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Could it be that a single mechanism underlies such diverse patterns such as the stripes on a zebra, the spots on a leopard, and the blobs on a giraffe? This model is a possible explanation of how the patterns on animals' skin self-organize. If the model is right, then even though the animals may appear to have altogether different patterns, the rules underlying the formation of these patterns are the same and only some of the values (the numbers that the rules work on) are slightly different.
Thinking of the formation of fur in terms of rules also helps us understand how offspring of animals may have the same type of pattern, but not the same exact pattern. This is because what they have inherited is the rules and the values rather than a fixed picture. The process by which the rules and values generate a pattern is affected by chance factors, so each individual's pattern is different, but as long as the offspring receive the same rules and values, their own fur will self organize into the same type of pattern as their parents'.
We model the animal skin by a square array of many melanocytes (pigment cells) that are each in either of two states: colorful ('D' for differentiated) or not-colorful ('U' for undifferentiated). The state of a cell can flip between D and U. The color cells (the D's) secrete two types of 'morphogens': activators (A) and inhibitors (I). Activators, on their own, cause a central cell to become colorful; inhibitors, on their own, cause the central cell to become not colorful. These competing morphogens are secreted in all directions so you can think of each color cell as creating a puddle that grows around it, spreading to other cells.
Each cell, whether or not it is colorful, is itself the center of its own neighborhood. For now, suppose the neighborhood is a circle. Say this circular neighborhood has a radius of 6 cells. This means that the cell in the center can be affected by other cells that are as far as 6 cells away from it in any direction. So if there is a D cell within this circle and it is secreting morphogens then these morphogens will diffuse as far as this central cell (but a D cell 7 cells away will not directly or immediately affect it). Also, each cells has an inner circle of radius, say, 3 cells.
D cells within the inner circle each contributes morphogens of type A (activator) to the central cell. Between the inner circle and the perimeter of the outer circle we have a ring of cells that are more than 3 cells away from the center but 6 or less cells away from the center. Every D cell in this outer ring contributes morphogens of type I (inhibitor) to the central cell. So at every moment each cell is affected both by activator and inhibitor cells in its circle and the question is will it ultimately be activated and become colorful or inhibited and lose its color (or just remain the way it was). The logic is that if the power of the activators is bigger than the power of the inhibitors then the cell will become colorful and vice versa (and if the power is balanced then nothing happens). The idea of "power" is that it's not enough to know how many morphogens there are of each type affecting a cell but one must multiply each cell by its "power" (or you can think of power in terms of the concentration of the morphogens in the inner and outer neighborhoods). Another idea is that since we'll be multiplying both types of morphogens by their power, we might as well just call the power of the activators "1" and the power of the inhibitors "w * 1" or just w. So w is the ratio between the power of the inhibitors and the activators. If w is bigger than 1 that means the power of the inhibitors is greater than that of the activators (for instance, if w = 2 then the inhibitors are each double as strong as each of the activators and if w = 0.5 then the inhibitors are half as strong as the activators). If w = 0.5 and if we have as many inhibitors as we have activators that are affecting the central cell, we would logically assume that the center cells would be more activated than inhibited and so would probably become (or remain) colorful on that step. (A tricky point to notice is that while a certain D-cell is activating a neighboring cell, this same D-cell can be inhibiting a different cell further away.)
Here are the rules that summarize what we've been discussing: count up all the D cells in the ring and call this number D*I (for instance 2 inhibitors), and count up all the D cells in the circle of radius three and call this number D*A (for instance, 5 activators). Then compute D*A - w*D*I, and:
- if it is > 0, set the central cell to D
- if it is < 0, set the central cell to U
- if it is = 0, leave the central cell unchanged
Note that this computation happens to all cells at the same time. After the first step and once the cells have been set accordingly, the entire business starts over at the next step. Once again, the cells are counted up according to the same rule. The rules have not changed but because some of the D cells are now U and vice versa we might get different counts and because of that -- different results of the "fight" between the A and I morphogens.
So what you see is that from step to step the individual cells often change from white (representing D or color cells) to black (representing U or no-color cells) and the overall impression is that the configuration of white and black changes as a whole. But these configurations are not random. You will see how these configurations often take form. Understanding how each cell behaves, as we have explained above, can help understanding how these global patterns take form.
All these explanations were for circular neighborhoods. In this model, the neighborhoods may be elliptical instead of circular. This is needed to produce stripes instead of spots.
In order that your first experiment will more-or-less match the explanations above, you should choose to set the initial-density slider to 50% (that gives each cell an equal chance of being white or black to start with and so the whole window will be roughly 50% white), set the INNER-RADIUS-X and INNER-RADIUS-Y sliders to 3 and the OUTER-RADIUS-X and OUTER-RADIUS-Y sliders to 6, and set RATIO to 0.35 (that means the I morphogens are 35% as powerful as the A morphogens). Now press SETUP. (In later experiments you are welcome to change those settings in various combinations.) It will take a while to complete. If you press STEP the model will advance a single step. If you press GO the model will keep stepping indefinitely.
It takes a while for the patches to determine their neighborhoods. Because of this, only press SETUP when you change the radius sliders. If you only change the INITIAL-DENSITY and RATIO sliders or if you'd like to run the model again with the same settings, press RESTART instead of SETUP. The RESTART button doesn't ask the patches to recalculate their neighborhoods.
DRAW BLACK and DRAW WHITE allow patches to be colored by hand, while the simulation is running or when it is stopped.
As the model runs, patterns may begin to emerge. Eventually, they stabilize. (Well, sometimes the model will run into an endless flip-flop between two states, but we could call that dynamic stability.) Even when it seems to come to a halt, the model is still running and executing the commands and carrying out the computations, but nothing is changing visibly. This is because for each and every cell the power of activators is equal to that of the inhibitors, so nothing changes.
Run the model with different INITIAL-DENSITY settings. How, if at all, does the value of the INITIAL-DENSITY affect the emergent pattern? Do you get the same pattern? Do you get a different pattern? Does it take longer?
Note how fragile the self organization of the cells is to slight changes in parameters. If you hold all other factors and slightly change just the RATIO, from trial to trial, you will note that for small ratios you will invariably get completely white fur and for high ratios you will invariably get completely black fur (why is that?). For ratios in between it fluctuates. That happens partially because the initial setting of black/white coloration has a random element to it (see the RESTART procedure in the code).
Try changing the sliders to have different values in the X and Y directions.
If you find a combination of slider and switch values that
consistently give you the fur patterns of a favorite animal, you could
create a button, for instance "Zebra," that sets the sliders to those
values. That way, if you make several of these, you can go on a virtual
safari tour by moving between your favorite animals. One such
combination that you could set in a single button could be:
You could call this, perhaps, Fish.
How about adding more colors? What could be the logic here? If you introduced, say, red, you would have to decide on specific conditions under which that color would appear. Also, you'd have to decide how that color influences other cells.
The Voting model, in the Social Science section, is based on simpler rules but generates patterns that are similar in some respects.
Modified 2007-09-05 by B. J. MacLennan to allow drawing of light and dark areas and to increase number of patches. Modified for NetLogo 4, 2008-07-02.
Various researchers have proposed similar theories of skin pattern formation. The particular variant presented in this model was proposed by David Young.
In building this model, we used information on this web site: http://classes.yale.edu/fractals/Panorama/Biology/Leopard/Leopard.html
Thanks to Seth Tisue and Dor Abrahamson for their work on this model.
To refer to this model in academic publications, please use: Wilensky, U. (2003). NetLogo Fur model. http://ccl.northwestern.edu/netlogo/models/Fur. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.