3-d Computational Model of
Water Movement in Plant
Root Growth Zone
Brandy Wiegers
University of California, Davis
Dr. Angela Cheer
Dr. Wendy Silk
2007 Joint Mathematics Meeting
January 8, 2007
New Orleans, LA
http://faculty.abe.ufl.edu/~chyn/age2062/lect/lect_15/MON.JPG
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
Osmotic Root Growth
Model Assumptions
 The tissue is cylindrical, 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.
Solving for 
L(z) =▼·(K·▼ )
(1)
L(z) = Kxxx+ Kyyy + Kzzz+ Kxxx +
Kyyy + Kzzz
(2)
Given Experimental Data
• Kx, Kz : 4 x10-8cm2s-1bar-1 - 8x10-8
8cm2s-1bar-1
• L(z) = ▼ · g
Erickson and Silk, 1980
Boundary Conditions (Ω)
zmax
rmax
  = 0 on Ω
 Corresponds to
growth of root in
pure water
 rmax = 0.4 mm
 Zmax = 10 mm
Solving for 
L(z) =▼·(K·▼ )
(1)
L(z) = Kxxx+ Kyyy + Kzzz+ Kxxx +
Kyyy + Kzzz
(2)
Known: L(z), Kx, Ky, Kz,  on Ω
Unknown: 
3D Osmotic Model Results
*Remember each individual element will travel through this pattern*
Analysis of 3D Results
Empirical Results
 Longitudinal 
gradient does exist
 No radial  gradient
Model Results
Boyer and Silk, 2004
Phloem Source
Gould, et al 2004
New Model Assumptions
• The tissue is cylindrical, with radius x,
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 the phloem sources.
http://home.earthlink.net/~dayvdanls/root.gif
3D Phloem Source Model
Comparison of Results
Osmotic 3-D Model Results
Internal Source 3-D Model Results
My Future Work…
• Sensitivity Analysis:
Looking at different
plant root anatomies,
source values,
geometry, and initial
value
• Plant Root MicroEnvironment
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 JMM 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