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%Coalescent time in the presence of inbreeding
\noindent{\bf Introduction.}
The genetic structure of a population depends on the ancestry of the genes in
the population, in particular the extent to which they have common
ancestry. The entire ancestral pedigree of all the genes in a population is
called the coalescent. Sometimes the term coalescent is used to refer to the
time since a common ancestor of the entire population, or the time since a
common ancestor of the two genes in an individual. The coalescent is well
defined for all populations, for all of the above definitions.
Kingman (1982) presented the first analysis of the coalescent. He determined
the distribution of the number of ancestral genes of the present population as
a function of time for a random mating population with a Poisson progeny
distribution. The distribution of the number of progeny is not critical to
his analysis, but random mating is. Subsequent studies of the coalescent have
incorporated selection (Kaplan, Darden, and Hudson 1988) and migration
(Takahata 1988, Slatkin 1991), but random mating has remained an essential
assumption of all analyses.
Several methods have been used to model nonrandom mating, some based on
phenotype (assortative mating), and others based on ancestry (inbreeding).
Inbreeding has been modelled with finite pedigrees (Thompson 1986), regular
systems of inbreeding (Wright 1921), and inbreeding coefficients (Cockerham
1970). Since finite pedigrees generally do not extend back to a common
ancestor, regular systems of inbreeding and inbreeding coefficients are
employed here.
The time since a common ancestor of the entire population is conceptually
essentially the same as the time until fixation of a gene, and in the case of
a random mating population with Poisson progeny distribution it is the same.
This allows us to use the diffusion approximation (Maruyama \& Kimura 1971)
to approximate the time since a common ancestor for the entire population when
inbreeding (e.g., selfing) is superposed on a random mating background. In
particular, in the absence of selection the time until fixation of a mutant
gene (conditioned on fixation) is given by
$$ \int_{\frac{1}{2N}}^{1} \frac{2\xi (1-\xi)}{V_{\delta \xi}} d \xi +
\frac{1 - \frac{1}{2N}}{\frac{1}{2N}} \int_{0}^{\frac{1}{2N}} \frac{2
\xi^{2}}{V_{\delta \xi}} d \xi $$
(Crow \& Kimura 1970, p.430), where $V_{\delta p}$ is the sampling variance of
the change in allele frequencies. Other methods are employed to study the
coalescent under regular systems of inbreeding.
{\bf N.B.:} Autozygosity is used below in the sense of identity by descent.
The Poisson progeny distribution is actually a binomial progeny distribution
since the population size is constant; the difference between the Poisson and
binomial progeny distributions will not significantly affect the results.
\newpage
\noindent{\bf Poisson progeny distribution.}
For random mating with a Poisson progeny distribution $V_{\delta p} = p(1-
p)/(2N)$, which produces the well known result that the expected time until
fixation of a mutant gene is $4N$ generations, the same as the coalescent time
found by Kingman.
If a fraction $s$ of the population selfs while the rest of the population
randomly mates (allowing the possibility of selfing in the random mating
fraction), $V_{\delta p} = (p(1-p)/2N)(1+ (\frac{s}{2}/(1 - \frac{s}{2})))$.
Hence the time until fixation is $4N/(1+(\frac{s}{2}/(1 - \frac{s}{2})))$. In
particular, if the population is entirely selfing ($s=1$), the time until
fixation is the $2N$, the same as for a population of half the size
(heuristically, autozygosity is soon obtained, after which both genes in an
individual are sampled as a unit).
The quantity $ (\frac{s}{2}/(1 - \frac{s}{2})))$ is equal to the asymptotic
probability of autozygosity in an infinite population which selfs $s$ of the
time. Denoting with $f$ the probability of autozygosity due to the mating
structure (as opposed to finite population size), it can be shown that if
inbreeding is superposed on a random mating background with Poisson progeny
distribution, the time until fixation is $4N/(1 + f)$.
\\
\noindent {\bf Two progeny per individual.} When the population
is constrained to two progeny per individual (one in the case of selfing), the
sampling variance of the change in allele frequencies is $V_{\delta p} =
p(1-p)/(4N)$ for a random mating population. This provides that the expected
time until fixation is $8N$ generations, twice the value for a population with
a Poisson progeny distribution.
If selfing with frequency $s$ is superposed on the random mating population
(which allows selfing), $V_{\delta p}$ becomes $(p(1-p)/(4N))(1 -
(\frac{s}{2}/(1-\frac{s}{2})))$. This provides that the time until fixation
is $8N/(1 - (\frac{s}{2}/(1-\frac{s}{2})))$. In the case of obligate selfing
($s = 1$), the time until fixation is infinite, which is obvious since a gene
cannot spread beyond a single lineage. Nonintersection of lineages also
assures that the time since a common ancestor is infinite. For values of $s$
less than one, the time until fixation increases with more selfing, which is
opposite to the effect found in the case of Poisson progeny distribution.
Analogous to the case of Poisson progeny distribution, $(\frac{s}{2})/(1-
\frac{s}{2})$ is the asymptotic probability of autozygosity. It can
be shown that the expected time until fixation is $8N/(1-f)$ for any mating
structure superposed on random mating with two progeny per individual.
\newpage
\noindent {\bf Regular Systems of Inbreeding} The regular systems of
inbreeding considered here have two progeny per individual, and half-sib and
circular pair mating have asymptotic probability of autozygosity equal to one
in an infinite population (the model of maximum avoidance of inbreeding is not
defined for infinite populations). However, they are not superposed on a
background of random mating, so the previous analysis is not valid. Half-sib,
circular pair, and maximum avoidance of inbreeding (Kimura \& Crow 1963) all
have the expected time since a common ancestor of the two genes in an
individual equals $4N-2$, the same value as for a random mating population
constrained to two progeny per individual.
(Another measure of inbreeding is the amount of genetic identity within
(homozygosity) and between individuals at equilibrium with mutation. These
values depend on the mutation rate. Campbell (1993) found numerically that
genetic identity was similar for random mating and avoidance of inbreeding,
but there is substantially more homozygosity and less identity between
individuals under half-sib and circular pair mating. Hence the
time since a common ancestor of two genes in an individual of $4N-2$
generations does not reflect that the populations have the same genetic
structure.)
The difference equation analyses of the above regular systems of inbreeding
characterize each pair of genes by their distance in the pedigree (i.e., the
time since a common ancestor of the individuals which contain the genes).
Under avoidance of inbreeding, the time since a common ancestor of two genes
in an individual ($4N-2$) is the largest time for any pair of genes. For
half-sib mating and circular pair mating, $4N-2$ is essentially the shortest
time among all pairs of genes (pairs of genes in half-sibs and sibs have a
slightly shorter expected time since a common ancestor). Of course, with
random mating the expected time since a common ancestor is the same for all
pairs of genes. Although the coalescent time for the entire population has
not been calculated, this suggests that it should be greater for half-sib and
circular pair mating than for random mating than for maximum avoidance of
inbreeding.
\newpage
\noindent {\bf Discussion.} The preceding preliminary results provide
initial insight into the effect of inbreeding on the coalescent. It is
demonstrated that inbreeding can either increase or decrease the coalescent
time depending on the progeny distribution. (This is related to the fact that
selfing decreases both heterozygosity and the number of segregating alleles
with a Poisson progeny distribution, but decreases heterozygosity and
increases the number of segregating alleles when there are two progeny per
individual.) The study of regular systems of inbreeding shows that the time
since a common ancestor of two genes in an individual does not indicate what
the coalescent time for the entire population is (although it does provide a
lower bound). But the above results raise more questions than they provide
answers.
The diffusion approximation raises the question: exactly what is
the difference between the time until fixation of a mutant gene and the time
since a common ancestor of the entire population; are they equivalent? Since
the inbreeding occurs in the diffusion equations in a factor which is
independent of allele frequencies, is the structure of the coalescent
unchanged, and only the time scale upon which it is measured altered? Since
inbreeding reduces the coalescent time for the Poisson progeny distribution
but increases the coalescent time with two progeny per individual, is there a
progeny distribution for which the coalescent is unaffected by inbreeding;
what is that progeny distribution?
For regular systems of inbreeding, the basic question of calculating the
coalescent time remains. Then the structure of the coalescent must be
investigated for the above question of whether only the time scale and not
the structure of the coalescent is affected by inbreeding. Also, regular
versus random sib mating, first cousin mating, etc. should be juxtaposed.
\vspace{10pt}\\
\begin{small}
Literature cited:\\
Campbell, R. B. 1993. {\it Th. Pop. Biol. 43\/}:129--140.\\
Cockerham, C. C. 1970. pp.104-127 in K. Kojima, ed. {\it Math. Tpcs. in
Popul.
Genet.} \\
Crow, J. F. \& M. Kimura. 1970. {\it Intro. Pop. Gen. Th.}\\
Kaplan, N. L., T. Darden, \& R. R. Hudson. 1988. {\it Genetics
120\/}:819-829.\\
Kimura M. \& J. F. Crow. 1963. {\it Genet. Res. 4\/}:399--415.\\
Kingman, J. F. C. 1982. {\it J. Appl. Prob. 19A\/}:27-43. \\
Maruyama, T. \& M. Kimura. 1971. {\it Jap. J. Genet. 46\/}:407-410.\\
Slatkin, M. 1991. {\it Genet. Res. 58\/}:167-175.\\
Takahata, N. 1988. {\it Genet. Res. 52\/}:1213-222.\\
Thompson, E. A. 1986. {\it Pedigree Anal. in Hum. Genet.}.\\
Wright, S. 1921. {\it Genetics 6\/}:111-178.\\
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