2 Example data set 1
2.1 Input
2.1.1 Data files
Both example data sets are provided in two formats: as a pair of haplotype & map files and as a single file in variant call format (vcf). They are copied (together with other files) into the current working directory by the command
Data set 1 contains the hypothetical variation at a particular genetic locus in 8 sequences of 4 diploid individuals. Eleven markers, which might be imagined as SNPs, have two alleles, coded as ‘0’ and ‘1’, for ancestral and derived allele, respectively.
The file example1.hap conforms to what in the main vignette is referred to as “standard” haplotype format, i.e. chromosomes are given in rows and each marker corresponds to a column. The first column is reserved for identifiers of the individual haplotypes.
> HG1_1 0 0 1 1 0 0 0 0 0 1 1
> HG1_2 0 0 0 0 1 0 1 0 1 0 0
> HG2_1 0 1 0 1 0 0 1 0 0 0 1
> HG2_2 0 0 1 0 0 0 0 1 1 0 0
> HG3_1 0 0 0 0 0 1 0 0 0 0 1
> HG3_2 0 0 0 0 0 1 0 0 0 0 1
> HG4_1 1 0 0 0 0 1 0 0 0 0 0
> HG4_2 0 0 0 0 0 1 0 0 0 0 0The calculation of the “integrated” statistics such as iHH and iES requires a measure for the distance between markers, which is provided by the file example1.map. It relates the markers with their chromosomal positions. The latter can represent physical positions (base pairs) or genetic positions derived from estimated recombination rates.
> rs1 chr1 10000 0 1
> rs2 chr1 20000 0 1
> rs3 chr1 30000 0 1
> rs4 chr1 40000 0 1
> rs5 chr1 50000 0 1
> rs6 chr1 60000 0 1
> rs7 chr1 70000 0 1
> rs8 chr1 80000 0 1
> rs9 chr1 90000 0 1
> rs10 chr1 100000 0 1
> rs11 chr1 110000 0 1The file in variant call format combines the information of haplotype and map files. However, vcf codes alleles with respect to a reference sequence, not with respect to ancestry status. Information about ancestry can be added using a key of the INFO field, conventionally named AA. For instance, in the file example1.vcf, the reference alleles of markers rs6 and rs11 differ from the ancestral alleles.
> ##fileformat=VCFv4.2
> ##reference=NCBI38
> ##contig=<ID=chr1,length=120000>
> ##INFO=<ID=AA,Number=1,Type=String,Description="Ancestral Allele">
> ##FORMAT=<ID=GT,Number=1,Type=String,Description="Genotype">
> #CHROM POS ID REF ALT QUAL FILTER INFO FORMAT HG1 HG2 HG3 HG4
> chr1 10000 rs1 G T 100 PASS AA=G GT 0|0 0|0 0|0 1|0
> chr1 20000 rs2 G A 100 PASS AA=G GT 0|0 1|0 0|0 0|0
> chr1 30000 rs3 T C 100 PASS AA=T GT 1|0 0|1 0|0 0|0
> chr1 40000 rs4 A C 100 PASS AA=A GT 1|0 1|0 0|0 0|0
> chr1 50000 rs5 T G 100 PASS AA=T GT 0|1 0|0 0|0 0|0
> chr1 60000 rs6 G A 100 PASS AA=A GT 1|1 1|1 0|0 0|0
> chr1 70000 rs7 C T 100 PASS AA=C GT 0|1 1|0 0|0 0|0
> chr1 80000 rs8 C T 100 PASS AA=C GT 0|0 0|1 0|0 0|0
> chr1 90000 rs9 T A 100 PASS AA=T GT 0|1 0|1 0|0 0|0
> chr1 100000 rs10 A G 100 PASS AA=A GT 1|0 0|0 0|0 0|0
> chr1 110000 rs11 C T 100 PASS AA=T GT 0|1 0|1 0|0 1|12.1.2 Input options
In order to work with the data, it has to be transformed to an internal representation, namely an object of class haplohh. Let us first use the pair of files example1.hap and example1.map. In this case the allele coding in the haplotype file is conform to the 01 format (defined in the main vignette), hence setting allele_coding = "01" is appropriate. This is also the format which is used internally.
> * Reading input file(s) *
> Map info: 11 markers declared for chromosome chr1 .
> Haplotype input file in standard format assumed.
> Assume that alleles in the haplotype file are coded as:
> NA or '.': missing value
> 0: ancestral allele
> 1, 2, ...: derived allele(s).
> * Filtering data *
> Discard markers genotyped on less than 100 % of haplotypes.
> No marker discarded.
> Data consists of 8 haplotypes and 11 markers.
> Number of mono-, bi-, multi-allelic markers:
> 1 2
> 0 11This data set is constructed such that setting allele_coding = "map" or allele_coding = "none" yields an identical haplohh object. In the first case, the fourth column of the map file, standing for the ancestral allele, contains always 0, hence for every marker in the haplotype file the “allele” 0 is assigned by the map file as ancestral and thus recoded by 0; the fifth column in the map file delineates the derived alleles of each marker, here again always 1, hence the “allele” 1 of the haplotype file is translated to the first derived allele, coded by 1.
In the second case, which causes a coding by alpha-numeric order, we end up again by just replacing 0 by 0 and 1 by 1.
hh_map <- data2haplohh(hap_file = "example1.hap",
map_file = "example1.map",
allele_coding = "map",
verbose = FALSE)
identical(hh, hh_map)> [1] TRUEhh_none <- data2haplohh(hap_file = "example1.hap",
map_file = "example1.map",
allele_coding = "none",
verbose = FALSE)
identical(hh, hh_none)> [1] TRUEFinally, we use the vcf file as input. No map file has to be specified. The option polarize_vcf is set by default to TRUE, assuming ancestral alleles are given in the INFO field by key AA. The function data2haplohh then recodes the markers where reference and ancestral alleles differ. If the option is turned off, coding with respect to the reference sequence is taken and the information about ancestry lost.
Again, one yields the same internal representation of the data:
hh_vcf <- data2haplohh(hap_file = "example1.vcf",
vcf_reader = "data.table",
verbose = FALSE)
identical(hh, hh_vcf)> [1] TRUE2.2 Calculations and visualizations
2.2.1 Visualizing the sequences
The haplohh-object can be visualized by a simple plot command:
Note, however, that this kind of plot is intended only for relatively small data sets.
2.2.2 EHH
We start with the computation of (allele-specific) EHH which is used for the detection of selection within a single, supposedly homogeneous, population.
We restrict our attention to the central marker with id rs6. We start by computing EHH for its alleles. We include the number of evaluated haplotypes into the output which tells us merely that 4 haplotypes are evaluated for each allele, in agreement with their frequencies. Note that the integral iHH_D is not computed, since the EHH_D is still above the threshold limehh (default value 0.05) at the borders of the chromosome and the option discard_integration_at_border is by default TRUE.
> An object of class "ehh"
> [[1]]
> [1] "rs6"
>
> [[2]]
> FREQ_A FREQ_D
> 0.5 0.5
>
> [[3]]
> POSITION EHH_A EHH_D NHAPLO_A NHAPLO_D
> rs1 10000 0.0000000 0.5000000 0 4
> rs2 20000 0.0000000 1.0000000 0 4
> rs3 30000 0.0000000 1.0000000 4 4
> rs4 40000 0.1666667 1.0000000 4 4
> rs5 50000 0.5000000 1.0000000 4 4
> rs6 60000 1.0000000 1.0000000 4 4
> rs7 70000 0.3333333 1.0000000 4 4
> rs8 80000 0.1666667 1.0000000 4 4
> rs9 90000 0.0000000 1.0000000 4 4
> rs10 100000 0.0000000 1.0000000 0 4
> rs11 110000 0.0000000 0.3333333 0 4
>
> [[4]]
> IHH_A IHH_D
> 18816.67 NAThe individual values of the table can be easily checked “by hand” using the internal data representation of the haplotypes:
> rs1 rs2 rs3 rs4 rs5 rs6 rs7 rs8 rs9 rs10 rs11
> HG1_1 0 0 1 1 0 0 0 0 0 1 1
> HG1_2 0 0 0 0 1 0 1 0 1 0 0
> HG2_1 0 1 0 1 0 0 1 0 0 0 1
> HG2_2 0 0 1 0 0 0 0 1 1 0 0
> HG3_1 0 0 0 0 0 1 0 0 0 0 1
> HG3_2 0 0 0 0 0 1 0 0 0 0 1
> HG4_1 1 0 0 0 0 1 0 0 0 0 0
> HG4_2 0 0 0 0 0 1 0 0 0 0 0The chromosomes HG1_1, HG1_2, HG_2_1 and HG_2_2 carry the ancestral allele of marker rs6 and form the “starting set” of shared haplotypes to be extended along the chromosome. By definition the starting set is homozygous and EHH equal to 1 at the focal marker. The shared haplotypes cover so far a single chromosomal position, namely that of the focal marker. We extend them to the next marker on the right, rs7, where two chromosomes carry one allele and the other two another allele (ancestry status matters only at the focal marker!). Hence the initial set of four can be subdivided into two sets of extended shared haplotypes, namely {HG1_1, HG2_2} sharing 00 and {HG1_2, HG2_1} sharing 01. They cover now the region from rs6 til rs7. Plugging the numbers into the formula for EHH (see main vignette) yields
\[
EHH^a_{rs6,rs7}=\frac{1}{n_a(n_a-1)}\sum_{k=1}^{K^a_{rs6,rs7}}n_k(n_k-1)=\frac{1}{4\cdot 3}(2\cdot1+2\cdot1)=\frac{1}{3}\;.
\]
One marker further to the right, at rs8, the first set can be partitioned into two, with HG1_1 having extended haplotype 000, and HG2_2 001. The second set {HG1_2,HG2_1} remains homozygous, now sharing the extended haplotype 010. Thus, at marker rs8 the corresponding EHH value yields
\[
EHH^a_{rs6,rs8}=\frac{1}{n_a(n_a-1)}\sum_{k=1}^{K^a_{rs6,rs8}}n_k(n_k-1)=\frac{1}{4\cdot 3}(1\cdot0+2\cdot1+1\cdot0)=\frac{1}{6}\;.
\]
At marker rs9, chromosome HG1_2 carries allele 1 and HG2_1 allele 0. Hence the extended haplotypes now differ between all four considered chromosomes and EHH becomes zero.
An analogous, but independent, calculation can be done for the markers to the left of the focal marker.
The starting set for the derived allele consists of {HG3_1, HG3_2, HG4_1 and HG4_2}. Extending to the right, the corresponding haplotypes remain homozygous and consequently the set is not split until marker rs11. In particular we have
\[EHH^d_{rs6,rs7}=EHH^d_{rs6,rs8}=\frac{1}{4\cdot3}\sum_{k=1}^14\cdot3=1\]
and essentially the same situation on the left side of the focal marker.
The corresponding plot (Figure 2.1) shows that EHH of the ancestral allele decays more rapidly than that of the derived allele.
Figure 2.1: Graphical output of the plot.ehh() function
Assume now that the haplotypes are not phased. That means, at each marker for which a diploid individual is heterozygous, it is unknown which allele belongs to chromosome ‘1’ and which to chromosome ‘2’. In this case the concept of extended haplotypes is not well-defined across individuals. However, we can still measure the decay of extended homozygosity within individuals. This is done by setting option phased to FALSE while assuming that the haplotypes in the input files are ordered as pairs belonging to individuals.
> An object of class "ehh"
> [[1]]
> [1] "rs6"
>
> [[2]]
> FREQ_A FREQ_D
> 0.5 0.5
>
> [[3]]
> POSITION EHH_A EHH_D NHAPLO_A NHAPLO_D
> rs1 10000 0.0 0.5 0 4
> rs2 20000 0.0 1.0 0 4
> rs3 30000 0.0 1.0 0 4
> rs4 40000 0.0 1.0 4 4
> rs5 50000 0.5 1.0 4 4
> rs6 60000 1.0 1.0 4 4
> rs7 70000 0.0 1.0 4 4
> rs8 80000 0.0 1.0 0 4
> rs9 90000 0.0 1.0 0 4
> rs10 100000 0.0 1.0 0 4
> rs11 110000 0.0 1.0 0 4
>
> [[4]]
> IHH_A IHH_D
> 13537.5 NAIndividuals which are heterozygous for the focal marker are excluded from the calculation. In our small sample all 4 individuals are homozygous at marker rs6; HG1 and HG2 for the ancestral allele and HG3 and HG4 for the derived allele, hence no haplotype is excluded. Note that in realistic data a substantial number of haplotypes is expected to belong to heterozygous individuals, cf. the Hardy-Weinberg principle.
Again we can retrace the calculations by hand keeping in mind that EHH is now estimated by the fraction of homozygous individuals at each marker, see the corresponding formula in the main vignette.
Extending the shared haplotypes to the right, we find that both individuals HG1 and HG2 become heterozygous already at marker rs7 and hence EHH at this position becomes 0. Extending to the left, HG1 becomes heterozygous at marker rs5, while HG2 is still homozygous in the region spanning from rs6 to rs5. Hence the proportion of homozygous individuals at this marker is \(\frac{1}{2}\). At marker rs4 the second individual becomes heterozygous, too, and EHH yields 0.
By contrast, the individuals carrying the derived focal allele are homozygous for the entire chromosome except for marker rs1 where HG4 becomes heterozygous.
Figure 2.2 shows again the difference between EHH of the two core alleles:
Figure 2.2: Graphical output of the plot.ehh() function
2.2.3 EHHS
Next, we calculate EHH per Site or EHHS which forms the basis for cross-population comparisons. The options of the corresponding function calc_ehhs() are essentially identical to those of the function calc_ehh(). The output contains EHHS and its normalized version nEHHS, distinguished merely by a factor such that the latter becomes 1 at the focal marker. We toggle the option discard_integration_at_border to FALSE in order to obtain the two corresponding integrals iES and inES even though the values EHHS and nEHHS have not fallen below the default threshold of 0.05 at the first resp. last marker. Note that elements 3 and 4 of the list can be addressed by res$IES and res$INES, too.
> An object of class "ehhs"
> [[1]]
> [1] "rs6"
>
> [[2]]
> POSITION EHHS NEHHS NHAPLO
> rs1 10000 0.10714286 0.2500000 8
> rs2 20000 0.21428571 0.5000000 8
> rs3 30000 0.21428571 0.5000000 8
> rs4 40000 0.25000000 0.5833333 8
> rs5 50000 0.32142857 0.7500000 8
> rs6 60000 0.42857143 1.0000000 8
> rs7 70000 0.28571429 0.6666667 8
> rs8 80000 0.25000000 0.5833333 8
> rs9 90000 0.21428571 0.5000000 8
> rs10 100000 0.21428571 0.5000000 8
> rs11 110000 0.07142857 0.1666667 8
>
> [[3]]
> [1] 19821.43
>
> [[4]]
> [1] 52916.67The calculations can be retraced easily. In fact, the only distinction to EHH is that all haplotypes are considered and not only those with a specific allele at the focal marker. Using visual inspection of the haplotype data file (see above), we can plug the numbers into the formula for EHHS (see main vignette): \[\mathrm{EHHS}_{rs6,rs6}=\frac{1}{n_s(n_s-1)}\left(\sum\limits_{k=1}^{K_{rs6,rs6}}n_k(n_k-1)\right)=\frac{1}{8\cdot7}(4\cdot3+4\cdot3)=\frac{3}{7}\] \[\mathrm{EHHS}_{rs6,rs7}=\frac{1}{n_s(n_s-1)}\left(\sum\limits_{k=1}^{K_{rs6,rs7}}n_k(n_k-1)\right)=\frac{1}{8\cdot7}(2\cdot1+2\cdot1+4\cdot3)=\frac{2}{7}\] \[\mathrm{EHHS}_{rs6,rs8}=\frac{1}{n_s(n_s-1)}\left(\sum\limits_{k=1}^{K_{rs6,rs8}}n_k(n_k-1)\right)=\frac{1}{8\cdot7}(1\cdot0+2\cdot1+1\cdot0+4\cdot3)=\frac{1}{4}\;.\]
By default, the following command shows the (un-normalized) EHHS values as in Figure 2.3. In order to draw the normalized values one can toggle the option nehhs to TRUE.
Figure 2.3: Graphical output of the plot.ehhs() function
2.2.4 “Genome-wide” scan
In order to find out whether marker rs6 is “special”, we compute the integrals over EHH and EHHS, yielding resp. iHH and iES, for all markers, hence performing a “genomic scan”. Since we are dealing with a single population, only the iHH values are of interest. We can see that IHH_A and IHH_D are particularly different for the central marker.
> CHR POSITION FREQ_A FREQ_D NHAPLO_A NHAPLO_D IHH_A IHH_D IES
> rs1 chr1 10000 0.875 0.125 7 1 21992.26 0.00 15295.238
> rs2 chr1 20000 0.875 0.125 7 1 36190.48 0.00 25892.857
> rs3 chr1 30000 0.750 0.250 6 2 45000.00 23512.50 22678.571
> rs4 chr1 40000 0.750 0.250 6 2 47666.67 28025.00 24285.714
> rs5 chr1 50000 0.875 0.125 7 1 33333.33 0.00 23750.000
> rs6 chr1 60000 0.500 0.500 4 4 18816.67 89166.67 19821.429
> rs7 chr1 70000 0.750 0.250 6 2 45333.33 28025.00 23035.714
> rs8 chr1 80000 0.875 0.125 7 1 38571.43 0.00 27678.571
> rs9 chr1 90000 0.750 0.250 6 2 50666.67 23512.50 25714.286
> rs10 chr1 100000 0.875 0.125 7 1 35000.00 0.00 25000.000
> rs11 chr1 110000 0.500 0.500 4 4 25075.00 25833.33 8032.143
> INES
> rs1 21992.26
> rs2 36190.48
> rs3 43437.50
> rs4 46250.00
> rs5 33333.33
> rs6 52916.67
> rs7 44062.50
> rs8 38571.43
> rs9 48750.00
> rs10 35000.00
> rs11 25416.67In order to evaluate the differences, the log ratio between IHH_A and IHH_D is calculated, yielding the, as yet, un-normalized values unihs. In general these should be normalized bin-wise, grouping markers with the same frequency of the derived allele. With so few markers, though, it is appropriate to normalize over all markers at once by setting the bin number to 1.
> $ihs
> CHR POSITION IHS LOGPVALUE
> rs1 chr1 10000 NA NA
> rs2 chr1 20000 NA NA
> rs3 chr1 30000 0.5813076 0.25101145
> rs4 chr1 40000 0.4464354 0.18357125
> rs5 chr1 50000 NA NA
> rs6 chr1 60000 -1.9389615 1.27979084
> rs7 chr1 70000 0.3890668 0.15662597
> rs8 chr1 80000 NA NA
> rs9 chr1 90000 0.7168779 0.32472642
> rs10 chr1 100000 NA NA
> rs11 chr1 110000 -0.1947262 0.07283128
>
> $frequency.class
> N_MRK MEAN_UNIHS SD_UNIHS LOWER_QT UPPER_QT
> [0.05,0.95) 11 0.1405648 0.8748648 -1.365018 0.7529103We can use calc_candidate_regions() to delineate regions with “extreme” scores hinting at selection. Because we have only a few markers, we choose an unrealistically low threshold of 1.5 and set the window size to 1, which means that no windowing is performed and only the positions of extreme markers returned.
> CHR START END EXTR_MRK
> rs6 chr1 60000 60000 -1.938961Under the assumption that most sites evolve neutrally, the standardized iHS values should follow a normal distribution with the sites under selection as outliers.
Obviously we do not have enough markers to fit a distribution, but a “genome-wide” plot of the ihs values shows clearly that the central marker is rather an outlier (as much as is possible for such a small sample), see Figure 2.4.
Figure 2.4: Graphical output of the manhattanplot() function
2.2.5 Furcations and haplotype length
A furcation plot represents a more fine-grained visualization of the homozygosity decay. In particular, individual haplotypes can be discerned which may instigate further investigations. The labels plotted in Figure 2.5 are set in bold face, if the branches with which they are associated encompass further haplotypes.
f <- calc_furcation(hh, mrk = "rs6")
# set equal plot margins on left and right side and save old ones
oldpar <- par(mar = (c(5, 3, 4, 3) + 0.1))
plot(f,
# increase line width
lwd = 1.5,
# set habplotype identifiers as labels
hap.names = hap.names(hh),
# find a place for the legend ...
legend.xy.coords = c(60000, 0.2)
)
# reset old margins
par(oldpar)
Figure 2.5: Graphical output of the plot.furcation() function
A furcation diagram consists of trees for each allele and both sides (“left” and “right”) of the marker. The individual trees can be exported into a string in Newick format to be rendered by external programs, e.g. the phylogenetic R-package ape, see Figure ??.
> [1] "(((HG2_2:0,(HG2_1:0,HG1_1:0):10000):10000,HG1_2:0):10000);"library(ape)
tree <- ape::read.tree(text = newick)
plot(tree,
direction = "leftwards",
edge.color = "blue",
underscore = TRUE,
no.margin = TRUE)
(#fig:newick1, )Graphical output of the plot.phylo() function of package ape
The end points of shared extended haplotypes can be defined as the “last split” in a furcation, i.e. the positions until which at least two different chromosomes of the sample are homozygous. Calculation of shared haplotype length and its visualization in Figure 2.6 are called by:
Figure 2.6: Graphical output of the plot.haplen() function
In case of unphased haplotypes, furcations can only occur within individuals which limits the informative value of furcation diagrams as in Figure 2.7.
f <- calc_furcation(hh, mrk = "rs6", phased = FALSE)
# set equal plot margins on left and right side and save old ones
oldpar <- par(mar = (c(5, 3, 4, 3) + 0.1))
plot(f,
# increase line width
lwd = 1.5,
# set haplotype identifiers as labels
hap.names = hap.names(hh),
# no place for a legend inside the plot
legend.xy.coords = "none")
# reset old margins
par(oldpar)
Figure 2.7: Graphical output of the plot.haplen() function
Nevertheless, the length of shared haplotypes, now identical to the ranges of individual homozygosity, can be calculated as before to yield Figure 2.8.
Figure 2.8: Graphical output of the plot.haplen() function
