrase into HCO₃ ⁻ that is used for PEP carboxylation into OAA in the McG cycle, the net CO₂ production is zero.

^h The energy cost for the three‐step reaction for synthesis of 1 mol betaine from glycine is not certain, nor was it given by Mo et al. (2025) on the imGS bypass. But the reaction is known as an energy‐intensive biosynthesis pathway in nature (Kimura et al., 2009). The ATP cost is high because each methylation step requires the methyl donor S‐adenosylmethionine (SAM). Since 3 mol SAM are required to methylate glycine to betaine and regenerating 1 mol SAM costs 1 ATP‐equivalent, here we set that 3 ATP are required for the betaine synthesis. Total energy requirements (including the cost for glyoxylate transamination to produce glycine) for this bypass are estimated in Table S1.

The next four bypasses are designed as carbon‐neutral pathways (Trudeau et al., 2018) but they remain to be tested in plants. In all of them, glycolate is first reduced to glycolaldehyde, thereby requiring 1 mol NADPH and 2 mol ATP. The difference among them is in ensuing glycolaldehyde‐condensing reactions with a sugar phosphate from the CBB cycle to form a longer‐chain sugar that is assimilated back to the cycle (Fig. 2).

• 14Ara5P: 1 mol glycolaldehyde is combined with 1 mol GAP into a 5‐carbon arabinose‐5P (the Ara5P bypass), and arabinose‐5P is converted by arabinose‐5P isomerase to Ru5P for RuBP regeneration.
• 15Ru1P: In this Ru1P bypass, 1 mol glycolaldehyde is combined with 1 mol DHAP into a 5‐carbon ribulose‐1P, which is for regenerating RuBP.
• 16Eryth: 1 mol glycolaldehyde is combined with a CBB‐cycle intermediate to form 4‐carbon erythrulose, which consumes 1 mol ATP for phosphorylation, and the phosphorylated intermediate joins the CBB cycle for RuBP regeneration. So, this Eryth bypass consumes 1 mol ATP more compared with Ara5P and Ru1P bypasses.
• 17Xyl: 1 mol glycolaldehyde is combined with a CBB‐cycle intermediate to form 5‐carbon xylulose, which consumes 1 mol ATP for being phosphorylated to Xu5P, and Xu5P joins the CBB cycle for RuBP regeneration. Like the Eryth bypass, this Xyl bypass consumes one extra mol ATP compared with Ara5P and Ru1P bypasses.

Overall, in terms of the amounts of CO₂ release, bypasses can be classified into four categories: full‐decarboxylating type (bypasses 3, 4, 5 and 7), partial‐decarboxylating type (bypasses 1, 2, 6, 8, 9 and 10), carbon‐neutral type (bypasses 12–17), and carbon‐positive or carbon‐fixing type (bypass 11).

CO₂ ‐diffusion equations

To analyse gas exchange data, the FvCB model has been coupled with CO₂‐diffusion models. Relative to the native photorespiratory pathway, most bypasses alter both the amount and location of photorespiratory CO₂ release, allowing mitigation of the diffusion barrier so that CO₂ released by bypasses could be more easily re‐fixed by Rubisco than the CO₂ released in the mitochondrion by the native pathway. The two contrasting intracellular CO₂‐diffusion scenarios have been described (Tholen et al., 2012; von Caemmerer, 2013). Here, algorithms extended by Yin & Struik (2017) and Ubierna et al. (2019) for intermediate scenarios are adapted to describe the bypass‐altered diffusion paths of CO₂ from intercellular airspaces (C _i) to the Rubisco‐carboxylating sites (C _c) via mesophyll cytosol (C _m):

$$C_{\mathrm{m}}=C_{\mathrm{i}}-A{r}_{\mathrm{wp}}$$ (Eqn 4(a))

$$C_{\mathrm{c}}=C_{\mathrm{m}}-\left(V_{\mathrm{c}}-{\lambda }_{\mathrm{F}}F-{\lambda }_{\mathrm{R}}{R}_{\mathrm{d}}\right)r_{\mathrm{ch}}$$ (Eqn 4(b))

where r _wp is the resistance of mesophyll cell‐wall and plasma membrane, and r _ch is the resistance of the chloroplast (r _wp and r _ch together constitutes the total mesophyll resistance r _m, i.e. r _wp + r _ch = r _m). In Eqn 4(b), $\left(V_{\mathrm{c}}-{\lambda }_{\mathrm{F}}F-{\lambda }_{\mathrm{R}}{R}_{\mathrm{d}}\right)$ represents the net CO₂‐fixing flux in the chloroplast, thereby, experiencing r _ch. Here, ${\lambda }_{\mathrm{F}}$ is the fraction of the total photorespiratory CO₂ that is released in the chloroplast, and ${\lambda }_{\mathrm{R}}$ is the fraction of the total nonphotorespiratory CO₂ (R _d) that is released in the chloroplast. Previously, CO₂ for the R _d term is implicitly assumed to be released predominantly in the mitochondrion; so, R _d has been set together with the F term as (F + R _d) in most CO₂ diffusion models for the native photorespiratory case (Tholen et al., 2012; Yin & Struik, 2017; Ubierna et al., 2019). As the release of CO₂ for the F term is relocated in various bypasses, F and R _d have to be treated separately. Furthermore, increasing evidence has been shown that R _d as the release of CO₂ from processes other than photorespiration could occur significantly in the chloroplast (e.g. Wieloch, 2022; Sharkey & Xu, 2026). For these two reasons, we introduce a separate parameter ${\lambda }_{\mathrm{R}}$.

Substituting V _c = A + F + R _d into Eqn 4(b) and then combining Eqns 4(a) and 4(b) gives:

$$C_{\mathrm{c}}=C_{\mathrm{i}}-\left\{A+\omega \left[\left(1-{\lambda }_{\mathrm{F}}\right)F+\left(1-{\lambda }_{\mathrm{R}}\right)R_{\mathrm{d}}\right]\right\}{r}_{\mathrm{m}}$$ (Eqn 4(c))

where ω represents the fraction of r _ch in the total mesophyll resistance r _m (i.e. ω = r _ch/r _m).

CO₂ ‐compensation points

CO₂‐compensation point in the absence of day respiration (i.e. the CO₂ level at which V _c = F or A + R _d = 0) is an important diagnostic parameter. For the native photorespiratory pathway, as can be seen from the canonical FvCB‐model equation, Supporting Information Eqn S1.1(b) in Notes S1, that the C _c level at which A + R _d = 0 (also called ‘the C _c‐based CO₂‐compensation point’) is Γ _*. For the bypass plants where the amount of CO₂ released as a result of oxygenation is altered, it can be seen from Eqn 3 that the C _c level, at which A + R _d = 0, is no longer Γ _* but is $\left[\left(1-x^{\prime }\right)+2g{x}^{\prime }\right]{\Gamma }_{*}$. We use Γ _*BP to denote the C _c‐based CO₂‐compensation point of the bypass plants:

$${\Gamma }_{*\mathrm{BP}}=\left[\left(1-x^{\prime }\right)+2g{x}^{\prime }\right]{\Gamma }_{*}$$ (Eqn 5)

In analogy to the native photorespiratory plants where the CO₂‐compensation point can also be based on the C _i level (C _i*), the equivalent CO₂‐compensation point for the bypass plants (C _i*BP) can be formulated from Eqn 4(c):

$${\Gamma }_{*\mathrm{BP}}=C_{\mathrm{i}*\mathrm{BP}}-\{-R_{\mathrm{d}}+\omega [\left(1-{\lambda }_{\mathrm{F}}\right)F+\left(1-{\lambda }_{\mathrm{R}})R_{\mathrm{d}}\right]\}{r}_{\mathrm{m}}$$

Combining this equation with Eqn 5 and expressing it for C _i*BP gives:

$$C_{\mathrm{i}*\mathrm{BP}}=\left[\left(1-x^{\prime }\right)+2g{x}^{\prime }\right]{\Gamma }_{*}-\left\{\left[1-\omega \left(1-{\lambda }_{\mathrm{R}}\right)\right]R_{\mathrm{d}}-\omega \left(1-{\lambda }_{\mathrm{F}}\right)F\right\}{r}_{\mathrm{m}}$$ (Eqn 6)

Can parameter x′ be estimated from conventional gas‐exchange data?

Knowing the fraction of glycolate‐carbon following bypass pathway (x′) is important both for physiological analysis and from modelling perspective. However, this parameter has hardly been reported as its quantification would need ¹²CO₂ vs ¹³CO₂ labelling analysis (e.g. Busch et al., 2013; Ma et al., 2017), which may not be available in a synthetic‐biology laboratory. With Eqns 5 and 6 presented above for the CO₂‐compensation point, we show two scenarios below, with which x′ could be quantified from conventional gas exchange data.

First, for the scenario where ω can be approximated as zero, x′ can be calculated as follows:

$$x^{\prime }=\frac{C_{\mathrm{i}*\mathrm{WT}}-C_{\mathrm{i}*\mathrm{BP}}}{{\Gamma }_{*}\left(1-2g\right)}$$ (Eqn 7(a))

where C _i*WT is the value of C _i* for the native photorespiratory plants (wild‐type (WT)). This ω = 0 scenario applies for plants like rice and wheat where mitochondria are observed to lie exclusively behind chloroplasts as if Rubisco carboxylation and GDC decarboxylation take place in the same compartment (Sage & Sage, 2009; Busch et al., 2013).

Second, for the more general scenario where parameter ω cannot be approximated as zero, x′ can be calculated as follows:

$$x^{\prime }=\frac{{\mathit{\text{Slope}}}_{\mathrm{WT}}-{\mathit{\text{Slope}}}_{\mathrm{BP}}}{{\mathit{\text{Slope}}}_{\mathrm{WT}}\left(1-2g\right)}$$ (Eqn 7(b))

where slope _WT and slope _BP are the values of slope for linear regression of C _i* vs the O₂ level for WT and bypass plants, respectively. Eqn 7(b) is for the general case, so it can be applied to the first scenario as well. However, as it relies on the slope value of O₂ dependence, applying Eqn 7(b) requires more gas exchange measurements across at least two (preferably more) O₂ levels. Derivation of both Eqn 7(a) and 7(b) is given in Notes S2.

For carbon‐neutral bypasses (g = 0), both Eqns 7(a) and 7(b) become somewhat simpler. However, neither Eqns 7(a) nor 7(b) works for the bypass pathways where the amount of CO₂ released per oxygenation by the bypass is equal to that by the native photorespiration (i.e. g = 0.5), because the denominator in both equations would become zero. For these bypasses (bypasses 1, 2 and 6 in Table 1), it is hard to determine x′ from conventional gas‐exchange data.

Coupling the basic model with the CO₂ ‐diffusion model

With x′ estimated beforehand and parameters g, p (and/or q) and ${\lambda }_{\mathrm{F}}$ determined according to the underlying biochemistry for each bypass (Table 1), it is necessary to combine Eqns 3 and 4(c) for expressing A as a function of C _i in order to estimate the common biochemical parameters V _cmax and J _max from fitting gas exchange data of A−C _i (or combined with light response) curves for bypass plants. To this end, the term F in Eqn 4(c) should first be expressed from Eqns 1 and 3 as:

$$F=\frac{{\Gamma }_{*\mathrm{BP}}{x}_1}{C_{\mathrm{c}}+x_2}$$ (Eqn 8)

Solving C _c from Eqn 3, substituting it into Eqns 4(c) and 8, and then combining them to solve for A gives the following quadratic equation:

$$A=\frac{-b-\sqrt{b^2-4\mathit{ac}}}{2a}$$ (Eqn 9)

where

$$a=r_{\mathrm{m}}\left\{x_2+\left[1-\omega \left(1-{\lambda }_{\mathrm{F}}\right)\right]{\Gamma }_{*\mathrm{BP}}\right\}$$

$$\begin{array}{c}b=2r_{\mathrm{m}}\omega \left(1-{\lambda }_{\mathrm{F}}\right)\left(x_1-R_{\mathrm{d}}\right){\Gamma }_{*\mathrm{BP}}\\ -\left\{C_{\mathrm{i}}+x_2+r_{\mathrm{m}}\left[x_1-R_{\mathrm{d}}-\omega \left(1-{\lambda }_{\mathrm{R}}\right)R_{\mathrm{d}}\right]\right\}\left({\Gamma }_{*\mathrm{BP}}+x_2\right)\end{array}$$

$$c=\begin{array}{l}\left\{\left[C_{\mathrm{i}}-r_{\mathrm{m}}\omega \left(1-{\lambda }_{\mathrm{R}}\right)R_{\mathrm{d}}\right]\left(x_1-R_{\mathrm{d}}\right)-{\Gamma }_{*\mathrm{BP}}{x}_1-x_2{R}_{\mathrm{d}}\right\}\left({\Gamma }_{*\mathrm{BP}}+x_2\right)-r_{\mathrm{m}}\omega \left(1-{\lambda }_{\mathrm{F}}\right){\left(x_1-R_{\mathrm{d}}\right)}^2{\Gamma }_{*\mathrm{BP}}\end{array}$$

Eqn 9 can be used for curve‐fitting to estimate V _cmax and J _max as well as mesophyll conductance g _m (= 1/r _m), provided that ω and ${\lambda }_{\mathrm{R}}$ are pre‐determined. Conversely, Eqn 9 can be used to show how A varies with either individual or combined change in parameters g, p (or q) and ${\lambda }_{\mathrm{F}}$, for a given set of ω and ${\lambda }_{\mathrm{R}}$.

Eqn 9 is generic, being able to calculate both Rubisco‐ or electron transport‐limited photosynthetic rate (denoted as A _c and A _j, respectively). One caveat when using Eqn 9 for modelling, as with the canonical FvCB model, is about using the min(A _c, A _j) function, which would choose the A _j value (instead of the biologically accepted rate A _c) when C _i is near or below the CO₂‐compensation point of A−C _i curves (e.g. Gu et al., 2010; Lochocki & McGrath, 2025). To circumvent this, the model is specifically formulated as:

$$A=\left\{\begin{array}{cc}{A}_{\mathrm{c}}\hfill & \text{if}{C}_{\mathrm{i}}<C_{\mathrm{i},\text{threshold}}\\ \min \left(A_{\mathrm{c}},A_{\mathrm{j}}\right)& \text{otherwise}\hfill \end{array}\right.$$ (Eqn 10)

where C _i,threshold is a threshold C _i value, below which A is limited surely by Rubisco's activity.

For the electron transport‐limited rate A _j, there are two forms, determined by NADPH or ATP balance. As most bypasses increased the ATP/NADPH requirement ratio (i.e. the q/p ratio) relative to 1.75 – the ratio for the native photorespiration (Table 1), ATP is conceivably more limiting and it is logical to use the ATP‐determined A _j for bypass plants. However, there is no guarantee that bypasses designed in the future all have a higher q/p ratio. Thus, for describing A _j, we set A _j = min(A _j,NADPH, A _j,ATP), where A _j,NADPH and A _j,ATP are the electron transport‐limited photosynthetic rate as set by NADPH and ATP supply, respectively.

Results

As bypass pathways are unlikely to replace the native photorespiratory pathway completely, we practically set x′ = 0.5 for model illustration. The fraction for the photorespiratory CO₂ released in the chloroplast (${\lambda }_{\mathrm{F}}$) is either x′ for the bypasses which relocate the CO₂ release in the chloroplast or 0 for other cases (Table 1). Unless specified, parameter ω is also set 0.5. Furthermore, to avoid uncertainties in the value of ${\lambda }_{\mathrm{R}}$ (the fraction for the CO₂ from the R _d term released in the chloroplast), we simply set R _d = 0 to focus our analysis on the photosynthesis parts. Input values of other parameters are given in Notes S3.

CO₂ response

Photosynthetic CO₂ response is characterized by CO₂‐compensation point and carboxylation efficiency (CE), and these parameters can be obtained directly from the model. The C _c‐ and C _i‐based CO₂‐compensation points (Γ _*BP and C _i*BP) can be calculated from Eqns 5 and 6, respectively. Compared with the native pathway, full‐decarboxylating bypasses had considerably higher Γ _*BP and C _i*BP, partial‐decarboxylating bypasses had comparable or somewhat lower values, whereas carbon‐neutral bypasses had much lower values (Table 2). Particularly striking is that the carbon‐positive bypass (bypass 11) had negative CO₂‐compensation points. This is because the reactions after RuBP‐oxygenation in this bypass fix carbon; as a result, only at a negative CO₂ level is V _c negative that compensates for the carbon‐fixation by the bypass reactions (i.e. V _c = F). Also being directly calculated is CE, defined as the initial slope of the curve (dA/dC _c) at Γ _*BP (Table 2). Values of Γ _*BP and C _i*BP increase, whereas CE decreases, in an order from the carbon‐positive, to carbon‐neutral, to partial‐decarboxylating and to full‐decarboxylating bypasses (Fig. 3a,b).

Table 2.

Values for various photorespiratory bypasses of diagnostic parameters, including C _c‐based CO₂‐compensation point (Γ _*BP), C _i‐based CO₂‐compensation point (C _i*BP), carboxylation efficiency (CE), transition point along the A−C _i curve from Rubisco to electron transport limitations (C _i,tr), transition point along the A−I _inc curve from electron transport to Rubisco limitations (I _inc,tr), quantum yield of CO₂ assimilation (Φ_CO2), and photosynthetic capacity under saturated light (A _sat) and under both saturated light and CO₂ conditions (A _max), modelled using the default input parameter values (see Supporting Information Notes S3).

Bypass (type) a	Γ *BP (μbar)	C i*BP (μbar)	CE b (mol m−2 s−1 bar−1)	C i,tr (μbar)	I inc,tr (μmol m−2 s−1)	ΦCO2 (mol mol−1)	A sat c (μmol m−2 s−1)	A max d (μmol m−2 s−1)
1. Kebeish (P)	34.7	41.5	0.156	495	600	0.0658	19.1	38.0
2. Carvalho (P)	34.7	48.3	0.156	503	600	0.0657	18.7	38.0
3. GMK (F)	86.9	102.5	0.144	503	615	0.0530	14.2	36.0
4. AP3 (F)	86.9	102.5	0.144	503	615	0.0530	14.2	36.0
5. GOC (F)	86.9	102.5	0.144	503	615	0.0530	14.2	36.0
6. GCGT (P)	34.7	41.5	0.156	495	615	0.0658	19.1	38.0
7. GMA (F)	86.9	102.5	0.144	503	600	0.0530	14.2	36.0
8. pBHAC (P)	26.1	36.4	0.158	512	585	0.0687	19.6	38.5
9. cBHAC (P)	26.1	36.4	0.158	512	585	0.0687	19.6	38.5
10. GCBG (P)	26.1	36.4	0.158	495	615	0.0676	19.6	38.3
11. TaCo (+)	−17.4	−21.1	0.170	401	840	0.0710	24.1	38.6
12. McG (0)	17.4	20.9	0.160	503	610	0.0706	20.7	38.8
13. imGS (0)	17.4	24.3	0.160	423	705	0.0642	20.5	37.4
14. Ara5P (0)	17.4	24.3	0.160	478	645	0.0683	20.5	38.4
15. Ru1P (0)	17.4	24.3	0.160	478	645	0.0683	20.5	38.4
16. Eryth (0)	17.4	24.3	0.160	444	705	0.0662	20.5	37.9
17. Xyl (0)	17.4	24.3	0.160	444	705	0.0662	20.5	37.9
Native	34.7	48.3	0.156	486	630	0.0646	18.7	37.8

^a Type in the brackets: F, full‐decarboxylating type; P, partial‐decarboxylating type; +, carbon‐positive type, and 0, carbon‐neutral type.

^b Defined as the initial slope of A−C _c curve at Γ _*BP; as $\frac{\mathrm{d}A}{\mathrm{d}{C}_{\mathrm{c}}}=\frac{\left({\Gamma }_{*\mathrm{BP}}+x_2\right)x_1}{{\left(C_{\mathrm{c}}+x_2\right)}^2}$, at C _c = Γ _*BP, $\frac{\mathrm{d}A}{\mathrm{d}{C}_{\mathrm{c}}}$ becomes $\frac{x_1}{{\Gamma }_{*\mathrm{BP}}+x_2}$, where x ₁ is V _cmax, x ₂ is K _mC(1 + O/K _mO) and O for the ambient atmosphere is c. 210 mbar.

^c Calculated as the CO₂‐assimilation rate at C _i = 286.5 μbar and I _inc = 1500 μmol m⁻² s⁻¹.

^d Calculated as the CO₂‐assimilation rate at C _i = 1000 μbar and I _inc = 1500 μmol m⁻² s⁻¹.

Fig. 3.

Modelled photosynthetic parameter values plotted against mol CO₂ released per oxygenation (g) in 17 photorespiratory bypasses. All parameters are normalized to 1 for the native photorespiratory pathway (see Table 2 for unnormalized values). Photosynthetic parameters: (a) CO₂‐compensation points Γ _*BP and C _i*BP; (b) photosynthetic efficiency parameters – carboxylation efficiency CE and quantum yield Φ_CO2; (c) transition point along A−C _i curve from Rubisco limitation to electron‐transport limitation (C _i,tr), and transition point along A−I _inc curve from electron‐transport limitation to Rubisco limitation (I _inc,tr); and (d) photosynthetic capacity parameters A _sat (net photosynthetic rate at saturating light and ambient CO₂ level) and A _max (net photosynthetic rate when light and CO₂ levels are both saturating). In each panel, the first parameter corresponds to filled symbols, full regression line and bold equation while the second parameter corresponds to open symbols, dashed regression line and nonbold equation.

For simulating the entire A‐C _i response curve, incident light intensity (I _inc) was fixed at 1500 μmol m⁻² s⁻¹. Differences in the simulated curve lie mainly in low CO₂ parts among bypasses (Fig. S1). Along the curve, the transition point from Rubisco to electron‐transport limitations (C _i,tr) and photosynthetic capacity under saturated light and ambient‐air CO₂ level (A _sat) can be identified (Table 2). As expected, the transition point roughly occurs at lower C _i for carbon‐positive (TaCo) and carbon‐neutral (imGS, Eryth, and Xyl) bypasses (Table 2; Fig. 3c) as these bypasses have high ATP‐requirements (Table 1). Values of A _sat decrease in an order from the carbon‐positive to full‐decarboxylating bypasses (Fig. 3d). The very high end of the A‐C _i curve is identified as A _max. All bypasses converge to a similar value of A _max (Table 2), which is expected as the RuBP oxygenation is low at high CO₂ levels. Thus, A _max decreases much less than A _sat with increasing CO₂‐release per oxygenation (Fig. 3d).

Light response

For simulating the photosynthetic light response curve, C _i was fixed at 286.5 μbar, roughly equivalent to the ambient‐air CO₂ level (C _a) of 420 μmol mol⁻¹. Main differences of the curve among bypasses lie in high light parts (Fig. S1). Along the curve, the transition point from electron‐transport to Rubisco limitations (I _inc,tr) can be identified (Table 2). This transition point roughly decreases with increasing CO₂ release per oxygenation (Fig. 3c). At or beyond I _inc,tr, A _sat can be identified (which equals A _sat identified from A‐C _i curves). The lowest part of the light‐response curves identifies quantum yield of CO₂ assimilation (Φ_CO2), and values of Φ_CO2 on the absorbed‐light basis (Table 2) indicate an overall light energy use efficiency. Overall, Φ_CO2 decreases in a similar order as for CE (Fig. 3b). Within the same bypass type, Φ_CO2 decreases with increasing ATP requirements.

The dependence of bypass benefits on cellular arrangements of organelles

Part of photorespiratory CO₂ can be refixed by Rubisco and the extent of refixation depends on the arrangement of mitochondria relative to chloroplasts (Sage & Sage, 2009; Busch et al., 2013; Yin et al., 2020). Theoretically, if they are arranged in a way as if Rubisco carboxylation and GDC decarboxylation take place in the same compartment, refixation would be greater. This can be the case if mitochondria are observed to lie in close proximity to, and exclusively behind, chloroplasts that distribute continuously with little gap. Such structure of organelles could be mimicked by setting parameter ω to nearly zero or ${\lambda }_{\mathrm{F}}$ to 1 (Yin & Struik, 2017; Ubierna et al., 2019). Because ${\lambda }_{\mathrm{F}}$ is reserved here to refer to the relocation of CO₂ release by bypasses, we choose ω and set it to 0.05 to assess how the benefit from bypasses would change.

As expected, compared with the default leaf structure where ω = 0.5, changing ω to 0.05 lowers the value of C _i‐based CO₂‐compensation point C _i*BP and the difference in C _i*BP between the two organelle‐arrangement scenarios increases largely with mol of CO₂ release per oxygenation (Fig. 4a). Also expectedly, high CO₂‐release bypasses increase the CO₂ level at the Rubisco‐carboxylating sites (C _c) and the refixation‐favoured organelle arrangement increases C _c more for high than low CO₂‐release bypasses (Fig. 4b). As a result, the loss of photosynthesis of high‐decarboxylating bypasses (relative to the native pathway) becomes somewhat smaller in the refixation‐favoured organelle arrangement than the default arrangement (Fig. 4c). Likewise, the benefit from the carbon‐positive bypass relative to the native pathway becomes somewhat smaller because the refixation is already relatively high in the native‐pathway plants with the refixation‐favoured organelle arrangement. However, overall, the impact of the organelle arrangements on the effect of bypass's photosynthetic rates is small (Fig. 4c).

Fig. 4.

Modelled results by setting parameter ω (chloroplast resistance to total mesophyll resistance ratio) either to 0.5 or to 0.05. (a) Difference in the calculated C _i*BP (CO₂‐compensation point of bypass plants) between ω = 0.5 and w = 0.05 scenarios; (b) Rubisco‐carboxylating site CO₂ level (C _c) under the A _sat‐measuring condition (A _sat refers to net photosynthetic rate at saturating light and ambient CO₂ level); (c) the relative A _sat value normalized to 1.0 for the native photorespiratory pathway. All are plotted as a function of model parameter g, i.e. molecules of CO₂ released per oxygenation, across 17 bypasses. The curve in panel (a) is drawn according to the theoretical relationship for the difference in C _i*BP (ΔC _i*BP) between the two scenarios, which is derived from Eqn 6 and Eqn S2.4(b) in Supporting Information Notes S2: $\Delta C_{\mathrm{i}*\mathrm{BP}}=\left({\omega }_1-{\omega }_2\right)\left(1-{\lambda }_{\mathrm{F}}\right)r_{\mathrm{m}}·\frac{V_{\text{cmax}}}{1+\frac{K_{\mathrm{mC}}\left(1+O/K_{\mathrm{mO}}\right)}{\left(1-x^{\prime }+2g{x}^{\prime }\right){\Gamma }_{*}}}$, where ω ₁ and ω ₂ represent the values of ω for the two scenarios (i.e. 0.5 and 0.05, respectively). Around the curve are scattered points resulting from different values of ${\lambda }_{\mathrm{F}}$ (the fraction of total photorespiratory CO₂ released in the chloroplast) for 17 bypasses and the curve is drawn using an intermediate ${\lambda }_{\mathrm{F}}$ = 0.35. Note that the above equation ignores the $\left(1-{\lambda }_{\mathrm{R}}\right)R_{\mathrm{d}}$ term in Eqn 6 as we used R _d (day respiration) = 0 for illustration (see the main text; ${\lambda }_{\mathrm{R}}$ is the fraction of total day respiratory CO₂ released in the chloroplast). There is no simple mathematical equation for describing C _c and relative A _sat, and an empirical linear regression line is thus drawn in panels b and c for each of the two scenarios (filled symbols and bold equations for ω = 0.5, and open symbols and nonbold equations for ω = 0.05).

Model fitting to gas exchange data

We fitted extended and canonical models to combined light‐ and CO₂‐response curves measured by Chen et al. (2025a) for their cBHAC‐ and GCBG‐bypass rice plants (see Notes S4 for fitting scripts). Required x′ values of the bypass plants for the extended model were derived from Eqn 7(a) – the equation held for the ω = 0 scenario, which applies for plants like rice where mitochondria lie exclusively behind chloroplasts (Sage & Sage, 2009; Busch et al., 2013; Yin et al., 2020). The estimated x′ values were low and ranged from 0.05 to 0.23 (Table 3), suggesting that a bypass pathway replaced the native photorespiratory pathway only to a small extent. As a result, the estimated V _cmax or J _max did not differ much between the two models (Table 3). However, one can envisage that the difference could be greater if x′ was higher. Nevertheless, the estimates were not identical – the canonical model overestimated V _cmax by c. 2.3% and J _max by 1.4%, and overestimations increased with increasing values of x′. These overestimations of V _cmax were mainly owing to the higher CO₂ release (parameter g), while those of J _max were owing to somewhat higher NADPH or ATP requirement (parameter p or q), in WT plants than in cBHAC‐ and GCBG‐bypass plants (Table 1). The canonical model (that has a higher value of parameters g, p, or q) calculates a lower photosynthetic rate; as such, it would require a higher estimated V _cmax or J _max to arrive at a good fit to the measured A. As the estimation of V _cmax relies on the lower end of CO₂‐response curves and higher end of light‐response curves, more overestimations of V _cmax than J _max by the canonical model are expected, given that bypasses alter mainly these ends of the curves (Fig. S1).

Table 3.

Values (SE of the estimate in brackets) of V _cmax (maximum rate of carboxylation by Rubisco) and J _max (light‐saturated rate of linear electron transport) estimated from fitting the extended model or the canonical FvCB model to light‐ and CO₂‐response curves measured on wild‐type (WT) plants, two cBHAC and two GCBG bypass rice plants of Chen et al. (2025a).

	x′ a	Extended model b		Canonical FvCB model c
		V cmax (μmol m−2 s−1)	J max (μmol m−2 s−1)	V cmax (μmol m−2 s−1)	J max (μmol m−2 s−1)
WT	0.00	103.7 (3.5)	203.2 (4.3)	103.7 (2.1)	203.2 (3.5)
cBHAC1	0.23	112.7 (3.9)	222.9 (4.8)	115.8 (2.1)	228.3 (3.9)
cBHAC2	0.19	117.9 (4.3)	228.3 (5.1)	122.1 (2.4)	231.6 (3.8)
GCBG1	0.05	121.8 (5.5)	222.3 (4.5)	122.3 (3.4)	223.3 (3.2)
GCBG2	0.14	119.8 (5.4)	220.1 (4.4)	122.8 (3.5)	222.8 (3.3)

^a x′ is the fraction of glycolate carbon following bypass pathway, and it was precalculated from Eqn 7(a) using C _i* reported by Chen et al. (2025a) for these plants. Another preestimated input parameter is day respiration (R _d), which was estimated using the method of Yin et al. (2011).

^b Simultaneously estimated parameters κ _2LL (efficiency of converting incident light into linear electron transport; see Eqn S1.4(b) in Supporting Information Notes S1) and g _m (mesophyll conductance) are not shown in the table. They did not differ among genotypes and their estimated common values (± SE) are κ _2LL = 0.32 ± 0.01 (mol mol⁻¹) and g _m = 1.36 ± 0.55 (mol m⁻² s⁻¹ bar⁻¹).

^c To ensure a proper comparison of V _cmax and J _max (the two common parameters synthetic biologists usually report) estimated by the two models, values of κ _2LL and g _m estimated by the extended model were used as input to this canonical model, so that only V _cmax and J _max were estimated at this step. As a result, the estimates of V _cmax and J _max were the same between the two models for the WT plants.

Discussion

Main differences among bypasses

Seventeen bypasses all alter the CO₂ release and NADPH or ATP requirement per oxygenation (Table 1). Their end products differ; some exit from the pathway in a certain C form, whereas others alter the glycolate‐C returned to the CBB cycle (Figs 1, 2). Our model analyses show that these bypasses perform most differently under low‐CO₂ and high‐light conditions (Fig. S1). While the results are obtained mainly using one set of physiologically relevant parameter values, relative differences in photosynthetic performance among bypasses vary little with parameter input. These differences are predominantly determined by amounts rather than the location of photorespiratory CO₂ release, as evidenced by that most photosynthetic parameters are linearly correlated almost exclusively with values of parameter g (the CO₂ release per oxygenation event) (Fig. 3). The higher the value of parameter g, the lower the CO₂‐assimilation rate. The TaCo pathway is the only one that fixes carbon, instead of releasing, and thus has the highest CO₂‐assimilation rate despite the highest energy costs per oxygenation event (Table 1). The carbon positive bypass concept was first proposed for cyanobacteria (Shih et al., 2014). Such a bypass was designed in vitro (Trudeau et al., 2018; Scheffen et al., 2021) and now implemented in rice (Chen et al., 2025b). Despite its lowest C _c values around Rubisco (Fig. 4b) modelled as a result of carbon being first fixed by glycolyl‐CoA carboxylase, advantages of the carbon‐positive bypass are theoretically illustrated for both photosynthetic efficiency and capacity parameters (Table 2; Fig. 3), meaning that it suits virtually all growth conditions.

Re‐fixing photorespiratory CO₂

A first idea behind developing photorespiratory bypasses is that CO₂ released in mitochondria in the native pathway, if being relocated to chloroplasts, can be more readily refixed by Rubisco. So, some bypasses were designed to fully decarboxylate glycolate carbon (i.e. 2 mol CO₂ release per oxygenation, Table 2), with a hope that the fully released CO₂ in chloroplasts can create a CO₂‐concentrating mechanism (CCM) around Rubisco. However, our simulation shows that although these bypasses do have somewhat higher C _c values around Rubisco (Fig. 4b), such CCM is not strong enough to compensate for the high rate of CO₂‐release. This can be seen in Fig. 3(d) that A _sat is lower in full‐decarboxylating than the WT plants.

Understandings of the Kranz anatomy‐based CCM in C₄ plants (Leegood, 2002) and the carboxysome‐based CCM in cyanobacteria (Nguyen et al., 2024) suggest that a strong CCM would need at least: (1) a higher rate of carbon delivery than the rate of Rubisco carboxylation; and (2) a very low conductance or high resistance to minimize CO₂ leakage from the compartment where Rubisco localizes. Neither of these requirements is satisfied in the full‐decarboxylating bypasses. For the case of the second requirement, various C₄ plants have a lower bundle‐sheath conductance of 1–8 mmol m⁻² s⁻¹ bar⁻¹ (Ouyang et al., 2024), which is c. 100 times as low as the chloroplast conductance of C₃ plants.

In fact, for C₃ plants, the fraction of (photo)respiratory CO₂ being refixed by Rubisco can be quantified by resistance components (Tholen et al., 2012). In this regard, a previous model in dealing with various chloroplast vs mitochondrion arrangements in WT plants can help to quantitatively assess the extent of refixation as the result of relocating photorespiratory CO₂ release to chloroplasts. Based on the algorithms of Yin & Struik (2017), the fraction of photorespiratory CO₂ being refixed by Rubisco (f _refix) in the context of analysing bypasses can be mathematically calculated as follows:

$$f_{\text{refix}}=\frac{\frac{{\lambda }_{\mathrm{F}}}{r_{\mathrm{cx}}}+\frac{1-{\lambda }_{\mathrm{F}}}{r_{\mathrm{ch}}+r_{\mathrm{cx}}}}{\frac{{\lambda }_{\mathrm{F}}}{r_{\mathrm{cx}}}+\frac{1-{\lambda }_{\mathrm{F}}}{r_{\mathrm{ch}}+r_{\mathrm{cx}}}+\frac{{\lambda }_{\mathrm{F}}}{r_{\mathrm{wp}}+r_{\mathrm{ch}}+r_{\mathrm{sc}}}+\frac{1-{\lambda }_{\mathrm{F}}}{r_{\mathrm{wp}}+r_{\mathrm{sc}}}}$$ (Eqn 11(a))

where r _wp and r _ch are defined earlier, which together constitute mesophyll resistance r _m (= r _wp + r _ch), and r _sc is the stomatal resistance to CO₂ diffusion. In Eqn 11(a), r _cx is the resistance from the Rubisco carboxylation itself, which can be defined as: (C _c + x ₂)/x ₁ (Tholen et al., 2012) and was similarly high as r _m in rice (Yin et al., 2020). For the decarboxylating bypasses where CO₂ release is relocated only to chloroplasts, ${\lambda }_{\mathrm{F}}$ = 1 and Eqn 11(a) is simplified to:

$$f_{\text{refix}}=\frac{r_{\mathrm{sc}}+r_{\mathrm{wp}}+r_{\mathrm{ch}}}{r_{\mathrm{sc}}+r_{\mathrm{wp}}+r_{\mathrm{ch}}+r_{\mathrm{cx}}}$$ (Eqn 11(b))

For the decarboxylating bypasses where CO₂ release is relocated to elsewhere (cytosol or peroxisome), ${\lambda }_{\mathrm{F}}$ = 0 and Eqn 11(a) becomes

$$f_{\text{refix}}=\frac{r_{\mathrm{sc}}+r_{\mathrm{wp}}}{r_{\mathrm{sc}}+r_{\mathrm{wp}}+r_{\mathrm{ch}}+r_{\mathrm{cx}}}$$ (Eqn 11(c))

It is clear from comparing Eqns 11(b) with 11(c) that relocating photorespiratory CO₂ release to chloroplasts increases its refixation than in the native pathway or than relocating it to elsewhere. It is important to note that unless stoma are completely shut (i.e. r _sc becomes infinite), f _refix calculated from Eqns 11(b), 11(c) is all <100%, meaning that not all photorespiratory CO₂, regardless of the organelle where CO₂ is released, will be refixed. In the case represented by Eqn 11(b), where f _refix is maximal, the estimated f _refix is not more than 60% (Yin et al., 2020) unless r _ch is substantially increased analogous to a very high bundle‐sheath resistance for the strong CCM in C₄ plants. On the other hand, there is a trade‐off because a high r _ch will not be in favour of the diffusion of CO₂ from the atmosphere via intercellular airspaces to chloroplasts. In fact, refixation of photorespiratory CO₂ reduces the chance of normally diffused CO₂ being fixed by Rubisco (Yin et al., 2021).

Given the above‐discussed incomplete re‐assimilation of photorespiratory CO₂, it is worthy to note that strictly speaking, the McG pathway (bypass 12) involving an equal photorespiratory carbon release and fixation (Fig. 2) is not carbon‐neutral. However, a much higher carbon affinity of PEPc than Rubisco means that our simplified handling of the McG bypass as the carbon‐neutral type does not yield an error.

Discrepancies between modelled and observed performance

Given our results that difference among bypasses is predominantly determined by amounts rather than the location of photorespiratory CO₂ release, there is an apparent discrepancy between theoretical and measured photosynthetic parameters for bypass plants. For example, theoretical modelling suggests that difference in the A−C _i curve lies mainly in the low‐C _i range (Fig. S1), whereas the measured A−C _i curves often still show significant differences at high‐C _i ranges between bypass and WT plants (e.g. South et al., 2019; Chen et al., 2025a, 2025b). Also, the model theory suggests that the full‐decarboxylating bypasses should have a higher value of C _i* than the WT plants (Fig. 3a; Table 2), whereas the measured C _i* was sometimes lower in full‐decarboxylating bypass than the WT plants (e.g. Shen et al., 2019; South et al., 2019) or was similar (Maier et al., 2012). This dilemma was also noticed by Evans (2019), who commented on the full‐decarboxylating AP3‐bypass of South et al. (2019), that ‘If 2 CO₂ were released per oxygenation instead of 0.5, then the CO₂‐compensation point would increase fourfold. This was not observed’. Evans (2019) raised questions about the fate of glycolate in AP3‐bypass plants.

Our modelling results are all shown for a reference temperature of 25°C. As photorespiratory loss in the native pathway increases with temperature (Jordan & Ogren, 1984), it is expected that full‐decarboxylating bypasses would perform worse under warming conditions. A discrepancy between the FvCB model's prediction and the observed advantage of a full‐decarboxylating bypass (AP3) at high temperature was also noted by Meacham‐Hensold et al. (2026). Under warm growth environments, photosynthesis of WT and bypass plants may differently acclimate to high temperature (e.g. Cavanagh et al., 2022).

In addition to discrepancies in terms of photosynthetic parameters, other unexpected effects were observed for bypass plants. For example, the pBHAC bypass (Roell et al., 2021), in principle, relies on peroxisomal NADH regeneration via peroxisomal NAD‐MDH. But OAA accumulation can exert feedback inhibition on NAD‐MDH, thereby reducing NADH regeneration and restricting redox turnover and pathway efficiency (Selinski & Scheibe, 2019), and this was assumed to ultimately lead to poor phenotypic performance of pBHAC plants (Chen et al., 2025a). Also reported is the lower seed set percentage in rice plants of the GCGT bypass (Wang et al., 2020), which might be attributed to high sugar contents (Li et al., 2024). By contrast, the imGS bypass improved panicle architecture of rice plants (Mo et al., 2025). All these unexpected mechanisms are beyond the scope of the FvCB modeling.

Uncertainties in modelling the impact of photorespiratory bypasses

While our model is formulated according to the FvCB‐model principles on main rate‐determining mechanisms, there are uncertainties for modelling bypass pathways.

First, the fate of products in some bypasses is uncertain. For example, pBHAC, cBHAC, and GCBG bypasses all have OAA as the end‐product of the bypass (Fig. 1), but there are different possibilities for OAA being utilized in the cellular metabolism that could go beyond photosynthesis. The assumption about the fate of OAA has a carry‐over effect on the estimated NADPH and ATP requirement per oxygenation, and thus, on the modelled performance of these bypasses. An isotopically nonstationary metabolic flux analysis (e.g. Ma et al., 2017) should be used to trace the actual flow of carbon fluxes. Also, bypasses 1, 4, 9, 10, and 12 as well as an incomplete bypass of Nölke et al. (2014) for potato use the GDH enzyme, but the identity of endogenous electron acceptors and donors associated with GDH is not certain, despite the recent indication by Meacham‐Hensold et al. (2026) that GDH may donate two electrons to plastoquinone, supporting proton accumulation via the Q‐cycle and yielding ATP synthesis. So far, we assume that electrons are considered NADPH‐equivalents (Table S1).

Second, we use the function min(A _j,NADPH, A _j,ATP) to model electron transport‐limited photosynthetic rate, similar to the logic of using min(V _c, V _j) in the FvCB model. von Caemmerer (2013) used A _j,NADPH only to model the electron transport‐limited photosynthetic rate of the Kebeish bypass, with which she concluded that the benefit from the bypass resulted more from energy savings than from CO₂ relocation. This is expected, given that the NADPH requirement per oxygenation in this bypass is only half of the requirement in the native pathway (Table 1). If A _j,ATP or min(A _j,NADPH, A _j,ATP) was used, the benefit from energy saving in this bypass would have been much lower as the ATP requirement is lowered marginally (3 for this bypass vs 3.5 for the WT, Table 1). Our use of min(A _j,NADPH, A _j,ATP) may not be in line with the concept of so‐called ‘total ATP cost at net zero CO₂’, which is often used (e.g. Peterhansel et al., 2013; Smith et al., 2025) to assess bypasses in energy use efficiency. For that, a conversion factor of 2.5 ATP per NAD(P)H‐equivalent is assumed. This assumption implies a highly flexible cellular energy mobilization between chloroplasts and mitochondria, via malate valve (Selinski & Scheibe, 2019), to adjust the ATP/NADPH requirement ratio in the stroma. As such, photosynthesis could overcome any ATP deficit from importing ATP generated from the mitochondrial electron transport chain. Our model can be adjusted to accommodate such cellular energy flexibility if this assumption is confirmed.

It is, nevertheless, worthy to note that the conversion factor of ATP per NAD(P)H‐equivalent is uncertain and could be lower than 2.5 (Amthor, 2023). Also, the calculation of ‘total ATP cost at net zero CO₂’ implicitly assumes that 100% of photorespiratory CO₂ can be recaptured by Rubisco. As discussed above, photorespiratory CO₂, even released within chloroplasts, cannot be completely refixed by Rubisco. We prefer to use the modelled photosynthetic quantum yield Φ_CO2 (Table 2) to assess differences in energy use efficiency among bypasses or relative to the native pathway (Fig. 3b).

Overall, our model provides a viable tool for conducting various thought experiments on how bypass parameters x′, p, q, and ${\lambda }_{\mathrm{F}}$ affect bypass performance in terms of Rubisco‐limited or electron transport‐limited photosynthesis. If a third limitation occurs (Harley & Sharkey, 1991; Zhou & Yin, 2024), the model can also be extended (see Notes S5) for assessing leaf photosynthesis of bypass plants as limited by subfoliar sink capacity.

Competing interests

None declared.

Author contributions

XY conceived the study, designed the model, conducted model analyses and wrote the draft, and KJ, GC and TL worked on bypasses that inspired this work, provided data for analyses, and gave significant feedback to improve the manuscript.

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Supporting information

Fig. S1 Simulated photosynthetic CO₂‐response and light‐response curves of 17 bypasses, in comparison with the native pathway.

Notes S1 The basic Farquhar–von Caemmerer–Berry model equations and their underlying energy budget.

Notes S2 Derivation of Eqns (Eqn 7(a)), (Eqn 7(b)) for calculating parameter x′.

Notes S3 Input values of the general Farquhar–von Caemmerer–Berry model parameters for model illustration.

Notes S4 Model fitting scripts as implemented in SAS.

Notes S5 Dealing with the TPU limitation.

Table S1 Estimation of NADPH‐equivalent and ATP requirements (per oxygenation) by individual reactions in 17 photorespiratory bypasses.

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