THREE DIMENSIONAL COMPUTATIONAL MODEL OF WATER MOVEMENT IN PLANT ROOT GROWTH ZONE

Transcription

1 THREE DIMENSIONAL COMPUTATIONAL MODEL OF WATER MOVEMENT IN PLANT ROOT GROWTH ZONE BRANDY WIEGERS Motivation Mathematical models of the plant root growth zone already exist, but there is a lack of experimental evidence to support the model results. It is our belief that these models, specifically the Osmotic Model presented by Silk and Wagner [7], need a component representing a pressure-driven bulk flow of water through the phloem to the region in the growth zone where phloem is just beginning to function. The combination of pressure driven conductive flow and water potential gradients could be bringing water from the functional phloem and the surrounding soil to facilitate growth. The new Internal Source Model that I am developing is expected to explain a body of empirical results on water relations. It will provide a mechanistic understanding of growth sustaining water relations. Biology of Root Growth As in animals, plant growth does not occur uniformly throughout the plant. Instead it is restricted to certain specialized zones. Primary root growth is contained in four overlapping zones (see Figure 1(b)). The main focus of this project is the elongation zone, where the cells are expanding. Expansive growth of plant cells is controlled principally by processes that loosen the wall and enable it to expand irreversibly [2]. It has been suggested that the phloem, an element in the vascular system of plants, may be providing an additional source of water for the growth process, beyond the water that is being provided by the surrounding soil, see Figure 1(b). The Internal Source Root Growth Model results from this observation and involves the addition of phloem sources to the Osmotic Model Root Growth. 245

2 246 WOMEN IN MATHEMATICS: MAY 18 20, 2006 (a) One-Dimensional Water Flux Through a Cell [1] (b) The Primary Root Growth Zone [5] (c) Generalized Coordinates [3] Figure 1. Graphics to demonstrate biological and mathematical concepts Governing Equations The main measure that is used to define primary root growth is the Relative Elemental Growth Rate (L). L is a measure of the spatial distribution of growth within a root organ and can be written in terms of the hydraulic conductivity (K) and the water potential (ψ): (1) L(z) = (g) = (K ψ). This is the equation that we use, given laboratory data for L and K, to solve for the needed growth sustaining water potential (ψ) in the root growth zone. The key to solving this equation is the assumptions, which are helpful in defining the problem setup, but before they can be used, we must address the fact that plant roots do not have box-like 2-D cross sections. To address this and to prevent the need to re-derive the equations with every new geometry, the model was put into generalized coordinates.

3 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 247 Generalized Coordinates provide a method to take any grid with components (x,y,z) and to convert it into an orthogonal, equally spaced grid with components (ξ, η, ζ) using the Jacobian matrix of the transformation. Define the following notation for the partial derivative, ψ x = ψ x is the x-partial derivative of ψ. Use of this notation in equation (1) results in the following. (2) L(z) = (Kξ x ξ x +Kη x η x +Kζ x ζ x )(ψ ξ ξ x +ψ η η x +ψ ζ ζ x )+K x (ψ ξ ξ x +ψ η η x +ψ ζ ζ x ) ξ ξ x + K x (ψ ξ ξ x + ψ η η x + ψ ζ ζ x ) η η x + K x (ψ ξ ξ x + ψ η η x + ψ ζ ζ x ) ζ ζ x + (K y ξ ξ y + Kη y η y + K y ζ ζ y)(ψ ξ ξ y + ψ η η y + ψ ζ ζ y ) + K y (ψ ξ ξ y + ψ η η y + ψ ζ ζ y ) ξ ξ y + K y (ψ ξ ξ y + ψ η η y + ψ ζ ζ y ) η η y + K y (ψ ξ ξ y + ψ η η y + ψ ζ ζ y ) ζ ζ y + (Kξ z ξ z + Kηη z z + Kζ z ζ z )(ψ ξ ξ z + ψ η η z + ψ ζ ζ z ) + K z (ψ ξ ξ z + ψ η η z + ψ ζ ζ z ) ξ ξ z + K z (ψ ξ ξ z + ψ η η z + ψ ζ ζ z ) η η z + K z (ψ ξ ξ z + ψ η η z + ψ ζ ζ z ) ζ ζ z. Approaching this equation using finite difference approximation results is a sparse matrix, [Coeff]. [Coeff] is used in equation (3) to solve for the unknown internal water potential values, ψ (i,j,k) using the known relative elemental growth rate, L (j) : (3) [Coeff]ψ = L Osmotic Model The key to the Osmotic Model is that the growth zone is hydraulically isolated, which allows us to define the boundary conditions by only the soil that surrounds the root. For the purposes of the Osmotic Model, the soil has been ignored and it has been assumed that the roots are grown in water which provides an external water potential, ψ = 0; this assumption matches the laboratory conditions. As with Silk and Wagner [7], the results of the Osmotic Model, see Figure 2(a), show a radial gradient in water potential that has not been verified empirically [7], [1]. Current empirical data for the plant root growth zone shows longitudinal water potential gradients with no radial water potential gradient [6], [1]. Internal Source Model After the simple Osmotic 3-D Model was created, the next step to test the present theory on water movement in the growth zone was to add sources within the interior of the root. These sources, with defined water potentials, have great effect on the neighboring values, and the results are very promising (see Figure 2(b)). The 3-D Internal Source model results are a better

4 248 WOMEN IN MATHEMATICS: MAY 18 20, 2006 representation of the current laboratory results but also show the need to further improve this model by improving the grid and better representing the physiology of the plant root sources. (a) Water Potential Distributions in 3-D plant root as predicted by Osmotic Model. A cross section, z = 5mm (left). A longitudinal section (right). (b) Water Potential Distributions in 3-D plant root as predicted by Internal Source Model. A cross-section, z = 5mm (left). A longitudinal section (right). Figure 2. 3-D Plant Root Growth Model Results Future Work Understanding of the physiology of plant root growth is the primary goal of this research. The current models of water movement in the root growth zone

5 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 249 cannot be verified experimentally. Achievement of a verifiable model will allow further study of soil/root relationships, looking specifically at uptake and efflux of water and nutrients in the growth zone. Understanding of these relationships will have far reaching effects, including understanding of hormone relationships, water stress, and bio-remediation applications. The next step in making this research useful is to understand how the plant-water relations affects plant-soil microenvironment. Such a model would have many applications including testing of soil moisture, effects of root growth, and further study of the soil-ph relationship [4]. With this in mind, the end goal for this research is to have a computational 3-D box of soil through which we can grow plant roots in real time while monitoring hydraulic conductivity, water potential, and relative elemental growth rates. References [1] John S. Boyer and Wendy K. Silk, Hydraulics of plant growth, Functional Plant Biology 31 (2004), no. 8, [2] D. J. Cosgrove and Z. C. Li, Role of expansin in cell enlargement of oat coleoptiles (Analysis of developmental gradients and photocontrol, Plant Physiol. 103 (1993), no. 4, [3] C. A. J. Fletcher, Computational Techniques for Fluif Dynamics: Specific Techniques for Different Flow Categories. 2nd ed. Springer Series in Computational Physics, vol. 2. Berlin: Springer-Verlag, [4] T. K. Kim and W. K. Silk, A mathematical model for ph patterns in the rhizospheres of growth zones, Plant, Cell Environ. 22 (1999), no. 12, [5] Peter H. Raven, Ray F. Evert, and Susan E. Eichorn, Biology of Plants. 5th ed. New York: Worth Publishers, [6] J. Rygol, J. Pritchard, J. J. Zhu, A. D. Tomos, and U. Zimmermann, Transpiration induces radial turgor pressure gradients in wheat and maize roots, Plant Physiol. 103 (1993), no. 2, [7] Wendy Kuhn Silk and Kit K. Wagner, Growth-sustaining water potential distributions in the primary corn root: A noncompartmental continuum model, Plant Physiol. 66 (1980), no. 5, Department of Mathematics; One Shields Ave.; University of California at Davis; Davis, CA USA address: wiegers@math.ucdavis.edu