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

wiegers@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

Grid Refinement & Grid Generation