Investigating Continuous Models of WUSCHEL Expression in the Shoot Apical Meristem of A.thaliana Dana Mohamed Mentor: Bruce Shapiro, Caltech

Computable Plant How do environmental factors and genetic makeup interact to shape plant meristem developmental processes that lead to plant flowers, leaves, shoots and stems? + =

Shoot Apical Meristem Stem cell equivalent Where new cells are created for the stem, leaves, and flowers Controlled by intercellular signaling of 2 genes WUSCHEL and CLAVATA3

WUSCHEL expression Side View Birds Eye View

Strategy Background In paper, model is discrete on extracted template Average WUS intensity for individual cells is obtained using confocal microscopy

Discrete Version Determined [WUS] Activator Model [WUS] Repressor Model [WUS]

Activator Model Vs. Repressor Model The activator model assumes there is an activator that activates WUS, located where WUS expression is high. The repressor model assumes there is always high WUS expression except where there is an inhibitor.

Equations, Repressor Model

Equations, Activator Model

Goal & Rationale Goal: –To extend the models of the gene expression to a continuous model to see if model still holds Rationale: –The models of this project were created as a way to describe and test several hypotheses –Further testing the models and extending their applicability simply furthers their research

Strategy To use Mathematica to extend the models. –1 Dimension, Line - 2 Dimensions, Square –2 Dimensions, Circle To test different initial conditions while holding the boundary conditions to 0, as set in the original paper.

Initial, Boundary Conditions Initial conditions- where the expression levels start Boundary conditions- where the edges of the model (of the expression levels) are held through time ↑↑ ↓ ↓

Basic Line, R-M, Standard Plot W[0,x] = Sin[Pi x] Plot[W(t,x)] from {t,0,1000}, {x,0,1}: Video: →

Basic Line, R-M, Standard Plot W[0,x] = Sin[Pi x](1+Sin[5 Pi x]) Plot[W(t,x)] from {t,0,1000}, {x,0,1}: Video: →

2D Basic Square, R-M, Standard Plot Plot3D[W(t,x,y)] at {t=0}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Cos[Pi 2 x])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=150}, from {x,0,1}, {y,0,1}

2D Basic Square, R-M, Standard Plot Plot3D[W(t,x,y)] at {t=0,150}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Cos[Pi 2 x])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=0,150}, from {x,0,1}, {y,0,1} W[0,x,y] = (1-Cos[Pi/2 x])(Sin[Pi x]Sin[Pi y])

2D Basic Square, A-M, Standard Plot Plot3D[W(t,x,y)] at {t=0,150}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Cos[1.5Pi x])(1+Cos[3Pi y])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=0,150}, from {x,0,1}, {y,0,1} W[0,x,y] = (1-Cos[.5Pi x])(Cos[.5Pi y])(Sin[Pi x]Sin[Pi y])

2D Basic Square, A-M, Time Difference Plot3D[W(t,x,y)] at {t=0,300}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Sin[4Pi x])(1+Sin[4Pi y])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=0,250}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Sin[6Pi x])(1+Sin[6Pi y])(Sin[Pi x]Sin[Pi y])

Polar Coordinates W[0,r,θ] = 0 Plot[W(t,r,θ)] from {t,0,150}, {θ,0,2Pi}: R-M Polar A-M Polar

Polar Coordinates Top ViewSide View

References, Acknowledgements (2005) Jönsson H, Heisler M, Reddy GV, Agrawal V, Gor V, Shapiro BE, Mjolsness E, and Meyerowitz E.M., Modeling the organization of the WUSCHEL expression domain in the shoot apical meristem. Bioinformatics 21(S1): i232-i240. Bruce Shapiro, Ph.D Computable Plant SoCalBSI

Basic Line, R-M, Standard Plot W[0,x] = Sin[Pi x Plot[W(t,x)] from {t,0,1000}, {x,0,1}: →

Basic Line, R-M, Standard Plot W[0,x] = Sin[Pi x](1+Sin[5 Pi x]) Plot[W(t,x)] from {t,0,1000}, {x,0,1}: →

2D Basic Square, R-M, Standard Plot Plot3D[W(t,x,y)] at {t=0}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Cos[Pi 2 x])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=150}, from {x,0,1}, {y,0,1}

Basic Square, A-M, Standard Plot Plot3D[W(t,x,y)] at {t=0}, from {x,0,1}, {y,0,1} W[0,x,y] = (1+Sin[6Pi x])(1+Sin[6Pi y])(Sin[Pi x]Sin[Pi y]) Plot3D[W(t,x,y)] at {t=150}, from {x,0,1}, {y,0,1}

Polar Coordinates W[0,r,θ] = 0 Plot[W(t,r,θ)] from {t,0,150}, {θ,0,2Pi}: R-M Polar A-M Polar

Polar Coordinates Top ViewSide View

References, Acknowledgements (2005) Jönsson H, Heisler M, Reddy GV, Agrawal V, Gor V, Shapiro BE, Mjolsness E, and Meyerowitz E.M., Modeling the organization of the WUSCHEL expression domain in the shoot apical meristem. Bioinformatics 21(S1): i232-i240. Bruce Shapiro, Ph.D Computable Plant SoCalBSI