splatPop: simulating population scale single-cell RNA sequencing data

Population-scale single-cell RNA sequencing (scRNA-seq) is now viable, enabling finer resolution functional genomics studies and leading to a rush to adapt bulk methods and develop new single-cell-specific methods to perform these studies. Simulations are useful for developing, testing, and benchmarking methods but current scRNA-seq simulation frameworks do not simulate population-scale data with genetic effects. Here, we present splatPop, a model for flexible, reproducible, and well-documented simulation of population-scale scRNA-seq data with known expression quantitative trait loci. splatPop can also simulate complex batch, cell group, and conditional effects between individuals from different cohorts as well as genetically-driven co-expression.


Background
Single-cell RNA-sequencing (scRNA-seq) has enabled the high-throughput quantification of gene expression at the level of the individual cell, making it possible to characterize cells in heterogeneous tissues by their cell-type and cell-state. As gene expression is an intermediate between DNA sequence and traits like response to stimuli and disease status, scRNA-seq can provide insights into the cellular context in which stimuli or diseases have an effect. Today, with decreases in costs and improvements in protocols for multiplexing and demultiplexing samples [1][2][3], scRNA-seq is being performed at larger and larger scales, including across population-scale cohorts.
An early focus for many scRNA-seq studies was to identify differentially expressed genes (DEGs) between cell types or cell states. Now, with population-scale scRNA-seq, cell-type/state specific DEGs can also be identified between individuals from different cohorts. For example, Lawlor et al. [4] identified 638 DEGs across eight different celltypes from the pancreatic tissue of non-diabetes and type 2 diabetes donors. Many of the DEGs were unique to a single cell-type and over half were not discovered when the comparisons were performed without accounting for cell-type, highlighting the importance of single-cell level resolution in deciphering the molecular basis of diseases [4].
Beyond characterizing the cellular context for DEGs, population-scale scRNA-seq data promises to improve our ability to study the regulatory basis for these differences. In recent efforts by the GTEx Consortium to map genetic regulatory effects in human tissues, only 43% of disease-associated genetic variants (i.e., hits from genome wide association studies; GWAS) co-localized with expression-associated genetic variants (i.e. expression quantitative trait loci; eQTL) [5]. It was further estimated that, across tissues, only an average of 11% of trait heritability could be explained by GTEX cis-eQTL [6]. One reason for the lack of disease-associated eQTL is that cellular context is critical for genetic regulation and that disease-associated eQTL are missed when eQTL are mapped using data from bulk tissue [7]. Pioneering efforts to use scRNA-seq data for single-cell eQTL (sc-eQTL) mapping have discovered novel cell-type and dynamic-state specific eQTL [8,9], suggesting population-scale scRNA-seq data could help uncover context-specific regulatory effects.
When scRNA-seq technologies were first becoming available, there was a rush to adapt bulk RNA-seq analysis methods and to develop new single-cell specific analysis methods to address challenges associated with single-cell expression data (e.g., noise, sparsity, high dimensionality). By September 2021, there were over 1050 software packages available for scRNA-seq analysis [10]. Many of these new and old methods have been benchmarked to critically assess their performance on a wide variety of data types and conditions, including benchmarks focused on batch-effect correction [11], normalization [12], differential expression [13], and trajectory inference [14]. While the gold standard for assessing performance of different methods is how well they perform on real datasets, such an assessment can be difficult for scRNA-seq because the ground truth is typically not known. One solution is to use simulated datasets where the ground truth is known. Splatter, a software package that implements a number of methods to simulate scRNA-seq data (including its own model, splat), has become a popular option for generating realistic simulated scRNA-seq data since its release in 2017 [15]. Splatter is flexible, fast, reproducible, and well maintained and, while all simulation frameworks have limitations, was considered one of the top performing models for simulating single-cell RNA-seq data in a recent independent benchmark [16]. However, Splatter simulates data for cells from a single individual, precluding its use for population-scale studies. Further, existing multi-sample single-cell simulation frameworks, such as Muscat [17], do not incorporate genetic effects (i.e., eQTL) into their simulations and therefore are limited in their downstream applications. Indeed, we are not aware of any existing scRNA-seq simulation tools that incorporate genetic effects.
Here, we present splatPop, an extension of splat for the simulation of population-scale scRNA-seq data with realistic population structure and known eQTL effects. The splat-Pop model utilizes the flexible framework of Splatter to simulate data with complex experimental designs, including designs with batch effects, multiple cell groups (e.g., celltypes), and individuals with conditional effects (e.g., disease status or treatment effects). These group and conditional effects are simulated by including both DEGs and eQTL effects, making these simulations a powerful tool for assessing downstream single-cell analysis methods. We first present the splatPop model, then demonstrate how splatPop can simulate data with similar properties to simple and complex empirical data sets. Finally, we demonstrate how these simulations can be used to assess single-cell analysis in a convenient object that, together with the genotype data, is sufficient to recreate the simulated population.
To simulate mean expression levels for every gene for every individual (Fig. 1, left panel), first population-wide means and variance levels are sampled for each gene. To account for the mean-variance trend (Additional file 1: Fig. S1), variance levels are sampled from gamma distributions parameterized from empirical genes in the same gene mean bin, with an option to manually tune the variance between individuals using the similarity.scale parameter. Baseline gene means for each individual are then sampled from a normal distribution using the mean and variance assigned to each gene. If populationscale parameters were estimated from bulk data, the baseline means for each individual are quantile normalized to match the desired single-cell distribution. Finally, eQTL and differential expression effects are added to the baseline gene means (see the "Methods" section). Flexible controls for defining the frequency, size, and context of eQTL and DE effects allow for the simulation of a wide range of complex datasets.
The final step is to simulate realistic expression counts for single-cells for each individual. This task is essentially done using the splat model, where counts are sampled from a Poisson distribution after adjusting means based on the expected library size and biological coefficient of variation (BCV) (Fig. 1, right panel). The splatPop model also uses the batch effect function from splat to simulate population-scale data from multiplexed experimental designs with technical replicates (i.e., where all or some individuals are simulated in more than one batch).

Simple splatPop simulations
Three empirical datasets are used as reference data throughout our study: induced pluripotent stem cells (iPSCs) captured on the SmartSeq2 platform (ss2-iPSC; [9]), floor plate progenitor and dopaminergic neuron cells from a panel of iPSCs differentiating toward a midbrain neural fate captured on the 10x platform (10x-Neuro; [18]), and fibroblasts from models of idiopathic pulmonary fibrosis (IPF) captured on the 10x platform (10x-IPF; [19]; see the "Methods"). For each reference, single-cell parameters were estimated using the individual with the most cells, and population parameters were estimated from mean aggregated counts, excluding individuals with less than 100 cells and individuals from the disease cohort for 10x-IPF.
First, using the ss2-iPSC data as a reference, we simulated scRNA-seq counts for genes on chromosome 22 (504 genes) for six individuals from the 1000 Genomes Project. This example simulation took less than 30 seconds to generate, but splatPop can be efficiently scaled up. For example, simulating 1000 genes for 500 individuals takes just 2 min (Additional file 1: Table S2). The similarity.scale parameter and the percentage of genes assigned with eQTL effects were manually adjusted to match data from six individuals from the reference that were sequenced in the same batch.
Since Zappia et al. and the independent benchmark by Cao et al. demonstrate splat's performance compared to other scRNA-seq simulation models in terms of various metrics at the individual-level [15,16], here we focus on population-level characteristics between splatPop simulations and empirical data. For example, visualizing the global relationship between cells in a lower-dimensional space (Fig. 2a), we see that both the empirical and simulated cells loosely cluster by individual, but with significant overlap. We can quantify the degree of clustering by individual by calculating the silhouette width The counts per gene were mean-aggregated (a.counts) across nCells from each individual and logged before calculating the population wide mean and variance. All cells were used from the empirical data (average = 52), while simulated data was down-sampled to nCells per individual of each cell, a measure of its similarity to other cells from the same individual compared to cells from its nearest neighbor individual. While simulations from a parametric model cannot perfectly replicate empirical data, we find a similar distribution of silhouette widths across simulated compared to empirical cells (Fig. 2b, left). Separating the silhouette widths by individual, we see that cells from some individuals cluster more distinctly than others (Fig. 2b, right). As the simulated individuals are not the same genotypes as the reference individuals and as the base gene means are sampled randomly for each individual, we do not expect the same inter-individual relationship pattern to be observed between simulated and empirical cells. However, splatPop can be used to repli-cate more exactly an empirical dataset (see the "Replicating empirical data with splatPop" section).
In addition to these cell-level comparisons, we can also compare our simulated data with the reference data in terms of gene-level characteristics. By calculating the percentage of variance in gene expression across cells explained by an experimental factor, such as individual, we see that for both the reference and simulated data, nearly 2% of genes have between 10-30% of their variance explained by individual (Fig. 2c). Another key aspect of scRNA-seq data is the mean-variance relationship. Here we compare the mean-variance relationship for each gene across individuals, with the counts per gene per individual calculated by mean aggregating counts across cells (see the "Methods" section; Fig. 2d. From left to right, the plots show that the mean variance relationship stabilizes as more cells are simulated per individual. We also demonstrate that splat-Pop can simulate a simple data set with the cell-level and gene-level properties of data generated on the 10x sequencing platform (Additional file 1: Fig. S2). This example better shows the stabilization of the mean variance trend as the number of simulated cells increases (Additional file 1: Fig. S2d). Together, these comparisons demonstrate that splat-Pop can approximate the cell and gene level characteristics of single-cell RNA-sequencing data.

Complex simulations
The simulations above demonstrate how splatPop can provide population scale scRNAseq data for a homogeneous cell population where the expression differences between individuals are only due to random sampling and genetic effects. Next, we demonstrate how splatPop can be used to simulate data with more complex effects (Fig. 3).

Simulating batch effects
Unwanted sources of variation due to technical differences during sample collection, processing, and sequencing are known as batch effects. Batch effects are especially important to consider for population-scale scRNA-seq simulations because these data are often generated by combining multiple individuals together by pooling or multiplexing. The splat model simulates batch effects by sampling a multiplicative effect for each gene for each batch and applying it to all cells in the batch. In splatPop, we use this same approach, but expand the function to allow the user to design complex multiplexing schemes.
We demonstrate this function by simulating five individuals in two batches using the ss2-iPSC data as a reference. Batch effect sizes are sampled from a log-normal distribution for each gene. Here, we adjust the location and scale parameters to simulate a data set where the batch effects are larger than the individual effects. Because we specified for three individuals to be included in each batch, splatPop randomly selected one individual (sample 4) to be simulated in both batches (Fig. 3a). As with the simple simulations, we can also tune the similarity.scale parameter, percentage of genes assigned as eQTL, and the location and scale parameter for the batch effects to simulate data that closely resembles empirical data. For example, Additional file 1: Fig. S3 shows a full comparison between ss2-iPSC reference data from 10 individuals sequenced across 3 batches. In this example, the batch effect parameters were set individually for each batch so that one batch (batch 3) had larger batch effects than the other batches, highlighting the flexibility of splatPop. The modular and reproducible nature of splatPop simulations also allows for the simulation of multiple populations of cells that share "biological" signals (i.e., the same DE and eQTL effects) but differ in their single-cell properties. This feature can be useful to generate datasets made up of cells sequenced using different chemistries for different batches (Additional file 1: Fig. S4).

Simulating cell groups
Multiple cell groups can be simulated to mimic heterogeneous cell populations (i.e., cell types) for each individual or to mimic treated versus untreated cells from the same individual. The splat model simulates cell groups by assigning group-specific multiplicative DE factors, sampled from a log-normal distribution, to a subset of genes. In splatPop, in addition to the DE factors, we also randomly designate a proportion of eQTL effects as cell group specific. In this way, different cell groups are defined by genetic and non-genetic DE. The proportions of genes with group-specific eQTL and DE and the level of DE can be set for each group, allowing for the simulation of highly complex cell populations.
We demonstrate this function by simulating two cell groups for six individuals using the 10x-Neuro data as a reference. Because the 10x-Neuro data had weak individual effects (i.e., cells from different individuals all clustered together), the group effects dominate this simulation (Fig. 3b). We can tune the splatPop parameters to simulate a dataset with cell group effects that closely resembles empirical data (Additional file 1: Fig. S5).

Simulating conditional effects
Another desirable feature for a population-scale scRNA-seq simulation framework is the ability to simulate differences between individuals that are due to different treatments or conditions (e.g., disease status). The splatPop model allows the user to define the number of conditional groups and the proportional of individuals assigned to each group, and then applies condition-specific DE and eQTL effects. This approach is similar to how cell groups are simulated, but effects are applied to all cells for all individuals in the condition group. For example, group effects can be used to simulate both treated and untreated cells for all individuals in the population, while conditional effects can be used to simulate cells for treated and untreated individuals. Again, the proportion of condition-specific-eQTL and the proportion of genes with condition-specific DE and the level of DE can be specified separately for each conditional group.
We demonstrate this function by simulating six individuals, three in each conditional group, using the 10x-IPF data as a reference. We adjust the location and scale parameters defining the DE effect sizes and proportion of condition-specific eQTL to simulate a data set where the conditional effects are larger than the individual effects (Fig. 3c). Again, we can tune these parameters to simulate datasets with conditional effects that closely resemble empirical data (Additional file 1: Fig. S6).

Simulating genetically-driven co-expression
By default, splatPop randomly pairs eSNPs with eGenes (within the designated window); however, splatPop can also be instructed to assign pairs of eGenes the same eSNP. This results in simulated scRNA-seq data with genetically-driven co-expression (i.e., coregulation) relationships between genes. This can be done by providing eQTL data either as summary statistics or as a formatted splatPop key with the desired eQTL associations. Alternatively, the eqtl.coreg parameter can be set to the desired proportion of co-regulated eGenes. For example, in a simulation using the 10x-Neuro data as reference, setting eqtl.coreg to 0.2 means that 20% of eGenes will be paired and assigned the same eSNP. These co-regulated eGenes have higher expression correlation across individuals than random pairs of eGenes (Fig. 4a).

Replicating empirical data with splatPop
Without the addition of co-regulation effects, the expression correlation between genes is centered around zero (Fig. 4a). This phenomenon occurs because the mean expression value for each gene for each individual is sampled randomly from a distribution. This sampling approach is fast, flexible (e.g., allowing for the simulation of any number of genes), and it maintains the population-level characteristics of empirical scRNA-seq data, all useful properties for many simulation studies. However, splatPop can also be used to simulate scRNA-seq data that more closely replicates the gene expression correlations seen in empirical data. As described above, splatPop estimates population-wide parameters from user provided population-scale bulk RNA-seq or aggregated scRNA-seq data. However, if this empirical data is provided directly to the splatPopSimulate function, it will be used as the base gene means (i.e., the "Gene mean per sample" Fig. 1, left panel). Used in this way, splatPop will generate simulated scRNA-seq data for the individuals and genes in the empirical data that replicates empirical inter-individual relationships (Fig. 4b, Additional file 1: Fig. S7a) and patterns of gene expression across individuals (Fig. 4c, Additional file 1: Fig. S7b) from the empirical data. We note that while cell-group, conditional, and batch effects can still be applied to simulations generated this way, adding simulated eQTL effects may compound with eQTL effects already present in the empirical gene mean data, inflating their effect size.

Using splatPop to evaluate downstream analysis tools
The simulations from splatPop can be used to evaluate new and existing scRNA-seq analysis methods [20]. They can also be used to assess power to detect known effects across experimental variables, such as number of cells sequenced per individual. Here, we demonstrate splatPop's capabilities with example evaluations of approaches to differential expression analysis and eQTL mapping.
For DE analysis, we simulated 500 cells for each of 12 individuals divided into two conditional groups (6 per group) using 10x-IPF as a reference. Conditional DE effects were assigned randomly to 40% of genes, with scaling factors sampled from a log normal distribution (location = 0.2, scale = 0.2). We then tested for DE genes between the two conditional groups by fitting zero inflated regression models to log normalized counts for each gene using MAST (see the "Methods" section) for down-sampled subsets of the cells ranging from 10 to 500 cells per individual. As we know which genes were simulated with conditional DE effects, we can calculate performance metrics such as true positive rate (TPR) and false discovery rate (FDR) (Fig. 5a) and assess the relationship between the estimated and simulated DE effect sizes. We can also look at the properties of simulated genes that were correctly identified as DE (true positives; TP) compared to genes that were missed (false negative; FN) or incorrectly identified as DE (false positives; FP). For example, with 80 cells simulated per individual, the correlation between the estimated and simulated DE effect sizes for TP genes was 0.64 (delta, Fig. 5b), and these TP genes tended to have higher mean expression and tended to have larger DE effect sizes (Fig. 5c). We note that this same approach could be used to test for DE genes between cell groups.
For eQTL analysis, we simulated 500 cells for each of 100 individuals in 10 batches with 10 individuals per batch using ss2-iPSC as a reference. We assigned eQTL effects to 70%  FN, FP, TN). The Pearson's correlation between simulated and estimated betas for TP genes is shown in blue. f The simulated gene mean and coefficient of variation, eQTL effect size (beta), and eSNP minor allele frequency (MAF) for TP, FN, and FP eGenes using 80 cells per individual. Statistical significance is reported for t tests testing for difference between TP and FN or FP categories (ns: p > 0.05, *: p <= 0.05, **: p <=, 0.01, ***: p <= 0.001, ****: p <= 0.0001) of simulated genes. Because splatPop samples eQTL effect sizes from a distribution estimated from empirical eQTL mapping data, many of these associations will have small effects that could only be successfully discovered if a large number of donors are simulated and used for mapping. We used the optimized single-cell eQTL mapping workflow defined by Cuomo, Alvari, Azodi, et al. [20] to map eQTL (see the "Methods" section) and assessed performance across a range of 10 to 500 cells simulated per individual (Fig. 5d). Focusing on eQTL mapping results with 80 cells simulated per individual, the correlation between estimated and simulated eQTL effect sizes was 0.95 (Fig. 5e) and eQTL tended to be missed for genes with higher coefficient of variation and for eQTL with small effect sizes and when the variant assigned to the gene had a low minor allele frequency (Fig. 5f). While a thorough benchmarking of approaches for DE analysis or eQTL mapping is beyond the scope of this paper, these examples demonstrate the utility of splat-Pop for assessing population-scale single-cell analysis methods and experimental design considerations.

Discussion
The expansion of single-cell RNA sequencing to population-scale cohorts has powerful implications for studies of functional genomics. New tools and methods need to be rapidly developed and rigorously tested to ensure researchers are able to make important new discoveries from these data. Independent simulation frameworks are a critical resource to this end.
Here, we present splatPop, a flexible framework for simulating population-scale singlecell RNA-sequencing data using the splat model. splatPop is implemented and available in the Splatter R package from Bioconductor, under a GPL-3 license. Because the splat-Pop model simulates genetic effects on gene expression (i.e., eQTL), realistic population structure can be achieved by providing real genotype data to the model. The simulation of genetic effects also means splatPop can be used to simulate eQTL mapping populations, greatly expanding the usefulness of this model. In addition to being able to simulate complex batch effects and cell group effects, splatPop allows for the simulation of conditional (e.g., treatment or disease status) effects for different cohorts of individuals. The modular framework of splatPop enables these functions to be combined to generate synthetic data with complex experimental designs. For example, splatPop could be used to simulate data for multiple cell-types for individuals from healthy, diseased, and disease-treated individuals, sequenced using a multiplex design with technical replicates.

Conclusions
We demonstrate that by estimating splatPop parameters from real data and manually adjusting control parameters, synthetic data can be generated that has properties resembling a wide range of real data sets. As a parametric simulation framework, splatPop cannot perfectly reproduce all aspects of empirical scRNA-sequencing data, but it has the benefits of flexibility, speed, and parameter interpretability. The simulation functions available in splatPop are well documented and reproducible. Code used to generate all simulations shown here and to generate the plots comparing the simulated to reference data are also available. Further, as improvements are made to the splat model, they will be adopted by splatPop.

Reference data
Empirical scRNA-seq data sets used as references are all previously described and available for download as processed, post-quality control counts per cell per gene (see Availability of data and materials).
Additional cell-level and gene-level filtering was performed as follows. For ss2-iPSC, only cells sequenced at day 0 were included (i.e., iPSCs). For 10x-Neuro, only cells from day 30 that were annotated as floor plate progenitors or dopaminergic neurons and were sequenced in the largest pool (pool 7) were included. Gene filtering was also done to remove genes with zero variance or a mean across cells equal to zero. For 10x-IPF, only fibroblast annotated cells from control and bleomycin 1.7 mg/kg fibrosis-induced mice were included. Annotation was done using SingleR v1.4.1 with the MouseRNAseqData cell type reference from CellDex v1.2.0 [21]. splatPop was estimating very large variance parameters (biological coefficient of variation common dispersion greater than 10) from 10x-IPF. This behavior was driven by the unusually deep sequencing of the libraries for this study, resulting in counts per gene values across cells that often reached into the hundreds while also exhibiting a considerable number of observed zeros. Filtering out genes with zero variance and zero mean as was done for the 10x-Neuro reference was not sufficient to remove this effect. Therefore, we performed additional filtering whereby cells were removed if they had zero counts for over 95% of genes and genes were removed if they had zero counts for over 60% of the remaining cells.
For each reference, the genes included were randomly down-sampled to n = 504 (after other gene filtering steps for the two 10x references) to match the number of genes being simulated (i.e., number of genes on chromosome 22). Single-cell parameters were estimated using the individual with the most cells (ss2-iPSC: joxm n = 383, 10x-Neuro: wihj n = 2268, 947170: n = 399) and population parameters were estimated from mean aggregated counts (i.e., taking the mean of the count values across the cells for each gene from each individual), excluding individuals with less than 100 cells and individuals from the disease cohort for 10x-IPF. The eQTL effect size parameters were estimated from eQTL mapping beta estimates for the top eQTL hit for each gene (for SNPs with MAF greater than 10%) from GTEx v7 using thyroid tissue [22]. These are the default splatPop eQTL effect size parameters.
Genotype data from the 1000 Genomes project for chromosome 22 was used for the simulations (hg19, phase3 [23]). The genotype data was filtered to include biallelic SNPs with no missing data, with minor allele frequency greater than 5%, Hardy-Weinberg equilibrium exact test p value greater than 0.00001, and to remove SNPs in high linkage disequilibrium (independent pairwise r2 less than 0.75, 1600 kb window).

splatPop simulation framework
The splatPop framework (Fig. 1) consists of (i) estimating key parameters from empirical data, (ii) simulating gene means for the population, and (iii) simulating single-cell counts for the population.

Step 1. Parameter estimation
The estimation of single-cell parameters is described in detail in the original Splatter manuscript [15]. Population parameters are estimated from population scale bulk RNAseq or aggregated scRNA-seq data, provided as a matrix of expression levels for each gene (rows) for each individual (column). The parameters that control population-wide mean expression of each gene (α m and β m ) are estimated by fitting a gamma distribution to the gene means across the population. Lowly expressed genes (expression less than 0.1 in greater than 50% of individuals) are excluded. To account for the mean-variance relationship (Additional file 1: Fig. S1), parameters that control expression variance across the population (α v and β v ) are estimated by fitting a gamma distribution to the coefficient of variation (cv) from genes in each expression bin (default n.bins=10). The parameters that control the eQTL effect sizes (α e and β e ) are estimated by fitting a gamma distribution to effect sizes from an empirical eQTL mapping study (bulk or single-cell).
The default values for the population and eQTL parameters were estimated from bulk data from GTEx (v7, thyroid tissue) [22]. The effect sizes from the top eSNP for each gene were used, after removing top eSNPs with a MAF less than 0.05.

Step 2. Simulating gene means for individuals
Population-wide expression means are modeled as λ i ∼ gamma(α m , β m ) for genes i = {i 1 , ..., i n.genes }. If specified by the single-cell parameters, expression outlier effects are added to the sampled values as described in Zappia et al. [15]. A variance is sampled for each gene as σ i ∼ gamma(α v , β v ), where α v and β v are specific to the gene mean bin. The variance can be manually adjusted with the similarity.scale s v parameter, which gets multiplied with the β v parameter. The baseline gene means per individual are then modeled as λ i,j ∼ N(λ i , λ i * σ i ) for every individual j = {j 1 , ..., j n.individuals }. We note, that if empirical gene mean data is provided directly to the splatPopSimulate function, the gene mean and variance sampling steps will be ignored, and the empirical gene means will be used directly. If the population-scale parameters were estimated from bulk RNA-seq data (or bulk RNA-seq data is provided directly to splatPopSimulate), for each individual j, the sampled means λ i are quantile normalized to match the distribution estimated from the single-cell data (i.e., gamma(α sc , β sc )).
The simulation of eQTL effects requires genotype information as input. Providing real genotype data or genotype information simulated using tools like HAPGEN2 [24] will ensure the simulated data reflects realistic population structure. Various control parameters can also be adjusted, including how many or what proportion of genes are assigned eQTL effects (i.e. eGenes), the minimum and maximum minor allele frequency (MAF) of the SNP assigned to each eGene (i.e. eSNP), the maximum distance between eGenes and eSNPs, and the percent of eGenes to have the same eSNP as another eGene (i.e., percent of genes with genetic co-regulation). For each eGene, an effect size is sampled as ω i ∼ gamma(α e , β e ). The eQTL effects are incorporated into the baseline means as in [25] using the equation: where G i,j is the minor allele dosage of the eSNP assigned to eGene i, coded as 0, 1, or 2, for individual j. To account for cell-group differences, a portion of eQTL effects (specified by eqtl.group.specific) are applied to the baseline means for the cells simulated as belonging to a specific cell group. To account for cohort differences, a portion of eQTL effects (specified by eqtl.condition.specific) are only applied to the baseline means for individuals assigned to a specific conditional cohort.
DE effects between cell-groups or between conditional cohorts are simulated as in [15]. Briefly, DE scaling factors are sampled as ω de i ∼ logNorm(μ de , σ de ) for the genes assigned DE effects (proportion specified by de.prob and cde.prob), where μ de and σ de can be adjusted separately for cell groups and cohorts to change the relative impact of these effects. A portion of DE effects can be randomly assigned as negative effects (specified by de.downProb and cde.downProb).

Step 3. Simulating gene means for individuals
Single-cell level expression counts are simulated using the splat model [15], with minor modifications to account for splatPop having sampled gene means and incorporated expression outliers in step 2. Batch effects are also incorporated into the simulations during this step, where multiplicative factors are added to genes for cells from the same batch. In splatPop, the batch effect function from splat is expanded to allow for the simulation of complex multiplexed experimental designs, where individuals are pooled where the multiplicative factors are applied to all cells from all individuals in that batch. The user can specify the number of batches (length of list provided to batchCells) and the number of individuals-per-batch (batch.size). By adjusting these parameters, splatPop can simulate populations where there are no batches, where all individuals are present in multiple batches, or where a subset of individuals are present in multiple batches as technical replicates.