In this tutorial, we will go step by step through the basic steps of
performing linkage analysis in polyploids using polymapR.
Most of the theory behind the functions described here can be found in
the Bioinformatics publication of Bourke et al. (2018) at https://doi.org/10.1093/bioinformatics/bty371. The
example dataset is a simulated dataset of an hypothetical tetraploid
outcrossing plant species with five chromosomes. Chromosomes have random
pairing behaviour. The population consists of 207 individuals
originating from an F1 cross between two non-inbred parents. Frequently
occurring data imperfections like missing values, genotyping errors and
duplicated individuals are taken into account.
7. Quality checks on marker data
A number of quality checks are recommended before we begin mapping.
For example, we do not want to include markers which have too many
missing values (“NA” scores) as this may be an indication that the
marker has performed poorly and may have many errors associated with it.
We would also like to screen out individuals that have many missing
values as this might be an indication that the DNA quality of that
sample was poor, or duplicate individuals which give no extra
information and should probably also be removed / combined. Because
there are no fixed rules for how stringent we should be with quality
checks (5% missing values ok? 10% missing values ok? 20%?), some visual
aids are produced in order to help with these decisions.
7.1 Missing value rate per marker
We first screen the marker data for missing values using
screen_for_NA_values. This function allows the user to
remove rows or columns with a certain rate of missing values from the
dataset. The dosage matrix, which is specified as
segregating_data contains markers in rows and individuals
in columns. In R, rows are usually specified by the number 1, and
columns by 2. The argument margin lets you specify by
margin number whether you want to screen markers
(margin = 1) or individuals (margin = 2) If
the argument cutoff is set to NULL, it takes
user input on the rate of markers to be screened. In the example below
cutoff is specified, but because decisions are made after
data inspection, we recommend to decide on the cutoff value after
inspection of the data by specifying cutoff = NULL. A
threshold of 10% missing values is sensible, as a compromise between
high-confidence marker data versus keeping a large proportion of the
markers for mapping and QTL analysis. In general, the most problematic
markers tend to have higher rates of missing values so any reasonable
threshold here should be fine. If problems are encountered later in the
mapping, it might be worthwhile to re-run this step with a more
stringent (e.g. 5%) threshold.
screened_data <- screen_for_NA_values(dosage_matrix = segregating_data,
margin = 1, # margin 1 means markers
cutoff = 0.10,
print.removed = FALSE)
##
## Number of removed markers : 1458
##
##
## There are now 1415 markers leftover in the dataset.
7.2 Missing value rate per individual
As in step 7.1 we can also visualise the rate of missing values per
individual, which can help us to make a decision about whether certain
individuals should be removed from the dataset before commencing the
mapping. In order to screen the individuals for missing values we
enter:
screened_data2 <- screen_for_NA_values(dosage_matrix = screened_data,
cutoff = 0.1,
margin = 2, #margin = 2 means columns
print.removed = FALSE)
##
## Number of removed individuals : 2
##
##
## There are now 205 individuals leftover in the dataset.
Again, it is not clear what an acceptable threshold for the rate of
missing values should be, but we recommend a cut-off rate of 0.1 – 0.15
(10-15%) missing values allowed.
7.3 Duplicate individuals
It is also desirable to be able to check whether there are any
duplicated individuals in the population as can sometimes happen (or due
to mix-ups in the DNA preparation and genotyping). The function
screen_for_duplicate_individuals does this for you. Note
that a duplicate can remain in the dataset, but they add no further
information for mapping or QTL analysis, and may even distort the
analysis (by giving an inaccurate population size). As for the function
screen_for_NA_values, cutoff can be set to
NULL to accept user input. Especially for
screen_for_duplicate_individuals it makes sense to make a
decision on the cut off after inspection of the data and figures. To
screen the data for duplicate individuals, enter:
screened_data3 <- screen_for_duplicate_individuals(dosage_matrix = screened_data2,
cutoff = 0.95,
plot_cor = TRUE)
##
## Combining F1_060 & F1_081 into F1_060
##
## Combining F1_046 & F1_085 into F1_046
## Warning in screen_for_duplicate_individuals(dosage_matrix = screened_data2, :
## Multiple duplicates of single genotype identified at this threshold. Attempting
## to merge...
##
## Combining F1_032 & F1_100 into F1_032
##
## Combining F1_032 & F1_114 into F1_032
##
## Combining F1_079 & F1_101 into F1_079
##
## Combining F1_106 & F1_113 into F1_106
##
## Combining F1_112 & F1_199 into F1_112
##
## #### 7 individuals removed:
##
## _ _ _ _
## [1,] "F1_081" "F1_085" "F1_100" "F1_114"
## [2,] "F1_101" "F1_113" "F1_199" ""
## |_ |_ |_ |_ |
## |:------|:------|:------|:------|
## |F1_081 |F1_085 |F1_100 |F1_114 |
## |F1_101 |F1_113 |F1_199 | |
Here we see that there were a number of duplicate pairs of
individuals identified (red points in the output graph). Each duplicate
pair is merged into a consensus individual (conflicts are made missing),
keeping the name of the first individual. This is summarised in the
output as well.
7.4 Duplicated markers
Finally, we can also check for duplicated markers (those which give
us no extra information about a locus) and remove these from the dataset
before mapping, using the screen_for_duplicate_markers
function. This function returns a list with 2 items - the
filtered_dosage_matrix which we are interested in now, and bin_list,
which we are possibly interested in again later.
It is a good idea to save the output of this function, as you might like
to add back the duplicate markers after mapping.
screened_data4 <- screen_for_duplicate_markers(dosage_matrix = screened_data3)
## | | | 0% | |======================================================================| 100%
## Marked and merged 3 duplicated markers
Note that increasingly, there are far too many markers generated in
comparison to the population size. It is possible to estimate the
genetic resolution of the population you are working with, and from that
estimate the expected bin-size in your dataset, assuming your markers
are approximately randomly-distributed across the genome (this may not
be a good assumption in your case, but as a rough guide this function
may still be useful). Considering only bivalent pairing, in a tetrasomic
species we can assume approximately four cross-over recombinations
occurred, taking the rule-of-thumb guide of one cross-over recombination
per bivalent, which has been observed in autotetraploid Arabidopsis
arenosa for example [2].
We can try to estimate the average bin size if we run the previous
function and specifying the argument
estimate_bin_size = TRUE. The function then does a quick
calculation on how many markers would be expected in an average marker
bin (under the tenuous assumptions of uniform distributions of both
markers and cross-over break-points). In a tetraploid, each offspring
would be expected to have 4 cross-over break-points per chromosomal
linkage group (2 maternal and 2 paternal), so across a population of
N individuals there are approximately 4N breaks per
linkage group, and 4NL such breaks across L linkage
groups. Supposing there are in total y distinct marker types
(like 1x0, 1x1, 2x0, 0x1 etc), then your M markers can
be split into M/4NLy segregation patterns (assuming no errors),
which is an approximate estimate for the average bin size. In datasets
with very large numbers of markers (e.g. genotyping using NGS),
it may be worthwhile to only include markers with a minimum
number of duplicates present, and that minimum number can be informed by
the calculation above. In this case, supposing we demand at least 7
duplicates (so a minimum bin size of 6), we would do something like the
following:
reliable.markers <- names(which(sapply(screened_data4$bin_list,length) >= 6))
reliable_data <- screened_data4$filtered_dosage_matrix[reliable.markers,]
Of course, this assumes we have a very large number of markers to
choose from, as this is a very severe demand in a more traditional,
well-balanced dataset. It also presumes there are no genotyping errors.
A genotyping error will make duplicate markers appear to be different
even when they are not (from a genetic mapping perspective). For now we
are ignoring this issue, but may return to it in future updates.
In the current data-set there is quite a good balance between the
number of markers and the number of individuals in the mapping
population, and so we extract the filtered_dosage_matrix that has the
duplicate markers merged:
filtered_data <- screened_data4$filtered_dosage_matrix
Before we start the mapping, it is a good idea to run the summary
function again, this time on the filtered_data object, so
we have a clear record of the numbers of markers going forward to the
mapping stage.
pq_screened_data <- parental_quantities(dosage_matrix = filtered_data)
## | | 1x0| 2x0| 0x1| 1x1| 2x1| 0x2| 1x2| 2x2| 1x3|
## |:---------|---:|---:|---:|---:|---:|---:|---:|---:|---:|
## |frequency | 175| 111| 194| 301| 183| 101| 198| 88| 61|
8. Simplex x nulliplex markers – defining chromosomes and
homologues
The first step in actually mapping the markers is to define linkage
groups for the markers. We use the simplex x nulliplex markers to define
the linkage groups (chromosomes) and following that, the homologues (4
per linkage group are expected in a tetraploid). We use the LOD scores
to cluster markers, which increases dramatically for tight
coupling-phase estimates.
To estimate the recombination frequencies, LOD and phasing between
all marker pairs of P1 enter:
SN_SN_P1 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
parent1 = "P1",
parent2 = "P2",
which_parent = 1,
ploidy = 4,
pairing = "random"
)
A note on
arguments and multi-core processing
There is much more to the linkage function. Checkout
?linkage to see the arguments that can be passed to it. One
important feature is that it can run in parallel on multiple cores. This
can be very time-efficient if you are working with a large number of
markers. The number of cores to use can be specified using the
ncores argument. Before doing so, first check the number of
cores on your system using parallel::detectCores(). Use at
least one core less than you have available to prevent
overcommitment.
Note that since polymapR version 1.1.5, some of the arguments of the
linkage function have been updated. The arguments
target_parent and other_parent have been
removed, and replaced with parent1 and
parent2, referring to the column names of the two parents.
A new argument which_parent has been introduced, to allow
you to specify which parent is being “targeted” for linkage analysis
(either 1 or 2). This was needed to accommodate an update for mapping in
triploid populations, but also makes it easier if you rename your
parental columns “P1” and “P2” at the start, as these are the default
values provided for the arguments parent1 and
parent2. This is the case here, so for subsequent calls to
linkage we will omit specifying the parental names each
time.
It is a good idea to first visualise the relationship between the
recombination frequency and LOD score, which gives us some insight into
the differences between the information content of the different phases,
as well as highlighting possible problems in the data. This can be done
by using the r_LOD_plot function:
r_LOD_plot(linkage_df = SN_SN_P1, r_max = 0.5)
## Warning: Computation failed in `stat_binhex()`.
## Caused by error in `compute_group()`:
## ! The package "hexbin" is required for `stat_bin_hex()`.
In some situations, it is possible to identify outlying datapoints on
this plot, which do not fit the expected relationship. We have
implemented a function to check this for the case of pairs of simplex x
nulliplex markers, as these are the markers that we use in the
subsequent steps of clustering and identifying chromosome and homologue
linkage groups. If you are unsure, you can run the
SNSN_LOD_deviation function anyway to see whether your data
fits the expected pattern:
P1deviations <- SNSN_LOD_deviations(linkage_df = SN_SN_P1,
ploidy = 4,
N = ncol(filtered_data) - 2, #The F1 population size
alpha = c(0.05,0.2),
plot_expected = TRUE,
phase="coupling")
Here we see there are a few “outliers” in the coupling pairwise data
(identified by the stars) from the expected relationship between r and
LOD (the bounds of which are shown by the dotted lines - these can be
widened as necessary using the alpha argument for the upper
and lower tolerances around the “true” line, usually in a trial and
error approach). In our example dataset, these highlighted points are
hardly outliers and we can proceed. In other datasets, you may want to
remove these starred pairwise estimates before continuing with marker
clustering, as it is possible they may cause unrelated homologue
clusters to clump together and prevent a clear clustering structure from
being defined. To do so, remove them from the linkage data frame as
follows:
SN_SN_P1.1 <- SN_SN_P1[SN_SN_P1$phase == "coupling",][-which(P1deviations > 0.2),]
We are now in a position to cluster the marker data into chromosomes
using the cluster_SN_markers function. This function
clusters markers over a range of LOD thresholds. In the example below
this is over a range from LOD 3 to 10 in steps of 1
(LOD_sequence = c(3:10)). If the argument
plot_network is set to TRUE this function will
plot the network of the lowest LOD threshold (LOD 3 in this case), and
overlays the clusters of the higher LOD scores. This setting is not
recommended with large number of marker pairs. Your computer will most
likely run out of memory.
P1_homologues <- cluster_SN_markers(linkage_df = SN_SN_P1,
LOD_sequence = seq(3, 10, 0.5),
LG_number = 5,
ploidy = 4,
parentname = "P1",
plot_network = FALSE,
plot_clust_size = FALSE)
## Total number of edges: 894
## 20 clusters were expected.
The function cluster_SN_markers returns a list of
cluster data frames, one data frame for each LOD threshold. The first
plot shows two things - on the normal y-axis, the number of clusters
over a range of LOD thresholds (x-axis) - and on the right-hand y-axis,
the number of markers that do not form part of any cluster, which is
shown by the blue dashed line. In the cluster_SN_markers
function, we specify the ploidy (4 for a tetraploid) and the expected
number of chromosomes (LG_number = 5) and therefore there are 4 x 5 = 20
homologue clusters expected. If we examine the second plot, we see how
these clusters stay together or fragment as the linkage stringency (LOD
score) increases. Here, we see clearly that at LOD 3.5, there are 5
clusters which all eventually divide into 4 sub-clusters each. This is
the ideal scenario - we have identified both the chromosomal linkage
groups and the homologues using only the simplex x nulliplex marker
data. Note that it is the coupling-phase estimates which define the
homologue clusters and the repulsion-phase estimates which provide the
associations between these coupling-clusters (homologues). In practice,
things might not always be so orderly. In noisy datasets with higher
numbers of genotyping errors, or at higher ploidy levels, the
repulsion-phase linkages between homologues will be lost among false
positive linkages between un-linked markers. This means that all markers
will likely clump together at lower LOD scores.
You can manually check how many clusters there are at a specific LOD
score, and how many markers are in a cluster:
P1_hom_LOD3.5 <- P1_homologues[["3.5"]]
t <- table(P1_hom_LOD3.5$cluster)
print(paste("Number of clusters:",length(t)))
## [1] "Number of clusters: 5"
t[order(as.numeric(names(t)))]
##
## 1 2 3 4 5
## 37 36 29 31 41
P1_hom_LOD5 <- P1_homologues[["5"]]
t <- table(P1_hom_LOD5$cluster)
print(paste("Number of clusters:",length(t)))
## [1] "Number of clusters: 20"
t[order(as.numeric(names(t)))]
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 13 9 10 4 12 7 9 8 3 13 6 7 4 10 6 11 17 6 6 12
Based on the plots (or the output as shown) you should be able to
decide which LOD threshold best separates the data. In this example, at
a LOD of 3.5 we identify chromosomes and anywhere between LOD 4.5 and
LOD 7 we have identified our homologues (4 x 5 = 20 clusters). In this
parent (P1) we have therefore solved the marker clustering problem
without any difficulty. It remains to gather this into a structure that
can be used at later stages. To do this, we need two LOD values which
define chromosomal groupings (here, LOD_chm = 3.5) and
homologue groupings (here, LOD_hom = 5). We can then use
the define_LG_structure function with these arguments as
follows (note the argument LG_number refers to the expected number of
chromosomes for this species):
LGHomDf_P1 <- define_LG_structure(cluster_list = P1_homologues,
LOD_chm = 3.5,
LOD_hom = 5,
LG_number = 5)
##
## #### SxN Marker(s) lost in clustering step at LOD 5:
##
## |_ |
## |:-----------|
## |Ls_rs099908 |
## SxN_Marker LG homologue
## 1 Ac_ns035629 1 1
## 3 Ac_ns045766 1 1
## 7 Ac_ns096474 1 1
## 22 Ac_ts072686 1 1
## 24 Ac_ws013795 1 1
## 42 Ap_rs075640 1 1
However, you may find that it is not possible to identify chromosomal
groups as simply as this. We therefore recommend that homologue groups
be first identified, and then another marker type be used to provide
associations between these clusters, and hence help form chromosomal
linkage groups. This can be done as follows:
Generate homologue clusters using the
cluster_SN_markers function at a high LOD.
Use the Duplex x Nulliplex or Simplex x Simplex marker
information to put the clusters back together into chromosomal
groups.
It makes sense to use only SxN marker pairs in coupling phase for the
homologue clustering (since these associations define the homologues).
Note that the most likely phase per marker pair (coupling or repulsion)
was already estimated by our previous call to the linkage
function.
SN_SN_P1_coupl <- SN_SN_P1[SN_SN_P1$phase == "coupling",] # select only markerpairs in coupling
P1_homologues_1 <- cluster_SN_markers(linkage_df = SN_SN_P1_coupl,
LOD_sequence = c(3:12),
LG_number = 5,
ploidy = 4,
parentname = "P1",
plot_network = FALSE,
plot_clust_size = FALSE)
## Total number of edges: 682
## 20 clusters were expected.
In the dot plot above, we see that all chromosomal associations have
now been removed with the repulsion information.
A note on nested lists
In the polymapR package, output is often given as a
nested list. Some basic information on lists is available on
e.g. r-tutor. We use
nested lists because we are often dealing with hierarchical data: marker
data > homologue > linkage group > organism. Some basic R
knowledge is needed to retrieve, visualize or filter data from such a
nested list. However, if you would like to visualize it without R
(e.g. in your favourite spreadsheet program), you can choose to
export a nested list using write_nested_list. It will keep
its hierarchical structure by writing it to directories that have the
same structure as the the nested list. Here’s an example:
write_nested_list(list = P1_homologues, directory = "P1_homologues")
Next, we use linkage to the DxN markers to provide bridging linkages
between homologue clusters, using the bridgeHomologues
function. The first time we run this function, we use a LOD threshold of
4 for evidence of linkage between a SxN and a DxN marker. However, there
may also be some false positive linkages at this level. It might
therefore be necessary to run the function again at a higher LOD (say
LOD 8) to avoid such false positives. In doing so, we might lose true
linkage information. Therefore, it’s a good idea to run the function at
LOD 4 first, then increase to a higher LOD if necessary, and build up a
picture of which clusters form part of the same chromosome. First
calculate linkage between SxN and DxN markers:
SN_DN_P1 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
markertype2 = c(2,0),
which_parent = 1,
ploidy = 4,
pairing = "random")
Then, use bridgeHomologues to use these SxN - DxN
linkages to find associations (“bridges”) between homologue clusters.
Note that we must provide a particular division of the Simplex x
Nulliplex data - in this example, we use the division at LOD 6 which
gave us 20 clusters (our 20 putative homologues):
LGHomDf_P1_1 <- bridgeHomologues(cluster_stack = P1_homologues_1[["6"]],
linkage_df = SN_DN_P1,
LOD_threshold = 4,
automatic_clustering = TRUE,
LG_number = 5,
parentname = "P1")
This results in an ideal situation of 5 linkage groups each with
exactly 4 homologues:
table(LGHomDf_P1_1$LG, LGHomDf_P1_1$homologue)
##
## 1 2 3 4
## 1 13 9 4 10
## 2 12 7 9 8
## 3 3 13 5 7
## 4 4 10 6 11
## 5 17 12 6 6
However, clustering of P2 is a bit more difficult. The following
re-does everything for P2. Note that for clarity, we include the
arguments parent1 and parent2 here, but we
could have just as well left them out (as they are the default column
names, see ?linkage):
SN_SN_P2 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
parent1 = "P1",
parent2 = "P2",
which_parent = 2,
ploidy = 4,
pairing = "random"
)
SN_SN_P2_coupl <- SN_SN_P2[SN_SN_P2$phase == "coupling",] # get only markerpairs in coupling
P2_homologues <- cluster_SN_markers(linkage_df = SN_SN_P2_coupl,
LOD_sequence = c(3:12),
LG_number = 5,
ploidy = 4,
parentname = "P2",
plot_network = FALSE,
plot_clust_size = FALSE)
## Total number of edges: 743
## 20 clusters were expected.
SN_DN_P2 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
markertype2 = c(2,0),
which_parent = 2,
ploidy = 4,
pairing = "random")
LGHomDf_P2 <- bridgeHomologues(cluster_stack = P2_homologues[["6"]],
linkage_df = SN_DN_P2,
LOD_threshold = 4,
automatic_clustering = TRUE,
LG_number = 5,
parentname = "P2")
table(LGHomDf_P2$LG,LGHomDf_P2$homologue)
##
## 1 2 3 4 5
## 1 11 9 3 5 0
## 2 13 13 7 11 0
## 3 13 11 11 8 2
## 4 9 9 13 11 0
## 5 10 10 6 9 0
Occasionally linkage groups possesses fewer than four “homologues”.
This can result in a loss of phase information in later steps, but will
not prevent us from proceeding with the mapping. However, it is also
possible that more than 4 homologues are grouped together, likely the
result of having a number of fragmented homologues. However, as we don’t
know which clusters are actual homologues, and which are fragments, we
must examine the affected linkage group in more detail.
polymapR currently offers two methods for this.
8.1 Using the function overviewSNlinks
Run overviewSNlinks as follows:
overviewSNlinks(linkage_df = SN_SN_P2,
LG_hom_stack = LGHomDf_P2,
LG = 3,
LOD_threshold = 0)
## Warning: Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Caused by error in `compute_group()`:
## ! The package "hexbin" is required for `stat_bin_hex()`.
In cases of fragmented homologues, it may be possible to identify
which fragments belong together by examining the phase of intra-cluster
marker pairs. Coupling-phase (green) linkage between fragments can be an
indication that separate homologue fragments belong together. This can
be then be used to merge these homologues (specified in the argument
mergeList - use ?merge_homologues for more
information). For example, if we wanted to merge homologues 4 and 5 of
linkage group 3 as the above data suggests, we would use the
following:
LGHomDf_P2_1 <- merge_homologues(LG_hom_stack = LGHomDf_P2,
ploidy = 4,
LG = 3,
mergeList = list(c(4,5)))
## #### Number of markers per homologue:
##
## | | 1| 2| 3| 4| 5|
## |:--|--:|--:|--:|--:|--:|
## |1 | 11| 13| 13| 9| 10|
## |2 | 9| 13| 11| 9| 10|
## |3 | 3| 7| 11| 13| 6|
## |4 | 5| 11| 10| 11| 9|
8.2 Using the function cluster_per_LG
cluster_per_LG does not rely on markers in repulsion,
but uses variation in LOD thresholds between markers in coupling. Run it
as follows:
cluster_per_LG(LG = 3,
linkage_df = SN_SN_P2[SN_SN_P2$phase == "coupling",],
LG_hom_stack = LGHomDf_P2,
LOD_sequence = c(3:10), # The first element is used for network layout
modify_LG_hom_stack = FALSE,
network.layout = "stacked",
nclust_out = 4,
label.offset=1.2)
## Total number of edges: 185
The network layout is based on the first LOD score you provide in
LOD_sequence (which is LOD = 3 in this example). The darker
the background, the more consistent the clustering is over the range of
LOD scores. Change the LOD_sequence if the network is not
as required (4 clusters in the case of a tetraploid) and re-run. If the
network looks as you would like it to, use the
cluster_per_LG function but with a
LOD_sequence of length 1 (the first LOD score, so again LOD
= 3 in this example), and specify
modify_LG_hom_stack = TRUE to make a modification to the
cluster numbering (the output is then returned and can be saved as a new
LG_hom_stack object):
LGHomDf_P2_1 <- cluster_per_LG(LG = 3,
linkage_df = SN_SN_P2[SN_SN_P2$phase == "coupling",],
LG_hom_stack = LGHomDf_P2,
LOD_sequence = 3,
modify_LG_hom_stack = TRUE,
network.layout = "n",
nclust_out = 4)
## Total number of edges: 185
table(LGHomDf_P2_1$homologue, LGHomDf_P2_1$LG)
##
## 1 2 3 4 5
## 1 11 13 10 9 10
## 2 9 13 13 9 10
## 3 3 7 11 13 6
## 4 5 11 11 11 9
8.3 Cross-ploidy populations (e.g. tetraploid x diploid to
give triploid F1)
polymapR is also able to map triploid populations, and
may be extended to other cross-ploidy populations if there is demand for
that in future. Tetraploid x diploid crosses are however the most common
(often done to induce seedlessness in the F1). Previously we had
implemented a mapping approach that assumes that the parent of higher
ploidy level is parent 1. However, since polymapR v.1.1.5 this is no
longer the case, so a 4x2 and 2x4 cross are both acceptable.
The next point to be aware of is that the segregation of markers in
the diploid parent is assumed to follow disomic inheritance (there is no
other option in a diploid), whereas the segregation of the tetraploid
parent is assumed to follow tetrasomic inheritance. Mixtures with
preferential pairing in the tetraploid parent are currently not
implemented for triploid populations.
Because of this disomic behaviour in the diploid parent, we are
unable to use the normal clustering function
(cluster_SN_markers) for identifying the homologues.
Proceed as normal with the steps described above for the tetraploid
parent. However in order to proceed correctly for the diploid parent 2,
the clustering proceeds a bit differently.
If you are confused about what marker types are possible in a
triploid population, run the following, which gives all possible marker
combinations (and the segregation ratios in the F1):
get("seg_p3_random",envir=getNamespace("polymapR"))
## dosage1 dosage2 f1_0 f1_1 f1_2 f1_3
## 1 0 0 1 0 0 0
## 2 0 1 1 1 0 0
## 3 0 2 0 1 0 0
## 4 1 0 1 1 0 0
## 5 1 1 1 2 1 0
## 6 1 2 0 1 1 0
## 7 2 0 1 4 1 0
## 8 2 1 1 5 5 1
## 9 2 2 0 1 4 1
## 10 3 0 0 1 1 0
## 11 3 1 0 1 2 1
## 12 3 2 0 0 1 1
## 13 4 0 0 0 1 0
## 14 4 1 0 0 1 1
## 15 4 2 0 0 0 1
To run the preliminary linkage analysis of 1x0 markers in the diploid
parent, we need to specify the ploidy of both parents,
e.g. ploidy = 4 and ploidy2 = 2 (or vice
versa). To demonstrate this, we need to load a special dataset with
triploid data, a sample of which is available from the
polymapR package also:
data("TRI_dosages")
# Estimate the linkage in the diploid parent (assuming this has been done for the 4x parent already):
SN_SN_P2.tri <- linkage(dosage_matrix = TRI_dosages,
markertype1 = c(1,0),
parent1 = "P1",
parent2 = "P2",
which_parent = 2,
ploidy = 4,
ploidy2 = 2,
pairing = "random"
)
Have a look at the plot of r versus LOD score - here you should
notice that both the coupling and repulsion phases are equally
informative now and cannot therefore be separated by LOD score as
before:
## Warning: Computation failed in `stat_binhex()`.
## Caused by error in `compute_group()`:
## ! The package "hexbin" is required for `stat_bin_hex()`.
If we try to cluster using the old function, we get the following
result - no separation of the homologues, but the chromosomal linkage
groups are identified:
P2_homologues.tri <- cluster_SN_markers(linkage_df = SN_SN_P2.tri,
LOD_sequence = seq(3, 10, 1),
LG_number = 3,
ploidy = 2, #because P2 is diploid..
parentname = "P2",
plot_network = FALSE,
plot_clust_size = FALSE)
## Total number of edges: 345
## 3 clusters were expected.
To separate the 0x1 markers into homologues, we use the phase
information from the linkage analysis directly, using the
phase_SN_diploid function. To learn more about how this
function works, input ?phase_SN_diploid first, to get an
idea of the arguments used.
LGHomDf_P2.tri <- phase_SN_diploid(linkage_df = SN_SN_P2.tri,
cluster_list = P2_homologues.tri,
LOD_chm = 4, #LOD at which chromosomes are identified
LG_number = 3) #number of linkage groups
## Total number of edges: 161
## Complete phase assignment possible using only coupling information at LOD 4
The rest of the analysis should be pretty much the same as for
tetraploids or hexaploids, although you should check each time you use a
new function whether that function has a ploidy2 argument,
and if it does, use it!
9. Assigning SxS and DxN markers and consensus linkage group (LG)
names
Assigning simplex x simplex markers to a homologue and linkage group
is done by calculating linkage between SxS and SxN markers and after
that assigning them based on this linkage using
assign_linkage_group:
SN_SS_P1 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
markertype2 = c(1,1),
which_parent = 1,
ploidy = 4,
pairing = "random")
P1_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P1,
LG_hom_stack = LGHomDf_P1,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
##
## #### Marker(s) showing ambiguous linkage to more than one LG:
##
## |_ |
## |:-----------|
## |Ap_ts069042 |
##
## In total, 296 out of 301 markers were assigned.
##
## #### Marker(s) not assigned:
##
## |_ |_ |_ |_ |
## |:-----------|:-----------|:-----------|:-----------|
## |Ac_ws033297 |Ap_ts089267 |St_ns048176 |St_ts030296 |
## |Zm_ws010615 | | | |
## Assigned_LG LG1 LG2 LG3 LG4 LG5 Hom1 Hom2 Hom3 Hom4 Assigned_hom1
## Ac_ns002135 4 0 0 0 7 0 0 7 0 0 2
## Ac_ns002510 2 0 7 0 0 0 0 0 0 7 4
## Ac_ns024650 2 0 11 0 0 0 11 0 0 0 1
## Ac_ns028519 5 0 0 0 0 5 0 5 0 0 2
## Ac_ns028533 5 0 0 0 0 3 0 0 3 0 3
## Ac_ns029178 3 0 0 12 0 0 0 12 0 0 2
## Assigned_hom2 Assigned_hom3 Assigned_hom4
## Ac_ns002135 NA NA NA
## Ac_ns002510 NA NA NA
## Ac_ns024650 NA NA NA
## Ac_ns028519 NA NA NA
## Ac_ns028533 NA NA NA
## Ac_ns029178 NA NA NA
SN_SS_P2 <- linkage(dosage_matrix = filtered_data,
markertype1 = c(1,0),
markertype2 = c(1,1),
which_parent = 2,
ploidy = 4,
pairing = "random")
P2_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P2,
LG_hom_stack = LGHomDf_P2_1,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
##
## In total, 300 out of 300 markers were assigned.
As simplex x simplex markers are present in both parents, we can
define which linkage groups correspond with each other between parents.
After this, one of the linkage groups of the parents should be renamed
and the SxS markers assigned again according to the new linkage group
names:
LGHomDf_P2_2 <- consensus_LG_names(modify_LG = LGHomDf_P2_1,
template_SxS = P1_SxS_Assigned,
modify_SxS = P2_SxS_Assigned)
##
## #### Original LG names
##
## | | 1| 2| 3| 4| 5|
## |:--|--:|--:|--:|--:|--:|
## |1 | 0| 0| 0| 47| 0|
## |2 | 64| 0| 0| 0| 0|
## |3 | 0| 0| 58| 0| 0|
## |4 | 0| 58| 0| 0| 0|
## |5 | 0| 0| 0| 0| 68|
##
## #### Modified LG names
##
## | | 1| 2| 3| 4| 5|
## |:--|--:|--:|--:|--:|--:|
## |1 | 47| 0| 0| 0| 0|
## |2 | 0| 64| 0| 0| 0|
## |3 | 0| 0| 58| 0| 0|
## |4 | 0| 0| 0| 58| 0|
## |5 | 0| 0| 0| 0| 68|
P2_SxS_Assigned <- assign_linkage_group(linkage_df = SN_SS_P2,
LG_hom_stack = LGHomDf_P2_2,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
##
## In total, 300 out of 300 markers were assigned.
Since we now have a consistent linkage group numbering, we can also
assign the DxN markers:
P1_DxN_Assigned <- assign_linkage_group(linkage_df = SN_DN_P1,
LG_hom_stack = LGHomDf_P1,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
##
## In total, 111 out of 111 markers were assigned.
P2_DxN_Assigned <- assign_linkage_group(linkage_df = SN_DN_P2,
LG_hom_stack = LGHomDf_P2_2,
SN_colname = "marker_a",
unassigned_marker_name = "marker_b",
phase_considered = "coupling",
LG_number = 5,
LOD_threshold = 3,
ploidy = 4)
##
## In total, 101 out of 101 markers were assigned.
15. Preferential pairing
Up to now, we have been working under the assumption of random
bivalent pairing in the parents. However, it has been shown that in
certain species the pairing behaviour is neither completely random nor
completely preferential (as we have in allopolyploids) but something
in-between, a condition termed segmental allopolyploidy [5]. A study on this topic in rose [6] found evidence of disomic behaviour in
certain regions of the genome, with tetrasomic behaviour everywhere
else. This sort of mixed inheritance can complicate the analysis and in
cases where this effect is particularly pronounced, it is probably
unwise to ignore [7].
The polymapR package can currently accommodate
preferential pairing in tetraploid species only, by correcting the
recombination frequency estimates once the level of preferential pairing
in a parental chromosome is known. In order to determine whether this is
necessary, we can run some diagnostic tests using the marker data after
an initial map has been produced which can inform our choice regarding
whether to include a preferential pairing parameter or not (in a
subsequent re-analysis).
First, a word of warning. There is a function to test for
preferential pairing among pairs of closely-linked simplex x nulliplex
markers in repulsion phase within polymapR. However, it is
certainly also advised to use multi-point methods to test for
preferential pairing (if possible). The only possible drawback is that
multi-point methods (using all markers across a chromosome) generally
assume uniform pairing behaviour across the length of a chromosome,
which may not always occur [6]). A robust
multi-point approach as described in [6]
used identity-by-descent probabilities for the population, estimated
using TetraOrigin [8], which estimates the
most likely pairing behaviour per individual and can therefore reveal
whether there are deviations from the assumption of random chromosomal
pairing and recombination across the population.
The function test_prefpairing examines closely-linked
repulsion-phase simplex x nulliplex marker pairs only, and is therefore
the results of this function may not be as robust as a
multi-point whole-chromosome approach. Be that as it may, it is probably
the simplest check we can perform post-mapping for the possibility of
unusual pairing behaviour. The method used here is described also in
Bourke et al.[6], which is a development
on ideas originally described by Qu and Hancock[9]. The idea is to look for signatures of
preferential pairing in repulsion-phase pairs which map to the same
locus.
We first need to define a minimum distance by which to consider
markers essentially at the same locus. This is somewhat arbitrary - in
our study of tetraploid rose, we used pairs of duplicated individuals to
determine an approximate error rate which, along with considerations of
the size of the population and rate of missing values, led us to
conclude that distances less than ~1 cM were close enough to be
informative (and for which we could assume 0 recombinations). The
smaller the distance we choose here the better, although by choosing a
very small distance we are limiting the number of repulsion-pairs
available. By default in test_prefpairing we use a value of
0.5 cM. This can be increased or decreased as required using the
min_cM argument.
P1.prefPairing <- test_prefpairing(dosage_matrix = ALL_dosages,
maplist = integrated.maplist,
LG_hom_stack = LGHomDf_P1_1,
min_cM = 1, #changed from default of 0.5 cM
ploidy = 4)
head(P1.prefPairing)
The output shows the repulsion-phase marker pairs tested. The most
important column is probably the column P_value.adj, which
gives the FDR-adjusted P-values from a Binomial test of the hypothesis
that the disomic repulsion estimate of recombination frequency differs
significantly from \(\frac{1}{3}\)
(actually it is a one-sided test, so we are testing that r
disom falls below \(\frac{1}{3}\)). In cases where this is
significant (with P_value.adj < 0.01, say), we might
have some evidence for preferential pairing (but from that marker pair
only!). The output of test_prefpairing also lists the
markers, their positions and on which homologues they have been mapped.
There is also an estimate of the pairwise preferential pairing parameter
pref.p. Note that there are two possible
(maximum-likelihood) estimators of a preferential pairing parameter in
the case of a pair of simplex x nulliplex markers, depending on whether
the marker alleles reside on homologous chromosomes, or homoeologous
chromosomes (i.e. whether the pair are contained within a
“subgenome”, or are straddled across “subgenomes”, to borrow the
language of allopolyploids for convenience). If there is strong evidence
of preferential pairing from multiple marker pairs, the approach we
recommend is to take the average of the pref.p values for a
particular chromosomal linkage group, which becomes one of the
parameters p1 or p2 in the
linkage function.
Here, we have purely random pairing in our parents, so the estimates
of pref.p should not be considered meaningful. The
preferential parameter p is defined such that \(0 < p < \frac{2}{3}\). Supposing this
were not the case, and we wanted to estimate the strength of
preferential pairing on LG1 of parent 1, we would do something like the
following:
mean(P1.prefPairing[P1.prefPairing$P_value.adj < 0.01 & P1.prefPairing$LG_a == 1,]$pref.p)
Note that again we require first that there be significant evidence
of disomic behaviour before including the estimate. If there is evidence
of preferential pairing, it is probably a good idea to go back and
reduce the min_cM argument even further, since we really
want this value to be as close to 0 as possible for an accurate estimate
of a preferential pairing parameter.
Finally, to re-run the linkage analysis with this estimate for parent
1 of linkage group 1, we would first have to subset our marker dosage
data corresponding to that linkage group only (since we do not need to
re-analyse any other chromosome!), and then re-run the linkage analysis.
Note that in cases with extreme levels of preferential pairing, the
clustering into separate homologues can also be affected. Supposing we
had estimated a value of 0.25 for the preferential pairing parameter we
could proceed as follows:
lg1_markers <- unique(c(rownames(marker_assignments_P1[marker_assignments_P1[,"Assigned_LG"] == 1,]),
rownames(marker_assignments_P2[marker_assignments_P2[,"Assigned_LG"] == 1,])))
all_linkages_list_P1_lg1 <- finish_linkage_analysis(marker_assignment = marker_assignments$P1[lg1_markers,],
dosage_matrix = filtered_data[lg1_markers,],
which_parent = 1,
convert_palindrome_markers = FALSE,
ploidy = 4,
pairing = "preferential",
prefPars = c(0.25,0), #just for example!
LG_number = 1 #interested in just 1 chm.
)
all_linkages_list_P2_lg1 <- finish_linkage_analysis(marker_assignment = marker_assignments$P2[lg1_markers,],
dosage_matrix = filtered_data[lg1_markers,],
which_parent = 2,
convert_palindrome_markers = FALSE,
ploidy = 4,
pairing = "preferential",
prefPars = c(0,0.25), #Note that this is in reverse order now.
LG_number = 1
)
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
