A Joint Batch Correction and Adaptive Clustering Method of Single-Cell Transcriptomic Data

Abstract

Clustering analysis for single-cell RNA sequencing (scRNA-seq) data is essential for characterizing cellular heterogeneity. However, batch information caused by batch effects is often confused with the intrinsic biological information in scRNA-seq data, which makes accurate clustering quite challenging. A Deep Adaptive Clustering with Adversarial Learning method (DACAL) is proposed here. DACAL jointly optimizes the batch correcting and clustering processes to remove batch effects while retaining biological information. DACAL achieves batch correction and adaptive clustering without requiring manually specified cell types or resolution parameters. DACAL is compared with other widely used batch correction and clustering methods on human pancreas datasets from different sequencing platforms and mouse mammary datasets from different laboratories. The results demonstrate that DACAL can correct batch effects efficiently and adaptively find accurate cell types, outperforming competing methods. Moreover, it can obtain cell subtypes with biological meanings.

1. Introduction

scRNA-seq is a technology measuring gene expression at a single-cell level [1]. The analysis of scRNA-seq data facilitates the discovery of biological processes in cells, the investigation of gene regulatory mechanisms that vary by cell subtypes [2], the deciphering of organism development processes [3], and many other biological discoveries. As an essential step in identifying cell types and revealing cellular heterogeneity, the clustering analysis of scRNA-seq data is critical in scRNA-seq data analysis.

However, the clustering analysis of scRNA-seq data is challenging. scRNA-seq data are sometimes compiled from different experiments. With the rapid development of single-cell RNA sequencing technologies, many options exist for different experiments, including different sequencing platforms, instruments, and reagents. Together with different operators and sample sources, all these factors will cause batch effects. The batch information caused by batch effects is often confused with biological information, which makes it challenging to decipher intrinsic biological heterogeneity by clustering analysis. Therefore, the batch effects are big barriers in the clustering analysis of scRNA-seq data.

Several methods have been proposed to perform clustering analysis on multi-batch scRNA-seq data. Most of these methods take batch correction and clustering as two separate steps, which remove batch effects first and then conduct clustering algorithms. Common approaches to alleviating the influence of batch effects include Seurat CCA [4], Seurat RPCA, LIGER [5], and Harmony. Seurat CCA achieves batch correction of scRNA-seq data by combining Mutual Nearest Neighbors (MNN) and diagonal Canonical Correlation Analysis (CCA). CCA can be replaced by Reciprocal PCA (RPCA) to improve efficiency. The LIGER (Linked Inference of Genomic Experimental Relationships) method uses integrative Non-negative Matrix Factorization (iNMF) to achieve the purpose of batch removal. Korsunsky et al. [6] proposed a cluster-guided batch alignment method named Harmony. It uses soft k-means clustering in a shared low-dimensional space of the original count matrix to constrain the batch removal process. In addition, adversarial learning has been used for batch correction in some studies. AD-AE [7] (Adversarial Deconfounding AutoEncoder) is a method based on Autoencoder (AE) [8]. It disentangles confounders from accurate signals and retains the accurate ones as the low-dimensional latent variable without batch effect. After batch correction, clustering analysis is performed to identify cell clusters using methods, such as Louvain [9] and Leiden [10].

However, since biological information is mixed with batch information; removing batch effects separately may impair biological information in scRNA-seq data and affect the performance of the clustering analysis. These methods might wrongly deplete or enrich certain cell types or accumulate errors in the training, leading to unsatisfactory clustering results [11].

A few methods have been presented to remove batch effects jointly with clustering analysis. The scCAEs algorithm combines convolutional autoencoder and soft k-means clustering [12]. The iterative training via Kullback–Leibler Divergence can ensure convergence while balancing biological and technical differences between clusters. It can remove batch effects while achieving clustering analysis. Li et al. [11] proposed the SCDC algorithm with a similar autoencoder structure. SCDC extracts biological and batch information from the original scRNA-seq data using two encoders and uses a soft clustering method on the low-dimensional representations of biological information. Compared with the methods that perform batch correction and clustering separately, these methods can perform much better [11,12]. However, these methods require users to specify the number of cell types (or resolution parameters) or to infer the parameters by traversing multiple values based on some metrics. Arbitrarily specifying cluster numbers often misguides clustering results, while traveling for optimal cluster numbers often requires significant computing resources. The methods that support batch effect removal and highly efficient adaptive clustering jointly are, therefore, in great need.

In this study, a deep adaptive clustering with an adversarial learning method is proposed. DACAL uses the AE architecture. It contains three modules: the AE module for dimensionality reduction, the Dirichlet Process Mixture Model (DPMM) [13] module for clustering, and the Adversarial module for batch correction. To achieve accurate analysis results, all the parameters of these modules are jointly optimized. In this study, DACAL is introduced in detail, and the robustness of the hyperparameter change in DACAL is evaluated using a simulated dataset. The performance of DACAL and other widely used batch correction and clustering methods are also evaluated on human pancreas datasets from different sequencing platforms and mouse mammary datasets from different laboratories. The results demonstrated that DACAL can adaptively obtain batch correction and clustering results and perform better than other methods. Moreover, DACAL can obtain meaningful subtypes for the input of scRNA-seq datasets containing multiple batches.

2. Materials and Methods

2.1. The Architecture of DACAL

DACAL is a deep batch correction and adaptive clustering algorithm for scRNA-seq data. The input of DACAL consists of two parts: a gene expression matrix $\mathit{X}{\in \mathbb{N}}^{N\times M}$, which contains $N$ cells and $M$ genes, and a vector $\mathit{s}{\in \mathbb{N}}^N$, which is composed of the corresponding batch IDs of these cells. The output of DACAL is the cluster label $\mathit{y}{\in \mathbb{N}}^N$.

The AE module is used to reduce the dimensionality of the input $\mathit{X}$. The latent variables $\mathit{C}$ of the biological information and $\mathit{B}$ of the batch information are obtained. $\mathit{C}$ and $\mathit{B}$ are used for inferring the parameter $\mathit{\Gamma }$ of the distribution of $\mathit{X}$, while $\mathit{B}$ is used for obtaining the parameter $\mathit{\Pi }$ of the distribution of batch IDs. To achieve clustering analysis for the input data, $\mathit{C}$ is used in the DPMM module. The predicted labels $\mathit{y}$ and the prior distribution to constrain $\mathit{C}$ are obtained. To remove the batch effect, $\mathit{C}$ is also used in the Adversarial module for inferring the Categorical distribution of batch IDs, which can lead to adversarial learning with the generation of $\mathit{C}$ in the AE module. The two processes are balanced via adversarial learning to achieve the distinction between $\mathit{C}$ and $\mathit{B}$. The workflow of DACAL is shown in Figure 1.

2.1.1. The AE Module

The input of the AE module is $\mathit{X}$ and $\mathit{s}$, where ${\mathit{x}}_n\in {\mathbb{N}}^M$ and $s_n\in \mathbb{N}$ are the $n$-th components of $\mathit{X}$ and $\mathit{s}$, which shows the gene expression and batch ID of cell $n,$ respectively. $s_n$ is converted into the one-hot encoding $G$-dimensional vector ${\mathit{\eta }}_n$, where $G$ is the number of batches in the dataset. The spliced ${\mathit{x}}_n$ and ${\mathit{\eta }}_n$ are converted into the low-dimensional representation ${\mathit{z}}_n$ using the encoder network $f$ which is parameterized by $\mathit{\varphi }$. ${\mathit{z}}_n$ consists of two parts: the low-dimensional latent variable ${\mathit{c}}_n\in {\mathbb{R}}^{D^c}$ of the biological information and the low-dimensional variable ${\mathit{b}}_n\in {\mathbb{R}}^{D^b}$ of the batch information. $f^c$ is the encoder network for ${\mathit{c}}_n$ and $f^b$ is the encoder network ${\mathit{b}}_n$ ($f=\{f^c,f^b\}$). The decoder network $g^{\mathit{\theta }}$, with parameter $\mathit{\theta }$, predicts the distribution of ${\mathit{x}}_n$ conditioned on ${\mathit{z}}_n$, which is assumed to be a Poisson distribution parameterized by ${\mathit{\gamma }}_n$ ($\mathit{\Gamma }$ = [${\mathit{\gamma }}_1;{\mathit{\gamma }}_2;\dots ;{\mathit{\gamma }}_N$]). The other decoder $g^{\mathit{\xi }}$ parameterized by $\mathit{\xi }$ can predict the distribution of $s_n$ conditioned on ${\mathit{b}}_n$, which is assumed to be a Categorical distribution parameterized by ${\mathit{\pi }}_n$, ($\mathit{\Pi }=[{\mathit{\pi }}_1;{\mathit{\pi }}_2;\dots ;{\mathit{\pi }}_N$]). The loss of AE $l_{\mathrm{A}\mathrm{E}}$ is defined as the reconstruction loss of ${\mathit{x}}_n$ and $s_n$. The loss consists of two parts: the negative log-likelihood of ${\mathit{\gamma }}_n$ given ${\mathit{x}}_n$, and the negative log-likelihood of ${\mathit{\pi }}_n$ given $s_n$, i.e.,:

$$\begin{array}{cc}\hfill -\ln p\left({\mathit{x}}_n|{\mathit{\gamma }}_n\right)& =-\ln p\left({\mathit{x}}_n|g^{\mathit{\theta }}\left({\mathit{z}}_n;\mathit{\theta }\right)\right)\hfill \\ & =-\ln \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left({\mathit{x}}_n|g^{\mathit{\theta }}(f\left({\mathit{x}}_n,s_n;\mathit{\varphi }\right);\mathit{\theta }\right)\hfill \\ \hfill -\ln p\left(s_n|{\mathit{\pi }}_n\right)& =-\ln p\left(s_n|g^{\mathit{\xi }}\left({\mathit{b}}_n;\mathit{\xi }\right)\right)\hfill \\ & =-\ln \mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\left(s_n\left|g^{\mathit{\xi }}\right(f^b\left({\mathit{x}}_n,s_n;\mathit{\varphi }\right);\mathit{\xi })\right)\hfill \\ \hfill -\ln p\left({\mathit{x}}_n|{\mathit{\gamma }}_n\right)-\ln p\left(s_n|{\mathit{\pi }}_n\right)& \triangleq l_{\mathrm{A}\mathrm{E}}(\mathit{\varphi },\mathit{\theta },\mathit{\xi };{\mathit{x}}_n,s_n)\hfill \end{array}$$

(1)

2.1.2. The DPMM Module

The input of the DPMM module is the latent variable $\mathit{C}$ of the biological information, where ${\mathit{c}}_n$ is the $n$-th component in $\mathit{C}$. The prior distribution provided by the DPMM module can constrain ${\mathit{c}}_n$ and infer the clustering label $y_n$ of cell $n$. In this module, ${\mathit{c}}_n$ is assumed to be generated from the Dirichlet process mixture model, and the loss function of this module is defined as the negative log-likelihood of ${\mathit{c}}_n$ and the distribution determined by DPMM [14]. The loss function $l_{\mathrm{D}\mathrm{P}\mathrm{M}\mathrm{M}}$ is defined as the following:

$$\begin{array}{cc}\hfill -\ln p\left({\mathit{c}}_n,\mathcal{G}\right)& =-\ln p\left(f^c\left({\mathit{x}}_n,s_n;\mathit{\varphi }\right),\mathcal{G}\right)\hfill \\ & \triangleq l_{\mathrm{D}\mathrm{P}\mathrm{M}\mathrm{M}}\left(\mathit{\varphi },\mathcal{G};{\mathit{x}}_n,s_n\right)\hfill \end{array}$$

(2)

where $\mathcal{G}$ is the set of latent variables in DPMM. The optimization of $l_{\mathrm{D}\mathrm{P}\mathrm{M}\mathrm{M}}\left(\mathit{\varphi },\mathcal{G};{\mathit{x}}_n,s_n\right)$ maximizes the item $\ln p\left({\mathit{c}}_n,\mathcal{G}\right)$, which has been given in [15,16]. In this module, as shown in [16], there is a hyperparameter $\alpha$ in the Dirichlet process, and the larger $\alpha$ leads to the more uniform distribution of weights, which means the greater number of clusters.

2.1.3. The Adversarial Module

The latent variable $\mathit{C}$ is the input of the Adversarial module. The module predicts the parameter ${\mathit{\pi }}_n^D$ of the Categorical distribution of $s_n$. The discrimination of $s_n$ using the Adversarial module and the generation of ${\mathit{c}}_n$ using the AE module are adversarially learned, which can help to separate the information in low-dimensional representation into two parts: biological information in ${\mathit{c}}_n$ and batch information in ${\mathit{b}}_n$. By using adversarial learning, the Adversarial module cannot determine the correct batch ID from ${\mathit{c}}_n$, which means that ${\mathit{c}}_n$ does not contain batch information anymore, thus completing batch correction.

There are two loss functions in this module: $l_{\mathrm{d}}$ is the negative log-likelihood of $s_n$ given ${\mathit{c}}_n$, which is used for updating the parameter $\mathsf{\Omega }$ in the Adversarial module; the other is the log-likelihood of $s_n$ given ${\mathit{c}}_n$, which is used for adversarial learning and updating the parameter $\mathit{\varphi }$ in the AE module. $l_{\mathrm{d}}$ and $l_{\mathrm{a}}$ are as the following:

$$\begin{array}{cc}\hfill -\ln p_{\mathsf{\Omega }}\left(s_n|{\mathit{c}}_n\right)& =-\ln p_{\mathsf{\Omega }}\left(s_n|f^c\left({\mathit{x}}_n,s_n;\mathit{\varphi }\right)\right)\hfill \\ & \triangleq l_{\mathrm{d}}\left(\mathsf{\Omega };{\mathit{x}}_n,s_n,\mathit{\varphi }\right)\hfill \\ \hfill \ln p_{\mathsf{\Omega }}\left(s_n|{\mathit{c}}_n\right)& =\ln p_{\mathsf{\Omega }}\left(s_n|f^c\left({\mathit{x}}_n,s_n;\mathit{\varphi }\right)\right)\hfill \\ & \triangleq l_{\mathrm{a}}(\mathit{\varphi };{\mathit{x}}_n,s_n,\mathsf{\Omega })\hfill \end{array}$$

(3)

2.1.4. Loss Function

The loss function of updating the parameters of encoders and decoders in the DACAL contains the loss of all the modules by using Equations (1)–(3), that is:

$$l\left(\mathit{\varphi },\mathit{\theta },\mathit{\xi },\mathcal{G};{\mathit{x}}_n,s_n,\mathsf{\Omega }\right)=l_{\mathrm{A}\mathrm{E}}\left(\mathit{\varphi },\mathit{\theta },\mathit{\xi };{\mathit{x}}_n,s_n\right)+l_{\mathrm{D}\mathrm{P}\mathrm{M}\mathrm{M}}\left(\mathit{\varphi },\mathcal{G};{\mathit{x}}_n,s_n\right)+l_{\mathrm{a}}(\mathit{\varphi };{\mathit{x}}_n,s_n,\mathsf{\Omega })$$

(4)

To minimize the loss function $l\left(\mathit{\varphi },\mathit{\theta },\mathit{\xi },\mathcal{G};{\mathit{x}}_n,s_n,\mathsf{\Omega }\right)$, an iterative training strategy is used to optimize the parameters of the AE, DPMM, and Adversarial modules alternatively. The training algorithm of DACAL is detailed in Algorithm 1.

Algorithm 1 The DACAL training algorithm

Input: A scRNA-seq dataset ${\left\{{\mathit{x}}_n,s_n\right\}}_{n\in \mathcal{N}}$

Output: The cluster label $\mathit{y}$

1: for $t=\mathrm{1,2},\dots ,T$ do:

2:   Sample a mini-batch ${\left\{{\mathit{x}}_n,s_n\right\}}_{n\in {\mathcal{N}}_{\mathrm{t}}}$ from the dataset, where ${\mathcal{N}}_{\mathrm{t}}\subset \mathcal{N}$

3:   for Step $k=\mathrm{1,2},\dots ,K$ do:

4:      $\mathrm{Freeze} \mathit{\varphi }$ and update $\mathsf{\Omega }$ with loss $\frac{1}{\left|{\mathcal{N}}_{\mathrm{t}}\right|}{\sum }_{{\mathcal{N}}_{\mathrm{t}}\subset \mathcal{N}}{l}_d(\mathsf{\Omega };{\mathit{x}}_n,s_n,\mathit{\varphi })$

7:   end for

$8: \mathrm{Freeze} \mathsf{\Omega }$ and $\mathcal{G}, \mathrm{update} \mathit{\varphi },\mathit{\theta },\mathit{\xi }$ with loss $\frac{1}{\left|{\mathcal{N}}_{\mathrm{t}}\right|}{\sum }_{{\mathcal{N}}_{\mathrm{t}}\subset \mathcal{N}}l\left(\mathit{\varphi },\mathit{\theta },\mathit{\xi },\mathcal{G};{\mathit{x}}_n,s_n,\mathsf{\Omega }\right)$

9:   if $t \mathrm{m}\mathrm{o}\mathrm{d} S=0$ then: 10:  Freeze $\mathit{\varphi }$ and update $\mathcal{G}$ with loss $\frac{1}{\left|{\mathcal{N}}_{\mathrm{t}}\right|}{\sum }_{{\mathcal{N}}_{\mathrm{t}}\subset \mathcal{N}}{l}_{\mathrm{D}\mathrm{P}\mathrm{M}\mathrm{M}}\left(\mathit{\varphi },\mathcal{G};{\mathit{x}}_n,s_n\right)$ 11:  end if

12: end for

*$T$ is the total number of iterations, $S$ is the number of iterations in each epoch, and $K$ is the number of updates for $\mathsf{\Omega }$ in each iteration.

Algorithm 1 shows the training process of DACAL. The parameters of the AE, DPMM, and Adversarial modules are alternatively optimized to minimize the loss function $l\left(\mathit{\varphi },\mathit{\theta },\mathit{\xi },\mathcal{G};{\mathit{x}}_n,s_n,\mathsf{\Omega }\right)$. Specifically, first, the parameter of encoder network $f$ in the AE module is frozen to update the parameter in the Adversarial module. Then, the parameters in the DPMM and Adversarial modules are frozen, and the parameters in the AE module are updated to remove the batch information in the low-dimensional representation. Finally, the parameters in the AE and Adversarial modules are frozen, and the parameters in the DPMM module are updated to obtain the clustering result.

2.2. Training the DACAL Model

The DACAL model is implemented using PyTorch [17] (v.1.12.0) in Python 3 (v.3.7.11). The BayesianGaussianMixture function using scikit-learn (v.1.0.2) is used in the DPMM module. The number of mixture components in the BayesianGaussianMixture function is set to 50, where 50 is the maximum number of cell types allowed in the model and far more than the true number of cell types. The parameter details of DACAL are as follows: the two hidden layer sizes of the encoder $f$ are set to 512 and 128, the hidden layer size of the decoder $g^{\mathit{\theta }}$ to 128 and 512, the hidden layer sizes of the decoder $g^{\mathit{\xi }}$ to 128 and 64, and the hidden layer sizes of the discriminator to 128 and 64. The dimensionality of ${\mathit{c}}_n$ is set to 32 and ${\mathit{b}}_n$ to 2. The number of updates used in $l_{\mathrm{d}}(\mathsf{\Omega };{\mathit{x}}_n,s_n,\mathit{\varphi })$ is 10. During the training process, the minibatch size is set to 512, and the AdamW optimizer [18] with a fixed learning rate of 1 × 10⁻⁴ is used for optimization. The hyperparameter $\alpha$ is set to 1 × 10⁻¹⁰ except for the experiment of the robustness of hyperparameter change. The AE and Adversarial modules are pre-trained during training until a stable low-dimensional representation is obtained; the entire model is then trained.

2.3. Datasets and Preprocessing

In order to demonstrate the performance of DACAL on multiple batches of scRNA-seq data, three datasets are used in this paper, including a simulated dataset and two real datasets. The simulated dataset is constructed as follows: assume that the data follows a Gaussian mixture distribution. The mean and log of variance are sampled from the Gaussian distribution, respectively. The weights are sampled from the Dirichlet distribution. The simulated dataset contains 32,000 cells from 4 batches with 10 cell types, available at https://github.com/omicshub/DACAL (accessed on 5 December 2023).

This article uses two real datasets from typical batch sources: the human pancreas cell dataset from 5 sequencing protocols and the mouse mammary cell dataset from 3 laboratories. The human pancreas cell dataset contains 14,890 cells from 8 batches with 13 cell types; the mouse mammary cell dataset contains 9288 cells from 3 batches including 3 cell types. These datasets can be obtained from https://figshare.com/articles/dataset/Batch_Alignment_of_single-cell_transcriptomics_data_using_Deep_Metric_Learning/20499630/2?file=36699252 (accessed on 5 December 2023) [19].

The Seurat package (v.4.1.0) [17] is used to preprocess the data. Specifically, UMI (Unique Multiple Index) count matrices are normalized and log-transformed using the NormalizeData function in Seurat; then, 4000 highly variable genes are selected using the FindVariableFeatures function; at last, the selected genes are scaled to obtain the gene expression matrix using the ScaleData function. In addition, batch information of the data needs to be extracted to obtain the batch IDs vector. Gene expression matrix and batch IDs vector are used together as different inputting methods.

2.4. Comparing Methods and Evaluation Metrics

DACAL is compared with commonly used batch correction and clustering methods, which includes combinations of four widely used batch correction methods and four commonly used cluster analysis methods, as well as a joint batch correction and clustering method scCAEs. The 4 batch correction methods are Harmony, LIGER, Seurat CCA, and Seurat RPCA. The input of these methods are gene expression matrix and batch IDs vector, which are the same as those of DACAL. The dimensionality of low-dimensional representation is set to 32 and implemented with the original code and the default hyperparameters in the experiments. The four clustering correction methods are the Louvain algorithm, the Louvain algorithm with multilevel refinement (MR Louvain), the Smart Local Moving (SLM) algorithm, and the Leiden algorithm. The input of these clustering methods is the output of the batch correction methods. In the experiments of this paper, these clustering methods are implemented using the Seurat (v.4.1.0) package with default parameters. The experiments on scCAEs use the original code and default hyperparameters; the hyperparameter of cluster numbers of scCAEs is set to the number of actual labels in each experiment.

In addition, PCA dimensionality reduction and four clustering methods are combined without batch correction method as a raw experiment to demonstrate the batch effect in the datasets.

Five metrics are used to evaluate the performance of these methods, i.e., four classical clustering performance indicators and one indicator defined by us, i.e., Normalized Mutual Information (NMI) [20], Adjusted Rand Coefficient (ARI) [21], F1 score [22], Silhouette Coefficient (SC) [23], and Deviation Ratio (DR). These classical indicators are implemented by using scikit-learn. As the metric scores of NMI, ARI, and F1 score range from 0 to 1, the SC score is scaled to $0–1$ for calculating the average score with NMI and ARI, where 0 and 1 correspond to disappointing and ideal results of the method, respectively.

To evaluate the degree of deviation between the number of predicted labels and the number of true ones, a DR indicator is put forward, that is:

$$DR=\frac{h_{predicted}-h_{true}}{h_{true}}$$

(5)

where $h_{predicted}$ is the number of predicted labels, $h_{true}$ is the number of true ones, respectively. The larger absolute value indicates a more significant deviation, and vice versa.

3. Results

3.1. DACAL Is Robust to Hyperparameter Changes on scRNA-seq Data

To verify the robustness of the hyperparameter $\alpha$ change in DACAL, the experiment by setting the hyperparameter $\alpha$ to 8 different values in the wide range of [1 × 10⁻²⁰⁰, 1 × 10⁻⁰] was designed. As shown in Figure 2a, the numbers of clusters obtained by DACAL are always very close to the numbers of the true one as parameter $\alpha$ increased, suggesting that DACAL can adaptively obtain accurate cluster numbers. As shown in Figure 2b, the NMI, ARI, SC, and F1 scores, and their average were also computed. The indicators are all greater than 0.7, and the average scores are greater than 0.8, suggesting the DACAL’s robustness of hyperparameter changes.

3.2. DACAL Can Jointly Remove Batch Effect and Cluster Adaptively on scRNA-seq Data

To verify the performance of the DACAL on the scRNA-seq data, DACAL and the other competing methods were applied to the human pancreatic cell dataset. The comparison methods include scCAES and the combinations of four widely used batch correction methods (Harmony, LIGER, Seurat CCA, and Seurat RPCA) and four commonly used cluster analysis methods (Louvain, MR Louvain, SLM, and Leiden). In addition, the combinations of PCA and four clustering methods are also demonstrated as the baseline.

The low-dimensional representations obtained by the above methods are visualized through UMAP and labeled with the true cell types, batch labels, and predicted labels, respectively. The UMAP plots of DACAL show that the cells in different batches mix well and the predicted labels accord the ground truth well (Figure 3a, left). Compared to DACAL, other competing methods either did not remove batch effects efficiently or did not predict cell-type labels accurately (Figure 3a, middle). The PCA dimensionality reduction result marked with the Batch label is depicted as a baseline, which shows that the batch effect is evident in this data set (Figure 3a, right). These results demonstrate that DACAL outperforms all the other batch correction and clustering methods.

In order to quantitatively measure the similarity degree between the cluster labels obtained by different methods and the true cell type labels, the indicators NMI, ARI, SC, and F1 scores and DR were calculated. Figure 3b shows the NMI, ARI, SC, and F1 scores and their average of different methods. DACAL obtained the highest scores in all four indicators, indicating that its clustering performance is better than the other 21 methods. The DR scores were also calculated to compare the deviation between the number of predicted labels from different methods and the number of true ones in this dataset. As shown in Figure 3c, the result of DACAL has the lowest DR score in different methods, which means that the predicted number of clusters is the closest to the number of cell types.

To show the relationship between the actual cell type label and the predicted label obtained by different methods, the Sankey plots are used to demonstrate the match between these labels. As shown in Supplementary Figure S1, the predicted labels by DACAL match the true label the best.

3.3. DACAL Can Provide Fine-Grained Clusters on scRNA-seq Data with Batch Effect

When comparing DACAL with different batch correction and clustering methods on the mouse mammary cell dataset, DACAL obtained far better results in NMI, ARI, SC, and F1 scores and DR than the competing methods (Figure 4a,b). It should be noted that the DR score of DACAL on this dataset is 2, which is a very high value. The high positive deviation indicates remarkably more clusters than the ground truth. The reasons for more predicted clusters by DACAL than true labels should be investigated.

To show the relationship between the actual cell type label and the predicted label obtained by DACAL, the Sankey plot demonstrated the match between these labels. As shown in Figure 4c, DACAL subdivides luminal progenitor and luminal mature into two clusters, respectively. Luminal mature cells are divided into cluster 2 and cluster 27. The literature shows that cells with a high expression of Prlr and S100a6 are luminal mature cells related to hormone changes [24]. The marker genes of these two clusters were analyzed to infer the meaning of the subtypes represented by cluster 2 and cluster 27. Figure 4d is a partial enlargement of the UMAP visualization of these two clusters marked by the genes of Prlr and S100a6. Compared to cluster 27, Prlr and S100a6 are specifically highly expressed in cluster 2. Cluster 27 is other luminal mature cells. The Sankey plots of other methods are shown in Supplementary Figure S2.

To determine why the luminal progenitor was divided into these two clusters, the marker genes of cluster 0 and cluster 6 are analyzed. Bhupinder et al. demonstrated that the early luminal progenitor subset can be marked by a high expression of the CD55 gene [25]. By calculating the average expression value of gene CD55 in cluster 0 and cluster 6, its expression level in cluster 0 is found to be twice that in cluster 6. Therefore, cluster 0 can be inferred as an early luminal progenitor subset through the specific high expression of CD55, and cluster 6 is a luminal progenitor subset at other stages. These results show that DACAL can achieve good batch correction and adaptive clustering results and adaptively identify accurate subtypes with biological meanings.

3.4. Comparisons of Runtime and Memory Usage of DACAL and Other Methods

The computing time and memory consumed by DACAL and 21 other approaches were compared on a SITONHOLY Linux server (Tianjin, China) with 72 CPU cores of two Intel Xeon Gold 6240 chips (Santa Clara, CA, USA), 503 GB of RAM, and NVIDIA RTX 2080 Ti GPUs (Santa Clara, CA, USA).

As shown in Figure 5a,b, the runtime of DACAL is higher than most other methods except scCAEs, where the mouse mammary cell dataset cannot provide the required number of genes for scCAEs, so the corresponding results are not obtained. DACAL and scCAEs are methods based on deep learning, which takes multiple epochs of training to learn stable model parameters and network parameters and usually takes longer than non-depth methods. However, when comparing the CPU and GPU requirements, DACAL needs less memory than all the comparing methods.

Notably, the clustering performance of DACAL is outstanding compared to these methods. Therefore, considering computing time, memory, and performance, DACAL is a very advantageous method.

4. Discussion

The joint batch effect correction and adaptive clustering methods are urgently needed and are still quite challenging in scRNA-seq data analysis. A deep DACAL model based on adversarial learning is proposed to solve this problem. The model consists of three modules: the AE module for nonlinear dimensionality reduction, the Adversarial module for adversarial learning to remove batches, and the DPMM module for clustering. Joint batch effect correction and adaptive clustering of multiple batches of scRNA-seq data are achieved by optimizing all modules. The comparisons of the two scRNA-seq dataset results show that DACAL outperforms other competing methods. In addition, DACAL can accurately identify fine-grained clusters corresponding well to biological meanings.

The ability to adaptively find accurate cluster numbers is novel and meaningful for batch effect correction and clustering methods. The results of commonly used methods are affected by the number of specified clusters, and arbitrarily setting cluster numbers often misguides results. DPMM is employed to realize the clustering function in the batch correction and clustering method. DPMM can automatically adjust the number of clusters based on data complexity. The results are obtained by using DPMM in the joint batch correction and clustering method without the need for cluster numbers. DACAL is the first method that jointly performs highly efficient batch correction and accurate adaptive clustering for scRNA-seq data.

As the transcriptome is not the sole determinant of cell type, state, and function, the analysis of single-cell sequencing data in different omics fields has provided possibilities for interpreting cellular heterogeneity from various perspectives. Moreover, the analysis of single-cell multi-omics sequencing data can help researchers obtain more comprehensive observational information and also aid in understanding the correspondence between different omics datasets. Our target in future studies is to extend the batch effect correction and adaptive clustering method to single-cell sequencing data from other omics or multi-omics.