Hydraulic Models

PhyTorch implements models for plant water relations, including hydraulic conductance (vulnerability curves) and hydraulic capacitance (pressure-volume curves).

Hydraulic Conductance (Vulnerability Curves)

Vulnerability curves describe the decline in hydraulic conductance as water potential becomes more negative.

Sigmoidal Model

The sigmoidal model describes vulnerability to cavitation:

$$K_{rel} = \frac{K_{max}}{1 + \left|\frac{\psi}{\psi_{50}}\right|^s}$$

where:

• $K_{rel}$ = Relative hydraulic conductance (0-1)
• $K_{max}$ = Maximum conductance (normalized to 1)
• $\psi$ = Water potential (MPa)
• $\psi_{50}$ = Water potential at 50% loss of conductance (MPa)
• $s$ = Slope parameter

Usage

from phytorch import fit
from phytorch.models.hydraulics import Sigmoidal
import numpy as np

# Vulnerability curve data
data = {
    'x': np.array([-0.5, -1.0, -1.5, -2.0, -2.5, -3.0, -3.5]),  # Water potential (MPa)
    'psi': np.array([0.98, 0.92, 0.78, 0.51, 0.28, 0.12, 0.05])  # Relative conductance
}

result = fit(Sigmoidal(), data)

print(f"P50: {result.parameters['x50']:.2f} MPa")
print(f"Slope: {result.parameters['s']:.2f}")
print(f"R² = {result.r_squared:.4f}")

# Plot vulnerability curve
result.plot()

Parameters

Parameter	Description	Typical Range	Units
Kmax	Maximum conductance	0.95-1.0	normalized
psi50	P50 value	-0.5 to -8.0	MPa
s	Slope parameter	10-100	-

Interpretation

• P50 (ψ₅₀): More negative values indicate greater drought tolerance
• Slope: Steeper slopes indicate rapid loss of conductance over a narrow water potential range

Hydraulic Capacitance (Pressure-Volume Curves)

Pressure-volume (P-V) curves characterize cell and tissue water relations.

SJB2018 Model

The Sack-John-Buckley (2018) model describes P-V relationships:

$$\psi = p + \pi$$

where:

$$p = \pi_o \cdot \max\left(0, \frac{w - w_{tlp}}{1 - w_{tlp}}\right)^{\epsilon}$$ $$\pi = -\frac{\pi_o}{w}$$

• $\psi$ = Water potential (MPa)
• $p$ = Turgor pressure (MPa, positive)
• $\pi$ = Osmotic potential (MPa, negative)
• $w$ = Relative water content (0-1)
• $\pi_o$ = Osmotic pressure at full turgor (MPa, positive value)
• $w_{tlp}$ = Relative water content at turgor loss point
• $\epsilon$ = Bulk modulus of elasticity (MPa)

Usage

from phytorch import fit
from phytorch.models.hydraulics import SJB2018
import numpy as np

# Pressure-volume curve data
data = {
    'w': np.array([1.00, 0.95, 0.90, 0.85, 0.80, 0.75, 0.70]),  # RWC
    'psi': np.array([-0.1, -0.3, -0.6, -1.0, -1.5, -2.1, -2.8])  # Water potential (MPa)
}

result = fit(SJB2018(), data)

print(f"Osmotic pressure at full turgor: {result.parameters['pi_o']:.2f} MPa")
print(f"Turgor loss point: {result.parameters['w_tlp']:.3f}")
print(f"Bulk modulus: {result.parameters['epsilon']:.2f} MPa")
print(f"R² = {result.r_squared:.4f}")

# Plot P-V curve
result.plot()

Parameters

Parameter	Description	Typical Range	Units	Default
pi_o	Osmotic pressure at full turgor	1.0-3.0	MPa	2.0
w_tlp	RWC at turgor loss point	0.70-0.90	fraction	0.85
epsilon	Bulk modulus of elasticity	5-30	MPa	1.0

Note: pi_o is the osmotic pressure (positive value). The corresponding osmotic potential at full turgor is -pi_o (e.g., default 2.0 MPa pressure = -2.0 MPa potential).

Key P-V Parameters

From the fitted curve, you can derive important physiological parameters:

Turgor Loss Point (TLP):

• Water potential at zero turgor: $\psi_{tlp} = -\pi_o / w_{tlp}$
• Lower (more negative) values indicate greater drought tolerance

Osmotic Pressure at Full Turgor ($\pi_o$):

• Higher values (representing more negative osmotic potential) indicate greater osmotic adjustment capacity
• Osmotic potential at full turgor = $-\pi_o$

Bulk Modulus of Elasticity:

• Higher values indicate stiffer cell walls
• Lower values indicate greater cell wall elasticity

Interpretation Example

# Calculate derived parameters
pi_o = result.parameters['pi_o']
w_tlp = result.parameters['w_tlp']
epsilon = result.parameters['epsilon']

# Turgor loss point
psi_tlp = -pi_o / w_tlp
print(f"Turgor loss point: {psi_tlp:.2f} MPa")

# Osmotic potential at full turgor (negative value)
pi_full = -pi_o
print(f"Osmotic potential at full turgor: {pi_full:.2f} MPa")

# Cell wall elasticity interpretation
if epsilon < 10:
    print("Elastic cell walls")
elif epsilon > 20:
    print("Rigid cell walls")
else:
    print("Intermediate cell wall elasticity")

Sign Conventions

PhyTorch follows standard plant physiology conventions:

Variable	Sign	Range	Meaning
Water potential (ψ)	Negative	0 to -10 MPa	More negative = drier
Turgor pressure (p)	Positive	0 to 3 MPa	Zero at TLP
Osmotic potential (π)	Negative	-0.5 to -5 MPa	More negative = higher solute concentration
Osmotic pressure (π_o)	Positive	0.5 to 5 MPa	Equal to -π in magnitude

Custom Parameter Bounds

Constrain parameters based on biological ranges:

from phytorch import fit, FitOptions

# P-V curve with constraints
options = FitOptions(
    bounds={
        'pi_o': (0.5, 5.0),    # Osmotic potential
        'w_tlp': (0.6, 0.95),  # TLP water content
        'epsilon': (1, 50)      # Bulk modulus
    }
)

result = fit(SJB2018(), data, options)

Data Requirements

Vulnerability Curves

• Measure hydraulic conductance at multiple water potentials
• Typically 6-10 points across the range
• Include points near P50 for accurate estimation
• Normalize conductance to maximum value

Pressure-Volume Curves

• Measure water potential at multiple relative water contents
• Start from full turgor (RWC ≈ 1.0)
• Continue past turgor loss point (RWC < w_tlp)
• Typically 7-12 measurements

Plotting

Both models support automatic visualization:

# 1:1 plot + model fit curve
result.plot()

# Save plot
result.plot(save='hydraulic_curve.png', show=False)

Model Comparison

Compare different hydraulic parameters across species or treatments:

# Fit multiple datasets
species_data = {
    'Species A': data_A,
    'Species B': data_B,
    'Species C': data_C
}

results = {}
for species, data in species_data.items():
    results[species] = fit(SJB2018(), data)

# Compare P50 values
for species, result in results.items():
    p50 = result.parameters['x50']
    print(f"{species}: P50 = {p50:.2f} MPa")

References

• Sack, L., John, G. P., & Buckley, T. N. (2018). ABA accumulation in dehydrating leaves is associated with decline in cell volume, not turgor pressure. Plant Physiology, 176(1), 489-495.
• Tyree, M. T., & Hammel, H. T. (1972). The measurement of the turgor pressure and the water relations of plants by the pressure-bomb technique. Journal of Experimental Botany, 23(1), 267-282.
• Pammenter, N. W., & Vander Willigen, C. (1998). A mathematical and statistical analysis of the curves illustrating vulnerability of xylem to cavitation. Tree Physiology, 18(8-9), 589-593.