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Correlations between relatives in small populations.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 71:377-380 (1986)
Correlations Between Relatives in Small Populations
ALAN R. ROGERS
Department of Anthropology, University of Pittsburgh,
Pittsburgh, Pennsylvania 15260
K E Y WORDS
Population genetics, Altruism
ABSTRACT
Correlations between relatives in small, closed populations
can be substantially smaller than predicted by the classical formulas of population genetics. This effect is especially pronounced for relatives whose most
recent common ancestor is several generations removed. When the effective
population size is small, correlations between even close relatives can be
negative. This implies that in small populations conventional estimators of
quantitative genetics parameters will be biased and that preferential treatment of close relatives will be less likely to evolve.
Correlations between relatives measure the
extent to which relatives tend to be more
similar than random individuals drawn from
some reference population. Classical population genetics theory predicts genetic correlations relative to a hypothetical infinite
foundation stock. Unfortunately, in many
contexts within agricultural genetics and evolutionary biology, the relevant reference
population is small. Although it is often noted
that these correlations are not equivalent
(Wright, 1969; Canning and Thompson,
1981), the distinction between them is usually ignored.
Correlations relative to a finite foundation
stock are smaller than predicted by classical
models, and correlations relative to the CUTrent generation are smaller still. In this paper, I investigate the magnitude of these
differences and discuss their implications
for quantitative genetics and evolutionary
biology.
CORRELATIONS RELATIVE TO A FINITE
ANCESTRAL POPULATION
Correlations between relatives are frequently predicted from pedigrees, using the
formulas of Wright (1969) and Malecot (1969).
As Jacquard (1975) has emphasized, these
correlations measure similarity of individuals relative to the earliest generation in the
pedigree (say generation 0). It is less widely
appreciated, however, that these formulas are
only approximate unless the population of
generation 0 is finite.
Consider two genes drawn a t random from
individuals, X and Y , in generation t in a
population of diploid individuals with dis-
0 1986 ALAN R. LISS, INC.
crete generations and constant size N . Let x
equal unity if the gene from X is a copy of
allele A l , or zero otherwise, and define y similarly for the gene drawn from Y . The correlation between x and y relative to generation
i is defined by
where piisthe frequency ofAi in generation i.
r; can be expressed in terms of &, the probability that genes drawn from individuals X
and Y are “identical by descent” from generation i, i.e., that both genes are copies of a
single gene inherited from a common ancestor in that generation. The possibility that X
and Y may be identical by descent from some
more remote generation does not contribute
to this probability.
For the moment, we are interested in ro,
the correlation relative to the earliest generation in the pedigree. In calculating it, three
possibilities need to be considered: The genes
drawn from X and Y may be (a)copies of the
same gene in generation 0, (b)copies of distinct genes from a common ancestor in generation 0, or (c) copies of genes from distinct
individuals in generation 0. ro is the average
of the correlations implied by (a), (b),and (c),
weighted by the corresponding probabilities,
which are 40, 40, and 1-240, respectively.
Putting all this together, we have
ro
=
40
+ +oro(b) + (1 - 24o)ro(~),
Received October 14, 1985;revision accepted May 27, 1986
378
A.R. ROGERS
where rkh)and r$c) are the correlations im- ents relative to the pool of gametes is thereplied by (b)and (c). To simplify matters, I will fore
assume that mating is at random, implying
that rJb) = r$) = r*, where r* is the average
correlation between distinct genes is generation 0, relative to generation 0. Thus
which is differs from the formula for correlations relative to generation 0 only in that N
is replaced by N,. Thus changing N to N, in
equation 2 produces a n expression for the
which is analogous to Crow’s (1980)equation correlation of genes drawn from X and Y
4.In Malecot’s theory, genes are taken to be relative to the gamete pool produced by genindependent if they are not identical by de- eration 0.
scent, yet distinct genes are not independent
in finite populations. In a popualtion of size
CORRELATIONS RELATIVE TO THE
N , the average correlation between distinct
CURRENT GENERATION
genes, relative to their own generation, is r*
Frequently, we are interested in rt, the cor= - l/(2N - 1).Substituting produces
relation relative to the current generation, t.
Equation 1 implies that
2N40 - 1
ro =
~ { x y =} PO ( 1 - p0)r-o + p20.
2N-1‘
When N is large, this reduces to ro = 40,the
familiar large-population result. Note that if
$0 < l / 2 N , ro is negative. Correlations between genes (relative to generation 0) are
negative if the genes are less likely to be
copies of the same gene in generation 0 than
two genes drawn with replacement from that
generation.
Although I have made ro a correlation relative to the population of generation 0, it
may be more useful to,use the correlation
relative to the pool of gametes produced by
that generation. Let l/2Ne be the probability
that two distinct gametes produced by generation 0 are copies of the same parental
gene. The probability that a gene drawn from
this gamete pool is A1 is PO.The probability
that a second gene is a copy of A l , but not a
copy of the same parental gene, is PO- 1/2N,.
If we condition on the fact that the genes are
derived from different parental genes, the
probability becomes
Substituting this back into equation 1
produces
rt
=
1
-
PO(1 - Po) - Pt - Po
( 1 - ro)
Pt(1 - PA’
PtU - Pt)
In the absence of selection, migration, and
mutation, the expectation of pt is PO, and
(Crow and Kimura, 1970:109).Taking a ratio
of expectations as a n approximation to the
expectation of a ratio, we have
To apply this formula, values are needed
for 40, t, and N,. If the complete pedigree of
X and Y is known back to generation 0, $0
can be obtained using Wright’s (1969:177)
well known formula. The value of t is the
This is the probability that both genes are number of generations separating X and Y
A1 given that the they are copies of different from their common ancestor. For example t
= 1 for sibs, since they share ancestors in
parental genes.
The correlation of genes from distinct par- the previous generation, and t = 2 for cou-
379
CORRELATIONS IN SMALL POPULATIONS
sins, since their common ancestors are
grandparents.
The correlations discussed here are between random genes drawn from two related
individuals. In many applications, interest is
focused instead on “genotypic values” of
these individuals, equal to 2, 1, and 0 for
genotypes AIA1, A1A2, and A2A2, respectively. When mating is at random, correlations between genotypic values are twice the
genic correlations studied here (Crow and Kimura, 1970:138).
DISCUSSION
Equation 3 indicates that rt is a decreasing
function oft, the number of generations separating individuals X and Y from their common ancestor. Thus the discrepancy between
the infinite-population and small-population
models depends on how closely the individuals are related. In the same population, the
infinite-population model may provide a good
approximation to the correlations between
sibs but a poor approximation to correlations
between more distant relatives. This effect is
important when correlations are inferred
from deep pedigrees.
Table 1 shows how correlations between
relatives are affected by effective population
size, N,. For comparison, the last row contains the correlations under the conventional
infinite-population model. Although the values of Ne in Table 1 are small, they are not
so small as to be irrelevant. Ne is about onethird of N in many human population and
would be much smaller in highly polygynous
species or in many breeds of domestic animals. For example, Wright (1951) concluded
that Ne was about 100 for British short-horn
cattle during the nineteenth century. Table
1shows that popualtion size can have a large
effect on correlations between relatives. Cor-
relations in small populations are substantially smaller than those predicted by largepopulation theory, particularly for distant
relatives. These results are accurate only for
populations that are completely isolated. In
other populations, correlations should lie
somewhere between the prediction of equation 3 and that of the infinite-population
model.
Estimates of quantitative genetics parameters are generally based on correlations obtained under the infinite-population model.
The results obtained here suggest that these
estimates will be biased in small populations, especially if based on similarity between distant relatives. Estimates based on
cousins would be biased by several percent
even in populations of effective size 1,000.
This bias should usually be negligible, since
most estimates are based on similarity between first-degree relatives (sibs and parentoffspring pairs) in fairly large populations.
With the advent of maximum-likelihood
methods for estimating quantitative genetics
parameters from deep pedigrees (Bulmer,
1980), however, the bias introduced by population size may become increasingly important.
Hamilton (1964) showed that altruism can
evolve if the benefit (B)to the recipient and
the cost (C)to the altruist satisfy CIB < 24.
In small populations, the 4 should be replaced by rt. Thus, Hamilton’s theory implies
that selection within a population of effective
size 10 should produce spite rather than altruism between cousins. On the other hand,
selection between such groups might favor
indiscriminate altruism toward members of
the group (Wade, 1978) but would not favor
selective altruism based on geneological reckoning. Subdivision of populations increases the opportunity for selection between
TABLE 1. Genic correlations between relatives as a function of effective population size*
First cousins
( t = 2)
Full sibs
( t = 1)
Ne
5
10
25
50
100
1,000
m
Third cousins
( t = 4)
)‘t
70
Tt
%
rt
5%
0.0741
0.1690
0.2191
0.2348
0.2424
0.2492
0.2500
30
68
88
94
97
100
100
-0.2860
-0.0935
0.0039
0.0338
0.0483
0.0611
0.0625
-458
- 150
6
54
77
98
100
-0.6869
-0.2873
-0.1020
-0.0474
-0.0214
0.0014
0.0039
-17,584
- 7,355
-2,610
- 1,214
-547
36
100
*Columns labeled “97” express E { r , ) as a percentage of its limiting large-population value.
A.R. ROGERS
380
groups. At the same time, it reduces the opportunity for kin selection within groups by
reducing correlations of relatives relative to
local groups.
This may be part of the reason why, among
Hoogland's (1985, 1986) black-tailed prairie
dogs, altruistic interactions are poorly correlated with pedigree kinship, and females
often kill the offspring of close relatives. His
study colony is polygynous and comprises
only about 130 individuals, so its effective
size is probably quite small. If so, selection
favoring preferential treatment of relatives
would be reduced.
CONCLUSIONS
The effects of finite population size on correlations between relatives are usually ignored. The results presented here make it
possible to determine when this procedure is
justified and when it is not. In small populations, correlations between relatives are
smaller than the predictions of the conventional infinite-population model, and this effect is much greater for distant relatives than
for close ones. For example, in a population
of effective size 100, the correlations between
sibs and third cousins are 98%and -547% of
their large-population values, respectively.
These results are approximate and refer to
completely isolated populations. When there
is immigration from the outside world, correlations will be larger than predicted by the
model used here.
These results imply that estimates of quantitative genetics parameters will be biased in
small populations, especially if based on deep
pedigrees Furthermore, they imply that seletion for altruism toward close kin will be
reduced. This may help explain the absence
of preferential treatment of close relatives
among Hoogland's (1985, 1986) prairie dogs.
ACKNOWLEDGMENTS
I thank James F. Crow, Randall Fitzgerald,
Steven Gaulin, Henry Harpending, Lynn
Jorde, Russell Lande, and Michael Siege1 for
their comments and suggestions.
LITERATURE CITED
Cannings, C, and Thompson EA (1981) Genealogical and
genetic structure. New York: Cambridge Univeristy
Press.
Crow, JF (1980) The estimation of inbreeding from isonymy. Hum. Biol. 52:l-14.
Crow, JF, and Kimura, M (1970) An introduction to
population genetics theory. New York:Harper and Row.
Hamilton, WD (1964) Genetical evolution of social behavior, I. J. Theor. Biol. 7:338-363.
Hoogland, JL (1985) Infanticide in prairie dogs: Lactating females kill offsping of close kin. Science 230:10371040.
Hoogland, JL (1986) Nepotism in prairie dogs (Cynomys
ludouicianus) varies with competition not with kinship. Anim. Behav. 34:263-270.
Jacquard, A (1975) Inbreeding: One word, several meanings. Theor. Population Biol. 7:338-363.
Malecot, G (1969) The mathematics of heredity. San
Francisco: H. Freeman.
Wade, MJ (1978) A critical review of the models of group
selection. Quart. Rev. Biol. 53:lOl-114.
Wright, S (1951) The genetical structure of populations.
Ann. Eugen. 15:323-354.
Wright, S (1969) Evolution and the genetics of Populations 11. The Theory of Gene Frequencies. Chicago:
University of Chicago Press.
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