Alan Turing was a British mathematician who was a cryptographer and a pioneer in computer science. As a side hobby, he was also a theoretical biologist who developed algorithms to try to explain complex patterns using simple inputs and random fluctuation. His "reaction-diffusion" model uses a two-protein system to generate a pattern of regularly-spaced spots, that can be converted to stripes with a third external force. In this model, there is one activating protein that activates both itself and an inhibitory protein, that only inhibits the activator1. By itself, transient expression of the activating protein would only produce a pattern of "both proteins off" or "spot of inhibitor on" since the activator would activate the inhibitor, thus turning off the expression of the activator (Figure 1 "No diffusion" case). Things get more interesting when the molecules can diffuse or be transported across the tissue. In this case, the activator gets randomly turned on and it begins to diffuse away from its point source, activating itself in nearby cells. At the same time, it activates the inhibitor, which also diffuses away from the point source, inhibiting the activator. Depending on the timing on activation and diffusion or transport, this can result in the formation of an expanding ring of activator expression (Figure 1 equal rates). To get spots, however, we need two more layers of complexity. First, there must be random fluctuations in expression that turn the activator on at low levels across a tissue. Second, the activator must diffuse more slowly than the inhibitor. In this case, random spots of activator can be stabilized when they are far enough away from each other. Each of the small spots activates the expression of activator (which does not diffuse away quickly) and inhibitor (which diffuses away too quickly to completely eliminate activator expression from the initial point source). This gradient of inhibitor diffusing from each spot keeps any nearby cells from making activator. The overall result of this is a regular pattern of spots (Figure 1 bottom and side panels). The exact patterning depends on the size and shape of the tissue, the speed of activator and inhibitor diffusion, as well as any other patterning elements that might be present.
Figure 1: Turing-type patterning with an Activator (red) and an Inhibitor (blue). The Activator activates itself and the Inhibitor. The Inhibitor only inhibits the Activator. If the inhibitor diffuses more quickly than the activator, this can set up a regular pattern of spots in a tissue. In the bottom panel, the activator is randomly expressed at low levels in a subset of cells over a tissue. It self-activates to increase it's own expression levels while also activating the inhibitor. The inhibitor quickly diffuses to the adjacent cells (second panel). Cells with both activator and inhibitor (purple) quickly convert to inhibtor-only as the inhibitor turns the activator off (third panel). Cells that are not exposed to the inhibitor (white) can randomly turn on the activator, turning on the diffusible inhibitor (fourth panel). This quickly sets up a stable pattern of "on" and "off" cells (fifth panel).
Can Math Explain How Animals Get Their Patterns? How Alan Turing's Reaction-Diffusion Model Simulates Patterns in Nature
In a very long and narrow tissue, there is only one direction diffusion can occur and this converts the Turing spot pattern into a stripe pattern (Figure 2). Similar forces, like directional growth and a morphogenic gradient, can also convert the spot pattern into stripes2. Without an external force, the default should be spots or a meandering labrinthine pattern, depending on the properties of the activator and inhibitor. Hiscock and Megason propose four main ways to get a stripe pattern. Besides making diffusion more likely in one direction than another, a tissue can be subject to a "production gradient." This gradient is a protein or transcriptional/translational cofactor that causes higher gene expression of both the activator and inhibitor on one side of the tissue. Computational models predict that this type of gradient causes stripes to orient themselves perpendicular to the gradient (Figure 2)2. Stripes will orient parallel to a "parameter gradient," where the activating and inhibitory properties of the two proteins are higher at one end of the tissue than the other. This type of modification could be produced by a gradient of a protein or cofactor that binds to the activator and both prevents it from activating gene expression and from being inhibited by the inihbitor (Figure 2)2. Finally, the tissue can grow directionally. For example, your limbs developed largely by growing away from your body (distally), with a much slower rate of growth in other directions. This is due to the AER at the distal-most part of the limb bud causing cell proliferation underneath it. A computational model shows that a reaction-diffusion Turing model will generate stripes parallel to the direction of tissue growth (Figure 2)2. A minilab helps us explore these models further with an online tool.
Figure 2: Turing stripes. Turing spots can become stripes under many circumstances including limited diffusion (stripes form perpendicular to the diffusion limit), a gradient of activator and inhibitor production (stripes form perpendicular to the gradient), a gradient of activator/inhibitor interaction strength (stripes form parallel to the gradient), or directional tissue growth (stripes form parallel to the direction of growth). Figure by Amanda Lo.