Stomatal Conductance Models

PhyTorch implements several empirical and semi-empirical models of stomatal conductance that link stomatal behavior to environmental conditions and photosynthesis.

Medlyn Model (MED)

The Medlyn model (also called the USO model) relates stomatal conductance to assimilation rate and vapor pressure deficit:

$$g_s = g_0 + 1.6 \left(1 + \frac{{g_1}}{{\sqrt{{D}}}}\right) \frac{{A}}{{C_a}}$$

where:

• $g_s$ = Stomatal conductance to water vapor (mol m⁻² s⁻¹)
• $g_0$ = Residual conductance (mol m⁻² s⁻¹)
• $g_1$ = Slope parameter (√kPa)
• $D$ = Vapor pressure deficit (kPa)
• $A$ = Net CO₂ assimilation rate (μmol m⁻² s⁻¹)
• $C_a$ = Atmospheric CO₂ concentration (μmol mol⁻¹)

Usage

from phytorch import *
import pandas as pd

# Load LI-COR data
df = pd.read_csv('your_gs_data.csv')
lcd = stomatal.initLicordata(df, preprocess=True)

# Initialize Medlyn model
med_model = stomatal.model(lcd, model_type='MED')

# Fit the model
fitresult = stomatal.fit(
    med_model,
    learn_rate=0.01,
    maxiteration=10000
)

# View fitted parameters
print(f"g0 = {fitresult.params['g0']:.4f} mol/m²/s")
print(f"g1 = {fitresult.params['g1']:.2f}")

Parameters

Parameter	Description	Typical Range	Units
g0	Residual conductance	0.0-0.05	mol/m²/s
g1	Slope parameter	2-8	√kPa

Ball-Woodrow-Berry Model (BWB)

The BWB model relates stomatal conductance to assimilation, relative humidity, and CO2:

$$g_s = g_0 + g_1 \frac{{A \cdot RH}}{{C_s}}$$

where:

• $g_s$ = Stomatal conductance to water vapor (mol m⁻² s⁻¹)
• $g_0$ = Residual conductance (mol m⁻² s⁻¹)
• $g_1$ = Slope parameter (dimensionless)
• $A$ = Net CO₂ assimilation rate (μmol m⁻² s⁻¹)
• $RH$ = Relative humidity (0-1)
• $C_s$ = CO₂ concentration at leaf surface (μmol mol⁻¹)

Usage

from phytorch import *
import pandas as pd

# Load LI-COR data
df = pd.read_csv('your_gs_data.csv')
lcd = stomatal.initLicordata(df, preprocess=True)

# Initialize BWB model
bwb_model = stomatal.model(lcd, model_type='BWB')

# Fit the model
fitresult = stomatal.fit(
    bwb_model,
    learn_rate=0.01,
    maxiteration=10000
)

# View fitted parameters
print(f"g0 = {fitresult.params['g0']:.4f} mol/m²/s")
print(f"g1 = {fitresult.params['g1']:.2f}")

Parameters

Parameter	Description	Typical Range	Units
g0	Residual conductance	0.0-0.05	mol/m²/s
g1	Slope parameter	5-15	dimensionless

Buckley-Mott-Farquhar Model (BMF)

The BMF model is a more mechanistic approach based on optimization theory.

from phytorch import *
import pandas as pd

# Load LI-COR data
df = pd.read_csv('your_gs_data.csv')
lcd = stomatal.initLicordata(df, preprocess=True)

# Initialize BMF model
bmf_model = stomatal.model(lcd, model_type='BMF')

# Fit the model
fitresult = stomatal.fit(
    bmf_model,
    learn_rate=0.01,
    maxiteration=10000
)

Ball-Berry-Leuning Model (BBL)

An extension of the BWB model that uses VPD instead of relative humidity.

Note: The BBL model is currently under development and will be available in a future release.

Fitting Stomatal Conductance Models

Example: Fitting the Medlyn Model

from phytorch.fitting import fit_model
from phytorch.models import Medlyn

# Your measured data
data = {
    'gs': torch.tensor([0.15, 0.25, 0.32, 0.40, 0.35, 0.28]),
    'A': torch.tensor([10, 18, 24, 28, 26, 20]),
    'VPD': torch.tensor([1.2, 1.5, 1.8, 2.0, 2.2, 2.5]),
    'Ca': 400  # Constant atmospheric CO2
}

# Fit the model
result = fit_model(
    model=Medlyn(),
    data=data,
    params_to_fit=['g0', 'g1']
)

print(f"g0 = {result.params['g0']:.4f} mol/m²/s")
print(f"g1 = {result.params['g1']:.2f} √kPa")

Coupled Photosynthesis-Stomatal Conductance

PhyTorch allows coupling photosynthesis and stomatal conductance models:

from phytorch.models import FvCB, Medlyn
from phytorch.coupled import CoupledModel

# Create coupled model
coupled = CoupledModel(
    photosynthesis=FvCB(Vcmax=100, Jmax=180),
    stomatal=Medlyn(g0=0.01, g1=4.0)
)

# Solve for coupled A and gs
result = coupled.solve(
    Ca=400,       # Atmospheric CO2
    temperature=25,
    ppfd=1500,
    VPD=1.5,
    gbw=0.3       # Boundary layer conductance (mol/m²/s)
)

print(f"A = {result['A']:.2f} μmol/m²/s")
print(f"gs = {result['gs']:.3f} mol/m²/s")
print(f"Ci = {result['Ci']:.1f} μmol/mol")

Model Comparison

Different models may be appropriate for different species or conditions:

from phytorch.models import Medlyn, BallWoodrowBerry
from phytorch.evaluation import compare_models

models = {
    'Medlyn': Medlyn(),
    'BWB': BallWoodrowBerry()
}

comparison = compare_models(
    models=models,
    data=validation_data,
    metrics=['RMSE', 'R2', 'AIC']
)

References

• Medlyn, B. E., et al. (2011). Reconciling the optimal and empirical approaches to modelling stomatal conductance. Global Change Biology, 17(6), 2134-2144.
• Ball, J. T., Woodrow, I. E., & Berry, J. A. (1987). A model predicting stomatal conductance and its contribution to the control of photosynthesis under different environmental conditions. Progress in Photosynthesis Research, 4, 221-224.
• Buckley, T. N., Mott, K. A., & Farquhar, G. D. (2003). A hydromechanical and biochemical model of stomatal conductance. Plant, Cell & Environment, 26(10), 1767-1785.