Download Presentation

Brandy Wiegers
University of California, Davis
3D Computational
Model of Water
Movement in Plant
Root Growth Zone
Under the Direction of Dr. Angela Cheer
email: [email protected]
website: http://math.ucdavis.edu/~wiegers
This material is based upon work supported by the National Science Foundation under Grant #DMS-0135345
Relationship between growth
and water potential
L(z) = ▼·(K·▼)
(1)
 Notation:
 Kx, Ky, Kz: The hydraulic conductivities
 fx = f/x
 In 3D:
L(z) = Kxxx+ Kyyy +Kzzz+ Kxxx Kyyy Kzzz (3)
Model Setup
Model Assumptions
Boundary Conditions ( Ω)
 The tissue is cylindrical, with radius x,
growing only in the direction of the long axis z
 = 0 on Ω
 Corresponds to
growth of root in
pure water
Osmotic Model: The distribution
of is axially symmetric.
Source Model: This assumption
no longer holds. Look at the
phloem structure. It’s not regularly
distributed. Add a set of
distributed sources within the
radial cross-section of the root.
zmax
r = z = 0.1 mm
rmax = 0.5 mm
Zmax = 10 mm
rmax
 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.
Experimental Data


Kx, Kz : 4 x10-8cm2s-1bar-1 - 8 x10-8cm2s-1bar-1
L(z) : 1/hr
2D Visualization of 3D
Resulting Models
Osmotic Model Known Values: L(z), Kx,Ky, Kz,
Source Model Known Values: L(z), Kx,Ky, Kz,
and at source cells.
L(z) = Kx
xx+K
y
yy+K
= f(Kξξ, Kηη,Kζζ,
L = [Coeff]
ξ,
z
x
zz+K x
η,
ζ,
ξξ,
y
x+K y
ηη,
z
y+K z
ζζ,
, Solve matrix equation for
ξη ,
Ω
on
Ω
on
z
ξζ,
(3)
ηζ)
The Research Problem
Motivation
Primary plant root growth is dependent on water movement within the
growth zone. Thus, an understanding of plant root growth can be helpful
in understanding crop draught and other water-soil-plant interactions.
History of the Problem
 1980: Silk and Wagner created the Osmotic Root Growth
Model, which predicated a radial water potential gradient in the
plant root growth zone.
 2004: Laboratory techniques/ equipment was finally available
to be able to test this hypothesis. Empirical evidence did not
support the 1980 theory.
 2004: Gould, et al find evidence that phloem is extending into
the plant root growth zone.
Problem Statement
It is our hypothesis that the phloem sieve cells that extend into
the primary plant root growth zone provide a water source to
facilitate the plant root growth process. This hypothesis is
tested using a computational model of the plant root growth
zone water potential.
Gould, et al 2004
Generalized Coordinates
Converts any grid (x,y,z) into a nice orthogonal grid (ξ,η,ζ) using Jacobian (J) and Inverse Jacobian (J-1)
Fletcher, 1991
Applying Generalized Coordinates to (3)
L(z) = Kx
xx+K
y
yy+K
z
x
zz+K x
y
x+K y
z
y+K z
z
(3)
Kxx = Kxξξx + Kxηηx + Kxζζx
x
=
/ x=
xx
=(
x)x
=(
=(
ξξx
+
ξξx
x)ξξx
ηηx
ηηx
+
+
+(
+
ζζx
x)ηηx +
ζζx )ξξx
(
+(
x)ζζx
ξξx
+
ηηx
+
ζζx )ηηx +
(
ξξx
+
ηηx
+
ζζx)ζζx
Osmotic Model Results
 : Growth Sustaining Water Potential
Analysis
of Results
This Model predicts:
 Radial and Longitudinal
gradient
Laboratory Tests Have
Shown:
 No radial gradient
 Longitudinal gradient
does exist
THE OSMOTIC MODEL
PREDICTED RADIAL
GRADIENT CAN NOT
BE PROVEN IN A
LABRATORY
Plant Physiology
Growth Zone Anatomy Plant Cell Growth
Zone of
Expansive growth of plant cells is controlled
principally by processes that loosen the wall and
enable it to expand irreversibly (Cosgrove, 1993).
Maturation
Plant Cell
Water Facilitated Cell
Growth
Zone of
Elongation
http://www.tro.k12.n.us/facult/smithda/Me
dia/Gen.%20Plant%20Cell%20Quiz.j
pg
http://sd67.bc.ca/teachers/northcote/biolog12/G/G1TOG8.html
Rules of Plant Cell Growth
Sieve Tube
Zone of
Cell Division
Apical Meristem
Root Cap

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
Numerical Methods
2nd Order Finite Difference
Approximations
Given general function G(i,j):


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
 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)
Source Model Results
 : Growth Sustaining Water Potential
Analysis
of Results
This Model predicts:
Decreased Radial and
Longitudinal gradient
THE CURRENT
SOURCE MODEL
PREDICTION OF
REDUCED RADIAL
GRADIENT IS
REASONABLE IN
TERMS OF THE
LABORATORY
EXPERIMENTATION
THE MODEL NEEDS TO
BE FURTHER
DEVELOPED
Growth Variables
Hydraulic Conductivity (K)
Water Potential ( )
 w gradient is the driving force
in water movement.
 Measure of ability of water to move
through the plant
 Inversely proportional to the
resistance of an individual cell to
water influx
 Typical values: Kr ,Kz = 8 x 10-8
cm2s-1bar-1

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/academic
s/
classes/soil2125/doc/s7chp3.htm
Relative Elemental Growth Rate (L)
 A measure of the spatial distribution of growth within the root
organ. Measured using a marked growth experiment.
 Co-moving reference frame centered at root tip.
 L(z) = lim(AB→0)(1/(AB) [V(A)- V(B)])
 Generalize this as: L(z) =  g
Erickson and Silk, 1980
Generalized 2-d
Coordinate Discretization
L(z) = (i+1,j) (Cξ/(2Δξ)+Cξξ/(Δξ)2 )
+ (i-1,j) (-Cξ/(2Δξ)+Cξξ/(Δξ)2 )
+ (i,j+1) (Cη/(2Δη)+ Cηη/(Δη)2 )
+ (i,j-1) (-Cη/(2Δη)+ Cηη/(Δη)2 )
- 2 (i,j) (Cξξ/(Δξ)2 + Cηη/(Δη)2 )
+(i+1,j+1) Cξη/(4ΔξΔη) - (i-1,j+1) Cξη/(4ΔξΔη)
- (i+1,j-1) Cξη/(4ΔξΔη) + (i-1,j-1) Cξη/(4ΔξΔη)
L = [Coeff] 
Future Work…
 Continued Work on Root Grid: Refinement and Generation
 Modification of Source Water Potential
Phloem contains many dissolved solutes,
Use ≠ 0 for the source terms.
 Looking at different plants
This work is done with a corn model, other plants need to be
examined
End Goal…
Computational 3-d box of soil in which the plant
roots grow in real time while changes in growth
variables are monitored.