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ARTICLE
Haplotype diversity and SNP frequency dependence
in the description of genetic variation
Michael PH Stumpf*
,1
1
Department of Biological Sciences, Biochemistry Building, Imperial College London, London SW7 2AY, UK
Haplotype diversity is controlled by a variety of processes, including mutation, recombination, marker
ascertainment and demography. Understanding the extent to which genetic variation at physically linked
loci is co-inherited is crucial for the design of the HapMap project and the correct interpretation of the
resulting data. In the absence of an analytical theory extensive coalescent simulations are used to
disentangle the influence of all of these factors on haplotype diversity. In addition to these qualitative
insights, this study also demonstrates (i) that marker spacing and frequency profoundly influence observed
levels of haplotype diversity; (ii) that the spectrum of haplotypes contains information about how
exhaustively genetic variation in a region is described by a given marker set; and (iii) that so-called
haplotype blocks can be generated due by the stochasticity inherent in the recombination process without
having to assume variation in the recombination rate.
European Journal of Human Genetics (2004) 12, 469–477. doi:10.1038/sj.ejhg.5201179
Published online 17 March 2004
Keywords: population genetics; HapMap project; haplotype tagging; haplotype blocks
Introduction
An increasing number of empirical
1–3
studies investigate
whether the inheritance of genetic variants occurs in a
block-like manner and the potential implications of this
for association studies.
4
Reported haplotype diversities
along extended stretches of DNA appear surprisingly
simple with most chromosomes belonging to one of
roughly a handful of different haplotypes.
5
The levels of
linkage disequilibrium (LD) are also reported to be con-
sistently high between markers that are in the same block
although LD can also extend beyond block-boundaries.
6,7
If such a picture were to prevail it would have obvious
consequences for the design of association studies.
8
In an important paper Jeffreys et al
1
showed that at least
sometimes blocks may be delimited by recombination
hotspots; the recombination rate in fairly localized regions
can exceed the background or block recombination rate by
up to four orders of magnitude and LD does not extend
beyond the block boundaries. In many cases, however,
there is as yet no conclusive evidence for block boundaries
to coincide with recombination hotspots.
9,10
If this
were generally the case then we could hope that block-
boundaries and possibly knowledge of haplotypes in one
population would allow us to make predictions of infe-
rences for other populations. Unfortunately, however, many
reports of blocks fail to show evidence for such a connection
with hotspots
11
and the methods by which blocks are
ascertained
3,8
may at least be partly to blame for this.
There are three main objectives of this study of haplo-
type diversity. On a fundamental level we gain insight (in
the absence of an analytical theory) into how physical
proximity between markers, the marker frequencies and
the intensity of recombination interact to determine the
complexity of the haplotype spectrum. Second, recent
theoretical work by Wiuf et al
12
is followed up. These
authors have shown that the number T of haplotype
tagging SNPs (htSNPs)
2
necessary to describe a given set of
M haplotypes defined by N SNPs is bounded by log
2
M
Revised 12 December 2003; accepted 21 January 2004
*Correspondence: Dr MPH Stumpf, Department of Biological Sciences,
Biochemistry Building, Imperial College London, London SW7 2AZ, UK.
Tel: þ 44 20 7594 5114; Fax: þ 44 20 7594 5789;
E-mail: m.stumpf@imperial.ac.uk
European Journal of Human Genetics (2004) 12, 469–477
&
2004 Nature Publishing Group All rights reserved 1018-4813/04
$30.00
www.nature.com/ejhg
oTomin(N,M-1). Here we determine, for different scenarios,
the relationship between M and N. Third, a simple block
definition is used to evaluate properties of blocks and how
inferred blocks (which do not correspond to recombina-
tion hotspots) depend on a marker characteristics and
recombination rate.
All three aspects of this work have implications in the
current run-up to the HapMap project.
8
The study provides
guidance into when and how the resulting SNP data are best
summarized in terms of haplotypes. Moreover, as will
become clear, haplotype diversity and the combinatorial
structure of haplotypes also hold information about how
exhaustively genetic variation in a region has been sampled.
Methods
Simulation procedures
In the following discussion we simulate the ancestral
recombination graph
13
assuming uniform recombination
and mutations rates r and m, respectively. Assuming
constant r allows the study of the behaviour in blocks of
low recombination rates, or the expected behaviour of
haplotype diversity under a Null model; both aspects will
be considered here. Throughout we assume a single
panmictic population with an effective population size of
N
e
¼ 10 000 diploid individuals. Throughout we use a
sample size of n ¼ 500 chromosomes and consider a stretch
of 50 kb length; the sample size is large compared to most
studies of LD performed today,
3,9
but smaller than the
population samples predicted for future case –control
studies.
14
The mutation rates are assumed to be 10
8
/
(nucleotide generation) and 10
9
/(nucleotide genera-
tion), whence the population mutation rates along the
whole stretch are m ¼ 50 and 5; the mutation model used
here is the infinite sites model. We consider two recombi-
nation rates which correspond to 1 and 0.1 cM/Mb in
addition to the case of no recombination; the correspond-
ing population recombination rates are thus: 50, 5 and 0,
respectively.
Human genetic diversity for a stretch of 50 kb corre-
sponds approximately to m ¼ 50 and r ¼ 50.
15
The other
values therefore correspond loosely to cases where the
recombination and/or average marker density (via the
mutation rate) is decreased. The case of m ¼ 5 and r ¼ 5,
however, can also be interpreted as the correct description
of a 5 kb stretch. We also use the m ¼ 5 case, which gives rise
to a sparser marker set, as a qualitative example for SNP
ascertainment.
16
Results for the lower mutation rate to
model as representing the case of a sparser set of markers.
In addition to the constant population size we also
investigate the effects of population growth on the result-
ing haplotype diversity but refrain from a more detailed
study of the effects of demography. For each scenario, 2000
independent runs of the ancestral recombination graph
were performed. Frequency cutoffs for the minor marker
allele (and not always the derived allele) are enforced by
counting the copies of each allele in the sample. Cutoff
frequencies considered are 1, 5, 10 and 20%.
Haplotype analysis and tagging approach
The minimum number of necessary tagging SNPs to tag a
given set of haplotypes is evaluated using a brute-force
implementation of the algorithm described in Wiuf et al
12
.
Starting from the k ¼ M, where M is the number of
haplotypes, we evaluate each possible combination of k
SNPs to see if it could be used as a basis for the set of
haplotypes. If one of the N!/(N-k)/k! possible SNP combina-
tions forms a valid basis then k is decreased by 1 until the
first time a basis cannot found. For large N and M the
number of SNP combinations can become enormous but in
smaller simulations it was observed that the distribution of
the minimum number of tags required to tag a given
number of haplotypes is relatively flat: many different
combinations can be used to tag haplotypes. Thus, for large
values of N and M it is possible to proceed heuristically
12
and investigate, for exmple, a maximum of 100 Million
combinations of candidate tags and an inferred minimal
basis will be close to optimal. Similarly, we have also
implemented a strategy where we start from k ¼ min(N,M-1)
and increment k until a basis has been found. Using either
approach (which of course yield identical results) when a
set of k SNPs is found the procedure stops and the number
of necessary and sufficient tagging SNPs is set to T ¼ k.At
most min(N,M-1) tagging SNPs are required to describe all
observed haplotypes in the sample. As the algorithm is in
the NP-complete class we only evaluate the number of
tagging SNPs for the case of SNP ascertainment outlined
above. Our heuristic approach can also be implemented
more formally in a Markov Chain Monte Carlo setting.
Results
Here we discuss how the number of haplotypes depends on
the number of SNPs, the recombination rate and the cutoff
frequency for the fraction of chromosomes that should be
included. Analytic results are only available for the case of
no recombination and free recombination, respectively,
and we therefore use coalescent simulations as outlined
above.
Determinants of haplotype diversity
In Figure 1 we show how the number of haplotypes depends
on the number of SNPs, their frequency and the recombina-
tion rate. The relationship between SNP number (for each
frequency cutoff) and the total number of haplotypes in a
sample already carries information about the recombina-
tion rate and how exhaustively a given set of SNPs repre-
sents or resembles underlying genetic variation. Large SNP
sets (with a low cutoff frequency) will contain correlations
among SNPs but if marker sets are sparse, recombination
Haplotype diversity
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European Journal of Human Genetics
will be more effective at breaking up associations between
markers; we therefore expect a lower value for the ratio
for y ¼ 5 than for y ¼ 50, irrespective of cutoff frequency
and recombination rate. The number of tags required
to adequately describe variation in a region will there-
fore be a function of both marker frequency and marker
density.
For y ¼ 50 we find that a 10-fold decrease in the
recombination rate from r ¼ 50 to 5 already brings the
observed number of haplotypes very close to the r ¼ 0
results. For lower values of r the average number of
haplotypes is virtually indistinguishable from the r ¼ 0
case. Note that for the decay of LD measured by the same
decrease in r from r ¼ 50 to 5 does not yield a behaviour
anywhere near the r ¼ 0 case (not shown). Haplotype
diversity and LD, although related, show somewhat
different dependence on the population recombination
rate r. This is also observed for growing and bottleneck
populations (data not shown).
The dependence of haplotype diversity on the minor
SNP allele frequency is further exemplified in Figure 2.
Here we show the number of haplotypes needed to describe
90, 95 and 99%, and all of the 500 chromosomes in the
sample. These numbers are displayed for five different
marker frequency cutoffs, three recombination and two
mutation rates. Such a table can either be used to assess the
genotyping cost necessary to capture a given amount of
variation or in case all the available genetic variation has
been characterized, to obtain an indication of the average
recombination/mutation rate ratio. While for high marker
density or mutation rates rare (fo1%) alleles give rise to a
large number of haplotypes we find that for fZ5% there
is no big reduction in genotyping effort as the cutoff
frequency is further increased. Also for fZ5% the fre-
quency distribution of haplotypes holds some information
about the recombination rate: a higher recombination rate
will lead to more rare haplotypes even at moderate to high
frequency cutoffs, as is also intuitively obvious.
The frequency distribution of haplotypes is displayed in
Figure 3. At the reported genome wide average of the
recombination rate a stretch of 50 kb is not expected to
have any haplotypes at a frequency greater than 10%,
irrespective of the cutoff frequency. If the marker spacing is
decreased, however, some haplotypes will gain in fre-
quency and at y ¼ 5 and r ¼ 50 we therefore observe some
haplotypes at moderate frequencies, especially at high
cutoffs. Low values of y result in a shift of weight to higher
haplotype frequencies. For ro1 (results not shown) the
resulting haplotype distributions are very similar to the
special r ¼ 0 case apart from the origin. At low recombina-
tion rates and for cutoffs fZ5 the haplotype distribution
obtains a mode at the cutoff frequency f.
The shift of the mode to the cutoff frequency is simply
a result of the fact that in the absence of excessive
Figure 1 Average SNP (grey) and haplotype numbers (black) versus minor allele frequency cutoff for y ¼ 5 and 50 and r ¼ 50,
5 and 0, respectively.
Haplotype diversity
MPH Stumpf
471
European Journal of Human Genetics
recombination, an SNP with a minor allele frequency
of x will define a haplogroup of frequency x if x is very
close to the cutoff frequency; thus an excess of haplo-
types with frequency x will be observed. If the recombina-
tion rate is high then haplotypes defined by the youngest
SNP can be broken up by recombination and here
r ¼ 50 appears to yield results that are very close to
the case of free recombination. As a result the mode of
the haplotype frequency distribution shifts back to the
origin.
Haplotype tagging
Only one tagging strategy is investigated here
12
and at the
moment it is by no means clear what tagging strategy is
best suited for association studies.
8
Rather than focusing
on tagging haplotypes, it may for example be better to
define tags that capture the patterns of LD and/or
association between SNPs. Simulation-based power analysis
along the lines taken here will help to assess such questions
in further detail. The tagging approach used here is quite
likely not optimal for association studies, but its easy
interpretation in terms of a geometric basis for the space
spanned by the SNP defined haplotypes nicely highlights
the combinatorial nature of haplotypes and the complexity
introduced by recombination. Other haplotype tagging
frameworks, however, are likely to behave qualitatively
similarly to the approach taken here.
We only consider an allele frequency cutoff of 5%. In
Table 1 we show mean values of the ratios T/N
5
(where N is
the number of SNPs with a minor allele frequency of 5%)
and the corresponding 5 and 95 percentiles. We find an
obvious dependence on r and for r ¼ 5 the results are
already quite similar to r ¼ 0. The results for r ¼ 50 are
discouraging: on average over 90% of SNPs need to be
typed in order to reliably distinguish between haplotypes.
This suggests that reports of low haplotype diversity
indicate regions of low recombination rate. We note,
however, that the majority of currently published studies
has marker density that is at least a factor of 5 lower than
the one obtained here.
9
Moreover our results concern true,
not inferred haplotypes. Haplotype inference may system-
atically bias tagging approaches.
Figure 2 Average number of haplotypes needed to explain 90, 95, 99 and 100% of observed chromosomes in a sample for
frequency cutoffs of 1, 5, 10 and 20%, respectively, for y ¼ 5 and 50 and r ¼ 50 and 5.
Haplotype diversity
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European Journal of Human Genetics
Dynamics of haplotype blocks
Notions and possible uses of extended haplotype blocks
that are characterized by high levels of pairwise LD
between SNPs within the same block (and accordingly
low haplotype diversity compared to the extreme case of
free recombination) have attracted considerable inter-
est.
4,6,17 – 19.
Here we follow Wang et al
17
and use probably
the simplest definition of a block: all SNP pairs that are
within the same block must fail the four-gamete test, that
is, at most three out of the possible four two-locus
haplotypes are observed for each pair of bi-allelic markers.
This definition has some shortcomings but is (i) easily
implemented, and (ii) we expect it to give at least some
insight into how SNP frequencies and ascertainment affect
the behaviour of blocks. Insights gained for this simple
model will be transferable to other, more involved, block-
ascertainment methods.
In Figure 4 we show how the average number and
average size of blocks, as well as the proportion of DNA and
SNPs that are found within blocks depend on minor allele
frequency cutoff and recombination rate r. We only
consider y ¼ 50 but in each case we show both the results
Haplotype distributions for ρ = 50
Relative HT AbundanceRelative HT Abundance
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2
Haplotype Frequency
Haplotype distributions for ρ = 5
0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.3 0.4
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2
Haplot
y
pe Frequenc
y
0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Figure 3 Frequency distribution of haplotypes and their dependence on y, r and minor allele frequency cutoff.
Table 1 Average fraction of SNPs (in percent) needed to capture x% of chromosomes in the sample (x ¼ 99, 95 and 90%,
respectively) for three different values of the recombination rate together with their 5 and 95 percentiles (in parentheses)
Population recombination rate r Average number of haplotypes Percentage of SNPs needed to explain fraction x of haplotypes
x ¼ 99% X ¼ 95% x ¼ 90%
0 7.3 49.7 (22.0 –92.5) 45.0 (17.9 –83.3) 37.5 (14.3 –66.7)
5 14.2 70.6 (41.2–100) 58.9 (32.0 –90.0) 49.4 (25.0 –78.6)
50 52.3 94.7 (83.3–100) 93.1 (80.0 –100) 91.7 (76.2 –100)
The average number of segregating sites is 33.9 of which 14.67 had a frequency Z5%; the corresponding 5 and 95 percentiles are 21 and 50 and 5
and 28, respectively.
Haplotype diversity
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European Journal of Human Genetics
for all blocks that adhere to our definition and of ‘long’
blocks. ‘Long’ blocks are blocks that contain at least four
SNPs while other blocks may also contain pairs of SNPs that
fulfil our four-gamete test criterion. Full symbols denote
results for r ¼ 50, empty symbols r ¼ 5; circles (full lines)
are for all blocks while boxes (dashed lines) refer only to
the long blocks.
We observe that for low frequency cutoffs there are many
more but shorter blocks for r ¼ 50 than for 5 where the two
curves are in very close agreement. At r ¼ 50 the average
block-size is determined largely by the long blocks but for
all other measures displayed in Figure 4 we observe
significant differences between long and short blocks.
The number of long blocks, the proportion of DNA in
long blocks, and perhaps most severely, the proportion of
SNPs that are found in long blocks decreases more
dramatically with minor allele frequency cutoff than the
same measures do for all blocks. At a minor allele frequency
of 20% only approximately 20% of DNA and 50% of SNPs
are found in long blocks. For all blocks these values
increase to 40 and 90%, respectively. It is obvious that
small blocks, containing only two or three SNPs, will offer
little or no reduction in genotyping effort. Long blocks, on
the other hand, account for only a small part of the total
sequence.
The average block-size remains approximately constant
for all allele frequency cutoffs. This result can be explained
by considering those pairs of SNPs that are the most likely
to give rise to four observed two-locus haplotypes. These
SNPs have to be old enough to have undergone at least one
recombination event and therefore will have reasonably
large minor allele frequencies. Pairs of younger markers,
which by and large will have a smaller minor allele
frequency, are less likely to give rise to four haplotypes
and therefore we expect SNPs with moderate to high minor
allele frequencies to determine block-size. Undersampling
of diversity (eg restricting the analysis to already known
SNPs such as those in dbSNP) could therefore system-
atically overestimates average block-lengths. This result
is in agreement with the study of Phillips et al
3
who
find that block-length increases with marker spacing; it is
likely to hold for other definitions as suggested by recent
studies of the effects of SNP ascertainment.
16
Thus,
interpretation of haplotype diversity (like LD and block
Figure 4 Average no. of blocks, average block-size, average of the total proportion of DNA in blocks and average of the total
number of SNPs in blocks calculated for a sample of 500 chromosomes drawn from a constant size population with y ¼ 50
versus frequency cutoff. Solid symbols represent the case r ¼ 50, empty symbols r ¼ 5. Circles (solid lines) represent results
obtained for all blocks, boxes (dashed lines) represent results for blocks containing at least four SNPs.
Haplotype diversity
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474
European Journal of Human Genetics
boundaries) is problematic if not supported by extensive
simulations.
3,7
Demography and haplotype diversity
Demography and population structure are known to have
profound effects on the frequency spectrum of segregating
sites, LD and thus also on haplotype diversity.
3,4,9.
Simula-
tions of population-growth scenarios suggest that the effect
of minor-allele frequency still persists. We only show
results for one particular demographic scenario where the
population has grown from 1% of its present size to its
present size over a time t ¼ 1 (in coalescent units); before
the onset of growth the population size is assumed to be
constant at 1% of the present size. Other cases are easily
assessed using coalescent simulations. Owing to the
problems associated with diversity discussed by Pritchard
and Przeworski
15
the mutation rate was adjusted such that
the number of segregating sites in the sample is the same in
the population growth scenario as in the constant popula-
tion scenario discussed above.
Comparing Figure 1 with the top row of Figure 5 shows
only quantitative differences that are easily explained by
the different SNP allele frequency distribution resulting
from a population growth scenario. We find at the higher
recombination rate that haplotype numbers exceed SNP
numbers already for lower frequency cutoffs (ie f45%
instead of f420%). At the same cutoff frequency the ratio
of [haplotype number]/[SNP number] is less for the growth
demography considered here than for the constant size
population. Comparison of Figure 2 with the bottom row
of Figure 5 shows only a minor vertical shift: the average
number of haplotypes needed to describe x%(x ¼ 90, 95,
99, 100) of the chromosomes in the sample is higher for
population growth than for constant population size.
Again this is easily understood because population growth
results in a relative excess of rare alleles compared to the
case of constant population size. These results suggest that
the basic patterns of haplotype dependence (on allele
frequency cutoff, marker spacing and recombination rate)
elucidated above may remain valid for a range of demo-
graphic scenarios.
Conclusions
In the search for the genetic components of complex
diseases or drug response phenotypes haplotype-based
Figure 5 Top row: average numbers of SNPs (grey) and haplotypes (black) resulting for yE 65 and r ¼ 50 and 5, respectively.
Bottom row: number of haplotypes that need to be considered in order to cover 90, 95 and 99%, and all of the chromosomes
in the sample. In each case the demographic model outlined in the text was used in the coalescent simulations.
Haplotype diversity
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475
European Journal of Human Genetics
approaches have recently been heralded as particularly
promising. A host of early studies suggested that relatively
few (eg 2 – 6) haplotypes may suffice to describe the genetic
variation along extended stretches of DNA.
3,5,9,10
The aim
of this study was to (i) gain some understanding of the
factors influencing observed haplotype diversities, (ii)
evaluate the behaviour of haplotypes expected for simple
population genetic models, and (iii) see to what extent
haplotype blocks can appear without underlying local
variation in the recombination rate.
Before discussing the application of the results presented
here to real world data, it is important to acknowledge the
limitations of the approach taken here. The population
model is of course incorrect and at best over-simplified.
While a quantitative interpretation of the results is thus
impossible they seem to reflect qualitative trends. For
example, for many if not all population models (including
the unknown true model), haplotype diversity will increase
with increased recombination rate and decrease dramati-
cally with increased SNP frequency cutoff. This is a general
result confirmed by simulations of a wide range of
demographic models (data not shown) and intuitively
obvious in the light of what is known about the ancestral
recombination graph.
The reported haplotype frequencies and diversities are
not easily reconciled with the standard neutral constant
size model of evolution although the generally small
sample sizes will result in overestimation of LD and of
haplotype frequencies. For the sample size considered here,
n ¼ 500, which is by no means large compared to what will
be required for genetic association studies,
14
the number of
segregating sites is very large for a region of 50 kb, SE330.
Even a moderate reduction of the recombination rate
brings haplotype diversities and the number of required
tSNPs into the range observed for r ¼ 0. This suggests that
at least some of the reported blocks may occur in regions
where the recombination rate r is less than the reported
genome wide average r ¼ 1 cM/Mb. The simulations also
show that haplotype diversity and block behaviour depend
on both allele frequency and marker spacing. A number of
reports of long-range disequilibrium and/or low haplotype
diversity, based on incomplete sampling of the genetic SNP
diversity, need to be reassessed in the light of this. A
detailed assessment of local recombination rate variation
becomes important and should provide crucial informa-
tion about the usefulness of blocks. Similarly, predictions
about the success/efficiency gains to be gained from the
HapMap project that are based on present studies may
systematically underestimate the number of tagging SNPs
required to describe human genetic diversity.
Generally, we find that for complete ascertainment of
segregating sites/SNPs haplotype diversity along a 50-kb
stretch is almost unmanageably large if all markers or those
with a minor allele frequency of fr1% are to be typed.
From a cutoff of ‘5%’ and above no big efficiency gains are
obtained and if the common variant/common disease
should turn out to be correct than 5% may be a reasonable
cutoff frequency. The genotyping effort, even if tagging
approaches are used, may be considerably more than had
been hoped.
2,9,10
There are considerable problems in interpreting current
experimental data sets and the simulation study presented
here gives some clues as to what factors may compromise
inferences drawn from summaries of the data such as LD
and/or haplotype diversity. Many of these problems could
be directly addressed if the underlying recombination rate
variation were known. In addition to approaches using
sperm-typing,
1,20
a number of inferential procedures has
recently developed that allow direct estimation of the
recombination rate.
21 – 25
These use mainly information
from informative sites with high minor allele frequency
and their inferences should be robust against the problems
associated with low marker density and bias in allele
frequencies. Knowledge of local recombination rate varia-
tion along the human genome will provide crucial
guidance in the setup of genetic epidemiology studies.
Acknowledgements
I thank Carsten Wiuf and Gil McVean for many discussions on this
topic and Monty Slatkin for his helpful comments on a earlier version
of this manuscript. This work was funded through a Wellcome Trust
Career Development Fellowship and a Royal Society Project Grant.
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