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Reaction-diffusion

Please note

The simulation needs a Java-Plugin (1.3.x) installed for your internet browser. If you do not already have one installed, the browser will prompt you to download the Plugin from “Sun”, who is the inventor of Java. Please download the JRE (=Java runtime environment) into a directory on your computer (e.g. “c:\temp”), execute the downloaded file for installation on your system (double-click on the file). Afterwards you will be able to reload the simulation page. Maybe you will have to restart your browser to succeed.

Run the simulation

Please click here.

Reaction-diffusion models

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.

Model definition

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:

User Interface

  • 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.

Screenshot

Implementation

The simulation was written in StarLogo by: Thomas Schmickl, Department for Zoology, Karl-Franzens-University Graz, Austria (Europe),
schmickl@nextra.at (preferred adress), thomas.schmickl@uni-graz.at
http://members.nextra.at/hfbuch/edoc
http://zool33.uni-graz.at/schmickl

Further readings

  • 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

 

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