Extension of a biochemical model for the generalized stoichiometry of electron transport limited C3 photosynthesis


Dr Xinyou Yin. Fax: 31-317-485572; e-mail: Xinyou.Yin@wur.nl


The widely used steady-state model of Farquhar et al. (Planta 149: 78–90, 1980) for C3 photosynthesis was developed on the basis of linear whole-chain (non-cyclic) electron transport. In this model, calculation of the RuBP-regeneration limited CO2-assimilation rate depends on whether it is insufficient ATP or NADPH that causes electron transport limitation. A new, generalized equation that allows co-limitation of NADPH and ATP on electron transport is presented herein. The model is based on the assumption that other thylakoid pathways (the Q-cycle, cyclic photophosphorylation, and pseudocyclic electron transport) interplay with the linear chain to co-contribute to a balanced production of NADPH and ATP as required by stromal metabolism. The original model assuming linear electron transport limited either by NADPH or by ATP, predicts quantum yields for CO2 uptake that represent the highest and the lowest values, respectively, of the range given by the new equation. The applicability of the new equation is illustrated for a number of C3 crop species, by curve fitting to gas exchange data in the literature. In comparison with the original model, the new model enables analysis of photosynthetic regulation via the electron transport pathways in response to environmental stresses.


With the availability of the overwhelming knowledge about contributing mechanisms in photosynthesis, several biochemical models have been developed to describe this important process. The steady-state model of Farquhar, von Caemmerer & Berry (1980) (hereafter, the FvCB model) has had the strongest impact and has become a standard model for photosynthesis of C3 species (see Farquhar, von Caemmerer & Berry 2001 for reflections). The model makes no attempt to treat all of the steps with dozens of enzymes and electron carriers in photosynthesis; rather it is a synthesis, a simplified view of the knowledge of the contributing mechanisms (Appendix A). The model predicts photosynthesis as the minimum of the ribulose-1,5-biphosphate (RuBP)-saturated rate of CO2 assimilation, which is a function of the  maximum  carboxylation  capacity  of  Rubisco  (Vcmax; see Table 1  for  a  full  list  of  model  variables),  and  the RuBP-regeneration  limited  rate  of  assimilation,  which  is a function of the maximum capacity of electron (e) transport (Jmax).

Table 1.  List of variables used in the models
ANet leaf photosynthesis rateµmol m−2 s−1
CaExternal air CO2 concentrationµmol mol−1
CiIntercellular or carboxylating-site CO2 concentrationµmol mol−1
DJEnergy of deactivation for JmaxJ mol−1
EActivation energy for Vcmax, or KmC, or KmOJ mol−1
EJmaxActivation energy for JmaxJ mol−1
EKmcActivation energy for KmCJ mol−1
EKmoActivation energy for KmOJ mol−1
EVcmaxActivation energy for VcmaxJ mol−1
fcycFraction of cyclic electron transport around PSI
fpseudoFraction of pseudocyclic electron transport
fQFraction of electron transport that follows the Q-cycle
hThe number of protons required to produce 1 ATPmol mol−1
IPPFD absorbed by chloroplasts, = σIincµmol PPFD m−2 s−1
I1Absorbed PPFD that is distributed to PSIµmol PPFD m−2 s−1
I2Absorbed PPFD that is distributed to PSIIµmol PPFD m−2 s−1
IincIncident photosynthetic photon flux density (PPFD)µmol PPFD m−2 s−1
JRate of linear whole-chain electron transportµmol e m−2 s−1
joAn intermediate variable in appendix C for electron transport rateµmol e m−2 s−1
J1Rate of electron transport through PSIµmol e m−2 s−1
J2Rate of electron transport through PSIIµmol e m−2 s−1
JcycRate of cyclic electron transport around PSIµmol e m−2 s−1
JcytRate of electron transport going to cytchrome and plastocyaninµmol e m−2 s−1
JmaxMaximum rate of electron transport through PSIIµmol e m−2 s−1
Jmax25Jmax at 25 °Cµmol e m−2 s−1
JNADP+Rate of electron transport to the end acceptor NADP+µmol e m−2 s−1
JpseudoRate of electron transport being pseudocyclicµmol e m−2 s−1
KmCMichaelis–Menten constant for CO2µmol mol−1
KmC25KmC at 25 °Cµmol mol−1
KmOMichaelis–Menten constant for O2mmol mol−1
KmO25KmO at 25 °Cmmol mol−1
OOxygen concentrationmmol mol−1
RUniversal gas constantJ K−1 mol−1
RdLeaf mitochondrial respirationµmol m−2 s−1
Rd25Rd at 25 °Cµmol m−2 s−1
SJEntropy term in JmaxT relationship formulaJ K−1 mol−1
TLeaf temperature°C
VCarboxylation rateµmol m−2 s−1
VcRuBP-saturated carboxylation rateµmol m−2 s−1
VcmaxMaximum rate of RuBP-saturated carboxylationµmol m−2 s−1
Vcmax25Vcmax at 25 °Cµmol m−2 s−1
VjRuBP regeneration-limited carboxylation rateµmol m−2 s−1
VomaxMaximum oxygenation rate of Rubiscoµmol m−2 s−1
xFraction of additional light to PSI to support cyclic e transport
αQuantum yield for whole-chain e flow on absorbed light basismol e mol−1 PPFD
α2Quantum yield for e flow of PSII on absorbed light basismol e mol−1 PPFD
ɛQuantum yield for CO2 uptake on absorbed light basismol CO2 mol−1 PPFD
σAbsorptance by chloroplasts
θConvexity factor for response of J to light
Φ2mMaximum electron transport efficiency of PSIImol e mol−1 PPFD
Γ*CO2 compensation point in the absence of dark respirationµmol mol−1

However, the rate of carboxylation when e transport is limiting (Vj) has not yet been described unambiguously in the FvCB model. The e transport equations in the model were derived on the basis of the non-cyclic (or linear) e transport in thylakoid reactions. However, linear e transport alone cannot give a balanced production of NADPH and ATP that matches the requirement by C3 stromal processes of photosynthesis. In applying the FvCB model, some workers (e.g. Harley et al. 1992; Leuning et al. 1995; Medlyn et al. 2002) use the equation where e transport limitation is assumed to be caused by insufficient NADPH (Eqn A3a). Others (e.g. Long 1991; Friend 2001) assume a limitation by insufficient ATP (Eqn A3b). Although the linear e transport assumption simplifies modelling procedures, various other pathways (Farquhar & von Caemmerer 1982) may occur in reality such that NADPH and ATP co-limit e transport. For example, Farquhar & von Caemmerer (1981) discussed how much cyclic e flow is required to provide the additional required ATP.

The operation of other pathways is especially relevant under suboptimal conditions when abiotic stresses such as low temperature, drought or mineral deficiency prevail. Under such conditions, non-assimilatory e flow may replace assimilatory e flow in order to act as a protective mechanism against photo-inhibition (Ort & Baker 2002). For example, chlorophyll fluorescence studies (e.g. Schreiber & Neubauer 1990) have revealed that O2-dependent pseudocyclic e transport can act as a major energy sink, and effective energy dissipation may occur due to a futile ‘water–water cycle’ through the coupling between the pseudocyclic e flow and the Mehler–ascorbate peroxidase reaction. Quantification of photosynthetic regulation in response to environmental stresses requires a model in which various e transport pathways are incorporated. The recent review of Allen (2003) gives an updated insight into the interplay of the linear chain, cyclic and other non-linear pathways of e flow (hereafter ‘non-linear’ is our collective name for pathways differing from linear whole-chain e transport). In the light of this new insight, we aim to develop a generalized steady-state model for the description of e transport, allowing for a co-limitation by NADPH and ATP. It will be shown that the algorithms in the FvCB model are embedded in our new model.


Our formulation is based on the original FvCB model for C3 species (Appendix A). All variables and their definitions are listed in Table 1.

The stoichiometry of linear electron transport in the FvCB model

Both carboxylation and oxygenation in C3 metabolic reactions require NADPH and ATP: each carboxylation requires 2 NADPH and 3 ATP, and each oxygenation requires 2 NADPH and 3.5 ATP. Given that the ratio of oxygenation to carboxylation is given by 2Γ*/Ci (Farquhar & von Caemmerer 1982), the rate of NADPH consumption can be expressed as (2 + 4Γ*/Ci)V, where V is the rate of carboxylation. Since the reduction of one NADP+ to NADPH requires two e(Fig. 1), the rate of e transport for satisfying the NADPH requirement is (4 + 8Γ*/Ci)V. This gives rise to Eqn A3a of the FvCB model for calculating Vj in terms of NADPH requirement.

Figure 1.

Various pathways of electron transport in the light reaction of photosynthesis. Thin arrows indicate electron transport. Thick curved arrows show the number of protons or NADPH produced per electron transported. The rates of electron transport (J1, J2, Jcyc, Jcyt, JNADP+, Jpseudo, JQ) are described in Appendix B.

The rate of ATP consumption in C3 reactions is (3 + 7Γ*/Ci)V. The movement of one e through the linear chain, if the Q-cycle is not operating, results in two protons (H+), one from the splitting of water in photosystem II (PSII) and one from the shuttle of reduced plastoquinone across the membrane (Fig. 1). The FvCB model assumes that 3 H+ are required for the photophosphorylation of 1 ADP to 1 ATP. Therefore, the flow of one e through the linear chain only produces 2/3 ATP. If ATP is produced by the linear e transport alone, the required rate of the linear e flow is (4.5 + 10.5Γ*/Ci)V. This gives rise to Eqn A3b for calculating Vj in terms of ATP requirement.

Non-linear pathways of electron transport

From the above stoichiometry, we see that the required rates of e transport in terms of ATP and NADPH consumption are conflicting, and ATP is limiting in the case of 100% linear e transport. There are various mechanisms (Fig. 1), by which chloroplasts may remove the disparity (Farquhar & von Caemmerer 1982; Heber 2002; Allen 2003). First, this could be met by a portion of the e transport following the Q-cycle, rather than directly going to plastocyanin in the whole-chain after the reduced plastoquinone. The Q-cycle activity (Mitchell 1975) refers to a mechanism for H+ translocation through the chloroplast cytochrome b6f complex and its mitochondrial counterpart (the cytochrome bc1 complex). The involved e transport that includes the two quinone-binding sites and the two cytochromes b6 effectively doubles the stoichiometry of H+ through the cytochrome complex from 1H+ to 2H+ per e transferred from PSII to photosystem I (PSI) (Furbank, Jenkins & Hatch 1990; Allen 2003). Second, the disparity could be removed by a portion of the e transport being cyclic around PSI. As for the non-cyclic e flow, the cyclic e transfer also passes through the ‘coupling site’ of ATP synthesis (Allen 2003), thereby eliminating the deficiency in ATP via cyclic photophosphorylation. This cycle contributes one (Farquhar & von Caemmerer 1982) or two (Allen 2003) H+ per e involved, depending on whether or not the Q-cycle is operated. The third likely mechanism is the pseudocyclic electron transport, whereby O2, as occurs in the Mehler reaction (Mehler 1951), or nitrate is the terminal whole-chain acceptor rather than NADP+.

A generalized stoichiometry of electron transport

All modes of e transport are likely involved, permitting flexibility in the ATP : NADPH ratio according to metabolic demands (Allen 2003). If all of the mechanisms indeed contribute to a balanced NADPH and ATP production, the required relation for the fraction of the e transport being cyclic around PSI (fcyc), being pseudocyclic (fpseudo), and being involved in the Q-cycle (fQ), can be derived as (Appendix B):


In Eqn 1, we introduced h, the number of H+ required to produce 1 ATP (h = 3 in the FvCB model) as an additional parameter, given the uncertainty in the stoichiometry of proton yield from e transport (von Caemmerer 2000).

The rate of e transport to NADP+, JNADP+, is (1 – fcyc – fpseudo)J1, where J1 is the total rate of e transport through PSI. The equation for Vj is therefore expressed in a form similar to Eqn A3a but as a function of JNADP+:


Substituting Eqn B6 into Eqn 2a, Vj can be reformulated as:


where J2 is the rate of e transport through PSII. Substituting Eqn 1 into Eqn 2b gives an equation for Vj, co-limited by both NADPH and ATP:


The relation between J2 and photon flux absorbed by chloroplast lamellae, I, can be expressed in a form similar to Eqn A4:


where Jmax is the upper limit to J2, equivalent to the maximum rate of whole chain transport while cyclic flow is occurring simultaneously; α2 is the e transport efficiency of PSII on the basis of light absorbed by both photosystems, and Φ2m is the maximum e transport efficiency of PSII on the basis of light absorbed by PSII alone. Equation 3a, expressing the dependence of α2 on fcyc and Φ2m, has a two-fold rationale. First, as indicated by Eqn B2, electrons energised by PSII only account for part (i.e. 1 – fcyc) of those energized by PSI if the cyclic e transport is engaged. Second, in order to carry more electrons, PSI captures more light quanta than PSII does (Albertsson 2001). The derivation of Eqn 3a is given in Appendix C, which shows that Eqn 3a holds regardless of how PSI receives the additional quanta.

Quantum yield for CO2 uptake

The new e transport model predicts the intrinsic quantum efficiency for CO2 uptake on the basis of light absorbed by chloroplast, ɛ, as:


The model predicts that the quantum yield increases markedly with increasing fQ, but very slightly with increasing fcyc. Furthermore, the quantum yield is a monotonically decreasing function of h, the required number of protons to synthesize 1 ATP.

The e flow assumption of Eqn A3a or Eqn A3b in the FvCB model yields:


Note that the α value in Eqns 5a and 5b allows the FvCB model to explicitly account for a difference in e transport efficiency between PSI and PSII, in contrast to an equal efficiency as implicitly assumed in many existing applications (cf. Appendix A).

It can be seen that Eqns 5a and 5b are special cases of Eqn 4 in the absence of cyclic e transport (fcyc = 0), as assumed in the FvCB model. Eqn 5a results from the additional condition that h is so small [≤ 2(4Ci + 8Γ*)/(3Ci + 7Γ*), see the next section] that the additional ATP is not required, or that no pseudocyclic mode occurs (fpseudo = 0), but the Q-cycle operates to provide the additional required ATP. Solving for fQ from Eqn 1 for such a condition (case 1 in Table 2) and substituting it into Eqn 4 results in Eqn 5a. Equation 5b results from additional conditions that the pseudocyclic mode operates in the absence of the Q-cycle, and that h = 3 as assumed in the FvCB model. The general form of Eqn 5b for any value of h is:

Table 2.  Special cases of Eqn 4, with their specific required fraction of nonlinear electron transport, and simplified equation for calculating Vj
CaseɛaRequired specific e flow bSimplified Vj
  • a

    Quantum yield for CO2 uptake;

  • b

    b Calculated using Eqn 1.

(1) inline imageEqn 5ainline imageinline image
(2) inline imageEqn 5cinline imageinline image
(3) inline imageEqn 6inline imageinline image
(4) inline imageEqn 7inline imageinline image

If cyclic e transport is the only non-linear mechanism for a balanced production of ATP and NADPH, quantum yield can be derived from Eqns 1 and 4 with fQ = fpseudo = 0:


A fourth case is that the cyclic mode and the Q-cycle operate to an equal degree whereas the pseudocyclic pathway does not run. Quantum yield then becomes:


where fcyc for this case can be solved from Eqn 1 as 1 − 2(4Ci + 8Γ*)/[h(3Ci + 7Γ*)].

The above distinctive special cases of our generalized equation are summarized in Table 2. Their required fractions of the corresponding non-linear pathway for a balanced production of NADPH and ATP can be easily derived from Eqn 1 and are also shown in Table 2. For these special cases, the equation for calculating Vj, Eqn 2c, can also be simplified accordingly (Table 2).

Clearly, the quantum yield for CO2 uptake (ɛ) depends on e transport mechanisms. It is most evident that the highest ɛ is given by Eqn 5a, resulting from the NADPH-limited mechanism in the FvCB model by the use of Eqn A3a. It is less evident that the lowest ɛ is given by Eqn 5c, resulting from the ATP-limited mechanism in the FvCB model, although relative to Eqn 6, Eqn 5c predicts a lower ɛ by a very small margin. It is algebraically provable (the algebra can be obtained upon request) that the magnitude of ɛ, for a given h, predicted by the above equations is in the order: ɛEqn5a ≥ ɛEqn7 ≥ ɛEqn6 ≥ ɛEqn5c, as long as h ≥ 2(4Ci + 8Γ*)/(3Ci + 7Γ*) which appears to hold always (see the next). For any lower h-value, quantum yield for CO2 uptake given by Eqn 5a then becomes valid.

Thresholds for parameter h

Because the fractions for the non-linear e transport pathway have to be ≥ 0, the equations for all these fractions in Table 2 mean that the value of h has to satisfy: h ≥ 2(4Ci + 8Γ*)/(3Ci + 7Γ*). For a wide range of Ci varying from Γ* to +∞ (infinity), this lower limit varies within a narrow range, namely from 2.4 to 2.667. Any h-value lower than 2(4Ci + 8Γ*)/(3Ci + 7Γ*) means that ATP supply is not limiting and a non-linear pathway is therefore not needed.

If the Q-cycle is operated alone, an upper limit for h also exists. Setting the condition that fQ ≤ 1 in the equation for fQ in Table 2 yields h ≤ 3(4Ci + 8Γ*)/(3Ci + 7Γ*). For a wide range of Ci from Γ* to +∞, this h limit ranges from 3.6 to 4.0. Any value more than 3(4Ci + 8Γ*)/(3Ci + 7Γ*) means that in addition to the full engagement of the Q-cycle, the cyclic or the pseudocyclic pathway is required as well for a balanced production of NADPH and ATP.


Like the FvCB model, the new model is, in principle, applicable to photosynthesis at the chloroplast level (Farquhar et al. 1980). To scale up to a whole leaf, it is necessary to sum the contributions of each chloroplast by considering the distribution of light within the leaf (e.g. Kull & Kruijt 1998). However, since von Caemmerer & Farquhar (1981) first applied the FvCB model to describe gas exchange data of the whole leaf, most applications of the model have been at the leaf level, showing that the model works well. In the next section, we apply our model to data obtained for leaf photosynthesis. For such application, units of model variables such as e transport rates are all expressed on a leaf area basis (Table 1).

Data sets, curve-fitting method and model specification

We will use data sets of ACi curves, reported in literature in several crop species, to compare the new e transport limited equation (Eqn 2c), with the two forms of the FvCB model. The summary information of the data sets is given in Table 3. Expressions for the Rubisco-limited photosynthesis in the FvCB model (cf. Appendix A) are also used for the new model. Assuming an infinite mesophyll conductance, each model was fitted to the combined data of net leaf photosynthesis measured at various levels of Iinc, Ci, and temperature (depending on data set), using non-linear least-squares regression of the DUD method in the PROC NLIN of the SAS software (SAS Institute Inc., Cary, NC, USA).

Table 3.  Details of experimental data sets used
(µmol m−2 s−1)
(mmol mol−1)
(µmol mol−1)
RiceMakino et al. (1988)1800,950,50025210various 29
WheatMakino et al. (1988)1800,950,50025210various 28
BarleyBunce (2000)150015,20,25,30,35210various 60
Common beanvon Caemmerer & Farquhar (1981)1400,95028210various 25
150028,2020various 29
150032,28,24,20210various 57
SoybeanHarley et al. (1985)200040,35,30,25,20600,400,200,10various342
various40,35,30,25,20,15,10 1000190

Our new e transport equation contains parameters for fractions for various non-linear e transports and parameter h, which is reportedly uncertain (Farquhar & von Caemmerer 1981; von Caemmerer 2000). We first tried to estimate all of these parameters by curve fitting. However, this resulted in over-fitting and unrealistic parameter estimates, even for the most comprehensive data set, namely that of Harley, Weber & Gates (1985) (Table 3). Therefore, we subsequently examined only the scenario specifically discussed by Allen (2003) with full operation of the Q-cycle (fQ = 1) while fpseudo is kept at zero. Allen (2003) indicated that h probably has a value of 14/3, based on the observation of Seelert et al. (2000) that the proton turbine of chloroplast ATP synthase has 14 subunits instead of the expected 12. Such a high value of h means that the linear e transport with the full Q-cycle alone cannot provide enough ATP per NADPH to drive CO2 reduction metabolism and has to be complemented by cyclic e flow; the required fcyc is calculated according to Eqn 1.

Parameterization of the models

For the temperature dependence functions, previous parameterizations of the FvCB model have mostly been based on in vitro measurements. We fitted the models, using recently available information about parameters based on in vivo measurements for transgenic Rubisco-deficient tobacco (Bernacchi et al. 2001) since intrinsic properties of the Rubisco enzyme are generally assumed constant among species (Harley et al. 1992; Medlyn et al. 2002). First, the Michaelis–Menten coefficients of Rubisco, KmC and KmO, vary with temperature. Their value at 25 °C and activation energies (cf Eqn A5) are taken from Bernacchi et al. (2001) (Table 4). Second, the key parameter Vcmax is not conserved, as it depends on Rubisco concentration and level of activation. On the other hand, the relative change in Vcmax with temperature is conserved since it depends on enzyme structure and not on the concentration; therefore, EVcmax is also set to the value of in-vivo measurement of Bernacchi et al. (2001). Third, the ratio of the maximum oxygenation rate to the maximum carboxylation rate of Rubisco (Vomax/Vcmax) in Eqn A7, was calculated as

Table 4.  Constants used in the models
  • a

    The dependence of σ on the ratio of red to blue light (which also varies with levels of Iinc), as considered by Bernacchi et al. (2003), is ignored here because of lack of information for the light wavelengths.

DJEqn A6J mol−1200000Medlyn et al. 2002)
EKmcEqn A5J mol−179430Bernacchi et al. (2001)
EKmoEqn A5J mol−136380Bernacchi et al. (2001)
EVcmaxEqn A5J mol−165330Bernacchi et al. (2001)
KmC25Eqn A5µmol mol−1404.9Bernacchi et al. (2001)
KmO25Eqn A5mmol mol−1278.4Bernacchi et al. (2001)
REqns A5, A6J K−1 mol−18.314Farquhar et al. (1980)
SJEqn A5J K−1 mol−1650Harley et al. (1992)
σI = σIinc0.80 aLong, Postl & Bolhár-Nordenkampf (1993)
Φ2mEqns 3, A4mol e mol−1 PPFD0.85Bernacchi et al. (2003)
θEqns 3, A40.7De Pury & Farquhar (1997)
Vomax/Vcmax = exp(− 3.3801 + 5220/[R(T + 273)])(8)

Equation 8 is also established from in vivo functions provided by Bernacchi et al. (2001).

Parameters associated with Jmax vary with species (Medlyn et al. 2002). The temperature dependence function of Jmax (Eqn A6) can be over-parameterized. Its entropy term (SJ) and deactivation energy (DJ) often cannot be unambiguously estimated even in recent specifically designed experiments for measuring parameters of e transport-limited photosynthesis (Bernacchi, Pimentel & Long 2003). They are fixed here at commonly used values (Table 4), in order to minimize the number of parameters to be fitted (Harley et al. 1992; Medlyn et al. 2002). The convexity of electron transport to irradiance, θ, has an effect on the estimation of Jmax. It could be determined from curve fitting if irradiance is also varied in measurements. However, such fitting could result in an unrealistic value of θ especially when data points related with varied irradiance are few, or did not yield significant improvements in fit to data points. Therefore, θ was taken here to be the common value 0.7 (De Pury & Farquhar 1997; von Caemmerer 2000) for all the models to ensure consistency. Last, Φ2m is set at the widely used value 0.85 (Bernacchi et al. 2003). All specified parameters or constants are listed in Table 4.

Although dark respiration rate, Rd, can be calculated using an Arrhenius equation (Bernacchi et al. 2001), related parameters for Rd cannot be reliably estimated from gas exchange data of net photosynthesis (Medlyn et al. 2002). Here Rd is estimated based on the Arrhenius equation, Eqn A5, with Rd25 given by 0.0089Vcmax25 (Watanabe, Evans & Chow 1994) and its activation energy as 46390 J mol−1 (Bernacchi et al. 2001). Thus, the parameters left to be estimated by curve fitting are Vcmax25, Jmax25 and EJmax.

Results of curve fitting

For the data sets of rice (Oryza sativa L.) and wheat (Triticum aestivum L.) that came from Makino, Mae & Ohira (1988), measurements were made only at 25 °C (Table 3), the reference temperature at which a temperature response function of photosynthetic parameters has a value of 1. For these two data sets, estimation of EJmax then becomes irrelevant. Table 5 shows the result in using each model for simultaneous fitting to the combined data of all the measured net photosynthesis within a crop. Not surprisingly, estimates of Vcmax25 did not vary much among the models, as all the models used the same expressions and parameters for calculating Rubisco-limited carboxylation rate Vc. The three models yielded a similar good fit. The model fit is visualized here for the new model in describing the rice data set (Fig. 2). The model may not describe an individual ACi curve well, since individual curves were fitted simultaneously to result in overall parameter values that give the best fit to the combined data.

Table 5.  Estimated values (standard errors in parentheses) of Vcmax25, Jmax25, and EJmax when different versions of model for electron transport limited carboxylation were used in curve fitting
(data points)
(µmol m−2 s−1)
(µmol m−2 s−1)
(J mol−1)
  • a

    Model of Eqns 1, 2c and 3 with incorporation of the e transport scenario discussed by Allen (2003), i.e. h = 14/3, fQ = 1, fpseudo = 0 (see text); na, not applicable because measurements were conducted only at the reference temperature, 25 °C (see Table 3).

Rice (29)New model a118.9 (5.3)194.5 (4.4)na0.973
FvCB with Eqn A3a119.1 (6.0)186.5 (4.5)na0.966
FvCB with Eqn A3b118.7 (5.2)226.8 (5.1)na0.973
Wheat (28)New model a224.9 (12.3)498.4 (27.7)na0.965
FvCB with Eqn A3a226.5 (8.9)449.0 (15.1)na0.979
FvCB with Eqn A3b224.6 (12.9)586.2 (35.1)na0.962
Barley (60)New model a182.5 (8.0)334.7 (7.0)31052.5 (2236.2)0.957
FvCB with Eqn A3a163.3 (4.4)331.4 (8.4)26898.1 (3166.8)0.953
FvCB with Eqn A3b182.3 (7.9)388.3 (8.1)32037.5 (2204.4)0.959
Common bean (111)New model a116.6 (2.8)191.1 (3.3)52970.3 (3098.6)0.945
FvCB with Eqn A3a116.6 (2.9)186.4 (3.2)51558.5 (3102.1)0.942
FvCB with Eqn A3b117.5 (2.8)219.2 (3.4)54253.7 (2787.1)0.953
Soybean (532)New model a 94.2 (1.5)214.9 (2.8)94078.3 (1061.4)0.925
FvCB with Eqn A3a 94.3 (1.6)209.2 (2.8)93310.4 (1103.0)0.914
FvCB with Eqn A3b 94.5 (1.5)244.0 (3.2)94377.5 (1087.0)0.922
Figure 2.

Measured (points) and fitted (curves) responses of the net rate of CO2 assimilation to intercellular CO2 concentration in the leaves of rice at three irradiance levels (data of Makino et al. 1988). Curves are drawn based on the new model with parameters in Table 5 (see text).

As the models differ in calculating the e transport-limited carboxylation rate Vj, estimates for Jmax25 and EJmax differed among the models (Table 5). The estimate of Jmax25 for the new model was between those estimated Jmax25 for the two versions of the original FvCB model. Therefore, the ratio of estimated Jmax25 to estimated Vcmax25 differed among the models (Fig. 3). Several studies (e.g. Wullschleger 1993; Leuning 1997, 2002; Medlyn et al. 2002) showed the conservative value of this ratio. Clearly, attention should be paid to the form of e transport equations and the assumed value of α in the FvCB model when citing a reported ratio for reducing the number of parameters to be estimated in modelling photosynthesis.

Figure 3.

Relationships between Jmax and Vcmax at 25 °C, estimated from using three model versions. Points represent fitted values of Jmax and Vcmax at 25 °C for five crops as shown in Table 5.

The dependence of Jmax on temperature is commonly described by the use of a modified Arrhenius equation, Eqn A6, allowing for an optimum response. The parameter EJmax characterizes the rate of exponential increase of the function below the optimum, whereas the parameter DJ describes the rate of decrease of the function above the optimum (Medlyn et al. 2002). If DJ and SJ in the equation are fixed as in our analysis, EJmax is the unique parameter for determining the optimum temperature: a high EJmax resulting in a high optimum temperature. Among the three crops for which EJmax was estimated (Table 5), soybean (Glycine max (L.) Merr.) had the highest, barley (Hordeum vulgare L.) had the lowest, and common bean (Phaseolus vulgaris L.) had an intermediate optimum temperature. Such information reflects crop adaptation to growth environments.


Various non-linear pathways may contribute to the removal of disparity in the required rates of the linear whole-chain e transport in terms of ATP and NADPH consumptions by the metabolism of photosynthetic dark reactions. On the basis of the popular FvCB model and updated insights of Allen's (2003) review, we presented a generalized algorithm for synthesizing the stoichiometries of these contributing mechanisms, which cover the Q-cycle, the cyclic e transport, and the pseudocyclic pathway. First, the relation for any engagement of these pathways for a balanced production of NADPH and ATP, Eqn 1, was presented. Generalized equations for e transport-limited carboxylation rate, Eqn 2c, and for quantum yield of CO2 uptake, Eqn 4, were derived. The involvement of the cyclic e transport is supported by the observation that more pigments are associated with PSI (the excess being in the range 14–20%), and the excess of PSI capacity is used for cyclic e transport (Albertsson 2001). The Q-cycle can increase quantum yields of CO2 uptake (ɛ) substantially, whereas the cyclic e transport increases ɛ only slightly because the effect of this pathway to increase ATP production is partly cancelled out by the fact that part of e at the ferredoxin are not directly coupled for producing NADPH.

The extent of any involvement of these non-linear pathways may depend on environmental conditions. As the pseudocyclic mechanism is energetically the least efficient, and carries hazards inherent in the formation of reactive oxygen species (Mehler 1951; Heber 2002; Ort & Baker 2002), the cyclic photophosphorylation and the Q-cycle are probably the major mechanisms that interplay for balancing the requirements in C3 plants under normal growth conditions. Allen (2003) discussed the general function of a combined cyclic and non-cyclic phosphorylation in order to permit flexibility in the ATP : NADPH ratio according to metabolic demands. We specifically tested his view with our generalized equation for e transport-limited carboxylation rate with no operation of the pseudocyclic flow. This combined cyclic and non-cyclic pathway may be used for modelling photosynthesis of non-stressed plants. Under stressed (e.g. chilling or drought) conditions, there is probably a need for operation of pseudocyclic e transport (Baker, Oxborough & Andrews 1995), whereby O2, as an e sink, not only defrays ATP deficiencies but also plays a crucial photoprotective role, substituting for CO2 in sustaining e flow (Ort & Baker 2002). The merit of our model is perhaps not in using gas exchange measurements to estimate the extent to which these non-linear pathways operate for a particular circumstance; rather, it offers a framework to assess photosynthetic regulatory mechanisms via the electron transport pathways in response to environmental stresses. When the extent of different non-linear pathways is determined (e.g. by advanced chlorophyll fluorescence measurements combined with other techniques, in particular, gas exchange measurements), our model can be used in a very straightforward manner to study the regulation of photosynthesis under various environmental stress conditions.


This work was supported partly by the STW part of the Netherlands Organization for Scientific Research through the PROFETAS programme of Wageningen University.


A biochemical C3 photosynthesis model

The model of Farquhar et al. (1980) and Farquhar & von Caemmerer (1982) describes the net rate of photosynthesis (A) by the minimum of the rate of carboxylation under the limitation of Rubisco activity (Vc) and the rate under the limitation of RuBP regeneration (Vj):


where the term (1 – Γ*/Ci) accounts for CO2 released through photorespiration.

Vc is calculated using a Ci-dependent function, as:


where KmC and KmO are the Michaelis–Menten constant for CO2 and O2, respectively.

The form of equations for Vj depends on whether electron transport limitation is assumed to be associated with insufficient NADPH or with insufficient ATP:


where J is the rate of linear whole-chain e transport.

The dependence of J on irradiance is still described empirically. Although it was originally described by a rectangular hyperbola equation (Farquhar & von Caemmerer 1981, 1982), J is now increasingly estimated as the smallest solution of a non-rectangular hyperbola:


where θ is the convexity of the response curve, and α is quantum efficiency of electron transport. For the linear e transport, the theoretical maximum value for α is 0.5 mol electron per mol photon absorbed (Farquhar et al. 1980), because one quantum must be absorbed by each of the two photosystems to move an electron from the level of H2O to that of NADP+. However, in actual applications (e.g. Harley et al. 1992; Leuning et al. 1995; Medlyn et al. 2002), α has been adjusted to a lower value to agree with a measured quantum yield for CO2 uptake that is often lower than that expected from the theoretical maximum.

Farquhar & von Caemmerer (1981) stressed that there is a failure rate of trapping centre in the photochemical conversion of PSII, which they called ‘biological misses’. This is consistent with the result of more recent chlorophyll fluorescence studies (e.g. Harbinson, Genty & Baker 1989; Genty & Harbinson 1996; Schapendonk et al. 1999) that the observed maximum electron transport efficiencies of PSII is lower than that of PSI, which has a value of 1.0 (no ‘biological misses’ in PSI). To this end, we make the estimation of α in Eqn A4 explicit as Φ2m/(1 + Φ2m) [cf. Eqn C1 in Appendix C], where Φ2m is the maximum e transport efficiency of PSII. Such an estimate for α allows the FvCB model to account for a difference in e transport efficiency between PSI and PSII, which may be preferable to the use of an equal efficiency as implicitly assumed in many existing applications.

The temperature dependence of kinetic properties of Rubisco, involving three parameters (Vcmax, KmC and KmO) in Eqn A2, is described by an Arrhenius function normalized with respect to 25 °C:

Parameter = Parameter25 exp((T − 25)E/[298R(T + 273)])((A5))

where E is the relevant activation energy for Vcmax, or KmC, or KmO, R is the universal gas constant (Table 4).

A modified Arrhenius function is used to describe an optimum response of Jmax to temperature as (De Pury & Farquhar 1997; Medlyn et al. 2002):


where Jmax25 is Jmax at the reference temperature 25 °C, SJ is an entropy term, EJmax and DJ are the energies of activation and de-activation, respectively (Table 4).

The value of Γ* can be related to KmC and KmO, and to Vomax and Vcmax:

Γ* = 0.5O(KmC/KmO)(Vomax/Vcmax)((A7))

where the factor 0.5 is mol CO2 released when Rubisco catalyses the reaction with 1 mol O2 in photorespiration (Farquhar et al. 1980).


Derivation of Equation 1

There are various mechanisms by which chloroplasts may remove the disparity between the requirements for ATP and NADPH (Fig. 1). From the reduced plastoquinone, a fraction (fQ) of e follows the Q-cycle, and a fraction (1 – fQ) is transferred directly to plastocyanin. From reduced ferredoxin, a fraction (fcyc) of e follows the cyclic mode, a fraction (fpseudo) follows the pseudocyclic mode, and the remaining fraction (1 – fcyc − fpseudo), are transferred to NADP+.

Let J2 be the rate of e transport through PSII, and J1 be the rate of e transport through PSI. The rate of cyclic e transport, Jcyc, is fcycJ1. The balance of inflow and outflow of e, both at plastoquinone and at PSI, means that:


where JQ is the rate of e following the Q-cycle and Jcyt is the rate of those going directly to the cytochrome complex and plastocyanin. Rewriting Eqn B1 gives


The rate of e transport being cyclic, Jcyc, is


Assuming that the Q-cycle, once engaged, is impartial for electrons of the whole chain and of the cyclic flow, the value for JQ and Jcyt can be expressed by


The rate of e transport being pseudocyclic, Jpseudo, is


The amount of e transport going to the end acceptor NADP+, JNADP+, is


Assuming that the required number of H+ to produce 1 ATP is h, the expression for ATP production can be derived from the number of H+ produced along various e pathways (Fig. 1):


Substituting Eqns B4a and B4b into Eqn B8a and further simplifying gives:


From Eqn B7 and Eqn B8b, the following equation can be written:


As said in the main text, the dark reaction of photosynthesis utilizes (2 + 4Γ*/Ci)V NADPH and 3 + 7Γ*/Ci)V ATP, where V is the rate of carboxylation. If the production of both NADPH and ATP exactly matches the requirements for their consumption by metabolisms in the dark reaction, then Eqn B9 becomes


Solving for 1 − fcyc − fpseudo from Eqn B10 results in Eqn 1.


Derivation of Equation 3a

The serial connection of the two photosystems in the linear whole-chain e transport as assumed in the FvCB model means that the rate of e transport between PSI and PSII must be equal, i.e. J1 = J2 (say, both have a value jo). If the maximum e transport efficiency of PSI is 1.0 and that of PSII is Φ2m (with Φ2m ≤ 1 to account for possible ‘biological misses’ of PSII), PSI needs jo mol photons whereas PSII needs jo2m mol photons to drive the equal amount of jo mol e flows passing the two photosystems. Thus, for the linear e transport assumption, the efficiency of e transport on the basis of light captured by both PSI (I1) and PSII (I2) is:


If cyclic e transport runs, more quanta of light energy need to be absorbed by PSI (Albertsson 2001). Let x be the fraction of additional light relative to the amount of light for PSI in the absence of cyclic e transport. We assume that the e transport efficiencies of PSI and PSII remain unchanged as those for the whole-chain flow. I1 and J1 in the presence of cyclic e transport would be:


As shown by Eqn B2, electrons energized by PSII only account for part of those energized by PSI if the cyclic e transport runs. Combining Eqn B2 with Eqn C2b gives:


The value of I2 depends on how PSI receives additional light. We assume that this additional light for PSI is achieved by a light ‘state transition’ from PSII to PSI because there is a mobile light-harvesting complex that serves to collect light for either PSI or PSII (Haldrup et al. 2001). In such a case, PSI seems to ‘steal’ some of light that otherwise goes to PSII. Therefore, I2 can be expressed by:


The expression for Φ2m can be written by definition as:


Solving for x gives:


The efficiency of e transport of PSII on the basis of light captured by both PSI and PSII is expressed by definition as:


Substituting Eqns C2a, C3 and C4 into Eqn C7a gives α2, the efficiency of e transport of PSII if cyclic e transport runs:


Substituting Eqn C6 into Eqn C7b gives α2 as expressed by Eqn 3a.

Equation 3a can also be derived without recourse to light ‘state transition’. For example, if the additional light comes from increased capture by PSI without sacrificing PSII's light, Eqns C5, C6 and C7b will become:


Combining Eqns C9 and C10 gives α2 expressed the same as Eqn 3a.

Similarly, Eqn 3a could also be derived by the same procedure but on the basis that more light captured by PSI (relative to PSII) is achieved by a reduced light going to PSII but the same amount of light to PSI as in the linear pathway (derivation not shown). Therefore, the e transport efficiency of PSII on the basis of light captured by both photosystems as a whole is independent of how PSI receives extra light to support a combined whole-chain and cyclic e transport mechanism.