Nonlinear Dynamic eQTL analysis on iPSC
Ziang Zhang
2025-02-21
Last updated: 2026-01-21
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Knit directory: fashr-paper-stephenslab/
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knitr::opts_chunk$set(fig.width = 8, fig.height = 6)
library(fashr)
library(biomaRt)
mart <- useEnsembl(biomart = "genes", dataset = "hsapiens_gene_ensembl")
result_dir <- paste0(getwd(), "/output/dynamic_eQTL_real")
data_dir <- paste0(getwd(), "/data/dynamic_eQTL_real")
code_dir <- paste0(getwd(), "/code/dynamic_eQTL_real")
log_prec <- seq(0,10, by = 0.2)
fine_grid <- sort(c(0, exp(-0.5*log_prec)))Obtain the effect size of eQTLs
We use the processed (expression & genotype) data of Strober et.al, 2019 to perform the eQTL analysis.
For the association testing, we use a linear regression model for each gene-variant pair at each time point. Following the practice in Strober et.al, we adjust for the first 3 PCs.
The code to perform this step can be found in the script
dynamic_eQTL_real/00_eQTLs.R from the code directory.
After this step, we have the effect size of eQTLs for each gene-variant pair at each time point, as well as its standard error.
Fitting FASH
To fit the FASH model on \(\{\beta_i(t_j), s_{ij}\}_{i\in N,j \in [16]}\), we consider fitting two FASH models:
A FASH model based on first order IWP (testing for dynamic eQTLs: \(H_0: \beta_i(t)=c\)).
A FASH model based on second order IWP (testing for nonlinear-dynamic eQTLs: \(H_0: \beta_i(t)=c_1+c_2t\)).
The code to perform this step can be found in the script
dynamic_eQTL_real/01_fash.R from the code directory.
We will directly load the fitted FASH models from the output directory.
load(paste0(result_dir, "/fash_fit2_all.RData"))We will load the datasets from the fitted FASH object:
datasets <- fash_fit2$fash_data$data_list
for (i in 1:length(datasets)) {
datasets[[i]]$SE <- fash_fit2$fash_data$S[[i]]
}
all_genes <- unique(sapply(strsplit(names(datasets), "_"), "[[", 1))
full_map <- getBM(
attributes = c("hgnc_symbol", "ensembl_gene_id"),
filters = "ensembl_gene_id",
values = all_genes,
mart = mart
)In this analysis, we will focus on the FASH(2) model that assumes a second order IWP and tests for nonlinear dynamic eQTLs.
Let’s take a quick overview of the fitted FASH model:
log_prec <- seq(0,10, by = 0.2)
fine_grid <- sort(c(0, exp(-0.5*log_prec)))
fash_fit2 <- fash(Y = "beta", smooth_var = "time", S = "SE", data_list = datasets,
num_basis = 20, order = 2, betaprec = 0,
pred_step = 1, penalty = 10, grid = fine_grid,
num_cores = num_cores, verbose = TRUE)
save(fash_fit2, file = "./results/fash_fit2_all.RData")fash_fit2Fitted fash Object
-------------------
Number of datasets: 1009173
Likelihood: gaussian
Number of PSD grid values: 52 (initial), 8 (non-trivial)
Order of Integrated Wiener Process (IWP): 2As well as the estimated priors:
fash_fit2$prior_weights psd prior_weight
1 0.00000000 9.890537e-01
2 0.02024191 8.838899e-03
3 0.03019738 3.058324e-04
4 0.03337327 7.904422e-04
5 0.06720551 6.899707e-04
6 0.27253179 2.240917e-04
7 0.30119421 4.862099e-05
8 1.00000000 4.841831e-05Problem with \(\pi_0\) estimation
The original MLE estimated \(\pi_0\) is 0.9890537. This could be under-estimated due to model-misspecification under the alternative hypothesis. To account for this, we will a conservative estimate of \(\pi_0\) based on the BF procedure:
fash_fit2_update <- BF_update(fash_fit2, plot = FALSE)
fash_fit2_update$prior_weights
save(fash_fit2_update, file = paste0(result_dir, "/fash_fit2_update.RData"))The conservative estimate is 0.9995927, which is much more conservative.
Detecting Nonlinear dynamic eQTLs
We will use the updated FASH model (2) to detect nonlinear dynamic eQTLs.
alpha <- 0.05
test2 <- fdr_control(fash_fit2_update, alpha = alpha, plot = F)44 datasets are significant at alpha level 0.05. Total datasets tested: 1009173. fash_highlighted2 <- test2$fdr_results$index[test2$fdr_results$FDR <= alpha]
test2_before <- fdr_control(fash_fit2, alpha = alpha, plot = F)159 datasets are significant at alpha level 0.05. Total datasets tested: 1009173. fash_highlighted2_before <- test2_before$fdr_results$index[test2_before$fdr_results$FDR <= alpha]How many pairs are detected?
pairs_highlighted2 <- names(datasets)[fash_highlighted2]
length(pairs_highlighted2)[1] 44length(pairs_highlighted2)/length(datasets)[1] 4.360006e-05What is the number before the BF adjustment?
pairs_highlighted2_before <- names(datasets)[fash_highlighted2_before]
length(pairs_highlighted2_before)[1] 159length(pairs_highlighted2_before)/length(datasets)[1] 0.0001575548How many unique genes are detected?
genes_highlighted2 <- unique(sapply(strsplit(pairs_highlighted2, "_"), "[[", 1))
length(genes_highlighted2)[1] 9length(genes_highlighted2)/length(all_genes)[1] 0.001414649Before the BF adjustment?
genes_highlighted2_before <- unique(sapply(strsplit(pairs_highlighted2_before, "_"), "[[", 1))
length(genes_highlighted2_before)[1] 37length(genes_highlighted2_before)/length(all_genes)[1] 0.005815781Visualize top-ranked pairs:
Some examples of null pairs:
Comparing with Strober et.al
We will compare the detected dynamic eQTLs with the results from Strober et.al.
Warning: package 'ggplot2' was built under R version 4.5.2Warning: package 'readr' was built under R version 4.5.2── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.6
✔ forcats 1.0.1 ✔ stringr 1.6.0
✔ ggplot2 4.0.1 ✔ tibble 3.3.0
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.2.0
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
✖ dplyr::select() masks biomaRt::select()
✖ tidyr::unite() masks ggVennDiagram::unite()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsLet’s take a look at the 4 pairs that are least significant from FASH:
Let’s also look at the genes that were missed by FASH, but detected by Strober et.al. In this case, we will pick the most significant pair for each gene in FASH:
Classifying nonlinear dynamic eQTLs
Following the definition in Strober et.al, we will classify the detected eQTLs into different categories:
Early: eQTLs with strongest effect during the first three days: \(\max_{t\leq3} |\beta(t)| - \max_{t> 3} |\beta(t)| > 0\).
Late: eQTLs with strongest effect during the last four days: \(\max_{t\geq 12} |\beta(t)| - \max_{t< 12} |\beta(t)| > 0\).
Middle: eQTLs with strongest effect during days 4-11: \(\max_{4\leq t\leq 11} |\beta(t)| - \max_{t> 11 | t< 4} |\beta(t)| > 0\).
Switch: eQTLs with effect sign switch during the time course: ${(t)+,(t)-}-c $ where \(c\) is a threshold that we set to 0.25 (which means with two alleles, the maximal difference of effect size is at least \(\geq 2\times\min\{\max\beta(t)^+,\max\beta(t)^-\}\times2 \geq 2 \times 0.25 \times 2 = 1\)).
We first take a look at the significant pairs detected by FASH (order 2), and classify them based on the false sign rate (lfsr):
smooth_var_refined = seq(0,15, by = 0.1)
functional_early <- function(x){
max(abs(x[smooth_var_refined <= 3])) - max(abs(x[smooth_var_refined > 3]))
}
testing_early_nonlin_dyn <- testing_functional(functional_early,
lfsr_cal = function(x){mean(x <= 0)},
fash = fash_fit2,
indices = fash_highlighted2,
smooth_var = smooth_var_refined)How many pairs and how many unique genes are classified as early dynamic eQTLs?
load(paste0(result_dir, "/classify_nonlin_dyn_eQTLs_early.RData"))
early_indices <- testing_early_nonlin_dyn$indices[testing_early_nonlin_dyn$cfsr <= alpha]
length(early_indices)[1] 0early_genes <- unique(sapply(strsplit(names(datasets)[early_indices], "_"), "[[", 1))
length(early_genes)[1] 0How many pairs are classified as middle dynamic eQTLs?
functional_middle <- function(x){
max(abs(x[smooth_var_refined <= 11 & smooth_var_refined >= 4])) - max(abs(x[smooth_var_refined > 11]), abs(x[smooth_var_refined < 4]))
}
testing_middle_nonlin_dyn <- testing_functional(functional_middle,
lfsr_cal = function(x){mean(x <= 0)},
fash = fash_fit2,
indices = fash_highlighted2,
num_cores = num_cores,
smooth_var = smooth_var_refined)load(paste0(result_dir, "/classify_nonlin_dyn_eQTLs_middle.RData"))
middle_indices <- testing_middle_nonlin_dyn$indices[testing_middle_nonlin_dyn$cfsr <= alpha]
length(middle_indices)[1] 19middle_genes <- unique(sapply(strsplit(names(datasets)[middle_indices], "_"), "[[", 1))
length(middle_genes)[1] 3Take a look at their results:
How many pairs are classified as late dynamic eQTLs?
functional_late <- function(x){
max(abs(x[smooth_var_refined >= 12])) - max(abs(x[smooth_var_refined < 12]))
}
testing_late_nonlin_dyn <- testing_functional(functional_late,
lfsr_cal = function(x){mean(x <= 0)},
fash = fash_fit2,
indices = fash_highlighted2,
num_cores = num_cores,
smooth_var = smooth_var_refined)load(paste0(result_dir, "/classify_nonlin_dyn_eQTLs_late.RData"))
late_indices <- testing_late_nonlin_dyn$indices[testing_late_nonlin_dyn$cfsr <= alpha]
length(late_indices)[1] 25late_genes <- unique(sapply(strsplit(names(datasets)[late_indices], "_"), "[[", 1))
length(late_genes)[1] 6Let’s take a look at the top-ranked late dynamic eQTLs:
How many pairs and how many unique genes are classified as switch dynamic eQTLs?
switch_threshold <- 0.25
functional_switch <- function(x){
# compute the radius of x, measured by deviation from 0 from below and from above
x_pos <- x[x > 0]
x_neg <- x[x < 0]
if(length(x_pos) == 0 || length(x_neg) == 0){
return(0)
}
min(max(abs(x_pos)), max(abs(x_neg))) - switch_threshold
}
testing_switch_nonlin_dyn <- testing_functional(functional_switch,
lfsr_cal = function(x){mean(x <= 0)},
fash = fash_fit2,
indices = fash_highlighted2,
num_cores = num_cores,
smooth_var = smooth_var_refined)load(paste0(result_dir, "/classify_nonlin_dyn_eQTLs_switch.RData"))
switch_indices <- testing_switch_nonlin_dyn$indices[testing_switch_nonlin_dyn$cfsr <= alpha]
length(switch_indices)[1] 1switch_genes <- unique(sapply(strsplit(names(datasets)[switch_indices], "_"), "[[", 1))
length(switch_genes)[1] 1Let’s take a look at the top-ranked switch dynamic eQTLs:
Gene Set Enrichment Analysis
library(clusterProfiler)
library(tidyverse)
library(msigdbr)
library(org.Hs.eg.db) # Assuming human genes
library(biomaRt)
library(cowplot)
# Retrieve Hallmark gene sets for Homo sapiens
m_t2g <- msigdbr(species = "Homo sapiens", category = "H") %>%
dplyr::select(gs_name, entrez_gene)
mart <- useMart("ensembl", dataset = "hsapiens_gene_ensembl")
## A function to check gene-enrichment
## Retrieve Hallmark gene sets for Homo sapiens (use Ensembl IDs)
m_t2g <- msigdbr(species = "Homo sapiens", category = "H") %>%
mutate(ensembl_use = dplyr::coalesce(ensembl_gene, db_ensembl_gene)) %>%
dplyr::filter(!is.na(ensembl_use)) %>%
dplyr::select(gs_name, ensembl_use) %>%
dplyr::distinct() # one row per (pathway, Ensembl) pair
## A function to check gene-enrichment using Ensembl IDs,
## forcing the universe to be exactly background_gene
enrich_set <- function(genes_selected,
background_gene,
q_val_cutoff = 0.05,
pvalueCutoff = 0.05) {
# ensure character vectors & unique
genes_selected_raw <- unique(as.character(genes_selected))
background_gene_raw <- unique(as.character(background_gene))
# we'll enforce that the test universe is exactly these:
universe_for_test <- background_gene_raw
# genes that are in Hallmark already
hallmark_genes <- unique(m_t2g$ensembl_use)
# background genes that are NOT in Hallmark (no annotation)
bg_not_in_hallmark <- setdiff(universe_for_test, hallmark_genes)
# extend TERM2GENE so that *all* background genes appear in it at least once
# add them to a dummy pathway that we will later drop
dummy_id <- "__DUMMY_BACKGROUND__"
if (length(bg_not_in_hallmark) > 0) {
dummy_t2g <- tibble(
gs_name = dummy_id,
ensembl_use = bg_not_in_hallmark
)
TERM2GENE_full <- bind_rows(m_t2g, dummy_t2g)
} else {
TERM2GENE_full <- m_t2g
}
# also make sure selected genes are a subset of the universe
genes_sel_used <- intersect(genes_selected_raw, universe_for_test)
# message("Original selected genes: ", length(genes_selected_raw),
# " ; used in enrichment: ", length(genes_sel_used))
# message("Universe size (background_gene): ", length(universe_for_test))
enrich_res <- enricher(
gene = genes_sel_used,
TERM2GENE = TERM2GENE_full,
universe = universe_for_test,
pAdjustMethod = "BH",
qvalueCutoff = q_val_cutoff,
pvalueCutoff = pvalueCutoff
)
if (is.null(enrich_res) || nrow(enrich_res@result) == 0L) {
return(enrich_res)
}
df <- enrich_res@result
# drop the dummy term from results
df <- df %>% dplyr::filter(ID != dummy_id)
# keep original ratios for reference
df$GeneRatio_orig <- df$GeneRatio
df$BgRatio_orig <- df$BgRatio
# now recompute ratios using exactly:
# - denominator for GeneRatio = number of selected genes in universe
# - denominator for BgRatio = number of background genes
n_sel_total <- length(genes_sel_used)
n_bg_total <- length(universe_for_test)
df$GeneRatio_fixed <- paste0(df$Count, "/", n_sel_total)
df$BgRatio_fixed <- paste0(df$setSize, "/", n_bg_total)
enrich_res@result <- df
enrich_res
}Among all the genes highlighted by FASH:
result <- enrich_set(genes_selected = genes_highlighted2, background_gene = all_genes)
result@result %>%
filter(pvalue < 0.05) %>%
dplyr::select(GeneRatio, BgRatio, pvalue, qvalue) GeneRatio BgRatio pvalue qvalue
HALLMARK_IL6_JAK_STAT3_SIGNALING 1/9 21/6362 0.02933661 0.05555675
HALLMARK_INTERFERON_ALPHA_RESPONSE 1/9 27/6362 0.03757675 0.05555675
HALLMARK_ALLOGRAFT_REJECTION 1/9 36/6362 0.04982039 0.05555675Among the genes highlighted by FASH that are classified as having middle dynamic eQTLs:
result <- enrich_set(genes_selected = middle_genes, background_gene = all_genes)
result@result %>%
filter(pvalue < 0.05) %>%
dplyr::select(GeneRatio, BgRatio, pvalue, qvalue) GeneRatio BgRatio pvalue qvalue
HALLMARK_INTERFERON_ALPHA_RESPONSE 1/3 27/6362 0.01267987 NA
HALLMARK_ALLOGRAFT_REJECTION 1/3 36/6362 0.01688255 NA
HALLMARK_INFLAMMATORY_RESPONSE 1/3 43/6362 0.02014305 NA
HALLMARK_INTERFERON_GAMMA_RESPONSE 1/3 56/6362 0.02617911 NA
HALLMARK_TNFA_SIGNALING_VIA_NFKB 1/3 58/6362 0.02710553 NA
HALLMARK_ESTROGEN_RESPONSE_LATE 1/3 79/6362 0.03679748 NAAmong the genes highlighted by FASH that are classified as having late dynamic eQTLs:
result <- enrich_set(genes_selected = late_genes, background_gene = all_genes)
result@result %>%
filter(pvalue < 0.05) %>%
dplyr::select(GeneRatio, BgRatio, pvalue, qvalue) GeneRatio BgRatio pvalue qvalue
HALLMARK_IL6_JAK_STAT3_SIGNALING 1/6 21/6362 0.01965004 NA
sessionInfo()R version 4.5.1 (2025-06-13)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.6.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.1
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Chicago
tzcode source: internal
attached base packages:
[1] stats4 stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] cowplot_1.2.0 org.Hs.eg.db_3.21.0 AnnotationDbi_1.70.0
[4] IRanges_2.42.0 S4Vectors_0.46.0 Biobase_2.68.0
[7] BiocGenerics_0.54.1 generics_0.1.4 msigdbr_25.1.1
[10] clusterProfiler_4.16.0 lubridate_1.9.4 forcats_1.0.1
[13] stringr_1.6.0 dplyr_1.1.4 purrr_1.2.0
[16] readr_2.1.6 tidyr_1.3.1 tibble_3.3.0
[19] ggplot2_4.0.1 tidyverse_2.0.0 ggVennDiagram_1.5.4
[22] biomaRt_2.64.0 fashr_0.1.42 workflowr_1.7.2
loaded via a namespace (and not attached):
[1] RColorBrewer_1.1-3 rstudioapi_0.17.1 jsonlite_2.0.0
[4] magrittr_2.0.4 ggtangle_0.0.9 farver_2.1.2
[7] rmarkdown_2.30 fs_1.6.6 vctrs_0.6.5
[10] memoise_2.0.1 ggtree_3.16.3 mixsqp_0.3-54
[13] htmltools_0.5.9 progress_1.2.3 curl_7.0.0
[16] gridGraphics_0.5-1 sass_0.4.10 bslib_0.9.0
[19] plyr_1.8.9 httr2_1.2.2 cachem_1.1.0
[22] TMB_1.9.19 igraph_2.2.1 whisker_0.4.1
[25] lifecycle_1.0.4 pkgconfig_2.0.3 gson_0.1.0
[28] Matrix_1.7-4 R6_2.6.1 fastmap_1.2.0
[31] GenomeInfoDbData_1.2.14 digest_0.6.39 numDeriv_2016.8-1.1
[34] aplot_0.2.9 enrichplot_1.28.4 colorspace_2.1-2
[37] patchwork_1.3.2 ps_1.9.1 rprojroot_2.1.1
[40] irlba_2.3.5.1 RSQLite_2.4.5 filelock_1.0.3
[43] timechange_0.3.0 httr_1.4.7 compiler_4.5.1
[46] bit64_4.6.0-1 withr_3.0.2 S7_0.2.1
[49] BiocParallel_1.42.2 DBI_1.2.3 R.utils_2.13.0
[52] rappdirs_0.3.3 tools_4.5.1 otel_0.2.0
[55] ape_5.8-1 httpuv_1.6.16 R.oo_1.27.1
[58] glue_1.8.0 callr_3.7.6 nlme_3.1-168
[61] GOSemSim_2.34.0 promises_1.5.0 grid_4.5.1
[64] getPass_0.2-4 reshape2_1.4.5 fgsea_1.34.2
[67] gtable_0.3.6 tzdb_0.5.0 R.methodsS3_1.8.2
[70] data.table_1.17.8 hms_1.1.4 xml2_1.5.1
[73] XVector_0.48.0 ggrepel_0.9.6 pillar_1.11.1
[76] babelgene_22.9 yulab.utils_0.2.3 later_1.4.4
[79] splines_4.5.1 treeio_1.32.0 BiocFileCache_2.16.2
[82] lattice_0.22-7 bit_4.6.0 tidyselect_1.2.1
[85] GO.db_3.21.0 Biostrings_2.76.0 knitr_1.50
[88] git2r_0.36.2 xfun_0.55 LaplacesDemon_16.1.6
[91] stringi_1.8.7 UCSC.utils_1.4.0 lazyeval_0.2.2
[94] ggfun_0.2.0 yaml_2.3.12 evaluate_1.0.5
[97] codetools_0.2-20 qvalue_2.40.0 ggplotify_0.1.3
[100] cli_3.6.5 processx_3.8.6 jquerylib_0.1.4
[103] dichromat_2.0-0.1 Rcpp_1.1.0 GenomeInfoDb_1.44.3
[106] dbplyr_2.5.1 png_0.1-8 parallel_4.5.1
[109] assertthat_0.2.1 blob_1.2.4 prettyunits_1.2.0
[112] DOSE_4.2.0 tidytree_0.4.6 scales_1.4.0
[115] crayon_1.5.3 rlang_1.1.6 fastmatch_1.1-6
[118] KEGGREST_1.48.1