Published online before print
March 12, 2003, 10.1101/gr.387103
Vol 13, Issue 4, 635-643, April 2003
METHODS
Novel Multilocus Measure of Linkage Disequilibrium to Estimate Past Effective Population Size
Ben J. Hayes1,4,
Peter M. Visscher2,
Helen C. McPartlan1 and
Mike E. Goddard1,3
1Victorian Institute of Animal Science, Department of
Natural Resources and Environment, Attwood, Victoria, 3049, Australia;2
University of Edinburgh, Edinburgh EH9 3JG, Scotland, UK;3
Institute of Land and Food Resources, University of
Melbourne, Parkville, Victoria, 3052, Australia
 |
ABSTRACT
|
|---|
Linkage disequilibrium (LD) between densely spaced, polymorphic
genetic markers in humans and other species contains information about
historical population size. Inferring past population size is of
interest both from an evolutionary perspective (e.g., testing the
"out of Africa" hypothesis of human evolution) and to improve
models for mapping of disease and quantitative trait genes. We propose
a novel multilocus measure of LD, the chromosome segment homozygosity
(CSH). CSH is defined for a specific chromosome segment, up to the full
length of the chromosome. In computer simulations CSH was generally
less variable than the r2 measure of LD, and
variability of CSH decreased as the number of markers in the chromosome
segment was increased. The essence and utility of our novel measure is
that CSH over long distances reflects recent effective population size
(N), whereas CSH over small distances reflects the effective
size in the more distant past. We illustrate the utility of CSH by
calculating CSH from human and dairy cattle SNP and microsatellite
marker data, and predicting N at various times in the past for
each species. Results indicated an exponentially increasing N
in humans and a declining N in dairy cattle. CSH is a valuable
statistic for inferring population histories from haplotype data, and
has implications for mapping of disease loci.
The large number of densely spaced, polymorphic genetic markers
generated by modern genomics is a powerful tool for
answering genetic questions. For instance, they are being used to
fine-scale-map trait genes (Pritchard and Przeworski et al. 2001 ) and
to infer the history of the human population (Reich et al. 2001).
Inferring past population size is of interest both from an evolutionary
perspective (e.g., testing the "out of Africa" hypothesis of human
evolution) and to improve models for the mapping of disease and
quantitative trait genes.
Under a neutral model with constant effective population size
(N), the homozygosity of a marker, the probability of sampling
two identical alleles from the population, can be used to estimate
N, provided the mutation rate is known (e.g., Kuhner et al.
1998; Slatkin and Bertorelle 2001). If N has changed in the
past, the homozygosity will estimate a form of average N. The
higher the mutation rate, the less events from the distant past remain
relevant and thus the average N estimated will reflect
N in the more recent past. Similarly, linkage disequilibrium
(LD) can be used to estimate N if the recombination rate is
known. As will be shown in this paper, LD over large recombination
distances estimates N in the more recent past than LD over
short recombination distances. Use of LD rather than individual marker
homozygosity has the advantage that the recombination rate is more
controllable than the mutation rate (by selecting the length of
chromosome segments), and more recent N can be estimated
because recombination rates can be much higher than mutation rates.
Therefore, although estimates of average past population size from
n unlinked loci can be more accurate than from n
linked loci (Kuhner et al. 1998), using LD between n linked
loci can provide additional information on historical changes in
population size.
Most measures of LD, such as r2 and related measures
(Devlin and Risch 1995; Weir 1996), quantify the association between a
pair of loci. Higher-order association coefficients analogous to
r2 can be defined for groups of 3, 4, or more loci,
but they have not been found to be a practical value (Hill 1981). Such
higher-order LD measures also ignore the essentially linear nature of
chromosomes and of recombination. An ideal multilocus measure of LD
would take account of this linearity and capture as much as possible of
the information content of the data (no single statistic could contain
all the information). In addition, it would be desirable if the measure
of LD used had a simple expectation, at least under standard models
such as the neutral model.
Definition of Chromosome Segment Homozygosity (CSH)
We propose a novel multilocus measure of LD, the chromosome segment
homozygosity (CSH). CSH is the probability that two chromosome segments
of the same size and location drawn at random from the population are
from a common ancestor, without intervening recombination. CSH is
defined for a specific chromosome segment, up to the full length of the
chromosome. The CSH cannot be directly observed from marker data but
has to be inferred from marker haplotypes for segments of the
chromosome. Consider a segment of chromosome with marker locus A at the
left-hand end of the segment and marker locus B at the other end of the
segment. The alleles at A and B define a haplotype. Two such segments
are chosen at random from the population. The probability that the two
haplotypes are identical by state (IBS) is the haplotype homozygosity
(HH). The two haplotypes can be IBS in two ways: - (1) The two segments are descended from a common ancestor without
intervening recombination, so are identical by descent (IBD); or
- (2) The two haplotypes are identical by state but not IBD.
The probability of (1) is CSH. Now let x = the
probability that A is homozygous when the chromosome segment is not
IBD, and let y = the probability that B is homozygous when
the chromosome segment is not IBD. Assuming the two loci behave
independently in this case, the probability of (2) is
Then the probability of observing homozygosity at A is
Solving for x,
And similarly,
Substituting the last two equations into the first, and summing over
(1) and (2) to get the probability of haplotype homozygosity, we get
This equation can be solved for CSH when the haplotype homozygosities
and individual marker homozygosities are observed from the data. For
more than two markers, the predicted haplotype homozygosity can be
calculated in an analogous but more complex manner (see Methods).
In this paper we show that CSH is generally a less variable statistic
than r2 when effective population size is constant.
We then derive the expectation of CSH under a neutral model with
changing N. Simulated data are used to validate the accuracy
of the expectation. Finally, we use CSH to estimate the N at
various times in the past in a human population (where we expect
N has been increasing) and in a cattle population (where we
expect N has been decreasing).
|
RESULTS AND DISCUSSION
|
|---|
In a population of constant effective size N, the
approximate expectation of CSH is 1/(4Nc + 1), which is the
same as the approximate expectation for r2, where
c is the length of the chromosome segment in morgans (Sved
1971). To test the agreement between expectation and observed results,
a chromosome segment of 10 cM containing 11 markers was simulated with
a mutation-drift model, with a constant N of 1000. The average
heterozygosity of markers was 0.65, and the number of alleles
segregating was  5–10 per marker. The simulation gave a total of 55
haplotype configurations: 10 different haplotype regions of 1 cM with
two markers, 9 different haplotype regions of 2 cM with three markers,
and up to a single haplotype region of length 10 cM with 10 markers. A
total of 200 replicate populations were simulated. The results (Fig.
1) indicate CSH and r2
have a similar mean, but different variance. The means of both
statistics were close to 1/(4Nc + 1), except for the 1-cM
haplotype, in which r2 was less than the
expectation. In our simulations, marker allele frequencies were the
result of drift and mutation, and follow a U-shaped distribution (e.g.,
Kimura and Crow 1964), often <0.05 or >0.95. Hudson (1985) showed in
simulations that r2 was lower than expected when
allele frequencies were <0.05 or >0.95. This may explain the
lower-than-expected r2 value we observed for the
1-cM chromosome segments. The CSH does not appear to be sensitive to
allele frequency.
The CSH had a lower coefficient of variation (CV) than
r2, provided there were more than three markers in
the haplotype (Fig. 1B), indicating that it is a less variable
statistic to estimate LD than pairwise measures.
Additional simulation indicated the decreasing variation of CSH was a
result of an increasing number of markers in the chromosome segment
rather than increasing haplotype length (data not shown). This is a
major advantage of using CSH to measure LD rather than two locus
measures such as r2 (for such measures variability
of LD for a given chromosome segment cannot be reduced using additional
markers).
With the infinite alleles model, all identical by state alleles are
also IBD. Although this is not one of our assumptions in the derivation
of CSH, we investigated estimates of CSH under an alternate mutation
model. With microsatellite markers, multiple mutations can occur in the
same marker, and two or more mutations can recover the initial allelic
state. A stepwise mutation model assumes an equal probability of
increasing the size of the allele by 1 and decreasing the size of the
allele by 1, and is suitable for modeling microsatellite markers (e.g.,
Shriver et al. 1993). We simulated a population similar to that used to
produce the results in Figure 1A, but with a stepwise mutation model.
The estimated CSH from these data was almost identical to estimates of
CSH using data from the infinite alleles model (Fig. 1C).
In Figure 2 CSH was recorded directly from
simulated data, rather than estimated from marker haplotypes as in
Figure 1. Again, where population size was constant over all
generations (CONS), CSH was very close to the values predicted by
1/(4Nc + 1).
A second set of data was simulated to illustrate the effect of past
N on CSH at different lengths of chromosome. In these data,
marker alleles were not simulated because the identity of chromosome
segments was tracked directly. We simulated four populations, a
population of constant N (CONST), a population with linearly
increasing N (LINI), a population with linearly decreasing
N (LIND), and a population with exponentially increasing
N (EXPI). When the population size was either linearly (LINI)
or exponentially (EXPI) increasing, or linearly decreasing (LIND), CSH
at small recombination rates agreed with the expected CSH based on
population size many generations ago, whereas CSH at large
recombination rates agreed with expected CSH based on more recent
population size (Fig. 2). These results concur with those of Hill
(1981), who found that estimates of N from LD for very tightly
linked genes were more dependent on long-term than on recent population
history.
When population size is changing linearly, the expectation of CSH is
1/(4Ntc + 1), where Nt was
the population size 1/(2c) generations ago. Effective
population size 1/(2c) generations into the past was predicted
from CSH (Fig. 3). Our method for
predicting N assumes constant linear population growth from
generation 1. Although this population growth model does not hold for
any of the populations we have simulated, the estimates of N
were in approximate agreement with the actual N for LINI and
LIND. For EXPI, the later estimates of N agree reasonably well
with actual N; however, N for 500, 200, and 100
generations ago was somewhat overestimated. The widths of the 95%
confidence intervals, averaged over time, on the estimates of
N for each population were 183, 260, 141, and 219 for CONST,
LINI, LIND, and EXPI, respectively. For example, 20 generations ago the
estimate of effective population size for CONST was 1062, and the 95%
confidence interval was 971–1154. Confidence intervals were generally
smaller with lower population sizes.
View larger version (25K):
[in this window]
[in a new window]
|
Figure 3. Simulated and estimated effective population size over time for four
populations; (CONST) constant population size from 0 to 6050
generations ago; (LINI) increase in population size in the last 50
generations from 1000 to 5000; (LIND) decrease in population size in
the last 50 generations from 1000 to 100; (EXPI) increase in population
size in the last 50 generations from 1000 to 11290. SIM and EST
identify the simulated and estimated population sizes for each
population.
|
|
CSH was calculated from a human haplotype data set including 24 SNPs
and 2 microsatellites in a 1-cM region (Moffat et al. 2000). To
validate that CSH was accurately estimated with the marker densities
and heterozygosities in this data set, we simulated a population of
constant N = 5000, using the mutation-drift model, with
similar marker density, marker heterozygosities, and haplotype lengths
to those observed by Moffat et al. (2000) in their data set. The value
of CSH observed from the simulated data sets was similar to the
expectation of CSH with N = 5000, and predictions of
Nt were reasonably accurate, although
Nt was somewhat overestimated in more recent
generations (Table 1). The coefficient of
variation for CSH was higher from this simulation compared with
simulations with more heterozygous markers. Initial investigation
showed that the values of CSH from the real data set were extremely
variable for similar lengths of haplotype. To clarify the extent and
variability of CSH, and Nt at different t
from CSH, we first averaged CSH values in 0.05-cM bins. The first bin
contained CSH for haplotypes 0–0.05 cM, and so on. The c
value used to calculate t was the midpoint of these bins, and
Nt was inferred from the average of CSH within a
bin. The CSHs in the human data set at large lengths of haplotype were
consistent with N = 15,000 (Fig.
4A). At short lengths of haplotype, CSH was
closer to that expected when N = 5000. The
Nt values indicated exponential growth in the human
effective population size (Fig. 4B). Chromosomal homozygosity at very
closely linked markers is needed to estimate effective population size
many generations into the past. The marker spacing in our data set
allowed us to calculate Nt up to 2000 generations
into the past, although the next oldest prediction of
Nt is many generations later. The situation improves
in more recent history, with less time between predicted values of
Nt. An attempt was made to assess the variability in
estimates of Nt at the different times in the past,
by calculating the 95% confidence of CSH within a bin, and calculating
Nt for the upper and lower confidence interval. This
95% confidence interval captures the variation in CSH due to the
process of gametic sampling of similar lengths of haplotype at
different chromosomal locations. The range of Nt
values for times in the recent past was extremely variable, with
Nt for times in the distant past less so. For
example, at 2000 generations ago, the lower limit of
Nt was 3749, and the upper limit was 8376, whereas
for 182 generations ago, the lower limit was 5146, and the upper limit
was 26,932.
We compared the results from our method for the human data set, in
which Nt has been increasing, with a species in
which Nt has been decreasing over time. Accordingly,
we sampled marker haplotypes from the Holstein-Friesian dairy cattle
population. The marker data were 16 microsatellites on Chromosome 20
covering a 65-cM chromosome segment, sampled from 264 Australian
Holstein-Friesian cows. In the dairy cattle data set, CSH at large
lengths was consistent with N = 250 (Fig. 4C). At short
lengths of haplotype, the observed CSH was more consistent with
N = 1000. The two large values of CSH at 7.9 cM and 11.2 cM
are for long chromosome segments with only two markers, the situation
in which CSH is most variable. When we calculated Nt
and t from CSH at c morgans, the results indicated a
recent decline in the effective population size of dairy cattle (Fig.
4D). Because of the wide spacing of markers in our sample, there is
little information on effective population size of dairy cattle more
than 100 generations (400 yr) in the past. The one data point 167
generations in the past certainly indicates that the historical
effective population size was much larger than the present effective
population size.
The simulation study confirmed that CSH could be used to predict
approximate effective population size at various times in the past. The
estimates of past N were qualitatively correct although not
numerically precise. In theory, more information could be extracted
from the data. For instance, the frequencies of each haplotype contain
information—many rare haplotypes imply an increasing population size
(Slatkin and Bertorelle 2001). However, in practice there may be no
method that is highly precise because of the need to make numerous
assumptions in any method. A strength of the CSH is its simplicity. It
also makes clear the close analogy between mutation affecting
homozygosity at individual loci and recombination affecting LD at
multiple loci. LD provides information equivalent to that from a
mutation rate that can be controlled (by choosing the length of
chromosome segment) and that can take much higher values than mutation
rates at individual loci.
The variation in LD arises from two sampling processes (Weir and Hill
1980). The first sampling process reflects the sampling of gametes to
form successive generations, and is dependent on finite population
size. The second sampling process is the sampling of individuals to be
genotyped from the population, and is dependent on the sample size,
n. Unless n is sufficiently large, the effect of
Nt on CSH is likely to be swamped by sampling
effects resulting from choosing only a fraction of individuals to be
genotyped from the population in the present generation (e.g., Weir and
Hill 1980). Hill (1981) discussed the sample size necessary to obtain
precise estimates of population size (from r2 in his
case), and showed CV(N) is approximately
where n is the number of haplotypes sampled and k is
the number of pairs of loci used in the estimate. Sample size must
therefore be large relative to 4Nc to precisely estimate
N. This conclusion is also likely to be true for estimates of
Nt from CSH (even though the variability of CSH is
reduced relative to r2 as the number of markers on
the chromosome segment increases). For a given sample size, Hill's
conclusion indicates that because the variability of the estimate of
Nt will increase as the length of the chromosome
segment used to estimate CSH (and then Nt)
increases, the recent population size will be estimated less accurately
than the population size many generations in the past. This concurs
with our results, in which the 95% confidence interval of
Nt in the recent past was much larger than for
Nt in the more distant past. For the human data, the
accuracy of estimates of recent population size is further eroded by
the rapidly increasing value of Nt. Unrealistically
large sample sizes would be necessary to obtain accurate estimates of
Nt in the very recent past.
As all chromosome segments within the genome are subject to the same
Nt, the variability of estimates of
Nt from LD caused by finite population size (the
history of sampling gametes) could be reduced by averaging LD over
chromosome segments of equal recombination length. An ideal data set
for estimating Nt from CSH would contain many
equally spaced markers across a number of chromosomes, so results for
haplotypes of the same recombination length in different parts of the
genome could be averaged to obtain more accurate estimates of
Nt. This is analogous to the result from Kuhner et
al. (1998) that a gain in precision is obtained by sampling multiple
unlinked loci.
Our estimates of past N for both cattle and humans agree
with what is historically known about these populations. Using a dense
genome-wide SNP map in humans, Reich et al. (2001) calculated
D‘, and then used simulation to infer what pattern of change
in N could give the observed results. They concluded that the
European population passed through a bottleneck 27,000–53,000 years
ago, and proposed the bottleneck led to an inbreeding coefficient of
0.2. This level of inbreeding could be caused by a bottleneck of 50
individuals for 20 generations, 1000 individuals for 400 generations,
or any other combination with the same ratio. We estimated that
N for the population ancestral to our sample 2000 generations
ago (30,000 yr ago) was 5000 and that this size lasted for
thousands of years.
Shifman and Darvasi (2001) compared the amount of linkage
disequilibrium at various distances in isolated populations (e.g.,
Finnish, Ashkenazi, and Sardinian) to that in an outbred population
(CEPH). They found that at short distances (<200 kb), there was a
similar amount of LD in isolated and outbred populations, whereas at
long distances (>200 kb), there was up to six times more LD in the
isolated populations. They concluded that LD was similar for all
populations at short distances because processes other than
recombination, such as mutation, determined the amount of LD at <200
kb. At >200 kb, recombination was the main determinant of LD and so LD
differed greatly between the different populations as a result of their
different N. Given our result of t = 1/2c,
LD at 200 kb would reflect the population size 213 generations or
5000 yr ago. Hence, another interpretation of Shifman and Darvasi’s
(2001) results is that LD at <200 kb reflects N of the common
ancestral population to both the isolated and outbred populations, and
is therefore similar regardless of the present population size.
Sabatti and Risch (2002) recently investigated the relationship between
two-locus haplotype homozygosity and linkage disequilibrium, and
illustrated how haplotype homozygosity can be used to measure and test
for multilocus LD. Their new measure is based on the population or
sample frequencies of haplotypes (like the HH in this study) but,
unlike our definition of CSH, does not take account of the linear
nature of chromosomes and recombination and does not model homozygosity
by descent.
Domestication, breed formation, and artificial breeding technologies
have all served to reduce the effective population size of the world
dairy cattle population. Because of the wide spacing of the markers in
our data set, we can only infer population sizes up to 167 generations
(700 yr) ago. This is prior to the emergence of Holstein-Friesians
as a separate breed, which is estimated to have occurred 200 yr
ago (Bradley and Cunningham 1998). Our results certainly indicate
that the effective population size of the ancestral dairy population
has declined sharply between 200 yr ago and the present. The data also
contain some long (0.25-M) haplotypes, the LD at which can be used to
infer recent effective population size. Our data indicate the recent
effective population size (in the last 5–6 generations) to be 150,
although there is variation around this value. An N of 150 is
larger than, but similar in magnitude to, estimates of 100 based on
the rate of inbreeding in Holstein populations (Young and Seykora
1996).
We have shown that our novel multilocus measure of linkage
disequilibrium can be used to estimate past effective population size.
A similar approach can be used for LD mapping of genes for complex
traits (Meuwissen and Goddard 2001). Chromosome segments that are IBD
contain the same allele, except for mutation, at any gene within the
segment. Therefore, the trait values of people that share IBD
chromosome segments will be correlated if there is a gene affecting the
trait located within the segment.
|
METHODS
|
|---|
Simulated Data Sets
Two types of simulated data were used. A diploid population, of
N = 1000, was simulated for 6000 generations with either an
infinite alleles or stepwise mutation model. Each individual in the
population consisted of a pair of chromosomes, and was either male or
female (probability 0.5). Each chromosome was 10 cM long, and had 11
marker loci. To create an offspring, a pair of parents of different sex
was randomly chosen from the population. For each parent in a mating
pair, a gamete was formed from its chromosome pairs by sampling the
number of crossovers for each chromosome pair from a Poisson
distribution, with mean of 0.1. Crossover points were randomly
positioned along chromosome pairs. The haploid gametes were mutated at
a rate of 5 x 10–4 per locus per gamete per
generation. In the infinite alleles model, if a locus was
mutated, a new allele was added. In the stepwise mutation model, the
allele was either increased by 1 or decreased by 1, with probability
0.5 of each occurrence. The results presented are the average of 200
replicate populations. This simulation model was also used to evaluate
CSH with other population sizes. The number of generations for which
the population was simulated was always 6N. The heterozygosity
of markers was decreased in some simulations by decreasing the mutation
rate.
In the second simulated data set, marker alleles were not used because
the identity of chromosome segments was tracked directly. To
demonstrate the effect of past N on CSH, we simulated four
populations, a population of constant N (CONST), a population
with linearly increasing N (LINI), a population with linearly
decreasing N (LIND), and a population with exponentially
increasing N (EXPI). Each population consisted of N
individuals, such that an individual comprises a pair of chromosome
segments. To form a new individual, two individual parents were
selected at random from the population. Each of these contributed a
chromosome to the progeny. There was a probability c that the
chromosome from a parent was a recombinant and hence a "new"
chromosome segment. Each population was simulated with seven values of
c, 0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, and 0.1. All
populations began with 6000 generations with N = 1000. Then
50 (CONS, LINI, LIND) or 100 generations (EXPI) of breeding with
changing N followed. Table 2
describes the change in N over generations for each
population. Fifty replicates of each population were simulated. Using
the observed CSH, the formula CSH = 1/(4Nc + 1) was solved
for N to estimate N at 500, 200, 100, 50, 20, and 5
generations into the past for each population, the times corresponding
to 1/2c for the values of c we have used.
Human Data Set
The data set of Moffat et al. (2000) was retrieved from the Web
site http://www.well.ox.ac.uk/asthma/public/TCR/index.html. The data
consisted of 24 SNPs and 2 microsatellites in an 850-kb section of the
TCR locus on Chromosome 14q. To derive haplotypes, 159
nuclear and extended families were genotyped, and the LD between
markers was investigated in 600 haplotypes from unrelated individuals
(the parents). CSH were summarized into 0.05-cM bins. The first bin
contained CSH for haplotypes 0–0.05 cM, and so on. The c
value used to calculate t was the midpoint of these bins, and
Nt was inferred from the average of CSH within a
bin. The 95% confidence interval for estimates within a bin was
calculated as the bin average CSH ± 2SE. The data set was
too small to reliably calculate CSH at >0.3 cM, as the value of the
CSH estimates became much smaller than the sampling variance caused by
sampling only 600 chromosomes from the population. Therefore, we only
considered haplotypes less than this length.
Cattle Data Set
The resource population for the dairy cattle data set consisted of
four Holstein-Friesian sire families. Sire A had 22 daughters, Sire B
38 daughters, Sire C 74 daughters, and Sire D 130 daughters. Each
daughter had a unique dam. Daughters were genotyped for 15
microsatellite markers on Chromosome 20. The markers were BM1225,
RM310, ILSTS068, BMS2361, AGLA29, BM4107, ILSTS072, BMS703, BM5004,
BMS3517, HEL12, RM106, BMS1282, TGLA304, and BMS1719(see http://www.thearkdb.org/browser?species=cow for details). The
markers bracketed a length of 65 cM, with various spacings between the
markers. As the daughters were from four sire families, the paternal
and maternal marker haplotypes could be determined. We estimated CSH
from the maternal haplotypes only, as CSH from the paternal haplotypes
would reflect CSH in each of the four sires, rather than in the wider
population. The data set was too small to reliably calculate CSH at
>25 cM, as the value of the CSH estimates became much smaller than the
sampling variance caused by sampling only 264 chromosomes from the
population. Therefore, we only considered haplotypes <25 cM.
Calculation of CSH and r2
We cannot observe CSH directly from the marker haplotypes. Instead,
we estimate CSH from the observed homozygosity of haplotypes (HH). HH
is defined as the probability that two chromosome segments drawn at
random from the population have identical marker haplotypes. The value
of HH in the population is estimated from the sample as
where there were n haplotypes in the population, and
pi was the frequency of the i-th haplotype.
This formula corrects HH for sampling effects (following Hill 1981).
The algorithm to calculate CSH with multiple markers proceeds as
follows. The number of markers in the chromosome segment is m,
and an array, CSH, stores CSH for the chromosome segment between
markers i and j. The algorithm calculates values of
CSHij for all possible combinations of
j > i.Step 1. For
i = 1 to m – 1, and
j = i + 1 (the case of two adjacent markers),
calculate CSHij using the definition given
above.Step 2. For
i = 1 to m – 2, and
j = i + 2,
k = j – i = 2 (three adjacent
markers), generate the 2k – 1 possible
recombination configurations for the segments between the markers
(k is the number of adjacent markers). Representing 0 as no
recombination, and 1 as a recombination, the four possible
recombination configurations are 00, 10, 01, 11. The recombination
configurations can be quickly found by writing 0 to
2k – 1 – 1 as binary numbers. The probability
of 00 is CSHij. To calculate the probability of the
other recombinations (Probl for
l = 1–3), the rules of Meuwissen and Goddard (2001) are
used, except the values of f(c...) are replaced by
the appropriate CSH for two markers calculated in Step 1. As some
Probl values also contain CSHij,
a search must be performed for the value of CSHij
that minimizes
 |
This value was taken as the value of
CSHij.Step
3. For increasing k (4, 5, ...), repeat Step 3
until j – i = m – 1.
The
value of r2 was calculated as described by Hudson
(1985), with a correction for sampling described by Hill (1981).
Estimation of Past N From CSH
We wish to determine the effective population size,
Nt, t generations ago from CSH. Let
Pt be the probability that two chromosome segments
of length c coalesce by generation t.
Pt is a cumulative probability, and time is measured
from the present t = 0 to generation t in the past.
Then the probability that coalescence occurs exactly in generation
t is
pt = Pt – Pt – 1.
Then, in the standard coalescence model,
pt = (1 – Pt – 1)/2Nt,
where (1 – Pt – 1) is the probability that
coalescence hasn't already happened and 1/2Nt is
the probability that the two random chromosomes have a common ancestor
in the previous generation (Kingman 1982). This can be expressed in a
continuous rather than discrete form as
pt = dPt/dt = (1 – Pt)/(2Nt).
The probability that there has been no recombination in either
chromosome over the t generations is
(1 – c)2t, which, for small c,
is approximately equal to e(–2ct).
Therefore, the probability that coalescence happens at
generation t and there has been no recombination is
The total probability that coalescence happens before recombination
(the CSH) is the sum of this expression over all t values from 0 to
infinity,
If we make the assumption that N is linear with time, such
that 2Nt =  +  t, then
Taking the logarithm of both sides of this equation gives
where K is a constant. Rearranging this formula gives
At t = 0, and Pt = 0,
eK = 1/. Substituting
1/ for eK and rearranging gives
which is differentiated to give
This expression is approximately equal to
Substituting
for dPt/dt in the second equation gives
and after integration,
Now
where 2Nt = + /2c. That is,
Nt is the effective population size at
t = 1/2c generations in the past.
|
WEB SITE REFERENCES
|
|---|
http://www.thearkdb.org/browser?species=cow; cattle data set.
http://www.well.ox.ac.uk/asthma/public/TCR/index.html; Moffat data set
from human Chromosome 14q.
|
Acknowledgements
|
|---|
The authors thank W.G. Hill, T.H.E. Meuwissen, and two referees for
useful comments on an earlier version of this manuscript. P.M.V.
acknowledges support from the UK Biotechnology and Biological Sciences
Research Council.
The publication costs of this article were defrayed in part by payment
of page charges. This article must therefore be hereby marked
"advertisement" in accordance with 18 USC section 1734 solely to
indicate this fact.
|
Footnotes
|
|---|
4 Corresponding author. 
E-MAIL Ben.Hayes{at}nre.vic.gov.au; FAX 61 39217 4359.
Article and publication are at
http://www.genome.org/cgi/doi/10.1101/gr.387103. Article published online before print in March 2003.
|
REFERENCES
|
|---|
Bradley, D.G. and Cunningham, E.P. 1998. Genetic aspects of domestication. In The genetics of cattle (eds. R. Fries & A. Ruvinski), pp. 15–32. CAB International, Oxon, UK.
Devlin, B. and Risch, N. 1995. A comparison of linkage disequilibrium measures for fine-scale mapping. Genomics 29: 311-322.[CrossRef][Medline]
Hill, W.G. 1981. Estimation of effective population size from data on linkage disequilibrium. Genet. Res. 38: 209-216.
Hudson, R.R. 1985. The sampling distribution of linkage disequilibrium under an infinite allele model without selection. Genetics 109: 611-631.[Abstract/Free Full Text]
Kimura, M. and Crow, J.F. 1964. The number of alleles that can be maintained in a finite population. Genetics 49: 725-738.[Free Full Text]
Kingman, J.F.C. 1982. On the genealogy of large populations. In Essays in statistical science: Papers in honour of P.A.P. Moran (eds. J. Gani & E.J. Hannan), pp. 27–43. Applied Probability Trust, Sheffield. J. Appl. Prob., special volume 19A.
Kuhner, M.K., Yamato, J., and Felsenstein, J. 1998. Maximum likelihood estimation of population growth rates based on the coalescent. Genetics 149: 429-434.[Abstract/Free Full Text]
Meuwissen, T.H.E. and Goddard, M.E. 2001. Prediction of identity-by-descent probabilities from marker haplotypes. Genet. Select. Evol. 33: 605-634.[CrossRef][Medline]
Moffat, M.F., Traherne, J.A., Abcasis, G.R., and Cookson, W.O.C.M. 2000. Single nucleotide polymorphism and linkage disequilibrium within the TCR / locus. Hum. Mol. Genet. 9: 1011-1019.[Abstract/Free Full Text]
Pritchard, J.K. and Przeworski, M. 2001. Linkage disequilibrium in humans: Models and data. Am. J. Hum. Genet. 69: 1-14.[CrossRef][Medline]
Reich, E.D., Cargill, M., Bolk, S., Ireland, J., Sabeti, P.C., Richter, D.J., Lavery, T., Kouyoumjian, R., Farhadian, S.F., Ward, R., et al. 2001. Linkage disequilibrium in the human genome. Nature 411: 199-204.[CrossRef][Medline]
Sabatti, C. and Risch, N. 2002. Homozygosity and linkage disequilibrium. Genetics 160: 1707-1719.[Abstract/Free Full Text]
Shifman, S. and Darvasi, A. 2001. The value of isolated populations. Nat. Genet. 28: 309-310.[CrossRef][Medline]
Shriver, M.D.L., Jin, L., Chakraborty, R., and Boerwinkle, E. 1993. VNTR allele frequency distributions under the stepwise mutation model: A computer simulation approach. Genetics 134: 983-993.[Abstract]
Slatkin, M. and Bertorelle, G. 2001. The use of intraallelic variability for testing neutrality and estimating population growth rate. Genetics 158: 865-874.[Abstract/Free Full Text]
Sved, J.A. 1971. Linkage disequilibrium and homozygosity of chromosome segments in finite population. Theoret. Pop. Biol. 2: 125-141.[CrossRef][Medline]
Weir, B.S., 1996. Genetic data analysis II. Sinauer Associates, Sunderland, MA.
Weir, B.S. and Hill, W.G. 1980. Effect of mating structure on variation in linkage disequilibrium. Genetics 95: 477-488.[Abstract/Free Full Text]
Young, C.W. and Seykora, A.J. 1996. Estimates of inbreeding and relationship among registered Holstein females in the Unites States. J. Dairy Sci. 79: 502-505.[Abstract]
Received April 28, 2002;
accepted in revised format December 30, 2002.
13:635-643 © by 2003 Cold Spring Harbor Laboratory Press ISSN 1088-9051/03 $5.00
 CiteULike  Connotea  Del.icio.us  Digg  Reddit  Technorati What's this?
This article has been cited by other articles:
|
|
|
|
 
A. P. W. de Roos, B. J. Hayes, R. J. Spelman, and M. E. Goddard
Linkage Disequilibrium and Persistence of Phase in Holstein-Friesian, Jersey and Angus Cattle
Genetics,
July 1, 2008;
179(3):
1503 - 1512.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
M. Sargolzaei, F. S. Schenkel, G. B. Jansen, and L. R. Schaeffer
Extent of Linkage Disequilibrium in Holstein Cattle in North America
J Dairy Sci,
May 1, 2008;
91(5):
2106 - 2117.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. J. Cai
PGEToolbox: A Matlab Toolbox for Population Genetics and Evolution
J. Hered.,
February 29, 2008;
(2008)
esm127v1.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
C. Andreescu, S. Avendano, S. R. Brown, A. Hassen, S. J. Lamont, and J. C. M. Dekkers
Linkage Disequilibrium in Related Breeding Lines of Chickens
Genetics,
December 1, 2007;
177(4):
2161 - 2169.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
M. Gautier, T. Faraut, K. Moazami-Goudarzi, V. Navratil, M. Foglio, C. Grohs, A. Boland, J.-G. Garnier, D. Boichard, G. M. Lathrop, et al.
Genetic and Haplotypic Structure in 14 European and African Cattle Breeds
Genetics,
October 1, 2007;
177(2):
1059 - 1070.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
W. G. Hill and J. Hernandez-Sanchez
Prediction of Multilocus Identity-by-Descent
Genetics,
August 1, 2007;
176(4):
2307 - 2315.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
A. Tenesa, P. Navarro, B. J. Hayes, D. L. Duffy, G. M. Clarke, M. E. Goddard, and P. M. Visscher
Recent human effective population size estimated from linkage disequilibrium
Genome Res.,
April 1, 2007;
17(4):
520 - 526.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
H. H. Zhao, R. L. Fernando, and J. C. M. Dekkers
Power and Precision of Alternate Methods for Linkage Disequilibrium Mapping of Quantitative Trait Loci
Genetics,
April 1, 2007;
175(4):
1975 - 1986.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
S. H. Lee and J. H. J. Van der Werf
Using Dominance Relationship Coefficients Based on Linkage Disequilibrium and Linkage With a General Complex Pedigree to Increase Mapping Resolution
Genetics,
October 1, 2006;
174(2):
1009 - 1016.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
B. Hayes, N. Hagesaether, T. Adnoy, G. Pellerud, P. R. Berg, and S. Lien
Effects on Production Traits of Haplotypes Among Casein Genes in Norwegian Goats and Evidence for a Site of Preferential Recombination
Genetics,
September 1, 2006;
174(1):
455 - 464.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
S. H. Lee and J. H. J. Van der Werf
Simultaneous Fine Mapping of Multiple Closely Linked Quantitative Trait Loci Using Combined Linkage Disequilibrium and Linkage With a General Pedigree
Genetics,
August 1, 2006;
173(4):
2329 - 2337.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
C. Sandor, F. Farnir, S. Hansoul, W. Coppieters, T. Meuwissen, and M. Georges
Linkage Disequilibrium on the Bovine X Chromosome: Characterization and Use in Quantitative Trait Locus Mapping
Genetics,
July 1, 2006;
173(3):
1777 - 1786.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
R. Fan, J. Jung, and L. Jin
High-Resolution Association Mapping of Quantitative Trait Loci: A Population-Based Approach
Genetics,
January 1, 2006;
172(1):
663 - 686.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. Nsengimana, P. Baret, C. S. Haley, and P. M. Visscher
Linkage Disequilibrium in the Domesticated Pig
Genetics,
March 1, 2004;
166(3):
1395 - 1404.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
![[Feedback]](/icons/banner/feedbackACT.gif){kind=icon}
![[For Subscribers]](/icons/banner/subscriberACT.gif){kind=icon}
![[Archive]](/icons/banner/archiveACT.gif){kind=icon}
![[Search]](/icons/banner/searchACT.gif){kind=icon}
![[Contents]](/icons/banner/tocACT.gif){kind=icon}
Top
ABSTRACT
RESULTS AND DISCUSSION
