How did the leopard get its spots?

How did tiger get stripes? One clue is that animal tails often have bands, even when the animal body has spots? It is as if spots need enough room to grow. In fact, the tails of some animal embryos are so tiny that the spots soon reach all the way around, forming bands. A thicker tail or leg may have spots, because there is room for the whole spots, but the skinny part of a tail has bands.

But this still leaves us with the question of how a patternless blob of cells develop patterns? Each cell of the early embryo is not much different from the rest. So, how do patterns start?

In an embryo, where there is very little room for expansion, such as in a narrow tail, spots become bands. Where there is more area, spots develop.

The positive feedback causes the color to spread in little clusters. The clusters expand until the negative feedback stops the expansion.

On a large sheet of cells, the positive and negative feedback result in complex patterns. Even if you start with a random mix of black and white pigment cells, you end up with spots or stripes, as originally proposed by Alan Turing. Turing’s model boils down to simple rules that are repeated to create a great variety of spots and stripes.

Complex patterns in nature form from simple rules, repeated over time and space.

Based on Turing’s model of animal fur pattern formation, we created this spots and stripes art project. Here’s a computer model of spots and stripes formation (hit “setup” and “go”) based on Turing’s model. Here’s a variation on the model, spots and stripes with color zones, that produces not only spots and stripes but a pattern of colors as seen on the Side-Blotched Lizard (featured in one of our Beautiful Discovery Boxes). See our page on lizards for more on spots on lizards, including a simple model of the dissolution of spots on certain lizards.

Similar to the Turing model is a reaction-diffusion model, a chemical oscillator, called the B-Z reaction (below).

This is our simulation of the “BZ” chemical reaction oscillating chaotically (never exactly repeating). 

Click green flag to restart. Beautiful scrolls and spirals usually emerge. 

Adjust “resistance” (setting resistance to 3 yields satisfying spirals after about 30 secs). Adjust incubation (default is 72). See how settings affect the patterns.   

Infection (and resistance) is a metaphor for the chemical dynamics. It’s like infection” by neighboring cells. Incubation is like the time to develop symptoms.  Negative feedback moves the state back to a state of zero once the threshold maximum state is reached (like recovering from illness).

The B-Z reaction models produce spots and stripes just with chemicals in a dish. Imagine a red chemical that says STOP, GO BACK and a green chemical that says GO. The green chemical gives positive feedback signals, saying “go further” in the direction it is already going. The negative feedback pulls the system back in the opposite direction once a threshold is reached. It starts out as a patternless soup of chemicals. Then we see a ring of green GO expanding, then a circle of red, then a circle of green, and so on, expanding like ripples in a pond. This mechanism generates expanding circles, then seemingly endless patterns of swirls and scrolls ( Click here if you want to see a NetLogo version of the BZ reaction. Compare to our Scratch version of the BZ reaction). 

There are yet more wonderful variations. Fern-like, striped patterns, similar to what you see when you drop dye into glue, are seen also in crystal formation, coral, fungi growth, liquid seeping through soil, and even lightning. When a thin liquid is pushed through a thick liquid, such fern patterns appear, due to a process called “viscous fingering .” A process called diffusion-limited aggregation happens to create the same patterns as seen in viscous fingering. Click the above “diffusion limited aggregation” to see the model in NetLogo. To see diffusion-limited aggregation in Scratch, click here.  See our movie of how to code this model in Scratch, with a few drag and drop coding blocks. 

Here’s a dramatic linear variation of the same diffusion model, where particles move straight down and the coral like branching grows upwards. 

Click green flag.

The particles start at random positions up high (red) and move down until they touch a stationary particle (green).

Coral like patterns appear.

Every time you run the model, it is a little different, but has the same general coral-like pattern.

Just for effect, we add a pen mark to each stationary particle. The color of new pen marks change over time. (We drew the background). 

The striped, finger-like pattern results when a more runny liquid spreads (”diffuses”) through a thicker (”viscous”) fluid. This process and the resulting pattern is called “viscous fingering.”

The tips of the skinniest “fingers” or stripes are most likely to continue to spread onwards, a kind of positive feedback. The long sides of the fingers are much less likely to spread, a kind of negative feedback. The pattern also shows up in crystal growth (”aggregation”). The crystal corners or tips stick out and so are more likely to pick up new molecules. For this same reason snowflake and frost tips grow faster, resulting in similar (but more symmetrical) fingering patterns.

You can adjust the model to get thinner or fatter “fingers.” When finger tips are stickier (more likely to grab floating particles), the fingers grow longer and thinner. The growth pattern is known to match that of the spreading liquid (viscous fingering).

These models, and the art projects we provide in in your subscription box, try to capture simple rules for generating spots and stripes.

Here is a snapshot from our  lesson on scaling, enlarging, a pattern. The general idea is to use a photo you like of any natural pattern, zoom in on a section using a grid and copy the contents, cell by cell, to a larger grid.

One way to study patterns from the point of view of the artist is to take a very closeup view. Contrast this closeup with a distant view. Above is Kathy’s paintings of closeup patterns on a flicker and a loon. A theme we pursue both with art and modeling lessons is abstraction, simplification that captures the essence. The simplicity of rules producing a complex diversity of patterns is elegance.