- Method
- Open Access

# SCRABBLE: single-cell RNA-seq imputation constrained by bulk RNA-seq data

- Tao Peng
^{1}, - Qin Zhu
^{2}, - Penghang Yin
^{3}and - Kai Tan
^{1, 2, 4, 5, 6, 7}Email authorView ORCID ID profile

**Received:**5 January 2019**Accepted:**27 March 2019**Published:**6 May 2019

## Abstract

Single-cell RNA-seq data contain a large proportion of zeros for expressed genes. Such dropout events present a fundamental challenge for various types of data analyses. Here, we describe the SCRABBLE algorithm to address this problem. SCRABBLE leverages bulk data as a constraint and reduces unwanted bias towards expressed genes during imputation. Using both simulation and several types of experimental data, we demonstrate that SCRABBLE outperforms the existing methods in recovering dropout events, capturing true distribution of gene expression across cells, and preserving gene-gene relationship and cell-cell relationship in the data.

## Keywords

- Single-cell RNA-seq
- Imputation
- Matrix regularization
- Optimization

## Background

Single-cell RNA sequencing (scRNA-seq) has revolutionized cell biology, enabling studies of heterogeneity and transcriptome dynamics of complex tissues at single-cell resolution. However, a major limitation of scRNA-seq data is the low capturing and sequencing efficiency affecting each cell, resulting in a large proportion of expressed genes with zeros or low read counts, which is known as the “dropout” phenomenon. Such dropout events lead to bias in downstream analysis, such as clustering, classification, differential expression analysis, and pseudo-time analysis. To address this critical challenge, two types of approaches have been developed. One approach adopts analysis strategies that take dropout into consideration. For instance, ZINB-WaVE generates weights for genes and cells using a zero-inflated negative binomial model which in turn is used to detect differential expression [1]. Lun et al. used a pool-and-deconvolute approach to deal with dropout events for accurate normalization of scRNA-seq data [2]. The second approach is direct imputation of scRNA-seq data. Among these methods, MAGIC imputes dropout events by data diffusion based on a Markov transition matrix that defines a kernel distance measure among cells [3]. scImpute [4] first computes dropout probability using a two-component mixture model. It then uses a LASSO model to impute dropout values. Similarly, SAVER [5] also uses a linear regression to impute the missing data. But, it differs from the scImpute by using a Bayesian model to compute the probability of dropout events. DrImpute [6] first conducts consensus clustering of cells followed by imputation by the average value of similar cells. VIPER uses a non-negative sparse regression model to progressively infer local neighborhood cells for imputation [7].

All imputation methods above recover dropout values using scRNA-seq only. Here, we describe the SCRABBLE algorithm for imputing scRNA-seq data by using bulk RNA-seq as a constraint. SCRABBLE only requires consistent cell population between single-cell and bulk data. The bulk data represent the unfractionated composite mixture of all cell types without sorting them into individual types. For many scRNA-seq data, there are usually existing bulk data on the same cell/tissue. And it is becoming increasingly common to collect matched bulk data when a new scRNA-seq experiment is performed. Bulk RNA-seq data allows SCRABBLE to achieve a more accurate estimate of the gene expression distributions across cells than using single-cell data alone. SCRABBLE is based on the framework of matrix regularization that does not impose an assumption of specific statistical distributions for gene expression levels and dropout probabilities. It also does not force the imputation of genes that are not affected by dropout events.

## Results

*λ*) and a Bernoulli process to introduce dropout events at the corresponding dropout rate [4, 11] (see the “Methods” section). Using the 2 strategies, we simulated data with dropout rates corresponding to 60 to 87% zeros in the data. Moreover, to evaluate the robustness of imputation methods, at a given dropout rate, we simulated 100 data sets. It is well known that real RNA-seq data tend to have a characteristic property of inverse relationship between mean and variance [12]. We confirmed that our simulated data also contains this property using the mean-variance plot (Additional file 1: Figures S1 and S3).

To evaluate the performance of each method, we define the imputation error as the *L*_{2} norm of the difference between the imputed and the true data matrices. Using both types of simulated data across a range of dropout rates, we found that SCRABBLE outperforms four state-of-the-art methods (DrImpute, scImpute, MAGIC, and VIPER) (Figs. 2d–f and 3e–h). More importantly, the performance gain is observed across the full spectrum of gene expression levels (Figs. 2c and 3d, Additional file 1: Figures S2, S4-S6). All other methods led to imputed values that were significantly lower than the true values for > 88% (Fig. 2c) and > 40% (Fig. 3d) of the genes. In contrast, SCRABBLE led to imputed values that were significantly higher than the true values for 1% (Fig. 2c) and 2% (Fig. 3d) of the genes. The imputed data by SCRABBLE also captures the data substructure (i.e., clusters) better as embedded in the true data (Figs. 2b and 3c, Additional file 1: Figures S2, S4-S6).

*Tmem208*and

*Naa25*(the rest of the genes are shown in Additional file 2: Figure S7). We observed the same performance gain by SCRABBLE in another set of 17 genes with dropout events in at least 39% of the cells (i.e., higher dropout rate, Additional file 2: Figure S9).

We further assess the performance of SCRABBLE using single-molecule RNA fluorescence in situ hybridization (smRNA FISH) data and scRNA-seq data measured on the same cell type, mouse embryonic stem cell line, E14 [17, 18]. We compared the distributions of the imputed expression and smRNA FISH measurements for the same set of 12 genes across single cells. Overall, the distributions of expression values imputed by SCRABBLE have the highest agreement with the smRNA FISH data (Fig. 4d), suggesting best performance by SCRABBLE. Figure 4c shows raw and imputed expression levels of two representative genes, *Esrrb* and *Tbp* (the rest of the genes are shown in Additional file 2: Figure S10).

For the gene-gene relationship, among the entire set of 5000 genes, data imputed by SCRABBLE results in the highest agreement with the gene-gene correlation pattern based on the true data (Fig. 5c, d). This higher agreement of gene-gene correlations is observed using the data simulated with both strategies and across a range of dropout rates (Additional file 2: Figures S14 and S15). Histograms of the correlation values are shown in Additional file 2: Figure S16.

The imputation procedure could inadvertently distort the clustering result. To evaluate this issue, we next computed the cell-cell and gene-gene correlations using cells/genes stratified based on their cluster membership (for cell-cell correlation) and on whether they are marker genes of a cluster (for gene-gene correlation). For cell-cell correlation, we computed the within- and between-cluster correlations across cells. For gene-gene correlation, we computed the correlations among marker genes and among marker and non-marker genes for a given cluster. For both cell-cell and gene-gene correlations, the distance between the two correlation distributions was quantified using the Kolmogorov-Smirnov (KS) statistic. Finally, the distortion of the clustering result is measured by comparing the KS statistic based on true data and imputed data. For both cell-cell (Additional file 2: Figures S17 and S18) and gene-gene (Additional file 2: Figures S19 and S20) correlations, SCRABBLE gives the smallest distortion compared to the other methods. The same performance gain is observed using the data simulated with strategy 1 (Additional file 2: Figures S21 and S22).

*K*-nearest neighbor clustering (Fig. 7a, Additional file 2: Figures S26-S32). This result further demonstrates that SCRABBLE can capture cell heterogeneity in complex tissues although it uses average gene expression values of the bulk data. To evaluate the clustering quality using either raw or imputed data, we used the Dunn index which computes the ratio of minimal inter-cluster distance versus maximal intra-cluster distance. A higher Dunn index indicates a better separation among clusters. We found that the use of imputed data by SCRABBLE results in improved clustering quality as compared to clustering without imputation and with imputed data by the other four methods (Fig. 7b, Additional file 2: Figures S26-S32).

SCRABBLE has three parameters (i.e., *α*, *β*, and *γ*). To evaluate the robustness of SCRABBLE over parameter setting, we varied the values of the three parameters by 0.1-, 0.5-, 2-, and 10-folds and performed imputation using data simulated using strategy 1 with the dropout rate of 83%. We found that the median percentage change in imputed data before and after changing the parameter is less than 5% for both *α* and *β* and less than 15% for *γ* (Additional file 2: Figure S33), suggesting SCRABBLE is very robust with regard to parameter setting. The sets of SCRABBLE parameters used in this study are provided in Additional file 3: Table S2. We also benchmarked the running time of SCRABBLE. The higher imputation accuracy of SCRABBLE comes with a price of slower running time. For dataset containing fewer than 2000 cells, SCRABBLE has a better or comparable speed as that of VIPER (Additional file 2: Figure S34). As the dataset size exceeds 5000 cells, SCRABBLE is twice as slow as VIPER, mostly due to the computationally expensive process of iterative single value decomposition.

## Discussion

SCRABBLE addresses several deficiencies of existing methods. First, several methods impute dropout events by using cell-cell distance, as quantified by either Euclidean distance or kernel distance. Such distance measures may not reflect the true relationship among cells. SCRABBLE relies on the framework of matrix regularization which does not use cell-cell distance measure. Second, SCRABBLE borrows information from bulk RNA-seq data to impute dropout data in order to reduce unwanted bias during imputation. Finally, since we transform the mathematical model of SCRABBLE to a convex optimization problem, the existence of the optimal solution is guaranteed mathematically. Our comprehensive analysis using both simulated and real experimental data suggests that SCRABBLE achieves significant improvement in terms of recovering dropout events and preserving cell-cell and gene-gene relationships in the samples. As an example of SCRABBLE’s utility to facilitate downstream analysis, we show that using SCRABBLE-imputed data leads to a better clustering quality and helps identify different cell types in complex tissues.

One caveat about our method is the use of average values of bulk RNA-seq data. It may reduce the ability of the method to capture biological heterogeneity in the data. However, we believe the advantage of using bulk data outweighs the disadvantage. Additionally, the other two terms of our model, projection and low rank, enable SCRABBLE to detect heterogeneity and covariation.

As other types of single-cell omics data become more abundant, such as single-cell DNA methylation and ATAC data, our method provides a general framework for imputing and integrating these data for new discoveries.

## Conclusions

Here, we describe the SCRABBLE algorithm and software package. SCRABBLE imputes single-cell RNA-seq data by using bulk RNA-seq data both as a constraint and as prior information. We show leveraging information in bulk RNA-seq data significantly improves the quality of imputed data. With SCRABBLE, existing or newly generated bulk RNA-seq data can be used to increase the utility of single-cell RNA-seq data.

## Methods

### The mathematical model of SCRABBLE

The input to SCRABBLE includes the scRNA-seq and bulk RNA-seq data on consistent cells/tissues. A matrix, *X*_{0}, represents expression values from scRNA-seq data with columns representing *m* genes and rows representing *n* cells. A vector, *D*, represents the average expression levels of all genes in the bulk RNA-seq data across *N* samples.

*X*

_{0}. The algorithm is based on the following mathematical model:

*P*

_{Ω}(·) is the projection operator that forces

*x*

_{ij}to be zeros (

*x*

_{ij}is the element at the

*i*th row and the

*j*th column of the matrix

*X*and (

*i*,

*j*) ∉ Ω); otherwise, the value of

*x*

_{ij}is kept as it is.

*Ω*is determined by

*X*

_{0}and (

*i*,

*j*) ∈ Ω if \( {x}_{ij}^0\ne 0 \), where \( {x}_{ij}^0 \) is the element at the

*i*th row and the

*j*th column of the matrix

*X*

_{0}. Rank(

*X*) is the rank of the matrix

*X*.

*a*is a row vector with the size 1 by

*n*and each element in

*a*is \( \frac{1}{n} \).

*α*and

*β*are the parameters of the mathematical model.

*α*is the weight for the rank of the imputed data matrix. Large

*α*results in reduced heterogeneity across the cells.

*β*is the weight for the agreement between the aggregated scRNA-seq and bulk RNA-seq data.

*β*is proportional to

*α*and the size of the imputed data matrix.

### Iterative optimization of the objective function during imputation

_{∗}is the nuclear norm, which is the convex envelope of the rank function. We use the following three steps to calculate \( \overset{\wedge }{X} \).

*Y.*

such that *X* − *Y* = 0.

where *χ*_{X ≥ 0} is the characteristic function which takes the value of 0 if *X* ≥ 0 and ∞ otherwise.

*X*to obtain the iteration scheme of Eq. (5).

*βa*

^{T}

*a*+

*γI*=

*W*and

*βa*

^{T}

*D*+

*X*

_{0}=

*T*

*Z*and

*τ*> 0 as follows:

Here, *Z* = *U* diag({*σ*_{i}}_{1 ≤ i ≤ r})*V*^{T} is the singular value decomposition of *Z*, and *σ*_{i}s are the positive singular values. Λ^{k}, *X*^{k}, and *Y*^{k} are the *k*th iteration matrix of Λ, *X*, and Y, respectively. In addition, *x*_{ij}, \( {y}_{ij}^k \), \( {\Lambda}_{ij}^k \), *w*_{ij}, and *t*_{ij} are the elements at the *i*th row and *j*th column in the matrices *X*, *Y*^{k}, Λ^{k}, *W*, and *T*, respectively. The convergence of ADMM for convex optimization problems has been extensively studied in the literature [25, 27]. Since the objective function in (2) is convex and non-negative, the problem has at least one global solution. This global structure of the objective function in Eq. (2) allows the above algorithm to converge more quickly compared to other evolutionary algorithms [28]. The penalty parameter *γ* plays an important role in solving the objective function in Eq. (9) using the singular value thresholding algorithm combined with the parameter *α*. Overall, *α*, *β*, and *γ* are the three necessary parameters of SCRABBLE.

### Generation of simulated data

We simulated the scRNA-seq data consisting of three cell types using the Bioconductor package Splatter (version 1.4.3) [9]. We used the *splatSimulateGroup* function to generate the simulation data with 1000 cells and 800 genes. Three clusters were embedded in each simulated dataset. The size of each cluster was controlled by the parameter “group.prob” to be 0.2, 0.35, and 0.45. The parameter controlling the probability that a gene is differentially expressed in each group was set equal to 0.045. The location parameter and the scale factor parameter of randomly generating multiplication factors from a log-normal distribution were set to be 0.1 and 0.4, respectively. Dropout midpoints (parameter “dropout_mid” in Splatter) were used to control the dropout rates in the simulated data. For instance, dropout midpoints of 4, 5, and 5.5 correspond to 71%, 83%, and 87% dropout rates in the simulated data, respectively. The corresponding bulk RNA-seq data were the mean values of genes in the true scRNA-seq data. The dropout RNA-seq and bulk RNA-seq data matrices are the inputs of the imputation methods. To determine the performance stability of the methods, we generated 100 datasets for each dropout midpoints.

### Generation of simulated data using bulk RNA-seq data

We used the bulk RNA-seq dataset of mouse hair follicles from [10]. In total, the dataset contains 20 different combinations of anatomic sites and developmental time points. We used the following procedures to generate the simulated datasets (Fig. 3a): (1) we randomly selected 8 out of the 20 conditions; (2) for each condition, we generated 100 resampled datasets. The means and standard deviations of genes were calculated for each condition based on the 100 resampled datasets; (3) 100 new datasets were generated based on the mean and the standard deviation of each gene; (4) in order to reduce the computation cost, we randomly selected 5000 genes from 20,721 genes in the above data matrices. The final data matrix represents 800 cells and 5000 genes; and (5) we made the dropout rate of each gene in each cell following an exponential function \( {e}^{-\lambda \bullet \mathrm{mean}\_{\mathrm{expression}}^2} \) [4, 11], where *λ* determines the dropout rate of scRNA-seq data. Zero values are introduced into the simulated data for each gene in each cell based on the Bernoulli distribution defined by the corresponding dropout rate. The corresponding bulk RNA-seq data are the mean values of genes in the scRNA-seq data without dropouts. To determine the performance stability of the methods, we generated 100 datasets for each dropout rate.

### Running of other imputation methods

We benchmarked DrImpute, scImpute MAGIC, and VIPER packages in this manuscript. For DrImpute (version 1.0), we used the following default parameter settings described in the Quick Start section of the user manual: ks = 10:15, dists = c(“spearman,” “pearson”), fast = FALSE, dropout.probability.threshold = 0, n.dropout = 10,000, n.background = 10,000, and mc.cores = 1. For scImpute (version 0.0.9), we used the following default parameter setting described in the Quick Start section of the user manual: labeled = FALSE, drop_thre = 0.5, and Kcluster = 1 in all analysis. For MAGIC (version 1.3.0 implemented in Python), we used the following default parameter setting, *k* = 10, *a* = 15, *t* = “auto”, n_pca = 100, knn_dist = “euclidean”, n_jobs = 1, and random_state = none. For VIPER (version 0.1.1), we used the following parameter setting: num = 5000, percentage.cutoff = 0.1, minbool = FALSE, and alpha = 1.

## Declarations

### Acknowledgements

We thank the Research Information Services at the Children’s Hospital of Philadelphia for providing computing support.

### Funding

This work was supported by the National Institutes of Health of the USA grants GM104369, GM108716, HG006130, HD089245, and CA233285 (to KT).

### Availability of data and materials

SCRABBLE is implemented using both R and MATLAB languages. The software packages are freely available under the MIT license. Source code has been deposited at the GitHub repository (https://github.com/tanlabcode/SCRABBLE) [29] and Zenodo with the access code DOI: https://doi.org/10.5281/zenodo.2585902 [30].

The datasets analyzed in this study are included in this published article and Additional file 3: Table S1. The analysis code used to analyze the datasets is available from the GitHub repository (https://github.com/tanlabcode/SCRABBLE_PAPER) [31] and Zenodo with the access code DOI: https://doi.org/10.5281/zenodo.2585885 [32].

### Authors’ contributions

TP and KT conceived and designed the study. TP designed and implemented the SCRABBLE algorithm with the help of QZ and PY. QZ and PY provided the additional analytical tools. TP and KT performed the data analysis. KT supervised the overall study. TP and KT wrote the paper. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

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## Authors’ Affiliations

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