Computational Model of Water Movement in Plant Root Growth Zone

1. Computational Model of Water Movement in Plant Root Growth Zone Brandy Wiegers University 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

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

3. Presentation Outline • Plant Biology • Existing (Osmotic) Root Growth Model • New (Internal Source) Model • Future Work

4. Presentation Outline • Plant Biology • Existing (Osmotic) Root Growth Model • New (Internal Source) Model • Future Work

5. Root Biology http://home.earthlink.net/~dayvdanls/root.gif http://www.emc.maricopa.edu/faculty/farabee/BIOBK/waterflow.gif http://www.resnet.wm.edu/~mcmath/bio205/

6. Photos from Silk’s lab

7. 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

8. 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

9. Growth 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

10. 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 • Typical values: Kx ,Kz = 8 x 10-8 cm2s-1bar-1 • Value for a plant depends on growth conditions and intensity of water flow

11. Relative Elemental Growth Rate, L(z) • A measure of the spatial distribution of growth within the root organ. • L(z) = ▼ · g Erickson and Silk, 1980

12. Water Potential, w • wgradient is the driving force in water movement. http://www.soils.umn.edu/academics/classes/soil2125/doc/s7chp3.htm

13. Presentation Outline • Plant Biology • Existing (Osmotic) Root Growth Model • New (Internal Source) Model • Future Work

14. Existing (Osmotic) 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.

15. Boundary Conditions (Ω) zmax •  = 0 on Ω • Corresponds to growth of root in pure water • Δx = Δz = 0.1 mm • Xmax = 0.5 mm • Zmax = 10 mm xmax

16. Solving for  Known: L(z), Kx, Kz,  on Ω Unknown:  L(z) =▼·(K·▼) (1) L(z) = Kxxx+Kzzz+ Kxxx + Kzzz (2)

17. Results Distribution of Water Fluxes Growth Sustaining  Distribution *Remember each individual element will travel through this pattern*

18. Analysis of Results Empirical Results • No radial  gradient • Longitudinal  gradient does exist Model Results

19. Presentation Outline • Plant Biology • Existing (Osmotic) Root Growth Model • New (Internal Source) Model • Future Work

20. Phloem Source • Adds internal known sources • Doesn’t change previous matrix: • L = [Coeff]  Gould, et al 2004

21. Model Results Preliminary Results New (Internal Source) Existing (Osmotic)

22. New 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

23. Presentation Outline • Plant Biology • Existing (Osmotic) Root Growth Model • New (Internal Source) Model • Future Work

24. 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.

25. Do you have any further 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

26. 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.