3-d Computational Model of Water Movement in Plant Root Growth Zone Brandy Wiegers University of California, Davis Dr. Angela Cheer Dr. Wendy Silk 2007 Joint Mathematics Meeting January 8, 2007 New Orleans, LA

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).

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).

Osmotic Root Growth Model Assumptions   The tissue is 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 (K x ) and longitudinal (K z ) 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.

Solving for  L(z) =▼·(K·▼  )(1) L(z) = K x  xx + K y  yy + K z  zz + K x x  x + K y y  y + K z z  z (2)

Given Experimental Data Kx, Kz : 4 x10 -8 cm 2 s -1 bar x cm 2 s -1 bar -1 L(z) = ▼ · g Erickson and Silk, 1980

Boundary Conditions (  Ω)    = 0 on  Ω   Corresponds to growth of root in pure water   r max = 0.4 mm   Z max = 10 mm r max z max

Solving for  L(z) =▼·(K·▼  )(1) L(z) = K x  xx + K y  yy + K z  zz + K x x  x + K y y  y + K z z  z (2) Known: L(z), K x, K y, K z,  on  Ω Unknown: 

3D Osmotic Model Results *Remember each individual element will travel through this pattern*

Analysis of 3D Results Empirical Results   Longitudinal  gradient does exist   No radial  gradient Model Results Boyer and Silk, 2004

Phloem Source Gould, et al 2004

New Model Assumptions The tissue is cylindrical, with radius x, growing only in the direction of the long axis z. The growth pattern does not change in time. Conductivities in the radial (K x ) and longitudinal (K z ) 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.

3D Phloem Source Model

Comparison of Results Osmotic 3-D Model Results Internal Source 3-D Model Results

My Future Work… Sensitivity Analysis: Looking at different plant root anatomies, source values, geometry, and initial value Plant Root Micro- Environment

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? Thank you! Do you have any further questions? Brandy Wiegers University of California, Davis My Thanks to Dr. Angela Cheer, Dr. Wendy Silk, the JMM organizers and everyone who came to my talk today. This material is based upon work supported by the National Science Foundation under Grant #DMS