Modeling Root Zone Effects on Preferred Pathways for the Passive Transport of Ions and Water in Plant Roots

We extend a model of ion and water transport through a root to describe transport along and through a root exhibiting a complexity of differentiation zones. Attention is focused on convective and diffusive transport, both radially and longitudinally, through different root tissue types (radial differentiation) and root developmental zones (longitudinal differentiation). Model transport parameters are selected to mimic the relative abilities of the different tissues and developmental zones to transport water and ions. For each transport scenario in this extensive simulations study, we quantify the optimal 3D flow path taken by water and ions, in response to internal barriers such as the Casparian strip and suberin lamellae. We present and discuss both transient and steady state results of ion concentrations as well as ion and water fluxes. We find that the peak in passive uptake of ions and water occurs at the start of the differentiation zone. In addition, our results show that the level of transpiration has a significant impact on the distribution of ions within the root as well as the rate of ion and water uptake in the differentiation zone, while not impacting on transport in the elongation zone. From our model results we infer information about the active transport of ions in the different developmental zones. In particular, our results suggest that any uptake measured in the elongation zone under steady state conditions is likely to be due to active transport.


GOVERNING EQUATIONS
For rigid and completely water-filled compartments with constant fluid density, mass conservation of solution requires the condition where Q ax in (Q ax out ) is the axial volume flow rate of solution into (out of) position (r, z), and Q rad in (Q rad out ) is the radial volume flow rate into (out of) position (r, z).
The concentration of solute species m at position (r, z) is given by a conservation equation: where V is the solution volume, C m is the concentration of species m, S ax in (S ax out ) is the axial flux of solute m into (out of) position (r, z) and S rad in (S rad out ) is the radial flux of solute m into (out of) position (r, z).
The flow rate of water in a plant root is driven by both hydraulic and osmotic pressure gradients. Hence, the radial flow rate of water is given by (Katchalsky and Curran, 1965): and similarly, the axial flow rate of water in all tissues except the region in which the xylem is conductive is given by, where L rad p (L ax p ) is the position-dependent, radial (axial) water permeability, A rad (A ax ) is the element of surface area through which the radial (axial) water flow is occurring, ∆p is the hydraulic pressure gradient across the region of interest, ρ is the fluid density, g is the acceleration due to gravity directed toward decreasing z, ∆z is a discrete height increment (see Section 2.4 for further details), R g is the Universal Gas Constant (8.314 J mol −1 K −1 ), T is the (constant) temperature, σ rad m (σ ax m ) is the position-dependent radial (axial) reflection coefficient of solute ion m = 1, . . . , N . Here, N is the total number of ions in the system; ∆C m is the concentration difference of ion type m across the region of interest. In Eqs. (S3) and (S4) we have assumed that solute concentrations are sufficiently low that use of the van't Hoff relation for osmotic pressure, ∆Π m = R g T ∆C m , is valid (Katchalsky and Curran, 1965). The appropriateness of this assumption is discussed in Foster and Miklavcic (2014).
In contrast to the other root tissue regions, axial transport in the functional xylem is not interrupted by cell membranes. Hence, the axial flow of water in the xylem is driven by hydraulic pressure gradients only, with no osmotic pressure gradients present. This water flow can be modeled as linearly proportional to the hydraulic pressure gradient, using Darcy's Law, where k ax is a position-dependent, axial water permeability, and µ is the (constant) dynamic viscosity of the fluid.
The transport of ions in the model plant root is governed by a chemical potential contribution (arising from concentration differences), an electric field contribution (due to an electric potential difference) and by convection. Hence, the radial flux of ions (S rad in,m ) and the axial flux of ions in all tissues except the functional xylem (S ax in,m ) are given by, where, k rad m (k ax m ) is the radial (axial) diffusive permeability of ion m, Z m is the valence of ion m, F is Faraday's constant (96 485 C mol −1 ), and ∆ψ is the electric potential gradient across the region of interest.
Due to the absence of membranes, the axial flux of solutes in the functional xylem is given by, where the convection term is not scaled by a solute reflection coefficient.
In order to implement the equations outlined above, the r and z root volume space was discretized into compartments identified by discrete positions (r α , z j ), where α = 1, . . . , 5 and j = 1, . . . , 100.