Photosynthesis Models

PhyTorch implements the Farquhar-von Caemmerer-Berry (FvCB) model for C3 photosynthesis, one of the most widely used biochemical models of leaf photosynthesis.

FvCB Model

The FvCB model calculates net CO2 assimilation rate ($A$) as the minimum of three potentially limiting rates:

$$A = \min(W_c, W_j, W_p) \left(1 - \frac{{\Gamma^*}}{{C_i}}\right) - R_d$$

where:

• Rubisco-limited rate ($W_c$): Limited by the maximum carboxylation rate ($V_{{cmax}}$)
• RuBP regeneration-limited rate ($W_j$): Limited by electron transport capacity ($J_{{max}}$)
• Triose phosphate utilization-limited rate ($W_p$): Limited by the capacity to use photosynthetic products
• $\Gamma^*$: CO2 compensation point in the absence of day respiration
• $C_i$: Intercellular CO2 concentration
• $R_d$: Day respiration

Rubisco-Limited Rate

$$W_c = \frac{{V_{{cmax}} \cdot C_i}}{{C_i + K_c(1 + O/K_o)}}$$

RuBP Regeneration-Limited Rate

$$W_j = \frac{{J \cdot C_i}}{{4C_i + 8\Gamma^*}}$$

where $J$ is the electron transport rate, calculated from light response

Triose Phosphate Utilization-Limited Rate

$$W_p = \frac{{3 \cdot TPU \cdot C_i}}{{C_i - \Gamma^*(1 + 3\alpha_G)}}$$

where:

• $TPU$ is the triose phosphate utilization rate
• $\alpha_G$ is the fraction of glycolate carbon not returned to the chloroplast (typically 0-0.5)

Basic Usage

PhyTorch's unified API makes fitting photosynthesis models simple and consistent:

from phytorch import fit
from phytorch.models.photosynthesis import FvCB
import pandas as pd

# Load LI-COR A-Ci curve data
df = pd.read_csv('your_aci_data.csv')

# Prepare data dictionary
data = {
    'Ci': df['Ci'].values,       # Intercellular CO2 (μmol/mol)
    'Q': df['PARi'].values,       # Light intensity (μmol/m²/s)
    'Tleaf': df['Tleaf'].values,  # Leaf temperature (°C)
    'A': df['Photo'].values       # Net photosynthesis (μmol/m²/s)
}

# Fit the FvCB model (that's it!)
result = fit(FvCB(), data)

# View fitted parameters
print(f"Vcmax25: {result.parameters['Vcmax25']:.2f} μmol/m²/s")
print(f"Jmax25: {result.parameters['Jmax25']:.2f} μmol/m²/s")
print(f"Rd25: {result.parameters['Rd25']:.2f} μmol/m²/s")
print(f"R² = {result.r_squared:.4f}")

# Generate comprehensive plots
# Includes: 1:1, A vs Ci, A vs Q, A vs T, and 3D surfaces
result.plot()

# Save the plot
result.plot(save='fvcb_fit.png')

Key Parameters

PhyTorch fits the following parameters at 25°C:

Parameter	Description	Typical Range	Units
Vcmax25	Maximum carboxylation rate at 25°C	50-150	μmol/m²/s
Jmax25	Maximum electron transport rate at 25°C	100-250	μmol/m²/s
Rd25	Day respiration at 25°C	0.5-2.0	μmol/m²/s
Tp25	Triose phosphate utilization at 25°C	5-15	μmol/m²/s

Biochemical Constants

Parameter	Description	Value at 25°C	Units
Kc	Michaelis constant for CO2	260	μmol/mol
Ko	Michaelis constant for O2	179	mmol/mol
Γ*	CO2 compensation point	~40	μmol/mol

Temperature Response

PhyTorch supports two temperature response types:

Type 1: Arrhenius Function

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

# Simple Arrhenius temperature response
model = FvCB(light_response=2, temp_response=1)
result = fit(model, data)

The Arrhenius function:

$$k = k_{{25}} \exp\left[\frac{{\Delta H_a}}{{R}}\left(\frac{{1}}{{298}}-\frac{{1}}{{T_{{leaf}}}}\right)\right]$$

Parameter	Description	Value/Range	Units
$k$	Parameter value (Vcmax, Jmax, or TPU)	Variable	μmol/m²/s
$k_{{25}}$	Parameter value at 25°C	Variable	μmol/m²/s
$\Delta H_a$	Activation energy	50,000-100,000	J/mol
$R$	Universal gas constant	8.314	J/mol/K
$T_{{leaf}}$	Leaf temperature	273-323 (0-50°C)	K

Type 2: Peaked Arrhenius Function

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

# Peaked Arrhenius (includes high-temperature deactivation)
model = FvCB(light_response=2, temp_response=2)
result = fit(model, data)

The peaked Arrhenius function:

$$k = k_{{25}} \exp\left[\frac{{\Delta H_a}}{{R}} \left(\frac{{1}}{{298}}-\frac{{1}}{{T_{{leaf}}}}\right)\right] \frac{{f(298)}}{{f(T_{{leaf}})}}$$

where:

$$f(T) = 1+\exp \left[\frac{{\Delta H_d}}{{R}}\left(\frac{{1}}{{T_{{opt}}}}-\frac{{1}}{{T}} \right)-\ln \left(\frac{{\Delta H_d}}{{\Delta H_a}}-1 \right) \right]$$

Parameter	Description	Value/Range	Units
$k$	Parameter value (Vcmax, Jmax, or TPU)	Variable	μmol/m²/s
$k_{{25}}$	Parameter value at 25°C	Variable	μmol/m²/s
$\Delta H_a$	Activation energy	50,000-100,000	J/mol
$\Delta H_d$	Deactivation energy	150,000-250,000	J/mol
$T_{{opt}}$	Optimal temperature	298-313 (25-40°C)	K
$R$	Universal gas constant	8.314	J/mol/K
$T_{{leaf}}$	Leaf temperature	273-323 (0-50°C)	K

Light Response

PhyTorch supports three light response types for electron transport rate ($J$):

Type 0: No Light Dependence

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

model = FvCB(light_response=0, temp_response=2)
result = fit(model, data)

$$J = J_{{max}}$$

No additional parameters are fitted. Electron transport is simply equal to $J_{{max}}$.

Type 1: Rectangular Hyperbola

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

model = FvCB(light_response=1, temp_response=2)
result = fit(model, data)

$$J = \frac{{\alpha Q J_{{max}}}}{{\alpha Q + J_{{max}}}}$$

Parameter	Description	Typical Range	Units
$J_{{max}}$	Maximum electron transport rate	100-250	μmol/m²/s
$\alpha$	Light use efficiency	0.2-0.4	mol e⁻/mol photons
$Q$	Photosynthetic photon flux density (PPFD)	0-2500	μmol/m²/s

Type 2: Non-Rectangular Hyperbola

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

model = FvCB(light_response=2, temp_response=2)
result = fit(model, data)

$$J = \frac{{\alpha Q + J_{{max}} - \sqrt{{(\alpha Q + J_{{max}})^2 - 4 \theta \alpha Q J_{{max}}}}}}{{2 \theta}}$$

Parameter	Description	Typical Range	Units
$J_{{max}}$	Maximum electron transport rate	100-250	μmol/m²/s
$\alpha$	Light use efficiency	0.2-0.4	mol e⁻/mol photons
$\theta$	Curvature parameter	0.7-0.9	dimensionless
$Q$	Photosynthetic photon flux density (PPFD)	0-2500	μmol/m²/s

A-Ci Curves

Fitting A-Ci (assimilation vs. intercellular CO2) curves:

from phytorch import fit
from phytorch.models.photosynthesis import FvCB
import numpy as np

# Your A-Ci curve data
data = {
    'Ci': np.array([50, 100, 200, 400, 600, 800, 1000]),
    'Q': np.full(7, 1500),  # Constant light at 1500 μmol/m²/s
    'Tleaf': np.full(7, 25),  # Constant temperature at 25°C
    'A': np.array([5.2, 10.5, 18.3, 24.1, 26.8, 28.2, 29.1])
}

# Fit the model
result = fit(FvCB(), data)

# View results
print(f"Vcmax25: {result.parameters['Vcmax25']:.2f} μmol/m²/s")
print(f"Jmax25: {result.parameters['Jmax25']:.2f} μmol/m²/s")
print(f"Rd25: {result.parameters['Rd25']:.2f} μmol/m²/s")

# Plot comprehensive results
result.plot()

Advanced Features

Custom Parameter Constraints

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

# Define custom parameter bounds
options = {
    'bounds': {
        'Vcmax25': (20, 200),
        'Jmax25': (40, 400),
        'Rd25': (0, 5)
    }
}

result = fit(FvCB(), data, options)

Custom Initial Guesses

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

# Provide initial parameter estimates
options = {
    'initial_guess': {
        'Vcmax25': 100,
        'Jmax25': 180,
        'Rd25': 1.5
    }
}

result = fit(FvCB(), data, options)

Model Configuration Options

The FvCB model constructor accepts several configuration parameters:

from phytorch import fit
from phytorch.models.photosynthesis import FvCB

model = FvCB(
    light_response=2,      # Light response type: 0, 1, or 2
    temp_response=2,       # Temperature response type: 1 or 2
    fit_gm=True,          # Fit mesophyll conductance (default: False)
    fit_gamma=False,       # Fit CO2 compensation point (default: False)
    fit_Kc=False,          # Fit Michaelis constant for CO2 (default: False)
    fit_Ko=False,          # Fit Michaelis constant for O2 (default: False)
    fit_Rd=True,           # Fit day respiration (default: True)
    preprocess=True,       # Preprocess data (default: True)
    verbose=True           # Print fitting progress (default: True)
)

result = fit(model, data)

Key configuration options:

• fit_gm=True: Enables fitting of mesophyll conductance, accounting for CO2 diffusion from intercellular spaces to chloroplast
• light_response: Controls the light response model (0=none, 1=rectangular hyperbola, 2=non-rectangular hyperbola)
• temp_response: Controls temperature response (1=Arrhenius, 2=peaked Arrhenius with deactivation)

Making Predictions

# Fit the model
result = fit(FvCB(), training_data)

# Make predictions on new data
new_data = {
    'Ci': np.linspace(50, 1000, 100),
    'Q': np.full(100, 1500),
    'Tleaf': np.full(100, 25)
}

predictions = result.predict(new_data)

References

• Farquhar, G. D., von Caemmerer, S., & Berry, J. A. (1980). A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species. Planta, 149(1), 78-90.
• Sharkey, T. D., et al. (2007). Fitting photosynthetic carbon dioxide response curves for C3 leaves. Plant, Cell & Environment, 30(9), 1035-1040.