Lecture notes for Friday, February 19, 2016


Two ways to break displacement symmetry:

Reaction-Diffusion Systems (Turing's, for example)

Liesegang Rings.



Turing's mechanism: Please memorize these three simple rules.
(Although these are not the only rules that can produce patterns, and they are not even the same as the rules Turing proposed, they are a good example of how simple the rules can be.

rule# 1) The concentration of chemical "A" causes more "A" and also more of Chemical "B" to be produced. (= Synthesized? Activated? Released from vesicles? Anything!)

rule# 2) Both "A" and "B" are destroyed (=Inactivated? Reabsorbed? Anything!)
in proportion to the concentration of "B" at any location.

rule# 3)"B"diffuses faster than "A"
(Or in any way produces effects at longer range than A)

The result of obeying these rules is formation of alternating waves of high and low concentration of both A and B. The peaks of A are a little higher than the peaks of B, but the B peaks are wider. Faster diffusion, and lower peaks result from faster diffusion.

Video of a computer obeying these three rules.

The peaks of A and B form at the same locations.
You can generate alternating waves by using different sets of rules.

The wave-length increase in proportion to the ratio of diffusion rates, and also changes if A or B reaction rates are changed.

These rules will generate evenly spaced waves, either when initiated by Brownian motion of the chemicals, or by local differences in concentrations of A or B, or other perturbations.

In two dimensions these rules will generate peaks or waves (depending on how you initiate them) In three dimensions, they generate regularly-spaced blobs of higher concentrations of A and B.

Textbooks tend to assume only irregular patterns can be produced when randomness is used to initiate the process. Computer simulations show this is not true. (A certain textbook has a graph showing the supposed irregularity.)

Another widely-believed fallacy is that B has to diffuse a lot faster than A.
(Twice as fast can be enough, which is good because 5 or ten-fold differences in diffusion rates are difficult to produce with molecules whose sizes are in the same ball-park.)

We used to post on the web some computer simulations of these and other sets of rules, and little control bars to change reaction rates, diffusion rates, initial concentrations, etc.

We would do that again if we could hire someone to rewrite our Pascal programs into JAVA, Python or some other language, in a form that can be run on current Macs and PCs. Please contact us if you would be willing to help us do this.


Nobody has yet proven that any particular spatial pattern of either animals or plants really uses the logical equivalent of this (or other) reaction-diffusion systems.

How would you prove (or conclusively DIS-prove) that something like stripes on a Zebra, or on a Zebra Fish, are generated by a reaction-diffusion system?

What if you mutated the gene that codes for the "Morphogen" chemical "A" or "B"? What would the phenotype of such a mutation be?

Suppose you inserted the amino acid sequence for GFP (green fluorescent protein) into the gene for protein A, or B (even assuming proteins?

What would happen if you isolated substance "A" or "B", and put drops of these substances on a developing embryo, where it was forming spots, or somites were splitting apart from each other? How would the spatial pattern be changed that would prove that the isolated chemical really functions like "A" or "B".

Suppose that you somehow knew for sure that chemicals A and B were among 10 or 20 chemicals that someone had isolated from body fluids of some kind of embryo;
please design an experiment by which you could prove which of the purified chemicals is "A" and which is "B".

Next, suppose that an embryo forms regularly-spaced spots.
Can we invent a method for extracting substances "A" or "B".
By that I mean, to extract them before you know what chemical they are.

NOTE: Auxin was purified from oats before it was known for sure to exist.
Serotonin was purified from blood when its existence was still hypothetical.

Shouldn't hypotheses include experimental criteria for detecting, not just whether they are capable (in principle) of producing a given effect, but also testing where and when and whether that category of mechanism is going on?

Not just a litmus test for acidity, but also a litmus test for "morphogen" chemicals, and even a litmus test for whether a reaction-diffusion is or is not the cause of some particular breaking of displacement symmetry.


Please graph concentrations of "A" on the x axis, and graph concentrations of "B" on the y axis, and for a few dozen points in A, B space, please graph little vector-like arrows that illustrate the changes on local concentrations of "A" and "B" as a result of their effects on their own and the other one's concentrations.

For example, the arrows in the lower right (corresponding to A being more concentrated than B) therefore point diagonally upward and further to the right.
That's because concentrations of both A and B would increase there.

Conversely, the arrows in the upper left would point diagonally downward and back further to the left. A diagonal band of very short or zero-length arrows would slope upward and rightward from the point that represents A and B being zero.

Such graphical representations of the net effects of combinations of equations are called "phase planes". By using phase planes, your brain can learn to recognize what net effects would be produced by any arbitrary combinations of equations.

Phase planes can also be used to do the reverse, which is to invent combinations of equations that will, acting in combination, produce any desired combination of net effects.

Incidentally, only those combinations of chemical reactions that produce diagonal boundaries between zones where arrows are pointed in opposite directions parallel to boundary.

Can you invent "rules" whose phase plane representation will have a diagonal boundary running from upper left down to lower right, with the arrows above this boundary pointing parallel to this boundary, and in opposite directions on one side of the border as compared with the other side of the border.

With effort, you can gradually become able to visualize what a phase plane has to look like in order to be able to break displacement symmetry.

An impressive but (I think) misleading article about Turing's theory is at the following URL http://rsfs.royalsocietypublishing.org/content/2/4/487

You are NOT assigned to read it. It is long and designed to impress instead of to be understood, or to help you understand.


LIESEGANG RINGS = "Liesegang Bands"
Another kind of Reaction Diffusion System.
discovered by accident, actually occurring in chemical reactions but not yet discovered in embryos or animals or plants

(And the mechanism of which is not yet known for sure!!)
(After a century of intensive research, and ~ 4,000 published research papers!)

Dissolve a low concentration of potassium bichromate in water;
also dissolve 2 or 3% by weight of gelatin in this same water.
(Be careful: bichromate ion is a serious cumulative poison, NEVER touch it! If by accident you do; then wash vigorously.)

Heat the water until the gelatin dissolves.

Pour this solution into a thin (millimeter) layer in a Petri dish, and let it gel.

Later, scatter a few crystals of silver nitrate on a small part of this gelatin surface.

Instead of forming solid blobs of insoluble silver bichromate, what happens instead is formation of many concentric rings of precipitated silver bichromate, with clear gaps between the rings of precipitate.

These gaps are what is unexpected and unexplained.

video of Liesegang rings forming from a string dipped in silver nitrate solution

another video of Liesegang rings forming. In this case, the silver nitrate was added at the left edge to bichromate gelatin under a cover slip.

computer simulation of Liesegang rings

Also surprising is that rings don't form if you dissolve silver nitrate in a gelatin solution, let it gel in thin layers, and then scatter crystals of potassium bichromate on top of that. That is some kind of asymmetry. However, rings do form if you put a small bichromate crystal on top of an area of bichromate gelatin gel where silver bichromate crystals are already formed.

The Wikipedia site on this subject is well done.

By using high power microscopes to making time lapse videos of Liesegang rings I have discovered several new and very unexpected phenomena. These videos will be shown in lecture.


Evidently, all self-organizing phenomena (even the simplest) resist human comprehension. Our brains expect shapes to be imposed by something internal, not as a net result of internal interactions.

How would Positional Information explain Liesegang Rings and reaction-Diffusion systems?

Could somites be formed by Reaction-Diffusion systems?

Could spatial patterns (Colinearity) of hox genes be produced by Reaction-Diffusion systems?

Patterns can also be generated without physical diffusion of any substance. In the following computer simulation, each location determines its value by averaging the values of its nearest neighbors.

video of computer simulation of number gradients

I wrote this and several other computer programs to simulate the formation of diffusion gradients. These programs taught me some unexpected lessons.

Lesson #1) The basis for the simulation linked above is as follows:

    * Visualize a square grid of mail-boxes, each box with a number in it.

    (Although I started by using hexagonal neighbors, it turns out not to matter).

    ** For each box, add up the numbers in the eight neighboring boxes.

    *** Divide this sum by 8, to get the average.

    **** Compare this sum with the number in the central mailbox.

    ***** Change the central number to make it closer to the sum of the neighbors.

    ****** Repeat...repeat...repeat



Lesson #2) Chemical concentrations aren't the only quantities that behave according to this algorithm.

The point is that you can get identical, indistinguishable gradients from any variable whose amounts change to become closer to the average amount at their neighboring locations.

Population densities of cells could obey the same algorithm, & make gradients. Tensions in extracellular matrix are also capable of "diffusing". Lots and lots of properties are able to behave as if diffusing.

Analogy to caloric: Until the mid 1800s, physicists firmly believed that heat must be a chemical substance, which they called "caloric". The evidence was that heat diffuses. What else diffuses besides chemicals?
(What else floats besides wood? A duck!)



Lesson #3) What else can make spatial gradations, as if diffusing?

-> Imagine a row of genes, adjacent to each other on a chromosome?

Gene_A; Gene_B; Gene_C; Gene_D ;Gene_E; Gene_F; Gene_G; Gene_H;

    -> Let each cell compare itself with its 8 adjacent cells.

       Cells can then obey any diffusion-like algorithm:
       Any cell transcribing the same genes as its neighbors, doesn't change.
      A cell transcribing genes adjacent to its neighbors, don't change.
      Cells transcribing non-adjacent genes, change to express adjacent genes.

The Big Lesson: Patterns that look exactly like diffusion gradients can be produced by repeated local interactions. Chemical diffusion is only one of many different methods for generating geometrical patterns.

Let's not make the same mistakes as physicists did with "caloric" and "the ether".






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