## Introduction

Since its publication in 1980, the model of Farquhar, von Caemmerer and Berry (FvCB; see Table 1 for list of abbreviations) has become the standard in understanding and quantifying the kinetics of carbon fixation by photosynthesis in terrestrial plants (Farquhar *et al*., 1980; von Caemmerer, 2000). For C_{3} plants, the model, in its simplest form, summarizes the dependence of carbon assimilation rate (*A*) on intercellular CO_{2} partial pressure (*C*_{i}) as determined by saturation of Rubisco with respect to carboxylation, electron transport as limited by ribulose bisphosphate (RuBP) regeneration, or triose phosphate export. At any given partial pressure of CO_{2}, *A* is modeled as the smallest value of the corresponding three functions, *A*_{c},* A*_{j}, and* A*_{p}, adjusted for nonphotorespiratory CO_{2} release, *R*_{d} (Fig. 2). In the currently prevailing notation:

Abbreviation | Definition | Units |
---|---|---|

RuBP, ribulose bisphosphate; TPU, triose phosphate utilization.
| ||

A | Assimilation rate | µmol m^{−2} s^{−1} |

A_{c} | Rubisco carboxylation-limited (i.e. RuBP-unsaturated) assimilation rate | µmol m^{−2} s^{−1} |

A_{j} | RuBP regeneration-limited assimilation rate | µmol m^{−2} s^{−1} |

A_{p} | TPU-limited assimilation rate | µmol m^{−2} s^{−1} |

V_{c max} | Maximum rate of Rubisco carboxylation | µmol m^{−2} s^{−1} |

J | Rate of electron transport | µmol e^{−} m^{−2} s^{−1} |

C_{c} | CO_{2} partial pressure at the site of carboxylation | µmol mol^{−1} or µbar |

C_{i} | Intercellular CO_{2} partial pressure | µmol mol^{−1} or µbar |

C_{i tr} | Value of C_{i} at the transition point between A_{c} and A_{j} | µmol mol^{−1} or µbar |

Γ_{*} | Photosynthetic compensation point | µmol mol^{−1} or µbar |

R_{d} | Nonphotorespiratory CO_{2} evolution | µmol m^{−2} s^{−1} |

K_{c} | Michaelis–Menten constant of Rubisco for CO_{2} | µmol mol^{−1} or µbar |

K_{o} | Michaelis–Menten constant of Rubisco for O_{2} | mmol mol^{−1} or mbar |

O | Partial pressure of O_{2} | mmol mol^{−1} or mbar |

T_{p} | Rate of TPU | µmol m^{−2} s^{−1} |

α | Nonreturned fraction of glycolate | µmol m^{−2} s^{−1} |

FvCB model | The Farquhar–von Caemmerer–Berry model of the response of carbon assimilation to CO_{2} concentration |

*A*= min{

*A*

_{c},

*A*

_{j},

*A*

_{p}} (Eqn 1)

with

Sets of measurements can be obtained with relative ease, in which *A* is recorded at varying values of *C*_{i}. In a majority of reports, eight to 12 measurements are taken on one leaf (or one branch), with eight to 12 values of *C*_{i} from 50 to 1500 µmol mol^{−1}, or some narrower range. An example is presented in Fig. 1. We will refer to such a set, consisting of a series of (*C*_{i}, *A*) points, as an ‘*A/C*_{i} set’. In consideration of the great rarity of studies where triose phosphate-limited data have been presented, the discussion will focus on the two-segment model (*A*_{c} and *A*_{j}), with a few applicable notes on the three-segment model.

Segments defined by the functions *A*_{c}, *A*_{j}, and *A*_{p} combine to form the continuous assimilation function *A* if, and only if, they intersect with one another.* A*_{c} and *A*_{j} intersect twice (Fig. 2), and the ordinate or* C*_{i} values of the two intersections are given by solving [*A*_{c} = *A*_{j}] for *C*_{i}. The first intersection is thus at *C*_{i} = Γ_{*} (and *A* = –*R*_{d}), and we define *C*_{i tr} as the second intersection, given by the other solution:

*C*_{i tr}, or *C*_{i} at the transition between *A*_{c} and *A*_{j}, is fully defined by the model, and does not introduce any new information or parameter. By definition of the model, this transition between *A*_{c} and *A*_{j} varies with every *A/C*_{i} set.

*A*_{c} and *A*_{j} are both asymptotic functions of *C*_{i}, with *V*_{c max} and *J* each a direct proportion of the respective horizontal asymptote. The maximum rate of electron transport,* J*_{max}, in turn, is a direct proportion of the asymptote of *J* under increasing light. As long as gas exchange is measured under saturating light, *J* can be assumed equal to *J*_{max}. If light was not saturating at the time of measurement, *J*_{max} must be calculated from *J*, either using the results from fitting a light–response function to data from the same plants, or borrowing a previous parametrization of that function. The coefficients of *C*_{i }and Γ_{*} in the denominator of Eqn 3 are modified according to whether ATP or NADPH shortage is believed to underlie electron transport limitation. The latter assumption is used most often, corresponding to 4*C*_{i}, and 8Γ_{*}.

Finally, the model was in fact reasoned for the response of *A* to *C*_{c}, or varying CO_{2} partial pressure at the chloroplast. *C*_{c} is related to* C*_{i}, *A*, and the conductance of the path from the substomatal cavity to chloroplasts (*g _{i}*) by the simple equation:

*C*

_{c}=

*C*

_{i}– (

*A*/

*g*

_{i})( Eqn 6)

Because *C*_{c} cannot be measured, usage has been to employ *C*_{i}, or intercellular CO_{2}, and in practice the two have thus been assumed to be equal. Problems with this assumption have often been raised, as reviewed in Ethier & Livingston (2004).

The two-segment model includes six parameters: *V*_{c max}, *J*, *R*_{d}, Γ_{*}, *K*_{c}, and *K*_{o}. Only three are usually computed from *A/C*_{i }data: *V*_{c max}, *J*, and *R*_{d}. Values for the others are taken instead from some other source, and ‘plugged in’. It is important to note that values for *V*_{c max}, *J*, and *R*_{d} are not measured, nor are they calculated algebraically from measurements. Instead, they are estimated through fitting the model to sets of measurements. They are not observations, but products of statistical estimation. For clarity, we will adhere to the following terminology: *V*_{c max}, *J*, *R*_{d}, Γ_{*}, *K*_{c}, and *K*_{o}, are all ‘parameters’. When some parameters are not included in the estimation process, and their value is ‘plugged in’, they will be referred to as ‘set’ or ‘fixed’. This is in contrast with the convention of reserving the term ‘parameter’ for coefficients that are the object of estimation, and ‘constant’ for all others. ‘Estimation’ and ‘estimate’ are used in their statistical sense, never as synonyms for ‘approximation’; for every plant part from which an *A/C*_{i} set is collected, there is a true, but unobserved value for the parameters of the model, and the process of fitting the model to those data should yield the best possible estimate of that true value, which remains unobserved.

The validity of the FvCB model has been abundantly corroborated, and it has been incorporated into most crop and ecosystem simulation models. Furthermore, its suitability for describing fundamental responses of photosynthesis is such that *V*_{c max} and *J*_{max} have become valuable metrics of photosynthetic performance in themselves. The task of computing them from gas exchange data is thus carried out primarily to either parametrize predictive models, or for its own sake, in order for example to characterize the effect of some factor of interest on photosynthesis.

Within the chain of procedures through which values for the parameters of the FvCB model are derived, the instruments and techniques used in measuring *A* and *C*_{i} have been the object of continuous optimization over the last 25 yr. Similarly, the methods used to obtain values for fixed parameters have seen important strides. Both aspects are summarized in Long & Bernacchi (2003), and efforts to optimize them continue to this day (see for example Flexas *et al*., 2007). In contrast, the methods used in fitting the model, that is, estimating the parameters of interest, have not received as much attention. Treatment of the statistical methods used in estimating parameters from *A/C*_{i} data has been both exceedingly rare and brief. The aim of this article is to suggest improvements to that link of the chain.

Multiple segments of models such as the FvCB model can be estimated simultaneously, with adjustments to all estimated parameters performed at once, based on the entire data set. All but a very few of the studies that have estimated the model have not been aware of this possibility, and have instead relied on estimating each segment separately. We review established estimation methods for the FvCB model, describe simultaneous estimation of multiple segments, and briefly provide some details of implementation. We also consider some of the properties of the FvCB model that can only be investigated once a method of estimating more than one segment at a time is available. Implementation requires care in avoiding some potential pitfalls, which are reviewed, along with suggested solutions. Programs, written for the SAS System (SAS Institute, Cary, NC, USA), are presented as Supplementary Material Appendix S1, and can be adapted to other software such as r.