LuisM Turing Patterns with Stochastic π Calculus

A global Context from the iGEM Mexico team and Turing Patterns

One of the most important problems in developmental biology is the understanding of how structures emerge in living systems. Several mechanisms have been proposed, depending on the observed patterns. The so called Turing patterns are based on the interaction of two effects: diffusion of some chemicals, called morphogenes, and the chemical interaction between them. It has been highly controversial whether some patterns observed in several organisms are of this type. In particular, although some systems have been identified to be of activator-inhibitor type (the most popular Turing system proposed by Gierer and Meindhart), it is still questioned if pattern formation and more generally, the appearance of functional structures can be understood by means of Turing patterns or more broadly, reaction-diffusion mechanisms.

Talking about Patterns

After the last jamboree(Nov, 2007) some members of team started a quest with the objetive of generate turing patterns at the lab and since a theoretical way. A teammate Juan A. investigated the theoretical part to produce Turing, and in a nice afternoon of january he explained to Federico and me his results:

There are many ways to produce the patterns, but essentialy we need an Activator-Inhibitor system.

The Activator-Inhibitor system is composed of two parts: An Autoctalizer and an Inhibitor. The Autocatalizer promotes its own synthesis (a positive feedback loop) and in turn induces the production of the inhibitors. When the Inhibitor is present, it generates a signal that repress the autocatalizer (Negative Control). The whole system might behave in an oscillatory way.

The autocatalizer and the Inhibitor produces diffusible signals that allow communication among other neighbors, in particular we need that the inhibitor signals spread faster than the autocatalizers' with the objective of getting syncronization in a global level and eventually generate oscillatory patterns. It is important to emphasize that no efective morphogen is needed: the internal dynamics and intracellular communication are sufficient to generate patterns.

The autocatalizer does a positive feedback loop autoinducing itself and inducing the production of the inhibitor, which will repress the catalizer untill the inhibitor signals degrade. And the cycle restarts.

We need too that the signals of both components get outside of the system and interacts with the neighbors.

In a first approach the neigborhood consist of a neighorhood of Von Neumann from the cellular automaton, which consist of 4 ortogonal cells interacting with a central one.

The model with Stochastic Pi Calculus