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The Story of Turing Pattern in Biology

The essay will represente the devolopment of turing pattern.

1. THE PREDICTION OF TRUING

In 1952, Alan Turing, a British genius best known for his work on code-breaking and artificial intelligence, was conviceted of engaging in homosexual acts an sdentenced to chmical castration. Amid that personal drama, he still found the time to finish a visonary paper THE CHEMICAL BASIS OF MORPHOGENESIS. In this paper, Turing attempt to discuss a possible mechanism by which the genes of zygote may determine the anatomical structure of the resulting organism. Just like what Schrdinger try to find mechian of hereditary, Turing suggested that only merely certain well-known physical laws are sufficient to account for many of the facts. So, morphogens, is created to representate the the system of chemical substancs, which reacting together and diffusing througn a tissue and is adeqate to account for the main phenomena of morphogensis. A very creative kernel idea that bring more thinking: although it may originally be quite homogeneous in morphogens, may later develop a pattern or structure due to an instabiliry of the homogeneous equilibrium, which is triggered off by random distrubances. The idea of morphogenesis could be applied to the stripes on tigers and zebar finsh, the spots on leopards and the the spacing in rows of alligator teetch, to name a few. After 60 years later, Jeremy Green , a developmental biologist at King's College London, said, "That structure is there, we just need to put the chemistry onto the mathematics to get the biology".

The most revolutionary feature of the turing model is its introduction of a "reaction" that produces the ligands (morphogens). If "diffusion" alone is at work, local sources of morphogens are needed to form the gradient. In such cases, the positional infomation made by the system is dependent on the prepatterns independent of the prepattern. From the generalized reaction-diffusion equations, during forming pattern, he insights: 1) more than 2 interacting chemicals are need. 2) Diffucsion in reacting systems can be a destabilizing influence. 3)Instability can cause growth of structure at a particular wavelength. 4). Diffusion coefficients of two reagents differ substantially. $$ U_t = D_U \bigtriangledown U + f(U,V),$$ $$ V_t = D_V \bigtriangledown V + g(U,V) .$$

Schematic drawing of gradient model and turing model Fig. 1 Schematic drawing showing the difference between the morphogen gradient model and Turing model.[1]

Turing revealed that such a system yields six potential states, depending on the dynamics of reaction term and wavelength of the pattern. Most attractive case is case six, which made stationary patterns. Itis the major achievement of Turing model, and these are usually referred to as Turing patterns. In other words, a turing pattern is a kind of nonlinear wave that is maintained by the dynamic equilibrium of the system. Its wavelength is determined by interactions between molecules and their rates of diffusion. This patterns arise autonomously, in spite of any preexisting positional information. Because of that, it is possible to explain how patterns arise percisely in , for example, fertilized oocytes, which present little in the way of positional information. The ability of Turing patterns to regenerate autonomously, even after experimentally induced disturbances, is also important and of great utility in explaining teh autonomy shown by pattern-forming developmental processes.

patterns by turing modelFig.2: Schematic drawing the mathermatical analysis of the turing model and the patterns generated by the simulation.[1]

2. Analysis of J.D Murray

J.D Murray thinked that the surface pattern of all mammal are cause by same reaction-diffusion mechians. When animal in embryo, morphogen will have a new spatial distribution during the development. Then morphogen can cause the product of melanin. Different melanin distribution vary on the distribution of morphogen. Although there was an explosion of theoretical work and computer modeling in the 1970s that successfully represente patterns like spots and stripes using Turing mechanisms, molecular biology was nowhere near the point where researchers could identify the specific molecules acting as acivators and inhibitors. So Murray said, the explation of using reaction-diffusion is attractive, but it maybe wrong in biology.

3. Turing model in BZ reaction

Because chemistry reaction is easier to control than reaction in life, Turing model is used to understand BZ reaction. In the 1960s, chemistist find the spiral wave and target wave in BZ reaction, which belong some solve of reaction-diffusion equtions. Until 1980s to 1990s, scientist designed a kind of open reaction container to sure only reaction and diffusion process. Then their results showed the frist sample of Turing pattern--CIMA(Chlorite-Iodide-Malonic Acid) Reaction.

4. Lastest research of Turing model

Now, the lastest research suggests that Turign-type mechanisms may be responsible for the spacing betwee hair follicles in mice and feather buds on a bird's skin, the ridges that form on a mouse's palate and the digits on a mouse's paw.

Reference

  1. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation, S Kondo and T mira, _science_329,1616(2010).

  2. https://www.quantamagazine.org/biologists-home-in-on-turing-patterns-20130325

  3. 生命的另一个奥秘──浅谈生物数学与斑图生成,史峻平,《数学文化》

Reading List: 1.http://guava.physics.uiuc.edu/~nigel/courses/563/Essays_2005/PDF/Mukherjee.pdf 2.https://biologydirect.biomedcentral.com/articles/10.1186/s13062-016-0124-7