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The maths behind a leopard's spots

It has been 60 years since Enigma codebreaker Alan Turing came up with a theory for why the leopard has its spots. As stem cell research proves him right, we look at how patterns in nature can be explained by mathematics.

It’s a rare sort of mind that looks at the visual hotchpotch of a spotted, speckled and striped leopard’s pelt and finds a means of applying order to the shapes through mathematics.

The apparent randomness of the patterns that appear in nature – a zebra’s zigzagging stripe or the labyrinthine mosaic of a giraffe’s skin – are accepted without question by most of us. Alan Turing, the prolific mathematician best known for helping to break the Enigma code at Bletchley Park during the Second World War, and for writing a scientific paper that would form the basis for modern computing – was not content with simply marvelling at the miracle of nature.

In 1952, Turing published a ground-breaking paper called ‘The Chemical Basis of Morphogenesis’ – his last work before his apparent suicide just before his 42nd birthday – containing an elegant mathematical schema for the formation of the patterns found in animals and plants. The patterns were far from random, he argued, and were the result of interacting chemicals that spread among groups of otherwise identical cells. He coined the term morphogen (morpho, from the Greek for ‘form’, and gen, from the Greek for ‘to beget’), meaning shape-formers. Turing was deliberately vague about what these morphogens were. They could be hormones, perhaps, or genes, the chemical nature of which, in the 1950s, was still to be unravelled. The main idea was that they diffuse and react with each other: this is now called the reaction-diffusion process. His theory – laid out in beguiling mathematics – had it that within tissue or cells there are two morphogens that act on one another: one creates the other, and makes more and more of itself, and the other, second chemical limits the creation of the first. Both diffuse, or in other words, spread out, at different rates. It has been compared to a predator-prey situation; the idea that, repeatedly in the development of a biological entity, two chemicals can work in tandem both independently and in competition. It’s as if one is chasing the other away.

At the most basic level, patterns emerge from randomness as varying concentrations of colour-activating and colour-inhibiting chemicals interact. Dark spots form and, in turn, the spots merge into stripes as the colour-activating tissues become more concentrated. Seen on creatures ranging from zebras to seashells to giraffes, they have come to be known as Turing patterns. In recent years, scientists have found Turing patterns in wind-blown sand, attributing the ripples to the same reaction-diffusion process, and even human communities, suggesting that phenomena such as crime hotspots can arise from social feedbacks on behaviour and movement that are similar to the push-me-pull-you process Turing identified.

Proving the theory

For three decades, Turing’s theory on morphogenesis was largely ignored by biologists. But the combined arrival of powerful computers and the dawn of modern molecular cell biology, as well as the work of two generations of scientists who took Turing’s theory seriously from the 80s onwards, have contributed to proving it.

Jeremy Green, professor of developmental biology at King’s College London, and his team provided one of the first non-theoretical proofs of Turing’s ideas in 2014, when they discovered that the ridges on the roof of a mouse’s mouth acted like stripes or spots in morphogenesis. They identified the two chemicals needed to produce this – and proved in mice that it works.

‘It’s been possible, but only in very recent times, to really test Turing’s theory,’ Professor Green says. ‘There are a relatively limited number of examples where people have done that in a rigorous way – skin being one, my research into the palate being another, and left-right asymmetry in fish being the third. Turing was right and we’ve now got the names [for the shape-formers].’ Broadly speaking, the morphogens in this case turned out to be protein growth factors, of which there are six main ones. ‘Each one of those six can come in a bunch of different flavours,’ says Green – so determining what the job of each morphogen is will be critical, as scientists come to programme stem cells to do certain things in the body.

It might be counterintuitive to many of us, but it makes sense that a mathematician would look at patterns in biology and try to formulate an equation for their existence. It’s only now that scientists are realising quite how far ahead Turing’s thinking was.

Photograph © Nick Johns

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