1. Pattern Formation by Reaction-Diffusion
Sertan Girgin
Ahmet Saçan
Reaction-Diffusion by A.Sacan & S.Girgin

2. Game-plan Reaction-Diffusion defined. Mathematical Model
Solution to RD
Simulations
Parameters
History of RD: models & applications
Reaction-Diffusion by A.Sacan & S.Girgin

3. 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

4. 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

5. Mathematical Model
Reaction-Diffusion by A.Sacan & S.Girgin

6. 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

7. 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

8. 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

9. 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

10. Reaction Diffusion Simulator
Available at:
Reaction-Diffusion by A.Sacan & S.Girgin

11.
Reaction-Diffusion by A.Sacan & S.Girgin

12. 3-D Simulation Da=0.125 Db=0.03125 β=0.1 k=0.0125 20030107
Reaction-Diffusion by A.Sacan & S.Girgin

13. Possible Trends Oscillating chemical concentrations
Unbounded increase (decrease)
A Steady State
Different diffusion rates
Random perturbation
Reaction-Diffusion by A.Sacan & S.Girgin

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27.
Reaction-Diffusion by A.Sacan & S.Girgin

28. Whitkin and Kass (1991). Emphasize anisotropy.
“diffusion map”: diffusion varies across a surface.
Reaction-Diffusion by A.Sacan & S.Girgin

29. Space Cookie
Reaction-Diffusion by A.Sacan & S.Girgin

30. 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

31.
Reaction-Diffusion by A.Sacan & S.Girgin

32. Xmorphia
Reaction-Diffusion by A.Sacan & S.Girgin

33. Xmorphia
Reaction-Diffusion by A.Sacan & S.Girgin

34. 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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37.
Reaction-Diffusion by A.Sacan & S.Girgin

38. 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

39.
Reaction-Diffusion by A.Sacan & S.Girgin

40. rock wood-grain wood AM-FM RD Level-line method 20030107
Reaction-Diffusion by A.Sacan & S.Girgin

41. 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

42. 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

43. 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

44. 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