Three-Dimensional Internal Source Primary Root Growth Model

Brandy WiegersUniversity 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

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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·) (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+ Krrr + Kz

zz+ Krrrr (2)

In 3d:L(z) = Kxxx+Kyyy+Kzzz

+Kxxx+Ky

yy+Kzzz

(3)

Experimental Data = -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

rmax

zmax

Solving for L(z) =·(K· ) (1)

Generalized Coordinates Finite Difference Approximations

Lijk = [Coeff] ijk (3)

Known: L(z), Kx, Ky, Kz, on ΩUnknown:

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.

*Remember each individual element will travel through this pattern*

External Source Theory

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 Multiple Source Model Results

3-D External 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.

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Thank you! Do you have any further questions?

Brandy WiegersUniversity of California, Daviswiegers@math.ucdavis.eduhttp://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 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

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

Fletcher, 1991

Numerical Methods2nd 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)

i , ji -1, j

i , j +1

i +1, j

i , j -1i -1, j -1 i +1, j-1

i +1, j +1i -1, j +1

ηξ

Grid Refinement & Grid Generation