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Joe Felsenstein joe at GENETICS.WASHINGTON.EDU
Sun Sep 8 15:35:48 EST 1991

Marc Rousell asks:

> >
> >     Prob(fixation)  =  (1 - exp(-4Nsp))/(1-exp(-4Ns))
> >
>      I would like to know more about this equation.  Is it an empirical
> law?  If not, what assumptions go into it?  Can anyone provide us with a
> reference that discusses and/or derives this law with a minimum of
> hocus-pocus?

It is a "diffusion approximation" from theoretical population genetics,
first employed for this purpose by Motoo Kimura in 1957.  His original papers

Kimura, M.  1957.  Some problems of stochastic processes in genetics.
   Annals of Mathematical Statistics  28: 882-901.
Kimura, M.  1962.  On the probability of fixation of mutant genes in a
   population.  Genetics  47: 713-719.

An accessible development will be found in section 8.8 of

Crow, J.F. and M. Kimura  1970.  An Introduction to Population Genetics
   Theory.  Harper and Row, New York (reprinted by Burgess, Minneapolis).

and you could also consult the treatment in

Ewens, W. J.  1979.  Mathematical Population Genetics. Biomathematics, volume
    9.   Springer-Verlag, Berlin.

The assumptions are the standard ones of population genetics: a constant
sized population with discrete generations reproducing according to a
"Wright-Fisher" model.  The parents reproduce an offspring generation of
infinite size by random mating, these offspring undergo a round of
natural selection, then N adults are chosen from the survivors to live to

The diffusion approximation is obtained formally by taking a limit as
N --> infinity and s --> 0 such that Ns = constant.  But whenever checked
numerically these approximations have been found accurate to a quite
amazing degree.  P.A.P. Moran showed in 1960 (I think in two papers in
J. Australian Math Society) that the true fixation probability for
finite N lies between the values of the above formula obtained by using
N and N+1 in the formula.  This means the approximation is VERY accurate.
An earlier approximation by J.B.S. Haldane in 1922 treated the case where
4Ns is large, in which case Prob(fixation) ~= 2s when p = 1/(2N).

I will be happy to provide more details if there are particular concerns.

Joe Felsenstein, Dept. of Genetics, Univ. of Washington, Seattle, WA 98195
 Internet:         joe at genetics.washington.edu     (IP No.
 Bitnet/EARN:      felsenst at uwavm
 UUCP:             ... uw-beaver!evolution.genetics!joe

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