3D Computational Model of Water

Movement in Plant Root Growth Zone

Under the Direction of Dr. Angela Cheer

email: wiegers@math.ucdavis.edu website: http://math.ucdavis.edu/~wiegers

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

Brandy WiegersUniversity of California, Davis

Notation: Kx, Ky, Kz: The hydraulic conductivities fx = f/x

In 3D:

L(z) = Kxxx+ Kyyy +Kzzz+ Kxxx Ky

yy Kzzz (3)

L(z) = ·(K·) (1)

Relationship between growth Relationship between growth and water potentialand water potential

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.

Model Model AssumptionsAssumptions Boundary Conditions (Boundary Conditions (Ω)Ω)

= 0 on Ω Corresponds to growth of root in pure water

ΔΔ

Experimental DataExperimental Data Kx, Kz : 4 x10-8cm2s-1bar-1 - 8 x10-8cm2s-1bar-1

L(z) : 1/hr

Model SetupModel Setup

2D Visualization of 3D

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

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.

Resulting Resulting ModelsModelsOsmotic Model Known Values: L(z), Kx,Ky, Kz, on Ω

Source Model Known Values: L(z), Kx,Ky, Kz, on Ω and at source cells.

L(z) = Kxxx+Kyyy+Kzzz+Kxxx+Ky

yy+Kzzz (3)

= f(Kξξ, Kη

η,Kζζ,ξ, η,ζ,ξξ, ηη,ζζ,ξη ,ξζ,ηζ)

L = [Coeff] , Solve matrix equation for

The Research ProblemThe Research Problem

1980: Silk and Wagner created the Osmotic Root Growth 1980: Silk and Wagner created the Osmotic Root Growth Model, which predicated a radial water potential gradient in the Model, which predicated a radial water potential gradient in the plant root growth zone.plant root growth zone. 2004: Laboratory techniques/ equipment was finally available 2004: Laboratory techniques/ equipment was finally available to be able to test this hypothesis. Empirical evidence did not to be able to test this hypothesis. Empirical evidence did not support the 1980 theory.support the 1980 theory. 2004: Gould, et al find evidence that phloem is extending into 2004: Gould, et al find evidence that phloem is extending into the plant root growth zone.the plant root growth zone.

MotivatioMotivationn

Primary plant root growth is dependent on water movement within the Primary plant root growth is dependent on water movement within the growth zone. Thus, an understanding of plant root growth can be helpful growth zone. Thus, an understanding of plant root growth can be helpful in understanding crop draught and other water-soil-plant interactions.in understanding crop draught and other water-soil-plant interactions.

History of the History of the ProblemProblem

It is our hypothesis that the phloem sieve cells that extend into It is our hypothesis that the phloem sieve cells that extend into the primary plant root growth zone provide a water source to the primary plant root growth zone provide a water source to facilitate the plant root growth process. This hypothesis is facilitate the plant root growth process. This hypothesis is tested using a computational model of the plant root growth tested using a computational model of the plant root growth zone water potential.zone water potential.

Problem StatementProblem Statement

Generalized CoordinatesGeneralized CoordinatesConverts any grid (x,y,z) into a nice orthogonal grid (ξ,η,ζ) using Jacobian (J) and Inverse Jacobian (J-1)

Applying Generalized Coordinates to (3)Applying Generalized Coordinates to (3)L(z) = Kxxx+Kyyy+Kzzz+Kx

xx+Kyyy+Kz

zz (3)

Kxx = Kx

ξξx + Kxηηx + Kx

ζζx

x = /x= ξξx + ηηx + ζζx

xx = (x)x = (x)ξξx + (x)ηηx + (x)ζζx

= (ξξx + ηηx + ζζx )ξξx + (ξξx + ηηx + ζζx )ηηx + (ξξx + ηηx + ζζx)ζζx

: Growth Sustaining Water Potential: Growth Sustaining Water Potential

Osmotic Model ResultsOsmotic Model ResultsAnalysisAnalysisof Resultsof 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 PhysiologyPlant Physiology

Δ

Growth Zone AnatomyGrowth Zone Anatomy Plant Cell GrowthPlant Cell GrowthExpansive growth of plant cells is controlled Expansive growth of plant cells is controlled principally by processes that loosen the wall and principally by processes that loosen the wall and enable it to expand irreversiblyenable it to expand irreversibly (Cosgrove, 1993).(Cosgrove, 1993).

Water must be brought into the cell to facilitate the Water must be brought into the cell to facilitate the growth (an external water source).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 additional Cells must shear to create the needed additional surface area.surface area.

The growth process is irreversibleThe growth process is irreversible

Plant CellPlant Cell Water Facilitated Cell GrowthWater Facilitated Cell Growth

Rules of Plant Cell GrowthRules of Plant Cell Growth

Numerical MethodsNumerical Methods

2nd Order Finite Difference

Approximations

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)

: Growth Sustaining Water Potential: Growth Sustaining Water Potential

Source Model ResultsSource Model ResultsAnalysisAnalysisof Resultsof 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 VariablesGrowth Variables 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

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)])

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).

Hydraulic Conductivity (K)Hydraulic Conductivity (K) Water Potential (Water Potential ())

Relative Elemental Growth Rate (L)Relative Elemental Growth Rate (L)

Generalized 2-d Generalized 2-d Coordinate DiscretizationCoordinate 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]

End Goal…End Goal…Computational 3-d box of soil in which the plant Computational 3-d box of soil in which the plant roots grow in real time while changes in growth roots grow in real time while changes in growth variables are monitored.variables are monitored.

Continued Work on Root Grid: Refinement and GenerationContinued Work on Root Grid: Refinement and Generation Modification of Source Water PotentialModification of Source Water Potential

Phloem contains many dissolved solutes, Phloem contains many dissolved solutes,

Use Use ≠ 0 for the source terms.≠ 0 for the source terms. Looking at different plantsLooking at different plants

This work is done with a corn model, other plants need to be examinedThis work is done with a corn model, other plants need to be examined

Future Work…Future Work…