Three Dimensional Computational Model of Water Movement in Plant Root Growth Zone Brandy Wiegers1,2,

1
Three Dimensional Computational Model of Water
Movement in Plant Root Growth ZoneBrandy
Wiegers1,2, Dr. Angela Cheer2, Dr. Wendy Silk31
wiegers_at_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)?
G(i1,j) G(i-1,j) /(2??)O(??2)G(i,j)??
G(i1,j) -2G(i,j) G(i-1,j) /(??2)
O(??2)G(i,j)?? G(i1,j1)
-G(i-1,j1) G(i1,j-1)
G(i-1,j-1) / (4????) O(????)
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.
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