BIOLOGY BACKGROUND

1. Zone of Maturation Zone of Elongation Sieve Tube Zone of Cell Division Apical Meristem Root Cap Three Dimensional Computational Model of Water Movement in Plant Root Growth ZoneBrandy Wiegers1,2, Dr. Angela Cheer2, Dr. Wendy Silk31 wiegers@math.ucdavis.edu 2 Department of Mathematics, University of California, Davis 3 Department of Land, Air, and Water ResourcesUniversity of California, Davis, One Shields Avenue, Davis, CA 95616 PRELIMINARY RESULTS ABSTRACTPrimary plant root growth occurs in the 10 mm root tip segment where cells expand by stretching the rigid cell wall that constricts their growth. Silk and Wagner (1980) provided an osmotic root growth model to describe this process. Their theory is expanded to a three-dimensional model with the addition of point source terms. This three-dimensional point source model is examined in terms of current plant physiology measurements which results in suggestions for future work. Osmotic Model Internal Source Model Model ObjectiveIn 2004, laboratory testing failed to find empirical evidence of the radial water potential gradient predicted by the Silk and Wagner osmotic root growth model. It is our hypothesis that the phloem sieve cells that extend into the primary plant root growth zone provide additional water to facilitate the plant root growth process. The Internal Source Root Growth Model tests this hypothesis by adding source terms to the Osmotic root growth model. THE MATHEMATICAL MODEL Defining the Relationship between growth and water potentialL(z) = ▼·(K·▼) Given Experimental DataK : Hydraulic ConductivityKx, Kz : 4 x10-8cm2s-1bar-1 - 8 x10-8cm2s-1bar-1L(z): Relative Element Growth Rate (1/hr) Unknown:: Water Potential = 0 Boundary Condition To model growing roots in pure water BIOLOGY BACKGROUND How do Plants Grow? Primary Root Anatomy(bottom 10 mm of plant root) Rules of Plant Cell Growth:* Water must be brought into the cell to facilitate the growth (an external water source).* The tough polymeric wall maintains the shape. * Cells must stretch to create the needed additional surface area.* The growth process is irreversible Model Assumptions Cross section of mature zone * The tissue is cylindrical, with radius r, growing only in the direction of the long axis z* Water potential distribution :* Osmotic Model: The distribution of  is axially symmetric. * Source Model: This assumption is not valid, biologically the  sources are not regularly distributed. * The growth pattern does not change in time.* Conductivities in the radial (Kx) and longitudinal (Kz) directions are independent. ANALYSIS of RESULTS * The Osmotic Model displays a radial gradient that can not be verified empirically.* The 3-d Internal Source model decreases the radial gradient and is a better representation of the empirical results.* Continued work, including further improvement and analysis of the model is recommended. Mathematical Methods Growth Variables Generalized CoordinatesGeneralized Coordinates allow for the most versatility in the grid. By using the Jacobian, any grid can be converted into a Cartesian grid that can be used to easily calculate numerical approximations. Grid Generation Hydraulic Conductivity (K)Measure of ability of water to move through the plant Water Potential ()The gradient in  the driving force in water movement.  Gradients in plants cause an inflow of water from the soil into the roots and to the transpiring surfaces in the leaves FUTURE GOALS * Continued Work on Root Grid Refinement and Generation.* Sensitivity analysis of K, geometry, physiology and source potential.* Examine different plant root anatomies and physiology.* Examination of plant root – soil microenvironment.End Goal: Computational 3-d box of soil in which the plant roots grow in real time while changes in growth variables are monitored. Numerical Method2nd Order Finite Difference Approximations Relative Elemental Growth Rate (L)A measure of the spatial distribution of longitudinal growth within the root organ. L is measured using a marked growth experiment. Given general function G(i,j):G(i,j)ξ = [G(i+1,j)–G(i-1,j)]/(2Δξ)+O(Δξ2)G(i,j)ξξ = [G(i+1,j) -2G(i,j)+G(i-1,j) ] /(Δξ2) +O(Δξ2)G(i,j)ξη = [G(i+1,j+1)-G(i-1,j+1)–G(i+1,j-1)+G(i-1,j-1)] / (4ΔξΔη) + O(ΔξΔη) A hybrid grid was created, using an h-grid for the radial cross-section (x-y) and a parametric grid with decreased curve for the longitudinal cross-section (r-z) ACKNOWLEDGEMENTS: We would like to acknowledge the NSF (Grant #DMS-0135345 ) for support of this project. Thank you also to the 2006 SIAM Meeting and AMS Workshop coordinators for the opportunity to present this work. References John S. Boyer and Wendy K. 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