Hydrodynamics of steady state phloem transport with radial leakage of solute

Abstract

Long-distance phloem transport occurs under a pressure gradient generated by the osmotic exchange of water associated with solute exchange in source and sink regions. But these exchanges also occur along the pathway, and yet their physiological role has almost been ignored in mathematical models of phloem transport. Here we present a steady state model for transport phloem which allows solute leakage, based on the Navier-Stokes and convection-diffusion equations which describe fluid motion rigorously. Sieve tube membrane permeability P s for passive solute exchange (and correspondingly, membrane reflection coefficient) influenced model results strongly, and had to lie in the bottom range of the values reported for plant cells for the results to be realistic. This smaller permeability reflects the efficient specialization of sieve tube elements, minimizing any diffusive solute loss favored by the large concentration difference across the sieve tube membrane. We also found there can be a specific reflection coefficient for which pressure profiles and sap velocities can both be similar to those predicted by the Hagen-Poiseuille equation for a completely impermeable tube.

Keywords: fluid dynamics; navier-stokes equation; phloem transport; sieve tube; solute exchange.

Figure 1

Sieve tube model. Solution enters the sieve tube with velocity U, pressure p_i and concentration c_i. There is radial exchange of water and solute across the sieve tube membrane along the pathway. The sieve tube is surrounded by a medium (“apoplast”) with pressure p_out(z) and concentration c_out(z). R and L are sieve tube radius and length, respectively. r and z are the directions of radial and axial flow, respectively.

Figure 2

Effect of solute permeability on the flow within a sieve tube limited by a permeable membrane. Pressure (A), average concentration (B), average axial velocity (C), radial velocity at sieve tube membrane (D), axial (E) and radial (F) solute fluxes with position ẑ for different reflection coefficients (and therefore solute permeability). Pressure and axial velocity profiles for the Poiseuille flow regime are also shown (••••••) in (A) and (C) respectively.

Figure 3

Comparison of zeroth and first order results. Average solute concentration for different values of the reflection σ and axial position ẑ between ẑ = 0 and ẑ = ẑ_max (see Figure 2). All curves start at ĉ₀ (ẑ = 0) = 1 and ĉ₁ (ẑ = 0) = 0. Parameters of Table 1 were used for the numerical calculation. The diagonal indicates same order of magnitude of zeroth and first order coefficients in the power series expansion.

Figure 4

Sensitivity analysis. Varying all relevant model parameters up to ±50% from the values of Table 1, the resulting relative change of the state variables (at the end of the tube, i.e., z = L) is shown as indicator of model sensitivity. State variables are pressure (A), average solute concentration (B), average axial velocity (C) and radial velocity at the sieve tube membrane (D). Dimensionless results were converted into original dimensions. Curves are not shown for parameters that give a boundary condition of the corresponding state variable.

Figure 5

Effect of membrane solute permeability on flow within long sieve tubes. Profiles of pressure (A), average concentration (B), average axial velocity (C) and radial velocity at sieve tube membrane (D) for same parameter set as in Figure 2, except for sieve tube length which was set to L = 3 m. The two curves are for semipermeability (σ = 1.0), and for the permeability which gives approximate equilibrium of radial water potential (σ = 0.73).