Table of Contents
- Introduction
- Installation
-
regionalpcs R Package Tutorial
- 3.1 Loading Required Packages
- 3.2 Load the Dataset
- 3.2.1 Overview
- 3.2.2 Inspecting the Data
- 3.3 Obtaining Methylation Array Probe Positions
- 3.4 Processing and Filtering Methylation Data
- 3.4.1 Introduction
- 3.4.2 Remove Low Variance CpGs
- 3.4.3 Normalize Methylation Values
- 3.5 Summarizing Gene Region Types
- 3.5.1 Introduction
- 3.5.2 Load Gene Region Annotations
- 3.5.3 Create a Region Map
- 3.5.3.1 Extract CpG Positions
- 3.5.3.2 Convert to GenomicRanges and Find Overlaps
- 3.5.4 Summarizing Gene Regions with Regional Principal Components
- 3.5.4.1 Compute Regional PCs
- 3.5.4.2 Inspecting the Output
- 3.5.4.3 Extracting and Viewing Regional PCs
- 3.5.4.4 Understanding the Results
regionalpcs
Tiffany Eulalio
The regionalpcs package aims to address the challenge of summarizing and interpreting DNA methylation data at a regional level. Traditional methods of analysis may not capture the biological complexity of methylation patterns, potentially leading to less accurate or less meaningful interpretations. This package introduces the concept of regional principal components (rPCs) as a tool for capturing more biologically relevant signals in DNA methylation data. By using rPCs, researchers can gain new insights into complex interactions and effects in methylation data that might otherwise be missed.
Installation
You can install the regionalpcs package from Bioconductor using the following command:
if (!requireNamespace("BiocManager", quietly=TRUE))
install.packages("BiocManager")
BiocManager::install("regionalpcs")You can install the development version of regionalpcs from GitHub with:
# install devtool package if needed
if (!requireNamespace("devtools", quietly=TRUE))
install.packages("devtools")
# download the regionalpcs package
devtools::install_github("tyeulalio/regionalpcs")
regionalpcs R Package Tutorial
Loading Required Packages
This tutorial depends on several Bioconductor packages. These packages should be loaded at the beginning of the analysis.
library(regionalpcs)
library(RNOmni)
library(GenomicRanges)
library(IlluminaHumanMethylation450kanno.ilmn12.hg19)
library(liftOver)
library(magrittr)
library(tidyr)
library(tibble)
library(dplyr)Here, we load the regionalpcs package, which is the main package we’ll be using in this tutorial. We also load RNOmni, which provides normalization functions, GenomicRanges, which provides tools for working with genomic intervals, and tidyverse, which provides a suite of tools for data manipulation and visualization.
It’s important to note that you need to have these packages installed on your machine before loading them. You can install them using the install.packages() function in R.
Once the packages are loaded, we can start using the functions provided by each package.
Load the dataset
Overview
The betas dataset in the regionalpcs package is a subset of 450k array methylation data from TCGA, containing 293 methylation sites and 300 samples. We’ll load this data into our R session and explore its structure.
data("betas", package = "regionalpcs")Inspecting the Data
We can take a quick look at the dimensions of the dataset and the first few rows to understand its structure.
head(betas)[, 1:3]
#> TCGA-EJ-7781-11A TCGA-BH-A1FE-11B
#> chr16_53434200_53434201_cg00000029 0.20361112 0.13654490
#> chr15_22838620_22838621_cg00000622 0.01311223 0.01024075
#> chr1_166989202_166989203_cg00001349 0.72180841 0.74037266
#> chr8_119416178_119416179_cg00002464 0.05881476 0.05834758
#> chr6_169751536_169751537_cg00005543 0.01868565 0.01808436
#> chr12_52069532_52069533_cg00006122 0.05748535 0.06136361
#> TCGA-BH-A0C3-11A
#> chr16_53434200_53434201_cg00000029 0.12996001
#> chr15_22838620_22838621_cg00000622 0.01847991
#> chr1_166989202_166989203_cg00001349 0.76361838
#> chr8_119416178_119416179_cg00002464 0.06189946
#> chr6_169751536_169751537_cg00005543 0.02085631
#> chr12_52069532_52069533_cg00006122 0.05805733
dim(betas)
#> [1] 293 300Note that the row names are CpG IDs and genomic positions, and the columns contain methylation beta values ranging from 0 to 1 for individual samples.
Obtaining Methylation Array Probe Positions
Introduction
To perform accurate and informative analyses on methylation array data, it is critical to have precise genomic positions for each probe. The IlluminaHumanMethylation450kanno.ilmn12.hg19 package contains annotations for 450k methylation arrays, which can be utilized for this purpose. This section will walk you through the steps to associate each probe in your dataset with its genomic position.
Extract Probe Names and Positions
First, we’ll extract the probe names from the betas data frame and use regular expressions to separate the CpG identifier from its genomic position.
# Extract probe names and CpG positions from row names of 'betas'
cpg_df <- data.frame(cpgs = rownames(betas)) %>%
separate(cpgs,
into = c("cpg_pos", "probe"), sep = "_(?=[^_]+$)",
extra = "merge"
)
head(cpg_df)
#> cpg_pos probe
#> 1 chr16_53434200_53434201 cg00000029
#> 2 chr15_22838620_22838621 cg00000622
#> 3 chr1_166989202_166989203 cg00001349
#> 4 chr8_119416178_119416179 cg00002464
#> 5 chr6_169751536_169751537 cg00005543
#> 6 chr12_52069532_52069533 cg00006122Load Illumina 450k Array Probe Positions
Next, let’s load the Illumina 450k array probe positions for further annotation.
data(Locations)
probe_positions <- data.frame(Locations)
head(probe_positions)
#> chr pos strand
#> cg00050873 chrY 9363356 -
#> cg00212031 chrY 21239348 -
#> cg00213748 chrY 8148233 -
#> cg00214611 chrY 15815688 -
#> cg00455876 chrY 9385539 -
#> cg01707559 chrY 6778695 +Merge Data Frames
Now, we merge the extracted probe names with the Illumina 450k array probe positions.
formatted_probe_positions <- probe_positions %>%
rownames_to_column("probe")
new_cpg_df <- cpg_df %>%
left_join(formatted_probe_positions, by = "probe")
head(new_cpg_df)
#> cpg_pos probe chr pos strand
#> 1 chr16_53434200_53434201 cg00000029 chr16 53468112 +
#> 2 chr15_22838620_22838621 cg00000622 chr15 23034447 +
#> 3 chr1_166989202_166989203 cg00001349 chr1 166958439 -
#> 4 chr8_119416178_119416179 cg00002464 chr8 120428418 +
#> 5 chr6_169751536_169751537 cg00005543 chr6 170151632 +
#> 6 chr12_52069532_52069533 cg00006122 chr12 52463316 +Addressing Genome Build Discrepancies
It’s critical to ensure that the genome builds match across datasets. In this example, we’ll use the GenomicRanges and liftOver packages to convert the genomic positions from hg19 to hg38. Here’s a quick example on how to lift over positions from one build to another. Always ensure that you are working with the correct genome build and that the build matches across all your datasets, or else you will run into big issues!
We need a chain file to lift the genomic positions. The chain file is an annotation file that links the positions between the genome builds. You can download this file from the (UCSC golden path download site)[https://hgdownload.cse.ucsc.edu/goldenpath/hg19/liftOver/]. Be sure to download the file that maps between the appropriate builds. We’ll be mapping from hg19 to hg38. We’ve included the chain used in this analysis as a part of the regionalpcs package, which can be accessed in the “extdata” folder as shown in the code below.
# Convert hg19 positions into a GenomicRanges object
hg19_pos <- new_cpg_df %>%
select("chr", "pos", "strand", "probe") %>%
mutate(start = pos, end = pos + 1)
hg19_pos_gr <- makeGRangesFromDataFrame(hg19_pos, keep.extra.columns = TRUE)
# Load chain file and liftOver positions
chain_file <- system.file("extdata", "hg19ToHg38.over.chain",
package = "regionalpcs"
)
print(paste("Using chain file for liftOver", chain_file))
#> [1] "Using chain file for liftOver C:/Users/tyeul/AppData/Local/R/win-library/4.3/regionalpcs/extdata/hg19ToHg38.over.chain"
print(file.exists(chain_file))
#> [1] TRUE
chain <- import.chain(chain_file)
hg38_pos <- liftOver(hg19_pos_gr, chain) %>%
as.data.frame()
# Merge the lifted positions back to the original data frame
formatted_hg38 <- hg38_pos %>%
select(chrom_hg38 = seqnames, pos_hg38 = start, probe)
lifted_cpg_df <- new_cpg_df %>%
left_join(formatted_hg38, by = "probe")
head(lifted_cpg_df)
#> cpg_pos probe chr pos strand chrom_hg38
#> 1 chr16_53434200_53434201 cg00000029 chr16 53468112 + chr16
#> 2 chr15_22838620_22838621 cg00000622 chr15 23034447 + chr15
#> 3 chr1_166989202_166989203 cg00001349 chr1 166958439 - chr1
#> 4 chr8_119416178_119416179 cg00002464 chr8 120428418 + chr8
#> 5 chr6_169751536_169751537 cg00005543 chr6 170151632 + chr6
#> 6 chr12_52069532_52069533 cg00006122 chr12 52463316 + chr12
#> pos_hg38
#> 1 53434200
#> 2 22838620
#> 3 166989202
#> 4 119416178
#> 5 169751536
#> 6 52069532Now that we have accurate genomic positions for each probe and have harmonized genome builds, we can proceed with preprocessing the methylation data.
Processing and Filtering Methylation Data
Introduction
Before conducting downstream analyses, it is essential to preprocess and clean the methylation data. In this section, we’ll walk you through the steps to remove low variance CpGs and normalize the methylation beta values.
Remove Low Variance CpGs
Firstly, we aim to filter out low variance CpGs. Variability is a crucial factor, as low variance CpGs may not provide much information for downstream analyses.
In this section, we’ll remove low variance CpGs and normalize our methylation beta values using the inverse normal transform.
# Remove CpGs with zero variance
var_betas <- betas[apply(betas, 1, var, na.rm = TRUE) != 0, ] %>%
na.omit()
dim(var_betas)
#> [1] 293 300We only remove CpGs that have zero variance in this example. You can adjust this threshold according to the requirements of your specific analysis.
Normalize Methylation Values
Methylation data often exhibit heteroscedasticity. Therefore, we’ll normalize the beta values using inverse normal transformation. For this, we’ll use the RankNorm function from the RNOmni package.
# Apply inverse normal transformation to methylation beta values
int_meth <- apply(var_betas, 1, RankNorm) %>%
t() %>%
as.data.frame()After these preprocessing steps, you will have a dataset ready for downstream analysis with the regionalpcs package. We’ll cover how to perform these analyses in subsequent sections of this tutorial.
Summarizing Gene Region Types
Introduction
Gene regions are significant functional units of the genome, such as promoters, gene bodies, and intergenic regions. We’ll focus on summarizing these regions to prepare for downstream analyses. We will use the regionalpcs package to perform these tasks.
Load Gene Region Annotations
First, let’s load the gene region annotations. Make sure to align the genomic builds of your annotations and methylation data.
All annotations included with the regionalpcs package are in build hg38.
# Load the gene region annotation file
data("gene_annots", package = "regionalpcs")
head(gene_annots)
#> # A tibble: 6 × 16
#> seqnames start end width strand tx_id type gencode_gene_id
#> <chr> <dbl> <dbl> <dbl> <chr> <chr> <chr> <chr>
#> 1 chr1 922928 923927 1000 + ENST00000420190.6 hg38_… ENSG0000018763…
#> 2 chr1 959584 960583 1000 + ENST00000338591.8 hg38_… ENSG0000018796…
#> 3 chr1 965482 966481 1000 + ENST00000379410.8 hg38_… ENSG0000018758…
#> 4 chr1 1000138 1001137 1000 + ENST00000624697.4 hg38_… ENSG0000018760…
#> 5 chr1 1019120 1020119 1000 + ENST00000379370.7 hg38_… ENSG0000018815…
#> 6 chr1 1172903 1173902 1000 + ENST00000379290.5 hg38_… ENSG0000016257…
#> # ℹ 8 more variables: gencode_gene_type <chr>, gencode_gene_name <chr>,
#> # transcript_type <chr>, transcript_name <chr>,
#> # transcript_support_level <dbl>, tag <chr>, is_canonical <lgl>,
#> # gencode_region <chr>The gene_annots dataset includes annotations for various gene regions.
Create a Region Map
Before summarizing gene regions using compute_regional_pcs, we need to create a region map that assigns CpGs to gene regions. This map enables us to identify which CpGs fall into each gene region.
Extract CpG Positions
Start by extracting the CpG positions from your methylation data frame’s row names.
head(int_meth)[1:4]
#> TCGA-EJ-7781-11A TCGA-BH-A1FE-11B
#> chr16_53434200_53434201_cg00000029 -0.38948253 -0.9465265
#> chr15_22838620_22838621_cg00000622 -0.03757696 -1.1171114
#> chr1_166989202_166989203_cg00001349 -0.87083034 -0.6274087
#> chr8_119416178_119416179_cg00002464 0.13818831 0.1045460
#> chr6_169751536_169751537_cg00005543 0.38049184 0.2488238
#> chr12_52069532_52069533_cg00006122 0.27474294 0.6274087
#> TCGA-BH-A0C3-11A TCGA-E9-A1N6-11A
#> chr16_53434200_53434201_cg00000029 -1.1171114 -1.4370361
#> chr15_22838620_22838621_cg00000622 1.2160129 -0.6376059
#> chr1_166989202_166989203_cg00001349 -0.3894825 0.3894825
#> chr8_119416178_119416179_cg00002464 0.2402219 -0.2574441
#> chr6_169751536_169751537_cg00005543 0.9080280 -1.5657713
#> chr12_52069532_52069533_cg00006122 0.3271598 -0.1129440
# Extract CpG information
cpg_info <- data.frame(cpg_id = rownames(int_meth)) %>%
separate(cpg_id,
into = c("chrom", "start", "end", "cpg_name"),
sep = "_", remove = FALSE
)
head(cpg_info)
#> cpg_id chrom start end cpg_name
#> 1 chr16_53434200_53434201_cg00000029 chr16 53434200 53434201 cg00000029
#> 2 chr15_22838620_22838621_cg00000622 chr15 22838620 22838621 cg00000622
#> 3 chr1_166989202_166989203_cg00001349 chr1 166989202 166989203 cg00001349
#> 4 chr8_119416178_119416179_cg00002464 chr8 119416178 119416179 cg00002464
#> 5 chr6_169751536_169751537_cg00005543 chr6 169751536 169751537 cg00005543
#> 6 chr12_52069532_52069533_cg00006122 chr12 52069532 52069533 cg00006122Convert to GenomicRanges and Find Overlaps
Next, we’ll use the GenomicRanges package to find overlaps between CpGs and gene regions.
# Convert to GenomicRanges
cpg_gr <- makeGRangesFromDataFrame(cpg_info, keep.extra.columns = TRUE)
annots_gr <- makeGRangesFromDataFrame(gene_annots, keep.extra.columns = TRUE)
# Find overlaps between the two GRanges objects
overlaps <- findOverlaps(query = cpg_gr, subject = annots_gr) %>%
as.data.frame()
head(overlaps)
#> queryHits subjectHits
#> 1 1 12678
#> 2 2 11904
#> 3 3 679
#> 4 4 7360
#> 5 5 5877
#> 6 6 10306
# Match overlaps
matched_cpg <- cpg_gr[overlaps$queryHits, ] %>%
as.data.frame() %>%
select(cpg_id)
# Select overlapped rows and just keep the columns we need
matched_annots <- annots_gr[overlaps$subjectHits, ] %>%
as.data.frame() %>%
select(gencode_gene_id)
# Combine the matched CpGs and gene annotations to form the region map
region_map <- cbind(matched_annots, matched_cpg)
head(region_map)
#> gencode_gene_id cpg_id
#> 1 ENSG00000103479.16 chr16_53434200_53434201_cg00000029
#> 2 ENSG00000140157.14 chr15_22838620_22838621_cg00000622
#> 3 ENSG00000143194.13 chr1_166989202_166989203_cg00001349
#> 4 ENSG00000136999.5 chr8_119416178_119416179_cg00002464
#> 5 ENSG00000130023.16 chr6_169751536_169751537_cg00005543
#> 6 ENSG00000123395.14 chr12_52069532_52069533_cg00006122
length(unique(region_map$gencode_gene_id))
#> [1] 52With these steps, you’ll have a region map that assigns CpGs to specific gene regions, which can be essential for downstream analyses.
Summarizing Gene Regions with Regional Principal Components
In this final section, we’ll summarize gene regions using Principal Components (PCs) to capture the maximum variation. We’ll utilize the compute_regional_pcs function from the regionalpcs package for this.
Compute Regional PCs
Let’s calculate the regional PCs using a subset of our gene regions for demonstration purposes.
# Display head of region map
head(region_map)
#> gencode_gene_id cpg_id
#> 1 ENSG00000103479.16 chr16_53434200_53434201_cg00000029
#> 2 ENSG00000140157.14 chr15_22838620_22838621_cg00000622
#> 3 ENSG00000143194.13 chr1_166989202_166989203_cg00001349
#> 4 ENSG00000136999.5 chr8_119416178_119416179_cg00002464
#> 5 ENSG00000130023.16 chr6_169751536_169751537_cg00005543
#> 6 ENSG00000123395.14 chr12_52069532_52069533_cg00006122
# Subset the region map
sub_region_map <- region_map %>%
filter(gencode_gene_id %in% unique(region_map$gencode_gene_id)[1:1000])
# Compute regional PCs
res <- compute_regional_pcs(int_meth, sub_region_map)
#> Using Gavish-Donoho methodInspecting the Output
The function returns a list containing multiple elements. Let’s first look at what these elements are.
# Inspect the output list elements
names(res)
#> [1] "regional_pcs" "percent_variance" "loadings" "info"Extracting and Viewing Regional PCs
The first element (res$regional_pcs) is a data frame containing the calculated regional PCs.
# Extract regional PCs
regional_pcs <- res$regional_pcs
head(regional_pcs)[1:4]
#> TCGA-EJ-7781-11A TCGA-BH-A1FE-11B TCGA-BH-A0C3-11A
#> ENSG00000103479.16-PC1 -0.8210157 -1.60659103 -1.0541882
#> ENSG00000140157.14-PC1 -0.1392372 -1.47241611 1.3317941
#> ENSG00000143194.13-PC1 -1.7895014 -3.07334567 -3.1270943
#> ENSG00000136999.5-PC1 0.4498687 0.11381324 -0.3555689
#> ENSG00000130023.16-PC1 -0.1838917 0.05055839 -0.8546051
#> ENSG00000123395.14-PC1 -1.1961240 0.36432710 0.7234163
#> TCGA-E9-A1N6-11A
#> ENSG00000103479.16-PC1 -1.6169547
#> ENSG00000140157.14-PC1 -0.6519472
#> ENSG00000143194.13-PC1 -1.0122026
#> ENSG00000136999.5-PC1 1.2443734
#> ENSG00000130023.16-PC1 2.1260186
#> ENSG00000123395.14-PC1 1.3592205Understanding the Results
The output is a data frame with regional PCs for each region as rows and our samples as columns. This is our new representation of methylation values, now on a gene regional PC scale. We can feed these into downstream analyses as is.
The number of regional PCs representing each gene region was determined by the Gavish-Donoho method. This method allows us to identify PCs that capture actual signal in our data and not the noise that is inherent in any dataset. To explore alternative methods, we can change the pc_method parameter.
# Count the number of unique gene regions and PCs
regions <- data.frame(gene_pc = rownames(regional_pcs)) %>%
separate(gene_pc, into = c("gene", "pc"), sep = "-")
head(regions)
#> gene pc
#> 1 ENSG00000103479.16 PC1
#> 2 ENSG00000140157.14 PC1
#> 3 ENSG00000143194.13 PC1
#> 4 ENSG00000136999.5 PC1
#> 5 ENSG00000130023.16 PC1
#> 6 ENSG00000123395.14 PC1
# number of genes that have been summarized
length(unique(regions$gene))
#> [1] 52
# how many of each PC did we get
table(regions$pc)
#>
#> PC1
#> 52We have summarized each of our genes using just one PC. The number of PCs depends on three main factors: the number of samples, the number of CpGs in the gene region, and the noise in the methylation data.
By default, the compute_regional_pcs function uses the Gavish-Donoho method. However, we can also use the Marcenko-Pasteur method by setting the pc_method parameter:
# Using Marcenko-Pasteur method
mp_res <- compute_regional_pcs(int_meth, sub_region_map, pc_method = "mp")
#> Using Marchenko-Pastur method
# select the regional pcs
mp_regional_pcs <- mp_res$regional_pcs
# separate the genes from the pc numbers
mp_regions <- data.frame(gene_pc = rownames(mp_regional_pcs)) %>%
separate(gene_pc, into = c("gene", "pc"), sep = "-")
head(mp_regions)
#> gene pc
#> 1 ENSG00000103479.16 PC1
#> 2 ENSG00000103479.16 PC2
#> 3 ENSG00000140157.14 PC1
#> 4 ENSG00000140157.14 PC2
#> 5 ENSG00000143194.13 PC1
#> 6 ENSG00000143194.13 PC2
# number of genes that have been summarized
length(unique(mp_regions$gene))
#> [1] 52
# how many of each PC did we get
table(mp_regions$pc)
#>
#> PC1 PC2
#> 52 26The Marcenko-Pasteur and the Gavish-Donoho methods are both based on Random Matrix Theory, and they aim to identify the number of significant PCs that capture the true signal in the data and not just the noise. However, these methods differ in how they select the number of significant PCs. The Marcenko-Pasteur method typically selects more PCs to represent a gene region compared to the Gavish-Donoho method. This may be due to the different ways in which the two methods estimate the noise level in the data.
Ultimately, the choice between the two methods depends on the specific needs and goals of the analysis. The Gavish-Donoho method tends to provide more conservative results, while the Marcenko-Pasteur method may capture more of the underlying signal in the data. Researchers should carefully consider their objectives and the characteristics of their data when selecting a method.
Session Information
sessionInfo()
#> R version 4.3.1 (2023-06-16 ucrt)
#> Platform: x86_64-w64-mingw32/x64 (64-bit)
#> Running under: Windows 10 x64 (build 19045)
#>
#> Matrix products: default
#>
#>
#> locale:
#> [1] LC_COLLATE=English_United States.utf8
#> [2] LC_CTYPE=English_United States.utf8
#> [3] LC_MONETARY=English_United States.utf8
#> [4] LC_NUMERIC=C
#> [5] LC_TIME=English_United States.utf8
#>
#> time zone: America/Los_Angeles
#> tzcode source: internal
#>
#> attached base packages:
#> [1] parallel stats4 stats graphics grDevices utils datasets
#> [8] methods base
#>
#> other attached packages:
#> [1] dplyr_1.1.2
#> [2] tibble_3.2.1
#> [3] tidyr_1.3.0
#> [4] magrittr_2.0.3
#> [5] liftOver_1.25.0
#> [6] Homo.sapiens_1.3.1
#> [7] TxDb.Hsapiens.UCSC.hg19.knownGene_3.2.2
#> [8] org.Hs.eg.db_3.17.0
#> [9] GO.db_3.17.0
#> [10] OrganismDbi_1.43.0
#> [11] GenomicFeatures_1.53.1
#> [12] AnnotationDbi_1.63.2
#> [13] rtracklayer_1.61.1
#> [14] gwascat_2.33.0
#> [15] IlluminaHumanMethylation450kanno.ilmn12.hg19_0.6.1
#> [16] minfi_1.47.0
#> [17] bumphunter_1.43.0
#> [18] locfit_1.5-9.8
#> [19] iterators_1.0.14
#> [20] foreach_1.5.2
#> [21] Biostrings_2.69.2
#> [22] XVector_0.41.1
#> [23] SummarizedExperiment_1.31.1
#> [24] Biobase_2.61.0
#> [25] MatrixGenerics_1.13.1
#> [26] matrixStats_1.0.0
#> [27] GenomicRanges_1.53.1
#> [28] GenomeInfoDb_1.37.2
#> [29] IRanges_2.35.2
#> [30] S4Vectors_0.39.1
#> [31] BiocGenerics_0.47.0
#> [32] RNOmni_1.0.1
#> [33] regionalpcs_0.99.1
#>
#> loaded via a namespace (and not attached):
#> [1] splines_4.3.1 BiocIO_1.11.0
#> [3] bitops_1.0-7 filelock_1.0.2
#> [5] preprocessCore_1.63.1 graph_1.79.0
#> [7] XML_3.99-0.14 lifecycle_1.0.3
#> [9] lattice_0.21-8 MASS_7.3-60
#> [11] base64_2.0.1 scrime_1.3.5
#> [13] limma_3.57.7 rmarkdown_2.24
#> [15] yaml_2.3.7 doRNG_1.8.6
#> [17] askpass_1.1 cowplot_1.1.1
#> [19] DBI_1.1.3 RColorBrewer_1.1-3
#> [21] abind_1.4-5 zlibbioc_1.47.0
#> [23] quadprog_1.5-8 purrr_1.0.2
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