Three-Dimensional Internal Three-Dimensional Internal SourceSource
Plant Root Growth ModelPlant Root Growth ModelBrandy WiegersUniversity 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
Photos from Silk’s lab
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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/farabee/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) = Kzzz+ Kxxx + Kz
zz+ Kxxxx
(2)
In 3d:L(z) = Kxxx+Kyyy+Kzzz
+Kxxx+Ky
yy+Kzzz
(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)
y = 0 on Ω Corresponds to
growth of root in pure water
rmax = 0.5 mm Zmax = 10 mm
rmax
zmax
Solving for L(z) =·(K· ) (1)
L(z) = Kxxx+ Kyyy + Kzzz+ Kxxx + Ky
yy + Kzzz (3)
Known: L(z), Kx, Ky, Kz, on ΩUnknown:
Lijk = [Coeff] ijk (4)
The assumptions are the key.
Osmotic Root GrowthModel 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 ResultsModel 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 ResultsOsmotic 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 Geometryr = 0.3mm:0.5mm:0.7mm
Plant Root GeometryProto-phleom Placement
2.1 mm from tip, 4.1mm, 6.1mm from tip, no source
Hydraulic ConductivityKr: 4 x10-8cm2s-1bar-1
Kr: 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
Source, 4.1 mm No Source
Hydraulic ConductivityKr: 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
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