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Around 1952, Alain Turing thought about the possible mechanisms behind
biological pattern formation and constructed a model that described a process that involved a low-range diffusing activator and a wide-range diffusing inhibitor.
The activator production is inhibited by the presence of inhibitors and enhanced by the presence of the activator. In contrast to that, the inhibitor is not
“self-enhancing”, that means its production is not linked to the presence of other inhibitors, but to the presence of activators.
Both substances are destroyed on a constant rate and diffused as described above.
Mathematically, we could call this:
The equation above describes the changes of the local activator concentration, where k1 regulates
the production, k2 the destruction of the activator, k3 the basic production without an activator present and Da the diffusion of the activator.
The equation above describes the changes of the local inhibitor concentration, where k4 regulates
the production, k5 the destruction of the inhibitor, k6 the basic production without an activator present and Dh the diffusion of the inhibitor.
This shows the ratio between the parameters to allow the overall process to reach a stable state
after some time, otherwise the generated pattern would always change.
This scheme again illustrates the overall processes from a local point of view:
- On the upper left side of the simulation applet, you have 3 monitors, that inform you about the
current time step and about the current (total) amount of activator and inhibitor present.
- Below the monitors, you have one slider that allows you to turn on (=1) or off (=0) the chemical
reaction between the two substances. This way you can see the difference between reaction-diffusion and diffusion. Try it !
- Below that section, you have a set of sliders that allow you to regulate the values of the parameters k1-k6, Da and Dh.
- The buttons on the left bottom of the simulation applet allow you to setup (restart) the
simulation run and to stop it for some time. As the simulation reacts very sensitive to changes of the parameters (note, they have very small values), a button restores some “default values”.
- Below the simulation area (the pattern field), there is a slider allowing you to switch between 3 different modes of view:
- The activator view: This view shows the activator concentration in different shades of
blue. Darker colors mean a higher concentration of the activator. The slider on the right hand (max-activator) lets you control the amount of activator that is shown in darkest
blue. This way you can adopt your view mode to different concentrations on the screen.
- The inhibitor view: The same as the activator view, just showing the local concentration
of inhibitor in shades of green. Also this view is adjustable by a slider called “max-inhibitor”.
- The pigment view: I assumed, that there is certain threshold of activator concentration
needed to let local pigments change the color of the skin. This threshold can be adjusted by the very right slider.
The simulation was written in StarLogo by: Thomas Schmickl, Department for Zoology, Karl-Franzens-University Graz, Austria (Europe),
firstname.lastname@example.org (preferred adress), email@example.com
- Camazine S., Deneubourg J.-L., Franks N.R., Sneyd J., Theraulaz G. and Bonabeau E. (2001) Self-Organization in biological systems.
Princeton University Press.
- Bonabeau E. (1997) From classical models of morphogenesis to agent-based models of pattern formation. Artificial Life 3:191-211
- Edelstein-Keshet L. (1988) Mathematical models in biology. Birkhäuser mathematics series. McGraw-Hill