BIOLOGY BACKGROUND

Three Dimensional Computational Model of Water Movement in Plant Root Growth Zone

Brandy Wiegers1,2, Dr. Angela Cheer2, Dr. Wendy Silk3

1 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

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.

Maturation

Zone of

Elongation

Cell Division

Root Cap

Meristem

Sieve Tube

PRELIMINARY RESULTS

FUTURE GOALS

THE MATHEMATICAL MODEL

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.

Given Experimental Data

K : Hydraulic ConductivityKx, Kz : 4 x10-8cm2s-1bar-1 - 8 x10-

8cm2s-1bar-1

L(z): Relative Element Growth Rate (1/hr)

Defining the Relationship between growth and water potentialL(z) = ▼·(K·▼)

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

Numerical Method2nd Order Finite Difference Approximations

Osmotic Model Internal Source Model

ANALYSIS of RESULTS

i , ji -1, j

i , j +1

i +1, j

i , j -1i -1, j -1 i +1, j-1

i +1, j +1i -1, j +1

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)

How do Plants Grow?

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.

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

* 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.

Growth Variables

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

* 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.Mathematical Methods

* 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.

Hydraulic Conductivity (K)Measure of ability of water to move through the plant

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.

Unknown:: Water Potential = 0 Boundary Condition

To model growing roots in pure water

References

Cross section of mature zone

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)

Model Assumptions

John S. Boyer and Wendy K. Silk, Hydraulics of plant growth, Functional Plant Biology 31 (2004), 761:773.C.A.J.Fletcher, Computational techniques for fluid dynamics: Specific techniques for different flow categories, 2nd ed., Springer Series in Computational Physics, vol. 2, Springer-Verlag, Berlin, 1991.Cosgrove DJ and Li Z-C, Role of expansin in developmental and light control of growth and wall extension in oat coleoptiles., Plant Physiology 103 (1993), 1321:1328.Ralph O. Erickson and Wendy Kuhn Silk, The kinematics of plant growth, Scientific America 242 (1980), 134:151.Nick Gould, Michael R. Thorpe, Peter E. Minchin, Jeremy Pritchard, and Philip J. White, Solute is imported to elongation root cells of barley as a pressure driven-flow of solution, Functional Plant Biology 31 (2004), 391:397.

Jeremy Pritchard, Sam Winch, and Nick Gould, Phloem water relations and root growth, Austrian Journal of Plant Physiology 27 (2000), 539:548. 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 Physiology 103 (1993), 493:500.W.K. Silk and K.K. Wagner, Growth-sustaining water potential distributions in the primary corn root, Plant Physiology 66 (1980), 859:863.T.K.Kim and W. K. Silk, A mathematical model for ph patterns in the rhizospheres of growth zones., Plant, Cell and Environment 22 (1999), 1527:1538.Hilde Monika Zimmermann and Ernst Steudle, Apoplastic transport across young maize roots: effect of the exodermis, Planta 206 (1998), 7:19.