Root Growth: 3D Hydraulic Model with Water from Phloem

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 from both 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 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 stress is predicted to involve more apical differentiation of phloem and/or higher phloem delivery rates to the growth zone.

x, y cartesian coordinates mm

Paper Text
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 water potential 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 hypocotyls. In this aerial tissue, water moves from the xylem both outward to the epidermis and inward to 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 growthsustaining 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 differs from shoot growth because of differences in xylem anatomy. Root xylem becomes functional perhaps a centimeter behind the tip and well behind the growth zone. To enter the growing cells near the maize root tip, externally supplied metabolites must move several millimeters without phloem ( Fig. 1 showing the apical four mm of the root growth zone); and any water supplied by functional xylem would need to move more than a centimeter. Silk and Wagner (1980) provided a theoretical framework for a two-dimensional treatment of the growth-sustaining Ψ gradients in maize roots.
They assumed the water source was external (the soil or root bathing medium) and 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 the theory, experimental studies have revealed that the root tip is not in equilibrium with the bathing medium (Pritchard et al, 2000Gould et al, 2004;Shimazaki et al, 2005). Pressure probes combined with osmotic potential determinations have shown that the water potential of exterior root cells ranges from -0.17 to -0.6 MPa, depending on environmental conditions. This range is more negative than 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, and other gramineous plants (e.g. 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, 1996Pritchard et al, , 2000Hukin et al, 2002;Gould et al, 2004).
The model described here follows the concepts of Pritchard andcolleagues (1996, 2000) in assuming a pressure driven bulk flow of solution through the phloem to the region where phloem is beginning to be functional (one to four mm from the apex, see Figure 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 water potential (-0.005 MPa to -0.05 MPa) and no conditions are placed on the phloem water potential.
This EE model assumes, as did Silk and Wagner, 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.
A "multiple source" model places boundary conditions on the water potential 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 (Phloem and External Water Sources) model.

Theoretical Background for the Multiple Source Model
Relationship Between Growth and Water Potential

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The relative elemental growth rate (L), equal to the divergence of the growth velocity ( g), is a measure of the local growth rate (see Abbreviations for variable definitions). Following Silk and Wagner (1980) it is assumed that water moves via a version of Darcy's Law relating water flux to the gradient in water potential. To cross a cell wall and inflate the cell, water velocity J must be greater than the velocity g with which the cell wall is being displaced during growth.
By the Reynolds transport theorem, in three dimensions ; and L can be related to the local water potential (Ψ) required to drive the growth sustaining water influx: The velocity with which a cell moves depends on the rate at which it is displaced by those cells behind it and its own expansion. If the divergence of velocity is 0, the cell is simply displaced and not actually expanding. Equation (1) predicts that in the absence of a water potential gradient there would be no expansion. More generally, the governing equation (1) can be used to calculate the growth sustaining water potential (Ψ) using experimental data for the relative elemental growth rate and hydraulic conductivity (Silk and Wagner, 1980;Boyer and Silk, 2004)

Assumptions
The PEWS model extends the previous external root growth model by including the assumption that some water is being supplied from non-growing tissue via the root phloem and protophloem. The phloem develops closer to the root tip than the xylem and channels water from the mature regions into the growth zone (Bret-Harte and Silk, 1994;Frensch and Hsiao, 1995;Pritchard, 1996;Pritchard et al, 2000).
We assume: 1. The root tissue is cylindrical beyond the parabolic root tip, with radius r. Growth is radial and longitudinal near the maize root tip and in the direction of the long axis z beyond z = 2 mm.
2. The growth pattern does not change in time.
3. Conductivities in the radial ( r K r r ) and longitudinal ( z K r r ) directions are independent so radial flow is not modified by longitudinal flow.
4. The water needed for primary root-growth is obtained from both the surrounding growth medium and the internal protophloem sources. In our PEWS model, this assumption is embodied by maintaining the bathing medium Ψ close to zero (Ψ = -0.05 MPa in many empirical studies) and the protophloem Ψ less negative than the interior cells (Ψ = -0.20 MPa) .
5. For both the EASE and the PEWS models, the cells at the periphery of the root growth zone have a total water potential gradient as measured by Pritchard et al (1996), increasing from -0.27 MPa below z = 2.5 mm to -0.15 MPa at z = 12 mm.

Numerical Approach
The governing equations (1) were converted to three-dimensional generalized coordinates, a method that converts grid components (x,y,z) into orthogonal, equally spaced grid components (ξ, η , ζ ) related by the Jacobian matrix of the transformation and a converted generalized equation (Fletcher, 1991) 1 . A second order finite difference approximation is used to solve the given generalized governing equations on the computational orthogonal grid. A literature review resulted in estimates of radial hydraulic conductivity in the range 7.3 x 10 -11 m 2 s -1 MPa -1 < r K r r < 5 x 10 -10 m 2 s -1 MPa -1 . We choose r K r r = 1.3 x 10 -10 m 2 s -1 MPa -1 as a reference value, from the empirical value of Frensch and Hsiao (1995). For more details, refer to the Materials and Methods.

Results
The three models predict different distributions of growth-sustaining Ψ in the root tip (Figures 2 and 3).  Figure 2C) the spatial pattern of Ψ no longer has the egg-shaped isopotential regions (compare Fig. 2B and 2C). The magnitude of the growth sustaining Ψ gradient is considerably less than required by the EASE model (Compare Fig. 3B and 3C). Comparing the flux reveals that for both EASE and PEWS, water movement (flux) decreases with distance from the source. However, in the area of maximum growth EASE predicts that the water moves mostly inward. In contrast, in PEWS there is both outward flux from the protophloem and inward radial water movement from the external water source (see flux arrows in Figs. 2B and 2C). The PEWS model results in a Ψ field closest to that found by Pritchard et al (2000) and Hukin et al (2002).

Sensitivity Analysis
The mathematical models were used to determine the sensitivity of the growth-sustaining water potential field to morphological, anatomical, and hydraulic parameters: root radius (r max ), the length and position of the phloem source and hydraulic conductivity ( K ) as described in Table I. Results demonstrate that the magnitude of the growth sustaining Ψ value becomes much less sensitive to root radius and hydraulic conductivity when phloem sources are included; that is, PEWS gives radial Ψ gradients that are not much influenced by root radius or a change in K . However, the growth sustaining Ψ is particularly sensitive to the location of phloem differentiation. To assess the importance of the phloem anatomy we first assumed that all phloem and protophloem within the growth zone acts as a source and we tested the effect of the source length, i.e. we solved eq. (1) assuming phloem differentiation at different distances from the tip. As the length of the phloem source is made shorter (the location that phloem supplies water is decreased from z = 1 mm to z = 6.1 mm), Ψ min becomes more negative (Fig. 4). The shorter the source is, the more closely the results of the PEWS model resemble the EASE model ( Fig 4E). It is not known whether all of the phloem in the growth zone provides water (extended phloem source) or whether a short zone of developing protophloem provides the osmotically dilute solution (limited phloem source). Figure 5 shows the effect of one-mm-long phloem sources located at different distances from the apex. If the source is found 1 mm from the tip, the most negative water potential is found in the region distal to 2 mm. If the protophloem acts as a water source 3 mm from the tip, the more negative Ψ is found in the region apical to the source. If the source extends from 4 to 5 mm, the PEWS model closely resembles the EASE model. These profiles show that the limited phloem sources have a very local influence over the water potential of the root. At the end of the source, the water potential values quickly (within less than 1 mm) come to approximate the values of the EASE model. PEWS. An increase of root radius from 0.3 mm to 0.7 mm causes a progressive decrease in interior Ψ with EASE but hardly changes the radial pattern of growth-sustaining Ψ for the PEWS model (Fig. 7B).

Discussion
The multiple source model developed here has more powerful numerics, full three-dimensional treatment, and more computational power than was available in older models. The results of the previous model of Silk and Wagner (1980) James et al. 2006). Our PEWS model also includes the new assumption that there is an additional water source that is transporting water into the growth zone via the protophloem. Within root growing regions it is commonly observed that turgor and osmotic gradients are rather uniform across the root radius, during steady growth (Pritchard et al, 2000;Spollen and Sharp, 1991). The results of PEWS are consistent with these empirical results.
The assumption of internal water sources is also consistent with work showing transport of water and sugars from protophloem to the more apical root tissue (Hukin et al, 2002;Gould et al, 2004) and effects of light intensity on sugar transport and associated growth rate patterns in roots (Muller et al, 1998;Nagel et al, 2006). The new results also support the concept of the hydraulic isolation of the growth zone, as our thermodynamic transport model replicates the empirical study showing that apical regions of corn roots are not affected by negative water potentials in mature, more distal regions (Zwieniecki et al, 2003).
The sensitivity analysis explains several adaptive morphological features if we assume that a small growth-sustaining water potential gradient facilitates growth under stressful conditions. As the phloem initiation sites become farther from the root tip (and the length of non-vascularized tissue in the growth zone becomes longer), the growth-sustaining water potential gradients become larger; and the PEWS solution approximates the EASE model. These results indicate the adaptive value of one of the known physiological responses to water stress: more apical vasculature development (Beauchamp and Lathwell, 1966). Surprisingly, the presence of phloem water sources makes the growth-sustaining water relations of the root rather immune to changes in root radius. Thus the reported thinning of roots under water stress (Sharp et al, 1988) may be adaptive as a way to produce osmotic adjustment or to economize carbon allocated for elongation, rather than as a way to permit growth with small Ψ gradients. A related insight is that the commonly observed thickening of roots growing in hard soils would not necessitate enormous changes in growth sustaining water potentials. Growth of the thicker root tips would be facilitated by increased flux to root protophloem and more apical differentiation of the phloem.
The PEWS model could be extended to explore the hydraulic interactions between root and soil if the present model is embedded in a porous matrix with appropriate properties: hydraulic conductivity that decreases with water content and flow governed by Darcy's Law. This is a complicated problem numerically but worthy of future study. The millimeter-to-meter scale of our approach would provide information on the relationships of soil hydraulic properties to growth and would be a useful complement to larger scale models that have shown complex time-dependent patterns of soil water depletion around root systems (Clausnitzer and Hopmans, 1994;Garrigues et al, 2006).

Root Grid
To facilitate the numerical approach to this problem, a body fitted grid was created that approximates an average corn root, see Figures 1 and 8. The outer grid surface was generated by averaging the boundary coordinates of corn roots in micrographs. The internal computational grids were created using a parabolic longitudinal grid, combined with a modified cross-sectional H-grid (See Figure 8).

Ψ
The governing equation (1) was converted to a three-dimensional generalized-coordinate partial differential equation. Finite difference approximation was used to convert the partial-differential equation into a linear system of equations, represented in matrix form by: This matrix equation is used to solve for the unknown internal water potential values (Ψ (i,j,k) ) using the known relative elemental growth rate (L (i,j,k) ) and the calculated sparse coefficient matrix [Coeff]. Matlab was used to solve the matrix system via Bi-Conjugate Gradient Method.
Details of the numerical approach are presented for two dimensions. The extension to three dimensions is straightforward.

Generalized Coordinates Applied to Equation (1)
Recall from Equation (1), the two-dimensional equation in cartesian coordinates for water potential, Ψ is given by: www.plantphysiol.org on August 16, 2017 -Published by Downloaded from Copyright © 2009 American Society of Plant Biologists. All rights reserved. Since the equations for other variable are similar we will only present the generalized coordinate transformation for x.
Denoting the Jacobian Matrix of Transformation by J and differentiating the right hand side of equation (8) by parts we get: r r r r r r r r r r r r r r r r (9) Proceeding in a similar manner with the other variables results in an additional sets of equations similar to (9) that can be substituted into (6) to get a formula for L.

Finite Difference Applied to Equation (9)
Second order finite different approximations are used to approximate the derivates resulting in: r r r r r r r r r r r r r r r r r r r r (10) where discretization in the x-direction is denoted by Ψ i , discretization in the y-direction is denoted by Ψ j and:

Three-Dimensional Model of Root Growth with Flux Boundary
Pritchard et al 1996 recorded an osmotic gradient that ran the length of the root, ranging linearly from 0.27 MPa at 2.5 mm to about 0.15 MPa at 12 mm from the tip . This gradient motivated an expansion of the three-dimensional External water source Equilibrium model to include a flux boundary condition. The resulting model was derived in two dimensions and then extended for three-dimensional implementation which consists of the following two equations.
Since the flux is in the radial direction (13b) is replaced by where r r y x r given y x r K K K r r r r r r = = .
Writing the flux boundary in generalized coordinates, recalling that Ψ r = Ψ ξ ξ r + Ψ η η r , we have: Substituting into (15) we obtain: Approximating Ψ with the 2nd Order Finite Difference Boundary Approximations we get: which results in: A similar process is applied at the other boarders to obtain: This flux equation is reflected in the following new coefficient matrix.

Comments on the Computational Flux Grids
Although the H-grid worked well for most of our simulations, it was not a good computational grid for the implementation of the flux condition. In order to address this a new flux computational grid was used (see Figure 8) with two radial rings added at a distance of Δr and 2Δr around the H-grid. In this gridding system, the outside ring was associated with the growth medium (Ψ soln ) and the inner ring was associated with the boundary of the growing root.

Solving the two-dimensional method
For the flux boundary condition, the resulting matrix equations are: Adding the two systems give This resulting matrix equation incorporates the flux boundary condition and can be solved for the water potential, Ψ . This system is again sparse and is very large. The Matlab implementation uses an iterative Bi-Conjugate Gradient Method to solve for Ψ . This approach was faster then the previous direct-solver and resulted in a solution within a tolerance of 10 -6 Trefethen and Bau (1997).

Experimental Data
A hydroponically grown corn root, Zea mays was modeled. The average primary growth zone for this root is 10 mm long, with an average root radius of 0.5 mm within the elongation zone. The average width of the phloem source tubes was a harder number to calculate. In reviewing vascular system physiology literature, many references refer to the difficulty of defining the protophloem radius due to the small number of cells involved. The best reference for source radius estimate was Beauchamp and Lathwell (1966), who estimated 17 to 24 sieve tubes per transverse section in the 1.5 mm root radius at 2.5 cm from the root tip. This information combined with the estimate of sieve element having a radius of 5 to 10 microns was used to estimate protophloem source radius of 0.16 to 0.06 mm. The sources were then placed, with given radius, in a penti-diagonal (non-axially symmetric) pattern around the root-cross section (see Fig 8) and modeled to extend to within 1 mm of the root tip. The source value water potential is Ψ = -0.2 MPa, which maintains the protophloem Ψ less negative than the interior cells (Pritchard, 1996).

Relative Elemental Growth Rates, L
Marking experiments can be found in the literature to establish L. Here the data was extended to the computational grid spacing using a cubic spline interpolation, see Figure 8D (Erickson and Sax, 1956;Boyer and Silk, 2004;Silk and Wagner, 1980).

Hydraulic Conductivity, K r r
A literature review resulted in estimates of radial hydraulic conductivity in the range 7.3 x 10 -11 m 2 s -1 MPa -1 < r K r r < 5 x 10 -10 m 2 s -1 MPa -1 (Ginsburg and Ginzburg, 1970;Bret-Harte and Silk, 1994;Frensch and Hsiao, 1995 used in the reference calculations (Frensch and Hsiao, 1995). The radial and longitudinal hydraulic conductivities are assumed to be independent, with possible spatial variation. Hydraulic conductivity of roots would be expected to vary with growth conditions and to be especially sensitive to plant water status and associated environmental conditions. Note that our transport coefficients are phenomenological rather than mechanistic, as we use a bulk tissue coefficient and have neglected cell structure. Nevertheless, the validity of the model is supported by early demonstrations that coupled flows in apoplasm and symplasm result in water movement that follows the simple transport law (Molz, 1976).

Root Growth Conditions
The root growth medium defines the model boundary conditions. Models assumed laboratory hydroponic growth, with a solution boundary condition (Ψ = -0.02 MPa) (Spollen and Sharp, 1991). For the internal boundary (at z = 10 mm), the governing equations (1) were used to solve for the cross-section boundary Ψ values.

Sensitivity Analysis
A physiology sensitivity analysis was conducted using the multiple source model. Refer to Table I to see range of physiological variables that were tested including maximum root radius, hydraulic conductivity and growth rate.             (Pritchard, 1996;Pritchard et al, 1996), -0.46 MPa (Warmbrodt, 1987  MPa at z = 12 mm .