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% Sex differences in coalescent times (a theoretical model)
% The nature of selection and coalescent differences among parts of the genome
{\center\bf Introduction \\}
Although the default model for population genetics is a panmictic haploid
mon\oe cious population, humans and other species are diploid and di\oe cious,
which allows several population genetic structures to simultaneously occur in
a population. In particular, the dynamics of autosomes, $y$-chromosomes, and
mitochondrial DNA may be rather different. We consider the effect that
selection acting on different portions of the population (male, female, or
both) based on various causes (cultural, autosomes, $y$-chromosomes, or
mitochondrial DNA) has on the population dynamics of autosomes,
$y$-chromosomes,
and mitochondrial DNA.
%The average coalescent time for the entire population (which is equal to the
%average fixation time) may be different for genes on autosomes, genes on the
%y-chromosome, and mitochondrial DNA. Indeed, the population structures
%governing y-chromosomes and mitochondrial DNA may be quite different, with no
%impact on mitochondrial DNA and y-chromosomes, respectively. But selection or
%other forces directed at one portion of the genome may impact the coalescent
%time at other portions.
\newpage
{\center\bf The Model \\}
The selection model employed here is a fertility selection model which
measures selection as an autocorrelation of the form $b \times r^n$ between
the numbers of progeny of lineal descendants $n$ generations apart. This could
be due to cultural selection in which case $b$ and $r$ are only constrained to
be less than or equal to $1$, or quantitative selection on the entire genome,
in which case $r$ would be $.5$ and $b$ would be the heritability of
fertility. This selection may not be due to genes at specific loci, but it is
a form of background selection which will affect the fixation and coalesent
times in the population. In particular, such selection divides the coalescent
time which would occur in a selectively neutral panmictic population by a
factor of $(1-b) + b \times \frac{1+r}{1-r}$. Random mating is assumed, except
for the possibility of permanent pair mating.
\newpage
{\center\bf Selection on All Individuals \\}
If the autocorrelation $ b \times r^n$ applies to all individuals, then it
will govern lineages containing autosomes and $y$-chromosomes or mitochondria
alike. It does not matter whether the autocorrelation has a cultural or
(autosomal) genetic basis. (If the autocorrelation is a result of averaging
over various components of the population, the population genetics of those
components might be different.) It should be noted that, since the
(effective) population size for $y$-chromosomes and mitochondria will be one
fourth that of autosomes, coalescent times and other quantities will vary
accordingly.
If selection occurs on the entire population, but is based on cytoplasmic DNA,
then transmission will be only from females, but to both sexes. The
autocorrelation along male lineages will of course be zero, and we may let the
correlation along female lineages be $b \times r^n$. Then the autocorrelation
for the population as a whole will be $(\frac{1}{2})^n \times b \times r^n$
because half of the lineages terminate with a male each generation, which
terminates correlation.
One cannot have selection on the entire population based on the $y$-chromosome
unless there is faithful pair mating. In that case an autocorrelation of
$b \times r^n$ along male lineages will be associated with no autocorrelation
along female lineages, and an autocorrelation of $(\frac{1}{2})^n \times b
\times r^n$ for the population as a whole. (The correlation is reduced by
one-half each generation because there is no transmission to females.) The
previous paragraph has transmission to both sexes, but from females only;
this paragraph has transmission from both sexes, but to males only.
\newpage
{\center\bf Selection on Males \\}
If selection acts on males, whether cultural or based on the $y$-chromosome,
the result will be the same. If the autocorrelation is $b \times r^n$ along
male lineages, that correlation will be lost as soon as autosomes leave the
male lineage, hence the autocorrelation for autosomes for the entire
population will be $\frac{1}{2} \times (\frac{1}{2})^n \times b \times r^n$.
This reflects that initially only one fourth of the possible relations are
male-male, and only one-half of the autosomes remain in male lineages each
generation. If there is faithful pair mating, the correlation will hold
between mothers and sons too, hence the contribution to correlation from the
first generation will be double, and the autocorrelation will be $
(\frac{1}{2})^n \times b \times r^n$.
If the selection which provides an autocorrelation $b \times r^n$ along
male lineages is based on autosomes, the genetic material will be
transmitted through females. One-quarter of all the pairings between
generations in the population as a whole will begin and terminate with males,
hence the autocorrelation for the population as a whole will be
$\frac{1}{4} \times b \times r^n$. If there is faithful pair mating, the
correlation will also be with the ancestral mother (even though there is no
genetic basis), and the autocorrelation will be $\frac{1}{2} \times b \times
r^n$.
If selection on males is based on cytoplasmic DNA, there will be no
autocorrelation, hence no selection.
Of course, there will be no autocorrelation of female (cytoplasmic DNA)
lineages in either case.
The cases of selection acting only on females can be treated analogously.
\newpage
{\center\bf Summary \\}
A diploid di\oe cious population contains several subpopulations (specifically
autosomes, $y$-chromosomes, and mitochondrial DNA) which may have different
population structure, even if the population is panmictic. Both the portion
of the population subject to selection (males, females, or both) and the
nature of the selection (cultural, autosomal, $y$-chromosome, or cytoplasmic
DNA based) will impact on the amount
of selection in the various components of the population, hence the coalescent
times within those subpopulations.
Fertility selection (autocorrelation of progeny number between generations)
has generally been measured between mothers and daughters. This may be larger
or smaller than the selection on the population as a whole, and bear little
relation to the selection along male lineages. Although an autocorrelation
of $ b \times r^n$ in female lineages could correspond to the same
autocorrelation in the population as a whole, it could also correspond to
$(\frac{1}{2})^n \times b \times r^n$ if selection acts on cytoplasmic DNA for
the entire population; if selection acts on autosomes, but only in females,
the autocorrelation for the entire population will be $\frac{1}{4} \times b
\times r^n$ if the autocorrelation along female lineages is $ b \times r^n$.
If $b = r = .5$, these would result in the random mating
coalescent time for the whole population being divided by 2, $\frac{4}{3}$,
and $\frac{5}{4}$, respectively, providing in the latter two cases larger
coalescent times than the autocorrelation of female lineages would predict.
If selection acts on males, there will be no autocorrelation along female
lineages, but positive autocorrelation for the population as a whole resulting
in a reduced coalescent time for the population as a whole relative to no
autocorrelation.
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