Introduction

Key photosynthetic parameters allow for the assessment of how biochemical and biophysical components of photosynthesis affect net carbon assimilation in response to environmental changes, phenotypic/genotypic differences, genetic modification, and the evolution of photosynthesis pathway. The changes in net assimilation (An) that occur along with the changes of intercellular CO2 concentration (Ci)—or A/Ci curves—are widely used to estimate photosynthetic parameters for C3 species. In particular, the method by Sharkey et al. (2007), based on the C3 photosynthesis model of Farquhar et al. (1980; FvCB model), has been one of the most widely used tools since it is based exclusively on A/Ci curves, which are easy to measure in both lab and field conditions.

Fewer estimates of photosynthetic parameters have been reported for C4 species, as there has been a lack of accessible C4 estimation methods. C4 photosynthesis enables the concentration of CO2 around Rubisco, thus reducing photorespiration. The concentration mechanism requires the carbonic anhydrase and PEP carboxylation/regeneration processes to operate. Furthermore, the enzymes which decarboxylate C4 acids in the PEP regeneration differ and result in three different C4 subtypes. The tight bundle sheath wall, which prevents CO2 from diffusing out from the bundle sheath, also limits the diffusion of gaseous CO2 into and O2 out of the photosynthesis spot. To model such a concentration mechanism requires accounting for distinct biochemistry and morphology, which leads to increased complexity in estimating parameters for C4 photosynthesis. Despite this, several recent studies used A/Ci curves to estimate photosynthesis parameters based on the C4 photosynthesis model of von Caemmerer (2000) (Ubierna et al. 2013; Bellasio et al. 2015). These studies use partial A/Ci curves; measuring assimilation rates for only a few CO2 concentrations coupled with ancillary measurements of chlorophyll fluorescence and/or 2% O2. While these estimation methods lead to estimates of photosynthetic parameters, the additional measurements they require make estimation more cumbersome for field work or large-scale sampling. Theoretically, it is possible to estimate photosynthetic parameters by exclusively fitting A/Ci curves to a C4 photosynthesis model. In this paper, we propose a method to estimate C4 photosynthesis parameters using only A/Ci curves.

There are several potential problems with A/Ci—based estimation methods for C3 plants that carry over to existing C4 methods (Gu et al. 2010); it is therefore important to develop a C4 estimation method with improvements to solve the general problems and drawbacks outlined below. First, the structure of the FvCB model makes it easy to be over-parameterized. Second, a general shortcoming for the estimation methods is that they require an artificial assignment of the RuBP regeneration and Rubisco carboxylation limitation states to parts of the A/Ci curves (Xu and Baldocchi 2003; Ethier et al. 2006; Ubierna et al. 2013; Bellasio et al. 2015), which has turned out to be problematic (Type I methods) (Gu et al. 2010). These methods assume constant transition points of limitation states for different species. Furthermore, Type I methods tend to minimize separate cost functions of different limitation states instead of minimizing a joint cost function. Some recent estimation methods for C3 species ameliorate these problems by allowing the limitation states to vary at each iterative step of minimizing the cost function (Type II methods; Dubois et al. 2007; Miao et al. 2009; Yin et al. 2009; Gu et al. 2010). However, for these type II methods, additional degrees of freedom in these “auto-identifying” strategies can lead to over-parameterization if limitation states are allowed to change freely for all data points. Gu et al. (2010) also pointed out that existing Type I and Type II methods fail to check for inadmissible fits, which happen when estimated parameters lead to an inconsistent identification of limitation states from the formerly assigned limitation states. More specifically to C4, the recently developed C4 estimation methods artificially assign limitation states for A/Ci curves (Ubierna et al. 2013; Bellasio et al. 2015) and also did not check for inadmissible fits.

Here, we present a method to estimate photosynthetic parameters for C4 species based solely on fitting intensive A/Ci curves to a C4 photosynthesis model (von Caemmerer 2000). Using intensive A/Ci curves (with 6–8 more sampling points than the commonly used for C3 species) for C4 plants is important for two reasons: First, at low Ci, the slope of A/Ci is very steep and the assimilation rate saturates quickly. Second, C4 species have more photosynthetic parameters as the carbon concentrating mechanism adds complexity. A further complication arises due to the fact that carbonic anhydrase catalyzes the first reaction step for C4 photosynthesis (Jenkins et al. 1989). It has been commonly assumed to not limit CO2 uptake in estimation methods and C4 models (von Caemmerer 2000; Yin et al. 2011b). Recent studies, however, showed evidence of potential limitation by carbonic anhydrase (von Caemmerer et al. 2004; Studer et al. 2014; Boyd et al. 2015; Ubierna et al. 2017).

To address these issues we first build an estimation method using two different fitting procedures of Sharkey et al. (2007) and Yin et al. (2011b) without considering carbonic anhydrase activity. Then, we add carbonic anhydrase limitation into the estimation method. This allows us to also use our method to examine how the carbonic anhydrase-limitation assumption impacts parameter estimation, and whether the modeling of C4 photosynthesis can be simplified by omitting it. All together, our method estimates five to six photosynthesis parameters: (1) maximum carboxylation rate allowed by ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco) (Vcmax), (2) rate of photosynthetic electron transport (J), (3) day respiration (Rd), (4) maximal PEP carboxylation rate (Vpmax), (5) mesophyll conductance (gm), and optionally (6) the rate constant for carbonic anhydrase hydration activity (kCA). Our approach eliminates common problems occurring in the previous C3 and C4 estimation methods in the following ways. First, we avoid over-parameterization, maximizing joint cost function, freely determining transition points instead of assigning in advance, and checking for inadmissible fits. Second, since both RuBP regeneration and PEP regeneration need ATP (Hatch 1987), we also examine two different assumptions about ATP distribution between RuBP regeneration and PEP regeneration in C4 photosynthesis models. Third, we validate the estimation methods in four independent ways, using: (i) simulation tests using A/Ci curves generated using our model with known parameters and adding random errors, (ii) out of sample test, (iii) existing in vitro measurements, and (iv) chlorophyll fluorescence measurements. Finally, we use the C4 photosynthesis model to perform sensitivity analyses and simulation analyses for important physiological input parameters.

Materials and methods

C4 mechanism

The CO2 concentrating mechanism of C4 pathway increases CO2 in the bundle sheath cells to eliminate photorespiration. Like the C3 pathway, the diffusion of CO2 starts from the ambient atmosphere through stomata into intercellular spaces, and then into the mesophyll cells. In the mesophyll cells, the first step is the hydration of CO2 into HCO3− by carbonic anhydrase. PEPC, then, catalyzes HCO3− and PEP into C4 acids and the C4 acids are transported to the bundle sheath cells. In the bundle sheath cell, C4 acids are decarboxylated to create a high CO2 environment for the C3 photosynthetic cycle, and PEP is regenerated. All the modeling equations and mechanistic processes used for our estimation method are from von Caemmerer (2000), Hatch and Burnell (1990), Boyd et al. (2015) and Ubierna et al. (2017) (Supplementary Methods).

Given the two limitation states of C4 cycle (PEP carboxylation (Vpc) and PEP Regeneration (Vpr)), and two limitation states of C3 cycle (RuBP carboxylation (Ac) and RuBP Regeneration (Aj)) in the C4 photosynthesis model, there are four combinations of limitation states (as Yin et al. 2011b, Fig. 1): RuBP carboxylation and PEP carboxylation limited assimilation (RcPc), RuBP carboxylation and PEP regeneration limited assimilation (RcPr), RuBP regeneration and PEP carboxylation limited assimilation (RrPc), and RuBP regeneration and PEP regeneration limited assimilation (RrPr) (Table 1). Since the C4 cycle operates before the C3 cycle and provides substrates for the C3 cycle, the determination process of An is as follows:

$${\text{If}}~\left( {{V_{{\text{pc}}}}<{V_{{\text{pr}}}}} \right),{A_c}=RcPc,~{A_j}=~RrPc,\,{\text{otherwise}}\,{A_c}=RcPr,~{A_j}=~RrPr$$
(1)
$${A_n}=\hbox{min} ({A_c},\,{A_j}),$$
(2)

which we used for our estimation method.

Fig. 1
figure 1

An introduction of how our estimation methods assign transition points between limitation states. RcPc represents RuBP carboxylation, and PEP carboxylation limited assimilation rate, RrPr represents RuBP regeneration and PEP regeneration limited assimilation rate. Transition states indicate assimilation could be limited by RcPc, RrPr, RcPr (RuBP carboxylation and PEP regeneration), and RrPc (RuBP regeneration and PEP carboxylation). Our algorithm allows the transition states to be freely limited by the above four conditions from a lower bound (CaL, 10 Pa for instance) and a higher bound (CaH, 65 Pa for example), indicated by the dashed vertical lines in the figure

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Table 1 Acronyms, detailed descriptions and applied conditions for the four limitation states of C4 photosynthesis process
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Plant material

We performed intensive A/Ci curves on nine different C4 species to develop and examine the efficacy of our estimation tools in the greenhouse of University of Pennsylvania (elevation of 13 m): Zea mays L., Eragrostis trichodes (Nutt.) Alph. Wood, Andropogon virginicus L., Schizachyrium scoparium (Michx.) Nash, Panicum virgatum L., Panicum amarum Elliott, Setaria faberi Herrm., Sorghastrum nutans (L.) Nash, and
 Tripsacum dactyloides (L.) L. The intensive A/Ci curves contain more sample points under more CO2 concentrations than the default curve used for C3 species. Here we set the CO2 concentrations as 400, 200, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 400, 500, 600, 700, 800, 1000, 1200, 1400 ppm under light intensity of 1500 µmol m−2 s−1 (light intensity encountered by the plants in greenhouse). At each point, data were recorded when the intercellular CO2 concentration equilibrated within 2–5 min. The datasets were obtained using a standard 2 ×  3 cm2 leaf chamber with a red/blue LED light source of LI-6400 (LI-COR Inc., Lincoln, NE, USA). If the stomatal conductance of a species does not decrease quickly at high CO2, then more points can be obtained by increasing the CO2 level. Fluorescence was measured along with A/Ci curves for seven C4 species (CO2 concentration is similar with above). After each change of CO2 concentration and A reached steady state, the quantum yield was measured by multiphase flash using a 2 cm2 fluorescence chamber head (Bellasio et al. 2014). All the measurements were conducted at a leaf temperature of 25 °C and VPD was controlled at 1–1.7 kPa with the flow rate of 500 µmol s−1. The cuvette was covered by Fun-Tak to avoid and correct for the leakiness (Chi et al. 2013).

Estimation protocol

We implemented the estimation methods using the non-linear curve fitting routine in MS Excel (Supplementary Material I, II, III) and independently in R (“C4Estimation”) to get solutions that minimize the squared difference between observed and predicted assimilation rates (A). Five (or six when considering carbonic anhydrase) parameters will be estimated by fitting the A/Ci curve: Vcmax, J, Rd, Vpmax, gm, and kCA. Other input parameters for C4 are in Table S1.

Input data sets and preliminary calculations

The input data sets are the leaf temperature during measurements, atmosphere pressure, two CO2 bounds (CaL and CaH discussed in the following section), and the assimilation rates (A) and the Cis (in ppm) in the A/Ci curve. Also, reasonable initial values of output parameters need to be given in the output section to initiate the non-linear curve fitting (Supplementary Material IV). Ci will be adjusted from the unit of ppm to the unit of Pa inside the program as suggested by Sharkey et al. (2007).

Estimating limitation states

We set upper and lower limits to the value of Ci between which the assimilation rates are freely determined by limitation states. We further avoid over-parameterization by pre-assigning limitation states at the lower and upper ends of the Ci range. Specifically, we assume that under very low Ci (CaL), CO2 is the limiting substrate; thus, Vp is limited by Vpc and A is given by Ac (RcPc); under very high Ci (CaH) electron transport is limiting, thus, Vp is limited by Vpr and A is given by Aj (RrPr) (Fig. 1). The points between CaL to CaH are freely determined by RcPc, RcPr, RrPc, or RrPr from Eqs. (3) to (4) to minimize the cost function. We recommend setting CaL as 10 Pa initially, then adjusting based on the preliminary results. The points of constant A at high Ci end can initially be set as being limited by RrPr primarily (based on the three points, we can CaH) or use 65 Pa as the first trial. The range of freely determined points can be adjusted by users by setting appropriate CaL and CaH. In the column of “Estimate Limitation”, whether the data points are limited by RcPc (represented by “1”), RrPr (represented by “4”), or freely vary (represented by “0” ), all the assignments of “1”, “4”, and “0” are determined automatically by the given values of CaL and CaH. One can input “− 1” to disregard a data pair. Users can adjust limitation states according to how many points and the range of Ci they have in their A/Ci curves.

We assume different processes in the C4 photosynthesis are coordinated with each other and co-limit the assimilation rate (Sharkey et al. 2007; Yin et al. 2011b; Ubierna et al. 2013; Bellasio et al. 2015). Thus, the estimation parameters allow the limitation states to be compactly clustered with each other (Fig. 1). However, if there were only a few points under CaL, the estimation results will depend heavily on the given initial values and unbalanced results would be more likely. Fig. S1 shows an example of unbalanced estimation results by deleting some points under 10 Pa or setting a very low CaL: in the estimation results, An is limited by RcPc at very low Ci and is mostly limited by Aj (shown by RrPc and RrPr) in the C3 cycle. In this case, Ac (shown by RcPc and RcPr) has a clear redundancy at higher Ci. Unbalanced results happened when there are not enough constraints points under CaL or above CaH. Such results explain why intensive A/Ci curves are preferred, especially more measuring points under the lower end and higher end of Ci. However, existing A/Ci data with 14 points might be used in the current estimation method if there are at least four points below CaL and three points above CaH.

Estimation algorithm and fitting procedures

The objective of our estimation methods is to minimize the following joint cost function (Eqs. 3 and 4) by varying the above five or six output parameters (Vcmax, J, Rd, Vpmax, gm, and kCA):

$$f=\sum\nolimits_{{(i=1)}}^{n} {{{({A_i} - {A_{mi}})}^2},}$$
(3)
$${A_i}=[If\left( {{C_i} \leq ~CaL} \right),RcPc;\,\,{\text{If}}~\left( {{C_i} \geq CaH} \right),~RrPr;\,{\text{IF}}\,~\left( {CaL \leq {C_i} \leq CaH} \right),\hbox{min} ({A_{ci}},{A_{ji}})]$$
(4)

n is the total number of observations, Aci is determined by RcPc and RcPr and Aji is determined by RrPc and RrPr from Eq. (1), Ami is the observed net assimilation rate.

In this calculation, we take Michaelis–Menten constant of Rubisco activity for CO2 (Kc), Michaelis–Menten constant of Rubisco activity for O2 (Ko), the specificity of Rubisco (γ*), Michaelis–Menten constants of PEP carboxylation for CO2 or HCO3− (Kp), the fraction of O2 evolution occurring in the bundle sheath (α), and bundle sheath conductance (gbs) as given (input parameters), similar to Sharkey et al. (2007). We conduct further sensitivity analyses in the following section to determine the effects of variability of these inputs parameters on the estimation results.

We used two fitting procedures in the current study: one was from Sharkey et al. (2007), which is an implicit minimization of error (Supplementary Material I, III), and the other one was based on the explicit calculations given by Yin et al. (2011b) (Supplementary Material II). For the method of Sharkey et al. (2007), “estimated” An was calculated using the above equations and observed An values. We call them “estimated”, because when we calculate An, observed An is used to calculate intermediate parameters, for example, the CO2 concentration in mesophyll cells (Cm), the CO2 concentration in bundle sheath (Cbs), which we then use to calculate Ac and Aj. The objective function is to minimize the sum of square errors between “estimated” An and observed An (Simulation Error in Supplementary Material I, III). For the model without carbonic anhydrase, Yin et al. (2011b) gave explicit solutions for RcPc, RcPr, RrPc, and RrPr. “Explicit” here means the assimilation rates are totally calculated by the estimated parameters without calculating the intermediates with observed An. These calculations give us the real estimation error of our fitting procedure for models without carbonic anhydrase and thus provide a validation for the goodness of fit (“True Error” in Supplementary Material I-III).

Checking inadmissible fits

We made it possible to check the inadmissible fits for limitation states in our estimation method. After the estimation process finishes, the limitation states based on the estimated parameters are calculated in the last column. If the calculated limitation states are inconsistent with the assigned ones in the estimation method, one needs to readjust the assignment of the “Estimate Limitation” (adjust CaL or CaH) and rerun the estimation method, until they are consistent with each other.

Results

Estimation results and assumptions

Estimation methods based on assumptions with and without carbonic anhydrase yield similar results (Supplementary material V). In Supplementary material III, carbonic anhydrase indeed shows limitation to Vpc, which confirms its potential role as a limiting step in the C4 cycle. However, Vpc calculated from CO2 are only a little higher than Vpc calculated from HCO3−, which resulted in the similar estimation results. In addition, the estimation errors and true errors from Yin’s equations are quite small (average < 1), and also similar between models with and without carbonic anhydrase.

Estimation methods based on the two equations of different assumptions about electron transport between RuBP regeneration and PEP regeneration yield consistent parameter estimates and assimilation- CO2 response curves (Fig. 2), but there were minor differences. The second assumption that ATP, resulting from electron transport, is freely allocated between PEP carboxylation-regeneration and RuBP regeneration leads to a bump at low CO2 when estimating RcPr. The two assumptions produce different RcPr under low CO2; but this is largely inconsequential because, under low CO2, assimilation is usually limited by RcPc.

Fig. 2
figure 2

Assimilation-CO2 response curves (A/Ci) generated using C4 photosynthesis of two different assumptions about electron transport. Photosynthetic parameters (Vcmax, J, Rd, Vpmax, and gm) are the same for both assumptions. RrPc_e−Assumption1 and RrPr_e−Assumption1 represent results of the assumption that no matter how much electron transport is used for PEP carboxylation/regeneration, a certain amount (×J) is confined for this use. RrPc_e−Assumption2 and RrPr_e−Assumption2 represent results of the assumption that electron transport can be freely distributed between PEP carboxylation/regeneration and RuBP regeneration. Parameters are estimated from A/Ci curve of T. dactyloides under the light intensity of 1500 µmol m−2 s−1. RcPc and RcPr are the same for both assumptions

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Sensitivity analysis

The parameters Kc, Ko, γ*, Kp, α, and gbs can vary among species in nature (Cousins et al. 2010; Galmés et al. 2016) and it is therefore important to know how sensitive our results are to variation in these parameters. We conducted a sensitivity analysis for variation in these parameters on the estimated Vcmax, J, Rd, Vpmax, gm, and kCA (Fig. 3). This analysis shows all the estimated parameters are robust under the variation of α (Fig. 3A) and showed little variation responding to the change of γ* (Fig. 3E) and Ko (Fig. 3C); however, the estimated parameters are sensitive to other input parameters to different extents (Fig. 3B, D, F). We calculate the average percentage change of estimated parameters along with the 50% decrease and 100% increase of the input parameters. Vcmax showed sensitivity for Kc and gbs with the average percentage change of 23.11 and 17.69% respectively but was relatively robust for Kp (7.54%). J is robust in the variations of Kc, and gbs (the average change is less than 2%) and with a medium 6.96% change for Kp. kCA is robust in the variations of Kc, Kp, and gbs (average change less than 5%). Vpmax is sensitive for Kp with the average change of 27.34%, moderately sensitive to the decrease and increase of gbs with 4.01% and 13.38% change respectively and is robust for Kc. Rd is sensitive to Kc, Kp, and gbs with the average change of 6.73, 43.88 and 13.38%. gm is strongly sensitive to Kc, Kp, and gbs with the average percentage changes of 22.95, 107.04 and 23.19%. These results suggest that Vcmax, J, Vpmax, and kCA estimated using our method are relatively robust.

Fig. 3
figure 3

Sensitivity analysis of six estimation parameters to the variation in six input parameters using the model with carbonic anhydrase. Relative changes in the estimated Vcmax, J, Rd, Vpmax, gm, and kCA in response to the relative change of six input parameters (Aα, BKp, CKo, DKc, Eγ*, and Fgbs) from the initial values in Table S1. The relative change of estimated parameters refers to the ratio of estimated values at a changed input parameter to the estimated value at the initial value of that input parameter. The symbols represent the average change of the nine C4 species and error bars represent standard error

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Physiological significance for assimilation rate of the input parameters

In addition to the sensitivity analysis, we performed a simulation analysis to illustrate the physiological importance of input parameters further, and to indicate further the importance of physiological properties in maintaining the efficiency of C4 photosynthesis pathway. We chose the estimation parameter set of T. dactyloides as an example, held photosynthetic parameters constant Vcmax (28 µmol m−2 s−1), J (134 µmol m−2 s−1), Rd (0.78 µmol m−2 s−1), gm (30.00 µmol m−2 s−1 Pa−1), and Vpmax (41.91 µmol m−2 s−1), while changing the values of α, γ*, gbs, and Kp (as half or twice of the original parameters) to see their effects on the assimilation rate, Cbs and the O2 concentration in bundle sheath (Obs) (Fig. 4; Table 2). Using photosynthetic parameter sets of other species to perform the simulation analysis yielded similar results (data not shown). The change of α did not lead to changes in assimilation rate (Fig. 4A) and led to small changes in Obs (Table 2). The decrease of γ* to half of the current value led to a small change of Cbs and assimilation rate (less than 0.5 µmol m−2 s−1) while doubling γ* led to a larger, but still not significant, change (less than 1 µmol m−2 s−1) (Fig. 4B; Table 2). Importantly, the changes of assimilation rates were less than 0.3 µmol m−2 s−1 when Ci was less than 20 Pa, which is the regular range of Ci under current ambient CO2. However, the change of gbs significantly changed the assimilation rate and Cbs (Fig. 4C; Table 2). The change of Kp significantly affected the assimilation rate and Cbs to a large degree under low Ci (Fig. 4D; Table 2).

Fig. 4
figure 4

Simulation results of assimilation rate along with different intercellular CO2 concentration (Ci) with the known photosynthetic parameters, but with the change of Aα, Bγ*, Cgbs and DKp. The original data set are Vcmax = 28 µmol m−2 s−1, J = 134 µmol m−2 s−1, Rd = 0.78 µmol m−2 s−1, gm = 30.00 µmol m−2 s−1 Pa−1 and Vpmax = 41.91 µmol m−2 s−1. The reference value of changing parameters at 25 °C: α0(25) = 0.15, γ*0(25) = 0.000244, gbs0(25) = 0.0295 and Kp(25) = 8.55 Pa

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Table 2 The average change of percentage of CO2 concentration (Cbs) and O2 concentration at bundle sheath (Obs) compared to the reference value of α0, γ*0, gbs0 and Kp
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Validating the estimation methods

In order to test our estimation methods, we first conducted a simulation test with manipulated error terms. We use the estimated results of the nine species as known parameters (the known values in Fig. 5) to generate new datasets using the C4 photosynthesis equations based the first assumption of electron transport and adding error terms to the assimilation rates. The error terms were randomly drawn from a normal distribution of mean zero and standard deviation of 0.1 or 0.2 in an effort to simulate the inevitable random errors in the real measurements. Estimating simulated data sets gave us an idea about how likely we can capture the real parameters of the species given unavoidable errors in measurements. The results show that robust estimation results for Vcmax, J, Vpmax, and Rd can be obtained (Fig. 5A, B, C, D). However, some estimation results of gm and kCA show some deviation from the real values (Fig. 5E, F).

Fig. 5
figure 5

Simulation tests for the estimated parameters (AVcmax, BJ, CVpmax, DRd, Egm, and FkCA) using estimation methods with and without carbonic anhydrase reaction (With CA and Without CA). Datasets are generated by adding random errors for the modeling results using the known photosynthesis parameters of nine species. These known photosynthesis parameters are the true values in the x-axis and are used to compare with the newly estimation parameters. The small error refers to error term randomly chosen with mean 0 and standard deviation of 0.1 and the bigger error refers to error term with randomly chosen mean 0 and standard deviation of 0.2. The line in the figure shows the 1:1 line

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To test whether our estimation method could give accurate predictions across typical prediction scenarios, (CO2 ranging from 20 to 60 Pa), we performed out of sample tests for our nine target species. To perform these tests, we removed five points of CO2 concentrations between 20 and 60 Pa range out of the A/Ci curves and used the rest of the A/Ci curves to estimate parameters. And then we used these parameters to predict the assimilation rate under the CO2 concentrations we took out before and calculated the estimation errors. In general, the estimation errors for all our species were small (Table 3).

Table 3 Out of sample test results
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We tried to compare our estimation methods with in vitro measurements or other estimation methods using isotopic analysis, especially for Zea. Our estimation results for Zea obtained similar Vcmax with the in vitro estimated Rubisco activity of Pinto et al. (2014) (33 µmol m−2 s−1); however, the estimated value for Vpmax is a little lower than the in vitro PEPC activity measurement (83 µmol m−2 s−1) with a difference of around 20 µmol m−2 s−1. This discrepancy could be related to the aqueous environmental differences in vitro versus in vivo. For species of the Panicum family with NAD-ME subtype, P. virgatum and P. amarum in the current study and P. coloratum in Pinto et al. (2014), the estimated Vcmax and Vpmax are quite consistent with the in vitro measurements (Vcmax of 33 µmol m−2 s−1 and Vpmax of 42 µmol m−2 s−1). Ubierna et al. (2017) reported the gm for Zea ranged from 1.69 ± 0.17 to 8.19 ± 0.80 µmol m−2 s−1 Pa−1 using 18O and in vitro Vpmax. Our estimation method fitted a gm for Zea of 7.34 µmol m−2 s−1 Pa−1, which falls into the range of their measurements. Barbour et al. (2016) reported a higher mesophyll conductance of 17.8 µmol m−2 s−1 Pa−1 for Zea using 18O measurements.

Validating transition point range

We used chlorophyll fluorescence measurements from seven C4 species to test whether the upper and lower boundary CO2 concentrations, CaL and CaH, are reasonable (Table 4). The apparent quantum efficiency of PSII electron transport was calculated with ΔF/Fm′ = (Fm′ − Fs) Fm′ (Genty et al. 1989). Fluorescence analysis (Baker et al. 2007) is a powerful tool for identifying the limitation states of C3 species (Sharkey et al. 2007). If Chlorophyll fluorescence is increasing with increasing CO2, An is limited by Rubisco carboxylation rate; when Chlorophyll fluorescence stays constant with increasing CO2, An is limited by RuBP regeneration. For C4 species, however, the situation is more complicated. Since Vp could be limited by Vpr and Vpc (Eq. (9) in Supplementary Methods). Part of the RuBP carboxylation limited condition and RuBP regeneration limited condition for the C3 cycle will mix together, leading to a linear increase of fluorescence with increasing of CO2, but of a small slope (Fig. S2). Thus, we can only obtain two boundaries of CO2 concentrations. Below the lower boundary, A and fluorescence increases with increasing Ci with a steep slope and A is RuBP carboxylation and PEP carboxylation limited (RcPc); above the higher boundary, A and fluorescence is relatively constant along with the increase of Ci and A is RuBP regeneration and PEP regeneration limited (RrPr). We measured fluorescence to test whether the upper and lower boundary CO2 concentrations, CaL and CaH, are reasonable. It seems all the CaL are above 14 Pa and all the CaH are below 65 Pa (Table 4). These results suggest that 10–65 Pa is a reasonable range for the transitional point.

Table 4 CO2 concentration boundaries result for assimilation-limited conditions from fluorescence measurements for seven species
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Discussion

The photosynthetic parameters from the estimation method are good indicators for the biochemical and biophysical mechanisms underlying the photosynthesis processes of plants. Together with photosynthesis models, they can provide powerful information for evolutionary and ecological questions in both physiological and ecosystem response to natural environmental variation and climate change, to illustrate evolutionary trajectory of C4 pathway, as well as in efforts to improve crop productivity (Osborne and Beerling 2006; Osborne and Sack 2012; Heckmann et al. 2013). Photosynthetic parameters represent different physiological traits, and comparison of these parameters within a phylogenetic background could help us to understand the further divergence of lineages and species through evolutionary time. Additionally, the response of productivity and carbon cycle of vegetation towards the future climate change depends heavily on photosynthesis parameter estimation as input parameters.

Each of the two different fitting procedures has advantages and disadvantages. Yin’s method (Supplementary material II) uses the explicit calculation of assimilation rate and consequently gives lower estimation error. However, it needs a more accurate assignment of limitation states, especially at the lower end. Thus, Yin’s method will be preferable if one has additional support (e.g., fluorescence measurements) to define the limitation states; otherwise, the Yin’s method may give unbalanced results (Fig. 3). However, Sharkey’s method (Supplementary material I) usually can avoid unbalanced results even without ancillary measurements. Thus, it is better to use both procedures to support each other to find more accurate results. For example, one can first use Sharkey’s method to get estimation results and limitation states, and then use them as initial values for Yin’s method.

Our estimation methods yielded similar results when using models with and without carbonic anhydrase reaction processes. Although carbonic anhydrase activity may well be a limiting step for C4 cycle (von Caemmerer et al. 2004; Studer et al. 2014; Boyd et al. 2015; Ubierna et al. 2017), its limitation did not greatly affect assimilation rates in this study. Including the carbonic anhydrase reaction makes the model more complex and difficult to get an explicit solution; therefore, the model without carbonic anhydrase could be used as a simplified form yielding flawed but ‘nearly correct’ predicted values as a part of larger models. However, carbonic anhydrase limitation of C4 photosynthesis needs the further assessment from physiological or biochemical perspectives, and our estimation method provides another way to derive carbonic anhydrase parameters, which were comparable with in vitro measurements (Boyd et al. 2015). It is possible if a machine with better low CO2 control (e.g., Li-cor 6800) is used, carbonic anhydrase may become more limiting at extremely low CO2 concentrations. In addition, our results for models with and without carbonic anhydrase activity support the proposition of Cousins et al. (2007) that carbonic anhydrase activity may not be a limiting factor for A/Ci curves of C4 plants.

Our results show that despite a clear difference between the assumptions of how the products of electron transport are distributed, the results were similar and comparable with studies using different models under measurements of high light intensity. The bump in the second model happens in RrPc. In RrPc, assimilation is limited by RuBP regeneration and PEP carboxylation; therefore, PEP regeneration is not reaching Vpr, and the extra electron transport in PEP regeneration could be freely assigned to RuBP regeneration. This effect will weaken as PEP carboxylation increases. However, under lower photosynthetic photon flux density, assimilation rate will be limited more by electron transport, and the separate assumptions concerning electron transport may start to show divergent results. Such divergence of predictions under low or high light could provide a way to test assumptions about electron transport further in the future. Such a test should be done for multiple species as different species may follow different assumptions.

It is worth highlighting other assumptions that upon which our estimation methods rely. First, our estimation methods share with previous methods an underlying assumption that dark and light reactions optimally co-limit the assimilation rate (Sharkey et al. 2007; Yin et al. 2011b; Ubierna et al. 2013; Bellasio et al. 2015). This requires that there is some kind of optimization of nitrogen allocation of RuBP carboxylation and RuBP regeneration. The optimality assumption is intuitive as there should be some mechanism to balance the resource distribution between dark and light reactions to avoid inefficiency. Nonetheless, it is possible that there is redundancy in nitrogen allocation in one reaction, which can cause the photosynthesis rate to be always limited by the dark or light reactions. Second, we did not differentiate between C4 subtypes in our model as we assumed electron transport is limited by ATP production and there is a similar ATP cost for different C4 subtypes (Hatch 1987). Such an assumption can be relaxed further to build exclusive estimation methods for different subtypes by considering mechanistic details. Third, we assumed the parameters, Kc, Ko, γ*, Kp, α, and gbs to be the same for different species. To get more accurate estimation results, one can use species-specific parameters obtaining from other measurements.

Researchers need to pay additional attention to interpret the estimated parameters. For the electron transport, J, our methods do not assume saturated light intensity. Instead, J is defined as maximal electron transport for the specific light intensity under which the A/Ci curve is obtained. Using our estimation methods, one can estimate “realized” J at different light intensities (e.g., a light intensity encountered by plants in natural habitat, but light intensity should not be very low because A/Ci curve may not be reliable). To estimate maximal electron transport rate for saturated light (Jmax), one can obtain the A/Ci curve under saturated light condition, where the “realized” J would be equal to the Jmax. A similar statement also applies to the estimation of Vcmax. It is possible that low light intensity may not maximize the Rubisco activation state. Thus, in such conditions, the estimation methods would estimate “realized Vcmax” under a specific light intensity, instead of the real Vcmax. Such interpretations for estimated parameters also pertain to the previous estimation methods. Similar with other C4 estimation methods (Yin et al. 2011b; Ubierna et al. 2013; Bellasio et al. 2015), we did not estimate the triose phosphate utilization (TPU). TPU has been found to limit assimilation rate when the A/Ci curve reaches a plateau and show a little decrease in C3 (Sharkey et al. 2007). Since we did not detect a decrease in our measurements and it is not clear how TPU affects C4 assimilation, we did not take it into consideration. However, TPU does deserve further consideration in future studies.

The photosynthetic parameters from the estimation method used together with photosynthesis models can provide information and inspiration about the evolutionary and physiological importance of different aspects of the C4 syndrome (Osborne and Sack 2012; Heckmann et al. 2013), which can be investigated by empirical measurements. Several examples emanate from our simulation analysis: (1) α represents the fraction of O2 evolution from photosynthesis occurring in the bundle sheath cells [Eq. (4) in Supplementary Methods] and any α > 0 means that O2 will accumulate in the bundle sheath cells, due to low gbs Both the sensitivity analysis and the simulation analysis showed the change of α did not affect the estimated parameters and assimilation rates, because the high Cbs created by C4 carbon concentrating mechanism overcame any increase of Obs and did not lead to high photorespiration. Thus, the compartmentation of O2 evolution may not have played an important role in the evolution of C4 photosynthesis. (2) A lower Rubisco specificity factor [γ*; Eq. (11) in Supplementary Methods] means lower specificity for O2, higher specificity for CO2, and lower photorespiration. In C3 species, selection for Rubisco with lower specificity to O2 and high specificity of CO2 can increase the carbon gain. However, there is a trade-off between the specificity of Rubisco for CO2 and its catalytic rate (Savir et al. 2010; Studer et al. 2014). Based on this trade-off, we can hypothesize that since C4 elevates CO2 around Rubisco relative to the O2 concentration, maintaining low specificity might be optimal, in order to get high catalytic rate of the enzyme to reach higher assimilation rate as shown by the empirical measurements of Sage (2002) and Savir et al. (2010). Our simulation analysis showed the increase of specificity for CO2 (decrease of γ*) did not increase the assimilation rate much, which indicates the selection upon Rubisco specificity in C4 plants should be relaxed. (3) gbs represents CO2 leakage from bundle sheath to the mesophyll cell, and changes in gbs significantly change the assimilation rate and Cbs. Therefore, avoiding CO2 leakage was of great importance for the evolution and efficiency of C4 photosynthesis pathway (Brown and Byrd 1993; Ubierna et al. 2013; Kromdijk et al. 2014).

Although we have shown that parameter estimation can be achieved solely with A/Ci curves, it is easy to combine our methods with ancillary measurements to yield more accurate estimation results by defining the parameters as estimated or known or add additional constraints (Supplementary Material IV). Yin et al. (2011b) proposed a method to obtain Rd from the fluorescence-light curve, since the method used for C3 species, the Laisk method, is inappropriate (Yin et al. 2011a). Additional measurement of dark respiration could be an approximation for Rd or could help to provide a constraint for estimating Rd in our estimating method. Ubierna et al. (2017) discussed the estimation method of gm using instantaneous carbon isotope discrimination. With external measurement results, one can change estimated parameters (such as Rd, gm and J) as input parameters, instead of output parameters, in this curve fitting method (Supplementary material IV). Additional methods, such as in vitro measurements (Boyd et al. 2015; Pedomo et al. 2015) and membrane inlet mass spectrometry (Cousins et al. 2010) of Vcmax, Vpmax, and carbonic anhydrase activity can also provide potential parameter values. Furthermore, if some output parameters are determined in the external measurements, one can also relax the input parameters (such as gbs) and make them estimated parameters (Supplementary material IV).

Conclusion

We have developed new, accessible estimation tools for extracting C4 photosynthesis parameters from intensive A/Ci curves. Our estimation method is based on an established estimation protocol for C3 plants and makes several improvements upon C4 photosynthesis models. External measurements for specific parameters will increase the reliability of estimation methods and are summarized independently. We developed estimation methods with and without carbonic anhydrase activity. The comparison of these two methods allows for an estimation of carbonic anhydrase activity, and further shows that the method that did not consider carbonic anhydrase activity was a sufficient simplification for C4 photosynthesis. We tested two assumptions related to whether the electron transport is freely distributed between RuBP regeneration and PEP regeneration or certain proportions are confined to the two mechanisms. They show similar results under high light. Simulation test, out of sample test, fluorescence analysis, and sensitivity analysis confirmed that our methods gave robust estimation especially for Vcmax, J, and Vpmax.