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Computer Models of Selection, Cambrian Explosion and Punctuated Equilibrium

Copyright 1995, 1997 Glenn R. Morton. This may be freely distributed so long as no charge is made and no changes are made to the text or pictures. (home.entouch.net/dmd/nonlin.htm)

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Living systems are generated from a DNA genome. During evolution, the DNA is mutated and new forms are generated by the new sequence of DNA. One of the things that modern research has shown is that some mutations have more morphological effect on the organism than others. This phenomenon has been well documented in nature. In the Rocky Mountains there are two different species of monkeyflower, one has flowers which attract bumblebees, the other has flowers which attract hummingbirds. The floral morphology between these two species is significantly different. What has been found is that three mutations accomplish most of the morphological change between two species of monkeyflower. (Bradley et al, 1995, p. 762) They write (1995, p. 765):

“Our mapping experiments show that for each of eight-floral traits likely to play a role in reproductive isolation there is at least one major QTL accounting for more than 25% of the phenotypic variance. This finding suggests that the evolution of reproductive isolation may involve genes of large effect and therefore that speciation may occur rapidly. “The floral syndrome associated with hummingbird pollination is found in 18 families and 39 genera in the flora of western North America, and in many cases has evolved from bee-pollinated ancestors. One plausible scenario for the initial steps in the evolution of hummingbird pollination in Mimulus would include a sequence of three major mutations affecting pollinator attraction, reward and efficiency. A mutation at the yup locus causes carotenoid pigment deposition through the flower, reducing attractiveness to bumblebees by eliminating contrast between the petals and nectar guides. A second mutation at the major ‘reward’ QTL leads to greatly increased nectar volume and visitation by hummingbirds. A third mutation at the major QTL for pistil length improves the efficiency of pollen deposition by hummingbirds. This hypothesis for the evolution of hummingbird pollination is testable in part by introgressing the M. cardinalis allele at each major QTL into a M.lewisi genetic background (singly and in combination), followed by assessment of pollinator visitation and its fitness consequences in nature.”

What kind of system can model such a complex system as DNA and its effect on morphology? As it turns out, certain very simple nonlinear dynamical systems are able completely mimic the behavior of morphological change observed in monkeyflowers, the rapid development of different morphological forms as was seen in the Cambrian Explosion and the stasis and rapid change suggested by the theory of punctuated equilibrium.

The Program Particulars

The program has a genome of 384 “nucleotides” A nucleotide in this context is a computer bit, a 0 or a 1. These nucleotides are mutated and a new screen form is produced. The variety of screen forms is almost infinite from this simple program. Selection is a program which mimics the process of natural selection. When the program starts, one can select the type of geometric form you want to select for. There are six choices. But any geometric shape can be can be selected for. The program was written to answer the standard argument against evolution which says that any given function in a protein is too unlikely to have been formed by pure chance. For instance, in cytochrome c, there are approximately 100 amino acids. Since there are 20 or so possible amino acids there are 20^100 (10^130) different possible combinations for a protein chain of this length. The argument continues that since a change in one of the amino acids can render the protein ineffective, only one permutation can perform the function. If this is true, then there has not been enough time in the age of universe (10^17 sec) for one to have found the correct sequence. If you search a quadrillion (10^15) different permutations each second the earth has existed, then today, you have only searched 10^33 different possibilities. This is only 10^-97 of all the possibilities. Thus the conclusion is that evolution could not have occurred. In the present program, with 384 bits arranged as 8 real numbers, there are 10^312 different combinations for the genome if the genome is restricted to numbers between 0 and 1 on each of the 8 numbers. Thus SINCE MY COMPUTER CAN ONLY ANALYZE ONE OF THESE OPTIONS EVERY 30 SECONDS OR SO, I CAN PREDICT WITH CERTAINTY THAT IF ONLY ONE COMBINATION FITS THE SELECTION CRITERIA, THAT MY PROGRAM WILL NEVER FIND IT.

In point of fact, my computer will always find a form that matches the selection criteria. The reason it can do this is that there are billions upon billions of combinations which produce a similar form. Every time the program is run a different solution to the same problem is found. What this means is that the genomes which produce a particular form are not collected together in the “genome phase space” but are spread like a mist throughout the 8 dimensional volume. Thus the genome for a / diagonal form may lie just next to a box-shaped-form's genome and next to innumerable other form's genomes. Thus, no matter where you start in genome space, you are not far from a solution to any arbitrary form. This is the nature of most non-linear system phase spaces. (see Thompson and Stewart 1985 p. 77, for a simple example of a complex phase space for Duffings equation) What this means is that a given evolutionary solution is not necessarily the best of all possible, but merely the best that was found. The probability argument against evolution is wrong because it wrongly assumes that only one possibility will perform the function.

Geometrically the program is an offshoot of Sierpinskis gasket. Where Sierpinskis gasket uses 3 fixed points and one moving point, my program uses 4 fixed points and 1 moving point. It was found when I was trying to code Sierpinski's gasket program from memory one day at lunch. I made a mistake, a mutation to the standard Sierpinksi code and found a wonderful world that mimics evolution on a computer. In Sierpinski's gasket, a dice roll determines the towards which point the moving point travels towards. After several hundred dice rolls, always moving according to the rules, these strange geometric forms are formed. Sierpinskis gasket forms a triangle.

My algorithm forms more complex shapes. After a specified number of dice rolls, the genome is mutated, the moving point reset to the starting position and the dice rolls resume producing a new morphology. Each form is compared to the form being selected for. Those genomes which produce the most similarity are retained, and then randomly mutated. Forms which are worse, are discarded. This is equivalent to the death of an unfit animal. The previously successful genome is then randomly mutated again. Over a period of 400-1000 mutations, the selected form begins to predominate. On some computers this could take 8-12 hours. Turn it on at the end of the day, go home and the results will await you in the morning. The mutations are performed in such a manner that 7/100 times a big mutation occurs. This is done to avoid having the solution be stuck in one small part of the genome phase space. This yields a better search algorithm.

The equations which produce these forms are

y:=abs(cos(2*(y-ya[a])/3))

x:=abs(cos(2*(x-xa[a])/3));

where cos is cosine, [a] goes from 1 to 4, and abs is the absolute value.

The options you can select for are a vertical or horizontal line, a either diagonal, or a cluster at a point or a repulsion from the point or a circle or a disk. With each screen, I draw the target form in yellow. This helps you remember what you are striving for. I have not fully tested the circle and disk and do not know if such forms exist. They may but need lots of time. With each form, I must scan the screen for pixels which are colored white and then determine the average pixel distance from the preferred shape. If a given genome produces a better match to the preferred form, the old genome is replaced by the new one. Thus, each final form, is a direct descendant of the single dot form. Also, most of the forms produced during the search for the preferred morphology are discarded since they are “worse” than the currently selected genome. Any form which can be programmed can be selected for. I start the diagonal programs with a randomly selected genome. The reason I do this is the starting location being close to the diagonal yields very good measures of similarity and the morphology never evolves. The single dot form fits the diagonal very well and never changes. All other programs start with the single dot form first. Remember, that a successful match may only fit part of the target curve. Thus, an arc could fit the circle very well even if no circle form exists. There is one final option and that is to have no selection at all. The richness of the geometric forms is best seen in this option. There are instructions for how to enhance the richness. One final thing. Choose one of the first 6 options with the identical parameters and run this several times. You will find similar solutions each with a different genome. One final thing. I do not want this to be taken as an attack on the Bible. I believe that God Created the world, and the design in mathematical systems such as these nonlinear systems are evidence of that. Belief in evolution does not preclude belief in the God of the Bible.

The program Cambrian Explosion takes the above code and runs it 48 times in parallel. All screen shapes start at the same point and within a few iterations, all the forms are different. It is like a Cambrian explosion occurred on the screen.

The program which made the pictures seen here was programmed by accident. I was not looking for such a system. One day at lunch I was trying to program Sierpinski's gasket from memory and I made a mistake in the code, a mutation.


Figure 1. Sierpinski's Gasket

What I saw on my screen was beautiful, but it was not Sierpinski's gasket. I saved the program and took it home to compare with the original code. The next day at lunch I tried to program Sierpinski's gasket again from memory and made another mistake. A different type of form appeared on my screen. This new system combined with the one from the previous day produced the system that produced these pictures.

Three programs were generated. The first shows the evolution of a given genome and the forms that it produces. In the pictures below, each small picture is generated from a genome. The evolution starts at the upper left and proceeds across the top row, then goes to the second row etc to the bottom of the page. The genome starts with the simplest form my system can produce, a dot. The pictures follow a single lineage of evolution. The last picture is a descendant of the first picture. That genome is mutated and a new form is created. the new genome is then mutated producing a third, etc. One can see that the complexity of the forms increases as time goes forward. The numbers in the upper left corner of each box is the crosscorelation coefficient between the picture and the previous picture. This is a measure of morphological change caused by the mutation in the genome. A correlation coefficient less than 50 means major morphological change.


Figure 2. Evolution of the Screen Morphology. First panel.

Figure 3. Evolution of Morphology. Panel 2.

One thing to notice in all of this is that there are periods of stability, like that along the top row of panel 2 and and then a sudden change as between the 3rd and 4th box on row 2 of panel 2. This is the behavior of punctuated equilibrium. Transitional forms occasionally occur but more often than not, the transition is rapid.

What happens if you follow lots of lineages and examine all the different lineages at the same time? When we find is something that is akin to the Cambrian explosion. In the pictures below, each box represents a separate lineage seen at a given time. This is like looking at today's fauna which are all descended from a common ancestor but are morphologically very different. We start as before from the very simple dot genome in each of the 48 boxes. Each of the 48 genomes is then mutated independently from the others. This procedure ensures that all 48 boxes began from the same common ancestor. The shapes produced by each box rapidly diverge from the others. This is exactly what we see in the Cambrian Explosion.


Figure 4.Starting point of all lineages. All genomes are the same.


Figure 5. Screen forms after one mutation. Some are vertical lines, some are horizontal lines, some are still dots.


Figure 6. Screen forms after 7 mutations. The diversity is amazing.


Figure 7. Screen forms after 12 mutational steps.

Natural Selection can also be mimicked in this process. A single lineage is followed and a selection criteria is determined. In the first sequence the selection criteria is for a vertical line a location x=200.


Figure 8. Starting point for selection. Once again the starting point was the simple dot genome.


Figure 9. Selection after 3 mutations.


Figure 10. After 5 mutational steps.


Figure 11. After 296 mutational steps.


Figure 12. Best fit after 684 mutations.

The same program can then be used to search for a horizontal line. After 2687 mutations, the best fit to a horizontal line at y=400 is shown below. the interesting thing is that the longer the search goes on, the closer is the match. After a long time, any mutation becomes detrimental to the fit of the screen form with the idealized form being selected for. This explains the fact that today most mutations in biological systems are detrimental.


Figure 13. Best fit genome after 2687 mutations.

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References

  • H.D. Bradshaw Jr., S. M. Wilbert, K. G. Otto and D. W. Schemske, “Genetic mapping of Floral Traits Associated with Reproductive isolation in monkeyflowers (Mimulus),” Nature, 376 Aug. 31, 1995:762-765
  • Thompson and Stewart, _Nonlinear Dynamics and Chaos_ John Wiley, 1986

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