Turing mechanism

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

From 1952 until his untimely death in 1954, Turing had dealt with problems in theoretical biology . In his 1952 work on The Chemical Basis of Morphogenesis , this process, now known as the Turing mechanism, was first described. Turing's later work, including on the importance of the Fibonacci numbers for the morphological structure of plants, remained unpublished. Because of Turing's previous involvement in intelligence projects, such as deciphering the Enigma code, his collected works were not released for publication until 1992.

Turing model for two chemicals

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

${\displaystyle \left.{\begin{matrix}u_{t}&=&\Delta u+\gamma f(u,v)\\v_{t}&=&d\Delta v+\gamma g(u,v)\end{matrix}}\right\}{\mbox{ auf }}B\subseteq \mathbb {R} ^{n}}$

with Neumann boundary conditions and the initial data , . ${\displaystyle n\cdot \nabla u=n\cdot \nabla v=0{\mbox{ auf }}\partial B}$ ${\displaystyle u(.,t=0)=u_{0}(.)}$ ${\displaystyle v(.,t=0)=v_{0}(.)}$

The vector is the outer unit normal vector an , the constant describes the ratio of the diffusion coefficients of the two substances, is the concentration of the activating substance (activator) and is the concentration of the deactivating substance (deactivator or inhibitor). The constant can be interpreted as the size of the area (a certain power of it, depending on the dimension ) or as the relative strength of the reaction terms compared to the diffusion effects. ${\displaystyle n}$ ${\displaystyle \partial B}$ ${\displaystyle d}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \gamma }$ ${\displaystyle B}$ ${\displaystyle n}$

The central idea of ​​the Turing instability is to consider a spatially homogeneous, linearly stable system that is unstable in the inhomogeneous case, when diffusion is described by suitably chosen diffusion coefficients (i.e. , “diffusion-driven instability”). This concept was novel because diffusion is generally seen as a stabilizing factor in the field of partial differential equations. ${\displaystyle d\neq 1}$

It is based on the inhibitor diffusing faster than the activator. First of all, where there is a lot of activator substance, there is also a lot of inhibitor. However, this does not lead to the disappearance of the activator substance, since the inhibitor evaporates quickly due to the rapid diffusion. Especially in model configurations in a restricted area with vanishing Neumann boundary conditions, comparatively high inhibitor concentrations arise at such remote points in the area where there is little activator. There they can successfully prevent the increase in the activator substance. At this point it can be seen that models in an unrestricted area show a qualitatively different behavior, since the inhibitor substance tends to diffuse away to infinity.

For the special choice with , we determine the Turing space , i.e. H. the set of parameter values ​​for which we can observe Turing instability . ${\displaystyle B=[0,L]}$ ${\displaystyle L>0}$

Linear stability of the homogeneous system

Let be a steady state, i.e. H. , then in the absence of diffusion effects the above system is linearly stable at this steady state if holds ${\displaystyle (u_{0},v_{0})}$ ${\displaystyle f(u_{0},v_{0})=g(u_{0},v_{0})=0}$

${\displaystyle f_{u}+g_{v}<0}$and , ${\displaystyle f_{u}g_{v}-f_{v}g_{u}>0}$

which is equivalent to the fact that the real parts of the eigenvalues ​​of the Jacobi matrix are all negative, whereby here and in the following we evaluate the partial derivatives of and at the stationary state. ${\displaystyle f}$ ${\displaystyle g}$

Instability of spatial disturbances

Let it be a solution of the eigenvalue problem , i. H. , where and is the so-called wave number. With the approach ${\displaystyle W_{k}(r)}$ ${\displaystyle \Delta W_{k}+k^{2}W=0}$ ${\displaystyle W_{k}(r)\propto \cos \left({\frac {n\pi }{L}}r\right)}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle k:={\frac {n\pi }{L}}}$

${\displaystyle w(r,t)=\sum _{k}c_{k}\exp(\lambda (k)t)W_{k}(r)\;}$

it turns out that the system is linearly unstable if holds for a . The expression is called the dispersion relation and only takes on positive values ​​if applies ${\displaystyle \Re \lambda (k)>0}$ ${\displaystyle k\neq 0}$ ${\displaystyle \lambda (k)}$

${\displaystyle df_{u}+g_{v}>0}$ and ${\displaystyle (df_{u}+g_{v})^{2}-4d(f_{u}g_{v}-f_{v}g_{u})>0\;.}$

The first inequality results in particular . Depending on the specific choice of parameters, the following interval of wavenumbers has positive real parts, ${\displaystyle d>1}$

${\displaystyle \gamma {\frac {(df_{u}+g_{v})-{\sqrt {(df_{u}+g_{v})^{2}-4d\det(A)}}}{2d}}<k^{2}<\gamma {\frac {(df_{u}+g_{v})+{\sqrt {(df_{u}+g_{v})^{2}-4d\det(A)}}}{2d}}\;,}$

whereby . Note that . The above interval is called the unstable interval . The amplitudes of the wavelengths corresponding to these wavenumbers increase over time, while other wavelengths are attenuated. These unstable modes describe the reinforced patterns. Since it only accepts discrete values, there is only a finite number of amplified wavelengths. ${\displaystyle A:={\begin{pmatrix}f_{u}&f_{v}\\g_{u}&g_{v}\end{pmatrix}}}$ ${\displaystyle \gamma \propto L^{2}}$ ${\displaystyle k}$

Examples of suitable reaction terms are

${\displaystyle f(u,v)=a-bu+{\frac {u^{2}}{v}}}$and (Gierer, Meinhardt; 1972), ${\displaystyle g(u,v)=u^{2}-v\;}$

such as

${\displaystyle f(u,v)=a-u-h(u,v)}$and , where , (Thomas, 1975). ${\displaystyle g(u,v)=\alpha (b-v)-h(u,v)}$ ${\displaystyle h(u,v):={\frac {\rho uv}{1+u+Ku^{2}}}}$

The constants , , , and are positive parameters that need to be suitably selected so that the system meets the above requirements. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \alpha }$ ${\displaystyle \rho }$ ${\displaystyle K}$

Pattern formation

Unconstrained areas correspond to models that are relevant for situations in which the embryo is far larger than the size of the pattern to be formed and therefore the edge of the area cannot contribute to the preference for certain wavelengths. The analysis is a little easier in this case. In general, there is no finite number of amplified wavelengths, but rather a certain wave number that has the greatest eigenvalue and whose pattern is ultimately formed. ${\displaystyle B}$

If the area becomes larger with the passage of time, for example as the embryo grows, then the value increases and at certain bifurcation points, amplified modes become attenuated, i.e. that is, they fall out of the unstable interval, or higher wavenumbers that were previously stable become unstable. This process is called mode selection and explains the complex development of patterns during morphogenesis. ${\displaystyle B}$ ${\displaystyle \gamma }$

See also

• Artificial life
• Autoregulatory model in limb development

literature

• JD Murray : Mathematical Biology. Volume 2: Spatial Models and Biomedical Applications. 3rd edition. Springer, New York NY et al. 2003, ISBN 0-387-95228-4 ( Interdisciplinary applied mathematics 18).

Individual evidence

1. ^ Alan Turing: The chemical basis of morphogenesis (PDF; 1.2 MB) . Phil. Trans. R. Soc. London B 237 pp. 37-72 (1952) . Original article.