A pipeline for analyzing single cell gene expression data

We have developed a pipeline for analyzing single cell qPCR data in order to determine the mechanism by which TFs were regulated during differentiation (Fig 2).

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Fig 2. Pipeline for the analysis of single cell gene expression data.

This paper develops several new tools for single cell analysis. (A) The kinetic parameters are estimated using a large look-up table. The likelihoods for each possible parameter set are found by multiplying the experimental data by the lookup table, and then the maximum likelihood is identified. (B) The SABEC algorithm is used to identify clusters of cells with uniform bursting kinetics– this iterative algorithm alternates between assigning cells to clusters and estimating bursting kinetic parameters for each cluster. This clustering algorithm is run 50 times and the results are summarized in a consensus matrix, which represents the frequency of each pair of cells being found in the same cluster. (C) Finally, the EPiK tool identifies the most likely set of parameters to have varied for a gene across two different cell populations, using a combination of the Bayesian Information Criterion (BIC), Marginal Probability (MP), and a Subsampling-based method, as described in (C). BIC calculates the likelihood for each possible set of parameters, penalised by the number of free parameters. The MP score calculates the likelihood of each parameter changing, independent of the behaviour of the other parameters. In the subsampling method, random sets of cells are selected, the kinetic parameters estimated for each population of cells; then, the distributions of estimated kinetic parameters are compared. (D) These new tools are combined to form a pipeline for analyzing single cell qPCR data. The details for each step of this pipeline are described in the Methods.

https://doi.org/10.1371/journal.pcbi.1005072.g002

The input data consists of single cell qPCR data for a set of genes, where the cells have been sorted into their expected cell populations, using fluorescence-activated cell sorting (FACS). There are two main outputs of the pipeline. Firstly, cells are divided into populations (clusters) based on their bursting behaviour– each gene has the similar rates of activation (K_on), rates of inactivation (K_off) and rates of transcription (K_t) across all cells within each cluster. Secondly, the pipeline compares between each pair of cell populations and predicts the mechanism by which each gene changes its expression– by changing the rate the gene turned on and off or by changing the rate of transcription once the gene is already active.

A key component in the pipeline is our strategy for estimating the kinetic parameters. Previous approaches for estimating the kinetic parameters were slow to compute, because they involved iterative algorithms, such as simulated annealing [13], expectation maximisation [11], or Gibbs Sampling [12]. Meanwhile, we have a pre-computed look-up table for P(x|K_on, K_off, K_t), the probability of seeing x mRNA transcripts, given a set of kinetic parameters (Fig 2A and Eq 1). This table can be used to compute the most likely set of kinetic parameters using a single matrix multiplication– a very efficient computation (Eq 2).

In the pipeline, first we normalize the measurements from the qPCR and transform them into estimated mRNA counts (See Eq 6). Next, these measurements are input into a newly developed unsupervised clustering algorithm (SABEC) which identifies populations of cells with uniform bursting kinetics. SABEC is an iterative algorithm that starts by randomly assigning cells to clusters; next, the algorithm alternates between estimating the kinetic parameters for each gene within each cluster (using Eq 2) and probabilistically re-assigning cells to clusters based on their likelihood of belonging to each cluster (using Eq 9), until the algorithm converges– i.e. when fewer than 5% of cells swap clusters. SABEC is run fifty times using different initial sets of randomly assigned clusters, and the results are summarised as a consensus matrix– a matrix that records the number of times two cells were grouped together. The number of clusters can be determined by one of the following methods: the proportion ambiguously clustered (PAC), Variable Information (VI) or the corrected Rand score (See Fig 2B).

Finally, each pair of populations is compared using a statistical tool we developed called EPiK, in order to identify whether each gene is regulated by modifying the rate the gene turns on (burst frequency) or the level of transcription once the gene is active (burst magnitude). EPiK is a compilation of three different methods: the Bayesian Information Criteria (BIC), a Marginal Probability (MP) Score, and a subsampling method. Since for each gene there can be between 0 and 3 parameters that can differ across a pair of populations, BIC identifies the set of parameters most likely to differ after penalising parameter sets that are larger (See Eq 12). The MP score separately calculates the likelihood of each parameter changing, independent of whether the other parameters change or stay the same (See Eq 15). In the subsampling method, the kinetic parameters are calculated for small random subsets of cells, then the distributions of estimated kinetic parameters for the subsets are compared by the Kolmagorov-Smirnov statistic(Fig 2C).

All together, our R data processing scripts, SABEC, and EPiK come together to form a pipeline that utilizes the mathematics of bursting gene expression to determine how genes are differentially regulated across populations of cells (Fig 2D).

Validation of methods with in silico datasets

These methods were all tested in silico using data sets that have similar structures to the experimental datasets. Each of the three methods introduced in this paper (ML approach for kinetic parameter estimation, SABEC and EPiK) required a different set of simulated data.

The ML method for kinetic parameter estimation was benchmarked populations of cells with known kinetic parameters to allow us to quantify the accuracy of the method. 3,000 parameter sets were randomly selected (uniformly distributed) with K_on between 0 and 5, K_off between 0 and 20, and K_t between 0 and 600. We randomly sampled 10% of the transcripts from each of the simulated cell, to represent the technical noise caused by the loss of 90% of the starting material during sample preparation. We included a stochastic loss of mRNA transcripts to account for material loss during cDNA library construction– Islam et al. [33] estimates that only 48% of transcripts are reverse transcribed into cDNA, and Wu et al. [34] could capture 42% of the total unique transcripts that were identified in bulk RNA-seq.

The simulated datasets for testing the SABEC method were chosen to be as similar to the experimental datasets as possible. 100 simulation sets were generated, with each completely parallel to the Moignard dataset; Each simulation set consisted of five populations of 124 cells with 18 genes each, with their kinetic parameters equal to those estimated by the ML method for the experimental data.

Finally, to test the final method of comparing kinetic parameters between two populations of cells, 1600 simulated datasets were generated, each one consisted of a pair of populations of 124 cells with 1 gene each. There are eight combinations of kinetic parameters that can change (none, all, three ways one parameter can change and three ways two parameters can change). Two hundred pairs of populations were selected for each of these eight scenarios, with 100 simulations with 0 < K_off < 5 and 100 simulations with 5 < K_off < 10. The range of the other parameters were 0 < K_on < 5 and 0 < K_t < 600. In these simulations we also simulated the random loss of 90% of the mRNAs.

Even after simulating 90% loss of biological material, the predicted parameters correlated well with their real values, although K_off was the most difficult parameter to predict (See Fig 3A–3C), especially at wider parameter ranges (S1 and S2 Figs), but it still performed better than the Kim and Marioni Method (S3 Fig). These in silico results emphasize that the cDNA library efficiency can have a large impact on the absolute values of predicted parameters, which suggests that raw parameter values are not comparable across different experiments. SABEC also accurately predicted clusters of simulated datasets, with similar population structures and parameter ranges to the experimental dataset (Fig 3D–3F). Finally, we found that the EPiK method was very conservative, with approximately a 0.05% false positive rate when the methods were intersected (Fig 3G and 3H). Further details about the validations and comparisons with alternative methods are described in depth in the Methods. Next, we applied these methods to experimental datasets.

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Fig 3. In silico validation of single cell analysis pipeline.

In order to estimate the accuracy of the new tools developed in this paper, each tool was evaluated against simulated datasets. (A-C) The kinetic parameter estimation method was tested against simulated data in which 90% of the data was randomly discarded, to simulate the loss of biological material through inefficient cDNA preparation. The known kinetic parameters and the estimated kinetic parameters were compared for K_on (A), K_off (B) and K_t (C). (D-F) Next, SABEC was tested against simulated datasets that had the same kinetic parameters as those estimated for the cell populations in the Moignard single cell dataset (124 simulated cells for each of the 5 cell populations)—SABEC was tested on 100 of such simulated datasets and the robustness of the clustering was measured by calculating the proportion of ambiguously clustered cells (PAC). This figure depicts a hierarchical clustering of the consensus matrices that come from (D) the dataset whose clustering had the worst PAC score, (E) a randomly selected dataset, and (F) the dataset whose clustering had the best PAC score. The coloured bars along the side and bottom represent the true class labels of each cell. (G-H) Next, the true positive and false positive rates were calculated for each proposed component of EPiK, the union of the MP and subset method, and the intersection of all three methods. For the MP and Subset methods, thresholds were set so the false positive rate would be approximately 2%– receiver operating characteristic (ROC) curves for these are found in S13 and S14.

https://doi.org/10.1371/journal.pcbi.1005072.g003

Single cell analysis of hematopoietic stem cells and progenitors

The dataset we use to test our pipeline comes from Moignard et al. [14], which is a high quality single cell qPCR dataset that includes approximately 124 cells each from five different populations of cells during hematopoeisis. Hematopoiesis is the process by which hematopoietic stem cells (HSC) in the bone marrow differentiate into different types of red and white blood cell types. This process can be depicted as a differentiation tree, in which each cell must make multiple “decisions” at each branching point in the tree that will determine its final cell fate. Moignard et al. [14] includes two key branching points: i) HSC cells can become lymphoid-primed multipotential progenitors (LMPPs) or premegakaryocytes (PreMegEs, also referred to as PreMs in figures) and ii) LMPP cells can become granulocyte-macrophage progenitors (GMPs) or common lymphoid progenitors (CLP).

The focus of this study is on the densely interconnected TF regulatory network that has been shown to contribute to cell fate decisions. There is strong evidence that at least seven of these TFs directly interact with one another, potentially forming a SCL/Lyl1/Gata2/Runx1/Lmo2/Fli-1/Erg heptad, with some TFs directly binding to the DNA (such as Gata2 and SCL) and other TFs serving a bridge between the DNA bound components (such as LMO2) [16]. Other key TFs that were profiled in Moignard et al. [14] were PU.1, Meis1, Hhex, Tel, Nfe2, Eto2, Mitf and Ldb1.

There are a number of open questions about the differentiation of HSC into the various progenitor cell populations. Firstly, although the cells were assigned to their populations via FACS, it is unclear whether these sub-populations are truly uniform. For instance, some HSC cells may be biased towards self-renewal or producing cells in the lymphoid or myeloid lineages.

Secondly, the specific mechanisms by which these TFs are being regulated are as yet uncharacterized. Influencing the rate at which genes turns on (K_on) could increase the proportion of cells with an active copy of a gene; whereas manipulating the transcription rate (K_t) would result in there being higher concentrations of mRNA, while maintaining the same proportion of cells with an active gene. In hematopoiesis, cellular TF concentrations help determine which branch the cell will take as it differentiates towards its final cell fate. Therefore, choosing to manipulate K_on instead of K_t could influence the proportion of cells that enter a certain differentiation trajectory. In addition, since the expression of these TFs is tightly linked, one gene’s bursting dynamics could have repercussions on the dynamics of the entire network.

Hematopoietic stem cell populations have heterogeneous bursting kinetics

Although the cells that we study have been sorted into distinct subpopulations using FACS, this does not guarantee that the populations are indeed transcriptionally uniform. Some cells may be misclassified by FACS (expected misclassification rate is 1% for the Moignard dataset), and some of the known populations may be composed of as yet unidentified subpopulations. In addition, extrinsic variability, such as having cells in different stages of the cell cycle, could cause the populations to be heterogeneous.

It can be difficult to accurately identify homogenous subpopulations using standard clustering approaches like K-means or hierarchical clustering (S5 Fig). For instance, K-means is most effective when each cluster is normally distributed and has similar variances. Due to bursting, gene expression is unlikely to come from such a distribution; gene expression distributions can look like a Poisson distribution, a negative binomial distribution, or even a bimodal distribution, depending on K_on, K_off, and K_t. Therefore, we developed a simulated annealing strategy (Simulated Annealing for Bursty Expression Clustering, SABEC), which takes into account bursting gene expression, in order to have more robust clustering. SABEC was rigorously tested against simulated datasets that were designed to be as close as possible to the structure of the experimental data (S6 Fig). Additionally, the algorithm was tested on a wide range of simulated datasets, to see if this method could perform well under varied conditions, such as on data with different numbers of genes measured in each cell and different numbers of cells in each population (S7 Fig).

Next, SABEC was applied to the experimental single cell qPCR data from Moignard et al. [14]. The final clustering is depicted in Fig 4A. Although the predicted hematopoiesis differentiation tree is expected to look like Fig 4Bi, we found that HSC is divided into two distinct subpopulations. One of these populations had 12.8% of cells expressing Gata1, while the other population had none. Since Gata1 is uncommon in HSC cells, but often found in multi-potent progenitor populations (MPP, the earliest progenitor population formed by HSCs [17]), we hypothesized that the populations of cells without Gata1 may be self-renewal HSCs and the other may be HSCs poised for differentiation.

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Fig 4. Application of SABEC to hematopoietic stem cells and progenitor populations.

SABEC was used to identify subpopulations of cells in the Moignard et al. dataset and to identify specific cells that do not cluster well with the other cells of the same type. (A) Hierarchical clustering was applied to the SABEC consensus matrix to reveal six subpopulations of HSC in subfigure. (Bi) The expected hematopoiesis differentiation tree is shown–the names of the cell populations are color-coded to match the vertical and horizontal bars in A. (Bii) We hypothesized a possible alternative differentiation tree, based on the clustering results. (C) Another application of this clustering method is to remove possible outliers, which may have arisen due to poorly sorted cells or extrinsic variability. Within each cell population (i.e. CLP (i), GMP (ii), HSC (iii), LMPP (iv) and PreM (v)), we sorted the cells by how often the cells are clustered with other cells from the same FACS label (% match). The vertical lines depict thresholds selected to approximately correspond to the region with the steepest slope– cells to the left are disregarded as outliers in the EPiK analysis.

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Our clustering with SABEC was based solely on the expression of 18 TFs. However, Moignard et al. [14] also profiled cell surface protein tyrosine kinase c-Kit. It is known that low levels of c-Kit correlate to greater self-renewal potential in HSCs [18]. Even though all of the HSC cells were positive for the c-Kit protein according to FACS, not all the cells had high levels of c-Kit transcripts. Our hypothesized self-renewal HSC population had significantly lower levels of c-Kit than our poised-to-divide population (p-value 9.117e-06 with Kolmogorov-Smirnov test). Even though c-Kit was not one of the genes used to cluster the cells, there was substantial difference in expression levels, suggesting the tree topology depicted in Fig 4Bii. Further evidence for this arrangement is in S9 Fig.

The SABEC algorithm has provided us with subpopulations that appear to have uniform bursting kinetics. In the next section we will identify which kinetic parameters were most likely adjusted across each pair of cell populations, using EPiK. EPiK works best when cell populations have uniform bursting dynamics; S11 Fig shows how having mixed cell types can influence estimates of kinetic parameters. Therefore, we remove cells that do not cluster well with other cells of the same type, with cutoff thresholds designated by the vertical lines in Fig 4C, which is determined by the region of the curve with the steepest slope. However, this pruning protocol could bias parameter estimation if it were to eliminate cells that are falsely identified as outliers (S10 Fig). Therefore, we run EPiK both on the pruned and unpruned datasets, and only consider parameters that are consistently found to have changed under both conditions. It is important to note that the proposed outlier cells may be of biological interest– for instance, they may be rare cell types or cells in the process of differentiation. The pruned dataset is only for use to boost the accuracy of predictions with EPiK, but all cells should be observed for other types of analysis.

Transcription factors involved in hematopoiesis are regulated by different mechanisms at each stage of differentiation

EPiK incorporates three different metrics for evaluating whether kinetic parameter changes are significant. By taking the intersection of these three prediction methods, the false positive rate decreases without a significant drop in the true positive rate. This gives us a very conservative list of probable kinetic parameter changes (a 0.05% false positive rate with our simulated dataset).

The first method is the Bayesian Information Criterion (BIC) (S12 Fig), and the second method is the marginal probability (MP), which is the log likelihood of a certain parameter being varied, independent of whether the other two parameters vary or stay the same (S13 Fig). In the third method, we repeatedly subsample cells from each population and estimate the kinetic parameters for each subset, comparing the maximum distance between the cumulative density functions of the distributions (S14 Fig).

These three methods were applied to the experimental data from Moignard et al. [14]. Each of the methods is based on slightly different assumptions, so they each have different distributions of parameters being varied (See S15 Fig). For instance, BIC predicts that K_off is adjusted in a few cases, but this is not deemed significant by either the MP or subset methods. In addition, there are more significant parameter changes in the case of the pruned populations compared to the complete populations, because these populations are more distinct from one another.

We can now take a closer look at a few examples of TFs that have predicted kinetic parameter changes during differentiation (Fig 5). The predictions from each method are drawn along the branches of the hematopoiesis differentiation tree (Fig 5A–5E). To demonstrate the magnitude and direction of these kinetic parameter changes, Fig 5E–5H illustrates the kinetic parameter estimates for each population of cells.

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Fig 5. Transcription factor-specific kinetic parameter changes.

For each specific gene, it is possible to use EPiK to predict which kinetic parameters vary throughout differentiation. (A-D) For each transition between populations of cells, the kinetic parameter changes are predicted for each TF using all six methods shown in the key, which we represent by six colored rectangles– the color corresponds to the parameters predicted to have changed, as designated in the key. If there are not enough cells with a certain gene expressed in order to calculate parameter changes, then the rectangles are depicted as white. The results are shown for a selection of TFs: Eto2 (A), Mitf (B), Nfe2 (C), and PU.1 (D). (E-H) To demonstrate that EPiK provides reasonable results, we depict the maximum likelihood estimates for K_t and K_on for each population of cells (pruned dataset). If there is a predominant change in K_on (red), K_t (yellow), or both variables (orange), then these populations are connected by lines of the corresponding color.

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In Fig 5A and 5F, four out of the six methods predict that Eto2 is up-regulated by increasing K_on (red) during the HSC to PreM transition, which is consistent with Eto2 having a role in increasing the proportion of cells that differentiate into PreM [19].

One striking feature of Fig 5A–5E is that Eto2, Mitf, and PU.1 are regulated by different kinetic parameters in different stages of blood differentiation. PU.1 has known cell-type-specific enhancer elements [20], and our results may suggest that each of these may regulate PU.1 through different mechanisms. On the other hand, Nfe2 is consistently regulated by K_t throughout differentiation.

In some instances, all six methods come to a consensus as to which kinetic parameter was the source of gene expression variability for a particular TF, and these are shown in Fig 6A. Throughout hematopoiesis, most of the differences in gene expression come from changes of K_on, but Lmo2, Nfe2 and Meis1 are regulated by K_t in the transition from LMPP to GMP. Recall that in the previous section, we identified that HSC forms two distinct sub-populations. We compared the two HSC sub populations with their child populations (LMPP and PreM) (see Fig 6). Between HSC1 and HSC2 only Tel seemed to consistently change its kinetic parameter (K_on). As expected, the cell population with higher Gata1 and c-Kit expression has more in common with LMPP and PreM cells than the other HSC subpopulation.

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Fig 6. Most significant kinetic parameters variations during hematopoiesis.

This figure illustrates kinetic parameter change predictions that are consistent across all six methods illustrated in the key of Fig 5– in other words, this is the intersection of BIC, MP, and the subset methods, as applied to both pruned and un-pruned datasets. Just as in Fig 5, red boxes designate that K_on varies, yellow boxes designate that K_t varies and orange boxes designate that both parameters change. A positive/negative signifies that gene expression increases/decreases along the line (top to bottom; or in the case of horizontal lines left to right). (A) The most significant kinetic parameter changes are shown for the expected hematopoietic tree. (B) Based on the SABEC clustering analysis, it appeared as if there were two distinct HSC subpopulations– the differences between these two subpopulations and LMPP and PreM subpopulations were also calculated.

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In summary, our methods have allowed us to not only identify which genes have changed their expression, but also what physical mechanism caused that change.

Gata2 and Gfi1 may regulate transcription by influencing the rate genes turn on

In the previous section, we identified TFs that were regulated by K_on or K_t during blood differentiation, but it is unclear which TFs were controlling these changes. It may be possible to discover the specific mechanistic role of a TF by manipulating its expression experimentally and then calculating the change in kinetic parameters of its downstream targets.

We decided to focus on the Gfi1-Gfi1b-Gata2 subnetwork that was identified as being important at the first branching point of HSCs to LMPP and PreMegEs [14]. Primary HSCs are difficult to isolate in large quantities, culture and manipulate, so we turned to HPC7 cells, a model cell line for hematopoietic stem and progenitor cells which has some differentiation potential towards more mature blood cells [16, 21]. Similarly to HSCs, HPC7 cells express Gata2 and Gfi1b, but little or no Gfi1. We therefore up-regulated Gfi1 expression and down-regulated Gata2 expression and performed single cell gene expression analysis for the gene set described by Moignard et al. [14], as well as some additional genes involved in HSC differentiation. We analysed 81 cells expressing an shRNA against Gata2 and 77 cells control cells expressing an shRNA against Luciferase, and 72 cells overexpressing Gfi1 and 45 control cells expressing an empty vector.

There are a number of strategies by which a TF could reduce gene expression: by decreasing the rate a gene turns on, by increasing the rate a gene turns off, or by decreasing the rate of transcription of an active gene, and each of these strategies would result in different temporal dynamics of gene expression bursting. All six methods came to a consensus that up-regulating Gfi1 seemed to significantly alter Erg, Gfi1b, Hhex and Mpl, by lowering K_on. Previous research suggests that Gfi1 is usually a repressor that either keeps the chromatin in a condensed state or actively competes for binding with activators [22]. Both of these mechanisms of action are consistent with Gfi1 down regulating its targets by lowering K_on. Based on ChIP-seq experiments from Sanchez-Castillo et al. [23], Gfi1 binds in or near all four of these potential targets (see S1 Table).

In the other experiment, Gata2 was down-regulated; however, this was not a complete knockdown, with only a slight overall decrease in expression (S20 Fig). All six methods suggest that Gfi1b expression was decreased via a change in K_on as Gata2 levels decreased. In the pruned population of cells, Procr (also known as EPCR, a known target of Gata2) had higher K_on after the knockdown [24].

When Gata2 and Gfi1 were down- and up-regulated, our methods could only detect changes in K_on. Therefore, we can hypothesize that this could be the mechanism of action of these two TFs.

Gene expression variations in leukemia

Our pipeline for the identification of differential kinetic parameter values can also be applied to compare healthy and diseased cell populations. In particular, we apply it to four of the cell populations isolated in Guo et al. [15]: two cell types from a healthy mouse (GMP, Lin+) and two from a leukemic mouse (LGMP, LLin+), whose leukemic cells came from a MLL-AF9 fusion protein [25]. The most significant kinetic parameter changes are shown in Fig 7. Some of the genes profiled by Guo et al. [15] were found in fewer than 10 cells in one or more cell population, and these were excluded from kinetic parameter comparisons.

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Fig 7. Kinetic parameter differences between leukemic and healthy cells.

This figure illustrates kinetic parameter change predictions for the single cell qPCR data in Guo et al. Just as in Figs 5 and 6, red boxes indicate that K_on varies, yellow boxes indicate that K_t varies and orange boxes signify that both parameters change. The blue box indicates that K_off varies. A positive/negative signifies that gene expression increases/decreases along the line (top to bottom; or in the case of horizontal lines left to right).

https://doi.org/10.1371/journal.pcbi.1005072.g007

Some of the genes are enriched in a single cell population; for instance, TLR9, a factor whose expression influences the prognosis of leukemia [26], is found predominantly in the LLin+ cells, and appears to be regulated by K_on. Most of the identified kinetic parameter changes were in K_on or a combination of K_on and K_t. However, MUC13 and SIGLEC5 were predicted to have been regulated by only K_t and IL3RA was predicted to be regulated by K_off. Interestingly, IL3RA is the only example where all methods predict changes in K_off, which suggests that this is a promising gene to focus on in future research.

Cell culture

HPC7 cells [21] were grown in suspension in Iscove’s Modified Eagle’s Medium (IMDM, Gibco) with 10% FCS, 10% stem cell factor-conditioned medium, 1% penicillin/streptomycin (Sigma) and 1.5 × 10⁻⁴ M monothioglycerol (MTG) at 37°C and 5% CO₂. Cells were passaged every two days to maintain a concentration of 0.5 − 2 × 10⁶ cells/ml.

Retroviral transduction

For knockdown experiments, shRNA fragments were cloned into pMSCV/LTRmiR30-PIG (pLMP, Open Biosystems): Luciferase (control, 5’ CACGTACGCGGAATACTTCGAA 3’, [30]), Gata2 (5’ CGCCGCCATTACTGTGAATATT 3’, [31]). For overexpression experiments, the mouse Gfi1 cDNA was inserted into pMSCV-ires-GFP (Addgene plasmid 20672), with the empty vector used as a control. Retrovirus was produced using the pCL-Eco Retrovirus Packaging Vector (Imgenex) in 293T cells.

HPC7 cells were infected with retrovirus by centrifugation at 800 xg at 32°C for 1.5 hours with 4 μg/ml polybrene (Sigma), after which the retroviral supernatant was replaced with fresh media and cells were cultured as normal. Transduction efficiency was monitored by flow cytometry for GFP.

Single cell gene expression analysis

Single GFP+ cells were sorted by FACS into individual wells of 96 well plates and single cell RT-qPCR was carried out as described previously [14]. Cells were captured 48 hours after retroviral transduction for Gfi1 overexpression and 72 hours after transduction for Gata2 knockdown (S1 Table).

Estimation of kinetic parameters

Bursting gene expression can lead to a number of different distributions of gene expression in a population of cells (See Fig 1B), ranging from a bimodal distribution (when K_on and K_off are low) to a Poisson distribution (when K_on is much higher than K_off) to an exponential decay-like distribution (when K_off is much higher than K_on). The probability of having x mRNA molecules in a cell with kinetic parameters K_on, K_off and K_t is given by the analytical solution developed by [11]: (1)

In this equation, 1F1 represents the confluent hypergeometric function of the first kind, a summation over an infinite series that is time intensive to compute. To improve our runtimes, we precomputed an extensive lookup table of values of P(x|K_on, K_off, K_t). Let us say that we have n cells, each with an mRNA molecule count (for a particular gene) of x_i, and let X = {x₀, x₁, …x_n}. Given this list of mRNA counts from a semi-uniform population, we can assess the log likelihood of each possible set of parameters: (2)

The set of kinetic parameters that has the maximum likelihood is chosen. However, it is important to note that some areas of the parameter space are more sensitive to parameter changes than others. For instance, [11] notes that at large values of K_off the equation of P(x|K_on, K_off, K_t) approaches: (3)

This equation depends of the ratio of K_t/K_off rather than K_t and K_off separately, which means that K_off and K_t are more difficult to distinguish as K_off increases. The practical implication of this observation is illustrated in Fig 9A and 9B, which depict the log likelihoods at different kinetic parameter values for GFI1b in HSC cells, with regions that have log likelihoods close to the peak value coloured in (specifically, within 0.5 of the maximum likelihood). Although there is only a narrow range of possible K_on values (A), there is a range of values where K_off and K_t can partially compensate for one another (B). A specific example is illustrated with simulated data in Fig 9C: while the original distribution of gene expression (grey) varies visibly when K_off (blue) or K_t (red) are varied, it is possible to change both variables (purple) and almost recover the original distribution. Furthermore, this analysis suggests that it would be difficult to incorporate any additional parameters into the system and find unique estimates for each of them.

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Fig 9. Properties of two-state model and kinetic parameter estimation strategy.

Kinetic parameters control the shape of the gene expression distribution. Kinetic parameters are estimated by calculating the maximum likelihood from Eq 2. (A-B) In order to illustrate some properties of the two-state model, the likelihoods of various kinetic parameter sets were calculated for a toy example–specifically GFI1b gene expression data in hematopoietic stem cells (HSC). (A) For each possible K_on and K_off value, we calculated the maximum log likelihood across all possible K_t values. The overall log-maximum likelihood was 603.66 and only values that were within 0.5 of this value were coloured in, with the maximum value in dark red. (B) The maximum log likelihood of all possible K_on values were also calculated, as K_off and K_t were varied. There is a region in subfigure B where K_off and K_t can compensate for one another. (C) An example of parameter compensation was simulated: the original distribution (grey) has visible changes when K_off (blue) and K_t (red) are varied, but the distributions are barely distinguishable when both parameters are varied (purple). (D) Finally, we show our method’s estimates of K_on and ln(K_off/K_t) for each TF in each population of FACS cells.

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In S1 Fig, we test the performance of this strategy of kinetic parameter estimation on simulated data, including artificial technical noise. Although K_off cannot be accurately identified when K_off > 5, the ratio of K_off to K_t is accurately predicted (S2 Fig). Our method also performs better than the Gibbs Sampling based approach developed by Kim and Marioni, 2012 (S3 Fig).

Next, we applied this maximum likelihood approach for kinetic parameter estimation on hematopoietic stem cell and progenitor populations (CLP, GMP, CLP, LMPP and PreM cells) from [14]. The values for K_on and ln(K_off/K_t) for the Moignard data are shown in Fig 9F. Note that many of the TFs were particularly chosen due to their importance in early differentiation of HSCs, so one would expect a lower level of gene expression (and therefore a higher K_off/K_t ratio) in later stage progenitor populations such as CLP and GMP. There is a wide range of estimated kinetic parameter values across the TFs in each cell population; however, we need to ensure that these kinetic parameter estimates are not being skewed by mis-classified cells before we can evaluate whether these differences are statistically significant.

Parameter range and look-up table

It is crucial to select a table of appropriate range and point density for applying to the experimental data. The criteria for selecting this table were: i) fewer than 5% of the experimental points were at the maximum parameter value for the K_on and K_t parameters. ii) K_off had to be high enough in order to include the parameter range in which only the ratio of K_t to K_off matter ii) the density of points was sufficient to minimise artifacts arising from the discretization of the parameter space.

Given these constraints, the range of parameters was chosen to be 0 < K_on < 5, 0 < K_off < 20 and 0 < K_t < 200 and possible mRNA counts as 0 < x < 200. The sampling density for K_on was every 0.1, for K_off every 0.4 and for K_t every 5 for a total of 395000 possible parameter sets.

We deposited the look-up table, the code used to generate the table in Mathematica and the code for calculating the maximum likelihood in R on Github: https://github.com/ezer/SingleCellPipelineOverview.

Scaling qPCR data to mRNA counts

The main source of data analysed in this paper comes from Moignard et al. [14], and contains single cell qPCR data from five populations of hematopoietic stem cell and progenitor cells (CLP, GMP, HSC, LMPP, PreM), as determined by FACS, with approximately 124 cells in each population. The genes profiled by the qPCR include 18 TFs with crucial roles in cell fate, which were normalised to two “housekeeping” genes (PolII and Ubc), as described in Moignard et al. [14].

The outcome of a PCR experiment is a normalised C_t value, which relates to mRNA molecules (x) as follows: (4) where a and b are constants. For each TF, we chose b: (5) where x_max = 200 is the maximum number of mRNA pre-calculated in our lookup table and X_TF is the set of mRNA counts for the particular TF. This choice of b_TF stretches the values of x to have as wide a range as possible. It also removes the need to set an a parameter, since the equation for calculating x can be simplified to: (6)

It is important to note that while a different b parameter was chosen for each TF, this b factor is consistent across all five populations of cells, which is crucial for our later attempts to compare kinetic parameter values between cell populations. Even though the choice of b changes the absolute value of the kinetic parameters that are estimated, it has minimal effects on the strength of the linear correlation between the known and estimated values.

Simulated Annealing for Bursty Expression Clustering (SABEC)

SABEC begins by assigning each cell to a random population 1 to K. Note that the total number of clusters K must be set at the start of the algorithm. Next, SABEC iteratively calculates the kinetic parameters of each of the K populations, and the cells are reassigned to new populations probabilistically.

In a standard Expectation Maximisation clustering algorithm, the probability of assigning a cell to a population is proportional to the ratio of the likelihoods of the cell coming from each population. Let the kinetic parameter set for a single population be S_i(g) = (K_on(g), K_off(g), K_t(g)), where g is the gene (between 1 and G) and i labels the population (between 1 and K), and let x(g) be the vector of mRNA counts for each gene, for a particular cell. The log likelihood for a certain population can be calculated as follows: (7)

If the sum of the likelihoods for each population is L_tot, this value can be scaled as such: (8)

Since it would take a long time for this algorithm to converge if the clusters are close to one another, we add a temperature parameter, so that initially it is easy for cells to be assigned to different clusters, but it becomes harder and harder to swap clusters over time: (9) where τ is the temperature parameter and t is the iteration number of the algorithm (a counter that increments each time the cells are reassigned to new clusters). A cell is probabilistically assigned to a new cluster based on the relative values of L″(S_i|x) for each population. The algorithm terminates when fewer than 5% of the cells swap clusters in an iteration, or after 100 iterations. S4 Fig shows how the accuracy of the algorithm depends of the temperature parameter, τ.

Since this algorithm is randomized and since it is possible for certain runs of the algorithm to fall into local optima, we run this algorithm 50 times and conduct a secondary consensus clustering step. In this step, a consensus matrix is produced, in which each cell of the matrix represents the number of times that two cells are found in the same cluster. A plot of the cumulative density function of the values of this matrix can help visualise the robustness of the clustering (See Fig 8A). For comparison and to show the necessity for SABEC, consensus clustering of K-means was also conducted (See Fig 8B).

Fig 8 shows the cumulative density functions for the number of times two cells cluster together for SABEC (A) and K-means (B). [32] determined that the most robust metric for comparing the consistency of consensus clustering methods is the Proportion Ambiguously Clustered (PAC) score, which is defined as the proportion of cell pairs that cluster together in 10% and 90% of the repeated runs of the algorithm. This corresponds to the proportion of cell pairs that lie between the vertical lines in the cumulative distribution functions in Fig 8(A) and 8(B). The PAC scores for SABEC and K-means are compared in C, illustrating that SABEC usually has more consistent outcomes than K-means, with the fewest ambiguously clustered cells at K = 7.

In addition, to estimate the accuracy of our method, we can assume that the expected cluster assignment (as determined by FACS) is our gold standard. By comparing the results of our clustering approach with the gold standard, we can estimate the accuracy of our method on the experimental data. To do this, we cluster each of our consensus matrices into five clusters using partitioning around medoids (PAM), a clustering approach similar to K-means (but more consistent since it uses data points as centres). We can then compare our results to the gold standard labels using metrics such as VI (See Fig 8D) and the corrected Rand index (See Fig 8E). SABEC performs better than K-means by these two metrics, with the estimated number of clusters equal to 6. The one exception is that the corrected Rand index suggests that K-means performs slightly better than SABEC when K = 5; however, SABEC provides substantially better outcomes at K = 6. Based on simulated datasets with values similar to the experimental data, we determined that these latter two methods provide more accurate estimates of the number of clusters than the PAC method, which can sometimes overestimate the appropriate number of clusters (S8 Fig).

Figs 8F and 2G compare the consensus matrices for K = 6 for SABEC (F) and K-means (G), with the cells sorted by their FACS-determined labels. SABEC results appear more consistent, with cells frequently clustering with other cells of the same type. In addition, any cells that are “misclassified” tend to cluster instead with cell populations of their parent or children populations. For instance, some HSC cells cluster with their child populations (PreM and LMPP), and some GMP and CLP cells cluster with their parent population (LMPP).

The R script for SABEC is available in Github: https://github.com/ezer/SingleCellPipelineOverview, including a sample input file to run in parallel on a Condor cluster, and appropriately merge the outputs.

Parameters chosen for K-means and hierarchical clustering

The SABEC method was compared to hierarchical clustering and K-means approaches. The hierarchical clustering approach used was the default one associated with the heatmap function in R (Euclidean distance metric and complete clustering). The K-means approach used the default algorithm in R (Hartigan and Wong), but the maximum iterations was increased to 100 in order to be more comparable to the SABEC approach. K-means was repeated 50 times and an additional consensus clustering step was taken, in order to provide a fairer comparison to SABEC. Note that the input to both the hierarchical clustering and K-means algorithms were the normalised C_t values, while the input to SABEC is the scaled mRNA counts. These algorithm and distance metrics were chosen since they are the most commonly used. Other variations of hierarchical clustering and K-means were tested, but none of the results were significantly better or different than the ones shown.

Estimation of Pairwise changes in Kinetics (EPiK)

The R script for determining which kinetic parameters vary across populations is available in Github: https://github.com/ezer/SingleCellPipelineOverview.

Three methods were tested on simulated datasets of genes with randomly selected kinetic parameters. We conducted these simulation tests for two different ranges of the kinetic parameters to illustrate that there is different sensitivity to parameter changes in different regions of the parameter space (S12, S13 and S14 Figs). These results suggest that changes in K_off cannot be accurately detected when K_off > 5. In addition, the methods often performs better when fewer kinetic parameters change at once (S12 and S17 Figs). The correctly identified simulated cases were those that have the largest magnitude of kinetic parameter change (S18 Fig) and had average kinetic parameter values closer to the origin, where the kinetic parameters are more accurately estimated (S19 Fig).

Two of the methods (MP and Subset methods), have continuous-valued outputs, and so a threshold must be set for determining whether or not a change in a kinetic parameter is likely significant. The thresholds were set to have approximately a 2% false positive rate, based on the in silico validation tests. The threshold values for the MP method when K_off < 5 are −6.3, −8.5 and −6.8 for K_on, K_off and K_t, and −4.9 and −6.0 for K_on and K_t when K_off > 5. For the subset method, the thresholds are 0.77, 0.91 and 0.86 (K_on, K_off and K_t, respectively) when K_off < 5, and 0.681 and 0.870 (K_on and K_t) when K_off > 5.