Summer 2011 UCUR Project on Reaction Diffusion

Reaction Diffusion: Turing Systems

In 1952, Alan Turing published a paper titled The Chemical Basis of Morphogenesis. Typically, diffusion is an averaging process that smoothes differences over space and time. In his paper, Turing presented a very novel idea by describing a mechanism in which diffusion amplifies small differences over space and time.

A Turing system has two properties. First, the homogeneous solution (in which there are no differences over space and time) is stable to small perturbations when there are reaction terms but no diffusion. Second, the homogeneous solution is not stable to small perturbation when the reaction terms are coupled with diffusion.

In other words, in a Turing system we can expect a pattern if there small random fluctuations (property 2) and this pattern is attributable to diffusion by property 1.

Note:  A reaction diffusion system is a system described by reaction and diffusion terms in the evolution of the concentration of some number of variables, and a Turing system is a special type of reaction diffusion system.

Homogeneous Solution Stable To Perturbation Without Diffusion, But

Unstable With Diffusion



Figure 1


Diffusion was turned off and a small perturbation was introduced to a center node, while all other nodes started with a morphogen concentration of zero. In the left graph, white represents a high concentration and black represents zero. The right graph corresponds to only the node at which the perturbation was introduced. Without diffusion the initial instability disappears




Figure 2  

Once diffusion was turned on, a small perturbation introduced to a center node grows …. the concentration diffuses out to the other nodes to result in more peaks.



 Movies of Patterns For Different Initial Conditions On a Growing Domain


Movies corresponding to one of three initial conditions:

1. a center point of initial concentration MOVIE

2. a random field of initial concentration MOVIE

3. center line of concentration MOVIE




Last updated July 28, 2011.