Re: We believe in design

On 5/31/05, Pim van Meurs <pimvanmeurs@yahoo.com> wrote:
>
> Again, that we can envision a 2-dimensional solution much easier than
> higher dimensions may make it harder to accept that neutrality works in
> higher dimensions.

It's harder to accept because the plain maths shows there is a problem. I
don't seem to be getting across this idea of "curse of dimension" which is
so well known to anyone who has worked in modelling of high-dimensional
data. So I'll make one last attempt to spell it out and then give up.

The 2-D diagrams in your link showed a population, via neutral mutations
spreading out like a gas over the 2D search space of equivalent phenotypes,
so that there were a number of differrent individuals that were poised very
near quite disparate regions of viable phenotypes, ready to adapt to
different changes in the environment. The geometry was complex; lots of
curved regions. To make it simple, let's assume a circular region of
interest (adaptation) and the neutral network being a square around the
circle. I don't think this simplification loses generality because I just
want to examine how the relative volumes change with increasing dimension. I
think it's easy to see that a unit circle has an area equal to pi, and the
enclosing square has an area equal to 4. Hence the chance that any randomly
chosen point lies in the circle is pi/4 or about 0.785. A pretty good
chance.

Now to see how this generalises to high dimension, we need the formula for
the volume of a unit hypersphere in N dimensions. For simplicity we take the
formula for even N (odd-N is a different formula unless you want to use the
gamma function which is a generalization of the factorial function). The
volume of an N-dimensional hypersphere is given by:

(pi/2)^(n/2) / (n/2)!

The volume of the enclosing hypercube is 2^n.

Working out this ratio for n=2 gives pi/4.
n=10 gives around 0.00249 (around 1/401)
n=100 gives around 1.87e-70 (around 1 in 5x10^69)
n=200 gives around 3.5e-169
n=1000 gives excel an overflow error ! :-)

As dimension of search space increases it becomes vanishingly unlikely that
you can find a region of interest via a random walk (which is all you can
manage with a set of neutral mutations which can't give selective
advantage). What scuppers it is the (n/2)! term.

That's why I personally don't think neutrality is the answer. In my opinion,
of more interest is looking at the gene regulation capabilities of the
intronic DNA. There was a fascinating article in Scientific American a few
months back which suggested that the so-called "junk" DNA may have been
pivotal in evolution. It showed a graph of the complexity of organism vs the
amount of junk DNA showing a monotonic increase in it. It suggested that we
don't yet know the function of the non-coding DNA, but this would be a
fruitful area of research.

Neutrality surely can lead to appearance of stasis while the genotype
> 'diffuses' until it reaches new possibilities. Stasis followed by rapid
> evolution (punctuated equilibria like...) is what we also observe in the
> fossil record for instance or even in simulation runs. There are some
> good examples for RNA space.

Yes, I've seen punctuated equlibria in simulation runs as well, while
waiting for the right mutation to occur, but thse weren't down to neutrality
as there wasn't any capability of neutrality. You also see punctuated
equilibria in the fossil record, but how do you know it's down to
neutrality?

Iain
Received on Tue May 31 16:55:22 2005