Getting started with the plyranges package
2019-09-17
Package
plyranges 1.4.4
1 Ranges revisited
In Bioconductor there are two classes, IRanges and GRanges,
that are standard data structures for representing genomics data.
Throughout this document I refer to either of these classes as Ranges if an
operation can be performed on either class, otherwise I explicitly mention if
a function is appropriate for an IRanges or GRanges.
Ranges objects can either represent sets of integers as IRanges
(which have start, end and width attributes)
or represent genomic intervals (which have additional attributes, sequence name,
and strand) as GRanges. In addition, both types of Ranges can store information
about their intervals as metadata columns (for example GC content
over a genomic interval).
Ranges objects follow the tidy data principle: each row
of a Ranges object corresponds to an interval, while each column will represent
a variable about that interval, and generally each object will represent a
single unit of observation (like gene annotations).
Consequently, Ranges objects provide a powerful representation for reasoning
about genomic data. In this vignette, you will learn more about Ranges objects
and how via grouping, restriction and summarisation you can perform common data
tasks.
2 Constructing Ranges
To construct an IRanges we require that there are at least two columns that represent at either a
starting coordinate, finishing coordinate or the width of the interval.
suppressPackageStartupMessages(library(plyranges))
set.seed(100)
df <- data.frame(start=c(2:-1, 13:15),
width=c(0:3, 2:0))
# produces IRanges
rng <- df %>% as_iranges()
rng## IRanges object with 7 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 2 1 0
## [2] 1 1 1
## [3] 0 1 2
## [4] -1 1 3
## [5] 13 14 2
## [6] 14 14 1
## [7] 15 14 0To construct a GRanges we require a column that represents that sequence name (
contig or chromosome id), and an optional column to represent the strandedness
of an interval.
# seqname is required for GRanges, metadata is automatically kept
grng <- df %>%
transform(seqnames = sample(c("chr1", "chr2"), 7, replace = TRUE),
strand = sample(c("+", "-"), 7, replace = TRUE),
gc = runif(7)) %>%
as_granges()
grng## GRanges object with 7 ranges and 1 metadata column:
## seqnames ranges strand | gc
## <Rle> <IRanges> <Rle> | <numeric>
## [1] chr2 2-1 - | 0.76255108229816
## [2] chr1 1 - | 0.669021712383255
## [3] chr2 0-1 + | 0.204612161964178
## [4] chr2 -1-1 - | 0.357524853432551
## [5] chr1 13-14 - | 0.359475114848465
## [6] chr1 14 - | 0.690290528349578
## [7] chr2 15-14 - | 0.535811153938994
## -------
## seqinfo: 2 sequences from an unspecified genome; no seqlengths3 Arithmetic on Ranges
Sometimes you want to modify a genomic interval by altering the width of the
interval while leaving the start, end or midpoint of the coordinates unaltered.
This is achieved with the mutate verb along with anchor_* adverbs.
The act of anchoring fixes either the start, end, center coordinates of the
Range object, as shown in the figure and code below and anchors are used in
combination with either mutate or stretch.
rng <- as_iranges(data.frame(start=c(1, 2, 3), end=c(5, 2, 8)))
grng <- as_granges(data.frame(start=c(1, 2, 3), end=c(5, 2, 8),
seqnames = "seq1",
strand = c("+", "*", "-")))
mutate(rng, width = 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 1 10 10
## [2] 2 11 10
## [3] 3 12 10mutate(anchor_start(rng), width = 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 1 10 10
## [2] 2 11 10
## [3] 3 12 10mutate(anchor_end(rng), width = 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -4 5 10
## [2] -7 2 10
## [3] -1 8 10mutate(anchor_center(rng), width = 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -2 7 10
## [2] -3 6 10
## [3] 1 10 10mutate(anchor_3p(grng), width = 10) # leave negative strand fixed## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 -4-5 +
## [2] seq1 -7-2 *
## [3] seq1 3-12 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsmutate(anchor_5p(grng), width = 10) # leave positive strand fixed## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 1-10 +
## [2] seq1 2-11 *
## [3] seq1 -1-8 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsSimilarly, you can modify the width of an interval using the stretch verb.
Without anchoring, this function will extend the interval in either direction
by an integer amount. With anchoring, either the start, end or midpoint are
preserved.
rng2 <- stretch(anchor_center(rng), 10)
rng2## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -4 10 15
## [2] -3 7 11
## [3] -2 13 16stretch(anchor_end(rng2), 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -14 10 25
## [2] -13 7 21
## [3] -12 13 26stretch(anchor_start(rng2), 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -4 20 25
## [2] -3 17 21
## [3] -2 23 26stretch(anchor_3p(grng), 10)## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 -9-5 +
## [2] seq1 -8-2 *
## [3] seq1 3-18 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsstretch(anchor_5p(grng), 10)## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 1-15 +
## [2] seq1 2-12 *
## [3] seq1 -7-8 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsRanges can be shifted left or right. If strand information is available
we can also shift upstream or downstream.
shift_left(rng, 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] -9 -5 5
## [2] -8 -8 1
## [3] -7 -2 6shift_right(rng, 10)## IRanges object with 3 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 11 15 5
## [2] 12 12 1
## [3] 13 18 6shift_upstream(grng, 10)## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 -9--5 +
## [2] seq1 2 *
## [3] seq1 13-18 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsshift_downstream(grng, 10)## GRanges object with 3 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] seq1 11-15 +
## [2] seq1 2 *
## [3] seq1 -7--2 -
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengths4 Grouping Ranges
plyranges introduces a new class of Ranges called RangesGrouped,
this is a similar idea to the grouped data.frame\tibble in dplyr.
Grouping can act on either the core components or the metadata columns of
a Ranges object.
It is most effective when combined with other verbs such as mutate(),
summarise(), filter(), reduce_ranges() or disjoin_ranges().
grng <- data.frame(seqnames = sample(c("chr1", "chr2"), 7, replace = TRUE),
strand = sample(c("+", "-"), 7, replace = TRUE),
gc = runif(7),
start = 1:7,
width = 10) %>%
as_granges()
grng_by_strand <- grng %>%
group_by(strand)## Warning: `as_quosure()` requires an explicit environment as of rlang 0.3.0.
## Please supply `env`.
## This warning is displayed once per session.grng_by_strand## GRanges object with 7 ranges and 1 metadata column:
## Groups: strand [2]
## seqnames ranges strand | gc
## <Rle> <IRanges> <Rle> | <numeric>
## [1] chr2 1-10 - | 0.889453538926318
## [2] chr2 2-11 + | 0.180407245177776
## [3] chr1 3-12 - | 0.629390850430354
## [4] chr2 4-13 + | 0.989564136601985
## [5] chr1 5-14 - | 0.130288870073855
## [6] chr1 6-15 - | 0.330660525709391
## [7] chr2 7-16 - | 0.865120546659455
## -------
## seqinfo: 2 sequences from an unspecified genome; no seqlengths5 Restricting Ranges
The verb filter can be used to restrict rows in the Ranges.
Note that grouping will cause the filter to act within each group
of the data.
grng %>% filter(gc < 0.3)## GRanges object with 2 ranges and 1 metadata column:
## seqnames ranges strand | gc
## <Rle> <IRanges> <Rle> | <numeric>
## [1] chr2 2-11 + | 0.180407245177776
## [2] chr1 5-14 - | 0.130288870073855
## -------
## seqinfo: 2 sequences from an unspecified genome; no seqlengths# filtering by group
grng_by_strand %>% filter(gc == max(gc)) ## GRanges object with 2 ranges and 1 metadata column:
## Groups: strand [2]
## seqnames ranges strand | gc
## <Rle> <IRanges> <Rle> | <numeric>
## [1] chr2 1-10 - | 0.889453538926318
## [2] chr2 4-13 + | 0.989564136601985
## -------
## seqinfo: 2 sequences from an unspecified genome; no seqlengthsWe also provide the convenience methods filter_by_overlaps and
filter_by_non_overlaps for restricting by any overlapping Ranges.
ir0 <- data.frame(start = c(5,10, 15,20), width = 5) %>%
as_iranges()
ir1 <- data.frame(start = 2:6, width = 3:7) %>%
as_iranges()
ir0## IRanges object with 4 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 5 9 5
## [2] 10 14 5
## [3] 15 19 5
## [4] 20 24 5ir1## IRanges object with 5 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 2 4 3
## [2] 3 6 4
## [3] 4 8 5
## [4] 5 10 6
## [5] 6 12 7ir0 %>% filter_by_overlaps(ir1)## IRanges object with 2 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 5 9 5
## [2] 10 14 5ir0 %>% filter_by_non_overlaps(ir1) ## IRanges object with 2 ranges and 0 metadata columns:
## start end width
## <integer> <integer> <integer>
## [1] 15 19 5
## [2] 20 24 56 Summarising Ranges
The summarise function will return a DataFrame because the information required
to return a Ranges object is lost. It is often most useful to use summarise()
in combination with the group_by() family of functions.
ir1 <- ir1 %>%
mutate(gc = runif(length(.)))
ir0 %>%
group_by_overlaps(ir1) %>%
summarise(gc = mean(gc))## DataFrame with 2 rows and 2 columns
## query gc
## <integer> <numeric>
## 1 1 0.675554534769617
## 2 2 0.6357951664831497 Joins, or another way at looking at overlaps between Ranges
A join acts on two GRanges objects, a query and a subject.
query <- data.frame(seqnames = "chr1",
strand = c("+", "-"),
start = c(1, 9),
end = c(7, 10),
key.a = letters[1:2]) %>%
as_granges()
subject <- data.frame(seqnames = "chr1",
strand = c("-", "+"),
start = c(2, 6),
end = c(4, 8),
key.b = LETTERS[1:2]) %>%
as_granges()
Figure 1: Query and Subject Ranges
The join operator is relational in the sense that metadata
from the query and subject ranges is retained in the joined range. All join operators in the plyranges DSL generate a set of hits based on overlap or proximity of ranges and use those hits to merge the two
datasets in different ways.
There are four supported matching algorithms: overlap,
nearest, precede, and follow. We can further restrict the
matching by whether the query is completely within the subject, and
adding the directed suffix ensures that matching ranges have the
same direction (strand).
The first function, join_overlap_intersect() will return a Ranges object where the start, end, and width coordinates correspond to the amount of any overlap between the left and right input Ranges. It also returns any metadatain the subject range if the subject overlaps the query.
intersect_rng <- join_overlap_intersect(query, subject)
intersect_rng## GRanges object with 2 ranges and 2 metadata columns:
## seqnames ranges strand | key.a key.b
## <Rle> <IRanges> <Rle> | <factor> <factor>
## [1] chr1 2-4 + | a A
## [2] chr1 6-7 + | a B
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengths
Figure 2: Intersect Join
The join_overlap_inner() function will return the Ranges in the query that
overlap any Ranges in the subject. Like the join_overlap_intersect() function
metadata of the subject Range is returned if it overlaps the query.
inner_rng <- join_overlap_inner(query, subject)
inner_rng## GRanges object with 2 ranges and 2 metadata columns:
## seqnames ranges strand | key.a key.b
## <Rle> <IRanges> <Rle> | <factor> <factor>
## [1] chr1 1-7 + | a A
## [2] chr1 1-7 + | a B
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengths
Figure 3: Inner Join
We also provide a convenience method called find_overlaps that computes the same result as join_overlap_inner().
find_overlaps(query, subject)## GRanges object with 2 ranges and 2 metadata columns:
## seqnames ranges strand | key.a key.b
## <Rle> <IRanges> <Rle> | <factor> <factor>
## [1] chr1 1-7 + | a A
## [2] chr1 1-7 + | a B
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsThe join_overlap_left() method will perform an outer left join.
First any overlaps that are found will be returned similar to join_overlap_inner(). Then any non-overlapping ranges will be returned, with missing values on the metadata columns.
left_rng <- join_overlap_left(query, subject)
left_rng## GRanges object with 3 ranges and 2 metadata columns:
## seqnames ranges strand | key.a key.b
## <Rle> <IRanges> <Rle> | <factor> <factor>
## [1] chr1 1-7 + | a A
## [2] chr1 1-7 + | a B
## [3] chr1 9-10 - | b <NA>
## -------
## seqinfo: 2 sequences from an unspecified genome; no seqlengths
Figure 4: Left Join
Compared with filter_by_overlaps() above, the overlap left join expands the
Ranges to give information about each interval on the query Ranges that
overlap those on the subject Ranges as well as the intervals on the left that do not overlap any range on the right.
7.1 Finding your neighbours
We also provide methods for finding nearest, preceding or following Ranges. Conceptually this is identical to our approach for finding
overlaps, except the semantics of the join are different.
join_nearest(ir0, ir1)## IRanges object with 4 ranges and 1 metadata column:
## start end width | gc
## <integer> <integer> <integer> | <numeric>
## [1] 5 9 5 | 0.780358511023223
## [2] 10 14 5 | 0.780358511023223
## [3] 15 19 5 | 0.780358511023223
## [4] 20 24 5 | 0.780358511023223join_follow(ir0, ir1)## IRanges object with 4 ranges and 1 metadata column:
## start end width | gc
## <integer> <integer> <integer> | <numeric>
## [1] 5 9 5 | 0.777584439376369
## [2] 10 14 5 | 0.603324356488883
## [3] 15 19 5 | 0.780358511023223
## [4] 20 24 5 | 0.780358511023223join_precede(ir0, ir1) # nothing precedes returns empty `Ranges`## IRanges object with 0 ranges and 1 metadata column:
## start end width | gc
## <integer> <integer> <integer> | <numeric>join_precede(ir1, ir0)## IRanges object with 5 ranges and 1 metadata column:
## start end width | gc
## <integer> <integer> <integer> | <numeric>
## [1] 2 4 3 | 0.777584439376369
## [2] 3 6 4 | 0.827303449623287
## [3] 4 8 5 | 0.603324356488883
## [4] 5 10 6 | 0.491231821943074
## [5] 6 12 7 | 0.7803585110232237.2 Example: dealing with multi-mapping
This example is taken from the Bioconductor support site.
We have two Ranges objects. The first contains single nucleotide positions
corresponding to an intensity measurement such as a ChiP-seq experiment,
while the other contains coordinates for two genes of interest.
We want to identify which positions in the intensities Ranges overlap the
genes, where each row corresponds to a position that overlaps a single gene.
First we create the two Ranges objects
intensities <- data.frame(seqnames = "VI",
start = c(3320:3321,3330:3331,3341:3342),
width = 1) %>%
as_granges()
intensities ## GRanges object with 6 ranges and 0 metadata columns:
## seqnames ranges strand
## <Rle> <IRanges> <Rle>
## [1] VI 3320 *
## [2] VI 3321 *
## [3] VI 3330 *
## [4] VI 3331 *
## [5] VI 3341 *
## [6] VI 3342 *
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsgenes <- data.frame(seqnames = "VI",
start = c(3322, 3030),
end = c(3846, 3338),
gene_id=c("YFL064C", "YFL065C")) %>%
as_granges()
genes## GRanges object with 2 ranges and 1 metadata column:
## seqnames ranges strand | gene_id
## <Rle> <IRanges> <Rle> | <factor>
## [1] VI 3322-3846 * | YFL064C
## [2] VI 3030-3338 * | YFL065C
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsNow to find where the positions overlap each gene, we can perform an overlap join. This will automatically carry over the gene_id information as well as their coordinates (we can drop those by only selecting the gene_id).
olap <- join_overlap_inner(intensities, genes) %>%
select(gene_id)
olap## GRanges object with 8 ranges and 1 metadata column:
## seqnames ranges strand | gene_id
## <Rle> <IRanges> <Rle> | <factor>
## [1] VI 3320 * | YFL065C
## [2] VI 3321 * | YFL065C
## [3] VI 3330 * | YFL065C
## [4] VI 3330 * | YFL064C
## [5] VI 3331 * | YFL065C
## [6] VI 3331 * | YFL064C
## [7] VI 3341 * | YFL064C
## [8] VI 3342 * | YFL064C
## -------
## seqinfo: 1 sequence from an unspecified genome; no seqlengthsSeveral positions match to both genes. We can count them using summarise
and grouping by the start position:
olap %>%
group_by(start) %>%
summarise(n = n())## DataFrame with 6 rows and 2 columns
## start n
## <integer> <integer>
## 1 3320 1
## 2 3321 1
## 3 3330 2
## 4 3331 2
## 5 3341 1
## 6 3342 17.3 Grouping by overlaps
It’s also possible to group by overlaps. Using this approach we can count the number of overlaps that are greater than 0.
grp_by_olap <- ir0 %>%
group_by_overlaps(ir1)
grp_by_olap## IRanges object with 6 ranges and 2 metadata columns:
## Groups: query [2]
## start end width | gc query
## <integer> <integer> <integer> | <numeric> <integer>
## [1] 5 9 5 | 0.827303449623287 1
## [2] 5 9 5 | 0.603324356488883 1
## [3] 5 9 5 | 0.491231821943074 1
## [4] 5 9 5 | 0.780358511023223 1
## [5] 10 14 5 | 0.491231821943074 2
## [6] 10 14 5 | 0.780358511023223 2grp_by_olap %>%
mutate(n_overlaps = n())## IRanges object with 6 ranges and 3 metadata columns:
## Groups: query [2]
## start end width | gc query n_overlaps
## <integer> <integer> <integer> | <numeric> <integer> <integer>
## [1] 5 9 5 | 0.827303449623287 1 4
## [2] 5 9 5 | 0.603324356488883 1 4
## [3] 5 9 5 | 0.491231821943074 1 4
## [4] 5 9 5 | 0.780358511023223 1 4
## [5] 10 14 5 | 0.491231821943074 2 2
## [6] 10 14 5 | 0.780358511023223 2 2Of course we can also add overlap counts via the count_overlaps() function.
ir0 %>%
mutate(n_overlaps = count_overlaps(., ir1))## IRanges object with 4 ranges and 1 metadata column:
## start end width | n_overlaps
## <integer> <integer> <integer> | <integer>
## [1] 5 9 5 | 4
## [2] 10 14 5 | 2
## [3] 15 19 5 | 0
## [4] 20 24 5 | 08 Data Import/Output
We provide convenience functions via rtracklayer and GenomicAlignments
for reading/writing the following data formats to/from Ranges objects.
plyranges functions |
File Format |
|---|---|
read_bam() |
BAM |
read_bed()/write_bed() |
BED |
read_bedgraph()/ write_bedgraph() |
BEDGraph |
read_narrowpeaks()/ write_narrowpeaks() |
narrowPeaks |
read_gff() / write_gff() |
GFF(1-3) / GTF |
read_bigwig() / write_bigwig() |
BigWig |
read_wig() / write_wig() |
Wig |
9 Learning more
- The Bioc 2018 Workshop book has worked examples of using
plyrangesto analyse publicly available genomics data. - The extended vignette in the plyrangesWorkshops package has a detailed walk through of using plyranges for coverage analysis.
- The case study by Michael Love using plyranges with tximeta to follow up on interesting hits from a combined RNA-seq and ATAC-seq analysis.
- The journal article (preprint here) has details about the overall philosophy and design of plyranges.
10 Appendix
sessionInfo()## R version 3.6.1 (2019-07-05)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.3 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.9-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.9-bioc/R/lib/libRlapack.so
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats4 parallel stats graphics grDevices utils datasets
## [8] methods base
##
## other attached packages:
## [1] ggplot2_3.2.1 plyranges_1.4.4 GenomicRanges_1.36.1
## [4] GenomeInfoDb_1.20.0 IRanges_2.18.2 S4Vectors_0.22.1
## [7] BiocGenerics_0.30.0 BiocStyle_2.12.0
##
## loaded via a namespace (and not attached):
## [1] SummarizedExperiment_1.14.1 tidyselect_0.2.5
## [3] xfun_0.9 reshape2_1.4.3
## [5] purrr_0.3.2 lattice_0.20-38
## [7] colorspace_1.4-1 htmltools_0.3.6
## [9] rtracklayer_1.44.4 yaml_2.2.0
## [11] XML_3.98-1.20 rlang_0.4.0
## [13] pillar_1.4.2 glue_1.3.1
## [15] withr_2.1.2 BiocParallel_1.18.1
## [17] plyr_1.8.4 matrixStats_0.55.0
## [19] GenomeInfoDbData_1.2.1 stringr_1.4.0
## [21] zlibbioc_1.30.0 Biostrings_2.52.0
## [23] munsell_0.5.0 gtable_0.3.0
## [25] evaluate_0.14 labeling_0.3
## [27] Biobase_2.44.0 knitr_1.24
## [29] highr_0.8 Rcpp_1.0.2
## [31] scales_1.0.0 BiocManager_1.30.4
## [33] DelayedArray_0.10.0 XVector_0.24.0
## [35] Rsamtools_2.0.0 digest_0.6.20
## [37] stringi_1.4.3 bookdown_0.13
## [39] dplyr_0.8.3 grid_3.6.1
## [41] tools_3.6.1 bitops_1.0-6
## [43] magrittr_1.5 RCurl_1.95-4.12
## [45] lazyeval_0.2.2 tibble_2.1.3
## [47] crayon_1.3.4 pkgconfig_2.0.2
## [49] Matrix_1.2-17 assertthat_0.2.1
## [51] rmarkdown_1.15 R6_2.4.0
## [53] GenomicAlignments_1.20.1 compiler_3.6.1