A hierarchical Bayesian approach for estimation of photosynthetic parameters of C3 plants



    Corresponding author
    1. Department of Biological Sciences, Texas Tech University, Lubbock, TX 79409, USA,
    2. Departments of Botany and Statistics, University of Wyoming, Laramie, WY 82071, USA and
    Search for more papers by this author

    1. Departments of Botany and Statistics, University of Wyoming, Laramie, WY 82071, USA and
    Search for more papers by this author

    1. Department of Biological Sciences, Texas Tech University, Lubbock, TX 79409, USA,
    2. Centre for Plants and the Environment, University of Western Sydney, Richmond, NSW 2753, Australia
    Search for more papers by this author

L. D. Patrick. Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721, USA. Fax: +1 520 621 9190; e-mail: Lpatrick@email.arizona.edu


We describe a hierarchical Bayesian (HB) approach to fitting the Farquhar et al.model of photosynthesis to leaf gas exchange data. We illustrate the utility of this approach for estimating photosynthetic parameters using data from desert shrubs. Unique to the HB method is its ability to simultaneously estimate plant- and species-level parameters, adjust for peaked or non-peaked temperature dependence of parameters, explicitly estimate the ‘critical’ intracellular [CO2] marking the transition between ribulose 1·5-bisphosphate carboxylase/oxygenase (Rubisco) and ribulose-1,5-bisphosphate (RuBP) limitations, and use both light response and CO2 response curve data to better inform parameter estimates. The model successfully predicted observed photosynthesis and yielded estimates of photosynthetic parameters and their uncertainty. The model with peaked temperature responses fit the data best, and inclusion of light response data improved estimates for day respiration (Rd). Species differed in Rd25 (Rd at 25 °C), maximum rate of electron transport (Jmax25), a Michaelis–Menten constant (Kc25) and a temperature dependence parameter (ΔS). Such differences could potentially reflect differential physiological adaptations to environmental variation. Plants differed in Rd25, Jmax25, mesophyll conductance (gm25) and maximum rate of Rubisco carboxylation (Vcmax25). These results suggest that plant- and species-level variation should be accounted for when applying the Farquhar et al. model in an inferential or predictive framework.


Mechanistic photosynthetic models based on phenomenological descriptions of the underlying biochemical reactions have broad applications in the field of plant ecophysiology. These models may be used to determine the impact of varying environmental conditions – including those predicted to be affected by climate change – on the biochemistry of photosynthesis and carbon acquisition at the leaf and plant levels (e.g. Wohlfahrt, Bahn & Cernusca 1999a). Further, mechanistic photosynthetic models are often used to parameterize vegetation components in process models applied at scales ranging from plant canopies (Baldocchi & Harley 1995; dePury & Farquhar 1997) to landscapes (Kimball et al. 2000; Williams et al. 2001) to continents (Sellers et al. 1996; Foley et al. 1998; Pitman 2003). As such, photosynthetic parameters associated with leaf-level models provide valuable, mechanistic information for predicting large-scale effects of future climate change on terrestrial ecosystems.

In the most commonly used mechanistic model of C3 photosynthesis, carbon assimilation is limited by one of three biochemical processes (Farquhar, von Caemmerer & Berry 1980). That is, the rate of photosynthesis is modelled as the minimum of three functions: (1) the saturation of ribulose 1·5-bisphosphate carboxylase/oxygenase (Rubisco) with respect to carboxylation; (2) electron transport limiting the regeneration of ribulose-1,5-bisphosphate (RuBP); and (3) the amount of triose phosphate exported from the chloroplast. Through fitting this ‘Farquhar et al.’ model (Farquhar et al. 1980) to photosynthetic gas exchange measurements (e.g. photosynthetic responses to changes in intercellular CO2 concentrations; ACi curve), the following parameters can be estimated: the maximum Rubisco carboxylation rate (Vcmax), the maximum rate of electron transport (Jmax), mitochondrial respiration in the light (Rd) and mesophyll conductance (gm). Because of the crucial role these parameters play in scaling photosynthesis, it is essential that accurate estimates of these parameters are obtained when fitting mechanistic photosynthetic models to leaf-level empirical data.

Importantly, estimates of the photosynthetic parameters of interest (e.g. Vcmax, Jmax) are sensitive to the statistical estimation methods used to fit the Farquhar et al. model (Manter & Kerrigan 2004; Dubois et al. 2007; Miao et al. 2009). These fitting methods are not yet consistent in the literature and can be categorized into six distinct methods (see Miao et al. 2009 for a comprehensive review). The primary difference among these methods is the statistical approach used to determine the transition Ci value (Ccrit; the value of Ci used to differentiate between Rubisco and RuBP limitations). In addition to the statistical fitting approach, the accuracy of fitting the Farquhar et al. model also relies on: (1) correct representation of the kinetic properties of Rubisco (Sharkey et al. 2007), often assumed to be relatively conserved in C3 plants (von Caemmerer 2000); (2) incorporation of temperature dependencies of parameters, which are described by either exponential or peaked exponential functions (Leuning 1997, 2002; Wohlfahrt et al. 1999b; Medlyn, Loustau & Delzon 2002; Kattge & Knorr 2007); (3) incorporation of gm (Niinemets et al. 2009a; Pons et al. 2009); and (4) accounting for species- and plant-level differences in both fixed parameters (e.g. kinetic constants), estimated parameters (e.g. Vcmax, Jmax) and temperature dependencies.

Indeed, empirical studies have shown that model parameters do vary by plant and species as a result of genetic or environmental variation. For example, mesophyll conductance (gm), which partially controls the transfer of CO2 from the mesophyll intercellular space to the site of carboxylation, has been shown to respond to light and leaf anatomy (Evans & von Caemmerer 1996; Tholen et al. 2008; Warren 2008; Loreto, Tsonev & Centritto 2009). This variability is important to recognize, because variation in gm has been linked to changes in photosynthetic capacity (von Caemmerer & Evans 1991; Lloyd et al. 1992; Loreto et al. 1992; Niinemets et al. 2009b). Rubisco kinetic constants (e.g. Kc, Ko) also change across diverse species and environmental conditions (Tcherkez, Farquhar & Andrews 2006). Paradoxically, while variability in such model parameters has been widely documented, many studies have not yet incorporated intra- and interspecific parameter variability into procedures for fitting the Farquhar et al. model. Subsequently, application of this model may incorrectly conclude that significant differences exist in parameter estimates for plants, species or treatments, thereby limiting the accuracy of this popular photosynthetic model.

In light of the need for accurate plant- and species-level estimates of photosynthetic parameters under varying environmental conditions, we describe a statistically rigorous method to estimate C3 photosynthetic parameters. Specifically, we implemented a hierarchical Bayesian (HB) framework that couples the Farquhar et al. model with multiple photosynthetic data sets, allowing estimation of plant- and/or species-level variability of kinetic constants and biochemical parameters. While other gas exchange data (e.g. AQ; light response curves) are often collected in conjunction with ACi curves, these data are rarely incorporated into the fitting procedure, although they may help to inform the biochemical processes regulating photosynthesis (von Caemmerer 2000). Here, we use both ACi and AQ data to simultaneously estimate all photosynthetic parameters, including a Ccrit value specific to each curve. Another attractive feature of the HB approach is that we can explicitly accommodate the nested sampling design such that the photosynthetic parameters are modelled hierarchically (e.g. curves/plants nested in species).

To illustrate and evaluate the HB approach, we used field data collected from four species of common North American desert shrubs. Desert plants were chosen because their photosynthetic responses differ greatly from temperate forest and agricultural species – which are most commonly studied with respect to parameterizing the Farquhar et al. photosynthetic model – based on sensitivity to water limitation and temperature (Ogle & Reynolds 2002). By comparing HB model parameter estimates for desert plants with estimates in the literature from temperate forest and crop plants, we highlight the potential importance of incorporating: (1) flexibility in defining kinetic and biochemical parameter values; (2) plant- and species-specific parameter variability when estimating photosynthetic parameters; and (3) more rigorous statistical methods for analyzing photosynthetic data in the context of mechanistic models such as the Farquhar et al. model.


Study sites, plants and field measurements

Photosynthetic data for shrub species used in this study were collected at three study sites, each within a distinct North American desert ecosystem. In the Great Basin Desert, the field site was located near the Sierra Nevada Aquatic Research Laboratory (SNARL) of the Valentine Eastern Sierra UC (University of California) Natural Reserve, in eastern California near the city of Mammoth Lakes (37°37′N, 118°50′W, elevation 2100 m). Mean annual precipitation (MAP) is approximately 386 mm, most of which is received between October and March as snow or convective rainstorms. For more detailed SNARL site characteristics, see Gillespie & Loik (2004) and Loik (2007). In the Mojave Desert, data were collected at the Mojave Global Change Facility (MGCF) located on the Nevada Test Site (36°49′N, 115°55′W, elevation 970 m). MAP at the MGCF is about 138 mm, occurring primarily during the winter months (Hunter 1995), with highly episodic summer precipitation and a low relative frequency of large rainfall events. For a more detailed MGCF site description, see Barker et al. (2006). In the Chihuahuan Desert, the study site was located in a sotol grassland ecosystem within the Pine Canyon Watershed in Big Bend National Park (BIBE), Texas (29°5′N, 103°10′W, elevation 1526 m). MAP is about 370 mm, with the majority of annual precipitation occurring during the summer months and arriving as monsoonal rains. For a more detailed BIBE site description, see Patrick et al. (2007, 2009) and Robertson et al. (2009).

At SNARL, measurements were collected on Artemisia tridentata (ARTR; Asteraceae, n = 5 plants) and Purshia tridentata (PUTR; Rosaceae, n = 5 plants), two native C3 woody shrubs in the sagebrush scrub ecosystem. These species are codominant in the ecosystem, representing about 80% of the plant cover (Loik 2007). At MGCF, measurements were made on the native, dominant C3 evergreen shrub, Larrea tridentata (LATR; Zygophyllaceae, n = 4 plants). At BIBE, measurements were made on the dominant C3 perennial shrub, Dasylirion leiophyllum (DALE; Liliaceae, n = 3 plants).

During the 2005 growing season (May–August), AQ curves (Supporting Information Fig. S1) were measured on each study plant at each site using a portable photosynthetic system (model Li-6400; Li-Cor, Lincoln, NE, USA). During the 2006 growing season (May–August), both ACi and AQ curves were measured on the same study plants as the previous year. ACi curves were measured at saturating irradiance (1500 µmol m−2 s−1) for 12 CO2 concentrations in the following order (first to last measurement): 0, 50, 100, 150, 200, 250, 380, 500, 700, 900, 1200 and 1500 µmol mol−1. To ensure steady-state conditions, the plants were allowed to acclimate to ambient CO2 (380 µmol mol−1) in the gas exchange chamber for approximately 5 min before beginning each ACi curve, and then logged. This logged measurement was then compared to the measurement at identical [CO2] in the middle of the curve sequence to confirm full enzyme activation. It took approximately 45 min to complete a single ACi curve. AQ curves were measured at ambient [CO2] (400 µmol mol−1) for 12 light levels in the following order (first to last): 2000, 1500, 1000, 800, 600, 400, 300, 200, 100, 70, 40 and 0 µmol m−2 s−1. The plants were allowed to acclimate to changes in light intensity for approximately 2–3 min before measurements were logged; it took about 35 min to complete an entire AQ curve.

Across all plants and curve types, leaf temperature and relative humidity were recorded and set to ambient values. Average temperature and relative humidity were 26.1 °C and 48.9%, respectively, and ranged from 11.2 to 44.4 °C, and from 7.2 to 94.1%, respectively. All curves were measured from the early morning to the early afternoon when day-time air temperature was near its daily minimum, vapour pressure deficit was relatively low and stomata were responsive to changes in CO2 and light. All measurements of photosynthesis and stomatal conductance were corrected for leaf area. A total of 17 ACi curves and 37 AQ curves were measured across all species, providing 696 observations of photosynthesis.

HB model of photosynthesis

Plant photosynthetic data were analysed within an HB framework (Clark 2005; Clark & Gelfand 2006). This approach has been successfully used to synthesize ecological data (e.g. Clark & LaDeau 2006; Ogle & Barber 2008), and it has proven to be exceptionally useful for making inferences about ecosystem and plant physiological responses (Cable et al. 2008, 2009; Ogle et al. 2009; Patrick et al. 2009). We propose that the HB approach is advantageous for modelling photosynthesis because it can: (1) simultaneously estimate unknown parameters – related to biochemical limitations of photosynthesis – while also accounting for uncertainty in measurements and parameters (Carlin, Clark & Gelfand 2006; Ogle & Barber 2008); (2) accurately estimate common photosynthetic parameters without the need for subjective determination of thresholds for limiting biochemical processes [e.g. it allows us to avoid setting a fixed and potentially arbitrary transition intracellular CO2 (Ccrit) value to separately estimate Vcmax and Jmax]; (3) simultaneously incorporate a variety of data types (e.g. ACi and AQ curve data) to arrive at parameter estimates and parameter variability; (4) allow for borrowing of strength between curves to help estimate population-level parameters of interest (e.g. species-specific biochemical parameters); and (5) incorporate prior information for biochemical parameters that may not be well informed by the ACi and AQ response curve data, but that reflect appropriate levels of uncertainty based on variation in these parameters as reported in the literature. We emphasize that the HB model provides the statistical framework for fitting a process-based model such as the Farquhar et al. model to observational data. That is, we do not present modifications to the Farquhar et al. model, but describe a rigorous and statistically consistent methodology for confronting such a model with diverse data.

The HB model has three primary components: (1) the observation equation that describes the likelihood of observed photosynthesis data; (2) the process equation that describes the ‘true’ or mean photosynthetic response, based on the Farquhar et al. model of C3 photosynthesis, as well as process uncertainty associated with random effects; and (3) prior distributions for process model parameters (e.g. species effects) and variance terms. These three parts are combined to generate posterior distributions of all unknown parameters (see Wikle 2003; Clark 2005), including photosynthesis-related parameters and all variance/covariance terms. All probability distributions are parameterized according to Gelman et al. (2004). Table 1 includes a list of abbreviations and units for parameters used in the model. The model was run with two different photosynthetic data sets: (1) observations for ACi curves only (n = 207); and (2) observations from both AQ and ACi curves (n = 696) to determine if AQ curve data can improve estimates of photosynthetic parameters.

Table 1.  List of abbreviations used in the coupled hierarchical Bayesian (HB)–photosynthetic model, their definitions and units
Observed data for model input
 AobsRate of CO2 assimilation measured by the Li-Cor 6400µmol m−2 s−1
 CiobsIntercellular airspace CO2 partial pressure measured by the Li-Cor 6400Pa
 QobsPhotosynthetically active radiation measured by the Li-Cor 6400µmol m−2 s−1
 TobsLeaf temperature measured by the Li-Cor 6400°C
 PobsPressure measured by the Li-Cor 6400Pa
HB model parameters associated with process model
 inline imagePredicted rate of CO2 assimilation (see Eqn 1)µmol m−2 s−1
 AcRubisco-limited rate of CO2 assimilationµmol m−2 s−1
 AjElectron transport limited rate of CO2 assimilationµmol m−2 s−1
 E (Eg, Em, Er, Ekc, Eko, Ev, Ej)Activation energy used in Arrhenius temperature functionkJ mol−1
 fSpectral light quality factor 
 gmConductance for CO2 diffusion from intercellular airspace to site of carboxylationµmol m−2 s−1 Pa−1
 H (Hgm, Hv, Hj)Deactivation factor used in Arrhenius temperature functionkJ mol−1
 JRate of electron transportµmol m−2 s−1
 Jmax (Jmax25)Maximal electron transport rate (standardized to 25 °C)µmol m−2 s−1
 Kc (Kc25)Michaelis–Menten constant for Rubisco for CO2 (standardized to 25 °C)Pa
 Ko (Ko25)Michaelis–Menten constant for Rubisco for O2 (standardized to 25 °C)kPa
 OPartial pressure of O2Pa
 Q2Photosynthetically active radiation absorbed by PSIIµmol m−2 s−1
 RUniversal gas constant (8.314 J K−1 mol−1)J K−1 mol−1
 Rd (Rd25)Mitochondrial respiration in the light (standardized to 25 °C)µmol m−2 s−1
 ΔSSgm, ΔSv, ΔSj)Entropy factor used in Arrhenius temperature functionJ mol−1 K−1
 TLeaf temperatureK
 ToptOptimum temperatureK (°C)
 Vcmax (Vcmax25)Maximum rate of Rubisco carboxylation (standardized to 25 °C)µmol m−2 s−1
 αFraction of PSII activity in the bundle sheath 
 Γ* (Γ*25)Chloroplastic CO2 photocompensation point (standardized to 25 °C)Pa
 θEmpirical curvature factor 
HB parameters associated with hierarchical priors and hyperpriors
 Y25Plant-level mean of any parameter (Y) standardized to 25 °C 
 µY25Species-level mean of any parameter (Y) standardized to 25 °C 
 µ*Y25Population-level mean of any parameter (Y) standardized to 25 °C 
 τPrecision (1/variance) parameter describing observation and measurement error 
 τYplantPrecision (1/variance) parameter describing plant-to-plant variation within species 
 τYsppPrecision (1/variance) parameter describing species-to-species variability 

The observation equation

The likelihood of all leaf-level photosynthesis data is based on the likelihood of individual observations of photosynthesis obtained from the Li-6400 (i.e. Aobs; µmol m−2 s−1). We assumed that the photosynthetic measurements could be described by a normal distribution, such that for observation i (i = 1, 2, . . . , N):


where inline image is the mean or predicted photosynthesis rate, and τ is the precision (1/variance) parameter that describes the variability in the photosynthetic observation or measurement errors.

The process model

The process model describes the predicted photosynthesis rate (inline image), which was specified according to the Farquhar et al. model of C3 photosynthesis (Farquhar et al. 1980). The dependence of potential electron transport rate on absorbed irradiance was specified according to Farquhar & Wong (1984). Triose phosphate limitation was not considered here because this process is expected to rarely limit photosynthesis and is not commonly included in models to estimate photosynthetic parameters (Wohlfahrt et al. 1999b; Medlyn et al. 2002; Dubois et al. 2007). Modifications for mesophyll conductance (gm) were included using quadratic equations as described by von Caemmerer & Evans (1991), von Caemmerer (2000) and Niinemets et al. (2009a). In addition, when values of Ci and internal oxygen concentration (O) were converted from µmol mol−1 to Pa, they were also corrected for pressure, because the pressure among measurement sites was different from standard pressure (range: 77.3–97.5 kPa). When both the ACi and AQ data sets were included, the Farquhar et al. model was still used to model the expected photosynthetic rate, and thus both data sets simultaneously informed parameters in the photosynthetic model. See Table 2 for a list of model equations and parameters used to describe inline image in Eqn 1.

Table 2.  List of equations used in the photosynthesis process model
Eqn no.Equation
  1. All equations are from Farquhar et al. (1980), Farquhar & Wong (1984) and von Caemmerer (2000). See Table 1 for abbreviations, definitions and units.

2.1inline image, if inline image
inline image, if inline image
2.2inline image
inline image
inline image
inline image
2.3inline image
inline image
inline image
inline image
2.4inline image
where θ = 0.7 (Evans 1989)
2.5inline image where α = 0.85 (von Caemmerer 2000) and f = 0.15 (Evans 1987)

Temperature dependencies of Rubisco's carboxylation and oxygenation rates affect photosynthesis (Bjorkman, Badger & Armond 1980), as do the temperature dependencies of Vcmax and Jmax (von Caemmerer 2000). Thus, temperature dependencies for all parameters (i.e. Kc, Ko, Γ*, gm, Rd, Vcmax and Jmax; Table 1) were chosen to follow an Arrhenius function (von Caemmerer 2000; Leuning 2002; Medlyn et al. 2002; Kattge & Knorr 2007) standardized to 25 °C. The general form of the Arrhenius function for parameter Y (where Y = Kc, Ko, Γ*, gm, Rd, Vcmax or Jmax) is:


where Y25 is the parameter at 25 °C, EY is the activation energy of Y, Tobs is the leaf temperature (K) measured by the Li-6400 and R is the universal gas constant (8.314 J mol−1 K−1). To include plant-level variation in the gm, Rd, Vcmax and Jmax parameters at 25 °C (i.e. the Y25s), these parameters were allowed to vary by curve (or plant). To account for potential species-level differences in temperature dependencies, the activation energies associated with the gm, Rd, Vcmax and Jmax parameters (Em, Er, Ev and Ej, respectively) were allowed to vary by species, such that for observation i, plant p and species s:


The notation s(p) is read as ‘s of p’, which represents the species identity associated with each plant (i.e. plant is ‘nested’ in species). Because the Rubisco kinetic properties at 25 °C (Γ*25, Kc25 and Ko25) and their associated activation energies (Eg, Ekc, Eko) have only been shown to vary by species (von Caemmerer 2000) and are generally not well informed by ACi or AQ data, we assume that these parameters vary at the species level such that:


While Eqns 2 and 3 were used for analyses of both data sets, it was also recognized that temperature response functions for gm, Vcmax and Jmax are commonly modelled (von Caemmerer 2000; Leuning 2002; Medlyn et al. 2002; Kattge & Knorr 2007) using the peaked Arrhenius function (Johnson, Eyring & Williams 1942). As such, we also ran the model using the combined ACi and AQ data set with the peaked Arrhenius functions to determine if model fit and parameter estimation were improved by incorporating this more flexible temperature response compared to the non-peaked function in Eqn 2. For this analysis, f1 in Eqn 2 was replaced with:


where EY is the activation energy, HY is the deactivation energy describing the rate of decrease for temperatures above the optimum temperature and ΔSY is an entropy factor. Once again, gm25, Vcmax25 and Jmax25 parameters were allowed to vary by plant, and because species-level variation has been observed in the temperature response parameters (i.e. E, H, ΔS) (Kattge & Knorr 2007), the associated parameters for gm (Em, ΔSgm, Hgm), Vcmax (Ev, ΔSv, Hv) and Jmax (Ej, ΔSj, Hj) also were allowed to vary between species, such that:


The prior model

The final stage in the HB modelling approach was the specification of the priors for the unknown parameters. Because many model parameters varied on a plant and/or species level, nested, hierarchical priors were chosen. The ability to have nested priors is another major advantage of the HB approach because it allows parameters within a given level (e.g. across plants within species or across species) to inform or ‘borrow strength’ from each other (Carlin et al. 2006). Moreover, the nesting of plants within species within an overall population of desert shrubs describes a natural hierarchy that reflects the sampling design. Thus, under this framework, plant-level parameters – which are directly related to individual curve data – are assumed to be nested within species, such that for any plant-level parameter at 25 °C (Y25 = gm25, Rd25, Vcmax25 and Jmax25):


where µY25s(p) is the species-level mean, and τYplant is the precision (1/variance) parameter that describes plant-to-plant variability within a species. The species-level parameters were then assumed to be nested within an overall population, such that:


where µ*Y25 is the population-level mean, and τYspp is the precision parameter that describes species-to-species variability within the desert shrubs studied here. Standard, independent and relatively non-informative (diffuse) priors were employed for the population-level mean parameters (the µ*Y25); that is, we used normal densities with large variances (small precisions). Folded-Cauchy (i.e. a Student's t-distribution with one degree of freedom) densities were assigned as priors for all standard deviations (σ = 1/√τ, where τ is the precision parameter of interest) as suggested by Gelman (2006). Another advantage of the HB framework is that we were able to incorporate informative priors for the Michaelis–Menten parameters (Kc and Ko), CO2 compensation point (Γ*) and the Arrhenius temperature function parameters (E, ΔS, H); that is, we assigned normal distributions centred on values reported in the literature (von Caemmerer 2000) and used small precisions based on coefficients of variation (CV = standard deviation/mean) around 15% (Supporting Information Table S1).

Finally, another advantage of this HB approach is that we were able to specify which parameters should be informed by which data set. For example, we do not expect the AQ data to contain sufficient information about Vcmax because the data were collected under ambient CO2, and Ci was approximately constant. Thus, within the HB model code (see WinBUGS implementation below), we employed the ‘cut’ function to sever the feedback between, for example, the AQ curve data and Vcmax. This resulted in posterior distributions for parameters describing Vcmax that were solely informed by the ACi data, but the uncertainty in the Vcmax values was propagated to the predicted photosynthetic values associated with the AQ data. Using this approach, we assumed that Kc, Ko, Γ*, gm, Vcmax and Jmax were solely informed by the ACi data, and Rd, Ccrit and temperature dependence parameters were informed by both data sets (i.e. did not use the cut function with these parameters). Although ACi curve data generally do not provide sufficient data on Rd (i.e. all Aobs measurements were made under constant, high light), we still allow the ACi data to inform Rd because many studies may only measure ACi curves in an effort to estimate photosynthetic parameters, including Rd.

The HB model defined by Eqns 1–8 was implemented in the Bayesian statistical software package WinBUGS (Lunn et al. 2000). WinBUGS code for the HB model has been provided as supplementary material. Three parallel Markov chain Monte Carlo (MCMC) chains were run for 30 000 iterations each, and the BGR diagnostic tool was used to evaluate convergence of the chains to the posterior distribution (Brooks & Gelman 1998; Gelman 2004a). The burn-in samples (first 4000) were discarded, and the remaining samples (after convergence) were thinned every 20th iteration, yielding an independent sample of 3000 values for each parameter from the joint posterior distribution (Gelman 2004b; Gamerman & Hedibert 2006). Model goodness-of-fit was evaluated by using Eqn 1 to generate replicated data for the observed values (Aobsi) (Gelman et al. 2004), yielding posterior predictive distributions for each observation. If the model perfectly predicted the data, all observed-versus-predicted (posterior means for replicated data) points would lie exactly on the 1:1 line. We compared models (e.g. ACi data model with non-peaked temperature functions versus ACi data model with peaked temperature functions) by computing the posterior predictive loss (D) for each model (Gelfand & Ghosh 1998). D is a model comparison statistic that accounts for model predictive ability (‘goodness-of-fit’) while penalizing for model complexity, and the model with the lower D value is preferred.


Model goodness-of-fit and model comparison

The coupled HB–photosynthetic model fit the data well for ACi data only and for the combined ACi and AQ data using both non-peaked and peaked temperature functions (Table 3). For example, points in the plots of observed-versus-predicted photosynthesis fell tightly along the 1:1 line (data not shown). When comparing among models (for a particular data set or data set combination), models that incorporated peaked temperature responses had lower D values compared to models that used non-peaked temperature functions (Table 3). Thus, the results reported below, unless otherwise specified, are from models that employed peaked temperature functions for photosynthetic parameters of interest (gm, Rd, Vcmax and Jmax).

Table 3. r2 Values for observed versus predicted photosynthesis, posterior predictive loss (D) and estimates of the uncertainty in the D values (i.e. approximate estimates of the 2.5th and 97.5th percentiles) obtained from the hierarchical Bayesian (HB) model using ACi data only and combined ACi and AQ data with either non-peaked or peaked temperature response functions for photosynthetic parameters
Data/model combinationr2D2.5%97.5%
  1. Note that comparison of the D values is only relevant within a given data set.

ACi only (non-peaked temp.)0.99204.7157.4265.7
ACi only (peaked temp.)0.99199.3150.3261.4
ACi and AQ (non-peaked temp.)0.7631 96022 82041 250
ACi and AQ (peaked temp.)0.8722 15013 51032 750

Utility of AQ data for estimates of biochemical parameters

While the model goodness-of-fit results were not statistically different for both ACi data only, and ACi and AQ data combined (Table 3), the addition of AQ data greatly improved estimation of Rd25. Using both data sets, there was a high frequency of positive posterior mean estimates of Rd25, while in the model that used only ACi data, the lower credible intervals and the posterior means for Rd25 were often negative (Fig. 1). While we did not estimate Rd25 using only AQ data in this study, we expect that our model estimates for ACi and AQ data combined would be similar to AQ only estimates given that ACi data collected here were not able to directly inform Rd25 because of measurement at high light. The inclusion of AQ data did not improve estimates of Vcmax25 or Jmax25, but this is expected because the AQ data were not allowed to inform these parameters. Importantly, the ability of AQ data to inform biochemical parameters other than Rd25 was limited by our AQ measurements at ambient [CO2]. By measuring AQ curves at above or saturating [CO2], AQ data could be used to inform additional biochemical parameters of interest (e.g. Jmax25). However, because both data sets did inform a subset of parameters, the inclusion of AQ data had a slight impact on parameter uncertainty such that the posterior estimates for Vcmax25 or Jmax25 varied more between plant/species and their credible intervals were smaller when using only ACi data (Fig. 2; Supporting Information Fig. S2). The use of AQ data produced more variability in posterior mean estimates and slightly wider posterior credible intervals for Ccrit, but the range of Ccrit values was similar using either data set (Fig. 2; Supporting Information Fig. S2).

Figure 1.

Posterior mean estimates and 95% credible intervals for the plant-level mitochondrial respiration rate standardized to 25 °C (Rd25) from the peaked temperature model using only ACi data compared to estimates using both ACi and AQ data. Rd25 estimates from the combined data sets are greater than zero, while many of the Rd25 estimates for the ‘ACi only’ data are negative.

Figure 2.

Posterior mean estimates and 95% credible intervals for the plant-level values (i.e. Y25p in Eqn 2) given by the hierarchical Bayesian (HB) model that incorporated ACi data only and that implemented the peaked temperature response functions. Estimates are shown for (a) maximum rate of carboxylation standardized to 25 °C (Vcmax25), (b) maximum rate of electron transport standardized to 25 °C (Jmax25) and (c) plant-level transition intercellular partial pressure of CO2 (Ccrit). Plant-level estimates are grouped by species where ARTR = Artemisia tridentata, PUTR = Purshia tridentata, LATR = Larrea tridentata and DALE = Dasylirion leiophyllum. Symbols correspond to species where bsl00066 = A. tridentata, ● = P. tridentata, inline image = L. tridentata, ◆ = D. leiophyllum. Plants are considered different if the posterior mean for one plant is not contained in the 95% CI for another plant.

Parameters poorly informed by photosynthetic data

While the HB model provided an updated estimate of the Michaelis–Menten constant of Rubisco for O2 (Ko25) that was informed by ACi data, this was not the case for all energy of activation parameters (Ekc, Eko, Eg, Em, Er, Ev, Ej) and the peaked temperature parameters (Hgm, Hv, Hj). That is, the posterior means for the species-level E and H parameters were very similar to the means specified by their informative prior distributions (Supporting Information Tables S1 & S2). This indicates that these parameters were poorly informed by the photosynthetic data used (Table 4), or the data are in close agreement with the priors. The first explanation is more likely because these parameters became less identifiable under less informative priors.

Table 4.  Classification of posterior estimates for parameters in the photosynthetic model obtained from the hierarchical Bayesian (HB) analysis using both ACi and AQ data, and peaked temperature functions
ParameterWell informedPoorly informedSpecies-level differencesPlant-level differences
  1. Posterior means and credible intervals (CIs) were evaluated and compared to the prior means and CIs to determine the degree to which parameter estimates were informed by the photosynthetic data. Estimated parameters were classified as well informed by observed data if the posterior means were different from the prior means, had narrow CIs and/or exhibited plant- or species-level variation. Estimated parameters were poorly informed if the posterior estimates were similar to the prior means and had wide CIs. Parameter estimates were also evaluated to determine if they differed between species. An × in a cell indicates how a parameter was classified; an asterisk (*) indicates the parameter was constrained by an informative prior distribution and may become well informed if the priors are relaxed. (For example, although the prior was informative, the posterior mean estimate was different from the prior mean on a plant or species level.)

Kc25× ×n/a
Ko25 × n/a
Γ*25×*  n/a
Rd25× ××
Vcmax25×  ×
Jmax25× ××
Ekc × n/a
Eko × n/a
Eg × n/a
Em × n/a
Er × n/a
Ev × n/a
Ej × n/a
gm×  ×
ΔSgm×*  n/a
Hgm × n/a
ΔSv× ×n/a
Hv × n/a
ΔSj× ×n/a
Hj × n/a

Parameters well informed by photosynthetic data

The HB model produced posteriors for curve-specific Ccrit values (used to differentiate between Rubisco- and RuBP-limited rates) that were well informed by the data, regardless of the data set used. These Ccrit posterior means ranged from 11.9 to 22.1 Pa across all curves analysed (Fig. 2; Supporting Information Fig. S2). On a plant level, the HB model estimates of gm25, Rd25, Vcmax25 and Jmax25 were also well informed by the photosynthetic data (Figs 1–3). This was demonstrated by narrow credible intervals, posterior means that were quite different from the corresponding prior means, and species differences for some of the parameters (Rd25, Jmax25; Fig. 4). Moreover, many studies have observed a strong, linear correlation between Vcmax25 and Jmax25 (i.e. Jmax25 tends to be two times Vcmax25) (Wullschleger 1993; Leuning 1997; Medlyn et al. 2002; Kattge & Knorr 2007). We did not impose any restrictions on the relationship between Vcmax25 and Jmax25 in our HB model, and we used the model results to evaluate the relationship between these two parameters. The average ratio between the posterior means for plant-level Jmax25 and Vcmax25 was estimated to be 1.74 ± 0.37 across all species, and Vcmax25 and Jmax25 were strongly correlated when using ACi data only (Fig. 5). This correlation, however, becomes weaker upon incorporation of the AQ data.

Figure 3.

Posterior mean estimates and 95% credible intervals for the plant-level values for mesophyll conductance standardized to 25 °C (gm25) given by the hierarchical Bayesian (HB) model that incorporated ACi data only and that implemented the (a) peaked and (b) non-peaked temperature response functions for the four study species (see Fig. 2 for species abbreviations and symbol codes). Plants are considered different if the posterior mean for one plant is not contained in the 95% CI for another plant.

Figure 4.

Posterior mean estimates and 95% credible intervals for the species-level values (i.e. µY25s in Eqn 8) given by the hierarchical Bayesian (HB) model that implemented the peaked temperature response function and that incorporated both ACi and AQ data for (a) mitochondrial respiration standardized to 25 °C (Rd25), and only ACi data for (b) maximum rate of carboxylation standardized to 25 °C (Vcmax25), (c) maximum rate of electron transport standardized to 25 °C (Jmax25) and (d) Michaelis–Menten constant of ribulose 1·5-bisphosphate carboxylase/oxygenase (Rubisco) for CO2 standardized to 25 °C (Kc25) for the four study species (see Fig. 2 for species abbreviations and symbol codes). Species are considered different if the posterior mean for one species is not contained in the 95% CI for another species.

Figure 5.

The relationship between plant-level posterior mean estimates for Vcmax25 (maximum rate of carboxylation) and Jmax25 (maximum rate of electron transport) standardized to 25 °C. That is, the points are the posterior means for plant-specific values of Vcmax25 and Jmax25 based on the hierarchical Bayesian (HB) model that incorporated peaked Arrhenius temperature functions and used either (a) ACi data only or (b) ACi and AQ data combined. Symbols correspond to species where bsl00066 = Artemisia tridentata, ● = Purshia tridentata, inline image = Larrea tridentata, ◆ = Dasylirion leiophyllum.

The potential for photosynthetic data to inform model parameters, which are not typically allowed to vary on a plant and species level, was demonstrated by significant species differences in posterior mean estimates (Kc25, ΔSv and ΔSj) and variability in the width of their credible intervals (Fig. 4; Supporting Information Fig. S3). Species-level estimates of Γ*25 and ΔSgm were constrained by informative prior distributions, but may become well informed if the priors are relaxed. For example, species-level posterior mean estimates of Γ*25 ranged from 4.7 to 5.9 Pa (Supporting Information Fig. S3), which is greater than the value currently used in most photosynthetic models (3.86 Pa, von Caemmerer et al. 1994). While the posterior credible intervals for the two species-level estimates of Γ*25 did contain the prior mean value of 3.86 Pa, estimates for the other two species were significantly different, thus supporting the potential for this parameter to become well informed with relaxed priors. Interestingly, plant-level model estimates of gm25 using the non-peaked temperature functions had narrower credible intervals than model estimates of gm25 using the peaked temperature functions (Fig. 3). All results are summarized in Table 4.


We used an HB framework to couple the Farquhar et al. model with photosynthetic data to estimate plant- and/or species-level variability in kinetic constants, biochemical and photosynthetic parameters. The HB approach was successful in that it explicitly estimated the uncertainty or variability in the photosynthetic parameters, many of which are often held constant in applications of the Farquhar et al. model. For example, variability in parameters associated with temperature dependence (e.g. E) and Rubisco properties (e.g. Kc25) was accounted for and estimated in the HB model (see Supporting Information Table S2), in addition to estimating parameters more directly linked with photosynthesis (gm, Rd, Vcmax, Jmax).

These parameters could be estimated via the rigorous HB statistical approach that accommodated multiple types of response curve data and that incorporated simultaneous plant-level estimates of Ccrit. Ccrit is a critical parameter in the model because it dictates the value of Ci used to differentiate between Rubisco and RuBP limitations. Usually, Ccrit values are manually set at approximately 20–25 Pa based on work with Phaesolus vulgaris (von Caemmerer & Farquhar 1981; Wullschleger 1993; Wohlfahrt et al. 1999b; Bunce 2000), but this relatively ad hoc approach to fitting the Farquhar et al. model to gas exchange data and the assumption of a fixed transition value across species has been contested (Ethier & Livingston 2004; Dubois et al. 2007). The need for a plant- and/or species-specific Ccrit value has been supported by recent studies on trees (e.g. Douglas fir trees, Ethier & Livingston 2004), and was even demonstrated by early studies that transformed relationships between photosynthesis and chloroplastic CO2 (Cc) into rates of RuBP regeneration or actual rates of electron transport, and then plotted these values against Cc to determine Ccrit (von Caemmerer & Farquhar 1981; Kirschbaum & Farquhar 1984). The HB approach described herein provides a feasible method for estimating this key parameter.

In addition to our statistical fitting approach, this study is unique in its simultaneous use of both ACi and AQ curve data to inform estimates of photosynthetic parameters. With the inclusion of AQ data, estimates of Rd25 were positive and therefore more biologically realistic. Other studies that fit the Farquhar et al. model to ACi data have had difficulty in obtaining accurate or biologically realistic estimates of Rd25 (but see Dubois et al. 2007), and as such, Rd25 is sometimes not reported (Medlyn et al. 2002). Additionally, when using the combined data set, species differences in Rd25 were observed where A. tridentata had a significantly greater Rd25 (5.7 µmol m−2 s−1) than the other three species (1.5–2.8 µmol m−2 s−1). Although knowledge about the regulation of Rd is generally limited (Nunes-Nesi, Sweetlove & Fernie 2007), recent work using isotopic techniques has shown that Rd plays a role in sustaining photorespiratory nitrogen cycling and perhaps nitrate assimilation (Tcherkez et al. 2008). This implies that A. tridentata may differ in nitrogen-use efficiency or ATP requirements for the sucrose synthesis and tricarboxylic acid (TCA) cycle intermediates compared to other desert species (Tcherkez et al. 2008). Furthermore, in some photosynthetic models, Rd is modelled as a function of Vcmax, where Rd set at 0.01–0.02 times Vcmax (von Caemmerer 2000); this relationship has been invoked to account for correlations between Rd and leaf nitrogen. At similar temperatures, the posterior means for Rd25 were on average 0.03 times Vcmax25 and were within the range of reported literature values from in vivo measurements (Bernacchi et al. 2001; Warren & Dreyer 2006). By assimilating both ACi and AQ data, the HB approach estimated this ‘difficult’ parameter on both a species and plant level, and the estimates were consistent with values reported in the literature based on direct measurements of Rd (Bernacchi et al. 2001; Warren & Dreyer 2006).

The type of temperature response function needed to accurately fit the Farquhar et al. model to photosynthetic data is often species dependent (e.g. Medlyn et al. 2002). Subsequently, we compared model fit between the standard, exponential Arrhenius function (Eqn 2) and the peaked exponential function (Eqn 5), and model goodness-of-fit improved with the peaked function model. Some studies suggest that the peaked function is over-parameterized, thereby increasing the difficulty in estimating photosynthetic parameters (Harley et al. 1992; Dreyer et al. 2001; June, Evans & Farquhar 2004). Conversely, we found that the peaked function was best suited for our native desert plants, which are often exposed to hot and highly variable temperatures. However, it should be noted that posterior mean estimates of gm25 had tighter credible intervals and more plant-level variability using the non-peaked model compared to the peaked model. Poor estimates of gm using the peaked model may be caused by incorporating prior knowledge of gm temperature dependencies based on plants from mesic ecosystems. While estimates of gm for C3 herbaceous annuals and woody perennials in mesic ecosystems have been shown to be affected by environmental conditions such as soil water deficit (Flexas et al. 2002, 2009; Galle et al. 2009; Perez-Martin et al. 2009), few studies report how temperature affects gm in desert species. Further study of the effect of environmental variation on gm in desert plants is needed to correctly parameterize temperature dependency functions for these species.

Utilization of peaked temperature functions was further supported in that species differences were observed for estimates of the temperature function parameters ΔSv (with the inclusion of AQ data) and ΔSj (using ACi data only; Supporting Information Fig. S3). Interestingly, both ΔSv and ΔSj were given informative prior distributions (Supporting Information Table S1), but the data resulted in posteriors that differed from the priors and that varied among species. To explore the implications of these differences for species-specific temperature responses, we recognize that the optimum temperature for Vcmax and Jmax (Topt) is inversely proportional to ΔS (Medlyn et al. 2002; Kattge & Knorr 2007). For example, P. tridentata in the Great Basin Desert had greater values for ΔSv (lower Toptv) than plants in the Mojave Desert (L. tridentata) and Chihuahuan Desert (D. leiophyllym). P. tridentata had significantly greater values for ΔSj (lower Toptj) than all other species. Further, these differences in Topt may reflect differences in plant growth temperatures (Hikosaka, Murakami & Hirose 1999; Medlyn et al. 2002; Bernacchi, Pimentel & Long 2003; Onoda, Hikosaka & Hirose 2005). Because the Great Basin Desert is a cold desert, and the Mojave and Chihuahuan Deserts are hot deserts, the optimum temperature for maximum carbon assimilation in our study species may be related to growing season temperature. Indeed, in an analysis of 36 species, Kattge & Knorr (2007) found that plant growth temperature did not significantly affect Vcmax at a given base rate temperature, but did affect Topt for Vcmax.

Species differences were also observed for model estimates of Kc25 and Jmax25. Because Kc25 was assigned an informative prior distribution based on literature data that were the same for all species (Supporting Information Table S1), the photosynthetic curve data, and not the prior distribution, were the primary determinants of the posterior estimates of Kc25. Few modelling studies have estimated Michaelis–Menten parameters (Kc25, Ko25) because of the difficulty in collecting field data directly related to these parameters (but see von Caemmerer 2000; Ethier & Livingston 2004); interestingly, observed data influenced the posterior distributions for Kc25 in this study. While this parameter describes intrinsic properties of Rubisco and is generally assumed constant across species (von Caemmerer 2000), our results show that Kc25 should not be held constant across plants, species and functional types when estimating Vcmax and Jmax. Importantly, we did not assume constant values for the kinetic constants or the temperature response parameters, but rather used informative priors to account for variability, thereby obtaining more accurate estimates for parameters directly related to Vcmax and Jmax.

Posterior estimates of species-level Jmax25 showed that it was significantly greater in P. tridentata compared to A. tridentata and L. tridentata. Given that Jmax25 may be influenced by environmental conditions (e.g. light, soil moisture), these results suggest that P. tridentata may have comparatively greater access to resources compared to the other species. This is partially supported by the observation that P. tridentata has a bimodal rooting distribution (Loik 2007), and thus it may utilize both stable (deep) and ephemeral (near-surface) water sources. Its surface roots may also facilitate uptake of nutrients from shallow soil layers. Greater access to water and nutrients is expected to increase the efficiency of electron transport. Because species have varying strategies for adapting to different environments, we suggest that photosynthesis should be measured under a wide range of environmental conditions (e.g. rooting depth, high-temperature stress, nutrient limitation, low soil moisture) and subsequently analysed using the flexible HB approach described herein. Using this approach, one can explicitly acknowledge important sources of uncertainty and accommodate variation in environmental drivers, existing knowledge about the photosynthetic process and parameters and diverse data sets to obtain more accurate estimates of species-level photosynthetic parameters (e.g. Vcmax25, Jmax25).

In addition to accounting for species-level variability, results from our HB fitting approach highlight the importance of recognizing plant-level variability when estimating photosynthetic parameters using the Farquhar et al. model. Significant plant-level variation was observed for gm, Rd, Vcmax25 and Jmax25 (Figs 1–3). Because there was greater plant-level variation observed than species-level variation, this highlights the potential importance of small-scale variation in environmental variables for understanding photosynthetic responses in desert plants. Indeed, it has been well documented that Vcmax25 exhibits high variation as a function of species identity, nutrient availability, season, leaf age and leaf position within the canopy (Medlyn et al. 1999; Wilson, Baldocchi & Hanson 2000; Misson et al. 2006). This study also indicates that plant-level variation must be accounted for when obtaining estimates of species-level photosynthetic parameters. Otherwise, if variability among plants is ignored, then the species-level parameter estimates and their associated uncertainties will be compromised.


The HB approach presented herein allowed us to rigorously fit fairly complicated photosynthetic models to fairly simple data sets via a probabilistic modelling approach that: (1) simultaneously analysed diverse data sources (both ACi and AQ curves) that informed the same underlying physiological processes; (2) explicitly accounted for and estimated parameter uncertainty for desert plant species, thereby filling a gap in our understanding of plant photosynthetic responses as most empirical studies have focused on mesic temperate species; (3) avoided ad hoc model tuning by incorporating informative prior information derived from the literature to help constrain parameters that are not well informed by the field data (e.g. activation energies); and (4) did not require fixed parameter specifications, but rather was able to accommodate different degrees of model flexibility and prior information, allowing for a rigorous evaluation of photosynthetic parameters and different sources of variability. As such, the HB model of C3 photosynthesis successfully predicted observed photosynthesis. In addition, it yielded explicit plant-level estimates for Ccrit, and plant- and species-level estimates for photosynthetic parameters (gm25, Rd25, Vcmax25 and Jmax25) for desert plants. In summary, the HB approach has great potential to improve the estimation of photosynthetic parameters across a wide range of C3 species, thereby extending the applicability and utility of process-based models such as the Farquhar et al. model. The ease of implementation and flexibility of the HB modelling approach make this an important tool that may be applied to a variety of ecosystems and experimental design settings.


We thank Travis Huxman, Michael Loik and Stan Smith for access to their study sites and use of their equipment. Assistance in the field and logistical support were provided by Holly Alpert, Greg Barron-Gafford, Topher Bentley, Jessie Cable, Dene Charlet, Dan Dawson, Earthwatch Mammoth Lakes SCAP students, Allison Ebbets, Alex Eilts, Lynn Fenstermaker, Alden Griffith, Danielle Ignace, Traesha Robertson, Joe Sirotnak, Anna Tyler and Natasja van Gestel. Victor Resco de Dios, Graham Farquhar and two anonymous reviewers provided valuable comments on earlier drafts of this manuscript. The research described in this paper has been funded in part by the United States Environmental Protection Agency (EPA) under the Greater Research Opportunities (GRO) Graduate Program (L.D.P.). EPA has not officially endorsed this publication, and the views expressed herein may not reflect the views of the EPA. This study was also supported by a US National Park Service grant (D.T.T.) and a US Department of Energy NICCR grant (K.O., D.T.T.).