Three Dimensional Computational Model of Water Movement in Plant Root Growth Zone
Brandy
1 wiegers@math.ucdavis.edu 2
1,2
Wiegers ,
Dr. Angela
2
Cheer ,
Dr. Wendy
3
Silk
Department of Mathematics, University of California, Davis 3 Department of Land, Air, and Water Resources
University of California, Davis, One Shields Avenue, Davis, CA 95616
PRELIMINARY RESULTS
ABSTRACT
Primary 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 threedimensional point source model is examined in terms of current plant physiology measurements which results in suggestions for future work.
Model Objective
In 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.
BIOLOGY BACKGROUND
Primary Root
Anatomy
(bottom 10 mm of plant root)
Cross
section of
mature zone
Zone of
Maturation
How do Plants Grow?
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
Growth Variables
Hydraulic Conductivity (K)
Measure of ability of water to move through the plant
Zone of
Elongation
Internal Source Model
THE MATHEMATICAL MODEL
Defining the Relationship between growth and water potential
L(z) = ▼·(K·▼)
Given Experimental Data
K : Hydraulic Conductivity
Kx, Kz : 4 x10-8cm2s-1bar-1 - 8 x10-8cm2s-1bar-1
L(z): Relative Element Growth Rate (1/hr)
Unknown:
: Water Potential
 = 0 Boundary Condition
To model growing roots in pure water
Model Assumptions
* 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
Generalized Coordinates
Generalized 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
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.
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
Sieve Tube
FUTURE GOALS
Numerical Method
2nd Order Finite Difference Approximations
Relative Elemental Growth Rate (L)
Zone of
Cell Division
Apical
Meristem
Osmotic Model
A measure of the spatial distribution of
longitudinal growth within the root organ. L is
measured using a marked growth experiment.
Root Cap
i -1, j +1
i , j +1
i +1, j +1
i -1, j
i,j
i +1, j
i -1, j -1
i , j -1
i +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)
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.
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 crosssection (r-z)
* 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.
References
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1991.
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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
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