Fits a C3 assimilation model to an A-Ci curve

Fits the Farquhar-von-Caemmerer-Berry model to an experimentally measured C3 A-Ci curve.

It is possible to fit the following parameters: alpha_g, alpha_old, alpha_s, alpha_t, gamma_star, gmc_at_25, J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25.

By default, only a subset of these parameters are actually fit: alpha_old, J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25. This can be altered using the fit_options argument, as described below.

Best-fit parameters are found using maximum likelihood fitting, where the optimizer (optim_fun) is used to minimize the error function (defined by error_function_c3_aci).

Once best-fit parameters are found, confidence intervals are calculated using confidence_intervals_c3_aci, and unreliable parameter estimates are removed.

For temperature-dependent parameters, best-fit values and confidence intervals are returned at 25 degrees C and at leaf temperature.

See below for more details.

Usage

fit_c3_aci(
    replicate_exdf,
    Ca_atmospheric = NA,
    a_column_name = 'A',
    ca_column_name = 'Ca',
    ci_column_name = 'Ci',
    gmc_norm_column_name = 'gmc_norm',
    j_norm_column_name = 'J_norm',
    kc_column_name = 'Kc',
    ko_column_name = 'Ko',
    oxygen_column_name = 'oxygen',
    rl_norm_column_name = 'RL_norm',
    total_pressure_column_name = 'total_pressure',
    tp_norm_column_name = 'Tp_norm',
    vcmax_norm_column_name = 'Vcmax_norm',
    sd_A = 'RMSE',
    atp_use = 4.0,
    nadph_use = 8.0,
    curvature_cj = 1.0,
    curvature_cjp = 1.0,
    optim_fun = optimizer_deoptim(200),
    lower = list(),
    upper = list(),
    fit_options = list(),
    cj_crossover_min = NA,
    cj_crossover_max = NA,
    relative_likelihood_threshold = 0.147,
    hard_constraints = 0,
    calculate_confidence_intervals = TRUE,
    remove_unreliable_param = 2,
    ...
  )

Arguments

replicate_exdf: An exdf object representing one CO2 response curve.
Ca_atmospheric: The atmospheric CO2 concentration (with units of micromol mol^(-1)); this will be used by estimate_operating_point to estimate the operating point. A value of NA disables this feature.
a_column_name: The name of the column in replicate_exdf that contains the net assimilation in micromol m^(-2) s^(-1).
ca_column_name: The name of the column in replicate_exdf that contains the ambient CO2 concentration in micromol mol^(-1).
ci_column_name: The name of the column in replicate_exdf that contains the intercellular CO2 concentration in micromol mol^(-1).
gmc_norm_column_name: The name of the column in replicate_exdf that contains the normalized mesophyll conductance values (with units of normalized to gmc at 25 degrees C).
j_norm_column_name: The name of the column in replicate_exdf that contains the normalized J values (with units of normalized to J at 25 degrees C).
kc_column_name: The name of the column in replicate_exdf that contains the Michaelis-Menten constant for rubisco carboxylation in micromol mol^(-1).
ko_column_name: The name of the column in replicate_exdf that contains the Michaelis-Menten constant for rubisco oxygenation in mmol mol^(-1).
oxygen_column_name: The name of the column in replicate_exdf that contains the concentration of O2 in the ambient air, expressed as a percentage (commonly 21% or 2%); the units must be percent.
rl_norm_column_name: The name of the column in replicate_exdf that contains the normalized RL values (with units of normalized to RL at 25 degrees C).
total_pressure_column_name: The name of the column in replicate_exdf that contains the total pressure in bar.
tp_norm_column_name: The name of the column in replicate_exdf that contains the normalized Tp values (with units of normalized to Tp at 25 degrees C).
vcmax_norm_column_name: The name of the column in replicate_exdf that contains the normalized Vcmax values (with units of normalized to Vcmax at 25 degrees C).
sd_A: A value of the standard deviation of measured A values, or the name of a method for determining the deviation; currently, the only supported option is 'RMSE'.
atp_use: The number of ATP molecules used per C3 cycle.
nadph_use: The number of NADPH molecules used per C3 cycle.
curvature_cj: A dimensionless quadratic curvature parameter greater than or equal to 0 and less than or equal to 1 that sets the degree of co-limitation between Wc and Wj. A value of 1 indicates no co-limitation.
curvature_cjp: A dimensionless quadratic curvature parameter greater than or equal to 0 and less than or equal to 1 that sets the degree of co-limitation between Wcj and Wp. A value of 1 indicates no co-limitation.
optim_fun: An optimization function that accepts the following input arguments: an initial guess, an error function, lower bounds, and upper bounds. It should return a list with the following elements: par, convergence, feval, and convergence_msg. See optimizers for a list of available options.
lower: A list of named numeric elements representing lower bounds to use when fitting. Values supplied here override the default values (see details below). For example, lower = list(Vcmax_at_25 = 10) sets the lower limit for Vcmax_at_25 to 10 micromol / m^2 / s.
upper: A list of named numeric elements representing upper bounds to use when fitting. Values supplied here override the default values (see details below). For example, upper = list(Vcmax_at_25 = 200) sets the upper limit for Vcmax_at_25 to 200 micromol / m^2 / s.
fit_options: A list of named elements representing fit options to use for each parameter. Values supplied here override the default values (see details below). Each element must be 'fit', 'column', or a numeric value. A value of 'fit' means that the parameter will be fit; a value of 'column' means that the value of the parameter will be taken from a column in replicate_exdf of the same name; and a numeric value means that the parameter will be set to that value. For example, fit_options = list(alpha_g = 0, Vcmax_at_25 = 'fit', Tp_at_25 = 'column') means that alpha_g will be set to 0, Vcmax_at_25 will be fit, and Tp_at_25 will be set to the values in the Tp_at_25 column of replicate_exdf.
cj_crossover_min: The minimum value of Cc (in ppm) where Aj is allowed to become the overall rate-limiting factor. If cj_crossover_min is set to NA, this restriction will not be applied.
cj_crossover_max: The maximim value of Cc (in ppm) where Wj is allowed to be smaller than Wc. If cj_crossover_max is set to NA, this restriction will not be applied.
relative_likelihood_threshold: To be passed to confidence_intervals_c3_aci when calculate_confidence_intervals is TRUE.
hard_constraints: To be passed to calculate_c3_assimilation; see that function for more details.
calculate_confidence_intervals: A logical value indicating whether or not to estimate confidence intervals for the fitting parameters using confidence_intervals_c3_aci.
remove_unreliable_param: An integer value indicating the rules to use when identifying and removing unreliable parameter estimates. A value of 2 is the most conservative option. A value of 0 disables this feature, which is not typically recommended. See below for more details.
...: Additional arguments to be passed to calculate_c3_assimilation.

Details

This function calls apply_gm and calculate_c3_assimilation to calculate values of net assimilation. The user-supplied optimization function is used to vary the values of alpha_g, alpha_old, alpha_s, alpha_t, gamma_star, gmc_at_25, J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25 to find ones that best reproduce the experimentally measured values of net assimilation. By default, the following options are used for the fits:

alpha_g: lower = 0, upper = 10, fit_option = 0
alpha_old: lower = 0, upper = 10, fit_option = 'fit'
alpha_s: lower = 0, upper = 10, fit_option = 0
alpha_t: lower = 0, upper = 10, fit_option = 0
Gamma_star: lower = -20, upper = 200, fit_option = 'column'
gmc_at_25: lower = -1, upper = 10, fit_option = Inf
J_at_25: lower = -50, upper = 1000, fit_option = 'fit'
RL_at_25: lower = -10, upper = 100, fit_option = 'fit'
Tp_at_25: lower = -10, upper = 100, fit_option = 'fit'
Vcmax_at_25: lower = -50, upper = 1000, fit_option = 'fit'

With these settings, the "new" alpha parameters are set to 0, values of Gamma_star are taken from the Gamma_star column of replicate_exdf, mesophyll conductance (gmc_at_25) is set to inifinity (so Cc = Ci), and the other parameters are fit during the process (see fit_options above). The bounds are chosen liberally to avoid any bias.

An initial guess for the parameters is generated by calling initial_guess_c3_aci as follows:

cc_threshold_rd is set to 100 micromol / mol.
If alpha_g is being fit, the alpha_g argument of initial_guess_c3_aci is set to 0.5; otherwise, the argument is set to the value specified by the fit options.
If alpha_old is being fit, the alpha_old argument of initial_guess_c3_aci is set to 0.5; otherwise, the argument is set to the value specified by the fit options.
if alpha_s is being fit, the alpha_s argument of initial_guess_c3_aci is set to 0.3 * (1 - alpha_g); otherwise, the argument is set to the value specified by the fit options.
if alpha_t is being fit, the alpha_t argument of initial_guess_c3_aci is set to 0; otherwise, the argument is set to the value specified by the fit options.
If Gamma_star is being fit, the Gamma_star argument of initial_guess_c3_aci is set to 40; otherwise, the argument is set to the value specified by the fit options.
If gmc_at_25 is being fit, the gmc_at_25 argument of initial_guess_c3_aci is set to 1; otherwise, the argument is set to the value specified by the fit options.

Note that any fixed values specified in the fit options will override the values returned by the guessing function.

The fit is made by creating an error function using error_function_c3_aci and minimizing its value using optim_fun, starting from the initial guess described above. The optimizer_deoptim optimizer is used by default since it has been found to reliably return great fits. However, it is a slow optimizer. If speed is important, consider reducing the number of generations or using optimizer_nmkb, but be aware that this optimizer is more likely to get stuck in a local minimum.

The photosynthesis model represented by calculate_c3_assimilation is not smooth in the sense that small changes in the input parameters do not necessarily cause changes in its outputs. This is related to the final step in the calculations, where the overall assimilation rate is taken to be the minimum of three enzyme-limited rates. For example, if the assimilation rate is never TPU-limited, modifying Tp_at_25 will not change the model's outputs. For this reason, derivative-based optimizers tend to struggle when fitting C3 A-Ci curves. Best results are obtained using derivative-free methods.

Sometimes the optimizer may choose a set of parameter values where one or more of the potential limiting carboxylation rates (Wc, Wj, or Wp) is never the smallest rate. In this case, the corresponding parameter estimates (Vcmax, J, or alpha_old & Tp) will be severely unreliable. This will be indicated by a value of 0 in the corresponding trust column(for example, Vcmax_trust). If remove_unreliable_param is 1 or larger, then such parameter estimates (and the corresponding rates) will be replaced by NA in the fitting results.

It is also possible that the upper limit of the confidence interval for a parameter is infinity; this indicates a potentially unreliable parameter estimate. This will be indicated by a value of 1 in the corresponding trust column (for example, Vcmax_trust). If remove_unreliable_param is 2 or larger, then such parameter estimates (but not the corresponding rates) will be replaced by NA in the fitting results.

The trust value for fully reliable parameter estimates is set to 2 and they will never be replaced by NA.

Once the best-fit parameters have been determined, this function also estimates the operating value of Cc from the atmospheric CO2 concentration atmospheric_ca using estimate_operating_point, and then uses that value to estimate the modeled An at the operating point via calculate_c3_assimilation. It also estimates the Akaike information criterion (AIC).

This function assumes that replicate_exdf represents a single C3 A-Ci curve. To fit multiple curves at once, this function is often used along with by.exdf and consolidate.

Value

A list with three elements:

fits: An exdf object including the original contents of replicate_exdf along with several new columns:
- The fitted values of net assimilation will be stored in a column whose name is determined by appending '_fit' to the end of a_column_name; typically, this will be 'A_fit'.
- Residuals (measured - fitted) will be stored in a column whose name is determined by appending '_residuals' to the end of a_column_name; typically, this will be 'A_residuals'.
- Values of fitting parameters at 25 degrees C will be stored in the gmc_at_25, J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25 columns.
- The other outputs from calculate_c3_assimilation will be stored in columns with the usual names: alpha_g, alpha_old, alpha_s, alpha_t, Gamma_star, gmc_tl, Tp_tl, Vcmax_tl, RL_tl, J_tl, Wc, Wj, Wp, Vc, Ac, Aj, and Ap.
fits_interpolated: An exdf object including the calculated assimilation rates at a fine spacing of Ci values (step size of 1 micromol mol^(-1)).
parameters: An exdf object including the identifiers, fitting parameters, and convergence information for the A-Ci curve:
- The number of points where An = Ac, An = Aj, and An = Ap are stored in the n_Ac_limiting, n_Aj_limiting, and n_Ap_limiting columns.
- The best-fit values are stored in the alpha_g, alpha_old, alpha_s, alpha_t, Gamma_star, gmc_at_25, Tp_at_25, J_at_25, RL_at_25, and Vcmax_at_25 columns. If calculate_confidence_intervals is TRUE, upper and lower limits for each of these parameters will also be included.
- For parameters that depend on leaf temperature, the average leaf-temperature-dependent values are stored in gmc_tl_avg, J_tl_avg, RL_tl_avg, Tp_tl_avg, and Vcmax_tl_avg.
- Information about the operating point is stored in operating_Cc, operating_Ci, operating_An, and operating_An_model.
- The convergence column indicates whether the fit was successful (==0) or if the optimizer encountered a problem (!=0).
- The feval column indicates how many cost function evaluations were required while finding the optimal parameter values.
- The residual stats as returned by residual_stats are included as columns with the default names: dof, RSS, RMSE, etc.
- The Akaike information criterion is included in the AIC column.

Examples

# Read an example Licor file included in the PhotoGEA package
licor_file <- read_gasex_file(
  PhotoGEA_example_file_path('c3_aci_1.xlsx')
)

# Define a new column that uniquely identifies each curve
licor_file[, 'species_plot'] <-
  paste(licor_file[, 'species'], '-', licor_file[, 'plot'] )

# Organize the data
licor_file <- organize_response_curve_data(
    licor_file,
    'species_plot',
    c(9, 10, 16),
    'CO2_r_sp'
)

# Calculate the total pressure in the Licor chamber
licor_file <- calculate_total_pressure(licor_file)

# Calculate temperature-dependent values of C3 photosynthetic parameters
licor_file <- calculate_temperature_response(licor_file, c3_temperature_param_bernacchi)

# For these examples, we will use a faster (but sometimes less reliable)
# optimizer so they run faster
optimizer <- optimizer_nmkb(1e-7)

# We can fit just one curve from the data set, although it is rare to do this
one_result <- fit_c3_aci(
  licor_file[licor_file[, 'species_plot'] == 'tobacco - 1', , TRUE],
  Ca_atmospheric = 420,
  optim_fun = optimizer
)

# We can fit the same curve, but also allow alpha_old and Gamma_star to vary
one_result_v2 <- fit_c3_aci(
  licor_file[licor_file[, 'species_plot'] == 'tobacco - 1', , TRUE],
  Ca_atmospheric = 420,
  fit_options = list(Gamma_star = 'fit', alpha_old = 'fit'),
  optim_fun = optimizer
)

# Fit all curves in the data set (it is more common to do this)
aci_results <- consolidate(by(
  licor_file,
  licor_file[, 'species_plot'],
  fit_c3_aci,
  Ca_atmospheric = 420,
  optim_fun = optimizer
))

# View the fitting parameters for each species / plot
col_to_keep <- c(
  'species', 'plot',                                       # identifiers
  'n_Ac_limiting', 'n_Aj_limiting', 'n_Ap_limiting',       # number of points where
                                                           #   each process is limiting
  'Tp_at_25', 'J_at_25', 'RL_at_25', 'Vcmax_at_25',        # parameters scaled to 25 degrees C
  'J_tl_avg', 'RL_tl_avg', 'Vcmax_tl_avg',                 # average temperature-dependent values
  'operating_Ci', 'operating_An', 'operating_An_model',    # operating point info
  'dof', 'RSS', 'MSE', 'RMSE', 'RSE',                      # residual stats
  'convergence', 'convergence_msg', 'feval', 'optimum_val' # convergence info
)

aci_results$parameters[ , col_to_keep, TRUE]
#>   species [UserDefCon] (NA) plot [UserDefCon] (NA)
#> 1                   soybean                     5a
#> 2                   tobacco                      1
#> 3                   tobacco                      2
#>   n_Ac_limiting [identify_c3_unreliable_points] ()
#> 1                                                8
#> 2                                                8
#> 3                                                9
#>   n_Aj_limiting [identify_c3_unreliable_points] ()
#> 1                                                5
#> 2                                                5
#> 3                                                4
#>   n_Ap_limiting [identify_c3_unreliable_points] ()
#> 1                                                0
#> 2                                                0
#> 3                                                0
#>   Tp_at_25 [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                             NA
#> 2                                             NA
#> 3                                             NA
#>   J_at_25 [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                      190.8864
#> 2                                      231.7749
#> 3                                      206.4820
#>   RL_at_25 [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                      0.5770718
#> 2                                      0.8490827
#> 3                                      0.3815412
#>   Vcmax_at_25 [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                          144.5533
#> 2                                          157.7597
#> 3                                          124.2033
#>   J_tl_avg [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                       268.1504
#> 2                                       319.2523
#> 3                                       285.9199
#>   RL_tl_avg [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                       0.8149572
#> 2                                       1.1742270
#> 3                                       0.5306264
#>   Vcmax_tl_avg [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                           232.0407
#> 2                                           245.8747
#> 3                                           195.1092
#>   operating_Ci [estimate_operating_point] (micromol mol^(-1))
#> 1                                                    264.3297
#> 2                                                    294.7032
#> 3                                                    301.2673
#>   operating_An [estimate_operating_point] (micromol m^(-2) s^(-1))
#> 1                                                         31.00316
#> 2                                                         37.51608
#> 3                                                         31.57904
#>   operating_An_model [fit_c3_aci] (micromol m^(-2) s^(-1))
#> 1                                                 33.29509
#> 2                                                 40.46328
#> 3                                                 33.05702
#>   dof [residual_stats] (NA) RSS [residual_stats] ((micromol m^(-2) s^(-1))^2)
#> 1                         8                                         149.01920
#> 2                         8                                          53.22942
#> 3                         8                                          49.78449
#>   MSE [residual_stats] ((micromol m^(-2) s^(-1))^2)
#> 1                                         11.463016
#> 2                                          4.094571
#> 3                                          3.829577
#>   RMSE [residual_stats] (micromol m^(-2) s^(-1))
#> 1                                       3.385708
#> 2                                       2.023505
#> 3                                       1.956930
#>   RSE [residual_stats] (micromol m^(-2) s^(-1)) convergence [fit_c3_aci] ()
#> 1                                      4.315947                           0
#> 2                                      2.579472                           0
#> 3                                      2.494607                           0
#>   convergence_msg [fit_c3_aci] () feval [fit_c3_aci] ()
#> 1          Successful convergence                   246
#> 2          Successful convergence                   214
#> 3          Successful convergence                   218
#>   optimum_val [fit_c3_aci] ()
#> 1                    34.30052
#> 2                    27.60900
#> 3                    27.17410

# View the fits for each species / plot
plot_c3_aci_fit(aci_results, 'species_plot', 'Ci')


# View the residuals for each species / plot
lattice::xyplot(
  A_residuals ~ Ci | species_plot,
  data = aci_results$fits$main_data,
  type = 'b',
  pch = 16,
  auto = TRUE,
  grid = TRUE,
  xlab = paste0('Intercellular CO2 concentration (', aci_results$fits$units$Ci, ')'),
  ylab = paste0('Assimilation rate residuals (', aci_results$fits$units$A_residuals, ')')
)


# In some of the curves above, there are no points where carboxylation is TPU
# limited. Estimates of Tp are therefore unreliable and are removed.