Path: utzoo!utgpu!watserv1!watmath!uunet!bionet!ARSUN.ANTHRO.UTAH.EDU!rogers
From: rogers@ARSUN.ANTHRO.UTAH.EDU (Alan R. Rogers)
Newsgroups: bionet.population-bio
Subject: Re: Estimation of Gene Flow
Message-ID: <9010191643.AA03866@arsun.anthro.utah.edu>
Date: 19 Oct 90 16:43:20 GMT
Sender: daemon@genbank.bio.net
Lines: 42

Kent Holsinger and Joe Felsenstein have commented on the difficulty of 
estimating gene flow from genetic data when equilibrium assumptions are not
satisfied.  Here is a suggestion that may work if you are willing to buy
the assumptions.  Let x denote a vector containing the frequencies of some
allele among newborns (or larvae, or whatever) in a variety of places, and
let y denote the corresponding vector for adults.  Even when the system is
far from equilibrium, we can write

     y = M x + e

where M is an unknown migration matrix, and e a vector of random displacements
due to genetic drift.  This formulation implies that population
regulation occurs after migration, i.e. that there is initially a large
number of juveniles who then migrate and whose number is then reduced to Ni
in group i.  If population regulation is mainly prior to migration, a
different formulation is needed.  Anyway, writing Ci for the covariance
matrix of vector i, and using "'" to denote matrix transpose, we have

     Cy = M Cx M' + Ce                                            (1)

where Ce is approximately diag[p(1-p)(1-Fst)/2Ni], p is the mean juvenile
allele frequency, and Fst is Var(juvenile allele freq)/p(1 - p).  If you know
the effective population sizes, Ni, then Cy, Cx, and Ce can all be estimated.
What we don't know, but would like to know, is the contents of M.
Unfortunately, as it stands, the system of equations that (1) represents has
more unknowns than equations.  If there are K groups, then there are K(K +
1)/2 equations (since the covariance matrices are symmetric) in K(K - 1)
unknowns (since the rows of M must sum to 1), so there are K(K - 1)/2 more
unknowns than equations.  You have to assume something about M to make any
progress.  But in some cases, perhaps that will be possible.  If you assume,
for example, that migration is symmetric, the number migrating from i to j
being the same as the number migrating from j to i, then (1) is no longer
underdetermined, and could presumably be solved.  Even better, if some
variant of a gravity model makes sense, or if you can build in information
about ocean currents, you may end up with a few extra degrees of freedom.
Thus, I see no reason why migration could not be estimated from genetic data
even where no equilibrium assumption is warranted.

Alan Rogers
 INTERNET: rogers@anthro.utah.edu
 USMAIL  : Dept. of Anthropology, Univ. of Utah, S.L.C., UT 84112
 PHONE   : (801) 581-5529