Surfing the Hyperbola Equations of the Steady-State Farquhar-von Caemmerer-Berry C3 Leaf Photosynthesis Model: What Can a Theoretical Analysis of Their Oblique Asymptotes and Transition Points Tell Us?

The asymptotes and transition points of the net CO2 assimilation (A/Ci) rate curves of the steady-state Farquhar-von Caemmerer-Berry (FvCB) model for leaf photosynthesis of C3 plants are examined in a theoretical study, which begins from the exploration of the standard equations of hyperbolae after rotating the coordinate system. The analysis of the A/Ci quadratic equations of the three limitation states of the FvCB model-abbreviated as Ac, Aj and Ap-allows us to conclude that their oblique asymptotes have a common slope that depends only on the mesophyll conductance to CO2 diffusion (gm). The limiting values for the transition points between any two states of the three limitation states c, j and p do not depend on gm, and the results are therefore valid for rectangular and non-rectangular hyperbola equations of the FvCB model. The analysis of the variation of the slopes of the asymptotes with gm casts doubts about the fulfilment of the steady-state conditions, particularly, when the net CO2 assimilation rate is inhibited at high CO2 concentrations. The application of the theoretical analysis to extended steady-state FvCB models, where the hyperbola equations of Ac, Aj and Ap are modified to accommodate nitrogen assimilation and amino acids export via the photorespiratory pathway, is also discussed.

Introduction

The steady-state Farquhar–von Caemmerer–Berry (FvCB) leaf photosynthesis model is broadly recognised by plant biologists and physiologists as one of the most useful models to assess in vivo the net CO2 assimilation rate (A) of plant leaves as a function of CO2 concentration (C) under different environmental cues. The initial FvCB model was first described in the 1980s for C3 plants (Farquhar et al. 1980), then modified to include the triose phosphate utilisation (Sharkey 1985a, b) and later extended to other works on C4 plants, antisense transgenic plants, the effect of bicarbonate pumps at the chloroplast envelope and global climate change, among others (Bellasio et al. 2016; Price et al. 2011; von Caemmerer 2000; Wullschleger 1993). Together with the basic rectangular hyperbolic FvCB model (Farquhar et al. 1980; Sharkey 1985a, b), other non-rectangular hyperbolic, exponential and empirical steady-state models have also been described (Duursma 2015; Ethier and Livingston 2004; Goudriaan 1979; von Caemmerer 2000). In the basic FvCB model, the steady-state CO2 assimilation rate proceeds at the minimum of three limitation rates denoted as Ac, Aj and Ap, which depend on the activity of the ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco), the ribulose-1,5-bisphosphate regeneration and the triose phosphate utilisation, hereafter abbreviated as states c, j and p.

In the basic FvCB model, the analysis of the net CO2 assimilation rate did not consider the (apparent) mesophyll conductance to CO2 diffusion (gm)—hereafter defined as the conductance for CO2 diffusion from the intercellular space to the site of Rubisco carboxylation assuming that photorespiratory and respiratory CO2 release occurs in the same compartment as Rubisco carboxylation (von Caemmerer 2013)—and thus, its value was assumed to be infinite. Consequently, the CO2 concentration in the intercellular (or substomatal) space (Ci) was set equal to the CO2 concentration at the site of Rubisco carboxylation (Cc) and both rate curves, A/Cc and A/Ci, were not distinguished between each other. The inclusion of a finite value for gm into the initial FvCB model transforms Ac and Aj into quadratic equations. This transformation was indeed demonstrated to provide a more accurate estimation of the values for the maximum carboxylation rate (Vcmax) of Rubisco and the maximum electron transport Jmax under steady-state conditions (Ethier and Livingston 2004; Niinemets et al. 2009; von Caemmerer 2000). From a mathematical point of view, the main difference between the equations of the A/Cc and A/Ci rate curves is that the latter are non-rectangular hyperbolae, whose curvature shape in the first quadrant of the Cartesian coordinate system depends on the magnitude of gm.

Nowadays, A/Ci, instead of A/Cc, rate curves are extensively used in the estimation of biochemical parameters from leaf photosynthesis, where gm is assumed to be finite and purely diffusional and not to depend on the CO2 concentration inside the leaf. However, the assumption that gm remains constant has been challenged in some studies and new extensions have been incorporated into the FvCB model (Flexas et al. 2007; Tholen et al. 2012). For instance, gm was proposed to depend on the ratio of mitochondrial CO2 release to chloroplast CO2 uptake and to decrease particularly at low Ci (Tholen et al. 2012), although other factors, such as the intracellular arrangements of chloroplasts and mitochondria in C3 leaves, were later included in a more generalised model to better explain the dependence of gm on the above ratio (Yin and Struik 2017). A decrease in the values for gm was also observed in response to an increase in Ci (Flexas et al. 2007). In this latter study, either modifications in chloroplast shape, which could prevent the chloroplast association with the cell surface, or the involvement of aquaporins, which could facilitate CO2 diffusion across cell membranes by a pH-dependent process, was proposed to regulate the variation of gm. Besides, gm is tightly co-regulated with the stomatal conductance (gs) (Flexas et al. 2008). gs varies with both the atmospheric CO2 concentration and the limitation state (Buckley 2017). The value of gs declines with increased atmospheric CO2 concentration under RuBP regeneration-limited photosynthesis, but, in contrast, it increases with increased atmospheric CO2 concentration under Rubisco-limited photosynthesis (Medlyn et al. 2011).

When the A/Ci rate curves of the FvCB model are analysed under steady-state conditions and the photorespiratory and respiratory CO2 release is also assumed to take place at the site of the Rubisco carboxylation, the quadratic equations for Ac, Aj and Ap (see “Appendix 1”, Eqs. A8–A10) can be fitted following different approaches, where gm is taken as a constant parameter (Duursma 2015; Gu et al. 2010; Sharkey 2016; Su et al. 2009). Some of the nonlinear fitting methods require starting from initial guessed parameters and letting the fit improve with successive iterations, while others constrain the Ci values at which the transition point between c and j occurs. A wealth of data on the transition point between the states c and j indicates that its value is species- and season-dependent, and so it should not be constrained in the fitting method (Duursma 2015; Miao et al. 2009; Zeng et al. 2010). The above fitting methods also take up that the A/Ci rate curves reach asymptotic values for A at supraoptimal CO2 concentration, when there is experimental evidence for the inhibition of the net CO2 assimilation rate by CO2 itself at high concentrations (Woo and Wong 1983). Also, some of these fitting methods make use of approximate estimations for Jmax and Tp—where Tp stands for the rate of phosphate release in triose phosphate utilisation—when Cc approaches infinity in an A/Cc rate curve (Su et al. 2009) or they reasonably assume that the order of the three limitation states along the Ci axis is the same as along the Cc axis (Gu et al. 2010). Dynamic models of photosynthesis are also suitable to analyse the leaf CO2 assimilation response under fluctuating environmental stimuli such as sunlight irradiance, atmospheric CO2 concentration or stomatal response to light (Bellasio 2019; Morales et al. 2018; Noe and Giersch 2004); however, they add complexity to the analysis or they have not been developed completely to date.

The simplicity of the quadratic equations for Ac, Aj and Ap still makes the steady-state FvCB model very useful in fitting approaches to estimate biochemical parameters from leaf photosynthesis (Duursma 2015; Gu et al. 2010; Sharkey 2016; Su et al. 2009). After nearly 40 years of research on the FvCB model, its quadratic equations still hide mathematical features of interest to establish when this model becomes short or when an extended FvCB model would be more suitable for the estimation of the biochemical parameters. On the mathematical analysis of the FvCB model we present here, the rotation of the coordinate system has been a key strategy to reach the conclusion that the quadratic equations of the FvCB model cannot explain the inhibition of the net CO2 assimilation rate at very high Ci. Also, the mathematical analysis of the limiting conditions for the transition points between Ac, Aj and Ap shows that they do not depend on the finite value of gm.

Computer Analysis

The computer algebra system Wolfram Mathematica v. 10.3 (Wolfram Research 2015) was used to program scripts to solve analytically the asymptotes and transition points of A/Cc and A/Ci rate curves of the three limitation states c, j and p. Comparative analyses were performed with hyperbolae in standard form after the rotation of the coordinates. The scripts were also run to plot representative A/Cc and A/Ci rate curves. The chosen and finite values for the kinetic constants for Rubisco and other biochemical parameters in the simulations are in the range of those experimentally determined for different C3 plant species (Jahan et al. 2014; von Caemmerer 2000). A list of definitions is given in Table 1 for the sake of clarity.

Table 1

List of biochemical parameters used with their definition and units

Symbol	Definition	Units
A	Net CO2 assimilation rate	µmol m−2 s−1
Ac	Net CO2 assimilation rate assuming Rubisco limitation	µmol m−2 s−1
Aj	Net CO2 assimilation rate assuming ribulose-1,5-bisphosphate regeneration limitation	µmol m−2 s−1
Ap	Net CO2 assimilation rate assuming triose phosphate use limitation	µmol m−2 s−1
A∗1xy	A at the transition point between any two limitation states (x and y) of the three states c, j and p in the fourth (or third) quadrant of the Cartesian coordinate system. The symbol * stands for chloroplast space (c), intercellular (i) or atmosphere (a)	µmol m−2 s−1
A∗2xy	As A∗1xy, except that the transition point takes place in the first quadrant of the Cartesian coordinate system.	µmol m−2 s−1
Cc	Chloroplast CO2 concentration	Pa
Ci	Intercellular CO2 concentration	Pa
Ca	Atmospheric CO2 concentration	Pa
C∗1xy	CO2 concentration at the transition point between any two limitation states (x and y) of the three states c, j and p in the fourth (or third) quadrant of the Cartesian coordinate system. * stands for chloroplast (c), intercellular space (i) or atmosphere (a)	Pa
C∗2xy	As C∗1xy, except the transition point takes place in the first quadrant of the Cartesian coordinate system	Pa
Cc2→1xy	Value of the CO2 concentration when Cc2xy approaches Cc1xy	Pa
Cc2→∞xy	Value of the CO2 concentration when Cc2xy approaches infinity	Pa
Ci2→1xy	Value of the CO2 concentration when Ci2xy approaches Ci1xy	Pa
Ci2→∞xy	Value of the CO2 concentration when Ci2xy approaches infinity	Pa
J	Electron transport rate	µmol m−2 s−1
Jmax	Maximal electron transport rate	µmol m−2 s−1
Kc	Rubisco Michaelis–Menten constant for carboxylation	Pa
Ko	Rubisco Michaelis–Menten constant for oxygenation	Pa
Kco	Apparent Michaelis–Menten constant	Pa
O	Oxygen concentration	Pa
Rd	Respiration rate in the light	µmol m−2 s−1
rm	Apparent mesophyll resistance to CO2 diffusion (see the Introduction section). Inverse of gm	Pa µmol−1 m2 s
rs	Stomatal resistance to CO2 diffusion. Inverse of gs	Pa µmol−1 m2 s
Tp	Triose phosphate export rate from chloroplasts	µmol m−2 s−1
Vcmax	Maximum carboxylation rate	µmol m−2 s−1
yasynx	Asymptote with horizontal slope of Ax, where x stands for c, j or p	µmol m−2 s−1Pa−1
yasypx	Asymptote with positive slope of Ax, where x stands for c, j or p	µmol m−2 s−1Pa−1
α	Fraction of glycerate that does not return to chloroplasts through the photorespiratory cycle	Dimensionless
β	Angle of rotation of the coordinate system	Dimensionless
Γ*	Chloroplast CO2 photocompensation point	Pa

Results and Discussion

Brief Description of the Asymptotes and Transition Points of the Rectangular Hyperbola Equations of the Basic FvCB Model

According to the basic FvCB model for leaf photosynthesis in C3 plants (Farquhar et al. 1980; Sharkey 1985a, b), the hyperbola equations of the dependence of the net CO2 assimilation rate on the CO2 concentration at the site of the Rubisco carboxylation (i.e. A/Cc) are as follows:withwhere A proceeds at a minimum of the three limitation rates Ac, Aj and Ap. The equations of the three rate curves in the basic FvCB model are branches of rectangular hyperbolae opening upwards and downwards or left and right, where the coordinate system has been rotated 45° (Appendix 1). The two asymptotes of each of the hyperbolic equations (Eqs. 2–4) are perpendicular to each other with slopes 0 and infinite (Table 2). An elemental analysis of the transition points between the rate equations of the three limitation states gives the following sets of solutions:

Table 2

Summary of the values and equations of the centre, asymptotes and bisecting lines describing the rectangular and non-rectangular hyperbolae of the FvCB model for C3 plants

	Vertical asymptote	Horizontal asymptote	Centre	Bisecting line
Rectangular hyperbolae
Ac	Cic=-K_co	Aasync=V_cmax-R_d	-K_co,V_cmax-R_d	Abisc=-C_i+K_co+V_cmax-R_d
Aj	Cij=-2Γ^∗	Aasynj=J/4-R_d	-2Γ^∗,J/4-R_d	Abisj=-C_i+2Γ^∗+J/4-R_d
Ap	Cip=1+3αΓ^∗	Aasynp=3T_p-R_d	1+3αΓ^∗,3T_p-R_d	Abisp=C_i-1+3αΓ^∗+3T_p-R_d

For ,for ,and for ,

Together with the transition points between any two limitation states of the three states c, j and p (superscripts x and y) in the first quadrant of the Cartesian coordinate system (subscript 2), there is a common transition point in the fourth quadrant (subscript 1) when (). Carbon and electron requirements for the assimilation of nitrogen and export of amino acids through the photorespiratory pathway (Busch et al. 2018) are not addressed here, and the standard definition for α in the basic FvCB model remains (see below for further discussion).

Dependence of the Oblique Asymptotes of the Non-rectangular Hyperbola (or Quadratic) Equations of the FvCB Model on rm

The mathematical analysis becomes more challenging if A/Ci, instead of A/Cc, rate curves are used. When steady-state conditions for CO2 diffusion are achieved, Ac, Aj and Ap can be determined after the substitution of Cc for Ci using the equation according to Fick’s diffusion law, where the finite and “constant” mesophyll resistance to CO2 diffusion is . Quadratic equations are obtained for Ac, Aj and Ap (Eqs. A8–A10). They are now non-rectangular hyperbolae opening upwards and downwards, for the case of Ac and Aj, and left and right, for the case of Ap, where the coordinate system has now been rotated anticlockwise an angle, here denoted β (Appendix 1). One of the two asymptotes from each non-rectangular hyperbola is parallel to the horizontal axis, but the other is now oblique with a slope exactly equal to the mesophyll conductance to CO2 diffusion, i.e. , a result which is valid for Ac (von Caemmerer 2000) and also for Aj and Ap. This conclusion is reached following the analysis of the coefficients of the quadratic equations obtained after the anticlockwise rotation of the coordinate system by β. When Eqs. A6 and A7 are compared with Eqs. A8–A10, some key features emerge: firstly, the summation of the coefficients of is equal to zero and, secondly, the second coefficient of the quadratic equations is, in fact, the summation of the two asymptotes of each hyperbola (Appendix 1). The equations of the two asymptotes for Ac, Aj and Ap are therefore summarised as follows:where and stand for the oblique and horizontal asymptotes of Ac, Aj and Ap, respectively.

It is worth noting that the use of directly in Eq. 4 is an oversimplification of Ap. The oblique asymptote of Ap is present, even when α is assumed to be equal to zero (Eq. 13a). The intersection between the two asymptotes of Ap (i.e. and ), in particular when , gives a limiting value below which Ci is meaningless. In fact, the approximation is not valid in the whole Ci domain between . The discontinuity is more obvious when because there is a Ci domain for which no real values for Ap can be obtained. The suitable Ci domain for the nonlinear fitting of Ap in the FvCB model is thus confined to the negative root of its branch opening right (Fig. 1), a result which is also in line with the study by Gu et al. (2010). When , the values of the negative root of the Ap branch opening right decrease as Ci increases.

Fig. 1

Representative Ap rate curves for the non-rectangular FvCB model for C3 plants with two different values for α and their corresponding oblique asymptotes. The negative root (thin solid line) and the positive root (thin dashed line) of the quadratic equation of Ap together with its oblique asymptote (thick solid line) are in grey for . The negative root (thin solid line) and the positive root (thin dashed line) of the quadratic equation of Ap together with its oblique asymptote (thick solid line) are in black for . The simulation was performed using the following values for the biochemical parameters: Tp, 12 μmol m−2 s−1; Rd, 2 μmol m−2 s−1; rm, 0.4 Pa µmol−1 m2 s; Γ*, 3.74 Pa. Note: The negative root of the branch opening left and the positive root of the branch opening right of Ap for are overlaid with its oblique asymptote

The Limiting Conditions for the Transition Points in the FvCB Model Do Not Depend on rm

The transition points for the negative roots of the quadratic equations for A/Ci (Eqs. A8–A10) can be solved mathematically and written in a simple form making use of the analytical solutions (Eqs. 5–10) for A/Cc as follows:

For ,for ,and for ,

The above solutions could be further extended to include the stomatal resistance to CO2 diffusion () in Ac, Aj and Ap (see Appendix 2). Figure 2 summarises the changes in the transition points in the first and fourth quadrants of the Cartesian coordinate system when the resistance(s) to CO2 diffusion is included in the net CO2 assimilation rate curves. For the sake of clarity, only Ac and Aj are shown. The analytical values for the assimilation rates ( and ) at the transition points remain constant, while the values for the CO2 concentration increase in the first quadrant (and decrease in the fourth quadrant) when the resistance(s) to CO2 diffusion increases. The solution for the transition point is restricted to the fourth quadrant of the Cartesian coordinate system when the rectangular hyperbolae (Eqs. 2–4) of the basic FvCB model are used. However, it should not be surprising to find out this analytical transition point in the third quadrant under conditions for which when dealing with the quadratic equations of the FvCB model.

Fig. 2

Transition points and between Ac (solid lines) and Aj (dashed lines) for the rectangular and non-rectangular hyperbolic equations of the FvCB model for C3 plants in the first (a) and fourth (b) quadrants of the Cartesian coordinate system. In the transition points, the obtained values for the net CO2 assimilation rate ( and ) remain constant (dotted-dashed lines), while the values for the CO2 concentration ( and ) at the transition points depend on the type of A/C rate curve (A/Cc, black lines; A/Ci, grey lines; and A/Ca, light grey lines). The asterisks stand for the CO2 concentration at the carboxylation site (c), intercellular space (i) or the atmosphere (a). The simulation was performed using the following values for the biochemical parameters: Vcmax, 100 μmol m−2 s−1; J, 150 μmol m−2 s−1; Rd, 2 μmol m−2 s−1; rm, 0.45 Pa µmol−1 m2 s; r, 0.4 Pa µmol−1 m2 s; Kco, 62.1 Pa; Γ*, 3.74 Pa

In the mathematical analysis, it can be observed that the common transition point between the three states c, j and p is always present; in contrast, the transition points depend on the biochemical parameters and might not be present in the net CO2 assimilation rate curves. Two limiting conditions can now be investigated in order to analyse all the possible combinations between the transition points between Ac, Aj and Ap regardless of the value of rm. One is that approaches (i.e. ) and another is that approaches infinity (i.e. ). The equation has to be solved for the analysis of the first limiting condition, whereas only the values for the denominator of the first summand (i.e. ) of (Eqs. 15a–19a) have to be inspected to analyse the second limiting condition. In the analysis, the constraint is imposed based on the values reported for C3 plants (von Caemmerer 2000); consequently, there are no finite values for the biochemical parameters of the second summand (i.e. ) of (Eqs. 15a–19a) that can make this summand approach infinity.

The ratios between the biochemical parameters to reach the above limiting conditions for the rectangular equations of an A/Cc rate curve (i.e. and ) can be derived straightforward from Eqs. 5a–10a. The results are as follows:

For and , respectively,for and , respectively,and for and , respectively,

Among the above ratios (Eqs. 20a–22b), the ratios between the biochemical parameters for and are of non-biochemical significance. They imply that there should be conditions for which one could expect triose phosphate import to chloroplasts (i.e. Tp < 0). In fact, if these transition points are analysed, particularly, in a non-rectangular A/Ci rate curve, one can observe that both transitions, and , occur with the negative root of the hyperbolic branch opening left of Ap, for which the Ci domain is not applicable as indicated above. (Further details are given in Fig. S1 of Online Resource.)

When the rest of the ratios between the biochemical parameters are now investigated in the non-rectangular hyperbola equations of the FvCB model—where is finite, —two solutions are in fact found for any equation like , of which only one also fulfils the condition (data not shown). The analysis indeed shows that the correct ratios between the biochemical parameters for and are the same as those found for and (Eqs. 20a–22b). This means that the ratios between the biochemical parameters in the two limiting conditions do not depend on the value of rm. The limiting conditions for the transition points can therefore be reduced as follows:

The graphic representation of Ac, Aj and Ap for A/Ci rate curves shows, firstly, that there are no experimental ratios for the biochemical parameter for which Ap (with ) can be the only limitation state along the domain and, secondly, there are ratios between the biochemical parameters for which Ac or Aj can be the only limitation state along the domain (Fig. 3a, b). Additional ratios between the biochemical parameters can be found for which there are one or two transition points in the first quadrant of Cartesian coordinate system (Fig. 3c–f). The latter ratios are equivalent to those discussed before for A/Cc rate curves (Gu et al. 2010). Regardless of the number of transition points (0, 1 or 2) that the ratios of the biochemical parameters can yield between the three limitation states in the first quadrant of the Cartesian coordinate system, the transition points in the fourth (or third) quadrant are always present.

Fig. 3

Ratios between the limiting values of the biochemical parameters of a representative A/Ci rate curve, where the summation of resistances to CO2 diffusion is included (i.e. ), for which there are no transition points between the three states c, j and p (a, b) and two transition points (c) or there is only one transition point (d–f) in the first quadrant of the Cartesian coordinate system . The symbols x and y stand for any of the three limitation states c, j and p. The simulation was performed using the following values for the biochemical parameters: Vcmax, 36–100 μmol m−2 s−1; J, 24–144 μmol m−2 s−1; Tp, 8–12 μmol m−2 s−1; Rd, 2 μmol m−2 s−1; rm, 0.4 Pa µmol−1 m2 s; Kco, 62.1 Pa; Γ*, 3.74 Pa

Analysis of the Inhibition of the Net CO2 Assimilation Rate at High CO2 Concentrations

If the steady-state FvCB model is strictly followed, one can state, first, that the slopes of the oblique asymptotes of the non-rectangular hyperbolae only depend on , while the slopes of the horizontal asymptotes of Ac, Aj and Ap remain unchanged regardless of the value for gm (Eqs. 11a–13b) and, second, the slopes of the bisecting lines (Table 2) of Ac, Aj and Ap correspond with the angle (or the perpendicular angle) of the rotation of the coordinate system that makes the summation of the coefficients of equal to zero (Eqs. A11 and A12). This means that there are no mathematical solutions for the quadratic equations of Ac, Aj and Ap (Eqs. A8–A10) in the FvCB model for which the slopes of the horizontal asymptotes can be modified to reach negative values. The fraction of glycerate () that does not return to chloroplasts through the photorespiratory cycle (Harley and Sharkey 1991) makes Ap decrease as Ci increases, but the slope of the horizontal asymptote of Ap remains unchanged, no matter what value α () has. This indicates that Ap must finally reach a constant value as Ci increases. This conclusion also applies to the extended FvCB model described in the study by Busch et al. (2018), where the parameter α of the basic FvCB model is replaced with two new parameters αG and αS that stand for the proportion of glycolate carbon taken out of the photorespiratory pathway as glycine, and the proportion taken out as serine, respectively. Although αG and αS might not be constant and depend on the photorespiratory pathway and the reduction of supplied nitrate (Busch et al. 2018), the new equations for the three limitation states remain, from a mathematical point of view, as rectangular hyperbolae with horizontal asymptotes equivalent to those summarised in Table 2. Likewise, the extension of the FvCB model using as a flux-weighted quantity that depends on mitochondrial respiration and photorespiration effects does not explain the inhibition of A/Ci at high Ci either (Tholen et al. 2012).

Despite what has been said above, there are lines of experimental evidence that indicate that negative slopes can be indeed observed in A/Ci rate curves. Woo and Wong (1983) showed that supraoptimal CO2 concentrations inhibited the net CO2 assimilation in cotton plants, and they proposed that an acidification mechanism mediated by CO2 could affect both the thylakoid electron transport and the activity of key enzymes of the Calvin–Benson–Bassham cycle (Kaiser and Heber 1983; Ögren and Evans 1993). At this point, one could speculate on some mathematical explanations for negative slopes in experimental A/Ci rate curves. In the first place, one could wonder whether other rotation of the coordinates—different from the one that yields Eqs. A28 and A29—would be possible under steady-state conditions, which rendered negative asymptotes instead of horizontal asymptotes. If this were possible, the summation of the coefficients of (Eqs. A11 and A12) would not be zero and so the chosen fitting method should start from extended quadratic equations as Eqs. A6 and A7, where at least four parameters defining the hyperbola equation and an angle of rotation had to be determined. In this case, the angle of rotation should depend on gm together with other biochemical parameters. Alternatively, one could wonder whether the steady-state conditions do not hold along the whole Ci domain, particularly at supraoptimal CO2 concentrations. If this were the case, one can assert that the equation is not always valid. So, A decreases at supraoptimal CO2 concentrations because either rm is not only diffusional and so it increases as Ci increases (Flexas et al. 2007) or the photosynthetic activity is indeed inhibited by CO2 acidification (Kaiser and Heber 1983; Ögren and Evans 1993; Woo and Wong 1983). Reliable nonlinear fittings of Ac and Aj of the FvCB model can thus be possibly obtained under steady-state conditions using standard approaches (Duursma 2015; Gu et al. 2010; Sharkey 2016; Su et al. 2009); however, the use supraoptimal CO2 concentrations to fit Ap might cast doubts on the fitted biochemical parameters if evidence for negative slopes in the experimental A/Ci rate curves is observed. Based on the variation of gm with Ci (Flexas et al. 2007), other nonlinear fitting approaches proposed the combination of gas exchange methods with chlorophyll fluorescence-based methods to estimate gm by using only data within the j state (Yin and Struik 2009).

Conclusions

The analysis of the steady-state FvCB model for C3 plants starting from the standard equations of hyperbolae after rotating the coordinate system has disclosed some features hidden in the quadratic equations of Ac, Aj and Ap of the A/Ci rate curves. In particular, academic interest has been the angle of the rotation of the coordinate system from which it has been established that the oblique asymptotes of the three limitation rate curves share a common slope whose value depends only on gm. Ap always has an oblique asymptote regardless of the value of α. The limiting conditions for the transition points in the FvCB model do not depend on gm. The hyperbola equations of Ac, Aj and Ap in the FvCB model or in some of the extended steady-state FvCB models here discussed can only provide horizontal asymptotes when the CO2 concentration approaches infinity when, in contrast, there is experimental evidence for negative slopes in A/Ci rate curves at high CO2 concentrations. This leads us to the conclusion that extended quadratic equations containing a term might be required for the analysis of Ac, Aj and Ap or, in contrast, that steady-state conditions do not hold, particularly, with increased CO2 concentrations. Dynamic modelling taking into account the decrease in the values for gm or the activity inhibition of key enzymes of the Calvin–Benson–Bassham cycle by CO2 acidification could alternatively provide suitable models for the estimation of the biochemical parameters from leaf photosynthesis.

Electronic supplementary material

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Supplementary material 1 (PDF 192 kb)