Question on Effective Population Size

henry harpending hxh5 at psu.edu
Thu Nov 30 22:22:36 EST 1995

We want the variance of a gene frequency in the offspring to look like

x(1-x)/2N_e

where x is the gene frequency in the parents.  This offspring gene frequency
is the sum of the frequency in  gametes from parental males, with variance
x(1-x)/2N_m, and of the frequency in gametes from parental females, with
variance x(1-x)/2N_f.  Add these, divide by two, and you get the expression
given by Li and Graur.  Note the assumption that the parental frequency is
the same in both sexes.

Henry Harpending, Penn State University

In message <bacon-2111951748120001 at slip-32-9.ots.utexas.edu> -
bacon at arlut.utexas.edu (Fred Bacon) writes:
:>
:>Recently, in a different newsgroup, the textbook _Fundamentals of
:>Molecular Evolution_ (by Wen-Hsiung Li and Dan Graur) was recommended as a
:>good introduction to molecular evolution.  Since I am a physicist and not
:>a biologist, my background in the subject is weak, but I have a personal
:>interest in the topic.
:>
:>To come to my point, I am having trouble understanding a formula presented
:>in the second chapter of FoME (page 30 in my copy).  The authors are
:>discussing the concept of effective population size, denoted N_e.  They
:>offer a formula to calculate N_e when the number of males and females in a
:>populations is different.
:>
:>                      4 N_m N_f
:>              N_e = -------------
:>                      N_m + N_f
:>
:>My problem is that they offer no argument for the validity of this
:>equation, and they give no reference for where it was first derived.  As a
:>consequence, I can't really understand the formula and what it
:>represents.  Furthermore, it would seem to me that there is not enough
:>freedom in this equation to accurately describe different species with
:>different mating behaviors.  For instance, they present the case in which
:>one breeding male is present in a population while many females are
:>available.  In the limit that N_f -> infinity (or at least really big),
:>the effective population goes to 4 (or so they say).  But this ignores the
:>fact that different species may behave differently.  Shouldn't we expect a
:>different result for swan's, which I am told mate for life, and mountain
:>gorillas which live in "harems" by nature?
:>
:>I admit that it gives the proper answer in the limit that N_m = N_f, but
:>then so will an equation of the form
:>
:>                      4 N_m N_f
:>              N_e = -------------  + (N_f - N_m) P(N_f - N_m)
:>                      N_m + N_f
:>
:>where P(N_f - N_m) is any polynomial in (N_f - N_m).
:>
:>Would some kind biologist either fill in the missing arguments from the
:>textbook or verify that the formula is nonsense?
:>
:>Fred Bacon
:>
:>===================================================================================
:>Most people believe they are thinking
:>when they really rearranging their prejudices.
:>                               ---Edward R. Murrow
:>===================================================================================

Henry Harpending
Penn State Anthropology Department
hxh5 at psu.edu