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Pattern Formation by Reaction-Diffusion
Sertan Girgin Ahmet Saçan Reaction-Diffusion by A.Sacan & S.Girgin
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Game-plan Reaction-Diffusion defined. Mathematical Model
Solution to RD Simulations Parameters History of RD: models & applications Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction Diffusion (RD)
A chemical mechanism for pattern formation. First described by Alan Turing (1952). Two chemicals diffusing across a surface and reacting with one another can form stable patterns of chemical concentration. A central issue in developmental biology: how cells organize themselves into particular patterns. cells in a worm become organized into segments How do amazing patterns occur in nature, starting from relatively simple, regular, and symmetric conditions? Non-linearity can cause complex patterns to form. One possible mechanism (Turing, 1952): two chemicals can diffuse through an embryo until forming stable patterns of chemical concentrations. Reaction-Diffusion by A.Sacan & S.Girgin
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RD in a line of cells The amount of chemical a in a cell changes based on the quantity of the chemicals a and b are already in the cell. If a particular cell has a higher concentration of chemical b than its neighbors, then that cell’s concentration of b will decrease over time by diffusion to its neighbors. Likewise, if the concentration of b is at minimum at a particular place along the row of cells, then more of b will diffuse from adjacent cells to this cell to raise the concentration of b at that cell. Reaction-Diffusion by A.Sacan & S.Girgin
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Mathematical Model Reaction-Diffusion by A.Sacan & S.Girgin
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Analytical Solution? Closed-form solution: difficult/impossible (except when F,G very simple). Therefore, Discretize and Solve numerically. Reaction-Diffusion by A.Sacan & S.Girgin
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Turing’s Solution ai : concentration of 1st morphogen at ith cell. (inhibitor) bi : concentration of 2nd morphogen at ith cell. (activator) Da : diffusion rate of a. Db : diffusion rate of b. β : random substrate k : reaction rate Initial concentrations of a, b: 4 Reaction-Diffusion by A.Sacan & S.Girgin
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Concentration of b over time.
1-D Simulation chemical concentrations form peaks & valleys. Concentration of b over time. Reaction-Diffusion by A.Sacan & S.Girgin
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2-D Simulation Da=0.1 Db=0.02 β=0.1 k=0.02 a b [a: black, b: yellow]
a diffuses more rapidly than b. a b [a: black, b: yellow] Da=0.1 Db=0.02 β=0.1 k=0.02 Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction Diffusion Simulator
Available at: Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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3-D Simulation Da=0.125 Db=0.03125 β=0.1 k=0.0125 20030107
Reaction-Diffusion by A.Sacan & S.Girgin
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Possible Trends Oscillating chemical concentrations
Unbounded increase (decrease) A Steady State Different diffusion rates Random perturbation Reaction-Diffusion by A.Sacan & S.Girgin
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No Reaction Case Da=0.1 Db=0.02 β=0.1 k=0.0 a b 20030107
Reaction-Diffusion by A.Sacan & S.Girgin
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No Diffusion Case Da=0.0 Db=0.0 β=0.1 k=0.01 n=1 n=1000 n=2000
Reaction-Diffusion by A.Sacan & S.Girgin
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Parameter-Game: k k=0.001 k=0.005 k=0.01 Da=0.1 Db=0.02 β=0.1 20030107
k: reaction rate. large-reaction rates cause more of a to be degraded, thus causing smaller regions where a is concentrated... k=0.001 k=0.005 k=0.01 Da=0.1 Db=0.02 β=0.1 Reaction-Diffusion by A.Sacan & S.Girgin
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Parameter: β Da=0.1 Db=0.02 k=0.005 β=0.05 β=0.1 β=3 20030107
beta=random perturbation. small values: more regular, circular patterns β=0.05 β=0.1 β=3 Da=0.1 Db=0.02 k=0.005 Reaction-Diffusion by A.Sacan & S.Girgin
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Da=0.1 Db=0.02 β=3 β=0.1 β=0.05 k=0.001 k=0.002 k=0.005 k=0.01
Reaction-Diffusion by A.Sacan & S.Girgin
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Parameter: Da / Db Db=0.02 β=0.1 k=0.005 Da=0.08 Da=0.1 Da=0.2
Reaction-Diffusion by A.Sacan & S.Girgin
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β=0.1 k=0.005 Da=0.2 Da=0.15 Da=0.1 Da=0.08 Da=0.06 Db=0.007 Db=0.01
Reaction-Diffusion by A.Sacan & S.Girgin
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β=0.1 Db=0.01 Da=0.2 Da=0.15 Da=0.1 Da=0.08 Da=0.06 k=0.002 k=0.005
Reaction-Diffusion by A.Sacan & S.Girgin
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Cascading Freeze b:[0-4] k=0.01 Cheetah Freeze b:[0-4]4 k=0.01
Leopard k=0.001, n=30000 Da=0.1 Db=0.02 β=0.05 Reaction-Diffusion by A.Sacan & S.Girgin
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History of RD Turing (1952) Bard and Lauder (1974)
RD system on a sphere may be responsible for triggering gastrulation in the embryo. Bard and Lauder (1974) Computer simulations Patterns generated by RD not regular enough to explain patterns in development. Can explain less regular patterns: leaf organization, distribution of hair follicles. Reaction-Diffusion by A.Sacan & S.Girgin
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Bard (1981), Murray (1981) independently
RD can explain the patterns on coats of animals. Bard (1981) Spot and stripe patterns. Small, white spots on a deer. Large, dark spots on a giraffe. Murray (1981) Spot-size dependent on size of animal. Paterns found on butterfly wings. Reaction-Diffusion by A.Sacan & S.Girgin
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Meinhardt and Klinger (1987)
Stripe patterns (by 5-morphogen RD) Veins on a leaf. Swindale (1980) Simulation by activation/inhibition between synapses. Young (1984) Irregular striped patterns Ocular dominance columns in mammalian visual system. Meinhardt and Klinger (1987) Patterns of pigment found on mollusc shells Reaction-Diffusion by A.Sacan & S.Girgin
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Kauffman et al. (1978), Lacalli (1990), Hunding et al. (1990)
Segmentation of fruit fly (Drosophila) embryos Turk (1991) Cascading Clusters of spots on leopards and jaguars (rosettes) Zebra’s pajamas. Mapping on arbitrary surfaces. Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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Whitkin and Kass (1991). Emphasize anisotropy.
“diffusion map”: diffusion varies across a surface. Reaction-Diffusion by A.Sacan & S.Girgin
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Space Cookie Reaction-Diffusion by A.Sacan & S.Girgin
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Pearson (1993) Gray-Scott Model
Well-Defined range of behavior for parameters. Du = 2E-5 and Dv = 1E-5 F: rate of the process that feeds U and drains U,V and P k: rate of conversion of V to P Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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Xmorphia Reaction-Diffusion by A.Sacan & S.Girgin
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Xmorphia Reaction-Diffusion by A.Sacan & S.Girgin
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M-Lattice Sherstinsky, Picard (1994)
State variables are guaranteed to be bound. Applied to image-restoration and half-toning. Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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Texture Completion Acton, Mukherjee, Havlicek, Bovik (2001).
Reconstruction of large missing regions of homogeneous oriented textures. RD seeded with noise identically distributed to surrounding region to match graylevel distribution. occluded stripe formation AM-FM RD Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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rock wood-grain wood AM-FM RD Level-line method 20030107
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Web-Resources Code Zebra (collection of RD links) Greg Turk’s page:
Greg Turk’s page: diffusion/reaction_diffusion.html Xmorphia: 3D images: Visual models of morphogenesis Reaction-Diffusion by A.Sacan & S.Girgin
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References A. Turing. “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society B, vol. 237, pp (August 14, 1952). Greg Turk. "Texture Synthesis on Surfaces", SIGGRAPH 2001, pp , (August 2001). A. Witkin and M. Kass. Computer Graphics (Proc. SIGGRAPH '91) Graphics, Vol. 25, No. 3, July, 1991. J.E. Pearson. Complex patterns in a simple system. Science, 261: , (July 1993). Reaction-Diffusion by A.Sacan & S.Girgin
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A. Sherstinsky, R. W. Picard
A. Sherstinsky, R. W. Picard. Restoration and Enhancement of Fingerprint Images Using M-Lattice. Proc. of the Internat. Conf. on Pattern Recognition (1994). S. T. Acton, D. P. Mukherjee, J. P. Havlicek, A. C. Bovik. Oriented Texture Completion by AM- FM Reactoin Diffusion. IEEE Transactions on Image Processing, Vol 10, No.6, (June 2001). Reaction-Diffusion by A.Sacan & S.Girgin
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J. Bard, I. Lauder, “How Well Does Turing’s Theory of Morphogenesis Work?,” Journal of Theoretical Biology, vol.45, no.2, pp (June 1974). P. Prusinkiewicz, “Modeling and Visualization of Biological Structures”, Proceeding of Graphics Interface ’93,pp (May 1993) Reaction-Diffusion by A.Sacan & S.Girgin
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Reaction-Diffusion by A.Sacan & S.Girgin
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