Transcript Presentation

Three-Dimensional Internal Source
Primary Root Growth Model
Brandy Wiegers
University of California, Davis
Dr. Angela Cheer
Dr. Wendy Silk
Joint Mathematics Meetings
January 2008
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
Presentation Outline
 Theoretical Background
Plant Biology
Governing Equations
Computational Approach
 Existing (External) Root Growth Theory
 Internal Source Root Growth Theory
 Future Work
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
What are the rules of plant
root 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 shear to create the needed
additional surface area.
 The growth process is irreversible
http://sd67.bc.ca/teachers/northcote/biology12/G/G1TOG8.html
Water Potential, 
  gradient is the driving
force in water movement.
  = 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).
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:
(Conductance = 1/ Resistance)
 A typical value:
Kr ,Kz = 1.3x 10-10 m2s-1MPa-1
 Value for a plant depends on growth
conditions and intensity of water flow
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.
Erickson and Silk, 1980
L(z) = ▼· (K·▼)
 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+ Krrr + Kzzz+ Krrrr (2)
 In 3d:
L(z) = Kxxx+Kyyy+Kzzz
+Kxxx+Kyyy+Kzzz
(3)
(1)
Experimental Data
zmax
rmax
  = -0.2 on Ω
 Corresponds to growth
of root in growth
solution
 rmax = 0.5 mm
 Zmax = 10 mm
 Kr, Kz :
1.3 x10-10m2s-1MPa-1
Solving for 
L(z) =▼·(K·▼ )
(1)
 Generalized Coordinates
 Finite Difference Approximations
Lijk = [Coeff] ijk
(3)
Known: L(z), Kx, Ky, Kz,  on Ω
Unknown: 
The assumptions are the key to the different
theories.
External Source Root Growth
Theory Assumptions
 The tissue is roughly 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 (Kr) 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.
External Source Theory
*Remember each individual element will travel through this pattern*
Multiple Source Root Growth
Theory
 Adds internal
known sources
 Doesn’t change
previous matrix:
L = [Coeff]
Gould, et al 2004
Multiple Source Root Growth
Theory Assumptions
 The tissue is roughly 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 (Kr) 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
Multiple Source Theory
Comparison of Results
3-D External Source Model Results
3-D Multiple Source Model Results
Sensitivity Analysis: Geometry
r = 0.3mm : 0.5mm :0.7mm
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 conductivity
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Thank you! Do you have any
further questions?
Brandy Wiegers
University of California, Davis
[email protected]
http://math.ucdavis.edu/~wiegers
My Thanks to Dr. Angela Cheer, Dr. Wendy Silk
and everyone who came to my talk today.
This material is based upon work supported by
the National Science Foundation under Grant
#DMS-0135345
References
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John S. Boyer and Wendy K. Silk, Hydraulics of plant growth, Functional Plant Biology 31 (2004),
761:773.
C.A.J.Fletcher, Computational techniques for fluid dynamics: Specific techniques for different flow
categories, 2nd ed., Springer Series in Computational Physics, vol. 2, Springer-Verlag, Berlin, 1991.
Cosgrove DJ and Li Z-C, Role of expansin in developmental and light control of growth and wall extension
in oat coleoptiles., Plant Physiology 103 (1993), 1321:1328.
Ralph O. Erickson and Wendy Kuhn Silk, The kinematics of plant growth, Scientific America 242 (1980),
134:151.
Nick Gould, Michael R. Thorpe, Peter E. Minchin, Jeremy Pritchard, and Philip J. White, Solute is
imported to elongation root cells of barley as a pressure driven-flow of solution, Functional Plant Biology
31 (2004), 391:397.
Jeremy Pritchard, Sam Winch, and Nick Gould, Phloem water relations and root growth, Austrian Journal
of Plant Physiology 27 (2000), 539:548.
J. Rygol, J. Pritchard, J. J. Zhu, A. D. Tomos, and U. Zimmermann, Transpiration induces radial turgor
pressure gradients in wheat and maize roots, Plant Physiology 103 (1993), 493:500.
W.K. Silk and K.K. Wagner, Growth-sustaining water potential distributions in the primary corn root, Plant
Physiology 66 (1980), 859:863.
T.K.Kim and W. K. Silk, A mathematical model for ph patterns in the rhizospheres of growth zones., Plant,
Cell and Environment 22 (1999), 1527:1538.
Hilde Monika Zimmermann and Ernst Steudle, Apoplastic transport across young maize roots: effect of
the exodermis, Planta 206 (1998), 7:19.
Generalized Coordinates
Fletcher, 1991
 Converts any grid (x,y,z) into a nice
orthogonal grid (ξ,η,ζ)
 Uses Jacobian (J) and Inverse
Jacobian (J-1)
Photo from Silk’s lab
Numerical Methods
2nd Order Finite Difference
Approximations
η
ξ
i -1, j +1
i , j +1
i +1, j +1
i -1, j
i,j
i +1, j
i -1, j -1
i , j -1
i +1, j-1
Given general function G(i,j):
 G(i,j)ξ = [G(i+1,j) – G(i-1,j)] / (2Δξ) + O(Δξ2)
 G(i,j)ξξ = [G(i+1,j) - 2G(i,j) + G(i-1,j)] / (Δξ2) + O(Δξ2)
 G(i,j)ξη = [G(i+1,j+1) - G(i-1,j+1) – G(i+1,j-1) + G(i-1,j-1)] /
(4ΔξΔη) + O(ΔξΔη)
Grid Refinement
& Grid Generation