1 Introduction

Recent studies associated the differences of cell-type proportions may be correlated to certain phenotypes, such as cancer. Therefore, the demand for the development of computational methods to predict cell type proportions increased. Hereby, we developed deconvR, a collection of functions designed for analyzing deconvolution of the bulk sample(s) using an atlas of reference omic signature profiles and a user-selected model. We wanted to give users an option to extend their reference atlas. Users can create new reference atlases using findSignatures extend their atlas by adding more cell types. Additionally, we included BSMeth2Probe to make mapping whole-genome bisulfite sequencing data to their probe IDs easier. So users can map WGBS methylation data (as in methylKit or GRanges object format) to probe IDs, and the results of this mapping can be used as the bulk samples in the deconvolution. We also included a comprehensive DNA methylation atlas of 25 different cell types to use in the main function deconvolute. deconvolute allows the user to specify their desired deconvolution model (non-negative least squares regression, support vector regression, quadratic programming, or robust linear regression), and returns a dataframe which contains predicted cell-type proportions of bulk methylation profiles, as well as partial R-squared values for each sample.

As an another option, users can generate a simulated table of a desired number of samples, with either user-specified or random origin proportions using simulateCellMix. simulateCellMix returns a second data frame called proportions, which contains the actual cell-type proportions of the simulated sample. It can be used for testing the accuracy of the deconvolution by comparing these actual proportions to the proportions predicted by deconvolute.

deconvolute returns partial R-squares, to check if deconvolution brings advantages on top of the basic bimodal profiles. The reference matrix usually follows a bimodal distribution in the case of methylation, and taking the average of the rows of methylation matrix might give a pretty similar profile to the bulk methylation profile you are trying to deconvolute. If the deconvolution is advantageous, partial R-squared is expect to be high.

2 Installation

The deconvR package can be installed from Bioconductor with:

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("deconvR")

3 Data

3.1 Comprehensive Human Methylome Reference Atlas

The comprehensive human methylome reference atlas created by Moss et al. 1 Moss, J. et al. (2018). Comprehensive human cell-type methylation atlas reveals origins of circulating cell-free DNA in health and disease. Nature communications, 9(1), 1-12. https://doi.org/10.1038/s41467-018-07466-6 can be used as the reference atlas parameter for several functions in this package. This atlas was modified to remove duplicate CpG loci before being included in the package as the dataframe. The dataframe is composed of 25 human cell types and roughly 6000 CpG loci, identified by their Illumina Probe ID. For each cell type and CpG locus, a methylation value between 0 and 1 is provided. This value represents the fraction of methylated bases of the CpG locus. The atlas therefore provides a unique methylation pattern for each cell type and can be directly used as reference in deconvolute and simulateCellMix, and atlas in findSignatures. Below is an example dataframe to illustrate the atlas format.

library(deconvR) 

data("HumanCellTypeMethAtlas")
head(HumanCellTypeMethAtlas[,1:5])
##          IDs Monocytes_EPIC B.cells_EPIC CD4T.cells_EPIC NK.cells_EPIC
## 1 cg08169020         0.8866       0.2615          0.0149        0.0777
## 2 cg25913761         0.8363       0.2210          0.2816        0.4705
## 3 cg26955540         0.7658       0.0222          0.1492        0.4005
## 4 cg25170017         0.8861       0.5116          0.1021        0.4363
## 5 cg12827637         0.5212       0.3614          0.0227        0.2120
## 6 cg19442545         0.2013       0.1137          0.0608        0.0410

3.2 Illumina Infinium MethylationEPIC v1.0 B5 Manifest Probes (hg38)

The GRanges object IlluminaMethEpicB5ProbeIDs contains the Illumina probe IDs of 400000 genomic loci (identified using the “seqnames”, “ranges”, and “strand” values). This object is based off of the Infinium MethylationEPIC v1.0 B5 Manifest data. Unnecessary columns were removed and rows were truncated to reduce file size before converting the file to a GRanges object. It can be used directly as probe_id_locations in BSmeth2Probe.

data("IlluminaMethEpicB5ProbeIDs")
head(IlluminaMethEpicB5ProbeIDs)
## GRanges object with 6 ranges and 1 metadata column:
##       seqnames              ranges strand |          ID
##          <Rle>           <IRanges>  <Rle> | <character>
##   [1]    chr19     5236005-5236006      + |  cg07881041
##   [2]    chr20   63216298-63216299      + |  cg18478105
##   [3]     chr1     6781065-6781066      + |  cg23229610
##   [4]     chr2 197438742-197438743      - |  cg03513874
##   [5]     chrX   24054523-24054524      + |  cg09835024
##   [6]    chr14   93114794-93114795      - |  cg05451842
##   -------
##   seqinfo: 25 sequences from an unspecified genome; no seqlengths

4 Example Workflow For Whole Genome Bisulfate Sequencing Data

4.1 Expanding Reference Atlas

As mentioned in the introduction section, users can extend their reference atlas to incorporate new data. Or may combine different reference atlases to construct a more comprehensive one. This can be done using the findSignatures function. In this example, since we don’t have any additional reference atlas, we will add simulated data as a new cell type to reference atlas for example purposes. First, ensure that atlas (the signature matrix to be extended) and samples (the new data to be added to the signature matrix) are compliant with the function requirements. Below illustrates the samples format.

samples <- simulateCellMix(3,reference = HumanCellTypeMethAtlas)$simulated
head(samples)
##          IDs   Sample 1   Sample 2   Sample 3
## 1 cg08169020 0.06735262 0.06962008 0.08139639
## 2 cg25913761 0.26468127 0.21970031 0.25952501
## 3 cg26955540 0.13995887 0.10318297 0.13411036
## 4 cg25170017 0.14590261 0.12977686 0.14244403
## 5 cg12827637 0.09064446 0.06361314 0.08834634
## 6 cg19442545 0.04161784 0.04258142 0.04374130

sampleMeta should include all sample names in samples, and specify the origins they should be mapped to when added to atlas.

sampleMeta <- data.table("Experiment_accession" = colnames(samples)[-1],
                         "Biosample_term_name" = "new cell type")
head(sampleMeta)
##    Experiment_accession Biosample_term_name
## 1:             Sample 1       new cell type
## 2:             Sample 2       new cell type
## 3:             Sample 3       new cell type

Use findSignatures to extend the matrix.

extended_matrix <- findSignatures(samples = samples, 
                                 sampleMeta = sampleMeta, 
                                 atlas = HumanCellTypeMethAtlas)
## CELL TYPES IN EXTENDED ATLAS: 
## new cell type 
## Monocytes_EPIC 
## B.cells_EPIC 
## CD4T.cells_EPIC 
## NK.cells_EPIC 
## CD8T.cells_EPIC 
## Neutrophils_EPIC 
## Erythrocyte_progenitors 
## Adipocytes 
## Cortical_neurons 
## Hepatocytes 
## Lung_cells 
## Pancreatic_beta_cells 
## Pancreatic_acinar_cells 
## Pancreatic_duct_cells 
## Vascular_endothelial_cells 
## Colon_epithelial_cells 
## Left_atrium 
## Bladder 
## Breast 
## Head_and_neck_larynx 
## Kidney 
## Prostate 
## Thyroid 
## Upper_GI 
## Uterus_cervix
head(extended_matrix)
##          IDs new_cell_type Monocytes_EPIC B.cells_EPIC CD4T.cells_EPIC
## 1 cg08169020    0.07278970         0.8866       0.2615          0.0149
## 2 cg25913761    0.24796886         0.8363       0.2210          0.2816
## 3 cg26955540    0.12575073         0.7658       0.0222          0.1492
## 4 cg25170017    0.13937450         0.8861       0.5116          0.1021
## 5 cg12827637    0.08086798         0.5212       0.3614          0.0227
## 6 cg19442545    0.04264685         0.2013       0.1137          0.0608
##   NK.cells_EPIC CD8T.cells_EPIC Neutrophils_EPIC Erythrocyte_progenitors
## 1        0.0777          0.0164           0.8680                  0.9509
## 2        0.4705          0.3961           0.8293                  0.2385
## 3        0.4005          0.3474           0.7915                  0.1374
## 4        0.4363          0.0875           0.7042                  0.9447
## 5        0.2120          0.0225           0.5368                  0.4667
## 6        0.0410          0.0668           0.1952                  0.1601
##   Adipocytes Cortical_neurons Hepatocytes Lung_cells Pancreatic_beta_cells
## 1     0.0336           0.0168      0.0340     0.0416              0.038875
## 2     0.3578           0.3104      0.2389     0.2250              0.132000
## 3     0.1965           0.0978      0.0338     0.0768              0.041725
## 4     0.0842           0.2832      0.2259     0.0544              0.111750
## 5     0.0287           0.1368      0.0307     0.1607              0.065975
## 6     0.0364           0.0222      0.1574     0.0122              0.003825
##   Pancreatic_acinar_cells Pancreatic_duct_cells Vascular_endothelial_cells
## 1                  0.0209                0.0130                     0.0323
## 2                  0.2249                0.1996                     0.3654
## 3                  0.0314                0.0139                     0.2382
## 4                  0.0309                0.0217                     0.0972
## 5                  0.0370                0.0230                     0.0798
## 6                  0.0378                0.0347                     0.0470
##   Colon_epithelial_cells Left_atrium Bladder Breast Head_and_neck_larynx Kidney
## 1                 0.0163      0.0386  0.0462 0.0264               0.0470 0.0269
## 2                 0.2037      0.2446  0.2054 0.1922               0.2045 0.1596
## 3                 0.0193      0.1134  0.1269 0.1651               0.1523 0.1034
## 4                 0.0187      0.0674  0.0769 0.0691               0.0704 0.0604
## 5                 0.0193      0.0432  0.0459 0.0228               0.0687 0.0234
## 6                 0.0193      0.0287  0.0246 0.0081               0.0098 0.0309
##   Prostate Thyroid Upper_GI Uterus_cervix
## 1   0.0353  0.0553   0.0701        0.0344
## 2   0.1557  0.1848   0.1680        0.2026
## 3   0.0686  0.0943   0.1298        0.1075
## 4   0.0369  0.0412   0.0924        0.0697
## 5   0.0508  0.0726   0.0759        0.0196
## 6   0.0055  0.0188   0.0090        0.0166

WGBS methylation data first needs to be mapped to probes using BSmeth2Probe before being deconvoluted. The methylation data WGBS_data in BSmeth2Probe may be either a GRanges object or a methylKit object.

Format of WGBS_data as GRanges object:

load(system.file("extdata", "WGBS_GRanges.rda",
                                     package = "deconvR"))
head(WGBS_GRanges)
## GRanges object with 6 ranges and 1 metadata column:
##       seqnames    ranges strand |   sample1
##          <Rle> <IRanges>  <Rle> | <numeric>
##   [1]     chr1     47176      + |  0.833333
##   [2]     chr1     47177      - |  0.818182
##   [3]     chr1     47205      + |  0.681818
##   [4]     chr1     47206      - |  0.583333
##   [5]     chr1     47362      + |  0.416667
##   [6]     chr1     49271      + |  0.733333
##   -------
##   seqinfo: 1 sequence from an unspecified genome; no seqlengths

or as methylKit object:

head(methylKit::methRead(system.file("extdata", "test1.myCpG.txt", 
                                     package = "methylKit"), 
                         sample.id="test", assembly="hg18", 
                         treatment=1, context="CpG", mincov = 0))
##     chr   start     end strand coverage numCs numTs
## 1 chr21 9764513 9764513      -       12     0    12
## 2 chr21 9764539 9764539      -       12     3     9
## 3 chr21 9820622 9820622      +       13     0    13
## 4 chr21 9837545 9837545      +       11     0    11
## 5 chr21 9849022 9849022      +      124    90    34
## 6 chr21 9853296 9853296      +       17    10     7

probe_id_locations contains information needed to map cellular loci to probe IDs

data("IlluminaMethEpicB5ProbeIDs")
head(IlluminaMethEpicB5ProbeIDs)
## GRanges object with 6 ranges and 1 metadata column:
##       seqnames              ranges strand |          ID
##          <Rle>           <IRanges>  <Rle> | <character>
##   [1]    chr19     5236005-5236006      + |  cg07881041
##   [2]    chr20   63216298-63216299      + |  cg18478105
##   [3]     chr1     6781065-6781066      + |  cg23229610
##   [4]     chr2 197438742-197438743      - |  cg03513874
##   [5]     chrX   24054523-24054524      + |  cg09835024
##   [6]    chr14   93114794-93114795      - |  cg05451842
##   -------
##   seqinfo: 25 sequences from an unspecified genome; no seqlengths

Use BSmeth2Probe to map WGBS data to probe IDs.

mapped_WGBS_data <- BSmeth2Probe(probe_id_locations = IlluminaMethEpicB5ProbeIDs, 
                                 WGBS_data = WGBS_GRanges,
                                 multipleMapping = TRUE,
                                 cutoff = 10)
head(mapped_WGBS_data)
##          IDs   sample1
## 1 cg00305774 1.0000000
## 2 cg00546078 0.8181818
## 3 cg00546307 0.5454545
## 4 cg00546971 0.5000000
## 5 cg00774867 0.8461538
## 6 cg00830435 0.9166667

This mapped data can now be used in deconvolute. Here we will deconvolute it using the previously extended signature matrix as the reference atlas.

deconvolution <- deconvolute(reference = HumanCellTypeMethAtlas, 
                             bulk = mapped_WGBS_data)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9963  0.9963  0.9963  0.9963  0.9963  0.9963
deconvolution$proportions
##         Monocytes_EPIC B.cells_EPIC CD4T.cells_EPIC NK.cells_EPIC
## sample1              0            0               0             0
##         CD8T.cells_EPIC Neutrophils_EPIC Erythrocyte_progenitors Adipocytes
## sample1               0                0                       0  0.5011789
##         Cortical_neurons Hepatocytes Lung_cells Pancreatic_beta_cells
## sample1        0.2917357           0          0                     0
##         Pancreatic_acinar_cells Pancreatic_duct_cells
## sample1                       0                     0
##         Vascular_endothelial_cells Colon_epithelial_cells Left_atrium Bladder
## sample1                          0            0.001012524           0       0
##         Breast Head_and_neck_larynx Kidney Prostate Thyroid Upper_GI
## sample1      0            0.2060729      0        0       0        0
##         Uterus_cervix
## sample1             0

5 Example Workflow For RNA Sequencing Data

Here, we will use RNA-seq example data from the granulator package. We will only use small sample of data. Notice that we set an IDs column. In order to run deconvR functions, you have to set the Gene names colname as IDs.

  • Note: It is possible to use simulateCellMix to simulate bulk RNA-seq data and, it is also possible to use findSignatures to extend RNA-seq reference atlas. In this example, we won’t extend the reference atlas or use a simulated data.
library(granulator)
#To load the data from granulator package
load_ABIS()
## [1] "ABIS dataset was loaded successfully."
#Read the bulk RNA-seq data
bulk_RNA <- bulkRNAseq_ABIS[1:50,] %>% 
  as.data.frame() %>% 
  mutate(IDs = rownames(bulkRNAseq_ABIS[1:50,])) %>%
  select("IDs", everything())

head(bulk_RNA[,1:5])
##             IDs      CYFZ      FY2H       FLWA      453W
## S1PR3     S1PR3  4.275496 11.544026 10.4491331 13.192052
## RXFP2     RXFP2  2.038530  3.434462  5.4659900  2.877763
## ADAMTS5 ADAMTS5  0.010506  0.000000  0.1700436  0.000000
## CLEC6A   CLEC6A  4.786615  9.357342  6.1288175  8.606988
## FXYD6     FXYD6 19.881627 29.860584 20.3102595 26.095918
## NRG1       NRG1 20.896435  5.769681 21.5369025  9.638740
#Read the reference RNAseq data
reference_RNA <- sigMatrix_ABIS_S0 %>%
  as.data.frame() %>% 
  mutate(IDs = rownames(sigMatrix_ABIS_S0))%>%
  select("IDs", everything())

head(reference_RNA[1:5])
##             IDs Monocytes.C        NK T.CD8.Memory T.CD4.Naive
## S1PR3     S1PR3   45.720735 0.2790023    0.1981103   0.3657506
## RXFP2     RXFP2   17.877398 0.0000000    0.0000000   0.0000000
## ADAMTS5 ADAMTS5    2.550237 0.0000000    0.0000000   0.0000000
## CLEC6A   CLEC6A   33.695996 0.0000000    0.0000000   0.0000000
## FXYD6     FXYD6  114.167642 0.4707691    0.1832934   0.2908456
## NRG1       NRG1  142.457347 0.1949568    0.1527024   0.2790682

We will use deconvolute to deconvolute cell type proportions of the bulk samples to their origin proportions using a quadric programming model.

deconv_RNA <- deconvR::deconvolute(reference = reference_RNA,
                             bulk = bulk_RNA,model = "qp")
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.2190  0.3287  0.5162  0.5047  0.6498  0.7602

Then we can access the deconvoluted proportions of bulk sample profiles.

head(deconv_RNA$proportions)
##      Monocytes.C           NK T.CD8.Memory  T.CD4.Naive  T.CD8.Naive
## CYFZ  0.06348309 7.147599e-16 0.000000e+00 0.000000e+00 0.000000e+00
## FY2H  0.12192004 0.000000e+00 0.000000e+00 2.242393e-15 3.335591e-15
## FLWA  0.14104057 7.055411e-16 1.179500e-15 6.386099e-15 0.000000e+00
## 453W  0.15737710 0.000000e+00 4.598167e-15 1.635336e-14 7.598590e-16
## 684C  0.11535642 0.000000e+00 0.000000e+00 2.139016e-15 1.090887e-15
## CZJE  0.12817739 0.000000e+00 6.059986e-15 2.404947e-15 0.000000e+00
##           B.Naive T.CD4.Memory         MAIT     T.gd.Vd2 Neutrophils.LD
## CYFZ 0.000000e+00 0.000000e+00 6.366268e-15 0.000000e+00      0.2687346
## FY2H 2.734682e-17 0.000000e+00 0.000000e+00 0.000000e+00      0.3144954
## FLWA 1.688036e-01 0.000000e+00 0.000000e+00 5.730223e-16      0.3559542
## 453W 3.024458e-01 4.845950e-17 0.000000e+00 0.000000e+00      0.4653581
## 684C 2.060142e-01 1.722048e-15 0.000000e+00 0.000000e+00      0.1898895
## CZJE 4.075307e-02 0.000000e+00 0.000000e+00 1.984425e-15      0.2649981
##      T.gd.non.Vd2 Basophils.LD Monocytes.NC.I    B.Memory        mDCs
## CYFZ 5.104239e-15 2.465226e-16     0.12302750 0.00000e+00 0.012795335
## FY2H 8.589163e-15 7.842739e-16     0.21114858 0.00000e+00 0.000000000
## FLWA 6.538782e-15 7.249774e-16     0.05502392 0.00000e+00 0.000000000
## 453W 0.000000e+00 0.000000e+00     0.03605163 0.00000e+00 0.000000000
## 684C 0.000000e+00 8.896594e-16     0.12257615 7.61391e-16 0.007300725
## CZJE 0.000000e+00 1.569145e-15     0.07846819 0.00000e+00 0.002757467
##              pDCs Plasmablasts
## CYFZ 4.803158e-01   0.05164364
## FY2H 1.625745e-01   0.18986147
## FLWA 2.401507e-17   0.27917772
## 453W 0.000000e+00   0.03876731
## 684C 2.078994e-01   0.15096372
## CZJE 2.361915e-01   0.24865428

6 sessionInfo

sessionInfo()
## R Under development (unstable) (2021-10-19 r81077)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.15-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.15-bioc/R/lib/libRlapack.so
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB              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] parallel  stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
## [1] granulator_1.3.0  dplyr_1.0.7       doParallel_1.0.16 iterators_1.0.13 
## [5] foreach_1.5.1     deconvR_1.1.1     data.table_1.14.2 knitr_1.36       
## [9] BiocStyle_2.23.0 
## 
## loaded via a namespace (and not attached):
##   [1] minqa_1.2.4                 colorspace_2.0-2           
##   [3] rjson_0.2.20                ellipsis_0.3.2             
##   [5] class_7.3-19                mclust_5.4.7               
##   [7] qvalue_2.27.0               XVector_0.35.0             
##   [9] GenomicRanges_1.47.1        proxy_0.4-26               
##  [11] Deriv_4.1.3                 fansi_0.5.0                
##  [13] mvtnorm_1.1-3               lubridate_1.8.0            
##  [15] codetools_0.2-18            splines_4.2.0              
##  [17] R.methodsS3_1.8.1           jsonlite_1.7.2             
##  [19] nloptr_1.2.2.2              Rsamtools_2.11.0           
##  [21] R.oo_1.24.0                 pheatmap_1.0.12            
##  [23] BiocManager_1.30.16         compiler_4.2.0             
##  [25] assertthat_0.2.1            Matrix_1.3-4               
##  [27] fastmap_1.1.0               limma_3.51.0               
##  [29] htmltools_0.5.2             tools_4.2.0                
##  [31] coda_0.19-4                 gtable_0.3.0               
##  [33] glue_1.4.2                  GenomeInfoDbData_1.2.7     
##  [35] reshape2_1.4.4              Rcpp_1.0.7                 
##  [37] limSolve_1.5.6              bbmle_1.0.24               
##  [39] Biobase_2.55.0              jquerylib_0.1.4            
##  [41] vctrs_0.3.8                 Biostrings_2.63.0          
##  [43] nlme_3.1-153                rtracklayer_1.55.0         
##  [45] rsq_2.2                     xfun_0.27                  
##  [47] stringr_1.4.0               fastseg_1.41.0             
##  [49] lme4_1.1-27.1               lpSolve_5.6.15             
##  [51] lifecycle_1.0.1             restfulr_0.0.13            
##  [53] gtools_3.9.2                XML_3.99-0.8               
##  [55] zlibbioc_1.41.0             MASS_7.3-54                
##  [57] scales_1.1.1                MatrixGenerics_1.7.0       
##  [59] SummarizedExperiment_1.25.0 RColorBrewer_1.1-2         
##  [61] yaml_2.2.1                  ggplot2_3.3.5              
##  [63] pander_0.6.4                yulab.utils_0.0.4          
##  [65] emdbook_1.3.12              sass_0.4.0                 
##  [67] bdsmatrix_1.3-4             stringi_1.7.5              
##  [69] S4Vectors_0.33.0            BiocIO_1.5.0               
##  [71] e1071_1.7-9                 BiocGenerics_0.41.1        
##  [73] boot_1.3-28                 methylKit_1.21.0           
##  [75] BiocParallel_1.29.0         epiR_2.0.38                
##  [77] GenomeInfoDb_1.31.1         rlang_0.4.12               
##  [79] pkgconfig_2.0.3             matrixStats_0.61.0         
##  [81] bitops_1.0-7                evaluate_0.14              
##  [83] lattice_0.20-45             purrr_0.3.4                
##  [85] GenomicAlignments_1.31.0    cowplot_1.1.1              
##  [87] tidyselect_1.1.1            plyr_1.8.6                 
##  [89] magrittr_2.0.1              bookdown_0.24              
##  [91] R6_2.5.1                    IRanges_2.29.0             
##  [93] generics_0.1.1              nnls_1.4                   
##  [95] DelayedArray_0.21.0         DBI_1.1.1                  
##  [97] pillar_1.6.4                survival_3.2-13            
##  [99] RCurl_1.98-1.5              tibble_3.1.5               
## [101] dtangle_2.0.9               crayon_1.4.1               
## [103] utf8_1.2.2                  rmarkdown_2.11             
## [105] grid_4.2.0                  digest_0.6.28              
## [107] tidyr_1.1.4                 numDeriv_2016.8-1.1        
## [109] gridGraphics_0.5-1          R.utils_2.11.0             
## [111] stats4_4.2.0                munsell_0.5.0              
## [113] ggplotify_0.1.0             BiasedUrn_1.07             
## [115] bslib_0.3.1                 quadprog_1.5-8
stopCluster(cl)