For more background information on the functions described here,
please refer to the 2021
article of Liao et al. [5].
1. Introduction
By their very nature, probabilistic datasets in polyploids are larger
than discrete ones: each marker-individual combination has
ploidy + 1 floating-point numbers associated with it, in
comparison to a single integer value for a discrete dataset. For
example, we might encounter the following probabilistic data for 5 SNP
markers in a single tetraploid individual:
| snp01 |
0.999849 |
0.000151 |
0 |
0 |
0 |
| snp02 |
0.000238 |
0.999762 |
0 |
0 |
0 |
| snp03 |
0.000000 |
1.000000 |
0 |
0 |
0 |
| snp04 |
0.000001 |
0.999999 |
0 |
0 |
0 |
| snp05 |
0.009729 |
0.990271 |
0 |
0 |
0 |
Using discrete dosages (e.g. by selecting the maximum
probability) would give the following table for the same individual:
| snp01 |
0 |
| snp02 |
1 |
| snp03 |
1 |
| snp04 |
1 |
| snp05 |
1 |
However, there are some advantages to probabilistic genotypes - in
cases where uncertainty exists in the genotype assignment, it might not
be obvious how to discretise the genotype scores. This may lead to
observations being marked as missing data or even being entirely removed
from the dataset, depending on what filtering thresholds are used.
Probabilistic genotypes are a standard output from many polyploid
genotype calling procedures. This vignette assumes the R package
fitPoly has been used (developed to score the output of SNP
arrays). polymapR also includes functions to convert the
output of other polyploid genotype callers developed for sequencing
reads into a compatible format. For example, output of the
multiflex function of updog can be
re-formatted using the convert_updog function, while
convert_polyRAD does the same with the
RADobject output of function
PipelineMapping2Parents in R package
polyRAD.
3. Initial mapping steps
We will assume you have already installed or updated the version of
polymapR to at least version 1.1.0. If not, please install
the latest version of polymapR from CRAN:
install.packages("polymapR")
library(polymapR)
Assuming you have not altered the filename of the sample dataset, we
will first load the genotype data into R:
geno <- read.csv("fitPoly_4x_output_2343_SNPs.csv")
Have a look at the layout of the data, as this is generally the
format that is expected for probabilistic genotypes:
| 1 |
LG1_88.17 |
F1_001 |
0.2548363 |
0.9998493 |
0.0001507 |
0 |
0 |
0 |
0 |
0.9998493 |
0 |
| 1 |
LG1_88.17 |
F1_002 |
0.3823194 |
0.0002380 |
0.9997620 |
0 |
0 |
0 |
1 |
0.9997620 |
1 |
| 1 |
LG1_88.17 |
F1_003 |
0.4570357 |
0.0000000 |
1.0000000 |
0 |
0 |
0 |
1 |
1.0000000 |
1 |
| 1 |
LG1_88.17 |
F1_004 |
0.4290638 |
0.0000006 |
0.9999994 |
0 |
0 |
0 |
1 |
0.9999994 |
1 |
| 1 |
LG1_88.17 |
F1_005 |
0.3535575 |
0.0097291 |
0.9902709 |
0 |
0 |
0 |
1 |
0.9902709 |
1 |
| 1 |
LG1_88.17 |
F1_006 |
0.2337918 |
0.9999929 |
0.0000071 |
0 |
0 |
0 |
0 |
0.9999929 |
0 |
For clarity, we’ll also define what the parental and offspring
samples are already. In this dataset there are two maternal replicates
(‘P1’ and ‘P1a’) but only a single paternal replicate (‘P2’). It is
quite common (recommended in fact) to have multiple parental replicates
to ensure parental calls are reliable.
parent1.reps <- c("P1","P1a")
parent2.reps <- "P2"
individuals <- setdiff(unique(geno$SampleName),c(parent1.reps,parent2.reps))
The first step in the mapping using probabilistic genotypes is to run
the checkF1 function (written by Roeland Voorrips). This
looks for concordance between parental and offspring genotypes, as well
as checking for marker skewness / distorted segregation under various
genetic models (disomic, polysomic, mixed) and returns a number of
different quality parameters that might be useful for flagging
problematic markers. See ?checkF1 for more details. We will
capture the output of checkF1, as this will be needed in
many subsequent function calls:
chk1 <- checkF1(input_type = "probabilistic",
probgeno_df = geno,
parent1 = parent1.reps,
parent2 = parent2.reps,
F1 = individuals,
polysomic = TRUE,
disomic = FALSE,
mixed = FALSE,
ploidy = 4)
This function returns a list with elements checked_F1
(the actual function output) and meta (which carries the
information about parameter settings used). Note that we have assumed
that we are dealing with a polysomic species here
(polysomic = TRUE). In cases where there is uncertainty
regarding the mode of inheritance, it can be useful to run
checkF1 multiple times, e.g. setting
polysomic = FALSE and disomic = TRUE. Quality
of the markers can be assessed by filtering on qall_mult
> 0 for example. In some cases, you may notice that parental scores
have been made missing:
## m MarkerName parent1 parent2 F1_0 F1_1 F1_2 F1_3
## 1 563 LG1_0.02 2 1 16.307194 91.47333 81.57135 10.648126
## 2 1315 LG1_0.03 1 0 91.105597 83.89326 22.70502 2.296116
## 3 437 LG1_0.13 0 NA 0.000000 101.97259 97.02741 0.000000
## 4 1130 LG1_0.21 1 2 17.758076 75.43383 91.64195 15.166141
## 5 512 LG1_0.64 4 NA 5.552898 32.96319 63.11849 72.637833
## F1_4 F1_NA P1 P1a P2 bestfit frqInvalid_bestfit Pvalue_bestfit
## 1 0.00000 0 2 2 1 1551_0 0.000 0.3895
## 2 0.00000 0 1 1 0 11_0 0.125 0.5856
## 3 0.00000 1 0 0 2 11_1 0.000 0.7259
## 4 0.00000 0 1 1 2 1551_0 0.000 0.6185
## 5 23.72759 2 4 4 1 1551_1 0.028 0.0000
## matchParent_bestfit bestParentfit frqInvalid_bestParentfit
## 1 Yes 1551_0 0.0000
## 2 Yes 11_0 0.1250
## 3 OneOK 11_1 0.0000
## 4 Yes 1551_0 0.0000
## 5 No 141_2 0.1945
## Pvalue_bestParentfit matchParent_bestParentfit q1_segtypefit q2_parents
## 1 0.3895 Yes 1 1
## 2 0.5856 Yes 0 1
## 3 0.7259 OneOK 1 1
## 4 0.6185 Yes 1 1
## 5 0.0000 OneOK 0 1
## q3_fracscored qall_mult qall_weights
## 1 1.0000 1.0000 1.0000
## 2 1.0000 0.0000 0.4444
## 3 0.9875 0.9875 0.9972
## 4 1.0000 1.0000 1.0000
## 5 0.9750 0.0000 0.4389
If you are interested, you can impute these values using the internal
function assign_parental_dosage within
polymapR as follows:
pardose <- polymapR:::assign_parental_dosage(chk = chk1,
probgeno_df = geno)
knitr::kable(head(pardose))
| LG1_0.02 |
2 |
1 |
| LG1_0.03 |
1 |
0 |
| LG1_0.13 |
0 |
3 |
| LG1_0.21 |
1 |
2 |
| LG1_0.64 |
4 |
2 |
| LG1_1.07 |
2 |
1 |
For now, we will just focus our attention on potentially problematic
markers that have had a qall_mult value of 0:
length(which(chk1$checked_F1$qall_mult == 0)) #859 markers
There appear to be quite a large number of markers with questionable
agreement between parental and offspring scores - we will remove these
as follows:
remove.mark <- chk1$checked_F1[chk1$checked_F1$qall_mult==0,"MarkerName"]
geno1 <- geno[!geno$MarkerName %in% remove.mark,]
If you are concerned about removing these markers, you can revisit
them later on and try to add them back to the map. For now we will
proceed with a cleaned-up dataset, as this makes the mapping a lot
easier.
6. Linkage analysis
The rest of the mapping pipeline is very similar to that of the
original polymapR pipeline. We will run quickly through the
steps here, starting with estimation of pairwise recombination
frequencies using the linkage.gp function. Note that there
are two possible methods / approaches here. The default method
(method = "approx") uses an approximation, by summing the
probabilities of each genotype class across the population, and using
these as estimates for different marker classes in the two-point
estimation of recombination frequency. This has been shown to lead to
biased estimates of recombination frequency (thanks to Marcelo
Mollinari, NCSU for highlighting this), but has the advantage that (a)
the calculation is much faster and (b) the bias is in general rather
small and can probably be ignored. The full likelihood, in which all
dosage probabilities are carried through in the calculation, has been
implemented in the mappoly function
est_pairwise_rf, the results of which can also be used in
polymapR. Therefore, although there can be quite a
computational penalty involved here, the results using
method = "mappoly" are of higher accuracy.
The function, like a number of others in polymapR allows
for parallel processing using the argument ncores. To
detect how many available cores your computer has, as well as
determining a reasonable number of them to use (max - 2 for example) we
first run the following:
nc <- parallel::detectCores() - 2
Note that because we have filtered the marker dataset, we need to
re-run checkF1 at this point to have a compatible genotype
and checkF1 object:
chk1 <- checkF1(input_type = "probabilistic",
probgeno_df = geno2,
parent1 = parent1.reps,
parent2 = parent2.reps,
F1 = individuals,
polysomic = TRUE,
disomic = FALSE,
mixed = FALSE,
ploidy = 4)
In what follows, we are using the approximate method for
recombination frequency estimation. If the full likelihood approach is
required, make sure to specify method = "mappoly in the
calls to the linkage.gp function.
6.1 Simplex markers - clustering homologues
We begin with the simplex markers in parent 1 in order to begin with
marker clustering:
SN_SN_P1 <- linkage.gp(probgeno_df = geno2,
chk = chk1,
markertype1 = c(1,0),
target_parent = "P1",
LOD_threshold = 3,
ncores = nc)
The output of this function is a linkage data.frame, with the
following format:
## marker_a marker_b r LOD phase
## 1 LG1_1.73 LG1_13.91 0.01695488 4.562119 repulsion
## 2 LG1_1.73 LG1_15.75 0.16375769 21.050858 coupling
## 3 LG1_1.73 LG1_16.6 0.09751892 3.133736 repulsion
## 6 LG1_1.73 LG1_22.73 0.21450923 14.975402 coupling
## 8 LG1_1.73 LG1_28.44 0.22455287 13.874338 coupling
## 10 LG1_1.73 LG1_31.52 0.23455153 12.827951 coupling
We can immediately begin to cluster this data into homologues using
the cluster_SN_markers function as follows (after first
setting par(mfrow) to 2 rows and 1 column to keep the 2
output plots together):
par(mfrow = c(2,1))
P1_homologues <- cluster_SN_markers(linkage_df = SN_SN_P1,
LOD_sequence = seq(3,12,0.5),
LG_number = 5,
ploidy = 4,
parentname = "P1",
plot_clust_size = FALSE,
min_clust_size = 3)
Here, we specify that we expect there to be 5 chromosomal linkage
groups (LG_number = 5) and we look at the clustering over a
range of LOD values (from 3 to 12 in steps of 0.5). We also demand that
the min_clust_size is 3, so we are not interested in
clusters with fewer than 3 markers (you can choose this value yourself).
Had we set plot_clust_size to be TRUE, we
would see instead the number of markers per cluster rather than a
visualisation of their size with a red dot.
Although we are looking for 20 clusters here (an auto-tetraploid with
5 chromosomes, so 5 x 4 = 20 homologue clusters are expected per
parent), we can immediately see that there is a clear splitting of the
data into 5 chromosomes at a lower LOD scores (as well as some “noise” -
small clusters of markers that are separate from the rest and will be /
should be discarded). It is less simple to see at which LOD score the
homologues split apart (in fact, there are variable LOD scores that are
relevant). If we look at the upper plot we see that there is a sort of
plateau in the data (black solid line) between LOD 5 - 8 approx, and so
we could take some LOD value in this range. It is useful to have a look
at the breakdown of cluster size in more detail:
sort(table(P1_homologues[['7']]$cluster),decreasing = T)
##
## 8 16 17 18 5 3 9 7 1 2 21 10 15 19 20 4 6 11 12 13 14
## 17 15 15 14 12 11 11 10 9 9 9 8 8 8 8 7 7 3 3 3 3
length(table(P1_homologues[['7']]$cluster))
## [1] 21
So there are 21 clusters at LOD 7 (we wanted 20, this is close
enough) and there seems to be a good distribution of markers across
these clusters (at most 17 markers and at least 3, but all with more or
less the same order of magnitude in terms of size). We’ll go with this
for now, and switch our attention to parent 2 in the same way:
SN_SN_P2 <- linkage.gp(probgeno_df = geno2,
chk = chk1,
markertype1 = c(1,0),
target_parent = "P2",
LOD_threshold = 3,
ncores = nc)
We cluster the parent 2 data as before:
P2_homologues <- cluster_SN_markers(linkage_df = SN_SN_P2,
LOD_sequence = seq(3,12,0.5),
LG_number = 5,
ploidy = 4,
parentname = "P2",
plot_clust_size = F,
min_clust_size = 3)
Here the plateau in the upper plot is less obvious, but it is
possible to see the chromosomal linkage groups splitting up nicely into
homologue clusters in the lower plot. At LOD 4.5 we seem to have
identified our chromosomes already. If we follow these we can see that
around LOD 6 we have identified for the most part 4 homologues (this
would have been better at LOD 5 for the first chromosome - we are
working in this direction but the function doesn’t handle it yet). For
the moment, we will use a LOD of 6:
sort(table(P2_homologues[['6']]$cluster),decreasing = T)
##
## 22 6 12 3 7 10 17 11 16 2 15 18 4 14 1 8 13 20 5 9 19 21
## 34 25 24 21 18 18 17 16 16 15 13 13 12 10 9 9 9 6 4 3 3 3
length(table(P2_homologues[['6']]$cluster))
## [1] 22
Here we see there are 22 clusters - again close enough to what we
want.
6.2 Higher-dose markers - connecting clustered homologues
In this dataset it would have been possible to define chromosomal
linkage groups from the 1x0 data alone - see the main vignette for how
to proceed if this is the case in your data also (using the function
define_LG_structure). However, for now we will assume
things are not so straightforward. We next turn to the simplex x simplex
markers (1x1) which can help provide bridging links between homologues
to assist in the clustering puzzle (duplex markers can also be used, see
the main vignette for more details here). We first calculate the linkage
between 1x0 and 1x1 markers, and then use this information for joining
our homologue clusters together into chromosomal linkage groups.
SN_SS_P1 <- linkage.gp(probgeno_df = geno2,
chk = chk1,
markertype1 = c(1,0),
markertype2 = c(1,1),
target_parent = "P1",
ncores = nc)
SN_SS_P2 <- linkage.gp(probgeno_df = geno2,
chk = chk1,
markertype1 = c(1,0),
markertype2 = c(1,1),
target_parent = "P2",
ncores = nc)
Using the function bridgeHomologues we can now look for
associations between the homologue clusters, first in parent 1:
LGHomDf_P1 <- bridgeHomologues(cluster_stack = P1_homologues[["7"]],
cluster_stack2 = P2_homologues[["7"]],
linkage_df = SN_SS_P1,
linkage_df2 = SN_SS_P2,
LOD_threshold = 5,
LG_number = 5)
One of the chromosomes has too many homologues while another one has
only 3. We can see this more clearly using the table
function:
table(LGHomDf_P1$LG,LGHomDf_P1$homologue)
##
## 1 2 3 4 5
## 1 9 9 11 7 0
## 2 12 7 10 0 0
## 3 17 11 8 3 3
## 4 3 8 15 15 0
## 5 14 8 8 9 0
However, we do have a clear clustering into 5 linkage groups, the
number of chromosomes we expected in this species.
We can perform a similar step in parent 2, however this time at LOD
6:
LGHomDf_P2 <- bridgeHomologues(cluster_stack = P2_homologues[["6"]],
cluster_stack2 = P1_homologues[["6"]],
linkage_df = SN_SS_P2,
linkage_df2 = SN_SS_P1,
LOD_threshold = 5,
LG_number = 5)
Again have a look more closely:
table(LGHomDf_P2$LG,LGHomDf_P2$homologue)
##
## 1 2 3 4
## 1 9 15 21 12
## 2 25 18 9 3
## 3 18 16 24 9
## 4 10 13 16 17
## 5 13 6 34 0
So again we are missing one homologue, but for the rest it seems
quite OK.
We have a number of options to merge homologues (or just delete the
small ones) - as it is not possible to proceed in a tetraploid mapping
with more than 4 homologues per parent. In this example dataset we’ll
use the function cluster_per_LG:
LGHomDf_P1a <-cluster_per_LG(LG = 3,
linkage_df = SN_SN_P1[SN_SN_P1$phase == "coupling",],
LG_hom_stack = LGHomDf_P1,
LOD_sequence = 3:10,
modify_LG_hom_stack = TRUE,
network.layout = "stacked",
nclust_out = 4,
label.offset=1.2)
In this example, it is not possible to merge homologues
satisfactorily using this function (even if we go down to LOD = 0).
Since we have specified that there should be at most 4 clusters in the
output (nclust_out = 4) and that we want to modify the
LG_hom_stack object (modify_LG_hom_stack = TRUE), the
function does the only thing it can in this situation - delete the
smallest homologue (here, it is a random decision as there were two
small homologues with only 3 markers)…
table(LGHomDf_P1a$LG,LGHomDf_P1a$homologue)
##
## 1 2 3 4
## 1 9 9 11 7
## 2 12 7 10 0
## 3 17 3 11 8
## 4 3 8 15 15
## 5 14 8 8 9
In parent 2 we don’t have to do any merging as we had no situation of
more than 4 homologues.
6.3 Marker assignments
Now that we have a structure with simplex markers that define both a
LG and a homologue, we are in a position to start assigning other marker
types - if you are unclear about what we just did, have a look at one of
the LGHomDf stacks, e.g.:
## SxN_Marker LG homologue
## 1 LG1_1.73 1 1
## 2 LG1_31.52 1 1
## 3 LG1_56.45 1 1
## 4 LG1_71.82 1 1
## 5 LG1_79.15 1 1
## 6 LG1_88.76 1 1
This is not quite a marker assignment object, but it is getting
close. We have already calculated linkage with 1x1 markers, so these can
be assigned directly as follows:
P1_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P1,
LG_hom_stack = LGHomDf_P1a,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
P2_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P2,
LG_hom_stack = LGHomDf_P2,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
We can now directly compare the assignments in both parents to get a
“consensus” naming of the chromosomal linkage groups across parents:
LGHomDf_P2c <- consensus_LG_names(modify_LG = LGHomDf_P2,
template_SxS = P1_SxS_Assigned,
modify_SxS = P2_SxS_Assigned)
##
## #### Original LG names
##
## | | 1| 2| 3| 4| 5|
## |:--|--:|--:|--:|--:|--:|
## |1 | 21| 0| 0| 0| 0|
## |2 | 0| 19| 0| 0| 0|
## |3 | 0| 0| 0| 20| 0|
## |4 | 0| 0| 0| 0| 15|
## |5 | 0| 0| 19| 0| 0|
##
## #### Modified LG names
##
## | | 1| 2| 3| 4| 5|
## |:--|--:|--:|--:|--:|--:|
## |1 | 21| 0| 0| 0| 0|
## |2 | 0| 19| 0| 0| 0|
## |3 | 0| 0| 20| 0| 0|
## |4 | 0| 0| 0| 15| 0|
## |5 | 0| 0| 0| 0| 19|
Overall, looking at the diagonal in the lower table (these are the
number of 1x1 markers that have been assigned to this linkage group in
both parents), it is clear which linkage groups correspond across
parents.
It is a good idea to save these objects (e.g. using
save or saveRDS):
save(LGHomDf_P1a,LGHomDf_P2c, file = "LGHomDf_stacks.Rdata") #for example..
We re-run the P2 assignment for the 1x1 markers, now that our naming
of linkage groups has changed in this parent:
P2_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P2,
LG_hom_stack = LGHomDf_P2c, #this is changed
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
We now run the complete marker assignment for both parents and the
rest of the markers, using linkage with our 1x0 markers:
marker_assignments_P1 <- homologue_lg_assignment(input_type = "probabilistic",
probgeno_df = geno2,
chk = chk1,
assigned_list = list(P1_SxS_Assigned),
assigned_markertypes = list(c(1,1)),
LG_hom_stack = LGHomDf_P1a,
target_parent = "P1",
other_parent = "P2",
ploidy = 4,
pairing = "random",
convert_palindrome_markers = FALSE,
LG_number = 5,
LOD_threshold = 3,
write_intermediate_files = FALSE)
marker_assignments_P2 <- homologue_lg_assignment(input_type = "probabilistic",
probgeno_df = geno2,
chk = chk1,
assigned_list = list(P2_SxS_Assigned),
assigned_markertypes = list(c(1,1)),
LG_hom_stack = LGHomDf_P2c,
target_parent = "P2",
other_parent = "P1",
ploidy = 4,
pairing = "random",
convert_palindrome_markers = FALSE,
LG_number = 5,
LOD_threshold = 3,
write_intermediate_files = FALSE)
Once this step has completed (it takes a couple of minutes) we can
compare chromosomal linkage group assignments for all markers across
parents and screen out those markers that give conflicting results, a
further quality check:
marker_assignments <- check_marker_assignment(marker_assignments_P1,marker_assignments_P2)
This is a very important object and it is recommended to save this
also (rather than relying on it being saved in a heap in the Global
Environment, which is a somewhat sloppy practice):
saveRDS(marker_assignments, file = "marker_assignments.RDS")
The final step in the linkage analysis is to compile linkage
estimates for all pairs of markers within a chromosomal linkage
group using the finish_linkage_analysis function. This step
may take a bit longer, so this is a good time to catch up on some other
work, having set both of the following function calls running:
all_linkages_list_P1 <- finish_linkage_analysis(input_type = "probabilistic",
marker_assignment = marker_assignments$P1,
probgeno_df = geno2,
chk = chk1,
target_parent = "P1",
other_parent = "P2",
convert_palindrome_markers = FALSE,
ploidy = 4,
pairing = "random",
LG_number = 5,
ncores = nc)
all_linkages_list_P2 <- finish_linkage_analysis(input_type = "probabilistic",
marker_assignment = marker_assignments$P2,
probgeno_df = geno2,
chk = chk1,
target_parent = "P2",
other_parent = "P1",
convert_palindrome_markers = TRUE,
ploidy = 4,
pairing = "random",
LG_number = 5,
ncores = nc)
We compile the results from both parents into a single list as
follows:
linkages <- list()
for(lg in names(all_linkages_list_P1)){
linkages[[lg]] <- rbind(all_linkages_list_P1[[lg]], all_linkages_list_P2[[lg]])
}
This (large) object is also worth saving. We will again use the
.RDS format:
saveRDS(linkages, file = "linkages.RDS")
This can be read back in using readRDS in future:
linkages <- readRDS("linkages.RDS")
8. References
1. Bourke PM et al: polymapR—linkage analysis and genetic map
construction from F1 populations of outcrossing polyploids.
Bioinformatics 2018, 34:3496–3502.
2. Voorrips RE, Gort G, Vosman B: Genotype
calling in tetraploid species from bi-allelic marker data using mixture
models. BMC bioinformatics 2011,
12:172.
3. Gerard D, Ferrão LFV, Garcia AAF, Stephens M: Genotyping
polyploids from messy sequencing data. Genetics 2018,
210:789–807.
4. Clark LV, Lipka AE, Sacks EJ: polyRAD: Genotype calling with
uncertainty from sequencing data in polyploids and diploids.
G3: Genes, Genomes, Genetics 2019, 9:663–673.
5. Liao Y, Voorrips RE, Bourke PM, Tumino G, Arens P, Visser RG,
Smulders MJ, Maliepaard C: Using probabilistic genotypes in
linkage analysis of polyploids. Theoretical and Applied
Genetics 2021, 134:2443--2457.
6. Voorrips RE, Maliepaard CA: The simulation of meiosis in
diploid and tetraploid organisms using various genetic models.
BMC bioinformatics 2012, 13:248.
7. Preedy KF, Hackett CA: A rapid marker
ordering approach for high-density genetic linkage maps in experimental
autotetraploid populations using multidimensional
scaling. Theoretical and Applied Genetics 2016,
129:2117–2132.
