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Turing Equation Explains how Leopard Spots Develop 109

BilZ0r writes "A slight modification of an equation developed by Alan Turing in 1952 has been used to show how the patterns of big cats change from kitten to adult markings. Sy-Sang Liaw of National Chung-Hsing University in Taichung, Taiwan, and colleagues set out to replicate these patterns using Turing's equations. But they found they had to do more than just tweak the parameters of the reaction-diffusion equation. Instead they had to assume two stages of spot growth with different rules: the first to get the baby cats their spots, and the second to create the final configurations. It took them a year to find a final solution."
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Turing Equation Explains how Leopard Spots Develop

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  • by nlogax ( 709388 ) on Sunday August 06, 2006 @01:48PM (#15855774)
    Damn, i thought "Spots" was some sweet new feature in OS X Leopard.
  • Turing test (Score:5, Funny)

    by Anonymous Coward on Sunday August 06, 2006 @01:50PM (#15855785)
    So how many questions were the leopard spots able to answer?

    • There was a clear response for various questions, such as:

          "How sexy is that female leopard's spots?"


          "Would you like to be my new coat?"
  • by Henriok ( 6762 ) on Sunday August 06, 2006 @01:52PM (#15855791)
    And this one day before Apple reveals features of Mac OS X 10.5 Leopard? What are the odds?
    • Posted by Jonathan at August 12, 2003 10:16 PM
      Considering that the blog post linked in the summary is very nearly three years old today, it does indeed look like someone did a bit of creative trawling...
  • by Anonymous Coward on Sunday August 06, 2006 @01:54PM (#15855794)
    Using Turing equations to model the growth of leopard spots reminds me of 2 other types of research. They are (1) analyzing the human navel and (2) extracting sunlight from cucumbers.
    • by Elemenope ( 905108 ) on Sunday August 06, 2006 @02:13PM (#15855848)

      Some researchers dicking around with orange molds accidentally discovered this little thing called PENICLLIN. Some Swiss mountain hiker got irritated with little seeds that kept sticking to his clothes, which upon further inspection led to the invention of VELCRO.

      On the other hand, researchers trying to solve a critical rubber shortage during World War II came up with an earth-shattering invention: SILLY PUTTY.

      Point is, you just never know. ;)

      • Come on!

        Everyone knows that Velcro was a Vulcan invention [].
      • silly putty, does have interesting properties.
        it can flow like a liquid and act like a solid when pushed rapidly.
        whats that good for polishing internal holes if you mix an abrasive in with it. Which might help get the best out of high performance engines.

        it's an interesting material wonder how it would cope with a leak in a pressurised container could it contain or slow that leak for a period of time.

        it also bounces and lifts ink off paper.
    • Considering that reaction-diffusion equations are thought describe (among many, many other things) the propagation of electric waves in the heart, I'd say research understanding the patterns they form is highly worthwhile even if you have no intellectual curiousity whatsoever.
    • by Pseudonym ( 62607 ) on Sunday August 06, 2006 @08:55PM (#15856968)

      And if you're an animation TD who has been assigned the task of creating a huge school of fish, each one of which should look different and yet still look like the right kind of fish, you'll be glad that someone has studied the problem of how to model animal markings.

      No, this is not hypothetical. It's real, and it's done today.

  • Link to article? (Score:3, Informative)

    by pacc ( 163090 ) on Sunday August 06, 2006 @01:58PM (#15855806) Homepage
    As interesting as the link may be it does not mention any of the new findings in the header.
  • Great ! (Score:3, Funny)

    by jfclavette ( 961511 ) on Sunday August 06, 2006 @01:58PM (#15855807)
    Awesome work guys ! Now on to the applications of this important discovery ! ... Lunch ?
  • New Adage (Score:5, Funny)

    by DumbSwede ( 521261 ) <> on Sunday August 06, 2006 @02:09PM (#15855827) Homepage Journal
    ... no more than a Leopard can change his Turing Equations
  • by agent dero ( 680753 ) on Sunday August 06, 2006 @02:09PM (#15855830) Homepage
    Sheesh, I'm really sick and tired of this WWDC, Mac OS X speculation... ;)

    But seriously, where in this silly blog posting does it ever talk about the Leopard spots? Is it just me, or is TFA missing here...
  • by posterlogo ( 943853 ) on Sunday August 06, 2006 @02:09PM (#15855832)
    This work is pretty interesting. My concern with complex mathematical models has always been that nearly any phenomena can be perfectly described given enough variables -- pretty much any curve, any pattern, any shape. In biology, when we try to fit models to data, we have to be very careful not to just keep trying to curve fit with more and more complex equations, because in the end we will be left with something that is not biologically very descriptive -- it leaves us with little understanding of the underlying biology. So when I hear these guys had to tweak parameters to make the reaction-diffusion equation fit the data, I am left wondering what biological factors those extra parameters are supposed to define? The original set of equations was meant to model a system with multiple morphogens that diffuse in two dimensions. When they act upon (or are acted upon) appropriate receptors, a particular "phenotype" emerges at that location. I did RTFA, but it doesn't actually say much about these things -- just makes up a dumb analogy with missionaries and cannibals in competition.
    • by QuantumFTL ( 197300 ) * <`justin.wick' `at' `'> on Sunday August 06, 2006 @02:33PM (#15855913)
      These types of nonlinear differential equations are usually very simple in form, and, most importantly, very local, as that is how most biological interaction is mediated. The parameter tweaking should not be considered too alarming when one considers that the number of biological parameters, in the sense of genetic material, involves thousands of degrees of freedom.

      A short (but good) web site about this can be found here []. The interpretation of these formulas is fairly trivial, as they describe a diffusion process (common in all biological systems) with a somewhat more complex reactive process, which could be mediated through all kinds of channels.

      This is not akin to fitting a polynomial to the shape of a bone and calling that a "model" - there are obvious interpretations which correspond to very well known processes.
    • This all look like Fractal Geometry, Cellular Automata, etc where the end result of a computation (a model) presents some resemblance to a naturally occuring phenomena or observation. It's interesting because there is some expressive power to modelize some phenomemon. It is also deceptive in that the model is not built directly from observations from the natural phenomena.

      In machine learning (really statistical modelization), people are interested in developing methods of representing relations. Express

    • My concern with complex mathematical models has always been that nearly any phenomena can be perfectly described given enough variables -- pretty much any curve, any pattern, any shape. In biology, when we try to fit models to data, we have to be very careful not to just keep trying to curve fit with more and more complex equations, because in the end we will be left with something that is not biologically very descriptive -- it leaves us with little understanding of the underlying biology.

      The objective of

  • I find this kind of research amazing. It's like nature has given us a hint at something, something on the tip a vastly larger and more profound realization. The ability to recognize these natural patterns, such as the Fibonacci sequence, is IMHO one of the fundamental qualities of intelligence and sentience. It seems to be something tied to the very basis of existence, upon which our human minds are a layer with a depth that may indeed have no bounds or may merely be a small slice. The potential infinity of it all is staggering, and yet beautiful, and is the primary reason I chose this handle which I use here.

    Here we witness the micro through the macro, through all scales of physical dimension, in an interplay of force, energy and motion, with the final result happening both all at once and forever spread over time. Incredible.


  • Sounds like (Score:3, Informative)

    by Black Parrot ( 19622 ) on Sunday August 06, 2006 @02:46PM (#15855948)
    cellular automata.
    • Re:Sounds like (Score:3, Informative)

      cellular automata.

      Actually, it's the precursor to cellular autonoma. There's a period of Alan Turrings life, that most people don't study and know about, which involves him studying a number of biological models. His 1952 paper 'On the Chemical Basis of Biological Morphogenesis' contains the foundations of what Wolfram would later go on and call 'cellular autonoma'. Go check it out, and form your own opinion. Having read both Wolfram and Turing, I have to give clear credit to Turing for coming up with t
      • Wolfram is a day late and a dollar short for the whole "creator of cellular automata" label.

        Von Neumann was the first. Conway popularized the idea with the Game of Life. Wolfram just did an intensive study of 1-d CAs. The applicability of CAs to biological patterns has been known since the early days though.

      • > [...] contains the foundations of what Wolfram would later go on and call 'cellular autonoma'.

        Actually the term predates Wolfram by a long time. Wikipedia gives some references [] from 1968.
        • Well... Wolfram did write a big fat book with lots of interesting pictures about the whole thing :-D

          And strangely, people really dislike him for it. I don't think he ever says -he- invented them (he just fails to provide a ton of references---like most scientific wr0ks... but then maybe his book is more of a pop-science type of book, rather than a serious research thing).

          (in any case, I found it interesting that the `random' number generator in Mathematica uses one of the automatas presented in the book :-D
      • Turing's research on pattern formation has nothing to do with celullar automata. He described reaction-diffusion systems using partial differential equations, which is a completeley different thing. The concept of cellular automata was introduced by John von Neumann.
    • It is, actually. Turing discretized PDEs to run his simulations, and that corresponds pretty closely to CAs, just with a continuous color level effectively. There is a bunch of stuff in Wolfram's book about this. Here is the section of A New Kind of Science (online) about pigmentation patterns from straight CAs, rather than reaction-diffusion PDEs -

      CA pigmentation models []

      First page shows sample animal patterns, next 3 show CA emulations of them.

      There is also this history note about Turing's work o

  • I remember my professor telling us in the class that anything which can be expressed/solved by any turing machine is an algorithm. Those equations seems different than the description of a turing machine[ To Me :( ]. I would say that it's a great finding. He was able to describe some natural phenomenon from a set of turing equations. It implies that we can simulate the whole world[or a part] on a computer.
    • It definitely entails(not implies) that we can simulate part of the world on a computer, as you say. It in no way implies that we can simulate the whole world. There are plenty of seemingly non-algorithmic processes going on - things that can't be reduced to a series of computational steps. Cognitivists and weak AI theorists, for instance, suggest that phenomenal consciousness and intelligence(the experience of what it's like to be X - X being you, for instance) are not algorithmic processes and thus cann
      • Clarification - weak AI theorists think that access consciousness(intelligence, concepts, memory, etc) can be simulated but *not reproduced.* They just don't believe that phenomenal consciousness can.
        • Cognitivists and weak AI theorists, for instance, suggest that phenomenal consciousness and intelligence(the experience of what it's like to be X - X being you, for instance) are not algorithmic processes and thus cannot be simulated or reproduced by a computer.

          They have only suggested, they haven't proved it. It's a big deal to prove or disprove these kind of things.
    • Those equations seems different than the description of a turing machine

      Turing was a very talented mathematician, and worked in several areas besides the theory of computation and cryptograhpy. The equations discussed in the article (reaction-diffusion equations) are partial differential equations that model chemical reactions. They don't have anything to do with Turing machines, other then the fact that you could use a Turing machine to solve them numerically.

    • Sorry, but you're confused. These are a set of differential equations and have 'nothing' to do with turing machines.
    • A Turing Space rather than a Turing machine. Turing was an excellent mathematician, and although currently most famous for his ideas on computation and the Halting Problem, this was a seperate area of research.

      This area of study (colourations on animals) is based on Reaction Diffusion Equations, of which a canonical example is the Belousov-Zhabotinskii equation derived from a chemical experiment, and a simpler one is the Heat Equation. These take the form of partial differential equations.

      As to simulation
    • This finding about leopard spots has nothing to do with Turing machines. It is less well known among computer scientists that Turing has also done fundamental research in the field of chemical pattern formation ( _ formation_and_mathematical_biology). BTW this is no news. It is well known that spotted patterns can be reproduced by reaction diffusion equations. These mechanisms have been thoroughly studied using computer simulations since the '80s. A really int
  • This is no big deal (Score:4, Informative)

    by crush ( 19364 ) on Sunday August 06, 2006 @03:00PM (#15855990)
    Since at least 1989 (with Dictyostelium) developmental and evolutionary biologists have used Turing's mechanism to explain pattern formation. Good site here []
  • You read a scroll titled BOOBIE FLETCH. --More--
    This is a scroll of pr0n. Read? [y/n] y
    You think impure thoughts, and start fapping. --More--
    Suddenly, a bolt of lightning hits the kitten! The kitten is killed!
  • Still no cure for leopard cancer!
  • Don't tease us with stories about Leopards [] the day before WWDC!!!
  • by Doc Ruby ( 173196 ) on Sunday August 06, 2006 @04:00PM (#15856183) Homepage Journal
    Apparently, old Jeremiah was teaching Turing's mathematics to homeless Israelites when he told one []

    "Can the Ethiopian change his skin, or the leopard his spots? then may ye also do good, that are accustomed to do evil."

    If one of the grad students working on this paper is an Ethiopian who's spent the year in a Taiwanese office rather than in the equatorial sun, we might have all the proof we need to test this ancient riddle.
  • There's a nice account of this in Ian Stewart []'s The Magical Maze [].
  • Turing's Paper (Score:3, Informative)

    by SirClicksalot ( 962033 ) on Sunday August 06, 2006 @04:26PM (#15856253)
    For those of you that don't know what this is about:

    This isn't related to Turing's work on early computer science, but concerns research he did shortly before his death.
    Turing proposed that under certain conditions diffusion can destabilize a chemical system and cause spatial patterns.

    His original paper on the subject can be found at the Turing Archive [].

    Mathematical biologists have been using these equations to model biological pattern formation for some time. If you want to read up on it, try googling for research by Gierer and Meinhardt on pattern formation
  • What actually is this useful for again?
  • by SimHacker ( 180785 ) * on Sunday August 06, 2006 @09:05PM (#15856996) Homepage Journal

    Collected Works of A.M. Turing
    P.T. Saunders, Editor


    Turing's work in biology illustrated just as clearly as his other work his ability to identify a fundamental problem and to approach it in a highly original way, drawing remarkably little from what others had done. He chose to work on the problem of form at a time when the majority of biologists were primarily interested in other questions. There are very few references in these papers, and most of them are for confirmation of details rather than for ideas which he was following up. In biology, as in almost everything else he did within science -- or out of it -- Turing was not content to accept a framework set up by others.

    Even the fact that the mathematics in these papers is different from what he used in his other work is significant. For while it is not uncommon for a newcomer to make an important contribution to a subject, this is usually because he brings to it techniques and ideas which he has been using in his previous field but which are not known in the new one. Now much of Turing's career up to this point had been concerned with computers, from the hypothetical Turing machine to the real life Colossus, and this might have been expected to have led him to see the development of an organism from egg to adult as being programmed in the genes and to set out to study the structure of the programs. This would also have been in the spirit of the times, because the combining of Darwinian natural selection and Mendelian genetics into the synthetic theory of evolution had only been completed about ten years earlier, and it was in the very next year that Crick and Watson discovered the structure of DNA. Alternatively, Turing's experience in computing might have suggested to him something like what are now called cellular automata, models in which the fate of a cell is determined by the states of its neighbours through some simple algorithm, in a way that is very reminiscent of the Turing machine.

    For Turing, however, the fundamental problem of biology had always been to account for pattern and form, and the dramatic progress that was being made at that time in genetics did not alter his view. And because he believed that the solution was to be found in physics and chemistry it was to these subjects and the sort of mathematics that could be applied to them that he turned. In my view, he was right, but even someone who disagrees must be impressed by the way in which he went directly to what he saw as the most important problem and set out to attack it with the tools that he judged appropriate to the task, rather than those which were easiest to hand or which others were already using. What is more, he understood the full significance of the problem in a way that many biologists did not and still do not. We can see this in the joint manuscript with Wardlaw which is included in this volume, but it is clear just from the comment he made to Robin Gandy (Hodges 1983, p. 431) that his new ideas were "intended to defeat the argument from design".

    This single remark sums up one of the most crucial issues in contemporary biology. The argument from design was originally put forward as a scientific proof of the existence of God. The best known statement of it is William Paley's (1802) famous metaphor of a watchmaker. If we see a stone on some waste ground we do not wonder about it. If, on the other hand, we were to find a watch, with all its many parts combining so beautifully to achieve its purpose of keeping accurate time, we would be bound to infer that it had been designed and constructed by an intelligent being. Similarly, so the argument runs, when we look at an organism, and above all at a human being, how can we not believe that there must be an intelligent Creator?

    Turing was not, of course, trying to refute Paley; that has been done almost a century earlier by Charles Darwin. But the argument from design had survived, and was, and indeed remains, still a potent force in biolog

  • So far, this is simply an equation that reproduces the visual appearance of spots. That's not an "explanation", it's merely a hypothesis or a model. Proving it will require actually identifying the substances and their concentrations over time that appear in the model.

"Kill the Wabbit, Kill the Wabbit, Kill the Wabbit!" -- Looney Tunes, "What's Opera Doc?" (1957, Chuck Jones)