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### Turing patterns in coats and sounds

One hundred years ago was born Alan Turing. He was known essentially for his role during the Second World War: he encrypted the Enigma machine. But He is also a brillant mathematichian and today I would try to describe one of his better model, that today biologists are applying to their research field.
A vibrating object tends to vibrate at certain preferred frequencies, called natural frequencies. These frequencies depend on properties such as the density and tension of the vibrating object. Mathematicians and physicists can determine the natural frequencies of an object when they know the values for these other properties. This article describes new work being done to solve the reverse problem - calculation of properties such as density when the natural frequencies of the object are known.
Elizabeth Veomett about Good Vibrations by Joyce R. McLaughlin. American Scientist, July - August 1998
Cymatics was the study of the waves' patterns. The first interested in this subject was Galileo Galilei:
As I was scraping a brass plate with a sharp iron chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound: on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by this sibilant note there was not the least trace of such marks.(1)
Some years after Galilei (1680), Robert Hooke
was able to see the nodal patterns associated with the modes of vibration of glass plates.
In 1787 Ernst Chladni repeated Hooke's experiments and published his results in Entdeckungen über die Theorie des Klanges (Discoveries in the Theory of Sound). Finally in 1967 Hans Jenny published Kymatik (Cymatics), a book based on Chladni's work, and cymatics became an interesting science, in particular for artists! For example, Jeff Volk, poet, writes an interesting article about Jenny and the pattern of sound: From Vibration to Manifestation (pdf). In particular he presents an image from Alexander Lauterwasser's Water Sound Images
Pay attention: following Lauterwasser and Volk we could explain the pattern of leopard's coat, but the first explanation come from one of the Alan Turing's paper The Chemical Basis of Morphogenesis(2). In this paper Turing is interested in the formation and development of path in biology (the so called phenomenon of morphogenesis).
Any pattern or shape observed in nature, even though governed by genetics, is most likely produced by an unknown mechanism. Thus, determining these mechanisms that generate pattern and shape in organisms is an important goal of theoretical biologists.(3)
The most used model for this type of systems is the reaction-diffusion system, described by the following formula: $u_t = d \Delta u + f (\gamma, u)$ where $u$is the position of the gene, $u_t$ the diffusion speed, $d$, $\gamma$ real constants. We can write two similar formulas for every morphogene in the system.
Reaction-diffusion models are particularly compelling with regard to their ability to capture complex evolving patterns.(3)
Similar equations are really complex to analyze, due to their local and general phenomena. In an intuitively way, we can see the pattern formation like a challenge between reaction mode and diffusion mode. In his paper, Turing
suggested that a system of reacting and diffusing chemicals (morphogens) can interact to produce stable patterns in concentration (Turing patterns).(3)
Or in a more simple way: we can imagine the presence of an activator molecule of the morphogenesis. This molecule will be produced more and more thanks an autocatalysis process, but the activator will produce also an inhibitor, that will limit the production of the activator. The dynamics between activator and inhibitor will generate the pattern observed in nature (for example tigers' stripes). Tipically the two diffusion velocities are different, so we can explain the great variety of patterns.
The story of Turing patterns received a great expansion in the last ten years, or more. First of all in 1995, on Nature, Kondo and Asai observed the presence of Turing patterns in the Pomacanthus fish(4).
Starting from a differential equation that describe the variation in time of the concentration of the activator molecule, the two researchers can to reproduce, with a computer software and the Turing dynamics, the pattern on Pomacanthus skin:

A similar mechanism was observed also in salmons: in this case Miyazawa, Okamoto and Kondo add to the model also the colors(5):
Another great success of Turing reaction-diffusion model is the expleanation of pattern in felids. For example a study published in 2011 about leopards
shows that, in general, detailed aspects of felid's flank patterns have evolved to match visual properties of the environments they inhabit and their behaviour within those environments(6)
The major advance reported here is the demonstration that evolved traits (pattern, habitat and behavioural) can be linked to a mathematical model of pattern development. We suggest that future comparative studies of animal coloration should also be linked to specific models of development, as these constrain the variety of pattern expressions as well as affording a parametric basis for analysis.(6)
More recently also a study about tigers' stripes confirmed the Turing theory:
We present direct evidence of an activator-inhibitor system in the generation of the regularly spaced transverse ridges of the palate. We show that new ridges, called rugae, that are marked by stripes of expression of Shh (encoding Sonic hedgehog), appear at two growth zones where the space between previously laid rugae increases. However, inter-rugal growth is not absolutely required: new stripes of Shh expression still appeared when growth was inhibited. Furthermore, when a ruga was excised, new Shh expression appeared not at the cut edge but as bifurcating stripes branching from the neighboring stripe of Shh expression, diagnostic of a Turing-type reaction-diffusion mechanism. Genetic and inhibitor experiments identified fibroblast growth factor (FGF) and Shh as components of an activator-inhibitor pair in this system. These findings demonstrate a reaction-diffusion mechanism that is likely to be widely relevant in vertebrate development.(7)
But the success of Turing patterns go away morphogenesis in biological systems: we can find a lot of thiese paths in a lot of real ecosystems(8):
For example the mussel beds (d):
Regular stripes are found in mussel beds occurring on sediments in the Wadden Sea, the Netherlands. The striped patterns, oriented perpendicular to the tidal flow, are suggested to be the outcome of scaledependent feedback. Mussels facilitate eachother over short range, because conspecifics are the main substrate for attachment on soft-bottom sediment. Competition between mussels occurs for algae, affecting mussel intake and growth, and can occur over long range because water depleted in algal stocks, as a result of the mussels, is carried over the mussel beds by tidal currents.(8)
Regular paths like these are possible only if there is a long-distance negative feedback:
ecological interactions resulting in a net negative feedback between organisms and their environment at a particular distance from the organisms.(8)
And could be an great principle in pattern formation:
which is that organisms modify their environment, inducing a net negative feedback at a certain distance. The strength of this feedback depends on the density of the organisms.(8)
So in a lot of ecological systems we can find a dynamics between an activator and an inhibitor, that it could be, for example the wind or water that act like short-range activator modifing the position of snow or water itself, but the accumulation of these elements creates a long-distange negative feedback(8). Furthermore
Ecosystems with regular patterns might be more resilient to disturbance and resistant to global environmental change as compared with homogeneous ecosystems.
Now we can imagine that this observation is obvious if we see the environment with the eyes of a countryman, but it is not so obvious if we think that the same behaviour is present also in natural environments and not only in human environments! And a simple explenation is that organisms, concentrating resources in their environment, activate a inhibitor mechanism like the long-distance negative feedback and create the condition for their survival. And this behaviour is common to all form of life in Earth. And, in conclusion:
The potential application and relevance of regular pattern formation for global environmental change, ecosystem adaptation and restoration involves transplanting organisms so that they reach a certain threshold density, to induce short-range facilitation, and arranging them spatially in a way to make optimal use of limiting resources.(8)
A good knowledge about this mechanism could provide us a sustainable way in order to change our environment and control the climate change. But the most incredible thing is that Turing patterns could explain also our propensity to create regular patterns (in our park, city and more).
The power of Turing model is not finished here: indeed this type of patterns seem to emerge also in electrochemical systems:
The electrochemical Turing structures are predicted to exist in systems with a bistable, S-shaped current-potential curve. In view of possible applications, as well as their relevance for modeling biological processes, the most important electrochemical systems of the S type seem to be those in which a first-order phase transition of an organic adsorbate is coupled with a faradaic reaction of some electroactive species.
As a representative of such systems, we used the periodate reduction on Au(111) electrodes in the presence of camphor.(9)
The camphor is the activator in the Turing dynamics.
Camphor adsorbed on Au(111) electrodes exhibits two first-order phase transitions upon variation of the electrode potential(9)
And finally, at least in research world, a team of researcher using tomography to study the Belousov-Zhabotinsky reaction in a microemulsion they found some three-dimensional Turing patterns(10):
Biological Turing patterns in 3D have not yet been reported, even though there has been considerable effort to find Turing-driven morphological phenomena. With the growing interest in Turing’s theory as a model for morphogenesis and emerging examples of 2D Turing patterns in living systems - for example, disposition of feather buds in chicks and hair follicles in mice - the near future seems likely to unveil evidence of 3D self-regulated patterns in living systems. Candidates include skeletal pattern formation in developing chick limbs and the head regeneration process in Hydra, a species that inspired Turing in developing his groundbreaking theory of morphogenesis. In addition, 3D Turing patterns may be useful vehicles for information storage.(10)
I think that this is the perfect conclusion of the post: I try to describe the versatility of Turing's theory of morphogenesis, also because it's the perfect example of Turing's versatility: he was not only the father of the modern computing, but also a great mathematician.
Happy birthday, Alan Turing!
On Chrome experiments you can find a fluid simulation with Turing patterns by Felix Woitzel.
On Wired there's also a great gallery about the existence of Turing patterns in nature (also in space!).
And finally there are two interesting article about Turing on Nature: Pattern of life by Philip Ball and A life computed by James Poskett.
(1) Galileo Galilei. Dialogue Concerning the Two Chief World Systems (1632)
(2) Turing, A.M. (1952). The Chemical Basis of Morphogenesis, Philosophical Transactions of the Royal Society B: Biological Sciences, 237 (641) 72. DOI: 10.1098/rstb.1952.0012 (pdf)
(3) Julijana Gjorgjieva, Jon Jacobsen. Turing patterns on growing spheres: the exponential case. Discrete and continuous dynamical systems, supplement 2007, pp.436-445
(4) Shigeru Kondo, Rihito Asai (1995). A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus Nature, 376 (6543), 765-768 DOI: 10.1038/376765a0 (ebookbrowser.com)
(5) Seita Miyazawa, Michitoshi Okamoto, Shigeru Kondo (2010). Blending of animal colour patterns by hybridization Nature Communications, 1 (6), 1-6 DOI: 10.1038/ncomms1071
(6) William L. Allen, Innes C. Cuthill, Nicholas E. Scott-Samuel, Roland Baddeley (2011). Why the leopard got its spots: relating pattern development to ecology in felids Proceedings of the Royal Society B: Biological Sciences, 278 (1710), 1373-1380 DOI: 10.1098/rspb.2010.1734
Read also: The evolution of cat coat patterns | Why the leopard got its spots
(7) Andrew D Economou, Atsushi Ohazama, Thantrira Porntaveetus, Paul T Sharpe, Shigeru Kondo, M Albert Basson, Amel Gritli-Linde, Martyn T Cobourne, Jeremy B A Green (2012). Periodic stripe formation by a Turing mechanism operating at growth zones in the mammalian palate Nature Genetics, 44 (3), 348-351 DOI: 10.1038/ng.1090
Read also: Proving Turing's tiger stripe theory | Scientists prove Turing's tiger stripe theory
(8) Rietkerk, M. & van de Koppel, J. (2008). Regular pattern formation in real ecosystems, Trends in Ecology & Evolution, 23 (3) 175. DOI: 10.1016/j.tree.2007.10.013
(9) Yong-Jun Li, Julia Oslonovitch, Nadia Mazouz, Florian Plenge, Katharina Krischer, Gerhard Ertl (2001). Turing-Type Patterns on Electrode Surfaces Science, 291 (5512), 2395-2398 DOI: 10.1126/science.1057830
(10) Tamás Bánsági Jr., Vladimir K. Vanag, Irving R. Epstein (2011). Tomography of Reaction-Diffusion Microemulsions Reveals Three-Dimensional Turing Patterns Science, 331 (6022), 1309-1312 DOI: 10.1126/science.1200815