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From: joe@evolution.u.washington.edu (Joe Felsenstein)
Newsgroups: sci.bio
Subject: Re: Eve
Summary: Wright-Fisher model
Keywords: Wright, Fisher, model, eve, population genetics
Message-ID: <15110@milton.u.washington.edu>
Date: 25 Jan 91 06:40:10 GMT
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The Wright-Fisher model (named after Sewall Wright and Ronald Fisher,
the two most important population geneticists ever), is a model in which
each gene in the offspring generation is copied from a random gene among
the 2N available in the parent generation, and this occurs randomly and
with replacement.  This is motivated by the idea that each parent
produces a very large and equal number of gametes, and the N surviving
offspring are the result of random collisions of these gametes, followed by
density-dependence trimming them randomly down to N survivors.

The haploid version, in which N things are randomly drawn from N, is the
one relevant to the Eve question, with N the number of females.  The
mathematics of the two is identical with 2N replaced by N.

You will find it discussed in almost any population genetics text, but a
reasonable starting point would be Dan Hartl and Andrew Clark's
"Principles of Population Genetics," second edition, published by
Sinauer Associates, 1988.  The W-F model dates from 1930-1931.

If you want something more mathematical that could be made to bear on
the Eve issue (or non-issue) try my 1971 paper, "The rate of loss of
multiple alleles in finite haploid populations" in Theoretical Population
Biology, volume 2, pages 391-403.

To (very) good approximation, if we have n females who we are
following in a W-F model where there are a total of N females in each
generation, the time in generations back until they have n-1 female parents is
approximately geometrically distributed with mean  N/(n(n-1)).
Applying this repeatedly starting with N=n and with n getting smaller and
smaller until it reaches 1 you get the required distribution, and can easily
calculate its mean and variance.  This is only an approximation but is a very
close one for large N and darn good even for moderate N.

(Asides: I have not posted this to soc.men or soc.women because it seemed more
appropriate here.  Also: Fisher was English, not German, and thus had no "c" in
his last name.)

-----
Joe Felsenstein, Dept. of Genetics, Univ. of Washington, Seattle, WA 98195
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