# Parameter Estimation of the Farquhar—von Caemmerer—Berry Biochemical Model from Photosynthetic Carbon Dioxide Response Curves

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## Abstract

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_{2}assimilation at various spatial scales from leaf to global, has been used to assess the impacts of climate change on crop and ecosystem productivities. However, it is widely known that the parameters in the FvCB model are difficult to accurately estimate. The objective of this study was to assess the methods of Sharkey et al. and Gu et al., which are often used to estimate the parameters of the FvCB model. We generated A

_{n}/C

_{i}datasets with different data accuracies, numbers of data points, and data point distributions. The results showed that neither method accurately estimated the parameters; however, Gu et al.’s approach provided slightly better estimates. Using Gu et al.’s approach and datasets with measurement errors and the same accuracy as a typical open gas exchange system (i.e., Li-6400), the majority of the estimated parameters—V

_{cmax}(maximal Rubisco carboxylation rate), K

_{co}(effective Michaelis-Menten coefficient for CO

_{2}), g

_{m}(internal (mesophyll) conductance to CO

_{2}transport) and Γ* (chloroplastic CO

_{2}photocompensation point)—were underestimated, while the majority of R

_{d}(day respiration) and α (the non-returned fraction of the glycolate carbon recycled in the photorespiratory cycle) were overestimated. The distributions of T

_{p}(the rate of triose phosphate export from the chloroplast) were evenly dispersed around the 1:1 line using both approaches. This study revealed that a high accuracy of leaf gas exchange measurements and sufficient data points are required to correctly estimate the parameters for the biochemical model. The accurate estimation of these parameters can contribute to the enhancement of food security under climate change through accurate predictions of crop and ecosystem productivities. A further study is recommended to address the question of how the measurement accuracies can be improved.

## 1. Introduction

_{3}plants [1,2] is fundamental for the prediction of leaf responses to environmental variation [3]. This model has been widely used to simulate CO

_{2}assimilation and the response of plants to climate change for different spatiotemporal scales [4,5,6,7,8,9,10,11], due to its solid theoretical basis and simplicity [12]. It is also frequently used in reverse to quantify the underlying biochemical properties (i.e., the model parameters) of leaves under different environmental conditions [13,14,15,16,17]. These parameters are often considered easier to estimate from gas exchange measurements rather than making the required in vitro measurements to quantify enzyme activity. This is because it is difficult to extract functional enzymes from many species [18], and in vitro conditions seldom represent those experienced in vivo [19]. According to the different versions of FvCB model [1,20,21,22,23], up to 8 parameters (V

_{cmax}, K

_{co}, J, T

_{p}, α, g

_{i}, R

_{d}, and Γ*) can be estimated from an analysis of the response of the net assimilation rate (A

_{n}) to intercellular CO

_{2}concentration (C

_{i}) if enough accurate data points are available [2].

_{n}/C

_{i}curves [2,12,24,25,26,27,28,29]. Each method relies on different assumptions and has technical limitations. Most approaches assume that α = 0 and that the K

_{co}and Γ* can be chosen a priori from estimates in previous studies to determinate V

_{cmax}, J

_{max}, T

_{p}, R

_{d}and g

_{m}[2,12,22,26,27]. Gu et al. [2] extended the method of Ethier and Livingston [12] to propose an exhaustive dual optimization (EDO) approach to estimate the parameters from fitting A

_{n}/C

_{i}curves. All of the curve-fitting methods minimize an objective function; e.g., the sum of the square of errors (SSE), based on the nonlinear FvCB model, with a limited number of measurements (typically 8–12, [2,24,26,30]) and the expected accuracy of a measured A

_{n}/C

_{i}curve (see [2,25,26] for comprehensive reviews). Depending on the equations used for fitting the parameters, two major groups of methods can be distinguished: Group I directly fits parameters with the FvCB model [14,25,26,27,31] and Group II fits parameters with a quadratic equation [2,12]. The implementation is sensitive to the methods used. The estimated parameters can be substantially different when using different A

_{n}/C

_{i}curve-fitting methods on the same dataset [2,25,26,29]. It is frequently difficult to determine which method is superior based on the measures of SSE because of the characteristics of the FvCB model [2], the assumption that the assigned values of the kinetic properties of Rubisco are the correct ones, and the number of parameters to be estimated [24].

_{n}/C

_{i}datasets from 4 shrubby indicator species, including highbush blueberry (Vaccinium corymbosum L.), dangleberry (Gaylusaccia frondosa L.), coastal fetterbush (Eubotrys racemosa L.) and sweet pepperbush (Clethra alnifolia L.). They concluded that the method developed by Sharkey et al. [27] was among the ‘best’, based on the lowest minimum of the root mean square error. Gu et al. [2] stated that their approach could estimate reliable FvCB parameters using error-free synthetic A

_{n}/C

_{i}curves and predicted limited states that matched chlorophyll fluorescence patterns from actual datasets.

## 2. Materials and Methods

#### 2.1. The FvCB Model and Characteristics

_{n}/C

_{i}curves are fitted with the FvCB model for C

_{3}leaves [1] that accounts for g

_{m}and whereby A

_{n}is given as

_{c}, W

_{j}and W

_{p}are the carboxylation rates limited by Rubisco (A

_{c}state), the Ribulose 1,5-bisphosphate (RuBP) regeneration (A

_{j}state) and triose phosphate utilization (TPU) (A

_{p}state), respectively; Equation (3) is one of the points where the A

_{j}state is equal to the A

_{n}state (see next the section for an explanation); C

_{c}is the chloroplastic CO

_{2}partial pressure and can be estimated by

_{n}/C

_{i}curves at saturating light levels is to assume that J approaches J

_{max}, the maximum rate of electron transport. If light is not saturating at the time of measurement, J

_{max}must be calculated from J [24]. In this study, we assume that J

_{max}= J.

_{n}and C

_{i}in the A

_{c}, A

_{j}and A

_{p}states can be expressed as three segment hyperbolic curves [2,12], respectively;

_{c}, A

_{j}and A

_{p}states, respectively; g

_{m}, p, q and u are 4 ‘coefficients’ in each segment of an A

_{n}/C

_{i}curve. The general form of Equation (9) is

_{c}state,

_{j}state,

_{p}state,

_{p}state is reduced to

_{n}as a function, C

_{i}[12]. Equation (16) consists of up to three segments (states). The three states share one common coefficient g

_{m}, while the other coefficients (p, q and u) are combinations of the common parameters g

_{m}, Γ* and R

_{d}and the state specific parameters V

_{cmax}and K

_{co}in the A

_{c}state, J

_{max}in the A

_{j}state, and T

_{p}and α in the A

_{p}state.

#### 2.2. Data Generation

#### 2.2.1. Constraints for the Parameters

_{c}and A

_{j}states and a monotonically decreasing function when α > 0, or it is a constant in the A

_{p}state when α = 0. The three states follow a fixed order along the C

_{i}axis [2].

_{n}/C

_{c}curve consists of both A

_{c}and A

_{j}states, there could, mathematically, be two conditions: (1) The two states could be exactly the same if J

_{max}/4 = V

_{cmax}and K

_{co}= 2Γ*. Since K

_{co}> 2Γ* [2,22], however, the two states cannot be the same. (2) There are two intersection points where A

_{nc}= A

_{nj}. The first point is at C

_{c}= Γ*; A

_{nj}< A

_{nc}when C

_{c}< Γ*. We define A

_{n}= A

_{nc}when C

_{c}< Γ* to create the fixed order of A

_{c}and A

_{j}states (Equation (3)). The second point is defined as the transition point C

_{cc_cj}by combining Equations (4) and (5),

_{co}< 1, the constraints for V

_{cmax}and J

_{max}can be expressed as:

_{j}and A

_{p}states coexist, the constraint for J

_{max}as a function of T

_{p}is [2]:

_{c}and A

_{p}states coexist, the constraint for V

_{cmax}as a function of T

_{p}is

#### 2.2.2. Criteria for Deriving Parameters

_{n}/C

_{i}dataset is three or higher, or at the lowest precision of the A

_{n}/C

_{i}dataset if the precision of A

_{n}/C

_{i}is less than three decimal places. Since there are limitations due to measurement accuracy, the number of data points of the A

_{n}/C

_{i}dataset, and the fitting methods used, an estimated parameter might be equal to its “true value” by chance. To stop this result from biasing our analysis, a correctly estimated parameter is defined here as one that is obtained only if all resolvable parameters in this dataset are equal to their “true values”. If the numbers of data points in A

_{c}, A

_{j}and A

_{p}states are x, y and z, respectively, the data point distribution is written as (x, y, z). More detailed information on the theoretical resolvability of parameters can be found in Gu et al. [2].

_{ci}, A

_{jj}and A

_{pk}are the calculated net assimilation rates at point i in the A

_{c}, A

_{j}, A

_{p}states respectively; A

_{cmi}, A

_{jmj}and A

_{pmk}are the measured counterparts, respectively; and subscripts nc, nj and np are the numbers of counterpart points, respectively. The objective equation used in the methods of Sharkey et al. and Gu et al. are similar to Equation (22); however, the calculation procedures are different.

#### 2.2.3. Generation of Datasets

_{max}varied from 20.000 to 160.000 μmol m

^{−2}s

^{−1}, J

_{max}from 20.000 to 250.000 μmol m

^{−2}s

^{−1}, T

_{p}from 5.000 to 15.500 μmol m

^{−2}s

^{−1}, R

_{d}from 0.010 to 5.000 μmol m

^{−2}s

^{−1}, g

_{m}from 0.100 to 30.000 μmol m

^{−2}s

^{−1}Pa

^{−1}, Γ* from 0.100 to 5.000 Pa, K

_{co}from 20.000 to 100.000 Pa, and α from 0.001 to 1.000. As a special case, α = 0, with the requirement of the constraints mentioned in the previous section (Equation (21)). C

_{i}ranges from 5 to 150 Pa in both cases. We assumed that the datasets were collected at a leaf temperature of 25 °C and an air pressure of 100 Pa. Each set of parameters (except for α) was used to generate datasets with either α = 0 or α > 0. There were 200 datasets within each accuracy level, of which 100 used α = 0 and 100 with α > 0.

_{n}and C

_{i}to eight decimal places. Secondly, a normal accuracy dataset was defined as a dataset which is rounded off from a high accuracy dataset. Normal precision is the same as a typical open gas exchange system, e.g., Li-6400 (Li-Cor, Inc., Lincoln, NE, USA). In this case, A

_{n}is rounded to three decimal places if A

_{n}< 1.000 μmol m

^{−2}s

^{−1}, to two decimal places if 1.00 μmol s

^{−1}m

^{−1}< A

_{n}< 10.00 μmol s

^{−1}m

^{−1}, and to one decimal place if A

_{n}> 10.0 μmol m

^{−2}s

^{−1}; C

_{i}is rounded to one decimal place if C

_{i}< 100 μmol mol

^{−1}and to an integer if C

_{i}> 100 μmol mol

^{−1}. Datasets can commonly have measurement errors. The errors in A

_{n}and C

_{i}were calculated according to Equations (A5) and (A6), respectively, in the appendix. The precision of a dataset from the measurements is the same as a normal accuracy dataset. Varied accuracy datasets were generated without measurement errors, either with varied accuracy or varied data points. The parameters used to generate varied accuracy datasets were: V

_{cmax}of 45.572 μmol m

^{−2}s

^{−1}, J

_{max}of 111.315 μmol m

^{−2}s

^{−1}, T

_{p}of 7.871 μmol m

^{−2}s

^{−1}, R

_{d}of 1.381 μmol m

^{−2}s

^{−1}, g

_{m}of 1.897 μmol m

^{−2}s

^{−1}Pa

^{−1}, Γ* of 4.396 Pa, K

_{co}of 43.616 Pa, and α of 0.352. The maximum number of data points in each state was nine. There were two subgroups in this group:

- (i)
- Datasets with varied data points. The accuracy of this dataset was eight decimal places; and the numbers of data points were 4, 5, 6, 7, 8, 9, and 12. The varied data point dataset was used to evaluate the impact of the number of data points on parameterization. It should be noted that these datasets are included in the high accuracy dataset.
- (ii)
- Datasets with varied accuracy. These datasets were with either eight or 12 data points, and accuracies were from one to eight decimal places. The varied accuracy dataset was used to identify the impact of accuracy on parameterization.

#### 2.3. Fitting Methods

#### 2.3.1. Gu et al.’s Method

- (1)
- The enumeration of all possible data point distributions of three states of a given dataset. The three limited states must follow a certain pattern along the C
_{i}axis in an order dictated by the FvCB model. The minimum numbers of data points (3 or 0, 2 or 0, 3 or 0) and the number of data points higher than the number of parameters to be derived are required for resolvable parameters. Under these conditions, the resolvable parameters are defined as Gu et al.’s resolvable parameters to differentiate them from resolvable parameters as a general case. Thus, the minimum numbers of data points (3, 2, 3) and a minimum number of nine observed data points are required for all eight parameters to be resolvable. We refer to these as Gu et al.’s requirements for all eight resolvable parameters. If only one or two states are Gu et al.’s resolvable states, the dataset is Gu et al.’s partially resolvable dataset and the resolvable parameters are Gu et al.’s partially resolvable parameters. If a state does not meet the minimum data point requirement of Gu et al., Gu et al.’s method forces the dataset to meet the requirements by moving data points from one state to another. If the number of observed data points is zero in the A_{p}state, α = 0 and T_{p}= (asymptote of A_{n}+ R_{d})/three. - (2)
- Fitting the FvCB model to each limited state distribution separately. In this step, the transition points are never calculated and the carboxylation rates in different states are never compared. The A
_{n}is calculated with the submodel of the limited state to which the data point is assigned. - (3)
- Detection and correction of inadmissible fits. Gu et al., [2] defined “inadmissible fits” as cases where the limitation states of the points in the A/C
_{i}curve have not consistently or correctly identified the derived parameters. This step is only used for a dataset that contains multiple limited states. If the calculated limited state distribution is the same as the assigned limited state distribution, then the fit is admissible; otherwise, the fit is inadmissible. If the fit is inadmissible, the fit will be corrected via a penalization strategy. - (4)
- Section of best fit. The best fit for an observed set of data is the method that gives the smallest value for the minimized objective function. If the values of the minimized objective function are equal when comparing across different limited state distributions, the one with fewer parameters is selected.

#### 2.3.2. Sharkey et al.’s Method

_{c}and A

_{j}states and A

_{j}and A

_{p}states; it then changes the values until the objective function is minimized. Since the parameters K

_{co}, α and Γ* are assigned a priori, a maximum of five parameters are estimated. The three states share common parameters—g

_{m}and R

_{d}—with state specific parameters of V

_{cmax}in the A

_{c}state, J

_{max}in the A

_{j}state, and T

_{p}in the A

_{p}state. Thus, the minimum data point distribution is (1, 1, 1) and the minimum number of points is five for all five parameters to be resolvable. If the data points are only distributed in one state or in two states, the minimum number of data points is three or four, respectively, for all involved parameters to be resolvable. It should be noted that, for the same dataset, the number of Gu et al.’s resolvable parameters is different from that of Sharkey et al.’s, and both are different from the resolvable parameters in the general case.

#### 2.3.3. Parameter Calculations

_{n}/C

_{i}curve fitting utility version 1.1) was used to test Sharkey et al.’s method.

## 3. Results

#### 3.1. High Accuracy Dataset

_{p}and α were non-involved parameters in this dataset, where the missing A

_{p}state was incorrectly estimated because T

_{p}was calculated by fitting the A

_{n}/C

_{i}curve with a sigmoid function and fixed α = 0. For example, one dataset (0, 15, 0) was identified as (5, 10, 0), and the non-involved parameters V

_{cmax}, K

_{co}, T

_{p}and α were incorrectly derived.

_{cmax}, J

_{max}and T

_{p}, when α > 0, the numbers of estimated parameters were 91, 74 and 100; the numbers of resolvable parameters were 71, 72 and 61; and the numbers of correctly estimated parameters were 52, 49, and 42, respectively (Table 1); when α = 0, the total numbers parameters were 86, 92 and 100 for estimated, 44, 62 and 45 for resolvable, and 37, 46 and 30 for correctly estimated parameters, respectively (Table 1). Obviously, g

_{m}was a resolvable parameter in any dataset, but could not always be correctly estimated.

_{cmax}, K

_{co}, J

_{max}, g

_{m}, T

_{p}and Γ* ranged within ±10% of error (Table 1). Some estimated values of g

_{m}could be very large; up to 100,000 μmol m

^{−2}s

^{−1}Pa

^{−1}(Figure 1E). The estimated R

_{d}values were more variable in comparison with other parameters, and many of the values were larger than the upper limit of 5.000 μmol m

^{−2}s

^{−1}used to generate datasets (Figure 1D). Most estimates of α were zero or very close to their “true values” when α > 0 (Figure 1H). The incorrectly estimated values of V

_{cmax}, K

_{co}, J

_{max}, Γ* and α showed a tendency to be underestimated, while R

_{d}tended to be overestimated. About half of the incorrectly estimated g

_{m}values (except for extreme values) were overestimated, while T

_{p}was evenly distributed around the 1:1 line (y = 1.002x, R

^{2}= 0.941). The uneven distributions of the estimated parameters V

_{cmax}, K

_{co}, Γ*, R

_{d}and α around the 1:1 line implied that the averages of these parameters may not be close to their “true values”.

_{cmax}, J

_{max}, g

_{m}, T

_{p}and R

_{d}—could be estimated for any dataset, and even for some noninvolved parameters in some datasets (Table 1), since Sharkey et al.’s method incorrectly forced data points into a missing state to minimize the objective function. Within ±10% of error ranges, the numbers for V

_{cmax}, J

_{max}, g

_{m}, T

_{p}and R

_{d}were 22, 65, 41, 7 and 6, respectively, when α > 0, and were 23, 64, 63, 8, and 5 when α = 0, respectively. V

_{cmax}and J

_{max}had extreme values in some datasets. Many values of g

_{m}reached their upper limit value of 30 μmol m

^{−2}s

^{−1}Pa

^{−1}. Overall, V

_{cmax}, and g

_{m}were overestimated, about 40% of the R

_{d}values were zero, J

_{max}was about evenly distributed around the 1:1 line (y = 1.016x, R

^{2}= 0.462) with a few extreme large values, and T

_{p}was about evenly distributed around the 1:1 line (y = 0.987x, R

^{2}= 0.866) (Figure 1).

#### 3.2. Normal Accuracy Datasets

_{n}/C

_{i}data was important for correctly determining the parameters. The total number of estimated parameters was more than the total number of resolvable parameters (except for g

_{m}) (Table 1). The number of estimated parameters was more than those utilizing high accuracy datasets, since more non-involved parameters were incorrectly estimated. Compared to the high accuracy datasets, more datasets with a missing state were incorrectly identified as datasets with all three states available by changing a missing state to an available one. For example, a dataset (3, 0, 12) was assigned to (4, 2, 9). The total number of values within ±10% of their “true values” for V

_{cmax}, K

_{co}, J

_{max}, T

_{p}, g

_{m}, R

_{d}, Γ* and α were 55, 40, 65, 70, 26, 16, 48 and 26 for α > 0; and were 43, 27, 64, 67, 20, 12, 42 and 99 for α = 0, respectively (Table 1). These numbers were fewer than the corresponding parameters in the high accuracy dataset. Overall, V

_{cmax}, K

_{co}, was underestimated, R

_{d}was overestimated and g

_{m}was generally underestimated. In some cases, g

_{m}was overestimated when the estimated value was higher than 20 μmol m

^{−2}s

^{−1}Pa

^{−1}. Most of the estimated parameters Γ*, J

_{max}and T

_{p}were evenly distributed around the 1:1 line (y = 0.952x and R

^{2}= 0.531 for J

_{max}, y = 1.001x and R

^{2}= 0.958 for T

_{p}) (Figure 2).

_{cmax}, J

_{max}, g

_{m}, T

_{p}and R

_{d}—were estimated for any dataset (Table 1) within ±10% of error. The values of V

_{cmax}, J

_{max}, g

_{m}, T

_{p}and R

_{d}were 24, 70, 6, 57 and 6, respectively, when α > 0; and 22, 59, 8, 51, and 8, respectively, when α = 0. V

_{cmax}and J

_{max}had extreme values in some datasets. Many values of g

_{m}reached their upper constrained value of 30 μmol m

^{−2}s

^{−1}Pa

^{−1}for both α = 0 and α > 0. Overall, the parameters V

_{cmax}and g

_{m}were underestimated, J

_{max}and T

_{p}were evenly distributed around the 1:1 line (y = 0.991x and R

^{2}= 0.572 for J

_{max}, y = 0.976x and R

^{2}= 0.800 for T

_{p}), and many values of R

_{d}were over their upper limit of 5.000 μmol m

^{−2}s

^{−1}when generated (Figure 2).

#### 3.3. Datasets with Measurement Errors

_{m}, R

_{d}and Γ*, and the specific parameters T

_{p}and α were estimated for all 100 datasets. The specific parameter V

_{cmax}, K

_{co}, or J

_{max}was estimated whenever its state was identified (Table 1). The number of estimated parameters was higher than that in normal accuracy datasets, indicating that more parameters would be estimated when using a less accurate dataset. The total numbers of values within ±10% of their “true values” for V

_{cmax}, K

_{co}

_{,}J

_{max}, T

_{p}, g

_{m}, R

_{d}, Γ* and α were 30, 10, 61, 64, 7, 7, 36 and 23 when α > 0; and 22, 7, 50, 51, 5, 8, 23 and 78 when α = 0, respectively (Table 1). The distributions of estimated parameters (except for T

_{p}) were more scattered than that of normal accuracy datasets. Most of the values for V

_{cmax}, K

_{co}, g

_{m}and Γ* were underestimated (Figure 3A,G,E,F), while most of R

_{d}and α were overestimated (Figure 3D,H). About half of the values of T

_{p}and J

_{max}were overestimated.

_{m}, R

_{d}, V

_{cmax}, J

_{max}, T

_{p}, within ±10% errors were 8, 5, 24, 70 and 60 when α > 0, and 4, 7, 26, 61 and 63 when α = 0, respectively. There were similar numbers of estimated parameters within ±10% errors between datasets with measurement errors having α > 0 and α = 0 and for normal accuracy and measurement error datasets. This observation suggests that the impacts of the value of α and measurement errors on the estimated parameters were insignificant in datasets with large errors when using Sharkey et al.’s method. Overall, the distributions of the estimated parameters were similar to those of a normal accuracy dataset (Figure 3).

#### 3.4. Datasets with Varied Data Points

_{m}, R

_{d}, Γ*, T

_{p}and α in any dataset (Table 2), though many of them were non-resolvable. Generally, the ratio of correctly estimated parameters to estimated parameters increased (except for K

_{co}and Γ*) with an increasing number of observed data points for the A

_{n}/C

_{i}curve.

_{max}and T

_{p}ranged from about 33% when the number of data points was four. When the number of data points was 12 for g

_{m}, R

_{d}and Γ*, this ranged from 0 to 20%. The estimated parameters were unevenly distributed around their “true values”.

#### 3.5. Datasets with Varied Accuracy

_{cmax}was underestimated with an error of about 13–17% for datasets with eight data points and 11–15% for datasets with 12 data points. J

_{max}was slightly overestimated, with an error of about 0–7% for datasets with eight data points and 1–4% for datasets with 12 data points. T

_{p}was overestimated, with an error of about 8–16% for for datasets with eight data points and 7–11% for for datasets with 12 data points. The estimated parameters g

_{m}, K

_{co}R

_{d}, α and Γ* had relatively large errors; for example, the errors of the mean values of R

_{d}were from 80 to 243%.

_{cmax}were overestimated, with errors of 18% for datasets with eight data points, and 12–16% for datasets with 12 points. The mean values of J

_{max}were close to the “true value”, with errors of 5% for the datasets with eight data points, and 5–7% for datasets with 12 points. The means of T

_{p}were overestimated with errors of 12–13% for datasets with eight data points, and 0–2% for datasets with with 12 points. Parameters g

_{m}and R

_{d}had relatively large errors. The ranges of the relative changes of the mean g

_{m}and R

_{d}were from 164 to 694%, and from −43 to 159%, respectively.

## 4. Discussion

_{n}is a random variable whose population mean is zero and variance is constant, and that C

_{i}is an independent variable without any error. Since only a few data points are available in each limited state, the sample error will vary considerably, simply by chance. A point with a larger error would tend to have a larger deviation from the curve and so would have a larger impact on the SSE. In contrast, a point with a smaller error would have a smaller influence. Minimizing the SSE would be inappropriate for datasets with a few data points with relative large errors.

_{n}/C

_{i}curve; (3) an accuracy of at least seven decimal places; and (4) α > 0. These requirements were necessary and sufficient conditions for Gu et al.’s method. If a dataset does not meet these conditions, Gu et al.’s method will be unable to guarantee a fit for any parameter. For example, in the dataset (2, 4, 2) with high accuracy, all the parameters were resolvable; however, Gu et al.’s method incorrectly identified it as (3, 5, 0), leading to all the parameters being incorrect (data not shown).

_{cmax}, J

_{max}, T

_{p}, R

_{d}and g

_{m}simultaneously, using all the data points of an A

_{n}/C

_{i}curve, by fitting Equations (1)–(5) and (14), with fixed K

_{co}and Γ*, and assuming α = 0. This approach simplifies the fitting method, but may introduce more errors to the estimated parameters if the wrong fixed values are used. Sharkey et al.’s method was unable to correctly estimate any parameters using all examined datasets. One of the major reasons for this could be the use of an incorrect value for the fixed parameters K

_{co}, Γ*, and α. There are different K

_{co}and Γ* values to choose from in the literature [12,27]. In addition, K

_{co}changes across diverse species and environmental conditions. One can see from Equations (9)–(13) that a change in one or more parameters may lead to changes in all the other parameters; for example, if a dataset is only in the A

_{c}state, by combining Equations (1), (4) and (8) and assuming independent of all involved parameters, one can have

_{co}on V

_{cmax}are

_{cmax}depend on the errors in K

_{co}and Γ* and the values of A

_{c}, C

_{i}, g

_{m}and R

_{d}. Thus, incorrect values for K

_{co}and/or Γ* will inevitably lead to incorrect parameter estimates. This result is in agreement with that of Ethier and Livingston [12].

_{2}partial pressure inside the chloroplast C

_{cs}was estimated by (a similar estimation is also in the methods of Dubois et al. [24] and Miao et al. [26]):

_{nm}is the measured net assimilation rate. Equation (26) is not identical to Equation (8); A

_{n}in Equation (8) is the calculated value in the algorithm. Thus, a minimization of SSE based on Equation (26) or Equation (8) is different. The parameterization of a four-point curve in the A

_{j}state (Table 3) illustrates this problem. The dataset with eight decimal places was generated using the same value of Γ* as in Sharkey et al.’s method to eliminate the effects of different values of Γ*. The two sets of estimated parameters were different from their “true values” and modeled A

_{jj}s were slightly different from the ‘measured’ values. One can see that Sharkey et al.’s method was unable to correctly derive the parameters, even using a dataset with a high accuracy and the same fixed value as in the method.

_{c}and A

_{j}states. If the data points were assigned to the same states as when it was generated (I), the SSE was larger than if the last three data points were assigned to the A

_{p}state, which was larger than if the transitional point between A

_{c}and A

_{j}was also adjusted. The estimated parameters were different among the three conditions (except for R

_{d}and g

_{m}) and were different from their “true values.” It is worth noting that this is an intrinsic problem in Sharkey et al.’s method, because of the limited accuracy and number of data points of an A

_{n}/C

_{i}curve. It is very easy to miss a state, especially the A

_{p}state [24]. This result can be explained by considering that the incorrect identification of the distribution of data points led to incorrect parameter estimation.

_{m}≤ 30 μmol m

^{−2}s

^{−1}Pa

^{−1}in Sharkey et al.’s method, and R

_{d}≤ 10 μmol m

^{−2}s

^{−1}, g

_{m}≤ 1,000,000 μmol m

^{−2}s

^{−1}Pa

^{−1}, and Γ* ≥ 0 Pa in Gu et al.’s method. There were also different constraints found in other methods in the literature, such as the constraint of −3 < R

_{d}< 50 μmol m

^{−2}s

^{−1}in Dubois et al. [24]. Firstly, if a parameter is estimated within the range of its constraint, the local minimum must be achieved in the range of the constraint. Secondly, the constraints are subjective choices which are probably not realistic; for example, we obtained R

_{d}as zero by Sharkey et al.’s method (data not shown), and Γ* as zero (data not shown) by Gu et al.’s method. Thirdly, if a parameter is equal to its constraint, which is likely to be incorrect, this incorrect parameter may substantially affect the estimates of other parameters. The problems of finding a local minimum and non-uniqueness of the parameters is intrinsic to nonlinear regression; for example, in Sharkey et al.’s method, the estimates of the parameters were sensitive to the initial values (Table 3). There are similar problems in the method of Dubois et al. [24] and Miao et al. [26], as stated by Gu et al. [2].

_{n}from C

_{i}if the SSE is small, as argued by Ethier and Livingston [12]. The four A

_{n}/C

_{i}curves modeled by the three sets of parameters with 15 data points (7, 6, 2) were compared. The three sets of parameters were high accuracy, including measurement errors, and derived from curves generated using the parameters fitted to the dataset with measurement errors by methods of Gu et al. and Sharkey et al., respectively. All curves are superposed.

_{p}) by both methods using a normal accuracy dataset and a dataset with measurement errors were very scattered. The majority of the estimated parameters were either overestimated or underestimated (Table 1, Figure 2 and Figure 3), implying that the mean values of these estimated parameters could not represent the “true values”. This can be explained by low data accuracies and by the intrinsic problems in both methods. The estimated T

_{p}was evenly distributed around the 1:1 line, indicating that the mean T

_{p}was close to its “true value”; this was because T

_{p}is almost equal to $\frac{{A}_{n}}{3}$ (Equations (1), (6), and (7)) since A

_{n}>> R

_{d}in the A

_{p}state.

## 5. Conclusions

_{n}/C

_{i}curve fitting method, and the method developed recently by Gu et al. [2], using datasets with a number of A

_{n}/C

_{i}curve points from four to 15 and accuracies from one to eight decimal places. The generated datasets were conservative. In the literature, the typical number of data points of an A

_{n}/C

_{i}curve is eight to 12, which is considered enough for estimation [2,24,26,30]. The accuracy of the measured data in a typical open gas exchange system is one decimal place; e.g., Li-6400. The error level of the generated datasets was lower than normally seen in practice, because only one source of measurement error, e.g., the random noise of CO

_{2}±0.2 μmol mol

^{−1}, was imposed as an error to the datasets; other error sources were not included [34].

_{n}/C

_{i}curve, we have to consider data accuracy and the number and distribution of data points as well as error distribution. Based on the results using different generated datasets, we concluded that Sharkey et al.’s method failed to correctly estimate the parameters, while Gu et al.’s method was unable to correctly estimate the parameters using a dataset with a number of data points fewer than five or with an accuracy of four or fewer decimal places. At least eight data points were required for Gu et al.’s method to correctly estimate all eight parameters. For the datasets with measurement errors and the same accuracy of a typical open gas exchange system—i.e., Li-6400—using Gu et al.’s approach, the parameters V

_{cmax}, K

_{co}, g

_{m}and Γ* were underestimated, while R

_{d}and α were often overestimated. The distributions of T

_{p}were evenly dispersed around the 1:1 line using both approaches. Using Sharkey et al.’s approach, the parameters J

_{max}was overestimated, V

_{cmax}and g

_{m}were underestimated, and many values of R

_{d}were over their upper limit of 5.000 μmol m

^{−2}s

^{−1}. The mean values of all estimated parameters, except for T

_{p}, were not close to their “true values”.

_{n}and C

_{i}). The other problem was the failure to identify correct parameter estimates using Gu et al.’s method, due to the unknown data point distribution.

_{n}/C

_{i}and enzyme kinetic measurements are required to correctly estimate these parameters, even when sufficient data points are provided. An accurate estimation of the parameters can contribute to the enhancement of food security under climate change by reducing potential errors when the biological and biophysical processes of CO

_{2}assimilation are correctly spatially and temporally scaled-up for ecosystem studies. This study does not address the question of how these measurement accuracies can be improved. It is recommended that this question be addressed in a further study.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A_{c} | Net assimilation rate assuming Rubisco-limited state, μmol m^{−2} s^{−1} |

A_{cmi} | Measured net assimilation rates in the A_{c} state point i, μmol m^{−2} s^{−1} |

A_{jj} | Calculated net assimilation rates in the A_{j} state point j, μmol m^{−2} s^{−1} |

A_{n} | Net assimilation rate, μmol m^{−2} s^{−1} |

A_{nj} | Net assimilation rate at A_{j} state, μmol m^{−2} s^{−1} |

A_{p} | Net assimilation rate assuming triose phosphate utilization (TPU) limited state, μmol m^{−2} s^{−1} |

A_{pk} | Calculated net assimilation rates in the A_{p} state point k, μmol m^{−2} s^{−1} |

C_{c} | Chloroplastic CO_{2} partial pressure, Pa |

C_{i} | Intercellular CO_{2} partial pressure, Pa |

g_{m} | Internal (mesophyll) conductance to CO_{2} transport, μmol m^{−2} s^{−1} Pa^{−1} |

J_{max} | Maximum rate of electron transport, μmol m^{−2} s^{−1} |

K_{o} | Michaelis–Menten constant for Rubisco for O_{2}, Pa |

R_{d} | Day respiration, μmol m^{−2} s^{−1} |

T_{p} | Rate of triose phosphate export from the chloroplast, μmol m^{−2} s^{−1} |

W_{c} | Maximal Rubisco carboxylation rate, μmol m^{−2} s^{−1} |

W_{p} | TPU-limited carboxylation rate, μmol m^{−2} s^{−1} |

A_{ci} | Calculated net assimilation rates in the A_{c} state point i, μmol m^{−2} s^{−1} |

A_{j} | Net assimilation rate assuming RuBP regeneration limited state, μmol m^{−2} s^{−1} |

A_{jmj} | Measured net assimilation rates in the A_{j} state point j, μmol m^{−2} s^{−1} |

A_{nc} | Net assimilation rate at A_{c} state, μmol m^{−2} s^{−1} |

A_{nm} | Measured net assimilation rate, μmol m^{−2} s^{−1} |

Γ* | Chloroplastic CO_{2} photocompensation point, Pa |

A_{pmk} | Measured net assimilation rates in the A_{p} state point k, μmol m^{−2} s^{−1} |

C_{cc_cj} | Chloroplastic CO_{2} partial pressure at transition point between A_{c} and A_{j} state, Pa |

C_{s} | Intercellular CO_{2} partial pressure, Pa |

J | Potential electron transport rate at the measurement light level, μmol m^{−2} s^{−1} |

K_{c} | Michaelis-Menten constant for Rubisco for CO_{2}, Pa |

K_{co} | Effective Michaelis–Menten coefficient for CO_{2}, K_{co} = K_{c}(1 + O/K_{o}), Pa |

O | Oxygen partial pressure, Pa |

V_{cmax} | Maximal Rubisco carboxylation rate, μmol m^{−2} s^{−1} |

W_{j} | RuBP regeneration-limited carboxylation rate, μmol m^{−2} s^{−1} |

α | Non-returned fraction of the glycolate carbon recycled in the photorespiratory cycle (dimensionless) |

## Appendix A. Errors Superimposed to an Ideal Dataset

_{n}and C

_{i}are subject to measurement errors. The NORMINV function in Microsoft Excel 2007 (Microsoft, Seattle, WA, USA) was used to generate these errors. There are three arguments for the functions (Probability, Mean, and Standard Deviation). For this study, the RAND function was used for probability and the Mean and Standard Deviation values were 0 and 0.2, respectively. The values of A

_{n}and C

_{i}are not directly measured in an open gas exchange system (e.g., Li-6400). They are calculated from the measured environmental conditions of a leaf enclosed in the leaf chamber, assuming the leaf is homogeneous, implying that the photosynthesis of every portion of the measured leaf is in the same limited state in the same time. The environmental conditions are provided by some signal processing algorithm; for example, CO

_{2}concentration was calculated from the signal of the infrared gas analyzer (IRGA). Under such conditions, the A

_{n}/C

_{i}curve can be described by the FvCB model if the measurements are without error. However, in practice, all physical measurements are subject to measurement errors. The accuracy of the measurements could be affected by many factors, including, but not limited to, random noise in the CO

_{2}and water vapor sensors, leaf biological and environmental heterogeneity, leaf temperature variation, gas leaks, dark transpiration and respiration from the leaf under the gaskets, signal processing algorithms, and the calibration of the instrument. For simplicity, we only focus on the errors from CO

_{2}sensors. Following von Caemmerer and Farquhar [23] and ignoring the dilution effects of water vapor, the gas exchange rate A

_{n}and C

_{i}in an open system can be simply expressed as:

_{tc}is the total conductance to CO

_{2}; F is the incoming air flow rate; C

_{r}and C

_{s}are reference IRGA and sample IRGA CO

_{2}concentrations, respectively; and S is the leaf chamber area. From Equations (A1) and (A2), one can see that C

_{i}is not independent of A

_{n}. The uncertainties of A

_{n}will transfer to C

_{i}.

_{tc}, F, S, Cr and C

_{s}are independent and randomly distributed, because the data were recorded at steady state. According to the principle of the propagation of errors, the uncertainties of A

_{n}and C

_{i}are given, respectively, by

_{2}difference between the reference and the sample, the smaller measurement errors of A

_{n}(Equation (A3)) and C

_{i}(Equation (A4)). Each gas analysis system has its own specific errors for F, C

_{r}and C

_{s}; for example, the Li-6400 we used had errors of C

_{r}± 0.36 μmol mol

^{−1}and errors of C

_{S}± 0.12 μmol mol

^{−1}at 370 μmol mol

^{−1}[34].

_{tc}are ignorable, and εC

_{r}= εC

_{s}= εC Equations (A3) and (A4) become to

^{2}, the flow rate is 400 μmol s

^{−1}and the δC = ±0.2 μmol mol

^{−1}[32], the error of A

_{n}is approximately ±0.2 μmol m

^{−2}s

^{−1}. The uncertainties of C

_{i}also depend on g

_{tc}, which is mainly derived from stomatal conductance. For generating the A

_{n}/C

_{i}points, we assume that g

_{s}is linear with CO

_{2}concentration and the range of g

_{s}is from 0.2 mol CO

_{2}m

^{−2}s

^{−1}at C

_{i}of 0 μmol mol

^{−1}to 0.1 mol CO

_{2}m

^{−2}s

^{−1}at C

_{s}of 2000 μmol mol

^{−1}.

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**Figure 1.**Comparison of the estimated parameters by Gu et al.’s method (black circle for α > 0 and black triangle for α = 0) and Sharkey et al.’s method (open circle for α > 0 and open triangle for α = 0) vs. true parameter values for synthetic A

_{n}/C

_{i}curves, 100 datasets with α > 0 and 100 with α = 0. The datasets consisted of 15 data points with high accuracy (eight decimal points). The points in each figure may have less than 100 values because some datasets do not contain all three states.

**Figure 2.**The same as in Figure 1; here applied to normal accuracy datasets.

**Figure 3.**The same as in Figure 2; here applied to datasets with measurement errors.

**Table 1.**Summary of the parameterization of Gu et al.’s and Sharkey et al.’s methods using 15 data points A

_{n}/C

_{i}curves. Resolvable: the parameter can be correctly estimated by an appropriate method; Correctly estimated: the estimated parameter by a specific method with the same value as “true value”; Total estimated: total estimated parameters including correctly and incorrectly estimated parameters; Error within ±10%: the ranges of the error of estimated parameters within ±10% of the “true value”.

α | Dataset | Method | Number of Parameters | g_{m} | Γ* | R_{d} | V_{cmax} | K_{co} | J_{max} | T_{p} | α |
---|---|---|---|---|---|---|---|---|---|---|---|

α > 0 | HDS | Gu et al. | Resolvable ^{a} | 100 | 90 | 90 | 71 | 80 | 72 | 61 | 61 |

Correctly estimated | 72 | 62 | 62 | 52 | 62 | 49 | 42 | 42 | |||

Total estimated | 100 | 100 | 100 | 91 | 91 | 74 | 100 | 100 | |||

Error within ±10% | 89 | 80 | 75 | 71 | 70 | 69 | 89 | 55 | |||

Sharkey et al. | Resolvable ^{a} | 100 | NA | 100 | 88 | NA | 79 | 72 | NA | ||

Correctly estimated ^{b} | 0 | NA | 0 | 0 | NA | 0 | 0 | NA | |||

Total estimated ^{d} | 100 | NA | 100 | 100 | NA | 100 | 100 | NA | |||

Error within ±10% | 41 | NA | 6 | 22 | NA | 65 | 7 | NA | |||

NDS | Gu et al. | Correctly estimated ^{e} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Total estimated | 100 | 100 | 100 | 98 | 98 | 76 | 100 | 100 | |||

Error within ±10% | 26 | 48 | 16 | 55 | 40 | 65 | 70 | 26 | |||

Sharkey et al. | Error within ±10% | 6 | NA | 6 | 24 | NA | 70 | 57 | NA | ||

DSE | Gu et al. | Total estimated | 100 | 100 | 100 | 98 | 98 | 86 | 100 | 100 | |

Error within ±10% | 7 | 36 | 7 | 30 | 10 | 61 | 64 | 23 | |||

Sharkey et al. | Error within ±10% | 8 | NA | 5 | 24 | NA | 70 | 60 | NA | ||

α = 0 | HDS | Gu et al. | Resolvable ^{a} | 98 | 62 | 62 | 44 | 72 | 62 | 45 | 45 |

Correctly estimated | 72 | 46 | 46 | 37 | 63 | 46 | 30 | 30 | |||

Total estimated | 100 | 100 | 100 | 86 | 86 | 92 | 100 | 100 | |||

Error within ±10% | 82 | 61 | 57 | 64 | 65 | 68 | 74 | 100 | |||

Sharkey et al. | Resolvable ^{a} | 100 | NA | 100 | 88 | NA | 95 | 79 | NA | ||

Error within ±10% | 63 | NA | 5 | 23 | NA | 64 | 8 | NA | |||

NDS | Gu et al. | Total estimated | 100 | 100 | 100 | 93 | 93 | 90 | 100 | 100 | |

Error within ±10% | 20 | 42 | 12 | 43 | 27 | 64 | 67 | 99 | |||

Sharkey et al. | Error within ±10% | 8 | NA | 8 | 22 | NA | 59 | 51 | NA | ||

DSE | Gu et al. | Total estimated | 100 | 100 | 100 | 98 | 98 | 93 | 100 | 100 | |

Error within ±10% | 5 | 23 | 8 | 22 | 7 | 50 | 51 | 78 | |||

Sharkey et al. | Error within ±10% | 4 | NA | 7 | 26 | NA | 61 | 63 | NA |

^{a}The number of resolvable parameters is the same for all datasets (HDS: high accuracy dataset, NDS: normal accuracy dataset and DSE: dataset with measurement errors) when α > 0 (or α = 0) using the method of Gu et al. [2], and using the method of Sharkey et al. [27], respectively;

^{b}The number of correctly estimated parameters is the same for datasets using the method of Sharkey et al. [27];

^{c}For values of α, the differences within 0.1 were listed;

^{d}The number of estimated parameters were the same for all datasets using the method of Sharkey et al. [27];

^{e}The number of correctly estimated parameters is the same for all datasets (using the method of Gu et al. [2]).

**Table 2.**Effects of the number of data points of A

_{n}/C

_{i}curves on the quality of the parameter estimates obtained from Gu et al.’s and Sharkey et al.’s methods. The “true values” of the parameters are V

_{cmax}= 45.572 μmol m

^{−2}s

^{−1}, J

_{max}= 111.32 μmol m

^{−2}s

^{−1}, R

_{d}= 1.381 μmol m

^{−2}s

^{−1}, g

_{m}= 1.897 μmol m

^{−2}s

^{−1}Pa

^{−1}, T

_{p}= 7.871 μmol m

^{−2}s

^{−1}, Γ* = 4.396 Pa, K

_{co}= 43.616 Pa and α = 0.352. The descriptions of the terms are the same as in Table 1.

Number of Data Points | Number of Datasets | Methods | Number of Values | g_{m} | Γ* | R_{d} | V_{cmax} | K_{co} | J_{max} | T_{p} | α |
---|---|---|---|---|---|---|---|---|---|---|---|

4 | 15 | Gu et al. | Resolvable | 3 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |

Correctly estimated | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | |||

Total estimated | 15 | 15 | 15 | 2 | 2 | 12 | 15 | 15 | |||

Sharkey et al. | Resolvable | 7 | NA | 7 | 5 | NA | 5 | 2 | NA | ||

Correctly estimated ^{a} | 0 | 0 | 0 | NA | 0 | 0 | NA | NA | |||

Total estimated | 15 | NA | 15 | 11 | NA | 13 | 10 | NA | |||

5 | 21 | Gu et al. | Resolvable | 9 | 3 | 3 | 0 | 3 | 3 | 0 | 0 |

Correctly estimated | 3 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | |||

Total estimated | 21 | 21 | 21 | 10 | 10 | 19 | 21 | 21 | |||

Sharkey et al. | Resolvable | 13 | NA | 13 | 10 | NA | 10 | 5 | NA | ||

Total estimated | 21 | NA | 21 | 21 | 21 | NA | 19 | NA | |||

6 | 28 | Gu et al. | Resolvable | 20 | 10 | 10 | 3 | 8 | 10 | 3 | 3 |

Correctly estimated | 4 | 3 | 3 | 2 | 3 | 3 | 0 | 0 | |||

Total estimated | 28 | 28 | 28 | 16 | 16 | 25 | 28 | 28 | |||

Sharkey et al. | Resolvable | 21 | NA | 21 | 17 | NA | 17 | 14 | NA | ||

Total estimated | 28 | NA | 28 | 27 | NA | 27 | 27 | NA | |||

7 | 36 | Gu et al. | Resolvable | 32 | 20 | 20 | 9 | 15 | 19 | 8 | 8 |

Correctly estimated | 7 | 5 | 5 | 3 | 4 | 5 | 1 | 1 | |||

Total estimated | 36 | 36 | 36 | 23 | 23 | 31 | 36 | 36 | |||

Sharkey et al. | Resolvable | 28 | NA | 28 | 23 | NA | 23 | 20 | NA | ||

Total estimated | 28 | NA | 28 | 28 | NA | 28 | 27 | NA | |||

8 | 45 | Gu et al. | Resolvable | 45 | 33 | 33 | 20 | 26 | 26 | 20 | 20 |

Correctly estimated | 7 | 6 | 6 | 4 | 4 | 5 | 3 | 3 | |||

Total estimated | 45 | 45 | 45 | 31 | 31 | 40 | 45 | 45 | |||

Sharkey et al. | Resolvable | 37 | NA | 37 | 31 | NA | 31 | 28 | NA | ||

Total estimated | 45 | NA | 44 | 45 | NA | 45 | 45 | NA | |||

9 | 55 | Gu et al. | Resolvable | 55 | 43 | 43 | 28 | 34 | 39 | 28 | 28 |

Correctly estimated | 15 | 13 | 13 | 10 | 11 | 11 | 8 | 8 | |||

Total estimated | 55 | 55 | 55 | 37 | 37 | 49 | 55 | 55 | |||

Sharkey et al. | Resolvable | 47 | NA | 47 | 40 | NA | 40 | 37 | NA | ||

Total estimated | 55 | NA | 55 | 53 | NA | 55 | 55 | NA | |||

12 | 73 | Gu et al. | Resolvable | 73 | 71 | 71 | 56 | 57 | 64 | 56 | 56 |

Correctly estimated | 30 | 30 | 30 | 25 | 25 | 26 | 24 | 24 | |||

Total estimated | 73 | 73 | 73 | 61 | 61 | 67 | 73 | 73 | |||

Sharkey et al. | Resolvable | 72 | NA | 72 | 64 | NA | 65 | 65 | NA | ||

Total estimated | 73 | NA | 73 | 73 | NA | 73 | 73 | NA |

^{a}The same values for all parameters using the method of Sharkey et al.

**Table 3.**Parameters estimated from three datasets fitted by Sharkey et al.’s method. The dataset was generated with the fixed value of Sharkey et al.’s method (Γ* = 3.743 Pa

^{−1}) at a leaf temperature of 25 °C and an air pressure of 100 Pa. A

_{jj}I and A

_{jj}II and estimated parameters I and II are values estimated using two different sets of the initial values of the parameters. A

_{jmj}and A

**were measured (generated) values using Equation (8) and modeled values using Equation (26), respectively.**

_{jj}C_{i} (μmol mol^{−1}) | A_{jmj} (μmol m^{−2} s^{−1}) | A_{jj}I (μmol m^{−2} s^{−1}) | A_{jj}II (μmol m^{−2} s^{−1}) | Parameter/SSE | True Parameter | Estimated Parameter I | Estimated Parameter II |
---|---|---|---|---|---|---|---|

373.56422385 | 20.52707868 | 20.53126975 | 20.52533184 | J_{max} (μmol m^{−2} s^{−1}) | 120.494 | 126.278 | 117.536 |

559.64521334 | 22.90931588 | 22.90327060 | 22.91250945 | R_{d} (μmol m^{−2} s^{−1}) | 1.674 | 3.014 | 1.039 |

672.56218564 | 23.76820169 | 23.76423592 | 23.7701254 | g_{m} (μmol m^{−2} s^{−1}Pa^{−1}) | 9.564 | 30.000 | 7.170 |

909.96541253 | 24.92146362 | 24.92728744 | 24.91809557 | SSE (μmol m^{−2} s^{−1})^{2} | ^{-} | 0.000 | 0.000 |

**Table 4.**Comparison of fitting results from a 12-point A

_{n}/C

_{i}using Sharkey et al.’s method by assigning a different transition point between the A

_{c}, A

_{j}and A

_{p}states, as indicated by I, II, III or IV. The dataset was generated as containing only A

_{c}and A

_{j}states, and the transitional point of C

_{i}is between 201 and 284 μmol mol

^{−1}.

A_{n} | C_{i} | Parameter/SSE | True Value | I ^{a} | II ^{b} | III ^{c} |
---|---|---|---|---|---|---|

2.17 | 38.9 | V_{cmax} (μmol m^{−2} s^{−1}) | 99.4 | 108.9 | 108.9 | 112.0 |

4.47 | 58.1 | J_{max} (μmol m^{−2} s^{−1}) | 136.2 | 143.1 | 146.1 | 145.5 |

9.60 | 99.4 | T_{p} (μmol m^{−2} s^{−1}) | - | 10.1 | 10.1 | |

15.3 | 150 | R_{d} (μmol m^{−2} s^{−1}) | 1.1 | 0.0 | 0.0 | 0.0 |

20.6 | 201 | g_{m} (μmol m^{−2} s^{−1} Pa^{−1}) | 3.7 | 30.0 | 30.0 | 30.0 |

25.3 | 284 | K_{co} (Pa) | 42.9 | - | - | - |

27.0 | 371 | Γ* (Pa) | 1.9 | - | - | - |

28.2 | 415 | SSE | - | 9.939 | 7.315 | 7.266 |

29.4 | 552 | - | - | - | - | - |

30.1 | 673 | - | - | - | - | - |

30.1 | 730 | - | - | - | - | - |

30.6 | 908 | - | - | - | - | - |

^{a}The transition point was the same as the corresponding error-free data set generated by true parameters;

^{b}is the same as

^{a}, except for the fact that the last 3 points were assigned to A

_{p}state;

^{c}minimum of SSE where the transition point of C

_{i}between A

_{c}and A

_{j}is 150 and 201 μmol mol

^{−1}

_{,}and between A

_{j}and A

_{p}it is between 552 to 673 μmol mol

^{−1}.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Chun, J.A.; Fleisher, D.; Reddy, V.; Timlin, D.; Resop, J.
Parameter Estimation of the Farquhar—von Caemmerer—Berry Biochemical Model from Photosynthetic Carbon Dioxide Response Curves. *Sustainability* **2017**, *9*, 1288.
https://doi.org/10.3390/su9071288

**AMA Style**

Wang Q, Chun JA, Fleisher D, Reddy V, Timlin D, Resop J.
Parameter Estimation of the Farquhar—von Caemmerer—Berry Biochemical Model from Photosynthetic Carbon Dioxide Response Curves. *Sustainability*. 2017; 9(7):1288.
https://doi.org/10.3390/su9071288

**Chicago/Turabian Style**

Wang, Qingguo, Jong Ahn Chun, David Fleisher, Vangimalla Reddy, Dennis Timlin, and Jonathan Resop.
2017. "Parameter Estimation of the Farquhar—von Caemmerer—Berry Biochemical Model from Photosynthetic Carbon Dioxide Response Curves" *Sustainability* 9, no. 7: 1288.
https://doi.org/10.3390/su9071288