The expression "random walk" has a special meaning for statisticians. I'd
like to speak of it in a more general way.

What if we were to take a three dimensional game court and divide it into a
3-D grid of, say, a hundred billion little cubicles (none would be a perfect
cube, as each would be expanded in direction away from center, to avoid
leaving any unfilled spaces. And let us agree on a little rule, that no
more than one pixel could migrate into any one cubicle, EXCEPT that in the
center cubicle we begin with 8 game pieces, one the color of each of the die
(singular of dice) we will mention momentarily.

In betwixt the frontal, sagital and horizontal planes we would have eight
subdivisions. Let there be no negative value of any cubicle, but merely
something fundamentally different from the cubicle directly opposite it.
Let each subdivision be a large class of critters; and let any cubicle
represent a different species from another if there are, let us say, 12
cubicles separating them.

We could have nine die (singular for dice), colored red, orange, yellow,
green, blue, white, black and ivory. (You thought I was going to say indigo
and violet somewhere in there, didn't you :>)?. And there would be a game
piece to match each differently colored (or shaded) die EXCEPT the ivory
one.

Each "move" of a colored game piece represents one MUTATION. And the object
of the game is to see just how far apart from one another the pieces get as
the game progresses, and which of the eight quadrants the pieces end up
in... just to get an idea what kinds of constellations of game pieces we
get, representing random mutations (within constraints of the rules we set
up).

The first move consists in our throwing the colored die (holding back the
ivory one). Each game piece can move only to an adjacent cubicle and, we
know that each cubicle -- having six sides -- is adjacent to exactly six
neighboring cubicles. Each direction to a neighbor is represented by a
number from one... to ...six (we would have to assign each of the axes a
number, such that, for example, the north pole would be one, the south pole
two, the east pole three, the west pole four... etc. Hence, the number
rolled would tell us which direction to move the game piece.

What is the ivory die for? Well if two pieces try to occupy the same
cubicle in a single move, then the ivory die is case once for each of the
two game pieces, to determine which get to move into that cubicle. In case
of a tie, the ivory die is rolled for each of the two "competing" game
pieces again, and the same again if necessary, until two different numbers
are rolled for the two "competing" game pieces, at which time the one
getting the larger number of the two on the ivory die gets to take the
cubicle being competed over.

(Now obviously this is GROSSLY less complex than a real life situation. We
just want to see how far we get with randomized mutational directions. If
it were a real world situation we would have to allow for reproduction,
predation, starvation ...etc. But we just want to see what randomization
does.)

In this little game, a piece can move right back to a cubicle it has
occupied previously. Statistically all the pieces could end up going
outward and outward and outward or could all end up competing for the center
of the spherical game world. More likely, however, they would zig and zag
all over the place.

What do you suppose would be the most likely patterning of the game pieces
after, say, 1,000 moves?

Well, if the boundaries are far enough away, they will approximately

Would at least one of the game pieces move in a fairly straight line by the
shortest possible route to the outer limit of the sphere? Not very likely,
is it?

We might want to simplify the game even further, by doing it on a flat game
board, using squares.

Modifications are needed to make the "competition" more meaningful, aren't
they?

Without that, it seems that there would not be much "change" in the sense of
getting more and more complex (as represented in moving out toward the outer
limits of the game board.

How might something similar to "natural selection" be introduced into this
game?

Could it be improved until it provides anything even remotely resembling
what may have happened in the history of living things on Earth?

Isn't that what Gavrilets is doing (I really ought to read his book!)?

We could have nine die (singular for dice), colored red, orange, yellow,
green, blue, white, black and ivory. [snip]

"g" <gillawton@earthlink.net> wrote in message
news:eqigf0$6bu$1@darwin.ediacara.org...
[repost]
[snip]
We could have nine die (singular for dice), colored red, orange, yellow,
green, blue, white, black and ivory. [snip]

Er... If there are nine of them, why do you use the singular 'die'? Does
it have something to do with the reason you only supplied 8 colors?

In any case, if you want to develop an intuitive feel for what mutation
can accomplish over time, I would suggest that you consult an evolutionary
genetics textbook in which you will find more realistic models presented
and analyzed. No need to re-invent the wheel, badly.

Well, if the boundaries are far enough away, they will approximately
have a 3D normal distribution: proof from central limit theorem.

If the boundaries are close enough, it will be a bit different, but
exactly what happens depends on the type of boundary.

Isn't that what Gavrilets is doing (I really ought to read his book!)?
Certainly, with no selection, what will happen is obvious from the
theories of stochastic processes, but is also the obvious place to start
building the theory from. With selection, the shape of the fitness
landscape is probably more important.

There are certainly models for this sort of evolution, on phylogenetic
trees. I can do no better than suggest that you read the chapter in
Felsenstein's Inferring Phylogenies on Brownian motion on trees (round
here we have to promote his book. ).

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org

If I gave you the impression I was trying to come up with a new theory, I
really, really blew it. My purpose was to convey to you and others that
random mutations would tend to move progeny around RANDOMLY, and I was (and
am) desiring input from others far more knowledgeable than I) as to how in
hell it could provide, as it were, the WHATEVER upon which "natural
selection" could EXERCISE ITS ALLEGED POWER to make sense out of
nonsense.

If anybody has worked out a theory of how natural selection has PERFORMED
its selection at the genetic, cellular and molecular level, then I shall
want to read it, after I have finished about forty others that lie waiting.
However, if they have formed any theory based on nothing other than POST
HOC analysis of paleontological evidence, then that is not going to tell me
ANYTHING that I want to know.

The intuitive problems I have with NSDI (natural selection did it) are AT
LEAST as meaningless to me as those based upon GDI (Gd did it)... by which I
mean that neither, so far as I know, has done more than take some
before-and-after snapshots, as it were, and given a name to something NOBODY
YET UNDERSTANDS whereby something occurred IN BETWEEN.

Well, if the boundaries are far enough away, they will approximately
have a 3D normal distribution: proof from central limit theorem.

If the boundaries are close enough, it will be a bit different, but
exactly what happens depends on the type of boundary.

Isn't that what Gavrilets is doing (I really ought to read his book!)?
Certainly, with no selection, what will happen is obvious from the
theories of stochastic processes, but is also the obvious place to start
building the theory from. With selection, the shape of the fitness
landscape is probably more important.

There are certainly models for this sort of evolution, on phylogenetic
trees. I can do no better than suggest that you read the chapter in
Felsenstein's Inferring Phylogenies on Brownian motion on trees (round
here we have to promote his book. ).

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org

"Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
news:eqqfs8$drt$1@darwin.ediacara.org...
Well, if the boundaries are far enough away, they will approximately
have a 3D normal distribution: proof from central limit theorem.

If the boundaries are close enough, it will be a bit different, but
exactly what happens depends on the type of boundary.


(snipped my words to leave, and respond to, your, Bob)
Isn't that what Gavrilets is doing (I really ought to read his book!)?
Certainly, with no selection, what will happen is obvious from the
theories of stochastic processes, but is also the obvious place to start
building the theory from. With selection, the shape of the fitness
landscape is probably more important.

Bob, I have not read Gavrilets.

Please understand that I am a layman who gets strong intuitive feelings that
there are flys in some ointments, involving not mathematics but the nature
of the mechanism whereby what is called "natural selection" physiologically
does its things. Spookily often, such intuitions of mine turn out to have
some actual basis ultimately, if I pursue and learn enough to confirm that.

My current intuition (which could be totally wrong) is that there are
numberous assumptions made by biologists and paleontologits on basis of a
simplistic view of "natural selection" which has never (so far) been borne
out at the DNA level. The details are BEGINNING to be mapped, which is
wonderful; but molecular biology impresses me as being in its infancy.

Perhaps you would be willing to help me get an accurate picture in my mind
of how distribution of mutations would radiate over time
in a spherical grid. The picture I intuit would tend to be ratable, that is
to simply expand out into the sphere fairly equably in all directions.

Indeed, that would be true under neutrality, hence my references to 3D

On the other hand, what little I have understood clearly from reading about
mutations indicates to me that most mutations, by far, are
deleterious.

Yes.

Follow my reasoning here, and correct me if you can... but I envision a lot
of mutations *RANDOM IN MORPHOLOGICAL IMPACT, AS WELL IN CAPABILITY OF
SUCCESS vs FAILURE POTENTIAL" to be enormously unlikely to produce lines of
genetic change that lead anywhere in particular.

A lot would just lead to the grave, yes.

Let me pick the example of the evolution into contemporary horses from a
dog-like creature. It is a popular misconception (I have read) that the
progression was a linear one or a smooth one.) The illusion it might have
been so is nothing more than what I might (with a weak attempt at humor)
term "paleontological selection," i.e., selection of specimins to connect
the dots, as it were, from a quadripedally five-toed, dog-like creature to a
creature with one toe contacting the ground, and running only on the toe
nail of that... which we call a "hoof." Actually (I have read) there were
many other variations and branchings off in the process which were somewhat
random... and which failed. To this extent, random mutation and the
possibility *realized* of four hooves do seem to have followed a course of
*PASSIVE* trials of different things MOST of which did not work.

It's more complex than that: the branches that survived did work

Okay, but what I intuit to be a fly in the ointment of a clear random
distribution of mutations, selected for by various natural factors (not
merely survival of the fleetist of foot but, also, survival as "screened" by
an enormous number of other factors... predation, disease, not having the
area one had to run on cut off from a main continental shelf and sink below
sea level, and at least a hundred more...

.... is that if most mutations are deleterious... then what are the odds that
anything -- even POST HOC capable of appearing to be linear or ratable over
time -- occur *AT ALL* ?

"Most" is not the same as "all".

For one thing, if a creature were experiencing *RANDOM MUTATIONS*, then it
is unlikely that it would have progressed from five toes being used on each
foot, to three in back and four in front, etc., etc., etc., to a hoof.
Random would not be simply foot mutations. It would be mutations ANYWHERE
in the morphology of descendants. How could so many *RANDOM* mutations have
a focus on the feet?

If you play the lottery, it's unlikely that you will win. BUT that does

Now, mind you, I am not saying they COULDN'T. My point is to say that the
statistical odds would seem to me to be enormously against it.

My current intuition (which could be totally wrong) is that there are
numberous assumptions made by biologists and paleontologits on basis of a
simplistic view of "natural selection" which has never (so far) been borne
out at the DNA level. The details are BEGINNING to be mapped, which is
wonderful; but molecular biology impresses me as being in its infancy.
My intuition tells me there had to have been something more going on than
*RANDOM* mutations and *selection by external influences, only*. (And rest
assured I am not even THINKING about any GDI (Gd did it) scenario any MORE
than I am thinking about any NSDI (natural selection did it) scenario.
NEITHER fills in the blanks as to what mechanisms, in detail, enabled a
sufficient number of
*FOCUSED* mutations and *FOC8UED* selection events even to enable the
development of one or more four-hoofed species.
more stuff snipped