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**Three-Dimensional Internal Source Plant Root Growth Model**

* Look at graphs- all fonts should be 24 (including plots) Three-Dimensional Internal Source Plant Root Growth Model Brandy Wiegers University of California, Davis Dr. Angela Cheer Dr. Wendy Silk 2007 RMA World Conference on Natural Resource Modeling June, 2007 Cape Cod, MA Presentation Reminders: ** BREATHE. Get your sighs out now. ** Keep it simple.** Look at the audience.** Repeat their questions Basic Plant Biology: * Where are the roots * Primary Plant Root * There is a vascular system used to transport from the leaves to the roots.- phloem and xylem (next slide- show the cross section) Plants have 3 tissue systems --- ground, dermal, and vascular tissues. Plant tissues make up the main organs of a plant --- root, stem, leaf, & flower. Ground tissue makes up most of the plant's body, dermal tissue covers the outside of the plant, & vascular tissue conducts water & nutrients Presentation ABSTRACT Preliminary report.Primary plant root growth occurs in the 10mm root tip segment. Primary growth is characterized by longitudinal cells expansion that uses water to stretch the rigid cell walls. Silk and Wagner provided an osmotic root growth model to describe the water potential necessary to sustain this process. The osmotic model assumes that the growth zone is hydraulically isolated from the rest of the root, with all water necessary for growth coming from the surrounding soil. Unfortunately the radial water potential gradient suggested by the osmotic model results cannot be verified empirically. We have expanded upon the original theory to create a three- dimensional model with the addition of leaky pipe point sources in the growth zone.It is our conjecture that these structures are acting as pipes for water to be pushed down into the growth zone from the mature section higher in the root. These pipes are providing the additional water necessary for growth to occur. Using data from corn, Zeamays, we are able to examine the three-dimensional point source model in terms of current water potential measurements and suggest future work to continue the development of this model.

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Research Motivation Field draught: Phytoremediation:

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Photos from Silk’s lab

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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). Expansive growth of plant cells is controlled principally by processes that loosen the wall and enable it to expand irreversibly (Cosgrove, 1993). 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

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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). Water Potential has units of pressure. This is because they measure free energy (potential) and divide by partial molal volume of water. Units: bar or Pascal (1 bar = 0.1 MPa). w of pure water is zero (by definition). w gradient is the driving force in water movement. w measures the ability to attract water: Water moves within plants from regions of high water potential to regions of lower water potential, i.e. Down gradient. Osmosis: water moves from a high concentration (of water) to a lower concentration (of water). w = p + s + m Solute Potential (s) = osmotic potential (o) : a measure of the number of dissolved particles in water The solute potential of pure water (no solutes) = 0. Solute potential (of a solution) is always negative (-). For living organisms, w will be negative. Ex: For pure water Ys= 0. For a sugar solution, Ys = -10. Pressure Potential (p) (aka: Hydrostatic pressure): In a turgid cell the pressure potential is +: The cell is full of water and dissolved solutes, and presses against the cell wall like an inflated balloon full of air. A xylem vessel cell is usually under – pressure: (negative pressure = tension). Under - pressure potential water tends to be sucked out of the cell (water moves up the xylem vessels). Matrix potential (m): pressure required to remove water from a surface, due to the adhesion of water molecules to a (charged) surface. Matrix potential is important in water absorption from the soil, A characteristic of soil type: The smaller the soil particles, the more surface area the soil particles have in a given volume of soil. particle size of sand > silt > clay. The presence of salts in the soil decreases the Ys of the soil. This in turn decreases the Yw of the soil, making it more difficult for a root to absorb water from the soil. Matrix potential inside cell Ym = 0 Matrix potential of dry soil Ym = -3 Example: Yw of soil: if Ys = -1, Yp = 0,Ym = -1 , Yw of soil = = -2 Yw of root cell : if Ys = -5,Yp = +1, Ym = 0, Yw of root = = -4 Water moves from a higher Yw (-2) to a lower Yw (-4). So Water moves from the soil into the root cell. Gradients in plants cause an inflow of water from the soil into the roots and to the transpiring surfaces in the leaves. (Refer to diagram) * The ultimate driving force of water movement in plants is the gas in the atmosphere. *A cell can lower its water potential by accumulating solutes in the cell or by decreasing the osmotic pressure (turgor pressure) of the cell wall (using apoplasts) (Pritchard, et. al. 2000) * Growing tissue was seen as a distributed sink for water (Silk and Boyer PPT).

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**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 A typical value: Kx ,Kz = 8 x 10-8 cm2s-1bar-1 Value for a plant depends on growth conditions and intensity of water flow (Pritchard, et al 2002). Units: length per unit of time. Relates flow velocity to hydraulic gradient: A property of the porous medium and the fluid (water content of the medium), models for very slow water flow. Darcy’s Law: q = Q/A = -K dh/dl , where h = the hydraulic head= p /pg + Z * the law is very similar to Ohm's law for electrical circuits I = 1/R * U (current = voltage divided by resistance) * Darcy's law has been found to be invalid for high values of Reynolds number and at very low values of hydraulic gradient in some very low-permeability materials, such as clays. Specific to Root Biology * Good Measures exist (see Previous Papers) * Hydraulic Conductivity of roots can be quite variable. It may depend on factors such as growth conditions and on the intensity of water flow (Steudle, Apoplastic page 1)

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**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. Streak photograph Marking experiments A stationary reference frame, examines growth relative to a stationary point, such as the soil surface. A co-moving reference frame uses a point that represents the peak acceleration, such as the root tip Using the root tip results in a measurement of the steady growth pattern that all growing cells experience as they move through the 10mm growth zone. In this reference frame, once a cell has existed the growth zone it will continue to move away from the root tip at a steady rate because the root is growing down at a steady rate and the cell is no longer moving, so relative to the tip the cell is moving at a steady rate away from the root tip. The only change to this rate would come if the growth conditions effected the rate of the tip's growth. Erickson and Silk, 1980

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**Relationship of Growth Variables 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+ Kxxx + Kzzz+ Kxxxx (2) In 3d: L(z) = Kxxx+Kyyy+Kzzz +Kxxx+Kyyy+Kzzz (3)

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**Given Experimental Data**

Kx, Kz : 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1 L(z) = ▼ · g A stationary reference frame, examines growth relative to a stationary point, such as the soil surface. A co-moving reference frame uses a point that represents the peak acceleration, such as the root tip Using the root tip results in a measurement of the steady growth pattern that all growing cells experience as they move through the 10mm growth zone. In this reference frame, once a cell has existed the growth zone it will continue to move away from the root tip at a steady rate because the root is growing down at a steady rate and the cell is no longer moving, so relative to the tip the cell is moving at a steady rate away from the root tip. The only change to this rate would come if the growth conditions effected the rate of the tip's growth. Erickson and Silk, 1980

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**Boundary Conditions (Ω)**

zmax y = 0 on Ω Corresponds to growth of root in pure water rmax = 0.5 mm Zmax = 10 mm rmax

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**Solving for L(z) =▼·(K·▼ ) (1)**

L(z) = Kxxx+ Kyyy + Kzzz+ Kxxx + Kyyy + Kzzz (3) Known: L(z), Kx, Ky, Kz, on Ω Unknown: Lijk = [Coeff] ijk (4) The assumptions are the key.

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**Osmotic Root Growth Model Assumptions**

The tissue is cylindrical beyond the root tip, 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 (Kx) 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. 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 (Kx) 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.

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**3D Osmotic Model Results**

WHAT DO THESE PICTURES REPRESENT??? REALITY?? CHECK THE PAPER Picture 1: Water movement is mostly radial. Velocity decreases with distance from the source (Silk and Boyer ppt). Water is being pumped from above, fairly actively, via the xylem/phloem system. The phloem develops closer to the tip then the phloem and could be channeling water from above all the way into the growth zone. Picture 2: Within root growing regions it is commonly observed that turgor and osmotic gradients are unchanged across the root radius, all cells are growing at the same rate and have the same demand for solute. More outer then inner cells so an increase in solute flux must occur, moving centipetally, through successive cell layers. (Pritchard, 2000) – in other words picture 2 does not represent what is observed experimentally. In a root median longisection, the spatial pattern of ψ has egg-shaped isopotential regions. The potential is most negative in the center of the root and at the region of fastest growth rate. ** Realize that individual tissue elements are displaced through the growth zone and experience in a temporal sequence the water potentials associated with the spatial pattern. *Remember each individual element will travel through this pattern*

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**Analysis of 3D Results Model Results Longitudinal gradient**

Radial gradient Empirical Results Longitudinal gradient has been measured No radial gradient has been measured

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Phloem Source Gould, et al 2004

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**Internal Source Root Growth Model Assumptions**

The tissue is cylindrical beyond the root tip, 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 (Kx) 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 from internal proto-phloem sources. 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 (Kx) 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.

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3D Phloem Source Model

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**Comparison of Results Osmotic 3-D Model Results**

Internal Source 3-D Model Results

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**My Current Work… Sensitivity Analysis**

Looking at different plant root anatomies, source values, geometry, and initial value conditions.

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**Plant Root Geometry r = 0.3mm:0.5mm:0.7mm**

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**Plant Root Geometry Proto-phleom Placement 2. 1 mm from tip, 4. 1mm, 6**

Plant Root Geometry Proto-phleom Placement 2.1 mm from tip, 4.1mm, 6.1mm from tip, no source

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**Hydraulic Conductivity Kr: 4 x10-8cm2s-1bar-1 Kr: 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1**

Source, 4.1 mm No Source

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**Hydraulic Conductivity Kr: 4 x10-8cm2s-1bar-1 Kr: 4 x10-8cm2s-1bar-1 - 8x10-8cm2s-1bar-1**

Source, 2.1 mm No Source

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**Growth Boundary Conditions Soil vs Water**

Source, 2.1 mm No Source

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**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 conducitivity decreases the radial gradient Growth Conditions: Soil vs Water Conditions play an important role in comparing source and non source gradients

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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 Wiegers University of California, Davis My Thanks to Dr. Angela Cheer, Dr. Wendy Silk, the RMA organizers and everyone who came to my talk today. This material is based upon work supported by the National Science Foundation under Grant #DMS

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**Grid Refinement & Grid Generation**

Grid Refinement and Grid Generation

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