Re: Percolation theory


<tchow@xxxxxxxxxxxxx> wrote in message
news:gpojgg$1f1m$1@xxxxxxxxxxxxxxxxxxxxxx
I am a mathematician and don't know a whole lot about biology.

I am a mathematician who has been learning biology recently. My first
impression is that you have a nice mathematical model and you are looking
for something to do with it. I don't think that is a good way of doing
biology (or science in general). Nevertheless...

[...]
Now for the vague idea of how this might be relevant to evolutionary
biology. In the simplest model, the points of the graph represent
genotypes. Edges represent mutations that could occur in a single
generation. The survival probability represents something like the
probability that one genotype will produce a viable organism, given
that the genotype at the other end of the edge produces a viable
organism. Finally, the phase transition would indicate what rate of
viable mutations are needed for the genotype to evolve to arbitrarily
distant genotypes, as opposed to getting "stuck" in some island and
being unable to evolve beyond certain limits.

Genotype for an individual or a set of similar genotypes in an interbreeding
population? I think the latter makes most sense, and will assume that.

I'd suggest you learn some populaton genetics. There's a good book available
online here

http://evolution.genetics.washington.edu/pgbook/pgbook.html

I'd recommend Chapters 1,2,3,6,7 for starters. You may then have a better
idea of how to relate your p to something biological. (I don't think 'viable
mutations' is good enough.)

Also look at work on 'fitness landscapes'. You could start here.

http://en.wikipedia.org/wiki/Fitness_landscape

It seems to me you need more structure than just connected components.

You might find this article interesting.

http://www.plosone.org/article/info:doi%2F10.1371%2Fjournal.pone.0002757

Roughly speaking this attempts to model the tree of life as if it were
trying to 'supply' all available niches with species, in analogy with the
way blood vessels supply all parts of the body. It seems to make more sense
to ask 'how efficiently can a high-dimensional space be explored by
such-and-such an algorithm?' than to ask 'can this region be reached at
all?'

Graham
www.indriid.com



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