The number is up: Maths provides part of the answer on zebra stripes

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This image shows simulations of what are known as Turing stripes. On the left, stripes are evenly spaced, but their direction is variable. On the right, a signaling gradient has made the stripes align in the same direction. Image: Tom Hiscock
This image shows simulations of what are known as Turing stripes. On the left, stripes are evenly spaced, but their direction is variable. On the right, a signaling gradient has made the stripes align in the same direction. Image: Tom Hiscock

Scientists have come up with a single mathematical equation to cover stripe formation in living things, including zebras. However, they have yet to unravel exactly how nature manages to apply the math.

Many questions remain over the development of stripes in animals. Scientists have pondered the reasons for the striking black-and-white markings found on zebras for more than a century.

Some have suggested the pattern might, in fast-moving zebras, confuse predators, while others propose the stripes help deter insects or play a role in keeping the animals cool.

Scientists have also puzzled for decades over exactly how stripes develop. Since the 1950s, mathematicians have been modeling possible scenarios.

Harvard University researchers have now assembled a range of these models into a single equation to identify which variables control stripe formation in living things. Their findings have been published in the monthly journal, Cell Systems.

“We wanted a very simple model in hopes that it would be big picture enough to include all of these different explanations,” said lead author Tom Hiscock, a doctoral student in Sean Megason’s systems biology lab at Harvard Medical School.

zebra-lewis“We now get to ask what is common among molecular, cellular, and mechanical hypotheses for how living things orient the directions of stripes, which can then tell you what kinds of experiments will – or won’t – distinguish between them.”

Stripes are surprisingly simple to model mathematically, with much of the early work on the subject carried out by mathematician Alan Turing, who cracked the code behind Germany’s Enigma machine during World War II. His exploits were chronicled in the 2014 movie, The Imitation Game.

These patterns emerge when interacting substances create waves of high and low concentrations of, for example, a pigment, chemical, or type of cell. However, what Turing’s model does not explain is how stripes orient themselves in one particular direction.

Hiscock’s investigation focused on orientation – for example, why tiger stripes are perpendicular to its body while zebrafish stripes are horizontal.

One surprise from his integrated model was that it took only a small change to the model to switch whether the stripes were vertical or horizontal. However, what has yet to be determined is how this translates to living things.

“We can describe what happens in stripe formation using this simple mathematical equation, but I don’t think we know the nitty-gritty details of exactly what molecules or cells are mapping the formation of stripes,” Hiscock says.

Genetic mutants exist that can’t form stripes or make spots instead, such as in zebrafish, but “the problem is you have a big network of interactions, and so any number of parameters can change the pattern”.

His master model predicts three main perturbations that can affect how stripes orient: one is a change in “production gradient,” which would be a substance that amplifies stripe pattern density; second is a change in “parameter gradient”, a substance that changes one of the parameters involved in forming the stripe; and the last is a physical change in the direction of the molecular, cellular, or mechanical origin of the stripe.

Although the paper is based in theory, Hiscock believes researchers are close to having the experimental tools that can decipher whether the math holds true in living systems.

The work was supported by the National Institutes of Health and the Herchel Smith Graduate Fellowship.

Cell Systems, Hiscock and Megason: “Orientation of Turing-like Patterns by Morphogen Gradients and Tissue Anisotropies,” http://dx.doi.org/10.1016/j.cels.2015.12.001

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