Deconvolution of bulk RNA-seq data with granulator
24 October 2023
Abstract
Heterogeneity in the cellular composition of bulk RNA-seq data may prevent or bias the results from differential expression analysis. To circumvent this limitation, in silico deconvolution infers cell type abundances by modelling gene expression levels as weighted sums of the cell-type specific expression profiles. Several computational methods have been developed to estimate cell type proportions from bulk transcriptomics data, and to account for cell type heterogeneity in the statistical analysis. The R package granulator provides a unified testing interface to rapidly run and benchmark multiple state-of-the-art deconvolution methods. We demonstrate its usage on published bulk RNA-seq data from peripheral blood mononuclear cells.
Package
BiocStyle 2.30.0
Contents
1 Introduction
Bulk transcriptomic data is often generated from heterogeneous samples composed of multiple cell types, where measured values represent an averaged gene expression across all cell types. This heterogeneity is a major hurdle in the statistical analysis, as differences in cell type proportions may prevent or bias the detection of cell type-specific transcriptional programs.
In silico deconvolution of bulk gene expression data allows to infer cell type composition of heterogeneous biological samples. Deconvolution methods are typically used to estimate cell type fractions in bulk RNA-seq data from whole blood, peripheral blood mononuclear cells or other complex tissues in healthy and diseased patients (Abbas et al. 2009; Shen-Orr et al. 2010). Estimated cell type proportions can then be used in subsequent analysis to correct for cell-type heterogeneity making in silico deconvolution an attractive alternative to the physical isolation of individual cell types or single cell RNA-seq.
Several deconvolution methods have been published in recent years, many of which use cell type-specific gene expression references. In this vignette, we present granulator, an R package that provides a unified testing interface to rapidly run and benchmark multiple state-of-the-art deconvolution methods (Table 1).
| Name | Function | Method | License | Reference |
|---|---|---|---|---|
| ols | stats::lsfit | Ordinary least squares | free (GPL-2) | |
| nnls | nnls::nnls | Non-negative least squares | free (GPL-2, GPL-3) | reimplemented based on (Abbas et al. 2009) |
| qprogwc | limSolve::lsei | Quadratic programming with non-negativity and sum-to-one constraint | free (GPL-2, GPL-3) | reimplemented based on (Gong and Szustakowski 2013) |
| qprog | limSolve::Solve | Quadratic programming without constraints | free (GPL-2, GPL-3) | |
| rls | MASS::rlm | Re-weighted least squares | free (GPL-2, GPL-3) | reimplemented based on (Monaco et al. 2019) |
| svr | e1071::svr | Support vector regression | free (GPL-2, GPL-3) | reimplemented based on (Newman et al. 2015) |
| dtangle | dtangle::dtangle | Linear mixing model | free (GPL-3) | (Hunt et al. 2018) |
Each deconvolution method takes as input bulk expression profiles of mixed tissue samples and a reference molecular profile of the individual cell types, which are used to estimate the abundance of cell types in each sample. In the following sections we show how to use granulator for the deconvolution of bulk RNA-seq data from peripheral blood mononuclear cells into the individual cellular components using public reference profiles (Table 2) and how to assess the quality of the obtained predictions.
| Name | Description | Reference |
|---|---|---|
| sigMatrix_ABIS_S0 | PBMCs reference profile (17 cell types) | (Monaco et al. 2019) |
| sigMatrix_ABIS_S1 | PBMCs reference profile (13 cell types) | |
| sigMatrix_ABIS_S2 | PBMCs reference profile (11 cell types) | |
| sigMatrix_ABIS_S3 | PBMCs reference profile (9 cell types) |
2 Installation
granulator can be installed from Bioconductor using:
if (!requireNamespace("BiocManager", quietly = TRUE))
install.packages("BiocManager")
BiocManager::install("granulator")The package can be loaded using:
library(granulator)3 Data
The datasets used in this vignette comprises bulk RNA-seq gene expression data
of peripheral blood mononuclear cells (PBMCs) from 12 healthy donors and bulk
RNA-seq data of 29 isolated immune cell types from 4 healthy donors (Monaco et al. 2019),
publicly available on the NCBI database under GEO accession number GSE107011.
For convenience, a subset of the data is included in the package and can be loaded by using the function load_ABIS():
# load datasets for deconvolution of PBMC RNA-seq data
load_ABIS()3.1 Bulk PBMCs RNA-seq
A subset of the PBMCs bulk RNA-seq data, stored in bulkRNAseq_ABIS, consists of a gene (rows) by
sample (columns) matrix with transcript-per-million (TPM) gene expression values:
# print TPM values in bulk RNA-seq
bulkRNAseq_ABIS[1:5, 1:5]## CYFZ FY2H FLWA 453W 684C
## S1PR3 4.275496 11.544026 10.4491331 13.192052 8.0657227
## RXFP2 2.038530 3.434462 5.4659900 2.877763 2.8150600
## ADAMTS5 0.010506 0.000000 0.1700436 0.000000 0.6866893
## CLEC6A 4.786615 9.357342 6.1288175 8.606988 4.4801144
## FXYD6 19.881627 29.860584 20.3102595 26.095918 22.77975533.2 Reference profiles
We use the reference profile from isolated cell types for 17 immune cell types. The PBMCs reference profile, stored in sigMatrix_ABIS_S0, consists of a gene (rows) by cell type (columns) matrix containing transcript-per-million (TPM)
gene expression values normalized for total mRNA abundance:
# print TPM values in reference profile matrix
sigMatrix_ABIS_S0[1:5, 1:5]## Monocytes.C NK T.CD8.Memory T.CD4.Naive T.CD8.Naive
## S1PR3 45.720735 0.2790023 0.1981103 0.3657506 0.1930285
## RXFP2 17.877398 0.0000000 0.0000000 0.0000000 0.0000000
## ADAMTS5 2.550237 0.0000000 0.0000000 0.0000000 0.0000000
## CLEC6A 33.695996 0.0000000 0.0000000 0.0000000 0.0000000
## FXYD6 114.167642 0.4707691 0.1832934 0.2908456 0.1365307Additionally, we provide a set of reference profile matrices stored in sigMatrix_ABIS_S1, sigMatrix_ABIS_S2, and sigMatrix_ABIS_S3, which were derived at different levels of cell type resolution by summing over similar cell types.
3.3 Measured cell type proportions
Cell type proportions were measured by fluorescence-activated cell sorting (FACS) for 29 immune cell types (Monaco et al. 2019). Additional cell type proportions were computed by summing over cell types with highly similar transcriptional profiles. For instance T.CD4.Naive proportions consist of the weighted sum of the subtypes Th1, Th2, Th1/Th17, Tregs, Tfh, Naive CD4 T cells and Terminal Effector CD4 T cells.
The measured cell type proportions, stored in groundTruth_ABIS, consists of a sample (rows) by cell type (columns) matrix with proportions expressed in percent:
# print measured cell type proportions (percentages)
groundTruth_ABIS[1:5, 1:5]## Monocytes.C NK T.CD8.Memory T.CD4.Naive T.CD8.Naive
## 453W 19.4 6.78 22.931 9.165 5.328
## 684C 19.6 8.45 7.078 10.051 8.411
## CR3L 14.0 10.80 3.597 10.871 11.532
## FLWA 19.6 19.50 4.530 6.084 3.815
## FY2H 26.8 2.60 12.008 8.098 5.2794 Workflow
The granulator workflow consists of four steps:
Reference profiles: Reference profiles for deconvolution are usually generated by differential expression analysis on bulk RNA-seq generated from isolated cell types or cell-type clusters identified by single cell RNA-seq;
Deconvolution: Bulk RNA-seq data from heterogeneous samples is than deconvoluted using one or more reference profiles and deconvolution methods;
Benchmarking: Estimated cell type proportions are benchmarked against measured cell type proportions to assess deconvolution performance. Measured proportions are usually generated from fluorescence-activated cell sorting or single cell RNA-seq data;
Correlation: Benchmarked reference profiles can be used to deconvolve bulk RNA-seq data where individual cell types abundances are unknown. The deconvoluted cell type proportions can be correlated with each other in order to assess the degree of similarity in predictions across methods.
4.1 Comparing multiple reference profiles
The performance of cell type deconvolution strongly depends on the choice and quality of the reference profile, and in particular on the degree of similarity between cell-type specific expression profiles (Vallania et al. 2018; Avila Cobos et al. 2020). It is therefore recommended to test multiple reference profile matrices generated at different cell type resolutions (Newman et al. 2019; Monaco et al. 2019). Here we use the published sigMatrix_ABIS_S0 reference profile, and additional signatures generated by collapsing highly similar cell types into single categories (sigMatrix_ABIS_S1, sigMatrix_ABIS_S2, sigMatrix_ABIS_S3).
# create list if multiple signature matrices to test simultaneously
sigList = list(
ABIS_S0 = sigMatrix_ABIS_S0,
ABIS_S1 = sigMatrix_ABIS_S1,
ABIS_S2 = sigMatrix_ABIS_S2,
ABIS_S3 = sigMatrix_ABIS_S3)We plot the cell-type similarity matrix of all reference profiles by computing their Kendall Rank Correlation Coefficient with plot_similarity(), highlighting clusters of transcriptionally related cell types:
# plot signature matrix similarity matrices
plot_similarity(sigMatrix=sigList)
A useful metric to evaluate the quality of reference profile matrices is to compute the
Condition Number k, which measures how sensitive the deconvolution is to variability in the input data. Generally, a matrix with low condition number (k close to 1) is well-conditioned, as it leads to a stable solution.
4.2 Deconvolution of bulk RNA-seq data
Once suitable reference profiles have been generated, we use deconvolute() to estimate cell type
proportions from the tissue bulk RNA-seq dataset. The function takes a matrix dataset
to be deconvoluted, a matrix or a list of reference profile matrices, and a vector of deconvolution methods.
All data matrices need to be normalized to TPM from raw counts with the function get_TPM(). By default, deconvolute() sequentially runs
all methods available. Optionally, we can provide a selected list of methods and the number of available processing cores
to minimize computation time. Every reference profile matrix is tested in combination with every selected method.
# deconvolute input data using all available methods by default
decon <- deconvolute(m = bulkRNAseq_ABIS, sigMatrix = sigList)For each reference profile and method combination, the function returns the estimated cell type coefficients and proportions (in percentage). Although there may be slightly negative proportions, significant negative values means that deconvolution mehtods fails to converge on a
biological meaningful solution, and the reference profile matrix should be further refined.
We can look at the cell type proportions computed by the support vector regression model (svr) using the sigMatrix_ABIS_S0 reference profile:
# print cell type proportions for svr model on ABIS_S0 reference profile
decon$proportions$svr_ABIS_S0[1:5, 1:5]## B.Memory B.Naive Basophils.LD MAIT Monocytes.C
## 36TS 2.73 3.18 1.82 0.00 25.91
## 453W 2.27 4.09 0.45 0.45 24.09
## 4DUY 4.09 5.00 1.36 1.36 15.91
## 684C 1.36 11.82 1.36 1.82 21.82
## 925L 2.27 17.73 2.27 0.91 22.73We can plot the estimated cell type proportions with the function plot_proportions(). Notice that while the sum of cell types proportions cannot exceed 100%, for some methods part of the bulk RNA-seq signal remains unassigned.
# plot cell type proportions for svr model on ABIS_S0 reference profile
plot_proportions(deconvoluted = decon, method = 'svr', signature = 'ABIS_S0')
To plot all estimated cell type proportions we use the function plot_deconvolute(), which allows to compare results across deconvolution methods and cell types. The option scale indicates whether cell type proportions should be transformed into standard scores. Scaling is useful to directly compare deconvolution output, as the absolute percentages may vary considerably across methods.
# plot cell type proportions
plot_deconvolute(deconvoluted = decon, scale = TRUE, labels = FALSE)
4.3 Benchmarking of deconvolution methods
The third step in the workflow is dedicated to assessing the
performance of deconvolution by taking advantage of available known cell types abundances.
We benchmark deconvolution methods by regressing the estimates against the measured
cell type proportions with the function benchmark():
# benchmark methods by correlating estimated to measured cell type proportions
bench <- benchmark(deconvoluted = decon, ground_truth = groundTruth_ABIS)Summary statistics of the regression models by method, signature, and cell type can be displayed as follows:
# print metrics
head(bench$summary)## signature method celltype pcc ccc adj.r2 rmse
## 1 ABIS_S0 dtangle B.Memory 0.6256 0.3488 0.330 0.0083
## 2 ABIS_S0 dtangle B.Naive 0.9620 0.9265 0.920 0.0062
## 3 ABIS_S0 dtangle Basophils.LD 0.9095 0.7520 0.810 0.0031
## 4 ABIS_S0 dtangle MAIT 0.8004 0.6796 0.600 0.0075
## 5 ABIS_S0 dtangle Monocytes.C 0.4210 0.3101 0.095 0.0320
## 6 ABIS_S0 dtangle Monocytes.NC.I 0.8373 0.8280 0.670 0.0085We can also print the average metrics by regression method and signature as follows:
# print metrics
head(bench$rank)## signature method mean_pcc mean_ccc mean_adj.r2 mean_rmse
## 1 ABIS_S2 nnls 0.8299 0.2093 0.6645 0.0118
## 2 ABIS_S2 ols 0.8298 0.2121 0.6636 0.0129
## 3 ABIS_S2 qprog 0.8298 0.2121 0.6636 0.0129
## 4 ABIS_S2 svr 0.8292 0.4746 0.6655 0.0161
## 5 ABIS_S3 svr 0.8045 0.4305 0.6211 0.0180
## 6 ABIS_S2 rls 0.7922 0.2129 0.6155 0.0124Evaluation metrics include the Pearson Correlation Coefficient (pcc),
the Concordance Correlation Coefficient (ccc), the Coefficient of Determination (adj.r2), and the Root Mean Square Error (rmse). When a cell type cannot be deconvoluted, it’s proportions are returned as NA, which causes the corresponding metric coefficients to be missing as well.
While pcc measures the linear correlation between relative changes in proportions across all samples, ccc measures the linear correlation between true and estimated proportions by taking the mean absolute percentages into account. Both pcc and ccc metrics can range between 1 and -1: a value of 1 represents a total positive correlation, 0 no correlation, and −1 a total negative correlation. adj.r2 represents the square of pcc adjusted for the number of predictors in the model and takes values between 0 and 1. Conversely the rmse measures the quadratic mean of the differences between predicted values and observed values. A value of 0.05 represent a difference of 5%.
The linear regression of estimated versus measured cell type proportions can be visualized
using plot_regress() on the benchmark() results. Here, we analyze the performance of the support vector regression model (svr) across the deconvoluted cell types using the sigMatrix_ABIS_S0 reference profile:
# plot regression for svr model on ABIS_S0 reference profile
plot_regress(benchmarked = bench, method = 'svr', signature = 'ABIS_S0')
Summary statistics across all methods are visualized using the
function plot_benchmark(). To do so, we provide the output from benchmark()
and the metric of interest (pcc,ccc,adj.r2,rmse). Below we show cell-type-specific
pcc scores for different deconvolution methods and reference profiles:
# plot pearson correlation between predictions and true proportions
plot_benchmark(benchmarked = bench, metric = 'pcc')
While there are differences among decononvolution methods, the biggest variability in deconvolution
performance is across different reference profiles. A number of cell types are accurately deconvoluted (mean pcc>0.75) when using the sigMAtrix_ABIS_S0 reference profile. In contrast, predictions for B.Memory, mDCs, Monocytes.C, Monocytes.NC.I, T.CD4.Naive, T.CD8.Naive, and T.CD4.Memory cell types are less accurate (mean pcc<0.75), likely reflecting the uncertainty in discriminating between closely related cell subtypes. A better deconvolution performance can be obtained when these cell type subpopulations are collapsed.
4.4 Correlation analysis of deconvoluted proportions
From the previous benchmark analysis, we select the reference profile sigMatrix_ABIS_S2 for subsequent deconvolution analysis. We exclude
the deconvolution methods dtangle and qprogwc, as they were underperforming other algorithms when comparing the pcc evaluation metric.
# deconvolute input data using selected methods and reference profile matrix
methods <- c('ols','nnls','qprog','rls','svr')
decon <- deconvolute(bulkRNAseq_ABIS, list(ABIS_S2 = sigMatrix_ABIS_S2), methods)When no ground truth data is available, we can assess the performance of the different deconvolution methods by computing the correlation between estimated cell type proportions generated by all methods using the correlate() function. By default
estimated cell type proportions are scaled to standard scores to correct for differences
in absolute estimated cell-type specific proportions across algorithms.
# correlation analysis
correl <- correlate(deconvoluted = decon)The plot_correlate() is used to visualize the results of correlate(), by plotting a heatmap, where estimated cell type proportions are clustered by collinearity across cell type and deconvolution models:
# correlation heatmap
plot_correlate(correlated = correl, method="heatmap", legend=TRUE)
We observe that estimated cell type proportions are highly correlated between methods for all cell types, indicating that the deconvolution methods agree on the assignment of cell type specific signals. The average correlations across methods by cell type can be obtained as follows:
# correlation mean summary statistics
head(correl$summary)## signature cellType mean_correlation
## 1 ABIS_S2 B.Cells 0.9905
## 2 ABIS_S2 Basophils.LD 0.9033
## 3 ABIS_S2 Dendritic.Cells 0.9101
## 4 ABIS_S2 MAIT 0.9568
## 5 ABIS_S2 NK 0.9803
## 6 ABIS_S2 Neutrophils.LD 0.9509Of particular use is also the computation of average correlations across cell types by method, which illustrate which methods are reporting similar estimated cell type proportions:
# deconvolution method ranking
head(correl$rank)## signature method mean_correlation
## 1 ABIS_S2 nnls 0.9261
## 2 ABIS_S2 ols 0.9251
## 3 ABIS_S2 qprog 0.9251
## 4 ABIS_S2 rls 0.8698
## 5 ABIS_S2 svr 0.8590For subsequent analysis, the estimated cell-type proportions can be now included in a linear model as covariates to account for cell type heterogeneity, or to impute cell-type specific gene expression profiles.
5 Session Info
Here is the output of sessionInfo() on the system on which this document was compiled:
# print session info
sessionInfo()## R version 4.3.1 (2023-06-16)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.3 LTS
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## Matrix products: default
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## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
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## time zone: America/New_York
## tzcode source: system (glibc)
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## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
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## [1] granulator_1.10.0 BiocStyle_2.30.0
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## [106] openssl_2.1.1 MASS_7.3-60References
Abbas, Alexander R., Kristen Wolslegel, Dhaya Seshasayee, Zora Modrusan, and Hilary F. Clark. 2009. “Deconvolution of blood microarray data identifies cellular activation patterns in systemic lupus erythematosus.” PLoS ONE 4 (7). https://doi.org/10.1371/journal.pone.0006098.
Avila Cobos, Francisco, José Alquicira-Hernandez, Joseph E. Powell, Pieter Mestdagh, and Katleen De Preter. 2020. “Benchmarking of cell type deconvolution pipelines for transcriptomics data.” Nature Communications 11 (1): 1–14. https://doi.org/10.1038/s41467-020-19015-1.
Gong, Ting, and Joseph D Szustakowski. 2013. “DeconRNASeq: a statistical framework for deconvolution of heterogeneous tissue samples based on mRNA-Seq data.” Bioinformatics 29 (8): 1083–5. https://doi.org/10.1093/bioinformatics/btt090.
Hunt, Gregory J, Saskia Freytag, Melanie Bahlo, and Johann A Gagnon-Bartsch. 2018. “Dtangle: Accurate and Robust Cell Type Deconvolution.” Bioinformatics 35 (12): 2093–9. https://doi.org/10.1093/bioinformatics/bty926.
Monaco, Gianni, Bernett Lee, Weili Xu, Seri Mustafah, You Yi Hwang, Christophe Carré, Nicolas Burdin, et al. 2019. “RNA-Seq Signatures Normalized by mRNA Abundance Allow Absolute Deconvolution of Human Immune Cell Types.” Cell Reports 26 (6): 1627–1640.e7. https://doi.org/10.1016/j.celrep.2019.01.041.
Newman, Aaron M., Chih Long Liu, Michael R. Green, Andrew J. Gentles, Weiguo Feng, Yue Xu, Chuong D. Hoang, Maximilian Diehn, and Ash A. Alizadeh. 2015. “Robust enumeration of cell subsets from tissue expression profiles.” Nature Methods 12 (5): 453–57. https://doi.org/10.1038/nmeth.3337.
Newman, Aaron M., Chloé B. Steen, Chih Long Liu, Andrew J. Gentles, Aadel A. Chaudhuri, Florian Scherer, Michael S. Khodadoust, et al. 2019. “Determining cell type abundance and expression from bulk tissues with digital cytometry.” Nature Biotechnology 37 (7): 773–82. https://doi.org/10.1038/s41587-019-0114-2.
Shen-Orr, Shai S., Robert Tibshirani, Purvesh Khatri, Dale L. Bodian, Frank Staedtler, Nicholas M. Perry, Trevor Hastie, Minnie M. Sarwal, Mark M. Davis, and Atul J. Butte. 2010. “Cell type-specific gene expression differences in complex tissues.” Nature Methods 7 (4): 287–89. https://doi.org/10.1038/nmeth.1439.
Vallania, Francesco, Andrew Tam, Shane Lofgren, Steven Schaffert, Tej D. Azad, Erika Bongen, Winston Haynes, et al. 2018. “Leveraging heterogeneity across multiple datasets increases cell-mixture deconvolution accuracy and reduces biological and technical biases.” Nature Communications 9 (1). https://doi.org/10.1038/s41467-018-07242-6.