A mathematical model for animal stripes

The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to its' spine. Scientists don't know exactly how stripes develop, but mathematicians since the 1950s have been modeling possible scenarios.

In Cell Systems on December 23, 2015, Harvard researchers assembled a range of these scenarios into one, single equation to identify what controls stripe formation in all living things.

"We wanted a very simple model in hopes it would be big picture enough to include all of the different explanations ... what is common among molecular, cellular, and mechanical hypotheses for how living things orient the directions of stripes ... what kinds of experiments will or won't distinguish between them," explains lead author Tom Hiscock, a PhD student in Sean Megason's systems biology lab at Harvard Medical School.

Stripes are surprisingly simple to model mathematically. Much of the early work on the subject was by Alan Turing a British pioneering computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. Recently, Turing was portrayed in the movie The Imitation Game. These patterns emerge when interacting substances create waves of high and low concentrations of a chemical, a pigment, or type of cell. What Turing's model doesn't explain is how stripes orient themselves in one particular direction.

Hiscock's investigation focused on stripe orientation, or why tiger stripes are perpendicular to its body while zebrafish stripes are horizontal. One surprise from his model is that it takes only a small change to switch whether the stripes are vertical or horizontal.

What we don't know is how this translates to living things. What is the variable for a tiger that pushes its' development of perpendicular stripes?

"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," says Hiscock. Genetic mutants exist that can't form stripes or make spots, he adds: "the problem is you have a big network of interactions, so any number of parameters can change that pattern."

Hiscock's master model predicts three deviations affecting stripe orientation:

(1) a substance changes "production gradient," amplifying stripe density

(2) a substance changes "parameter gradient" changing one of the parameters involved in forming the stripe

(3) a physical change in the cellular, molecular, or mechanical origin of a stripe can affect stripe direction.

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

Abstract Highlights
• A simple model predicts three ways to orient the direction of Turing stripes
• These are gradients in production rates or in model parameters and anisotropies
• The simple model predicts stripe orientation in a range of more complex models

Summary
Patterning of periodic stripes during development requires mechanisms to control both stripe spacing and orientation. A number of models can explain how stripe spacing is controlled, including molecular mechanisms, such as Turing’s reaction-diffusion model, as well as cell-based and mechanical mechanisms. However, how stripe orientation is controlled in each of these cases is poorly understood. Here, we model stripe orientation using a simple, yet generic model of periodic patterning, with the aim of finding qualitative features of stripe orientation that are mechanism independent. Our model predicts three qualitatively distinct classes of orientation mechanism: gradients in production rates, gradients in model parameters, and anisotropies (e.g., in diffusion or growth). We provide evidence that the results from our minimal model may also apply to more specific and complex models, revealing features of stripe orientation that may be common to a variety of biological 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

Cell Systems (@CellSystemsCP), published by Cell Press, is a monthly journal featuring papers that provide, support, or apply systems-level understanding in the life sciences and related disciplines. Research describes novel discoveries, milestone achievements, applied research, translational findings, broadly useful tools or resources, or insights into the use of technology. For more information, please visit http://www.cell.com/cell-systems. To receive media alerts for Cell Press journals, contact press@cell.com.

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How does development program striped patterns to always point in the same direction?
What makes fingers form parallel to the arm? Researchers propose a simple model of
Turing stripe formation to predict three distinct stripe orientations which apply
to a wide variety of biological mechanisms.
Image Credit: Sean Megason Laboratory, Harvard Medical School,