Modeling the Hydraulics of Root Growth in Three Dimensions with Phloem Water Sources

Primary growth is characterized by cell expansion facilitated by water uptake generating hydrostatic (turgor) pressure to inflate the cell, stretching the rigid cell walls. The multiple source theory of root growth hypothesizes that root growth involves transport of water both from the soil surrounding the growth zone and from the mature tissue higher in the root via phloem and protophloem. Here, protophloem water sources are used as boundary conditions in a classical, three-dimensional model of growth-sustaining water potentials in primary roots. The model predicts small radial gradients in water potential, with a significant longitudinal gradient. The results improve the agreement of theory with empirical studies for water potential in the primary growth zone of roots of maize (Zea mays). A sensitivity analysis quantifies the functional importance of apical phloem differentiation in permitting growth and reveals that the presence of phloem water sources makes the growth-sustaining water relations of the root relatively insensitive to changes in root radius and hydraulic conductivity. Adaptation to drought and other environmental stresses is predicted to involve more apical differentiation of phloem and/or higher phloem delivery rates to the growth zone.Plant growth involves water uptake by the cells and expansion of the cell walls under the resultant turgor (internal hydrostatic pressure). The water uptake and increase in cell volume are accompanied by nutrient and metabolite deposition. Thus, hydraulics of growth (i.e. the energies, conductivities, and fluxes of water in growing tissue) are fundamental to understanding primary plant growth. Quantitatively, the driving force for water movement in the plant, as in other porous media, is considered to be the gradient in water potential (Ψ), an energy per unit volume given in MPa. Thus, primary growth can be modeled by considering plant tissue to be a distributed sink for water, with low Ψ and/or high hydraulic conductivity driving water deposition into rapidly expanding regions. Molz and Boyer (1978) developed the theoretical basis for predicting the radial water flux in one dimension within the intercalary meristem of growing soybean (Glycine max) hypocotyls. In this aerial tissue, water moves from the xylem both outward to the epidermis and inward to the pith. Thus, in the growing hypocotyls, Ψ is predicted to be least negative in the xylem and to decrease toward the epidermis and the pith. These predictions for growth-induced or growth-sustaining Ψ were confirmed when the experimental technology became sensitive enough to detect the gradients in Ψ (Nonami and Boyer, 1993). Passioura and Boyer (2003) expanded the theory to incorporate anatomical detail and corresponding spatial patterns of hydraulic conductivity. Their model explains experimental results on water relations during growth transients for many areas of the plant.The hydraulics of root growth differ from shoot growth because of differences in xylem anatomy. Root xylem becomes functional perhaps 1 cm behind the tip and well behind the growth zone. To enter the growing cells near the maize (Zea mays) root tip, externally supplied metabolites must move several millimeters without phloem (Fig. 1), and any water supplied by functional xylem would need to move more than 1 cm. Silk and Wagner (1980) provided a theoretical framework for a two-dimensional treatment of the growth-sustaining Ψ gradients in maize roots. They assumed that the water source was external (the soil or root-bathing medium) and that the root surface was in equilibrium with the soil or bathing medium, so that the flow path to growing cells in the root was predicted to be primarily inward. As in the shoot model, growing tissue was seen as a distributed sink for water. However, since the publication of that theory, experimental studies have revealed that the root tip is not in equilibrium with the bathing medium (Pritchard et al., 1996, 2000; Gould et al., 2004; Shimazaki et al., 2005). Pressure probes combined with osmotic potential determinations have shown that the Ψ of exterior root cells ranges from −0.17 to −0.6 MPa, depending on environmental conditions. This range is more negative than in the nutrient medium. Furthermore, evidence has accumulated that at least some water for root growth comes from the phloem. The most obvious evidence is perhaps the growth of nodal (adventitious) roots of maize, rice (Oryza sativa), and other gramineous plants (Westgate and Boyer, 1985). This growth is a normal part of crop development. The nodal roots grow through air and then dry layers of surface soil, making it unlikely that the expanding root cells obtain water from the dry media surrounding the root. Empirical and theoretical studies have concluded that the phloem probably provides water for growth of the primary maize root (Bret-Harte and Silk, 1994; Frensch and Hsiao, 1995; Pritchard, 1996; Pritchard et al., 1996, 2000; Hukin et al., 2002; Gould et al., 2004).Open in a separate windowFigure 1.Primary root growth zone. The tip of the seedling root of maize showing the meristem as part of the apical third of the elongation zone. The boundary of this root section was digitized to provide the computational body-fit grid used for the model. [See online article for color version of this figure.]The model described here follows the concepts of Pritchard and colleagues (1996, 2000) in assuming a pressure-driven bulk flow of solution through the phloem to the region where phloem is beginning to be functional (1–4 mm from the apex; Fig. 1). Water movement can occur from both the surrounding soil and the developing phloem. Henceforth, we refer to the “external water source equilibrium” or EE model, for which the boundary condition is solely an exterior medium of fairly high Ψ (−0.005 to −0.05 MPa) and no conditions are placed on the phloem Ψ (Silk and Wagner (1980), that the exterior of the root is in equilibrium with its bathing solution. Empirical studies have shown that this model is not realistic, because the root maintains peripheral cells at more negative Ψ than the bathing medium. Since this is hypothesized to occur by deposition of apoplastic solutes, we will refer to a model with external water source and apoplastic solutes near the exterior as the EASE model.

Table I.

Acronyms for models and definitions of symbols used in mathematical modeling

Acronym		Boundary Condition
EE		External water source Equilibrium
EASE		External water source and Apoplastic Solutes near the Exterior
PEWS		Phloem and External Water Sources
Symbol	Physical Significance	Units
L	Relative elemental growth rate	h−1
	Growth velocity vector	mm h−1
	Water flux vector	mm h−1
	Hydraulic conductivity tensor	mm2 s−1 MPa−1
Ψ	Total water potential	MPa
	Unit normal to the surface
s	Control surface	mm2
V	Control volume	mm3
r	Radial coordinate	mm
z	Longitudinal coordinate	mm
x, y	Cartesian coordinates	mm
J	Jacobian Matrix of Transformation

Open in a separate windowA “multiple source” model places boundary conditions on the Ψ of both the bathing medium and the phloem to simulate both external and internal source activity, so we will refer to this model as the PEWS (for phloem and external water sources) model.