During the 1930s and 1940s a group of scientists tried to reconcile the newly rediscovered Mendelian genetics with Darwin's theory of evolution. This program, now called the Modern Synthesis, has been wildly successful and dominates modern biology to this day. There were also a few biologists that were dissatisfied with the modern synthesis. Most developmental biologists who felt that the data didn't reflect the gene centrist view of the Modern Synthesis. One of these was Conrad Waddington, Waddington was a developmental biologist at Cambridge and then the University of Edinburgh. Waddington was one of the first to look at development through the lens of complex systems and to poke at one of the weak points of the Modern Synthesis: how the environment directed evolution. Waddington's work stands as a precursor to the Evo-Devo movement that started in the late 1970s and is now considered an important adjustment to the Modern Synthesis. (12)
In 1950, Alan M Turing took on one of the premier problems of biology, how a single cell can turn into a complex creature. Turing's group had broken the German secret codes during World War II. This alone made Turing a war hero and his work in cryptology set the stage for modern breakthroughs. After the war, his work in the theory of computation, logic, philosophy, and the study of Artificial Intelligence formed the theoretical foundation of our modern digital world. In November of 1951 he turned in his first paper on morphogenisis and on August, 15th, 1952, 68 years ago today, it was published. In January 1952 Turing was arrested for practicing homosexuality and in March 1952 he was convicted under an 1885 law that they had used to persecute Oscar Wilde. Instead of jail time, Turing volunteered to submit to chemical castration, a procedure that was then used to “cure” homosexuality. On June 7th, 1954 Turing died after eating an apple laced with cyanide. Whether this was suicide or an accident (he was dabbling in chemistry at home) or worse remains a mystery but none of his planned papers on the mathematics of development were ever written. (2)
Turing's paper, The Chemical Basis of Morphogenesis, has only six citations in the bibliography. One is a book by Conrad Waddington (12). Waddington and Turing were both at Cambridge at the same time but there is no evidence that they had met or that Turing had attended any of his lectures. Another book among the citations is an old favorite of mine, D'Arcy Thompson's On Growth and Form (11). Turing specifically wrote in the paper that he was keeping the math simple so biologists could understand it. His problems and untimely death meant no follow-up from the author and the publication of the structure of DNA in 1953 cemented biological research for decades so his paper was pretty much ignored.
The idea that different patterns of flow can exist in dynamical systems was well known as Turing's reference to the elasticity and electromagnetism book (13) attests. The Lotka-Volterra model was developed between 1910 and 1926 first for chemical reactions and then later as a model of predator-prey interactions. This was one of the early nonlinear dynamical systems studied. It is unknown whether Turing was familiar with this model. Chemical reactions with standing chemical waves were first discovered in 1951 by Boris Belousov. At that time Soviet scientists were not allowed to attend international conferences so it is unlikely that Turing knew about this. Reaction-diffusion instabilities as an explanation for morphogenisis were first proposed in 1938 by Nicholas Rashevsky (17). Turing not only proposed this but demonstrated how this was possible.
As digital computers started becoming a growing tool for research in the 1970s and as numerical methods for solving nonlinear dynamics improved, a theory of what is now called complex or self-organizing systems started to emerge. In 1981, the 1977 winner of the Nobel Prize in chemistry (14), Ilya Prigogine, published From Being To Becoming (15). Prigrogine was the first to study how nonlinear reactions worked as thermodynamic systems and wrote about the philosophical implications of this. In 1984, the Sante Fe Institute, dedicated to the study of such systems, was founded. During the 1980s, research on Turing patterns continued to be stalled as no chemical example could be found. In fact, sometime in the late 80s, I remember reading an article saying that Turing patterns were a dead end. This changed in the 1990s when these reactions were discovered (16).
It is hard to imagine now how off the wall Turing's ideas were to biologists in 1952. Biological mathematics was in its infancy and to assert that complex biology such as morphogenisis and even behavior can emerge out of a set of physical conditions was pretty extreme. Even today, one of the most potent criticisms of complex models is that the parameters of the model have little to do with anything biological. Maybe the geometry or the pattern of the data model and the parameter model might loosely match but the parameters used have little biological value. Thus the model might show similarities but tells nothing about the underlying biological processes. This is slowly changing as more biologists embrace complexity theory and design experiments around it and the models, software and mathematics become more sophisticated. (7)
Turing's paper contains many leaps in mathematics that make it hard to follow but I have found helpful commentaries and explanations in the literature. (5)(6)(18) Basically you have two reacting chemicals, A and B, A changing into B. Also, you have two substances (Turing calls them morphogens) that are diffusing across a surface, Turing uses a ring of connected cells, a ring of continuous substance, and a sphere in his examples. One of these substances increases the rate of reaction, the other slows it down. Without the diffusing morphogens, the reaction will burn out with all of A turning into B. Even with diffusion, the reaction will either take slower or faster to complete or it will stop leaving some of B. To produce patterns there must be two conditions:
This must be an open system as opposed to a closed one, This means a continuous supply of chemical A and the morphogens.
The diffusion of morphogens across the surface changes from homogeneous (everywhere the same) to heterogeneous (patches of difference) through some sort of random or non-random imperfections. This drives the system to a different state where patterns form across the surface.
While Turing uses linear dynamic equations to explain his model, he is really talking about nonlinear dynamic systems. At the end of the paper, he promises to go on to nonlinear models to study phyllotaxix, the patterned growth of leaves and stems in plants. He also says he wants to use a digital computer at Manchester to gain mathematical insights:
The difficulties are, however, such that one cannot hope to have any very embracing theory of such processes, beyond the statement of the equations. It might be possible, however, to treat a few particular cases in detail with the aid of a digital computer. This method has the advantage that it is not so necessary to make simplifying assumptions as it is when doing a more theoretical type of analysis. (1)
This is extraordinary in that it wasn't until the mid-1970s that this technique started being used in mathematics, first off, in the study of nonlinear dynamics. The study of nonlinear systems was pioneered by Poincaré between 1891 and 1896. Poincaré first showed that analytic solutions were not possible for systems with a degree of freedom (number of dynamic variables) of n>=2. He then devised a geometric means to study nonlinear systems with two degrees of freedom.
Poincaré Phase Portraits (4)
This is what Turing is alluding to. There was sporadic work during the 1930s and 1940s and some work with regard to radar which Turing may have been familiar with. During the 1940s to 1960s, a group of mathematicians and physicists in Moscow made many contributions to nonlinear theory. Whether Turing was familiar with what is now called the Moscow School is unknown as I mentioned earlier.
Turing talks about these small changes in the rates of diffusion of the morphogens as moving the system from a stable to an unstable state. What does he mean by this? Try to balance a cane or a walking stick on a hard floor. No matter how perfect the stick is or how carefully you balance it, it is going to fall over. Once on the ground, it will stay on the ground unless moving. This lying on the ground would be the stable state of what is called an inverted pendulum. No matter where you start it will end up lying on the ground. Now try to balance the stick on your hand. With a bit of practice, you can achieve this for a minute or so. The end of the stick doesn't remain still but wobbles back and forth around that unstable point. This can be modeled graphically in two dimensions with the inverted pendulum mounted on a cart that rolls back and forth on a rail. As the cart moves back and forth at a certain distance and speed the symmetry of the system will break and the stable point of the pendulum will become unstable and the unstable point will turn into what is called a limit cycle. (10)
Inverted Pendulum (10)
The Fitzhugh-Naguma Model was a model proposed by Richard Fitzhugh in 1961 and an equivalent electrical circuit for this model was published in 1962 by J. Naguma. Fitzhugh developed this model as a two-variable simplification of the four-variable Hodgkin-Huxley model of the giant squid axon. This is very powerful and successful which can be calculated numerically but a nonlinear model with over two variables cannot be analyzed while the simplified version can. The Fitzhugh-Naguma model can also be thought of as a reaction-diffusion model and thus a candidate for Turing patterns. The images below show the evolution of the pattern, the final pattern, and Turing's drawing of the hand-compiled pattern from his paper. (20) This is just one of several reaction-diffusion models I have found online in which I can run the code and play with parameters.
Evolution of Turing Patterns in the Fitzhugh-Nagumo Model (21)
Final Pattern in the Fitzhugh-Nagumo Model (21)
Pattern Hand Drawn and Hand Calculated by Turing (1)
I have barely scratched the surface here and every time I do a little more research I find more results. There are several more questions I would like to find answers to:
Turing shows that two stable linear systems can be combined to create an unstable system. Then he makes the leap to instabilities in the diffusion part of a reaction-diffusion equation. It has been known since Newton that to solve the solution of diffusion over a patch it is only necessary to define the boundary, to reduce the two-dimensional surface to a one-dimensional line. Adding the reaction term is where Turing instability enters the Poincaré world of nonlinearity. Just what are these conditions on the boundary and how does Turing Instability fit into the general theory of bifurcations? (4)(5)(6)(18)
What is the parameter space for Turing patterns and how does this differ for stripes and grid patterns (hexagons)?
Do all reaction-diffusion models form Turing patterns?
How do Turing patterns fit within a general theory of patterns? Cellular automata create patterns including stripes. Are these Turing patterns?
Turing, AM. “The Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 237, no. 61 (August 14, 1952): 37–72.
Ball, Philip. “Forging Patterns and Making Waves from Biology to Geology: A Commentary on Turing (1952) ‘The Chemical Basis of Morphogenesis.’” Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 370, no. 1666 (April 19, 2015). https://doi.org/10.1098/rstb.2014.0218.
Jenesis. “Exponential Stability.” In Wikipedia, August 4, 2020. https://en.wikipedia.org/w/index.php?title=Exponential_stability&oldid=971090204.Jenesis. “Exponential Stability.” In Wikipedia, August 4, 2020. https://en.wikipedia.org/w/index.php?title=Exponential_stability&oldid=971090204.
freesodas. “Stability Theory.” In Wikipedia. By Freesodas - Gimp, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=67990034, April 8, 2020. https://en.wikipedia.org/w/index.php?title=Stability_theory&oldid=949841045.
Diamond, Patrick H. “Physics 221A,” 2017. https://courses.physics.ucsd.edu/2017/Spring/physics221a/lecture.html.
Rauch, J. “The Turing Instability,” http://www.math.lsa.umich.edu/~rauch/256/turingexample.pdf
Camazine, Scott, Jean-Louis Deneubourg, Nigel R. Franks, James Sneyd, Guy Theraula, and Eric Bonabeau. Self-Organization in Biological Systems. Princeton University Press, 2020.
Grenander, Ulf. Elements of Pattern Theory. JHU Press, 1996.
“Numerical Methods for Engineers.” Accessed August 8, 2020. http://folk.ntnu.no/leifh/teaching/tkt4140/._main000.html.
Kuse, Manohar. Inverted_pendulum. Python, 2020. https://github.com/mpkuse/inverted_pendulum.
Thompson, D’Arcy Wentworth. On Growth and Form, 1942. Cambridge University Press.
Waddington, C. H. 1940 Organisers and genes. Cambridge University Press.
Jeans, J. H. The Mathematical Theory of Elasticity and Magnetism. Cambridge University Press, 1927.
Prigogine, I. “Time, Structure, and Fluctuations.” Science 201, no. 4358 (September 1, 1978): 777–85. https://doi.org/10.1126/science.201.4358.777.
Prigogine, Ilya. From Being to Becoming: Time and Complexity in the Physical Sciences. 1st edition. San Francisco: W H Freeman & Co, 1981.
Boissonade, J., E. Dulos, and P. De Kepper. “Turing Patterns: From Myth to Reality.” In Chemical Waves and Patterns, edited by Raymond Kapral and Kenneth Showalter, 221–68. Understanding Chemical Reactivity. Dordrecht: Springer Netherlands, 1995. https://doi.org/10.1007/978-94-011-1156-0_7.
Rashevsky, Nicolas. “Mathematical Biophysics,” 1938. https://agris.fao.org/agris-search/search.do?recordID=US201300352824.
Diamond, Patrick H, and Xuang Li (notes). “Lecture 16: Spatial Pattern Formation by Turing Instability,” 2017. https://courses.physics.ucsd.edu/2017/Spring/physics221a/lecture.html.
Aubin, David, and Amy Dahan Dalmedico. “Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures.” Historia Mathematica 29, no. 3 (August 1, 2002): 273–339. https://doi.org/10.1006/hmat.2002.2351.
Izhikevich, Eugene M., and Richard FitzHugh. “FitzHugh-Nagumo Model.” Scholarpedia 1, no. 9 (September 23, 2006): 1349. https://doi.org/10.4249/scholarpedia.1349.
Rossant, Cyrille. IPython Cookbook - 12.4. Simulating a Partial Differential Equation — Reaction-Diffusion Systems and Turing Patterns. 2nd ed., 2018. https://ipython-books.github.io/124-simulating-a-partial-differential-equation-reaction-diffusion-systems-and-turing-patterns/.
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