The Chemical Basis of Morphogenesis

The Turing mechanism is one of the British mathematician Alan Turing, one of the most influential theorists of the early computer science, described mechanism, such as reaction-diffusion systems can spontaneously form structures. This process is also still the focus of many chemical-biological structure formation theories, he explains, for example, the morphogenesis of colored patterns on the fur of animals like zebra, giraffe or kudu.

Turing had been employed from 1952 until his untimely death in 1954, with problems of theoretical biology. In his 1952 published work on the topic of The Chemical Basis of Morphogenesis this was described today as a Turing mechanism known process for the first time. Later work of Turing, among other things, the importance of Fibonacci numbers for the morphological structure of plants remained unpublished. Because Turing's earlier collaboration on intelligence projects such as the deciphering of the Enigma code, were released in 1992 his collected works for publication.

Turing model for two chemicals

Turing a model for two chemicals in a dimensionless form, for example, given by the system of partial differential equations

With Neumann boundary conditions and initial data.

The vector is at the outer unit normal vector, the constant is the ratio of the diffusion coefficients of the two substances, the concentration of the activating agent (activator) and the concentration of the deactivating agent ( inhibitor or deactivator ). The constant can be used as the size of the area ( a certain power of a function of the dimension) or be interpreted as the relative intensity of the reaction compared to the terms in the diffusion effects.

The central idea of the Turing instability is to consider a spatially homogeneous, linearly stable system in the inhomogeneous case when diffusion described by suitably chosen diffusion coefficient effect (that is ), is unstable ( " diffusion-driven instability "). This concept was novel, since diffusion is understood in the field of partial differential equations in general as a stabilizing factor.

It is based on that the inhibitor diffuses faster than the activator. First of all, arises where a lot of activator is also much inhibitor. However, this does not lead to disappearance of the activator, since the inhibitor evaporates rapidly due to the rapid diffusion. Especially in model configurations in a restricted area with vanishing Neumann boundary condition rather relatively high inhibitor concentrations accumulate in such remote points of the area where little is activator. There, they can in fact prevent the increase in activator successful. At this point shows that models on unrestricted area show a qualitatively different behavior because the inhibitor substance diffuses tends to infinity.

For the particular choice we determine the Turing space, that is, the amount of parameter values ​​for which we can observe Turing instability.

Linear stability of the homogeneous system

Is a steady state, i.e., then the above system in the absence of diffusion effects linear stable at this steady state, if and only if

Where we evaluate the partial derivatives of and to the stationary state, here and below.

Instability of spatial interference

It is a solution of the eigenvalue problem, that is, where and is the so-called wave number. With the approach

Shows that the system is linear unstable, if for a. The expression is referred to as dispersion relation and only assumes positive values ​​, if and only if

From the first inequality follows in particular. Depending on the specific choice of the parameters has the following interval of wave numbers positive real part,

Said. Note that. The above interval is called unstable interval. The wave numbers corresponding to these wavelengths are increasing over time development of the amplitude, while other wavelengths are attenuated. These unstable fashion describe the reinforced pattern. Since only assumes discrete values ​​, there are only a finite number of wavelengths amplified.

Examples of suitable reaction terms are

As well as

The constants, and are positive parameters that must be suitably selected so that the system satisfies the above conditions.

Pattern formation

Unbounded domains correspond to models that are relevant for situations in which the embryo is far greater than the magnitude of trainees pattern and therefore the edge of the area can not contribute to the preference for certain wavelengths. The analysis is somewhat easier in this case. Generally, there is not a finite number increased wavelengths, but a certain wave number, which has the largest eigenvalue and its pattern is ultimately formed.

If the area is greater in the course of time, for example, when the embryo is growing, then the value increases, and to be attenuated in some bifurcation enhanced modes, i.e., they fall out of the unstable interval or higher wavenumbers that were stable yet, be unstable. This process is called mode selection and explains the complex development of patterns during morphogenesis.