Optimizing the statistical estimation of the parameters of the Farquhar–von Caemmerer–Berry model of photosynthesis

Authors

  • Jean-Jacques B. Dubois,

    1. United States Department of Agriculture, Agricultural Research Service, Plant Science Research Unit, 3127 Ligon Street, Raleigh, NC 27607, USA;
    2. Crop Science Department, North Carolina State University, 3127 Ligon Street, Raleigh, NC 27607, USA,
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  • Edwin L. Fiscus,

    1. United States Department of Agriculture, Agricultural Research Service, Plant Science Research Unit, 3127 Ligon Street, Raleigh, NC 27607, USA;
    2. Crop Science Department, North Carolina State University, 3127 Ligon Street, Raleigh, NC 27607, USA,
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  • Fitzgerald L. Booker,

    1. United States Department of Agriculture, Agricultural Research Service, Plant Science Research Unit, 3127 Ligon Street, Raleigh, NC 27607, USA;
    2. Crop Science Department, North Carolina State University, 3127 Ligon Street, Raleigh, NC 27607, USA,
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  • Michael D. Flowers,

    1. United States Department of Agriculture, Agricultural Research Service, Plant Science Research Unit, 3127 Ligon Street, Raleigh, NC 27607, USA;
    2. Department of Crop and Soil Science, Oregon State University, 109 Crop Science Building, Corvallis, OR 97331-3002, USA;
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  • Chantal D. Reid

    1. United States Department of Agriculture, Agricultural Research Service, Plant Science Research Unit, 3127 Ligon Street, Raleigh, NC 27607, USA;
    2. Department of Biology, Box 90338, Duke University, Durham, NC 27708, USA
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Author for correspondence: Jean-Jacques B. Dubois
Tel: +1 (919) 515 9496
Fax: +1 (919) 515 3593
Email: Jean-Jacques.Dubois@ars.usda.gov

Summary

  • • The model of Farquhar, von Caemmerer and Berry is the standard in relating photosynthetic carbon assimilation and concentration of intercellular CO2. The techniques used in collecting the data from which its parameters are estimated have been the object of extensive optimization, but the statistical aspects of estimation have not received the same attention.
  • • The model segments assimilation into three regions, each modeled by a distinct function. Three parameters of the model, namely the maximum rate of Rubisco carboxylation (Vc max), the rate of electron transport (J), and nonphotorespiratory CO2 evolution (Rd), are customarily estimated from gas exchange data through separate fitting of the component functions corresponding to the first two segments. This disjunct approach is problematic in requiring preliminary arbitrary subsetting of data into sets believed to correspond to each region.
  • • It is shown how multiple segments can be estimated simultaneously, using the entire data set, without predetermination of transitions by the investigator.
  • • Investigation of the number of parameters that can be estimated in the two-segment model suggests that, under some conditions, it is possible to estimate four or even five parameters, but that only Vc max, J, and Rd, have good statistical properties. Practical difficulties and their solutions are reviewed, and software programs are provided.

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 C3 plants, the model, in its simplest form, summarizes the dependence of carbon assimilation rate (A) on intercellular CO2 partial pressure (Ci) 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 CO2, A is modeled as the smallest value of the corresponding three functions, Ac, Aj, and Ap, adjusted for nonphotorespiratory CO2 release, Rd (Fig. 2). In the currently prevailing notation:

Table 1.  Definition of abbreviations
AbbreviationDefinitionUnits
  1. RuBP, ribulose bisphosphate; TPU, triose phosphate utilization.

AAssimilation rateµmol m−2 s−1
AcRubisco carboxylation-limited (i.e. RuBP-unsaturated) assimilation rateµmol m−2 s−1
AjRuBP regeneration-limited assimilation rateµmol m−2 s−1
ApTPU-limited assimilation rateµmol m−2 s−1
Vc maxMaximum rate of Rubisco carboxylationµmol m−2 s−1
JRate of electron transportµmol e m−2 s−1
CcCO2 partial pressure at the site of carboxylationµmol mol−1 or µbar
CiIntercellular CO2 partial pressureµmol mol−1 or µbar
Ci trValue of Ci at the transition point between Ac and Ajµmol mol−1 or µbar
Γ*Photosynthetic compensation pointµmol mol−1 or µbar
RdNonphotorespiratory CO2 evolutionµmol m−2 s−1
KcMichaelis–Menten constant of Rubisco for CO2µmol mol−1 or µbar
KoMichaelis–Menten constant of Rubisco for O2mmol mol−1 or mbar
OPartial pressure of O2mmol mol−1 or mbar
TpRate of TPUµmol m−2 s−1
αNonreturned fraction of glycolateµmol m−2 s−1
FvCB modelThe Farquhar–von Caemmerer–Berry model of the response of carbon assimilation to CO2 concentration 
Figure 2.

Photosynthetic assimilation (A) as a function of intercellular partial pressure of CO2 (Ci). Ac, Aj, and Ap are three assimilation functions (as described in the text and Table 1). The solid line is A, as modeled by the Farquhar, von Caemmerer and Berry (FvCB) model.

A = min{Ac, Aj, Ap} (Eqn 1)

with

image(Eqn 2)
image(Eqn 3)
image(Eqn 4)

Sets of measurements can be obtained with relative ease, in which A is recorded at varying values of Ci. In a majority of reports, eight to 12 measurements are taken on one leaf (or one branch), with eight to 12 values of Ci 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 (CiA) points, as an ‘A/Ci 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 (Ac and Aj), with a few applicable notes on the three-segment model.

Figure 1.

Typical set of A/Ci data (response of photosynthetic assimilation (A) to varying intercellular partial pressure of CO2 (Ci)) for soybean ‘Essex’ (Glycine max‘Essex’).

Segments defined by the functions Ac, Aj, and Ap combine to form the continuous assimilation function A if, and only if, they intersect with one another. Ac and Aj intersect twice (Fig. 2), and the ordinate or Ci values of the two intersections are given by solving [Ac = Aj] for Ci. The first intersection is thus at Ci = Γ* (and A = –Rd), and we define Ci tr as the second intersection, given by the other solution:

image(Eqn 5)

Ci tr, or Ci at the transition between Ac and Aj, is fully defined by the model, and does not introduce any new information or parameter. By definition of the model, this transition between Ac and Aj varies with every A/Ci set.

Ac and Aj are both asymptotic functions of Ci, with Vc max and J each a direct proportion of the respective horizontal asymptote. The maximum rate of electron transport, Jmax, 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 Jmax. If light was not saturating at the time of measurement, Jmax 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 Ci 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 4Ci, and 8Γ*.

Finally, the model was in fact reasoned for the response of A to Cc, or varying CO2 partial pressure at the chloroplast. Cc is related to Ci, A, and the conductance of the path from the substomatal cavity to chloroplasts (gi) by the simple equation:

Cc = Ci – (A/gi)( Eqn 6)

Because Cc cannot be measured, usage has been to employ Ci, or intercellular CO2, 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: Vc max, J, Rd, Γ*, Kc, and Ko. Only three are usually computed from A/Ci data: Vc max, J, and Rd. Values for the others are taken instead from some other source, and ‘plugged in’. It is important to note that values for Vc max, J, and Rd 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: Vc max, J, Rd, Γ*, Kc, and Ko, 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/Ci 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 Vc max and Jmax 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 Ci 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/Ci 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.

Disjunct segments estimation methods

Fitting the FvCB model to gas exchange data is customarily accomplished by fitting each component function separately (disjunct segments estimation). This requires that each A/Ci set be first divided into two subsets, each believed to comprise samples of one segment or the other. Preliminary subsetting has been implemented in several ways. It is necessarily arbitrary, in the sense of being at the discretion of the investigator.

Subsetting is usually subjective: each A/Ci set is examined individually, and a different subsetting point (cut-off) is chosen for each. The subset of observations below the cut-off is fitted using Eqn 2, with Γ*, Kc, and Ko fixed, yielding estimates of Vc max and Rd. The estimated value of Rd is then used as a fixed value in estimating J by fitting Eqn 3 to the subset defined by Ci being greater than the cut-off. It should be noted that the value of Ci tr that can be calculated from the resulting parameter estimates is not necessarily the same as the cut-off value. The A/Ci set in Fig. 1 suggests a discontinuity in overall curvature (change point): the observer must choose, without knowledge of the curve, the two observations between which the set is to be divided. This type of subsetting may be accomplished by visual inspection of the plotted observations or, alternatively, by first fitting only Eqn 2, often linearized for convenience (Long & Bernacchi, 2003), to the entire A/Ci set. Beginning with the highest values, observations are gradually removed from the set, simultaneously monitoring changes in the coefficient of determination r2 with each new fitting of Ac. Upon reaching some chosen value of r2, the observations still remaining are regarded as belonging in the Rubisco-limited range. The estimates of Vc max and Rd obtained from fitting Ac to the subset thus defined are accepted, and Aj is fitted to the other subset, with Rd fixed. This method is not valid and, unfortunately, does not provide the objectivity that is hoped for. Maximization of r2 may be supported in considering parameters for inclusion in a model, but it cannot be used to choose data to be included or excluded from estimation. There is also no objective way to parse the proportion of the observed change in r2 attributable to approach of the true change point. The upward progression of the r2 statistic toward unity with the withdrawal of observations is inherent to its definition, and because the progression is nonmonotonic, setting a critical value for r2 cannot provide an objective criterion.

Arbitrary subsetting can be also be accomplished objectively, as long as every set is subsetted blindly. This is normally accomplished using a single cut-off value of Ci, chosen a priori. Separate fitting of Ac and Aj to the two subsets then proceeds as in the first method. Examples include, but are not limited to, Harley & Tenhunen (1991), Wullschleger (1993), Cai & Dang (2002), Schultz (2003), and Ethier & Livingston (2004). von Caemmerer & Farquhar (1981) have been cited in recommending a single cut-off at approx. 200–250 µbar partial pressure of CO2, but in the original paper it is made clear that this value is applicable only to Phaesolus vulgaris, and is only tentative. Similarly, Wullschleger (1993) only called on his personal subjective experience. Others then referenced these authors’ values.

Whether objective or subjective, arbitrary subsetting of the data is less than optimal for two reasons. First, it does not proceed from the data themselves, and from the fit of the model to them. Being arbitrary, it creates an entry for systematic deviation from the true parameter values. This is far from a trivial consideration, because, as illustrated in Fig. 3, the choice of where to perform the subsetting has a substantial influence on parameter estimates. When performed subjectively, subsetting undermines reproducibility, and provides ready leverage for influencing results. Irrespective of whether this ever actually occurs, and of whether or not it is inadvertent if it does, this is an exceedingly undesirable situation.

Figure 3.

Effect of alternate choices in preliminary subsetting of A/Ci data (see Table 1 for definition of abbreviations). Data for soybean ‘Essex’ (Glycine max‘Essex’) are the same as in Fig. 1. (a) Results from simultaneous estimation. (b–d) Result of changing the cut-off point and performing estimation separately on each subset. Ac was fitted to the lower subset of points, and Aj to the higher. In (b–d), open symbols are used to represent data used to fit Ac, i.e. estimate Vc max and Rd, and closed symbols for data used to fit Aj, i.e. estimate J with Rd taken from the fitting of Ac. Ci tr is computed from Vc max, J, and Rd, after they have been estimated. Resultant estimates and fits are shown. Changing subsets affects all parameter estimates, and Ci tr. Subsetting data between the two observed points that bracket the value of Ci tr obtained from simultaneous estimation (a) leads to the closest results for both methods. Note that Ci tr, the transition point, is different from the cut-off point at which data were subsetted; in (c), Ci tr is between the fifth and sixth data, as in (a) and (b), even though subsetting was performed between the sixth and seventh.

Secondly, another consequence of arbitrary subsetting of the data is that each segment is estimated using only partial data. Instead of all parameters being adjusted simultaneously based on all data, and each other, Vc max and Rd are estimated from one subset, and J from the other. Vc max and J are proportional to the asymptote of the two hyperbolas, Ac and Aj, and estimation of a parameter of an hyperbola is especially problematic when few or no observations are available in the region of the curve over which it has control (Seber & Wild, 1989; Ratkowsky, 1990). As long as data are available in the flatter region of A, the estimate of J is thus inherently less sensitive to misspecification of the cut-off. In contrast, by definition of the model, data are never available in the asymptotic region of Ac. Therefore, the estimate of Vc max from disjunct estimation is necessarily more erratic than that of J, as illustrated by Manter & Kerrigan (2004). Where estimation of Rd is concerned, small differences in the other parameters, and in data, can have large proportional effects, as a result of the high slope of both Ac and Aj at the (Γ*, –Rd) point. By using information from the Rubisco-limited region only to estimate Rd, less accuracy is achieved than is possible. As this estimate of Rd, based on fitting Ac only, is used as a fixed parameter in estimation of J from the RuBP regeneration-limited region, some avoidable bias (in the statistical sense) may be introduced in the estimate of J.

An additional problem is the potential for artifactual inflation of the relatedness of Vc max and Jmax, as a result of the narrowing of the constraints placed on these parameters. The potential for artifactual inflation of the relatedness of Vc max and Jmax stems from the structure of the model itself, and is a particular concern when using the universal cut-off method. The proportionality of Vc max and Jmax has been observed in pools of estimates (Wullschleger, 1993; Leuning, 1997; Wohlfahrt et al., 1999; Medlyn et al., 2002; Onoda et al., 2005). Taking into account the fact that, as long as light is saturating, J = Jmax, the proportionality can be seen as a property of the model itself, by rearranging Eqn 5:

image(Eqn 7)

As defined in Eqn 7, and illustrated for four values of Ci tr in Fig. 4, Vc max and J (Jmax under saturating light) are a function of each other and of Ci tr, for any given values of Γ*, O, Kc, and Ko. At any single value of Ci tr, the relationship of Vc max and J is a line. The space in which (Vc maxJ) points are possible (Fig. 4, shaded areas) is bounded by the range of Ci tr. Any restriction on that range, brought about for instance by estimation methodology, restricts the domain of the relationship between Vc max and J (Jmax under saturating light). The larger the number of A/Ci sets under consideration, and the more diverse their source, the more likely variability in the ratio of Vc max and Jmax is to be underestimated; the less felicitous the cut-off value, the more biased the estimate of that ratio. Underestimation of the variability in the ratio of Vc max and Jmax would be reflected in inflated r2 for the regression of Vc max on J, while a poor choice of cut-off would affect the slope of that regression. The results of Medlyn et al. (2002), having been obtained through simultaneous estimation, are therefore preferable to those of Wullschleger (1993). Interestingly, however, those results suggest that, in practice, this concern may not be warranted. It could be that the biological constraints on Vc max and Jmax are already such that constraining Ci tr only hems in the (Vc maxJ) point in a very small proportion of A/Ci sets.

Figure 4.

The role of Ci tr in defining the relationship of Vc max and J (see Table 1 for definition of abbreviations). 0 < Vc max < 200 µmol m−2 s−1; 0 < J < 575 µmol m−2 s−1; 100 < Ci tr < 1500 µmol mol−1; Kc = 405 µmol mol−1; Ko = 278 µmol mol−1; Γ* = 45 µmol mol−1; O = 210 mmol mol−1. At a single value of Ci tr, the relationship of Vc max and J is a line. For any range of Ci tr, all possible (Vc maxJ) points fall within the shaded area bounded by the corresponding lines.

Finally, when using disjunct segment estimation, we can ensure that the fit of Ac and Aj to the respective subsets chosen to estimate them is optimal. The overall fit of the entire model may also be quantified, but we cannot be sure that it is optimal. More generally, it has not been possible to characterize the estimation behavior of the model, and, in particular, to study how many parameters are in fact estimable.

Simultaneous estimation method

The conditional submodels of the FvCB model (Ac, Aj, and Ap) can all be fitted simultaneously, generating parameter estimates from the entire data set at once. While disjunct approaches never fit more than one segment at a time, simultaneous estimation, sometimes referred to as ‘segmented regression’, provides the best objective joint fit of multiple segments to data (Seber & Wild, 1989; Rawlings et al., 1998). Simultaneous estimation of the FvCB model can be accomplished readily by any statistical software with the capability to perform nonlinear regression, and to process conditional syntax. At least three studies have made use of simultaneous estimation for the FvCB model, but, to our knowledge, no discussion of this method has been published (Dreyer et al., 2001; Medlyn et al., 2002; Nippert et al., 2007; P. Montpied & B. Medlyn, pers. comm.).

Estimation of nonlinear models is accomplished by minimizing an objective function, using a minimization algorithm. Both elements are chosen independently of one another. Regression methods may differ in the objective function they seek to minimize: ordinary least squares and nonlinear ordinary least squares minimize the error sum of squares (SSE), while maximum likelihood, seemingly unrelated regression, and other methods each minimize a distinct corresponding objective function. In nonlinear models, as opposed to linear ones, minima cannot be found analytically. Nonlinear regression therefore uses iterative algorithms such as Gauss–Newton, steepest descent, or Levenberg–Marquardt algorithms. The value of the objective function is returned for successive combinations of values of the parameters being estimated, until it converges to a minimum, where parameter values are accepted (Seber & Wild, 1989; SAS Institute Inc., 2005). As long as the model is continuous, segmented nonlinear models can be estimated through this process in the same way as other nonlinear models. The only additional requirement is that the software have the capability to process the conditional statements that are required, into a form that is tractable to the minimization algorithm. Most current statistical software has this capability, although not all provide diagnostics and other important options. We limit our discussion to the two-segment model (Ac and Aj), but, should data be encountered that are thought to include data from the triose phosphate utilization (TPU)-limited segment (Ap), this approach can be extended to include it, and thus test for its presence.

Programming

Segmentation can be achieved through a variety of programming syntaxes. Segmented models are characterized by the presence of conditional submodels: if the independent variable is smaller than the join point, the response takes one functional form, and another if it is larger. Alternatively, the overall segmented function takes on the value of one function if its value is smaller than the other at a given value of the independent variable, and of the other function otherwise.

Programming examples in the text are generic, and full programs, written for the SAS System, are presented as Appendix S1. Programming statements are denoted by the font Courier New. In the following, ‘rubisco’ stands for Eqn 2, and ‘RuBP’ for Eqn 3. It is assumed that the values of Vc max, J, and Rd are being estimated, and that Kc, Ko, and Γ* are set. Depending on the software used, an explicit expression may or may not be required for Ci tr, the upper join point. The practical consequence of avoiding an explicit expression for Ci tr is that this allows the procedure to modify the set values of Kc, Ko, and Γ* associated with each data point. When fluctuations in temperature were recorded within the course of the A/Ci set, this allows temperature adjustments to the fixed parameters within the procedure. Two forms are possible:

. . .

if rubisco < RuBP then A = rubisco;
else A = RuBP;

. . .

or the equivalent:

. . .

A = min(rubisco, RuBP);

. . .

If software does not permit the adjustment of set parameters within the procedure, Ci tr must first be defined explicitly (using Eqn 5), and the following syntax may then be used instead:

. . .

if Ci < Citr then A = rubisco;
else A = RuBP;

. . .

This requires that the values of fixed parameters be kept constant for all points within a single A/Ci set, for example by averaging temperature over the course of measurements, but all three options produce identical parameter estimates and statistics.

Differences between disjunct and simultaneous estimates

Despite their evident relatedness, the models estimated by disjunct and simultaneous estimation are different, in that the former fits two separate models to two separate data sets, while the latter fits a single model to one data set. Parameter estimates will therefore differ, but the magnitude of the difference is expected to vary widely. Insofar as simultaneous estimation fits the more complete model, and uses more information and statistically sound methods, its estimates are expected to be more reliable. They are also entirely reproducible. However, that is not to say that estimates obtained by the two methods cannot be very close to one another, on a biologically meaningful scale.

The magnitude of the difference in parameter estimates depends on both data and observer, and is thus unpredictable. The following reasoning may help frame the question of comparing methods. The value of Ci tr computed from parameters estimated through simultaneous estimation is the best statistical estimate of the true, but unknown change point between Ac and Aj, or the point of discontinuity in curvature (Seber & Wild, 1989; Rawlings et al., 1998). If an A/Ci set is subsetted between the two observations that bracket the value of Ci tr estimated by simultaneous fitting, and disjunct estimation is used on those two subsets, estimates of Vc max and J obtained by the two methods are expected to produce the closest possible agreement between the methods. However, accurate subjective identification of the change point is often precarious. The presence of distinct segments is often far from evident (Harley & Tenhunen, 1991), and even when it is discernible from the data, the location of the change point generally is not. The magnitude of the difference between methods depends both on data and on how far from Ci tr the observer places the cut-off. Different data and observers will produce varying degrees of agreement between methods, and the more difficult the change point is to identify, the more likely results are to diverge. The data in Figs 1 and 3 are a clear instance of how close the estimates obtained by the two methods can be, if the cut-off for disjunct estimation is chosen close to the estimate of Ci tr obtained by simultaneous estimation. They also illustrate the impact the choice of subsetting point has on estimates derived by disjunct estimation. Our informal experience with this A/Ci set is that, without knowledge of the various fits, some observers place the cut-off point after the fifth observation, some after the sixth, and very few after the seventh. Other sets are more challenging, some less.

It is likely that, for many of the published studies that include estimates of Vc max and Jmax, the change that would result from using simultaneous estimation would be minor, but this can only be assessed by processing each data set concerned. It is also likely that, for at least some published studies, the difference between previous estimates and simultaneous ones would not be inconsequential. However, a survey is outside the scope of this article. More generally, an adequate test of either method would entail applying it to a large number of simulated A/Ci sets. In simulated data, the true value of the parameters is known, as it is controlled when generating the data, as is the nature of the error, or noise, that is added to them. The performance of each estimation procedure is measured by its success in retrieving those true values, under various configurations of both parameters and noise, with more than a thousand data sets for each configuration (D. Dickey, pers. comm.). Applying this approach to disjunct estimation of the FvCB model is probably unfeasible, considering that subsetting would have to be performed on every A/Ci set by several independent observers. In addition, it is uncertain whether enough is known of the correlations among parameters in real gas exchange data to generate an accurate representation of real-world data.

Some statistical properties of the FvCB model

With the ability to estimate the whole model comes the possibility of characterizing some of its estimation properties. We first address limits to the number of estimable parameters, in an attempt to determine how many of the six parameters of the model can be estimated reliably from A/Ci data. We then look for the presence of local minima in the objective function, to establish whether the FvCB model is subject to that most common difficulty in fitting nonlinear models.

Estimable parameters

For linear models with k levels of an independent variable, up to k – 1 parameters can be estimated. This property does not extend to nonlinear models. The number of parameters that can be estimated in nonlinear models, given some number of discrete levels of the independent variable(s), is difficult or impossible to predict analytically. Should an A/Ci set include enough levels of Ci, it is not impossible that all six parameters might become estimable, but gauging how many are, in reality, is difficult. Joint estimation of Γ*, Kc, and Ko along with Vc max, J, and Rd warrants investigation because, as Ethier & Livingston (2004) have pointed out, the suitability of using generic values for those parameters is not settled, and the value of the fixed parameters has a strong impact on estimates of the parameters of interest. We approached the question both empirically and using Hougaard's measure of skewness. Hougaard's measure of skewness, or g1i, is a statistic that may be computed for each parameter, as it is estimated from a specific data set. It changes with every data set, and with the addition of other parameters to the model. It is used to assess the closeness of individual parameters to linearity (Ratkowsky, 1990; Haines et al., 2004): a close-to-linear parameter is one whose statistical reliability is close to that of parameters of linear models. An estimator is very close to linear if | g1i | < 0.1, and reasonably so if 0.1 < | g1i | < 0.15, but strongly skewed if | g1i | > 0.15 (Haines et al., 2004), making inference all but invalid. Its utility is twofold. First, when estimation fails in some proportion of data sets, Hougaard's measure for the successful sets helps to identify the parameter or parameters whose removal is most likely to make estimation possible. Secondly, it provides an indication of the chance that the true value of a parameter may be captured at all.

Two collections of A/Ci sets were processed using simultaneous estimation, with varying numbers and combinations of parameters to be estimated, among Vc max, J, Rd, Γ*, Kc, and Ko. The proportion of A/Ci sets for which estimation failed altogether, and the range of the estimates and their standard error, are presented in Tables 2 and 3. Hougaard's measure of skewness was computed for all estimated parameters in all cases. The criteria for failure of the procedure were failure to converge, and missing statistics for one or more parameter. In those cases, an estimate may have been produced, but the procedure could not compute a standard error and test of significance. Failure of the procedure was almost always traceable to multicollinearity, as a result of excessive correlation among parameters. Estimates, standard errors, and the proportion of A/Ci sets for which skewness was > 0.15 are for successful sets only.

Table 2.  Effect of varying number of estimated parameters on the quality of the estimates, for 231, 8–12 observation A/Ci sets (see Table 1 for definitions of abbreviations)
Parameters estimated% failedestimation% skewness> 0.15, one ormore parameters Vc maxJRdΓ*KcKo
  1. ‘% failed estimation’ is the proportion of A/Ci sets for which estimation failed to converge, or for which some estimates or statistics could not be computed. Estimation did not necessarily fail for the same sets under all combinations of parameters. ‘% skewness > 0.15, one or more parameters’ is the proportion of sets for which the absolute value of Hougaard's measure of skewness exceeded 0.15. ‘Min’ and ‘Max’ are the minimum and maximum estimated values of a parameter, for the sets that did not fail. ‘min SE’and ‘max SE’ are the smallest and largest standard errors for the estimated value of the parameters.

Vc max, J, Rd 0  0Min4253–1   
Max23937214   
Min SE320   
Max SE63449   
% skewness > 0.15000   
Vc max, J, Γ*16 60Min6287 3  
Max256358 68  
Min SE21 1  
Max SE3526 31  
% skewness > 0.15220 49  
Vc max, J, Kc 30100Min3597  104 
Max1016336  3538 
Min SE21  7 
Max SE158714  5764 
% skewness > 0.151000  100 
Vc max, J, Ko51100Min59122   1
Max10591339   107146
Min SE51   26
Max SE26867213   25916666
% skewness > 0.151000   100
Vc max, J, Rd, Γ*40100Min4777–30  
Max28335718112  
Min SE2101  
Max SE646420183  
% skewness > 0.1510114898  
Vc max, J, Rd, Kc28100Min391120 55 
Max621437319 63971 
Min SE131 10 
Max SE4027186815 4166953 
% skewness > 0.15968694 100 
Vc max, J, Rd, Ko67100Min541120  7
Max190937313  20708
Min SE1361  48
Max SE168676815  3714101
% skewness > 0.151009199  100
Table 3.  Effect of varying number of estimated parameters on the quality of the estimates, for 32, 22-to-39-observation A/Ci sets (see Table 1 for definitions of abbreviations)
Parametersestimated% failedestimation% skewness> 0.15, one ormore parameters Vc maxJRdΓ*KcKo
  1. ‘% failed estimation’ is the proportion of A/Ci sets for which estimation failed to converge, or for which some estimates or statistics could not be computed. Estimation did not necessarily fail for the same sets under all combinations of parameters. ‘% skewness > 0.15, one or more parameters’ is the proportion of sets for which the absolute value of Hougaard's measure of skewness exceeded 0.15. ‘Min’ and ‘Max’ are the minimum and maximum estimated values of a parameter, for the sets that did not fail. ‘Min SE’ and ‘Max SE’ are the smallest and largest standard errors for the estimated value of the parameters.

Vc max, J, Rd22  0Min621360   
Max19835711   
Min SE571   
Max SE26304   
% skewness > 0.15000   
Vc max, J, Γ*25 72Min64135 5  
Max199338 49  
Min SE47 5  
Max SE1621 29  
% skewness > 0.1580 65  
Vc max, J, Kc22100Min62164  157 
Max1306326  4293 
Min SE94  38 
Max SE547820  18535 
% skewness > 0.151000  100 
Vc max, J, Ko28100Min62164   12
Max1305326   1652
Min SE114   44
Max SE547714   7797
% skewness > 0.151000   100
Vc max, J, Rd, Γ*63100Min135217–46  
Max1913661093  
Min SE98310  
Max SE20341047  
% skewness > 0.1517058100  
Vc max, J, Rd, Kc28100Min971752 150 
Max128337214 5431 
Min SE10112 68 
Max SE232324910 102204 
% skewness > 0.15100100100 100 
Vc max, J, Rd, Ko38100Min971752  9
Max128437212  10371
Min SE10112  77
Max SE232424910  349731
% skewness > 0.15100100100  100

We first used 231 A/Ci sets, collected in the course of three experiments conducted between 1994 and 2003 at the US Department of Agriculture (USDA)/Agricultural Research Service (ARS) Plant Science Research Unit in Raleigh, North Carolina, on soybean ‘Essex’ (Glycine max‘Essex’) (Reid et al., 1998; Booker et al., 2004) and snap bean (Phaesolus vulgaris) (S331 line) (Flowers et al., in press) grown in various atmospheres and at various temperatures. In all sets, A was recorded at nine to 12 levels of Ci, ranging between 26 and 1350 µmol mol−1. Measurements were collected using three different instruments, and included A/Ci sets recorded from plants under stress, and well into the senescence phase. It is important to note that many of these sets were far from the ideal presented in Fig. 2 and, for example, did not show any evidence of distinct segments under visual examination. The aim of including such sets is to avoid using only data that have been chosen for their conformity to model expectations. However, only sets that supported estimation of at least Vc max, J, and Rd were included. In other words, simultaneous estimation using the SAS program included as Appendix S1 succeeded in identifying two segments in all 231 sets, even though two segments were not always obvious from examining the data. Sets that did not fail, but for which one of the parameters did not pass the significance test (t-test with α = 0.05), are included also. Even though, strictly speaking, nonsignificant parameters should be removed from the model, many plant physiologists consider the function of the FvCB model as a source of values for photosynthesis parameters paramount, and choose to accept them.

Thirty-two A/Ci sets, each with between 22 and 39 levels of Ci, were then created by aggregating 93 sets corresponding to multiple experimental subsamples (multiple leaves on the same plant, recorded at the same time). Besides increased dispersion of the data with respect to A, some of the 32 sets were also far from the ideal presented in Fig. 2. Seven of them failed when only Vc max, J, and Rd were included.

Tables 2 and 3 include results from estimation of seven combinations of either three or four parameters. It is immediately apparent that, although the FvCB model is nonlinear by virtue of its segmentation, Vc max, J, and Rd have the same excellent statistical properties as parameters of linear models. This is only true, however, when they are the only parameters to be estimated. As shown by the proportion of A/Ci sets for which skewness exceeded 0.15 for one or more parameters, and by consideration of the magnitude of standard errors relative to parameters, the introduction of any other parameter creates serious difficulties. From combinations of three parameters, it appears that reliable estimation of Kc and Ko from A/Ci data is always problematic, as is, to a somewhat lesser extent, that of Γ*. Four parameters cannot be estimated reliably, and while estimation succeeded with up to five parameters in some of the 32 sets with increased levels of Ci, adding further parameters can only worsen these problems. The procedure failed with all A/Ci sets when estimation of six parameters was attempted. There was no clear improvement when data included more levels of Ci than customary. This is important, because it strongly suggests that the difficulty in estimating Γ*, Kc, and Ko from A/Ci data is an intractable limitation of the model.

As pointed out by Long & Bernacchi (2003), once all parameters except Vc max, J, and Rd have been fixed, both Ac and Aj are in fact linear functions. Combining them into a segmented model produces a nonlinear model, but it is possible that having the parameter Rd included in both segments improves the overall estimation properties of the joint model.

These results suggest that, given reliable data, and provided that reliable values are available independently for Kc, Ko, and Γ*, very good estimates of Vc max and J can be obtained from gas exchange data through simultaneous estimation. The estimate of Rd produced by simultaneous estimation is expected to be more reliable than that obtained through disjunct estimation, because it is not based only on Ac, and on part of the data. However, its standard error tends to be greater, relative to its estimate, than that of Vc max or J, and its worth remains diminished by the necessity to fix the value of Γ*. (Because Ci = Γ* at A = –Rd, setting Γ* imposes a constraint on Rd.)

Internal conductance gi

Although assessing the reliability of various methods of measuring gi has proved difficult (Warren, 2006), it is nonetheless very likely that in many cases, possibly most, the assumption that Ci = Cc is untenable. Variation in gi has recently been shown to correspond to species, leaf age, and various environmental factors (Niinemets et al., 2005, 2006; Warren & Dreyer, 2006). The effect of a nonnegligible resistance on estimates of the parameters of the FvCB model can easily be quantified: Cc values can first be calculated from Ci, using values of gi that are either hypothetical or measured independently. Parameter estimates can then be obtained and compared using either Cc or Ci as the independent variable. Ethier & Livingston (2004) have done this on a small scale, but they unfortunately used disjunct estimation, with a single, universal cut-off value for subsetting their data.

They also proposed that gi can be estimated (regressionally) from conventional A/Ci data. Beyond some algebraic operations, the method they describe requires arbitrary subsetting, the exclusion of some data altogether, and the estimation of four parameters, in addition to gi. Whether or not ordinary A/Ci data are sufficient to support estimation of gi can be tested by adding Eqn 6 to the FvCB model in order to define Cc from Ci, and fitting the joint model with Cc as the independent variable, instead of Ci. The model thus includes gi as a parameter to be estimated, in addition to Vc max, J, and Rd. This produces reasonable-seeming estimates of gi from many A/Ci sets. Estimation may fail when gi makes no contribution to the model, suggesting that, in those data sets, the assumption that Ci is equal to Cc would be inconsequential. The main difficulty with estimating gi from a single A/Ci set is similar to that affecting estimation of Γ*: both parameters relate to variation in the independent variable. Not surprisingly, introducing gi in the estimation results in values for Hougaard's coefficient of skewness > 0.15 for at least one parameter in all cases. Therefore, while it is quite feasible to estimate gi from a single A/Ci set, careful validation would be required in the context of prediction, irrespective of the regressional method used. Issues regarding the estimation of gi require more extensive treatment, and, as indicated in ‘Extensions of simultaneous estimation’, simultaneous estimation may be a preliminary step in achieving better estimation of gi.

Local minima and flat regions in the objective function

The objective function of nonlinear models is sometimes marked by the presence of multiple minima, only one of which is the global one, and corresponds to the authentic best estimate of the parameters. Minimization algorithms require that starting values be supplied, from which they will efficiently converge to a minimum. Whenever a local minimum exists in the objective function, and a single combination of starting parameter values is given to the minimization algorithm such that the initial evaluation is closer to the local minimum than to the global one, the likelihood of convergence to the local one is very high (Seber & Wild, 1989; SAS Institute Inc, 2005). Examination of SSE for matrices of fixed values of subsets of the parameters showed the frequent presence of multiple local minima for the FvCB model. Some choices of starting values are therefore very likely to lead to incorrect estimates.

Another difficulty arises when the objective function presents unbounded flat areas, where changing the value of a parameter has no effect, or very little, on the objective function. In that case, the parameter should be eliminated. In the two-segment FvCB model, this does not occur when Vc max, J, and Rd are the only parameters estimated, but gi is affected by this problem in some data sets.

Estimation problems and solutions

Local minima

The difficulty posed by local minima is readily overcome by choosing starting values for the parameters through a preliminary high-density grid search, and it is imperative that one be used. In a grid search, several starting values are supplied for one or all parameters to be estimated, and an SSE is computed using every combination of those starting values, but without minimization. The combination of starting values that yields the smallest SSE is then used to start the minimization and estimate the model. Use of high grid density, and wide ranges of values in constructing the grid, ensures that the starting combination will be in sufficiently close proximity to the global minimum to avoid convergence to a local one. The programs included as Appendix S1 incorporate a preliminary grid search.

Biologically implausible estimates

Under the assumption that the FvCB model constitutes a good representation of the response of A to varying CO2, grossly implausible estimates should prompt a close review of the particular data, and of the suitability of the model. If problematic parameter estimates can be held as accidents of normal data variability, as with occasional small negative values of Rd, and do not affect many replicate A/Ci sets in any systematic manner, they should be allowed to stand, lest some bias be introduced. It is important to note that this only applies to contexts in which a large number of A/Ci sets are being used, with the objective of analyzing the effect of some factor of interest on photosynthesis. It does not apply when there are few sets, and the objective is to parametrize a predictive model.

It is possible to prohibit the estimation procedure from reaching impossible values by constraining the range of a parameter (without necessarily setting it to a unique value), but great caution must be exercised in adding ad hoc bounds to estimation procedures. Often, the procedure simply converges to the bound value, doing little more than setting the parameter at that value, while obscuring the fact that the parameter is indeed being set. Wide bounds may have some utility in keeping the minimization from drifting into nonsensical ranges, but, for A/Ci sets that do result in a bound being reached, the suitability of the segmented model must be reconsidered, and the quality of the data closely reviewed.

Single-phase data

Regardless of what segments were sampled when measuring A, the procedure will attempt to estimate as many segments as have been requested. If the data being fitted comprise samples of only one segment, the procedure will use one of the component functions to model A over the entire recorded range of Ci, while extending the other segment solely to the extrapolated range, resulting in meaningless estimates for that function or functions. In the process, Ci tr is placed at one extreme of the range of Ci values actually recorded, or even well outside of it. Depending on the software used, confidence intervals may also be missing for the meaningless parameter(s). The procedure may also fail to converge. This problem can be readily identified by examining Ci tr, and confidence intervals for the parameters. A plot of the data and the predicted curve will also show that the transition point is at the edge, or even outside of the recorded data, and that at least one entire segment does not overlap with any observation. In these cases, a model with fewer segments is obviously more appropriate, and should be used. Data that support estimation of the Ap segment are rare, and adding it to the model will usually lead to this problem, but fitting two segments to data that only support one should also be avoided.

In very rare circumstances, data may include samples of both segments, but random errors result in an A/Ci set for which one of the two functions alone happens to provide a better fit than the two functions combined, despite the underlying presence of two phases. In those rare A/Ci sets, the same symptoms may appear, including extreme values of Ci tr. The presence of this condition can be judged by consideration of A/Ci sets corresponding to closely related experimental units, such as replicates, recorded closely in time. If the values of A in those related sets are similar, and if fitting both segments to them proceeds normally, then it is likely that the isolated problem set does comprise two phases. In these circumstances, constraining Ci tr, for that single set only, becomes justifiable. Values for the constraint are best obtained from the unconstrained fitting of the model to the related sets. Note that, when Ci tr is constrained, parameters are still estimated from the entirety of the data.

Multicollinearity and excessive correlation between parameters

For some data, any number of different combinations of parameter values all result in optimal fits, because of multicollinearity. Some data may also involve correlation among parameters that exceeds the tolerance of the procedure. Either type of data may result in failure to converge, grossly outsized standard errors, singularity in the Hessian, or estimates identified as biased, all indicative of failure of the procedure. Various statistical software may report different errors for the same underlying problem, which can be viewed as a special case of overparametrization. Correlation between parameters is largely inherent to the model, as reviewed in ‘Disjunct segments estimation methods,’ and can be examined by outputting the correlation matrix, or collinearity diagnostics as available. The Levenberg–Marquardt minimization algorithm is reported to have greater tolerance of excessive correlation than other algorithms (Schabenberger et al., 1999; SAS Institute Inc., 2005; D. Dickey, pers. comm.), but reparametrization is generally recommended. In view of the value of the FvCB model as a standard, this option may not be desirable. Assuming that suitable starting values for the estimation have been supplied, the addition of a single mild constraint, such as a constraint on Ci tr, is sufficient to resolve it, and should be preferred.

Extensions of simultaneous estimation

The method outlined herein can be extended to estimating models that require more information than a single A/Ci set. The objectives of including additional information include: modeling of environmental effects on photosynthesis, improving estimates of Vc max, J, and Rd, and estimating more than three parameters.

For example, knowing that theory suggests that values of Γ* and Rd at the intersection point (Γ*, –Rd) remain invariant with variable irradiance, we can speculate that separate A/Ci sets obtained at different irradiances might support simultaneous estimation of the various corresponding Vc max and J, and common Γ* and Rd. Preliminary results indicate that estimation of Γ* and gi may be strengthened too. In a further extension, estimation of the FvCB model and of an accepted empirical model of the dependence of A on irradiance may be combined into estimation of a single, higher dimensional model. Finally, simultaneous estimation may be integrated into systems of regression equations, in which estimation is conducted from separate data that do not share variables, but whose models share at least one parameter. A prominent example in photosynthesis research is provided by gas exchange and chlorophyll fluorescence data, both of which yield estimates of J. Systems of regression equations may yield improved estimates of parameters of both models based on gas exchange and models based on chlorophyll fluorescence data.

Conclusion

Segmented regression, which is easily implemented using statistical software that processes conditional syntax, obviates the need for arbitrary determination of transition points and subsetting of the data before analysis, and thus removes bias while improving accuracy and precision. The method lends itself to rapid fitting of the FvCB model to large numbers of A/Ci sets, and dramatically improves the efficiency of processing gas exchange data. Estimates of Vc max, J, and Rd have excellent statistical properties when they are the only parameters to be estimated, but estimation of more than these three parameters from a single, 8–12 observation A/Ci set may be difficult. Because of the prevalence of local minima, implementations that identify starting values for minimization in proximity to the global minimum are indispensable. Some specific difficulties in estimation can be resolved by placing bounds on parameters, or by the carefully considered application of other mild constraints. The methods described may be extended to fitting higher dimensional models, and to systems of regression equations.

Acknowledgements

We thank David Dickey (Department of Statistics, NCSU), Pierre Montpied (Unité Mixte de Recherche INRA-UHP 1137), Kevin Tu (Department of Integrative Biology, UCB), and Patrick B. Morgan (Li-Cor Biosciences, Inc.), for repeated discussions and comments that contributed to this paper in its present form.

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