Jennifer Ouelette, science writer extraordinaire at Ars Technica, is our guide for the odd intersection of mathematics and biochemistry, ushered in by Alan Turing:
Scientists create new class of “Turing patterns” in colonies of E. coli
Computer science pioneer Alan Turing first proposed the patterning mechanism in 1952.
JENNIFER OUELLETTE / Ars Technica
2/23/2021
Shortly before his death, Alan Turing published a provocative paper outlining his theory for how complex, irregular patterns emerge in nature—his version of how the leopard got its spots. These so-called Turing patterns have been observed in physics and chemistry, and there is growing evidence that they also occur in biological systems. Now a team of Spanish scientists has managed to tweak E. coli in the laboratory so that the colonies exhibit branching Turing patterns, according to a recent paper published in the journal Synthetic Biology…
As we've reported previously, Turing was attempting to understand how natural, nonrandom patterns emerge (like a zebra's stripes or a leopard's spots), and he focused on chemicals known as morphogens in his seminal 1952 paper. He devised a mechanism involving the interaction between an activator chemical that expresses a unique characteristic (like a tiger's stripe) and an inhibitor chemical that periodically kicks in to shut down the activator's expression…
Scientists have tried to apply this basic concept to many different kinds of systems. For instance, neurons in the brain could serve as activators and inhibitors, depending on whether they amplify or dampen the firing of other nearby neurons—possibly the reason why we see certain patterns when we hallucinate. There is evidence for Turing mechanisms at work in zebra-fish stripes, the spacing between hair follicles in mice, feather buds on a bird's skin, the ridges on a mouse's palate, as well as the digits on a mouse's paw. Certain species of Mediterranean ants will pile the dead bodies of ants into structures that seem to exhibit Turing patterns, and there is evidence of Turing patterns in the movement of Azteca ant colonies on coffee farms in Mexico.
In essence, it's a type of symmetry breaking. Any two processes that act as activator and inhibitor will produce periodic patterns and can be modeled using Turing's diffusion function...
When Math Meets Nature: Turing Patterns and Form Constants
Jennifer Ouellette / Scientific American
March 27, 2013
Turing patterns are a perennial favorite among science writers, especially in light of the 100th anniversary of Turing's birth last year. Also? Pretty! And narrative angles like why the tiger has stripes play to broad audiences, so editors love them too.
Scientists, on the other hand, have mixed feelings about Turing's little foray into mathematical biology. Even Murray, who has done seminal work in biological patterning, confessed that he was a little burnt out on Turing after the centenary, pronouncing the mathematician's contributions to biology rather over-rated. Turing was a mathematician, first and foremost, and his proposed mechanism is (by his own admission) a highly simplified and idealized take on a messy, complicated system.
Nor was he the first to tackle this sort of thing: in 1917, for example, D'Arcy Thompson published On Growth and Form, which also talked about chemical morphogens contributing to periodic patterns. And Boris Belousov independently came up with a closely related model to that proposed by Turing, probably also in the early 1950s, although Belousov struggled to get his work published; it did not appear until 1959, in an obscure journal. Per Murray (in a 2012 paper), Belousov showed "how a group of three reacting chemicals could spontaneously oscillate between a colorless and a yellow solution."
That doesn't mean reaction-diffusion and other proposed models for pattern formation can't be useful: they may lead to breakthroughs later on. This certainly seems to be the case with Turing mechanisms, and the Simons Science News article gives a bit more detail on two recent papers in particular that are generating interest among mathematical biologists: one on how the ridges form on the roof of the mouth in mice, and another on the formation of digit patterns in mouse paws, and why polydactylism may occur.
Old math reveals new secrets about these alluring flowers
A model developed by Alan Turing can help explain the spots on these astoundingly diverse flowers—and many other natural patterns as well.
KATHERINE J. WU / National Geographic
MAY 11, 2020
Scientists who study monkeyflowers sometimes feel as though the plants are looking back at them. The blooms are said to resemble the faces of playful monkeys—hence the name—complete with a speckled central region that looks like a gaping mouth, helping bees zero in on their nectar-rich targets.
“It's like a friendly smile indicating safe harbor for pollinators,” says Benjamin Blackman, a plant biologist at the University of California, Berkeley. By attracting these pollinating insects, the speckled petals help ensure the plants will go on to bloom another day.
“The color contrast makes pollination more efficient, more effective,” says Yaowu Yuan, a biologist at the University of Connecticut…
In a recent study, Yuan and his colleagues showed that the dots that dapple monkeyflower petals are the result of a war between just two genes. Tussling over control of each plant cell’s pigments, these genes can seed wild amounts of diversity within even a single species—a find that may validate a decades-old theory first cooked up by British mathematician Alan Turing, who proposed a common template for many of nature’s most enigmatic designs.
“Pigmentation patterns are complex and ubiquitous in the natural world,” says Blackman, a co-author of the new research. “This study tells us that a relatively simple system can give rise to this complexity.”
The Community Ecology of Herbivore Regulation in an Agroecosystem: Lessons from Complex Systems
BioScience, Volume 69, Issue 12, December 2019
The Turing process and the spatial distribution of the ant Azteca sericeasur
The ant is oddly nonrandom in its spatial distribution: When you find a nest (almost always in a shade tree), you frequently find another nest nearby, but large sections of shaded farms have no nests at all. Quantitative sampling verifies this simple observable fact (Vandermeer et al. 2008, Jackson et al. 2014, Li et al. 2016), an important feature of the regulation of all three of the herbivores. The question first arises as to where this pattern comes from. There is now substantial evidence that the spatial pattern of the ants is self-organized, which is to say that it emerges from the internal dynamics of the ant population itself, not from any underlying forces such as moisture or temperature or particular vegetation formations (Vandermeer et al. 2008, Liere et al. 2014, Li et al. 2016). The pattern is formed in a complicated fashion by a process similar to that described by Alan Turing in 1952. Turing was interested in chemicals, especially morphogens (proteins involved in the creation of patterns during biological development). In chemistry, a reaction is frequently assumed to be stabilized by the balance between an activation process and a repression process. However, a completely different form of chemical process occurs in a spatially constrained space, the process called diffusion. Inject a drop of black ink into a beaker of water, and the instability of the drop isolated from the water gradually turns into a beaker of grey—very stable indeed. It would be natural to think that these two stabilizing processes, activation or repression and diffusion, when combined, would also be stable. What Turing demonstrated was that if the repression force diffused at a rate greater than the activation force, a nonrandom pattern of some sort would develop. The basic idea is that the activating chemical starts the reaction at a specific point in the space but begins its diffusion away from that point immediately. The repressive chemical is eventually produced by the reaction and cancels the effect of the activator, but, because it diffuses at a rate that is greater than that of the activating chemical, it eventually occupies a space where the activator had not yet arrived, therefore canceling the effect of the activator at that point. The results could be spots (e.g., the spots on a leopard's coat) or stripes (e.g., the stripes on a tiger's coat) or some other more complicated form, but the point is simply the qualitative one that a nonrandom pattern would spontaneously develop when these two stabilizing forces were combined. The two stabilizing forces, reaction (activation or repression) and diffusion, combine to form an instability; the whole system is therefore referred to as diffusive instability or sometimes Turing instability.
Something very similar happens in ecological systems. Evident at a qualitative level but also explored several times mathematically (Alonso et al. 2002), a predator–prey system distributed in space is such a system. The prey as activator and the predator as repressor is the reaction, and the migration or dispersal of predator and prey is the diffusion. However, an important difference between the ecological and chemical metaphor is that, in the predator–prey situation, it is frequently the case that, at a very local level, the predator and prey form an unstable relationship (perhaps usually), whereas adding diffusion to the mix may result in stabilizing the system. The classic experiments of Huffaker (1958) illustrated this point many years ago (although the insights of Turing were apparently unknown to Huffaker), with two species of mites, one a forager on the surface of oranges, the other a predator on that forager. Huffaker devised a spatial system in which a part of the surface of an orange provided a substrate for the two mites, and the oranges could be arranged in an array to represent a spatial matrix. As frequently cited in ecology textbooks, when predator and prey were isolated on a single orange, the predator would inevitably overeat, first driving the prey locally extinct (on a single orange) and then dying of starvation. In contrast, if a spatial pattern of a number of oranges was presented to the mites, with the possibility of dispersal from one orange to another, a seeming stabilization of the predator–prey system over the whole space occurred.
Although the take-home message of Huffaker's experiment is that environmental heterogeneity can stabilize an inherently unstable system…
In the coffee agroecosystem, we have argued (Vandermeer et al. 2008, Li et al. 2016) that the Azteca nest pattern is formed in a similar fashion. The repression agent is thought to be a parasitic fly, Pseudacteon spp., in the family Phoridae (Philpott et al. 2009). The fly oviposits on the back of the ant's head, and its larva penetrates the head capsule, there developing to the point that the head of the ant falls off of the body (hence the name “decapitating fly”) and a new fly emerges to mate and repeat the cycle. As a local population of ant nests builds up from single (or a few) queens taking a part of the colony to a new shade tree (the activator), spatial clusters are formed. As the clusters become larger, they are targets for the phorid flies, either because the flies are attracted from far away or they build up local populations within the area of the nest cluster. Either way, the flies act as the repressor in the system. The result is a patchy distribution of ants. In figure 2, we show the nest distribution for 3 of the 12 years of the study and enlarge one section (about 1.5 hectares) of the data to illustrate how clusters form and dissipate over time. We note that the general qualitative dynamics of the formation and dissolution of patches, such as the Huffaker mite example above, resonates quite well with the basic expectations of the Turing process.
How do patterns in nature form?
Turing Patterns and the Remarkable Mathematical Similarities Across Nature
Skanda Vivek/ Medium
Sept. 5, 2020
In 1952, Alan Turing (yes The Alan Turing, the same one who invented modern computing!) made a bold hypothesis on the origin of morphogenesis (the processes by which spatial order is created in developing organisms). In his paper “The Chemical Basis of Morphogenesis,” Turing hypothesized that chemicals known as morphogens are generated, that react and diffuse leading to the emergence of coherent patterns…
…slowly but certainly, there have been validations of the Turing mechanism for pattern formation in various other systems, such as periodic stripes in the mammalian palate. This takes us back to the second sentence in Turing’s original paper (It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.)
This statement has held true. Turing's contribution has been validated across multiple systems.
How Alan Turing used mathematical biology to find secret patterns hidden in nature
04 Jul 2019
Alan Turing may be best known for decrypting German messages created by their enigma machine in World War II. But the influential scientist thought about the interaction between nature and mathematics in great depth before his untimely death in 1954. In fact, his last published paper became one of the founding theories of mathematical biology, a subject devoted to understanding how nature’s mechanisms work by finding equations that describe them, from species population changes to the way cancerous tumours grow.
Turing proposed that two biological chemicals moving and reacting with each other in a mathematically predictable way could explain shapes and patterns across nature. For example, imagine that a cheetah’s coat is a dry forest with chemical “fires” breaking out all over. Simultaneously, firefighting chemicals of a second type work to surround and contain these fires, leaving charred patches – or spots – in the furry landscape.
Importantly, the speed of the firefighting inhibitor chemical must be faster than that of the spot-creating activator chemical for patterns to be created. Too slow, and the activator chemical will dominate, leading to uniform colour.
The story of Turing’s work on the mathematics of morphogenesis would not be complete without recognizing how he died from state-sponsored persecution for being who he was: