Module 3

Phloem Transport & Carbon Allocation

Münch pressure flow, sieve tube anatomy, sucrose loading, sink–source dynamics, and the pipe model

3.1 Münch Pressure-Flow Hypothesis

Ernst Münch (1930) proposed that phloem transport is driven by an osmotically generated turgor pressure gradient from source (leaves, where sugars are loaded) to sink (roots, growing tissues, where sugars are unloaded). This is the pressure-flow hypothesis, now supported by extensive experimental evidence.

Physical Derivation

At the source, sucrose loading raises osmotic pressure \(\pi_s\), water enters by osmosis, raising turgor \(P_s\). At the sink, sucrose unloading lowers \(\pi_{sink}\), water leaves, reducing turgor \(P_{sink}\). The turgor difference drives Hagen–Poiseuille flow through the sieve tube lumen:

\[ Q_{phloem} = \frac{\pi r^4}{8\eta} \frac{\Delta P}{L} = \frac{\pi r^4}{8\eta L}(P_s - P_{sink}) \]

More completely, the driving force includes both pressure and osmotic potential differences across the entire phloem path (Tyree et al. formulation):

\[ Q_{phloem} = L_p A (\Delta\Psi_p - \Delta\Psi_s) = L_p A \left[(P_s - P_{sink}) - (\pi_s - \pi_{sink})\right] \]

where \(L_p\) is membrane hydraulic conductivity and \(A\) is cross-sectional area. The flow velocity in the sieve tube:

\[ v = \frac{r^2}{8\eta} \frac{dP}{dx} \approx 0.5\text{--}2 \,\text{mm s}^{-1} \]

in agreement with direct measurements (NMR velocimetry, aphid stylectomy). Sieve tube radius in trees: \(r \approx 10\text{--}20\,\mu\)m; phloem sap viscosity \(\eta \approx 2\text{--}5\) mPa·s (twice that of water, due to sucrose).

Scaling with Tree Height

For a tree of height \(H\), the minimum pressure difference to sustain transport against gravity and viscous resistance:

\[ \Delta P_{min} = \rho g H + \frac{8\eta v L}{r^2} \]

For a 100 m Sequoia with \(v = 1\) mm s\(^{-1}\), \(r = 15\,\mu\)m: gravity term = 1.0 MPa; friction term = \(\approx 0.6\) MPa. Total \(\approx 1.6\) MPa, achievable with a sucrose concentration difference of \(\Delta C \approx 0.6\) M (\(\Delta\pi = RT\Delta C \approx 1.5\) MPa at 25°C).

Münch Flow Diagram

3.2 Sieve Tube Anatomy & Phloem Sap

Mature sieve tube elements are enucleate cells connected end-to-end via sieve platesperforated by pores of radius 0.5–5 \(\mu\)m lined with callose. Key anatomical features:

Companion cells: retain nucleus; connected to sieve element by plasmodesmata; supply ATPs, proteins, ribosomes via symplastic connection.
Callose (\(\beta\)-1,3-glucan): rapidly deposited at sieve plates upon wounding (wound response); reversible via callose synthase/glucanase.
P-proteins (phloem proteins): form filaments that may plug sieve pores upon stress; formerly called “slime bodies.”

Phloem Sap Composition

Sucrose10–25% (0.3–0.9 M)

Amino acids50–200 mM (mainly Gln, Asn)

HormonesIAA, GA, CK, ABA (nM–µM)

mRNA & small RNASystemic signalling molecules

pH7.3–8.5 (alkaline)

K+50–150 mM

3.3 Sucrose Loading Mechanisms

Apoplastic Loading: H\(^+\)/Sucrose Symport

In apoplastic loaders (many trees including poplar), sucrose exported from mesophyll cells crosses the cell wall space and is actively loaded into the phloem by SUT (Sucrose Transporter) proteins, particularly SUT1 (AtSUC2 in Arabidopsis). These are H\(^+\)/sucrose symporters: one proton co-transported per sucrose molecule, driven by the H\(^+\) gradient generated by H\(^+\)-ATPase.

The free energy available from the proton gradient per symport event:

\[ \Delta G_{transport} = RT \ln\frac{[Suc]_{SE}}{[Suc]_{apo}} + \Delta\tilde{\mu}_{H^+} \]

where SE = sieve element. Sucrose is concentrated \(\sim\)10× from apoplast to sieve tube. The proton motive force (\(\approx -180\) mV plus \(\Delta\)pH of 2) provides\(\approx 25\) kJ mol\(^{-1}\) per proton, sufficient to drive this uphill concentration.

Symplastic Loading: Plasmodesmata

Some tree species (e.g., willow) load via plasmodesmata that connect mesophyll to companion cells. The companion cells are specialised as intermediary cellswith many branched plasmodesmata. Raffinose family oligosaccharides (RFOs: raffinose, stachyose) may be synthesised in the companion cells to create a polymer trap—too large to diffuse back through plasmodesmata.

3.4 Sink–Source Dynamics & the Pipe Model

Michaelis–Menten Sink Uptake

Carbon uptake rate at a sink (root meristem, fruit, storage) follows saturable kinetics:

\[ F_{sink} = \frac{F_{max} [Suc]}{K_m + [Suc]} \]

where \(F_{max}\) is maximum unloading flux and \(K_m \approx 10\)–\(20\) mM for SUT transporters. Sink strength is defined as \(SS = F_{max} \times V_{sink}\) (activity × size). Fruit growth, wood formation, and root growth compete for carbon as sink strengths.

Pipe Model Theory (Shinozaki 1964)

The pipe model states that each unit of leaf area is supported by a fixed cross-sectional area of conducting tissue (phloem + xylem). For a tree with total leaf area \(A_L\):

\[ A_{sapwood} = \eta_s \cdot A_{leaf} \]

where \(\eta_s\) (cm\(^2\) leaf cm\(^{-2}\) sapwood) varies by species but is remarkably constant within a species across environments. This has been interpreted as the minimum sapwood area needed to supply the evaporative demand of supported leaves. The model predicts tapering of stems with height and is the basis of allometric scaling in functional ecology (metabolic scaling theory).

Non-Structural Carbohydrate Dynamics

NSC (starch + soluble sugars) buffers carbon supply. A simple two-pool model:

\[ \frac{d[\text{sugar}]}{dt} = A_{gross} - R_m - G - k_{synth}[\text{sugar}] + k_{mob}[\text{starch}] \]

\[ \frac{d[\text{starch}]}{dt} = k_{synth}[\text{sugar}] - k_{mob}[\text{starch}] \]

where \(k_{synth}\) and \(k_{mob}\) are synthesis and mobilisation rate constants (both sugar-concentration dependent). Under drought, \(A_{gross} \rightarrow 0\) but \(R_m\) continues, depleting NSC over weeks to months. NSC depletion is a key mechanism of carbon-starvation mortality in drought-stressed trees.

3.5 Python: Münch Flow Simulation

We model steady-state Münch pressure flow in a 30 m tree: sucrose concentration and hydrostatic pressure along the phloem path, with loading at the top (source) and Michaelis–Menten unloading at the bottom (sink).

Python

script.py102 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Parameters
L = 30.0        # tree height (m)
r = 15e-6       # sieve tube radius (m)
eta = 3e-3      # phloem sap viscosity (Pa s, ~3 mPa s for 20% sucrose solution)
Lp = 5e-14      # membrane hydraulic conductivity (m Pa^-1 s^-1)
rho_w = 1000.0  # kg/m^3
g = 9.81        # m/s^2
R = 8.314       # J/(mol K)
T = 298.0       # K (25 C)
# Van't Hoff osmotic pressure: pi = RT*C (C in mol/m^3)
# Sucrose MW = 342 g/mol

# Loading rate at source (top, z=L)
C_source = 800.0  # mol/m^3 (0.8 M sucrose at leaf phloem)
C_sink_0 = 100.0  # mol/m^3 (0.1 M sucrose at root phloem)

# Hydraulic conductance per unit length
k_hp = np.pi * r**4 / (8.0 * eta)   # Hagen-Poiseuille conductance (m^4 / Pa s)

# Set up spatial grid
nz = 200
z = np.linspace(0, L, nz)
dz = z[1] - z[0]

# Assume linear sucrose gradient from source (top) to sink (bottom)
C = C_source - (C_source - C_sink_0) * (L - z) / L

# Osmotic pressure (Pa) = RT * C
pi_osm = R * T * C

# Turgor pressure: P(z) = pi(z) - psi_leaf + rho*g*(L-z) + P_sink
# For a simple model: P(z) = pi(z) - pi_ref where pi_ref adjusted to match boundary
# At source (z=L): P_s = pi_s - pi_xylem_source, assume xylem psi ~ -0.5 MPa at top
psi_xylem_source = -0.5e6  # Pa
psi_xylem_base = -0.2e6    # Pa
# Xylem water potential varies with height
psi_xylem = psi_xylem_source + (psi_xylem_base - psi_xylem_source) * (L - z) / L

# Turgor pressure = osmotic - (xylem water potential) -> osmosis driven
P_turgor = pi_osm + psi_xylem   # P_turgor = pi_osm - |psi_xylem|

# Gravity correction
P_phloem = P_turgor - rho_w * g * (L - z)

# Flow velocity from pressure gradient
dP_dz = np.gradient(P_phloem, z)
v_flow = (r**2 / (8.0 * eta)) * (-dP_dz)   # positive = downward flow

# Mass flux
J_sucrose = C * v_flow  # mol m^-2 s^-1

# Michaelis-Menten unloading at sink end
Fmax = 0.05     # mol m^-2 s^-1 (maximum unloading flux per sieve tube area)
Km = 50.0       # mol/m^3
F_unload = Fmax * C / (Km + C)

# Plotting
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0f0a')
axes_flat = axes.flatten()

plot_data = [
    (z, C / 1000, 'Sucrose conc. (mol/L)', 'Sucrose Concentration Profile', '#34d399'),
    (z, P_phloem / 1e6, 'Turgor pressure (MPa)', 'Phloem Turgor Pressure Profile', '#60a5fa'),
    (z, v_flow * 1e3, 'Flow velocity (mm/s)', 'Sieve Tube Flow Velocity', '#fbbf24'),
    (z, F_unload * 1e3, 'Unloading flux (mmol m-2 s-1)', 'Michaelis-Menten Unloading', '#f87171'),
]

for ax, (x, y, ylabel, title, color) in zip(axes_flat, plot_data):
    ax.set_facecolor('#0a0f0a')
    for sp in ax.spines.values():
        sp.set_color('#34d399')
    ax.tick_params(colors='#a7f3d0')
    ax.set_xlabel("Height (m)", color='#a7f3d0')
    ax.set_ylabel(ylabel, color='#a7f3d0')
    ax.set_title(title, color='#34d399', fontsize=12)
    ax.plot(x, y, color=color, lw=2.5)
    ax.axhline(0, color='#4b5563', lw=0.7)

# Annotate source and sink
for ax in axes_flat:
    ax.axvline(L, color='#34d399', lw=1, linestyle='--', alpha=0.5)
    ax.axvline(0, color='#6ee7b7', lw=1, linestyle='--', alpha=0.5)

plt.tight_layout()
plt.savefig("output.png", dpi=120, facecolor="#0a0f0a")

print("Munch Flow Model - 30 m tree")
print(f"  Source sucrose: {C_source/1000:.2f} mol/L = {C_source*342/1000:.0f} g/L")
print(f"  Sink sucrose:   {C_sink_0/1000:.2f} mol/L")
print(f"  Osmotic pressure gradient: {(R*T*(C_source-C_sink_0))/1e6:.3f} MPa")
print(f"  Mean flow velocity: {np.mean(v_flow)*1e3:.3f} mm/s")
print(f"  Gravity term (30 m): {rho_w*g*L/1e6:.3f} MPa")
print(f"  Max unloading flux: {Fmax*1e3:.1f} mmol m-2 s-1")
print(f"  Km for sucrose: {Km/1000:.3f} mol/L")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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