For details on the concepts presented here, consider having a look at our publication:

Crowell HL, Soneson C*, Germain P-L*, Calini D,
Collin L, Raposo C, Malhotra D, and Robinson MD:
muscat detects subpopulation-specific state transitions from
multi-sample multi-condition single-cell transcriptomics data.
Nature Communications 11, 6077 (2020).
DOI: 10.1038/s41467-020-19894-4

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Data description

To demonstrate muscat’s simulation framework, we will use a SingleCellExperiment (SCE) containing 10x droplet-based scRNA-seq PBCM data from 8 Lupus patients obtained befor and after 6h-treatment with IFN-\(\beta\) (Kang et al. 2018). The complete raw data, as well as gene and cell metadata is available through the NCBI GEO, accession number GSE96583.

1 Simulation framework

muscat’s simulation framework comprises: i) estimation of negative binomial (NB) parameters from a reference multi-subpopulation, multi-sample dataset; ii) sampling of gene and cell parameters to use for simulation; and, iii) simulation of gene expression data as NB distributions of mixtures thereof. See Fig. 1.

Let \(Y = (y_{gc})\in\mathbb{N}_0^{G\times C}\) denote the count matrix of a multi-sample multi-subpopulation reference dataset with genes \(\mathcal{G} = \{ g_1, \ldots, g_G \}\) and sets of cells \(\mathcal{C}_{sk} = \{ c^{sk}_1, ..., c^{sk}_{C_{sk}} \}\) for each sample \(s\) and subpopulation \(k\) (\(C_{sk}\) is the number of cells for sample \(s\), subpopulation \(k\)). For each gene \(g\), we fit a model to estimate sample-specific means \(\beta_g^s\), for each sample \(s\), and dispersion parameters \(\phi_g\) using ’s function with default parameters. Thus, we model the reference count data as NB distributed:

\[Y_{gc} \sim NB(\mu_{gc}, \phi_g)\]

for gene \(g\) and cell \(c\), where the mean \(\mu_{gc} = \exp(\beta_{g}^{s(c)}) \cdot \lambda_c\). Here, \(\beta_{g}^{s(c)}\) is the relative abundance of gene \(g\) in sample \(s(c)\), \(\lambda_c\) is the library size (total number of counts), and \(\phi_g\) is the dispersion.

Figure 1: Schematic overview of muscat’s simulation framework
Given a count matrix of features by cells and, for each cell, pre-determined subpopulation identifiers as well as sample labels (0), dispersion and sample-wise means are estimated from a negative binomial distribution for each gene (for each subpopulation) (1.1); and library sizes are recorded (1.2). From this set of parameters (dispersions, means, library sizes), gene expression is sampled from a negative binomial distribution. Here, genes are selected to be “type” (subpopulation-specifically expressed; e.g., via marker genes), “state” (change in expression in a condition-specific manner) or equally expressed (relatively) across all samples (2). The result is a matrix of synthetic gene expression data (3).

For each subpopulation, we randomly assign each gene to a given differential distribution (DD) category (Korthauer et al. 2016) according to a probability vector p_dd \(=(p_{EE},p_{EP},p_{DE},p_{DP},p_{DM},p_{DB})\). For each gene and subpopulation, we draw a vector of fold changes (FCs) from a Gamma distribution with shape parameter \(\alpha=4\) and rate \(\beta=4/\mu_\text{logFC}\), where \(\mu_\text{logFC}\) is the desired average logFC across all genes and subpopulations specified via argument lfc. The direction of differential expression is randomized for each gene, with equal probability of up- and down-regulation.

Next, we split the cells in a given subpopulations into two sets (representing treatment groups), \(\mathcal{T}_A\) and \(\mathcal{T}_B\), which are in turn split again into two sets each (representing subpopulations within the given treatment group.), \(\mathcal{T}_{A_1}/\mathcal{T}_{A_2}\) and \(\mathcal{T}_{B_1}/\mathcal{T}_{B_2}\).

For EE genes, counts for \(\mathcal{T}_A\) and \(\mathcal{T}_B\) are drawn using identical means.For EP genes, we multiply the effective means for identical fractions of cells per group by the sample FCs, i.e., cells are split such that \(\dim\mathcal{T}_{A_1} = \dim\mathcal{T}_{B_1}\) and \(\dim\mathcal{T}_{A_2} = \dim\mathcal{T}_{B_2}\). For DE genes, the means of one group, \(A\) or \(B\), are multiplied with the samples FCs. DP genes are simulated analogously to EP genes with \(\dim\mathcal{T}_{A_1} = a\cdot\dim\mathcal{T}_A\) and \(\dim\mathcal{T}_{B_1} = b\cdot\dim\mathcal{T}_B\), where \(a+b=1\) and \(a\neq b\). For DM genes, 50% of cells from one group are simulated at \(\mu\cdot\text{logFC}\). For DB genes, all cells from one group are simulated at \(\mu\cdot\text{logFC}/2\), and the second group is split into equal proportions of cells simulated at \(\mu\) and \(\mu\cdot\text{logFC}\), respectively. See Fig. 2.

Figure 2: Schematic of the various types of differential distributions supported by muscat’s simulation framework
Differential distributions are simulated from a NB distribution or mixtures thereof, according to the definitions of random variables X, Y and Z.

1.1 prepSim: Preparing data for simulation

To prepare a reference SingleCellExperiment (SCE) for simulation of multi-sample multi-group scRNA-seq data, prepSim will

  1. perform basic filtering of genes and cells
  2. (optionally) filter for subpopulation-sample instances with a threshold number of cells to assure accurate parameter estimation
  3. estimate cell (library sizes) and gene parameters (dispersions and sample-specific means)

Importantly, we want to introduce known changes in states across conditions; thus, only replicates from a single condition should go into the simulation. The group to be kept for simulation may be specified via group_keep, in which case samples from all other groups (sce$group_id != group_keep) will be dropped. By default (group_keep = NULL), prepSim will select the first group available as reference.

Arguments min_count, min_cells, min_genes and min_size are used to tune the filtering of genes, cells and subpopulation-instances as follows:

  • only genes with a count > min_count in >= min_cells will be retained
  • only cells with a count > 0 in >= min_genes will be retained
  • only subpopulation-sample instances with >= min_size cells will be retained; min_size = NULL will skip this step
# estimate simulation parameters
ref <- prepSim(example_sce, verbose = FALSE)
# only samples from `ctrl` group are retained
## ctrl101 ctrl107 
##     200     200
# cell parameters: library sizes
sub <- assay(example_sce[rownames(ref), colnames(ref)])
all.equal(exp(ref$offset), as.numeric(colSums(sub)))
## [1] "names for target but not for current"
## [2] "Mean relative difference: 0.4099568"
# gene parameters: dispersions & sample-specific means
## DataFrame with 6 rows and 4 columns
##                  ENSEMBL      SYMBOL                           beta      disp
##              <character> <character>                    <DataFrame> <numeric>
## ISG15    ENSG00000187608       ISG15 -7.84574:-0.24711310:-1.039480 4.6360532
## AURKAIP1 ENSG00000175756    AURKAIP1 -7.84859: 0.00768812:-1.171896 0.1784345
## MRPL20   ENSG00000242485      MRPL20 -8.31434:-0.58684477:-0.304827 0.6435324
## SSU72    ENSG00000160075       SSU72 -8.05160:-0.17119703:-0.793222 0.0363892
## RER1     ENSG00000157916        RER1 -7.75327: 0.10731331:-1.261821 0.5046166
## RPL22    ENSG00000116251       RPL22 -8.03553:-0.03357193: 0.143506 0.2023632

1.2 simData: Simulating complex designs

Provided with a reference SCE as returned by prepSim, a variery of simulation scenarios can be generated using the simData function, which will again return an SCE containg the following elements:

  • assay counts containing the simulated count data
  • colData columns cluster/sample/group_id containing each cells cluster, sample, and group ID (A or B).
  • metadata$gene_info containing a data.frame listing, for each gene and cluster
    • the simulationed DD category
    • the sampled logFC; note that this will only approximate log2(sim_mean.B/sim_mean.A) for genes of the de category as other types of state changes use mixtures for NBs, and will consequently not exhibit a shift in means of the same magnitude as logFC
    • the reference sim_gene from which dispersion sim_disp and sample-specific means beta.<sample_id> were used
    • the simulated expression means sim_mean.A/B for each group

In the code chunk that follows, we run a simple simulation with

  • p_dd = c(1,0,...0), i.e., 10% of EE genes
  • nk = 3 subpopulations and ns = 3 replicates for each of 2 groups
  • ng = 1000 genes and nc = 2000 cells, resulting in 2000/2/ns/nk \(\approx111\) cells for 2 groups with 3 samples each and 3 subpopulations
# simulated 10% EE genes
sim <- simData(ref, p_dd = diag(6)[1, ],
    nk = 3, ns = 3, nc = 2e3,
    ng = 1e3, force = TRUE)
# number of cells per sample and subpopulation
table(sim$sample_id, sim$cluster_id)
##             cluster1 cluster2 cluster3
##   sample1.A      111      119      117
##   sample2.A      113      104      103
##   sample3.A       92      123      103
##   sample1.B      124      108      105
##   sample2.B      131       82       99
##   sample3.B      113      130      123

By default, we have drawn a random reference sample from levels(ref$sample_id) for every simulated sample in each group, resulting in an unpaired design:

##         A         B        
## sample1 "ctrl101" "ctrl107"
## sample2 "ctrl107" "ctrl101"
## sample3 "ctrl101" "ctrl107"

Alternatively, we can re-run the above simulation with paired = TRUE such that both groups will use the same set of reference samples, resulting in a paired design:

# simulated paired design
sim <- simData(ref, paired = TRUE, 
    nk = 3, ns = 3, nc = 2e3,
    ng = 1e3, force = TRUE)
# same set of reference samples for both groups
ref_sids <- metadata(sim)$ref_sids
all(ref_sids[, 1] == ref_sids[, 2])
## [1] TRUE

1.2.1 p_dd: Simulating differential distributions

Argument p_dd specifies the fraction of cells to simulate for each DD category. Its values should thus lie in \([0,1]\) and sum to 1. Expression densities for an exemplary set of genes simulated from the code below is shown in Fig. 3.

# simulate genes from all DD categories
sim <- simData(ref, p_dd = c(0.5, rep(0.1, 5)),
    nc = 2e3, ng = 1e3, force = TRUE)

We can retrieve the category assigned to each gene in each cluster from the gene_info table stored in the output SCE’s metadata:

gi <- metadata(sim)$gene_info
##  ee  ep  de  dp  dm  db 
## 982 192 216 190 218 202