Three-Dimensional Internal
Source
Plant Root Growth Model
Brandy Wiegers
University of California, Davis
Dr. Angela Cheer
Dr. Wendy Silk
2007 RMA World Conference
on Natural Resource Modeling
June, 2007
Cape Cod, MA
http://faculty.abe.ufl.edu/~chyn/age2062/lect/lect_15/MON.JPG
Research Motivation
http://www.wral.com/News/1522544/detail.html
http://www.mobot.org/jwcross/phytoremediation/graphics/Citizens_Guide4.gif
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Photos from Silk’s lab
How do plant cells grow?
Expansive growth of plant cells
is controlled principally by
processes that loosen the wall
and enable it to expand
irreversibly
(Cosgrove, 1993).
http://www.troy.k12.ny.us/faculty/smithda/Media/Gen.%20Plant%20Cell%20Quiz.jpg
Water Potential, w
 w gradient is the driving
force in water movement.
 w = s + p + m
 Gradients in plants cause
an inflow of water from
the soil into the roots and
to the transpiring
surfaces in the leaves
(Steudle, 2001).
http://www.soils.umn.edu/academics/classes/soil2125/doc/s7chp3.htm
Hydraulic Conductivity, K
 Measure of ability of water to move
through the plant
 Inversely proportional to the resistance of
an individual cell to water influx
 Think electricity
 A typical value:
Kx ,Kz = 8 x 10-8 cm2s-1bar-1
 Value for a plant depends on growth
conditions and intensity of water flow
http://www.emc.maricopa.edu/faculty/far
abee/BIOBK/waterflow.gif
Relative Elemental Growth
Rate, L(z)
 A measure of the spatial
distribution of growth
within the root organ.
 Co-moving reference
frame centered at root tip.
 Marking experiments
describe the growth
trajectory of the plant
through time.
 Streak photograph
 Marking experiments
Erickson and Silk, 1980
Relationship of Growth Variables
L(z) = ▼· (K·▼)
(1)
 Notation:
 Kx, Ky, Kz: The hydraulic conductivities in x,y,z
directions
 fx = f/x: Partial of any variable (f) with respect to
x
 In 2d:
L(z) = Kzzz+ Kxxx + Kzzz+ Kxxxx
(2)
 In 3d:
L(z) = Kxxx+Kyyy+Kzzz
+Kxxx+Kyyy+Kzzz
(3)
Given Experimental Data
 Kx, Kz : 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
 L(z) = ▼ · g
Erickson and Silk, 1980
Boundary Conditions (Ω)
zmax
 y = 0 on Ω
 Corresponds to
growth of root in
pure water
rmax
 rmax = 0.5 mm
 Zmax = 10 mm
Solving for 
L(z) =▼·(K·▼ )
(1)
L(z) = Kxxx+ Kyyy + Kzzz+ Kxxx + Kyyy + Kzzz
(3)
Known: L(z), Kx, Ky, Kz,  on Ω
Unknown: 
Lijk = [Coeff] ijk
The assumptions are the key.
(4)
Osmotic Root Growth
Model Assumptions
 The tissue is cylindrical beyond the root tip, with radius r,
growing only in the direction of the long axis z.
 The growth pattern does not change in time.
 Conductivities in the radial (Kx) and longitudinal (Kz)
directions are independent so radial flow is not modified by
longitudinal flow.
 The water needed for primary root-growth is obtained only
from the surrounding growth medium.
3D Osmotic Model Results
*Remember each individual element will travel through this pattern*
Analysis of 3D Results
Model Results
 Longitudinal  gradient
 Radial  gradient
Empirical Results
 Longitudinal  gradient
has been measured
 No radial  gradient
has been measured
Phloem Source
Gould, et al 2004
Internal Source Root Growth
Model Assumptions
 The tissue is cylindrical beyond
the root tip, with radius r, growing
only in the direction of the long
axis z.
 The growth pattern does not
change in time.
 Conductivities in the radial (Kx)
and longitudinal (Kz) directions are
independent so radial flow is not
modified by longitudinal flow.
 The water needed for primary
root-growth is obtained from the
surrounding growth medium and
from internal proto-phloem
sources.
3D Phloem Source Model
Comparison of Results
Osmotic 3-D Model Results
Internal Source 3-D Model Results
My Current Work…
Sensitivity Analysis
Looking at different plant root anatomies, source values, geometry,
and initial value conditions.
Plant Root Geometry
r = 0.3mm:0.5mm:0.7mm
Plant Root Geometry
Proto-phleom Placement
2.1 mm from tip, 4.1mm, 6.1mm from tip, no source
Hydraulic Conductivity
Kr: 4 x10-8cm2s-1bar-1
Kr: 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
Source, 4.1 mm
No Source
Hydraulic Conductivity
Kr: 4 x10-8cm2s-1bar-1
Kr: 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
Source, 2.1 mm
No Source
Growth Boundary Conditions
Soil vs Water
Source, 2.1 mm
No Source
Summary: Growth Analysis
 Radius: increase in radius results in increase
of maximum water potential and resulting
gradient
 Phloem Placement: The further from the root
tip that the phloem stop, the more the solution
approximates the osmotic root growth model
 Hydraulic Conductivity: Increased conducitivity
decreases the radial gradient
 Growth Conditions: Soil vs Water Conditions
play an important role in comparing source
and non source gradients
End Goal…
Computational 3-d box of soil through which
we can grow plant roots in real time while
monitoring the change of growth
variables.
Thank you! Do you have any
further questions?
Brandy Wiegers
University of California, Davis
wiegers@math.ucdavis.edu
http://math.ucdavis.edu/~wiegers
My Thanks to Dr. Angela Cheer, Dr. Wendy Silk,
the RMA organizers and everyone who came to
my talk today.
This material is based upon work supported by the National Science
Foundation under Grant #DMS-0135345
Grid Refinement
& Grid Generation