Three-Dimensional%20Internal%20Source%20Plant%20Root%20Growth%20Model

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Three-Dimensional Internal SourcePlant 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
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Research Motivation
http//www.wral.com/News/1522544/detail.html
http//www.mobot.org/jwcross/phytoremediation/grap
hics/Citizens_Guide4.gif
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Photos from Silks lab
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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/Ge
  n.20Plant20Cell20Quiz.jpg

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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/soil212
  5/doc/s7chp3.htm

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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
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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
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Relationship of Growth VariablesL(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?xxKy?yyKz?zz
• Kxx?xKyy?yKzz?z (3)

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Given Experimental Data

• Kx, Kz 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
• L(z) ? g

Erickson and Silk, 1980
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Boundary Conditions (??)
zmax

• y 0 on ??
• Corresponds to growth of root in pure water
• rmax 0.5 mm
• Zmax 10 mm

rmax
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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 ??
• Unknown ?
• Lijk Coeff ?ijk (4)
• The assumptions are the key.

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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.

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3D Osmotic Model Results
Remember each individual element will travel
through this pattern
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Analysis of 3D Results

• Model Results
• Longitudinal ? gradient
• Radial ? gradient
• Empirical Results
• Longitudinal ? gradient has been measured
• No radial ? gradient has been measured

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Phloem Source
Gould, et al 2004
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Internal Source 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 from the surrounding growth medium and
  from internal proto-phloem sources.

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3D Phloem Source Model
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Comparison of Results
Osmotic 3-D Model Results
Internal Source 3-D Model Results
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My Current WorkSensitivity Analysis

• Looking at different plant root anatomies,
  source values, geometry, and initial value
  conditions.

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Plant Root Geometryr 0.3mm0.5mm0.7mm
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Plant Root GeometryProto-phleom Placement2.1 mm
from tip, 4.1mm, 6.1mm from tip, no source
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Hydraulic ConductivityKr 4 x10-8cm2s-1bar-1
Kr 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
Source, 4.1 mm
No Source
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Hydraulic ConductivityKr 4 x10-8cm2s-1bar-1
Kr 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1
Source, 2.1 mm
No Source
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Growth Boundary ConditionsSoil vs Water
Source, 2.1 mm
No Source
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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

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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.

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Thank you! Do you have any further questions?

• Brandy Wiegers
• University of California, Davis
• wiegers_at_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

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Grid Refinement Grid Generation
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