Computational Model of Water Movement in Plant Root Growth

Zone

Brandy WiegersUniversity of California, Davis

Angela Cheer

Wendy Silk

2005 World Conference on Natural Resource Modeling

June 17, 2005

http://www.uic.edu/classes/bios/bios100/labs/plantanatomy.htm

Research MotivationResearch Motivation

http://www.wral.com/News/1522544/detail.html http://www.mobot.org/jwcross/phytoremediation/graphics/Citizens_Guide4.gif

Presentation OutlinePresentation Outline

Plant BiologyPlant Biology Existing (Osmotic) Root Growth ModelExisting (Osmotic) Root Growth Model New (Internal Source) ModelNew (Internal Source) Model Future WorkFuture Work

Presentation OutlinePresentation Outline

Plant BiologyPlant Biology Existing (Osmotic) Root Growth ModelExisting (Osmotic) Root Growth Model New (Internal Source) ModelNew (Internal Source) Model Future WorkFuture Work

Root BiologyRoot Biology

http://www.emc.maricopa.edu/faculty/farabee/BIOBK/waterflow.gifhttp://www.resnet.wm.edu/~mcmath/bio205/

http://home.earthlink.net/~dayvdanls/root.gif

Photos from Silk’s lab

How do plant cells grow?How do plant cells grow?

Expansive growth of Expansive growth of plant cells is plant cells is

controlled controlled principally by principally by

processes that processes that loosen the wall loosen the wall and enable it to and enable it to

expand expand irreversibly irreversibly

(Cosgrove, 1993).(Cosgrove, 1993).

http://www.troy.k12.ny.us/faculty/smithda/Media/Gen.%20Plant%20Cell%20Quiz.jpg

What are the rules of plant What are the rules of plant root growth?root growth?

Water must be brought into the cell to facilitate Water must be brought into the cell to facilitate the growth (an external water source).the growth (an external water source).

The tough polymeric wall maintains the shape.The tough polymeric wall maintains the shape. Cells must shear to create the needed Cells must shear to create the needed

additional surface area.additional surface area. The growth process is irreversibleThe growth process is irreversible

http://sd67.bc.ca/teachers/northcote/biology12/G/G1TOG8.html

Growth VariablesGrowth Variables

g : growth velocity, mm/hr K : hydraulic conductivity,

cm2/(s bar) L : relative elemental

growth rate (REG) , 1/hr : water potential, bar

Silk and Wagner, 1980

Hydraulic Conductivity, KHydraulic Conductivity, K

Measure of ability of water to move Measure of ability of water to move through the plantthrough the plant

Inversely proportional to the resistance of Inversely proportional to the resistance of an individual cell to water influxan individual cell to water influx

Typical values: Typical values: KKxx ,K ,Kzz = 8 x 10 = 8 x 10-8-8 cm cm22ss-1-1barbar-1-1

Value for a plant depends on growth Value for a plant depends on growth conditions and intensity of water flowconditions and intensity of water flow

Relative Elemental Relative Elemental Growth Rate, L(z)Growth Rate, L(z)

• A measure of the spatial distribution of growth within the root organ.

• L(z) = · g

Erickson and Silk, 1980

Water Potential, Water Potential, ww

w gradient is the driving force in water movement.

http://www.soils.umn.edu/academics/classes/soil2125/doc/s7chp3.htm

Presentation OutlinePresentation Outline

Plant BiologyPlant Biology Existing (Osmotic) Root Growth ModelExisting (Osmotic) Root Growth Model New (Internal Source) ModelNew (Internal Source) Model Future WorkFuture Work

Existing (Osmotic) Model Existing (Osmotic) Model AssumptionsAssumptions

The tissue is cylindrical, with radius x, growing only in the direction of the long axis z.

The distribution of is axially symmetric. 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.

Boundary Conditions (Boundary Conditions (Ω)Ω)

= 0 on Ω Corresponds to

growth of root in pure water

Δx = Δz = 0.1 mm Xmax = 0.5 mm Zmax = 10 mmxmax

zmax

Solving for Solving for

Known: L(z), Kx, Kz, on Ω

Unknown:

L(z) =·(K·) (1)

L(z) = Kxxx+Kzzz+ Kxxx + Kz

zz (2)

ResultsResults

*Remember each individual element will travel through this pattern*

Distribution of Water Fluxes

Growth Sustaining Distribution

Analysis of ResultsAnalysis of Results

Empirical Results No radial gradient Longitudinal

gradient does exist

Model Results

Presentation OutlinePresentation Outline

Plant BiologyPlant Biology Existing (Osmotic) Root Growth ModelExisting (Osmotic) Root Growth Model New (Internal Source) ModelNew (Internal Source) Model Future WorkFuture Work

Phloem SourcePhloem Source

Adds internal known sources

Doesn’t change previous matrix:

L = [Coeff]

Gould, et al 2004

Model ResultsModel Results

Preliminary ResultsNew (Internal Source)

Existing (Osmotic)

New Model AssumptionsNew Model Assumptions

• The tissue is cylindrical, with radius x, growing only in the direction of the long axis z.

• The distribution of is axially symmetric.

• 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.

http://home.earthlink.net/~dayvdanls/root.gif

Presentation OutlinePresentation Outline

Plant BiologyPlant Biology Existing (Osmotic) Root Growth ModelExisting (Osmotic) Root Growth Model New (Internal Source) ModelNew (Internal Source) Model Future WorkFuture Work

End Goal…End Goal…

Computational 3-d box of soil through Computational 3-d box of soil through which we can grow plant roots in which we can grow plant roots in real time while monitoring the real time while monitoring the change of growth variables.change of growth variables.

Do you have any further Do you have any further questions?questions?

Brandy Wiegers

Graduate Group of Applied Mathematics (GGAM)University of California, Davis

Email:wiegers@math.ucdavis.edu

This material is based upon work supported by the National Science Foundation under Grant #DMS-0135345

ReferencesReferences

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.