Re: Percolation theory
- From: "Perplexed in Peoria" <jimmenegay@xxxxxxxxxxxxx>
- Date: Wed, 18 Mar 2009 12:31:23 -0500 (EST)
<tchow@xxxxxxxxxxxxx> wrote in message news:gpojgg$1f1m$1@xxxxxxxxxxxxxxxxxxxxxx
I am a mathematician and don't know a whole lot about biology. Recently I
had a vague but seemingly (to me at least) promising idea about the possible
application of percolation theory to evolutionary biology. I thought I'd
post it here to see if the idea is worth anything, and if so, whether it
has already been pursued.
I'll start by describing a mathematical result. Let Z^n denote the set of
points in n-dimensional space with integer coordinates. Connect two points
in Z^n with an edge if the distance between them is 1 (i.e., the two points
differ by 1 in some coordinate and are equal in all other coordinates).
This gives us an infinite graph, which we shall also call Z^n.
Now fix some number 0 < p < 1, which we call the "survival probability."
We consider the edges in the graph Z^n one at a time and, with probability p,
allow an edge to survive; with probability 1-p we delete the edge. The edge
survivals are all probabilistically independent.
The mathematical result is that for sufficiently large values of p, the
graph will, with probability 1, have a single "giant connected component."
As p decreases, however, it reaches a critical value---a "phase transition"
---below which the graph disintegrates into "dust" (i.e., a whole lot of
small components that are disconnected from each other).
What I've just described for Z^n holds for a variety of other random graph
models. The result is remarkably robust. Under a wide variety of
conditions we observe a sharp phase transition from dust to a giant
component.
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.
Obviously, the model is very crude, but this doesn't worry me all that
much since as I said, the results of percolation theory are rather
robust. What I'm wondering is whether this line of thinking seems
worth pursuing, and if so, what the next step should be.
For some prior applications of percolation theory to biology, see the
recent book by Sergey Gavrilets - "Fitness Landscapes and the
Origin of Species". There you will also find many references to
papers on percolation, many of which apply the topic to biology.
It seems to me that all such efforts suffer from one major flaw. No one
knows how well random graphs model fitness landscapes in DNA
sequence space. Another flaw is that the existence of a path
connecting two points in sequence space is not a guarantee that
evolution could follow that path. The probability that a member of a
population find a path through a maze depends upon the population
size and the time available. Both are finite. And it also depends
inversely upon the number of false trails. Which depends (exponentially)
upon the dimensionality of the search space. Which is huge.
.
- References:
- Percolation theory
- From: tchow
- Percolation theory
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