We use the probability of identity by descent
(*F _{ST}*) in subpopulations relative to the total metapopulation
as our measure of the progress of genetic drift. The probability of identity
by descent from a previous generation (

However, in the island model, we assume
that migration brings in entirely unrelated genes from infinitely many
other islands or from an infinitely large mainland, with probability *m*.
Thus, the chance that any allele in a particular generation has not immigrated
is (1-*m*), and the probability that neither of the two alleles chosen
have immigrated in that generation is (1-*m*)^{2}. Thus, after
accounting for immigration, the overall probability of identity by descent
at time *t* is:

It is fairly clear that the combined effects
of drift and migration should tend to some sort of equilibrium value of
*F*;
if *N* and *m *are small, drift outweighs migration, and *F*
will be nearly 1; if *m* and *N *are large, migration will homogenize
gene frequencies faster than drift, then
*F* will be nearly zero.
Intermediate values of
*N* and *m* should give intermediate levels
of *F*.

At equilibrium, , so:

At the risk of boring the algebraically unchallenged, here are a number of steps of simplification of this formula to give the result in the lecture on population structure:

(OK, OK, I made a number of steps there, but you can handle it!!)

Now we make an approximation: that, if
*m*
is small, say 0.1 or less, then terms in *m*^{2} are very
small, nearly zero, and can be ignored. In other words, (1-*m*)^{2}~1-2*m*.
Then:

Now, provided that 2*m* is small relative
to *Nm* and 1, we have the approximate relationship:

*Nm* is interesting; because as it
is the fraction of the population x the number within that population,
it represents the "number of migrants" into a population. For this reason,
some people have taken to viewing estimates of *Nm* as measures of
"gene flow".

*A problem:
why *Nm* is not "gene flow."*

In fact *Nm* does not measure the
sort of gene flow we want at all; this is given by *m*. For instance,
if we are interested in the homogenizing effect of actual gene flow that
works *against* diversity-producing processes like disruptive selection
or drift, then we are interested in
*m*. All
*Nm* really measures
is the "tendency to produce *F _{ST}*" under random drift,
provided the rather unrealistic assumptions of Wright's island model are
met.

*Another problem:
the island model assumptions may not be met*

Another problem is that, in many real
situations, the assumptions of the island model are usually *not*
met. For example, some people have used
*Nm* to measure "gene flow",
for example between populations/subspecies of butterflies on real islands
in the Caribbean. But here we must make the assumption that the islands
are at equilibrium between drift and migration. This assumption is dubious
because the islands may in fact be slowly diverging since colonization,
and there may be no migration at all. In other words drift and migration
are not at equilibrium. (For a good review of this problem, see Michael
C. Whitlock & David E. McCauley 1999 Indirect measures of gene flow
and migration: .
Heredity 82: 117-126).

Even worse, some people have assumed that
one can measure "gene flow" between genetically rather divergent species
or geographic races that do or do not hybridize. Here, it is likely that
selection (or a complete absence of genetic contact) is keeping the two
populations apart; so again, the assumption that the populations are at
drift/migration equilibrium is simply wrong.

Wright's equation is one of the most influential theories in the history of population genetics, and rightly so. But it must be used with care.

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© Jim Mallet 1999