The model can be summarized by a couple of equations describing the carbon balance in the mesophyll and in the bundle sheath cells (Fig. 1) based on the simplifying assumption that oxygen concentration in the bundle sheath cells is equal to oxygen concentration in mesophyll cells. This assumption is particularly true for maize, an NADP–ME species, because bundle sheath cells produce little or no O_{2} (Hatch 1987). However, for other applications, O_{2} concentration in the bundle sheath cells could be modelled according to Berry & Farquhar (1978).

In the mesophyll cells, the net carbon assimilation (*A*_{n}) is the carbon fixed to PEP, defined by the rate of PEP carboxylation (*V*_{p}), minus the leakage (*L*) from the bundle sheath cells and minus the carbon released by mitochondrial respiration of the mesophyll cells (*R*_{m}) (see Appendix for a full list of abbreviations and symbols):

In the bundle sheath cells, the net carbon assimilation is the carbon fixed by Rubisco defined by the rate of Rubisco carboxylation (*V*_{c}) minus the carbon released by the photorespiratory cycle (half the rate of Rubisco oxygenation, *V*_{o}) and minus mitochondrial respiration (*R*_{d}):

where *g*_{bs} is the bundle sheath cell conductance to CO_{2}, *C*_{s} and *C*_{m} the CO_{2} partial pressure in the bundle sheath cells and in the mesophyll cells, respectively, and *γ** half of the reciprocal of Rubisco specificity (*S*_{c/o}). The original version of the model is forced with values of light intensity, temperature and CO_{2} partial pressure in the mesophyll cells (*C*_{m}), and it calculates the resulting net photosynthesis (*A*_{n}). *V*_{cmax}, *V*_{pmax} and *J*_{max} are given parameters in the model.

In this model, photosynthesis can either be enzyme limited or electron transport limited. In each case, the rate of PEP (*V*_{p}) and Rubisco (*V*_{c}) carboxylation are described as a function of the maximal rates of Rubisco (*V*_{cmax}) and of PEP (*V*_{pmax}) carboxylation for the former and of the maximal electron transport rate (*J*_{max}) for the latter.

For the inversion model concerning experimental gas measurements, we tried to have conditions of either enzyme-limited or electron transport-limited photosynthesis. By doing so for the calculation of *V*_{cmax} and *V*_{pmax}, we used the enzyme-limited expressions of *V*_{c} and *V*_{p}, whereas for the calculation of *J*_{max} we used the electron transport limited expressions of *V*_{c} and *V*_{p}. The inversion method consists of forcing the model with values of lighting (PPFD), measurement temperature, intercellular air space CO_{2} partial pressure (*C*_{i}) and photosynthesis (*A*_{n}) and calculating *V*_{cmax}, *J*_{max} and *V*_{pmax}.

##### Calculation of *V*_{cmax} and *V*_{pmax}

From the experimental gas measurements, under saturating light conditions, we obtained carbon dioxide assimilation rate (*A*_{n}) as a function of substomatal CO_{2} concentration (*C*_{i}). Because in the von Caemmerer & Furbank (1999) photosynthesis model, calculations are based on CO_{2} concentrations in the mesophyll cells (*C*_{m}), we can add an equation to the two equations (Eqns 3 & 4) described earlier that allows us to calculate the corresponding *C*_{m} for every *C*_{i} measured, assuming a value for the mesophyll cell conductance (*g*_{i}) is given.

- (5)

Considering that the measurements were made in conditions where photosynthesis is enzyme limited, we can use the equations given as follows for *V*_{c} and *V*_{p}.

- (6)

- (7)

where *K*_{c}, *K*_{o} and *K*_{p} are the Michaelis–Menten constants for Rubisco carboxylation, oxygenation and PEP carboxylation, respectively, and O is the oxygen concentration in the mesophyll and bundle sheath cells.

- (8)

- (9)

Considering the three Eqns 5, 8 and 9: *g*_{i}, *g*_{bs}, *R*_{m}, *K*_{p}, *K*_{c}, *K*_{o}, O, *R*_{d} and *γ** are constant parameters at a given temperature (see Table 1 for values at 25 °C); *A*_{n} and *C*_{i} are measured values; and *C*_{m}, *C*_{s}, *V*_{cmax} and *V*_{pmax} are unknowns.

In this system, we have only three equations and four unknowns; there is therefore no unique solution. To overcome this problem, we take advantage of the fact that *V*_{cmax} and *V*_{pmax} are constant for different *C*_{i} and thus different *C*_{s} and *C*_{m}. Taking two sets of experimental points of the *A*_{n} versus *C*_{i} curve (*A*_{n1}; *C*_{i1}) and (*A*_{n2}; *C*_{i2}) and replacing in Eqns 5, 8 and 9, we end up with six equations and six unknowns (*V*_{cmax}, *V*_{pmax}, *C*_{s1}, *C*_{s2}, *C*_{m1} and *C*_{m2}).

Our experiments were based on nine different external CO_{2} concentrations (400, 350, 300, 240, 180, 120, 80, 40 and 0 *µ*mol mol^{−1}). Because a pair of experimental points is needed for every calculation of *V*_{cmax} and *V*_{pmax}, we end up with four independent values for a given temperature. We therefore chose to replace the experimental points by an equation that best fits those points. A fourth-degree polynomial was chosen as a best fit according to the least square methods using Stanford graphics. Figure 2a illustrates an example of experimental points measured at 25 °C together with the fitted polynomial. This polynomial allows us to have values of *A*_{n} for any value of *C*_{i} at chosen intervals. Two couples of (*A*_{n}, *C*_{i}) are then used in two sets of Eqns 5, 8 and 9 to calculate *V*_{cmax}, *V*_{pmax}, *C*_{s1}, *C*_{s2}, *C*_{m1} and *C*_{m2}. With this system of resolution for every experimental *A*_{n} versus *C*_{i} curve at a given temperature, we can get a *V*_{cmax} versus *C*_{i}, *V*_{pmax} versus *C*_{i} and two confounded *C*_{s} versus *C*_{i} curves as illustrated by an example in Fig. 2b–d.

Upon analyzing the *V*_{cmax} and *V*_{pmax} values, we notice that those are not constant especially for small values of *C*_{i}. This can be explained by instability in the model calculation because calculations are done numerically and not analytically. Furthermore, small values of *C*_{i} might hold some errors because of limitations in measurement techniques. As a result, *V*_{cmax} and *V*_{pmax} were taken in the constant part of the *V*_{cmax} versus *C*_{i}, and *V*_{pmax} versus *C*_{i} curves, respectively; that is for values of *C*_{i} limited between 40 and 80 *µ*mol mol^{−1}. Although the parameters were rather constant at high *C*_{i} values, we chose to limit ourselves by an upper boundary because we were not sure to be in saturating light conditions.

##### Calculation of *J*_{max}

For the calculation of *J*_{max}, we consider the measurements made with a constant *C*_{i} and a variable PPFD which correspond to electron transport-limited photosynthesis and saturating CO_{2} partial pressures. We therefore use the electron transport-limited expressions of *V*_{c} and *V*_{p}.

- (10)

- (11)

where *x* is a partitioning factor of electron transport and *J*_{t} is the electron transport rate and is modelled according to Farquhar & Wong (1984) as an empirical non-rectangular hyperbolic function:

- (12)

where *I*_{2} is the total absorbed irradiance and is a function of the incident irradiance *I*, and *θ* is an empirical curvature factor.

- (13)

- (14)

Similar to *V*_{cmax} and *V*_{pmax} calculations, *C*_{m} can be obtained from Eqn 5.

Considering these two equations (Eqns 13 & 14) together with Eqns 12 and 5: *θ*, O, *x* and *γ** are constant parameters at a given temperature (see Table 1 for values); *A*_{n}, *C*_{i} and *I* (PPFD) are measured values (*C*_{i} is a fixed value for all measurements); and *J*_{max}, *C*_{m} and *C*_{s} are unknowns.

In this case, we have a set of three equations and three unknowns. We can therefore, for a given temperature and using one value of *A*_{n} and PPFD, obtain a unique value of *J*_{max}, *C*_{m} and *C*_{s}. It is simple in this case to calculate values of *J*_{max}, *C*_{m} and *C*_{s} for every measurement point of the *A*_{n} versus PPFD experimental curves.