scTsI: an effective two-stage imputation method for single-cell RNA-seq data

Abstract

Single-cell RNA-seq facilitates the understanding of cell types and states and the revealing of the cellular heterogeneity in developmental processes and disease mechanisms. However, the dropout events in single-cell RNA-seq data, in which genes are not detected due to technical noise or limited sequencing depth, seriously affect downstream analyses. Imputation is an effective way to relieve the impact of dropout events. However, the current methods may introduce new noise or modify the high expression values in the imputation process and their performance may be lower than expected when dealing with data with a high dropout rate, facing with different types of data, and aiming at various downstream analyses. We propose a two-stage imputation algorithm, scTsI, for single-cell RNA-seq data. In the first stage, scTsI imputes the zero values using the information of neighboring cells and genes. In the second stage, scTsI transforms the expression matrix into a vector, performs row transformation, and adjusts the imputed values through ridge regression and leveraging bulk RNA-seq data as a constraint. scTsI ensures that the original highly expressed values are unchanged, avoids introducing new noise, and allows sparse matrix input to accelerate imputation. We conduct experiments on a variety of simulated and real data with different dropout rates and compare scTsI with the commonly used imputation methods. The results show that scTsI can restore gene expression and maintain cell–cell similarity across different data dimensions and dropout rates. scTsI can also improve the performance of data visualization, clustering, and cell trajectory inference.

Introduction

Cellular transcriptome analysis plays an important role in characterizing the state of cells and describing the specificity of physiological conditions or developmental stages [1, 2]. Traditional bulk RNA sequencing (bulk RNA-seq) technology can detect average gene expression level of a cell group. However, there are differences in the expression levels of genes between different cells and bulk RNA-seq cannot quantify the heterogeneity [3–5]. With the breakthrough development of single-cell RNA sequencing (scRNA-seq), researchers are able to capture the heterogeneity of gene expression and study cell–cell variability at the cellular level, paving the way for RNA-level measurements at single-cell resolution [6, 7]. Nevertheless, the dropout events in scRNA-seq lead to a high proportion of zeros in scRNA-seq data, caused by biological reasons or the loss of information about the expressed genes during the process of sequencing [8, 9]. Dropout events bring major challenges to the subsequent analysis of scRNA-seq data, such as data visualization, clustering, and trajectory inference, especially as the data dimension and dropout rate increase [3, 10]. Therefore, a solution to address the dropout events is imperative for preprocessing scRNA-seq data, especially for those corrupted by a high dropout rate or noise. Developing statistical or computational methods for dropout events contributes to improving the power of scRNA-seq data in deciphering pathological and biological processes.

Imputation is a common approach to relieve the influence of dropout events [3]. Many scRNA-seq data imputation methods have been proposed in recent studies. These imputation methods can be categorized into three broad approaches: smoothing-based, model-based, and reconstruction-based imputation methods [11]. The smoothing-based methods impute data by averaging the gene expression information of similar cells. For example, DrImpute [12] performs clustering multiple times and imputes according to the averaged gene expression of similar cells. MAGIC [13] imputes by data from a Markov matrix that defines a measure of intercellular distance. SAVER [14] uses the Bayesian approach to estimate the true level of gene expression based on information between genes. Instead of only using information of similar cells or genes, scTSSR [15] uses a two-sided sparse self-representation to leverage the information of both genes and cells. The model-based methods employ probabilistic models for imputation. For instance, scImpute [16] uses a mixture model of two components to calculate the missing probability and uses the LASSO model to impute data. VIPER [17] applies a sparse non-negative regression model to populate the cells that need to be imputed. The reconstruction-based methods identify the potential representation of cells by capturing linear or nonlinear relationships and reconstruct the expression matrix from the potential space. The methods capturing linear relations are low-rank matrix-based methods, while the methods capturing nonlinear relationships are mostly deep learning–based methods. Among the low-rank matrix-based methods, PBLR [18] groups cells into subpopulations and achieves low-rank matrix recovery in each subpopulation. ALRA [19] reconstructs data by setting thresholds to achieve singular vector decomposition and obtaining a low-rank approximation of gene expression matrix. SCRABBLE [20] uses the alternating direction method of multipliers [21] algorithm to optimize the objective function and obtain the reconstructed matrix. It leverages the external bulk RNA-seq data information. scWMC [22] adopts the imperfect prior information of genes and cells to estimate the corresponding true information and then incorporates it into a low-rank matrix completion–based framework. Among the deep learning–based methods, deep count autoencoder network (DCA) [23] applies a deep count autoencoder network for capturing nonlinear correlations between genes to impute missing data. DeepImpute [24] trains deep neural networks with a dropout layer and loss function to perform imputation. Bis [25] uses an optimal transport-based autoencoder with cell embedding regularization and leverages bulk RNA-seq data for recovering scRNA-seq data. scMGCA [26] employs a graph convolutional autoencoder with a multinomial-based decoder to characterize the multimodal distribution, which simulates the high probability dropout events of single-cell data. scGCOT [27] uses attention-based graph autoencoders to learn cell correlations and gene correlations, a ZINB decoder to reconstruct the expression matrix, and optimal transport theory for joint representation alignment to enhance cell type identification.

Although existing imputation methods can reduce the influence of dropout events to a certain extent, their performance may be lower than expected [28]. Some methods perform well on simulated data but not on real data. For example, SCRABBLE and DeepImpute show a good imputation effect on simulated data but poor performance on real data [28]. Some methods perform well on data with a low dropout rate but fail to recover missing values effectively in a high dropout rate. For example, the effects of scImpute, SAVER, and DCA are significantly reduced in the case of high dropout rate [29]. Many current methods will change the originally expressed values and introduce new noise during the imputation process. For example, MAGIC changes the high expression values of genes, leading to the introduction of new biases to the data and the possible elimination of meaningful biological variations [30]. knn-smooth adjusts all expression values, generating many false positives [31]. scImpute imputes some lower and moderate expression values at some probability thresholds [31]. Many reconstruction-based methods would also change the original high expression unless they perform postadjustments. Additionally, deep learning methods are “black box” models due to their end-to-end training and lack of interpretability and transparency. Compared with only borrowing information from similar cells, leveraging information from similar genes and external bulk RNA-seq data would improve the reconstruction performance.

To this end, we propose a two-stage imputation algorithm scTsI for scRNA-seq data. scTsI first imputes the missing data by the information of neighboring cells and genes through K-nearest neighbors (KNN) [32]. Next, scTsI uses bulk RNA-seq data as a constraint to adjust the imputed values in the first stage by ridge regression. scTsI distinguishes the expressed values and zero values to ensure that the highly expressed levels remain unchanged and reduce the introduction of noise. scTsI is straightforward to increase interpretability and allows sparse input to accelerate imputation. We demonstrate the performance of scTsI using various simulated and real scRNA-seq datasets and compare it with multiple existing methods. The results show that scTsI performs well across different types of data and varying degrees of dropout rates. scTsI enhances the recovery of missing values, maintains cell–cell similarity, and improves the performance of visualization, clustering, and the construction of cell trajectories.

Materials and methods

scTsI algorithm

We propose a two-stage scTsI algorithm for single-cell gene expression imputation. In the first stage, scTsI finds neighboring cells and genes for initial imputation through KNN, and, in the second stage, scTsI leverages bulk RNA-seq data as a constraint and uses ridge regression for adjusting the initial imputed values. During the imputation process, scTsI only imputes the zero values and keeps the high expression values unchanged. For this, scTsI transforms the expression matrix into a vector, performs row transformation, and conducts imputation for zero values. Nonzero values are excluded from the subsequent updates to preserve their original information.

The flow chart of scTsI algorithm is shown in Fig. 1. Denote the original input data matrix as , with rows representing genes and columns representing cells. In the first stage, the zero value in at the position of row i and column j, is imputed as by:

Figure 1.

Flow chart of scTsI. In the first stage, scTsI uses neighboring cells and genes for initial imputation through KNN. In the second stage, scTsI leverages bulk RNA-seq data as a constraint and uses ridge regression for adjusting the initial imputed values.

(1)

where and are the expression of gene i in the u-th cell among the k₁ nearest neighbor cells of cell j and the expression in the cell j of the u-th gene among the k₂ nearest neighbor genes of gene i, respectively. scTsI performs the initial imputation by calculating the average value of the nearest neighbor genes and cells.

In the second stage, the scTsI algorithm leverages bulk RNA-seq data to adjust the initial imputed values by:

(2)

where the matrix represents the data after the first stage; is the data after the second stage, i.e. the final imputed matrix; column vector d of length m represents the averaged gene expression of each gene calculated from the bulk RNA-seq data; a is a column vector of length n with each value being 1/n, which is used to calculate the average of each row of ; λ is a regularization parameter. The first term represents the difference between the gene expression matrices before and after the second stage, and the second term represents the difference between the gene expression averages calculated from the bulk RNA-seq data and the imputed scRNA-seq data.

In order to maintain high expression values unchanged, we further transformed the objective function in the second stage. Let be a column vector of length , which represents the expression values of each column of followed by the next column, i.e. the flattened . We used a matrix denoted as representing a row transformation on to arrange the zero values of the column vector in the front and the nonzero value part in the back, obtaining a vector . Let the number of zero values be and the number of nonzero values be . It is noted that scTsI only imputes the zero-value part of . Let be a column vector of length , which represents the imputation of the zero-value part by KNN in the first stage. Let be a column vector of length , which represents the adjustment made by the second stage to the result of first stage, and then correspond exactly to the imputed values for the zero-value part of . Therefore,

(3)

represents the final imputed column vector. represents the zero vector of length .

Let represents a row vector of length n with each value being 1/n, and is the identity matrix of dimensions m by m. The Kronecker product of and yields a matrix with dimensions m by mn. The product of this matrix and vector y represents the average value of genes calculated from the final imputed matrix , which plays a similar role to the vector a in the original objective function (2). Therefore, the objective function becomes:

(4)

The first item represents the difference before and after imputing in the second stage, and the second item represents the difference between the average value of genes calculated from the imputed values and the bulk RNA-seq data.

Let the Kronecker product of and be matrix B, then the function (4) is:

(5)

Let the product of B and be matrix D, which is a matrix of dimension m by mn. We divided the D matrix into matrix with dimension m by and the matrix with dimension m by . Then the function (5) becomes:

(6)

The simplified formula is as follows:

(7)

Let vector c be , and the function (7) becomes:

(8)

Now we can find that the function (8) has the form of ridge regression. The regularization parameter λ controls the relative degree of importance between bulk RNA-seq data and the initial imputed values obtained in the first stage. We applied the R package of glmnet [33] to optimize the objective function and return a list of suggested λ values. We chose the middle value among the list of λ to balance the two sources of information, allowing for meaningful refinement in the second stage while retaining the essential features captured in the first stage. Besides, the middle value almost would not be changed by the setting of “nlambda” (the number of returned λ values) in the glmnet and often provides a relatively better performance in our experiments. The detailed usage of glmnet package can be seen in the Supplementary Materials. After obtaining by solving (8), we added to as the imputed values for the zero part of . Then, we calculated (3) to obtain the imputed column vector y. Last, vector y is transformed into a matrix with dimension m by n as the final imputation result .

Analyzed datasets

In the analysis of simulation data, we generated simulation datasets with different dimensions and different dropout rates through the Splatter [34] package (Supplementary Materials), and the numbers of genes and cells in these datasets are 800 × 1000, 1000 × 3000, 3000 × 1000, 1000 × 5000, and 5000 × 1000, respectively. For each dimension, we generated data sets with dropout rates of 20%, 30%, 40%, 50%, 60%, and 70%. It is noted that the dropout rate here denotes the rate of values changing to zeros. After obtaining the true and dropout data matrices using Splatter, both in the form of counts per million, we calculated the average of each gene in the true gene expression data to represent the corresponding bulk RNA-seq data. The dropout data and bulk data (if needed) are the inputs for imputation.

In the analysis of experimental data, we obtained preprocessed and normalized single-cell gene expression data and corresponding bulk RNA-seq data from a benchmark study [11] to test the influence of imputation methods on downstream analysis, including data visualization, cell clustering, and trajectory inference. For data visualization, we used the scRNA-seq data of CellBench [35], including the sc_10x dataset derived using the 10x Chromium protocol, the sc_celseq2 dataset derived using CEL-seq2, and the sc_dropseq dataset derived using the Drop-seq Dolomite protocol [36]. For these three datasets, consisting of three human lung adenocarcinoma cell lines (HCC827, H1975, and H2228), we used bulk RNA-seq samples from GSE86337 [37] to calculate the average expression levels of genes. For clustering analysis, in addition to the three datasets above, we also used two simulated datasets generated based on the real bulk RNA-seq dataset of mouse hair follicles [20, 38], including the Kidney dataset with 29 cell types and the Spleen dataset with 11 cell types. For trajectory inference, we analyzed five RNA mixture datasets from GSE118767 [35] derived using the CEL-seq2 protocol, including cellmix1, cellmix2, cellmix3, cellmix4, and RNAmix_celseq2 datasets [35]. These datasets include three human lung adenocarcinoma cell lines, HCC827, H1975, and H2228. The corresponding bulk RNA-seq data were calculated from GSE86337 [37]. These scRNA-seq data involve different species and tissues, multiple sequencing protocols, and different experimental designs with gene numbers ranging from thousands to tens of thousands and sample sizes ranging from hundreds to thousands (Supplementary Materials).

Benchmarking

We compared scTsI with 11 common imputation methods. Among them, DrImpute [12] and MAGIC [13] are smooth-based methods; SAVER [14], scImpute [16], and VIPER [10] are model-based methods; SCRABBLE [20] and CMF [39] are low-rank matrix-based methods; and DCA [23], DeepImpute [24], Bubble [40], and SAUCIE [41] are deep learning–based methods. Among these methods, Bubble and SCRABBLE are imputation methods leveraging bulk RNA-seq data. We used the default parameters of these methods to impute in general, but when these methods did not perform well on some data, we adjusted the parameters. For the imputation methods that accept normalized data or contain a normalization procedure, we provided the preprocessed data for imputation methods in a unified way and skipped the gene and cell filtering and normalization steps within each method’s pipeline to ensure that the output imputed matrices have the same dimensions. For the methods that only accept the count data matrix, we met the input requirement, providing the corresponding count data, and skipped the gene and cell filtering steps within the method’s pipeline to ensure that the output imputed matrices have the same dimensions.

Results

scTsI recovers gene expression levels

To evaluate the ability of imputation methods in recovering gene expression levels, we tested on simulated datasets and calculated the Pearson correlation coefficient (PCC) and root mean square error (RMSE) between the flattened imputed data and the flattened real data. Figure 2A shows the PCC of all methods. From the bar chart, it can be seen that most methods can recover gene expression levels in data of different dimensions under the case of a low dropout rate. In a high dropout rate, such as the data of 3000 × 1000 with 60% of dropout rate, whose proportion of zero values is 86%, the PCC of scTsI, VIPER, and SCRABBLE is still very close to one. This result shows that these methods still have a good recovery level in the case of a high dropout rate. The results of CMF, SAUCIE, Bubble, and SAVER at a high dropout rate are not ideal. scTsI has a prominent ability to recover gene expression levels across data with different dimensions and different dropout rates.

Figure 2.

(A) PCC and (B) RMSE between the flattened imputed gene expression data by every method and the flattened real data for all simulated data and all dropout rates.

At the same time, we also calculated the RMSE metric between the gene expression data after imputation and the real data to evaluate the ability of restoring gene expression levels (Fig. 2B. DrImpute, DeepImpute, and scImpute have good imputation results at low dropout rates but are poor at high dropout rates, which indicates that these methods are affected by dropout rates. MAGIC, Bubble, and SAVER do not perform well. scTsI and SCRABBLE show gene expression levels close to the real data at different dropout rates in each data set.

Leveraging information from neighboring cells and genes and external bulk RNA-seq data facilitates the reconstruction of single-cell gene expression accurately. A good imputation method should recover the missing biological signals in scRNA-seq data to the greatest extent [40], and, at the same time, it is also necessary to minimize the errors caused by the imputation process, especially the changes of the original expressed values. scTsI reduces this error by dividing the input data into a nonzero part and a zero part and only imputes the zeros to keep the original expressed values unchanged.

scTsI maintains cell–cell similarity

To assess whether imputation methods could keep cell–cell similarity, we tested on simulated datasets and calculated the PCC between the flattened cell–cell similarity matrices obtained from the true and the imputed single-cell gene expression data. Figure 3A shows that scTsI has the best cell–cell correlation and is not affected by the dropout rate, and SCRABBLE also has a good performance. The results of CMF, SAUCIE, Bubble, DeepImpute, and SAVER are affected by the dropout rate. DCA performs even worse than dropout data in preserving cell–cell similarity. scImpute and DrImpute perform well at low dropout rates, but the results are not ideal at high dropout rates, especially at 70% of dropout rate. Overall, compared with other methods, scTsI can better retain the true cell–cell relationship. In addition, scTsI has a more stable performance and is less affected by the dropout rate.

Figure 3.

(A) PCC between the flattened cell–cell similarity matrices obtained from the true and the imputed single-cell gene expression by every method. (B) t-SNE visualization results of all imputation methods on simulated data of 1000 genes × 3000 cells with a 60% dropout rate. (C) t-SNE visualization of all imputation methods on real dataset sc_10x.

Moreover, to further evaluate whether the imputation methods can accurately reflect the cell interaction and cell community structure, we tested on all simulated datasets and real datasets including sc_10x, sc_celseq2, and sc_dropseq datasets for the experiments. We used t-SNE [42] to reduce the dimension of data to a 2D space for visualization. Figure 3B shows the simulated data of 1000 genes and 3000 cells with 60% of dropout rate as an example, where the proportion of zero values is 75%. We can see that scTsI, SCRABBLE, DrImpute, and DeepImpute can well distinguish between different cell clusters, and the visualization results are close to the real data. MAGIC, SAUCIE, SAVER, Bubble, and VIPER methods are not ideal for clustering cells. DCA roughly distinguishes three cell clusters, but each cluster is intermixed with a number of cells from other clusters. The visualizations of other dropout rates of the dataset are presented in Supplementary Fig. 1, and the visualizations of other simulated datasets are presented in Supplementary Figs 2–5. In addition to the simulated data, we also show the visualization results of sc_10x dataset in Fig. 3C. It can be seen that scTsI, CMF, DeepImpute and Bubble can better distinguish three cell clusters, while other methods divide H1975 cell line into two far cell clusters, which cannot accurately reflect the cell community structure. The visualizations of other experimental datasets are presented in Supplementary Fig. 6.

ScRNA-seq technology helps to study cell–cell similarity or dissimilarity, exploring the heterogeneity between cells [40]. Therefore, imputation methods need to restore and maintain cell–cell correlation. In terms of the results, scTsI can better maintain cell–cell similarity and more accurately reflect the community relationships among differentiated cells during the process of data imputation, facilitating the subsequent cell-level analysis.

scTsI improves cell clustering performance

To evaluate whether the imputation methods can improve clustering performance, we tested on all simulated data from Splatter, two simulated datasets based on real bulk (Spleen and Kidney data), and three real datasets (sc_10x, sc_celseq2, and sc_dropseq data). To avoid the influence of the randomness of k-means clustering method [43], we performed clustering on the data for 100 times and calculated the averages of adjusted Rand index (ARI) [44] and normalized mutual information (NMI) [45] of the clustering results as metrics to evaluate the clustering performance.

Figure 4A shows the boxplots of the clustering metrics ARI and NMI, for the five simulated datasets, each dot denoting a dropout rate. The heat maps of the clustering metrics on the simulated datasets are presented in Supplementary Figs 7 and 8. It can be seen that scTsI and SCRABBLE show stable and prominent metric results in all five datasets. Compared with SCRABBLE, scTsI performs better in datasets of 3000 genes × 1000 cells and 5000 genes × 1000 cells. Figure 4B shows a heat map of the averages of the clustering metrics across all dropout rates for the five simulated datasets. It can be clearly seen that the results of scTsI and SCRABBLE are very close to the real data. In addition, the clustering metrics of scTsI basically perform the best across different datasets.

Figure 4.

(A) Boxplots of ARI and NMI values on data imputed by all methods for each simulated dataset. Each dot represents a specific dropout rate. (B) Averages of ARI and NMI across all dropout rates for each simulated dataset. (C) ARI and NMI on imputed data by all methods for the five tested experimental datasets.

Moreover, we also measured clustering metrics on the two simulated datasets based on real bulk data and the three real datasets for evaluation. Figure 4C shows that scTsI generally provides better results across the five datasets. Some methods stand out in some datasets but perform poorly in other datasets. For example, DCA performs best on the sc_10x dataset but poorly on the remaining datasets. Although SCRABBLE performs well in simulated data, scTsI performs much better than SCRABBLE in real datasets. Overall, scTsI improves the cell clustering performance well across both simulated and real data.

Performing imputation for a specific gene in a specific cell using the expression of similar cells and similar genes helps to maintain cell–cell similarity, keep gene expression heterogeneity, and improve cell cluster identification. Accurate cell clustering can more effectively facilitate the revealing of the heterogeneity among cell populations and facilitate the identification of rare cell types or functional states, such as transitional cells during development or specific pathological subpopulations in disease contexts.

scTsI facilitates cell trajectory inference

To evaluate whether the imputation methods can facilitate the analysis of trajectory inference, we analyzed a total of five real datasets: cellmix1, cellmix2, cellmix3, cellmix4, and RNA_celseq2. Monocle2 [46] was used for the construction of trajectories on the data, which used DDRT tree (Discriminative DRT tree) [47] for dimensionality reduction and tree construction. We selected the H2228 cell line as the root trajectory state [11]. To evaluate the trajectory inference performance, we calculated three metrics [11]: correlation, overlap, and percentage. Correlation is the Pearson correlation between the order of the inferred trajectory and the order of the true trajectory of cells. Overlap refers to the proportion of cells whose inferred branches overlap the branches in the true trajectory. Percentage is the percentage of cells whose trajectory inference is correctly sorted.

Figures 5A and B shows the visualization of time and developmental stages of the RNA_celseq2 dataset, respectively. We can see that scTsI accurately shows the cell development process, making the inferred trajectory more accurate and clearer. DCA, DeepImpute and DrImpute exhibit mixed differentiation states in their inferred differentiation trajectories. SCRABBLE and VIPER diverge in a disorderly manner. Figure 5C shows the trajectory inference metrics on the five data sets, and it can be seen that scTsI shows better metric results across all five data sets. In addition, although DrImpute performs well in percentage on the RNA_celseq2 dataset, it performs worse than scTsI on other datasets. In general, scTsI facilitates the downstream analysis of trajectory inference and accurately shows the cell differentiation process.

Figure 5.

The inferred trajectory based on the imputed RNAmix_celseq2 dataset, showing different (A) pseudotimes and (B) cell types. (C) Trajectory inference metrics, correlation, overlap, and percentage, calculated based on the five real datasets.

This improvement in trajectory inference performance carries significant biological importance. Precise trajectory inference can more faithfully reflect the dynamic changes during cell differentiation, aiding in the identification of critical turning points in cell fate determination, such as lineage divergence in embryonic development or aberrant differentiation pathways in disease. Furthermore, optimized trajectory analysis facilitates the identification of key genes or regulatory networks driving differentiation, providing more accurate data support for studies in developmental biology and disease mechanisms.

scTsI is effective across different datasets and downstream tasks

To provide a unified comparison of imputation methods across all evaluation aspects, we summarized the performance of these methods by using the eight metrics above: ARI, NMI, PCC, RMSE, cell–cell similarity, correlation, overlap, and percentage across all simulated and experimental datasets. We performed one-sided paired t-tests to assess whether scTsI significantly outperforms each compared method. Figure 6 shows that scTsI achieves leading or competitive performance across all metrics on both simulated and experimental datasets. On simulated datasets, scTsI significantly outperforms most competing methods. Although SCRABBLE performs similarly on simulated datasets, scTsI exhibits a clear advantage on experimental datasets in terms of clustering and trajectory inference accuracy. scTsI shows stable performance across simulated datasets with different data dimensions and dropout rates, accurately recovering gene expression, maintaining cell–cell correlation, and improving cell clustering. scTsI also improves the practical downstream analyses on experimental data, including cell clustering and trajectory inference.

Figure 6.

Violin plots of eight evaluation metrics, ARI, NMI, PCC, RMSE, cell–cell similarity, correlation, overlap, and percentage across all simulated and experimental datasets. One-sided paired t-tests were conducted to evaluate whether each method significantly underperforms scTsI. Significance levels are indicated as follows, ^*: P < .1, ^**: <.01, ^***: <.001, ^****: <1e-4, ^*****: <1e-5.

Runtime analysis of scTsI

Although scTsI performs imputation based on a vector, it allows a sparse format of input to achieve imputation. To measure the running time of scTsI, we first tested on the simulated data of 1000 genes × 5000 cells and 5000 genes × 1000 cells with all dropout rates. Supplementary Figure 9A shows the running time of scTsI and other methods on these datasets. SAUCIE runs the fastest on the simulated data, followed by scTsI, which maintains a stable and relatively short runtime across different dropout rates. In contrast, scImpute and SAVER exhibit variable runtimes as dropout rate changes. Additionally, VIPER and SCRABBLE require significantly longer imputation times than other methods.

Next, we used the simulated data of 1000 genes × 5000 cells and 5000 genes × 1000 cells with a dropout rate of 50% to generate multiple subdata matrices of different dimensions for testing. From the data with 1000 genes and 5000 cells, we performed stratified sampling based on the known cell populations to generate data with the same gene number and cell numbers of 100, 500, 1000, 1500, and 2000. From the data with 5000 genes and 1000 cells, we selected from the top differentially expressed genes to the nondifferentially expressed genes in each cell group to generate data with the same cell number and gene numbers of 100, 500, 1000, 1500, and 2000. As illustrated in Supplementary Fig. 9B, the runtimes of VIPER and SCRABBLE obviously increase as the numbers of genes or cells increase. SAUCIE still demonstrates stable runtimes across different numbers of genes and cells. The runtime of MAGIC is more affected by the number of genes, while DCA is more affected by the sample size. scTsI remains relatively fast and stable across all tested dimensions.

In addition, we evaluated the running time of each method on experimental datasets whose gene number ranges from thousands to tens of thousands and sample size ranges from hundreds to thousands. As shown in Supplementary Fig. 9C, SAUCIE runs the fastest on the experimental data, showing its runtime advantage. scTsI achieves a relatively low and stable computational time cost compared with half of the other methods across different sizes of experimental data. VIPER and SCRABBLE still require significantly longer imputation times than other methods.

Combining with the overall evaluation results in Fig. 6, although several methods run faster than scTsI, such as SAUCIE and DeepImpute, the imputation accuracy of scTsI significantly outperforms them across different data sizes and dropout rates, maintaining cell–cell correlations and facilitating various downstream analyses. scTsI balances efficiency with accuracy across different kinds of data and usage scenario.

Performance comparison between the two stages of scTsI

Since scTsI contains two stages, whether the second stage can improve the imputation performance was explored. We compared PCC, RMSE, cell–cell similarity, and clustering metrics of the results between the first and the second stages on the simulated data (Supplementary Fig. 10). It can be seen that the data points of PCC, cell–cell similarity, and clustering metric plots are located at the upper left of the diagonal, indicating that the results of the second stage are all better than those of the first stage. The data points of the RMSE plot are located in the lower right of the diagonal, which illustrates that the second stage can adjust the result of the first stage to recover the data more accurately. Overall, the second stage can significantly improve the performance of scTsI in recovering expression values, maintaining cell–cell similarity and clustering performance, which can justify the need for the second stage of scTsI.

Parameter analysis of scTsI

To examine the influence of key parameters in scTsI, we evaluated the sensitivity of performance to the settings of the number of nearest cells (k₁), the number of nearest genes (k₂), and the threshold of high expression (the expression values greater than which would be unchanged by scTsI). First, we tested various combinations of k₁ and k₂ used in the first imputation stage of scTsI, using the sampled subdatasets introduced in the Runtime Analysis of scTsI section, as the number of total cells and genes in the data may affect the choice of k₁ and k₂. Supplementary Figure 11 shows that the performance of scTsI remains stable across a broad range of (k₁, k₂) values. Notably, setting both k₁ and k₂ as 25 usually yields better performance in clustering-related metrics such as ARI and NMI and also achieves competitive results in expression-level metrics like PCC and RMSE. Therefore, we set (k₁, k₂) = (25, 25) as the default configuration for scTsI and the results above were also generated using these default values.

We used scTsI to impute the zero values and keep the nonzero values unchanged, generating relatively conservative results, while users can change the threshold according to a specific gene expression distribution. Therefore, we used the simulated data of 1000 genes × 3000 cells and 3000 genes × 1000 cells with all dropout rates to investigate the effect of the threshold, considering the threshold of high expression may be affected by the distribution of gene expression in the data and the dropout rate (Supplementary Fig. 12). We tested different thresholds and only imputed the values ranking the last 0.1%, 0.2%, 0.5%, 1%, 2%, and 5% among all nonredundant expression values. For PCC, RMSE, and cell–cell similarity, thresholds set at or below 2% result in minimal variation, while a 5% threshold leads to a noticeable performance degradation, especially under low dropout rates. As to clustering, the threshold of 1% or 2% leads to better performance. Therefore, 1% or 2% would be a suggested choice of high expression threshold.

Discussion

Dropout events in scRNA-seq can seriously affect downstream analyses. The imputation methods proposed in recent years are not particularly effective in imputing data with high dropout rates and do not perform consistently satisfactorily across simulated data and experimental data and across different downstream tasks. Moreover, many methods change the original highly expressed values, hence introducing new noise during the imputation process. In this study, we proposed a new imputation method, scTsI. scTsI has three advantages. First, scTsI does not change the originally expressed values during the imputation process, which reduces the introduction of noise by transforming the expression matrix into a vector, performing row transformation, and imputing only for the zero values. Second, scTsI leverages the information of neighbor cells and genes and matched bulk RNA-seq data. These strengthen the ability to recover missing data even in a high dropout rate and a high zero ratio and improve the subsequent analyses across different types of data and different downstream tasks. Third, scTsI is easy to interpret and effective. It balances efficiency and accuracy across varying data dimensions and dropout rates.

We used simulated data with different sizes and dropout rates and also experimental datasets to demonstrate the performance of scTsI compared with different types of imputation methods. We evaluated from the aspects of recovering expression level, improving cell–cell similarity and facilitating data visualization, cell clustering, and trajectory inference. scTsI and SCRABBLE can accurately recover the gene expression level and show stable and excellent results in maintaining and recovering cell–cell similarity even in data with a high dropout rate. scTsI can accurately distinguish cell clusters and shows stable and outstanding results on both simulated data and real data, while SCRABBLE performs poorly on real data as its global optimization framework may tend to over-smooth expression signals. In terms of trajectory inference, some methods such as DeepImpute perform well on some data using specific metrics but do not perform well on other data or using other metrics. However, scTsI shows stable and excellent metric results. Combining all evaluated aspects, DrImpute exhibits inconsistent performance, and its imputation accuracy tends to degrade under high dropout conditions or in datasets with complex cell-type structures, likely due to its reliance on clustering-based averaging, which may fail to preserve fine-grained expression variation. Overall, scTsI performs consistently well across different kinds of data and various subsequent tasks.

In the future, scTsI can be further studied from the following aspects. The zeros that need to be imputed can be further identified before imputing to retain meaningful biological information and reduce the introduction of noise. For this, a probabilistic model can be applied first to identify false zeros. In addition, scTsI can introduce more prior knowledge to improve the imputation accuracy, such as the spatial location of cells and the cell cluster annotation. In scTsI, the information of similar cells according to expression level is involved in the first stage of imputation, while spatial location can be used to identify neighboring cells, and then, their expression can be borrowed. When searching for similar cells, we could limit the searching range to the same cell cluster, which can be a known label or annotated using a clustering tool beforehand.

Key Points

• A two-stage single-cell RNA-seq data imputation algorithm, scTsI, is proposed, in which the information of similar cells and genes is borrowed for the initial imputation and then external bulk RNA-seq data are leveraged to adjust the initial imputed values.

• scTsI imputes dropouts and keeps the high expression levels unchanged simultaneously by vector transformation and calculation.

• scTsI restores gene expression accurately and maintains cell–cell similarity across different data dimensions and dropout rates.

• scTsI improves the performance of downstream analyses, including data visualization, clustering, and cell trajectory inference.

Author contributions

J.G. conceived and designed the study. H.Z. and W.L. conducted the analyses. J.G., H.Z., and W.L. wrote the manuscript. All authors approved the final manuscript.

Conflict of interest: None declared.

Funding

This work was supported by the National Science and Technology Major Project (No. 2021ZD0112600), Natural Science Foundation of Fujian Province of China (No. 2022J05012), and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, China (No. Scip20240104).

Data availability

scTsI is available at: https://github.com/JGuan-lab/scTsI. The simulated datasets and the analyzed experimental single-cell datasets can be accessed at: https//zenodo.org/uploads/13859610. The experimental datasets were downloaded from the study of Hou et al. at: https://github.com/Winnie09/imputationBenchmark.