Constructing GRN for each cell

To construct the gene regulatory network for each cell (Fig 1A (II)), we first perform gene pooling [32], which involves dimensionality reduction by clustering genes based on their correlations across all cells. That is, for genes x and y, the Pearson correlation coefficient r_xy can be calculated by (1) where x_i represents the expression of gene x in cell i, and represents the mean expression of gene x in all cells. A supergene can be defined as a group of the k most correlated original genes. As shwon in Fig 1B, typically the number of supergenes l decreases with increasing k, and the rate of change gradually diminishes. Using sliding average algorithm, when the difference between the average values of two consecutive windows, and , is less than 2, the and values of the latter window are chosen as the optimal k and l. For each supergene in a cell, the mean expression level of all genes within the class is considered as the supergene’s expression level. Compared to other dimension reduction methods such as the PCA, gene pooling retains better interpretability while reducing dimensions.

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Fig 1.

(A) Starting from single-cell data (I), we obtain the distance matrix of cells in PCA space (III) and the GRN of individual cells (II). By solving the Euclidean distance of cells in the space characterized by the out-degree and in-degree of all genes, we obtain the cell distance matrix (IV). The Hadamard product of the distance matrix in PCA space and the distance matrix in GRN space, with a constraint on the number of neighbors, results in the cell differentiation graph G (V). Simultaneously, we solve the rough pseudotime of cells based on their relationships in GRN space (VI). We use the rough pseudotime to correct the cell differentiation graph G and then find the minimum spanning tree of G to obtain the cell differentiation trajectory graph (VII). (B) The curve of the number k of supergenes obtained during gene pooling with respect to the number l of original genes in each group. We select the optimal supergenes and the number of original genes per group by calculating the mean of supergenes in the preceding and following sliding windows. (C) The function graph of the tick function, which attains its minimum value at x = 1. We consider that when the pseudotime difference between clusters is 1 at the cluster scale, the distance between the two clusters is minimized, meaning the probability of a connecting edge between the two clusters is maximized.

https://doi.org/10.1371/journal.pcbi.1012638.g001

To obtain the GRN in a single cell, we extend the method for solving GRNs from [33] to scenarios with only gene expression data. Based on the expressions of supergenes in each cell, we further construct a GRN for the cell using a ordinary differential equation (2) Here, we assume that the generation rate of gene i is a result of its regulation by all other genes X_i (the set of all genes excluding gene i), approximated as a function of the expression levels of these other genes, f_i(X_i), the first-order Taylor expansion of which can be written as (3) where m is the number of supergenes, is the higher order infinitely small quantity of , and A_ij is the coefficient to be solved. Consider that in the steady state , then we can get (4) since and are constants, can be denoted as A_i0. The degradation rate γ is considered as constant. The x_i can be further expressed as: (5) Subject to the conservative condition , the coefficient matrix A is calculated by minimizing the mean square error: (6) For the solved GRN (Fig 1A (II)), represented by the matrix A = (A_ij), we define: Intensity of the regulation of gene i of cell k on other genes and the intensity of the regulatory gene i of other genes respectively, where and . The regulatory intensity vectors and regulated intensity vectors of all genes in single cell k are denoted as and .

Constructing the K-nearest neighbor graph of cells

In the linear space (GRN space) composed of I and O, cell k can be represented as (I_k, O_k). We consider the binormal of I and O between cells to measure the distance D^GRN between cells (Fig 1A (IV)), which reflects the dissimilarity of gene regulatory network between cells, that is, the dissimilarity of state changes between cells: (7) The distance between cells in the GRN space reflects to some extent the differences in gene regulatory relationships within cells. In addition, the static distance between cells in the expression space, D^PCA, can also be calculated by performing PCA on the gene expression matrix and then computing the Euclidean distance between cells in the PCA-transformed space (Fig 1A (III)).

Based on the similarity of cells at the current time and their similarity in future state transitions, the distance between cells k and l in the differentiation space can be obtained by combining the above two measures, i.e., D = D^PCA ⋅ D^GRN, where “⋅” represents the Hadamard product. Furthermore, based on the distances between cells in the differentiation space, the k-nearest neighbor graph of cells can be constructed (Fig 1A (V)), i.e., for each cell, edges are established with its K nearest neighboring cells, where the edge weights correspond to the distances between them. Here, k = 0.1n, where n is the number of cells.

Calculation of pseudotime

Based on the similarity of state changes between cells k and l, i.e., , we define the probability of cell k transferring to another cell (e.g., l) as (8) Thus, the transfer entropy S_k of the cell k can be calculated according to the transition matrix, (9) which is used to measure the uncertainty of disorder in the state transitions of cell k. Generally, cells with higher transfer entropy will transition to cells with lower entropy, e.g., stem cells typically have a more chaotic system state compared to terminal cells [34]. Due to the continuity of this process, the probability of transition and the difference in entropy between the two states should be a decreasing function. Thus, we construct the following function (10) to correct the transition probability from cell k to cell l in the KNN graph. We consider the infinite step transition probability of the cell in graph G, (11) For root cell M₍₀₎, we calculate rough pseudotime [21] RPT_k = ‖M₍₀₎ − M_k‖₂ for other cells, the rough pseudotime RPT will be used as one of the features to construct the cell differentiation trajectory (Fig 1A (VI)).

Trajectory inference

To simplify the computation process, we will calculate the cell differentiation trajectory at the cluster level. Therefore, we need prior knowledge of the cell clusters, which can either be predetermined labels or obtained through clustering algorithms. For each cell cluster Q, we consider the connectivity of all nodes (cells) in the cluster, select the largest connected region in each cluster, and merge the cells in the connected region of a single cell cluster. The merging process follows the following rules. For the combination of nodes and in cell cluster A, node will be removed from graph G and its edges will be spliced onto node . For node C_k, if and , the old edge between and C_k is deleted and the weight of the new edge is (12) If C_k exists edge with only or , then (13) Here we consider that the nodes (cells) with more edges to cluster A should be closer to the merged cluster A. When all the nodes (cells) of the cluster are merged, we get a new relationship graph G′ between the clusters, the edges between the nodes (clusters) are derived from the merged result, and here we modify the edges according to the rough pseudotime. The pseudotime of the cluster is the mean of the rough pseudotime of all cells in the cluster. Here we use tick function , where x represents the difference in pseudotime between clusters and f(x) represents the distance between clusters (Fig 1C). Considering that f(x) takes the minimum value at x = 1, that is, the distance between the two clusters calculated according to the pseudotime distance is the minimum, this is consistent with the continuity of the state transition of the cell. We normalized the calculated cluster pseudotime to the range of 0–10, then the distance D^PT of the pseudotime difference between clusters can be obtained, and the final adjacency matrix of graph G′ between cell clusters can be obtained, (14) and then the transfer relationship between cell clusters can be obtained by calculating the minimum spanning tree for W (Fig 1A (VII)), that is, the differentiation trajectory graph G^MST on the scale of cell clusters. According to G^MST, we delete the wrong edges on graph G to obtain graph G″, and run the minimum spanning tree algorithm on graph G″ to obtain the differentiation trajectory on a single cell scale.

Visualization of cell differentiation trajectories

By following the above steps, we can obtain the differentiation relationships between cell clusters. We utilize the differentiation relationships between clusters to modify the edges of the KNN graph at the single-cell level. We do not change the edges between cells within the same cluster, as cell differentiation is a continuous process and such transitions between cell states in different clusters are reasonable. If there is an edge between two cells but no edge between their corresponding clusters, we will remove the edge between the cells. This way, we can obtain the cell differentiation relationships at the single-cell level.

Based on the eigen decomposition of the matrix, we select the eigenvectors corresponding to the two largest eigenvalues other than 1. We then combine these with the results of UMAP dimensionality reduction to perform a two-dimensional visualization of the cell relationships. The method of combination involves normalizing the positional information corresponding to the eigenvectors and the positional information from UMAP to the same scale, for example, between -1 and 1, and then summing them to obtain the coordinate information for visualizing the cell positions.

The depiction of the differentiation trajectory relies on the information of cell clusters. We select the center of each cell cluster as the point through which the trajectory polyline needs to pass, and we draw the differentiation trajectory polyline based on the obtained differentiation relationships between cell clusters.

Datasets and evaluation metrics

To assess the performance of scGRN-Entropy in inferring cell differentiation trajectories, we selected 14 real datasets from [14], where Saelens et al. developed a benchmark consisting of 110 real and 229 synthetic datasets for trajectory inference methods. Since the 110 real datasets include 8 different types of cell differentiation trajectories, we randomly selected a proportional number of datasets for each trajectory type (S1 Fig). Additionally, The datasets we used include four species (i.e. fly, human, macaque, and mouse) and covered six different sequencing technologies (S1 Table). Each dataset includes both gene expression counts for each cell and the cellular groupings, along with the connections (directed edges) between these groups (e.g., group 1 → group 2 → group 3). These directed edges serve as the ground truth. Accordingly, accuracy, defined as the ratio of correctly predicted edges to the total number of edges, was used as a metric to evaluate predictive performance.

The overview of scGRN-entropy

scGRN-Entropy is a method designed for inferring cell differentiation trajectories and calculating pseudotime from scRNA-seq data, utilizing cell type labels and root cell information. Initially, we apply two dimensionality reduction techniques to the scRNA-seq data. The first method involves gene pooling, where resulting components, termed supergenes, retain biological interpretability while reducing dimensionality and grouping co-expressed genes. The second method is PCA, which determines static relationships between cells based on their gene expression values.

Following gene pooling, we construct a GRN composed of supergenes for each cell. Within the GRN space, we compute cell similarity metrics based on out-degree and in-degree, reflecting dynamic cell similarities. Integrating this dynamic similarity matrix with diffusion pseudotime allows us to infer the pseudotime of cell differentiation.

Additionally, combining dynamic similarity with a static similarity matrix yields a K-Nearest Neighbor (KNN) graph that describes cell differentiation. By merging nodes based on cell cluster information and edge attributes, we derive differentiation relationships between cells at the cluster level.

Finally, we visualize differentiation trajectories and pseudotime results through matrix decomposition and UMAP dimensionality reduction.

Tested on real datasets

We verify the correctness and effectiveness of our method on multiple real datasets [14], which contain 8 types of trajectory (S1 Table). For the acyclic_graph trajectory type dataset, i.e. neonatal-inner-ear-all_burns, our method can establish most of the differentiation relationships between cell clusters (S2 Fig). Although it cannot fully construct the acyclic differentiation trajectory. For the two bifurcation trajectory datasets, i.e. cellbench-SC1_luyitian and distal-lung-epithelium_treutlein (S3 Fig), in the cellbench-SC1_luyitian dataset (Fig 2A and S3A Fig), the trajectory lines based on the centers of cell clusters effectively illustrate the differentiation pathways except the pseudotime coloring is somewhat chaotic. Similarly, in the distal-lung-epithelium_treutlein dataset (Fig 2B and S3B Fig), the pseudotime coloring is more distinct between different clusters. For the convergence trajectory type dataset, i.e. olfactory-projection-neurons-DC3_VA1d_horns (S4 Fig), which has two different starting points, DC3_24hAPF and VA1d_24APF, and a common endpoint, adPN adult, our method’s main error is establishing differentiation relationships between the highly similar DC3_72hAPF and VA1d_72hAPF clusters. For the cycle trajectory dataset, i.e. cell-cycle_buettner (S5 Fig), our method fails to identify the differentiation relationship between G2M and S2, which we attribute to the reliance on the minimum spanning tree algorithm. For the disconnected graph trajectory type datasets, i.e. mouse-cell-atlas-combination-1 (S6A Fig) and mouse-cell-atlas-combination-5 (Fig 2C and S6B Fig), our method can accurately infer the differentiation trajectories. However, the visualization does not perfectly distinguish between the internal cell clusters of the two disconnected parts. We attribute this to the limitations of UMAP and eigendecomposition methods in handling bipartite graphs. For the linear trajectory type datasets, i.e. germline-human-female_li, hematopoiesis-gates_olsson, mESC-differentiation_hayashi, and neonatal-inner-ear-SC-HC_burns (Fig 2D–2G and S7 Fig), our method successfully infers the differentiation trajectories, maintains good trajectory continuity and appropriate distances between cell clusters. For the germline-human-female_li dataset (Fig 2D and S7A Fig), we reconstructed the developmental process of human female fetal germ cells (FGCs). The individual embryo contains several subpopulations simultaneously, highlighting the asynchrony and heterogeneity in the development of FGCs. In the mESC-differentiation_hayashi dataset (Fig 2F and S7C Fig), we replicated the developmental trajectory of mouse embryonic stem cells from 0 hours to 72 hours. We accurately identified cells at different time points and constructed a continuous cell developmental trajectory. Moreover, For the multifurcation trajectory datasets placenta-trophoblast-differentiation_mca (Fig 2H and S8 Fig), we accurately identified three branch lineages, including the complete differentiation process from Progenitor trophoblast Gjb3 to Spongiotrophoblast Phlda2 and then to Spongiotrophoblast Hsd11b2. For the tree trajectory type datasets, i.e. epiblast-monkey_nakamura and hematopoiesis-clusters_olsson (S9 Fig), due to the complexity of classifying cell clusters, it is hard to perfectly infer the cell trajectories at the cluster level.

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Fig 2.

Differentiation trajectory plots for several datasets: (A) cellbench-SC1_luyitian, (B) distal-lung-epithelium_treutlein, (C) mouse-cell-atlas-combination-5, (D) germline-human-female_li, (E) hematopoiesis-gates_olsson, (F) mESC-differentiation_hayashi, (G) neonatal-inner-ear-SC-HC_burns, and (H) placenta-trophoblast-differentiation_mca.

https://doi.org/10.1371/journal.pcbi.1012638.g002

Comparison with existing methods

We compared our method with Slingshot [35], Monocle3 [36], and DBCTI [31] on these 14 datasets (S1 Table). For the 14 datasets, all three methods were run using default script parameters. We used accuracy, defined as the true positive rate of the inferred trajectories relative to the ground truth, to evaluate the performance of all methods (Table 1). The average prediction accuracy of scGRN-Entropy is 84.1%, significantly outperforming Slingshot (78.4%), Monocle3 (58.9%), and DBCTI (48.8%). It should be noted that for 7 of the datasets, Slingshot failed to produce valid inference results, with the R script returning the error “the system computation is singular”. These cases were marked as “None,” and the average accuracy was calculated based on the other 7 datasets where valid predictions were returned. Since scGRN-Entropy, Slingshot, and Monocle3 rely on the minimum spanning tree algorithm for trajectory inference, they cannot accurately predict cycle-type trajectories (e.g., for acyclic graph and cycle types), although they can infer all connections except for those connected to the closed cell groups, such as the cell transition between the G2M and S2 phases in the cell-cycle_buettner dataset (S5 Fig). Accurately predicting complex tree-type trajectories remains challenging for existing methods. For example, in the epiblast-monkey nakamura and hematopoiesis-clusters olsson datasets, the prediction accuracy of scGRN-Entropy, Monocle3, and DBCTI is below 50%. Although scGRN-Entropy achieves an accuracy of 43% on the epiblast-monkey nakamura dataset, significantly higher than the 14% accuracy of the other two methods (Table 1), the inferred trajectory still deviates substantially from the experimental results; see S9 Fig. S2–S9 Figs in the Supplementary Materials also show the comparisons between the trajectories inferred by scGRN-Entropy and Monocle3 and the ground truth across the 14 datasets, suggesting that scGRN-Entropy provides more accurate predictions of cell differentiation trajectories, especially for linear types.

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Table 1. Comparison of scGRN-Entropy with existing state-of-the-art methods in terms of accuracy across 14 real datasets encompassing 8 types of differentiation trajectories.

https://doi.org/10.1371/journal.pcbi.1012638.t001

To demonstrate the importance of incorporating dynamic cell relationships, we also tested the scGRN-Entropy method without considering the dynamic cell relationships obtained from the GRN space (i.e., only using the cell relationship matrix in PCA space, treated as a baseline) on both linear and non-linear trajectory types. As shown in S10 Fig in the Supplementary Materials, for mESC-differentiation_hayashi (linear type), the baseline model predicted an evolution direction opposite to the experiments at 48h and 72h. For distal-lung-epithelium_treutlein (bifurcation type) dataset, the baseline predictions incorrectly inferred the direction of terminal differentiation path, suggesting that BP first differentiates into AT1 and then into AT2. For the multifurcation-type dataset splacenta-trophoblast-differentiation_mca, the predicted progenitor trophoblast Gjb3 cluster directly differentiated into the spongiotrophoblast hsd11b2 cluster, bypassing the spongiotrophoblast phlda2 cluster. In contrast, for all the three datasets, the scGRN-Entropy method accurately predicted the differentiation trajectories, fully aligning with experimental results (i.e., 100% accuracy; see Table 1). Moreover, compared to the baseline results, scGRN-Entropy better distinguished the sequential order of differentiation among individual cells in terms of pseudotime (S10 Fig). This demonstrates that the inclusion of GRN information significantly improved prediction accuracy across both linear and non-linear trajectory types, suggesting that GRNs may serve as an intrinsic driving force of cell differentiation. We solve for the transition probability matrix, transfer entropy, and cell pseudotime through the D^PCA matrix, construct the k-nearest neighbors (KNN) graph for the cells, and obtain the differentiation trajectory using the minimum spanning tree algorithm. It can be observed that for the hematopoiesis-gates olsson dataset, the baseline method does not infer the cell differentiation trajectory well, and the pseudotime values show no distinction, meaning that the differentiation relationships between cells cannot be discovered (S10 Fig).

Case study

We analyzed the germline-human-female li dataset [45], selecting the top 20 most and least correlated genes with pseudotime based on gene expression correlations (Fig 3A). We observed that during the transitional period, the expression of the selected genes in the Female FGC2 cell cluster was highly disordered. Some genes only exhibited high expression at the early and late stages of differentiation. Based on the analysis of the in-degree and out-degree of genes in the gene regulatory network, we found that the pooled supergenes, similar to specifically expressed genes, exhibited significant expression differences across different differentiation stages (Fig 3B).

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Fig 3.

(A) The expression patterns of the 20 genes most positively correlated with pseudotime and the 20 least correlated genes clearly reflect the characteristics of cells at different stages: Female FGC1, FGC2, and FGC3. Genes highly expressed in the early stage (Female FGC1) gradually decrease in expression over time, whereas genes highly expressed in the later stage (Female FGC3) show a gradual increase. During the transitional stage (Female FGC2), all 40 genes display varying levels of expression. (B) The in-degree and out-degree plots for supergenes, shown on the horizontal and vertical axes, respectively, reveal that supergenes highly expressed during the transition period exhibit an increase in in-degree followed by a gradual decline. In contrast, supergenes highly expressed in the early and late stages demonstrate opposite trends in both in-degree and out-degree.

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Furthermore, we performed enrichment analysis on this dataset to explore how accurate trajectory inference contributes to understanding the roles of these genes in cellular signal transduction, metabolism, and other complex biological processes. First, we calculated the correlation between each gene’s expression level across all cells and the inferred pseudotime. Using a significance threshold of 0.05, we selected genes that showed a significant positive or negative correlation with pseudotime. Enrichment analysis was then conducted on these two gene sets separately using EnrichR [53, 54] (Fig 4 and S2–S4 Tables). Among the genes positively correlated with pseudotime, KEGG analysis [55–59] revealed pathways related to the development of female reproductive organs, such as Progesterone-mediated oocyte maturation (Homo sapiens hsa04914), Oocyte meiosis (Homo sapiens hsa04114), Cell cycle (Homo sapiens hsa04110), and FoxO signaling pathway (Homo sapiens hsa04068) (Fig 4A). As shown in Fig 4B, GO analysis of the positively correlated genes identified biological processes related to gonadal development and the cell cycle (with p-values/adjusted p-values < 0.001). These processes include homologous chromosome pairing at meiosis (GO:0007129), piRNA metabolic process (GO:0034587), homologous chromosome segregation (GO:0045143), meiosis I (GO:0007127), meiotic DNA double-strand break formation (GO:0042138), meiotic sister chromatid cohesion (GO:0051177), and ncRNA metabolic process (GO:0034660); see S3 Table in the Supplementary Materials. In the top 10 pathways with the smallest p-values, there are many molecular function pathways related to cell growth and development. For genes negatively correlated with pseudotime, KEGG analysis only identified the Focal adhesion (Homo sapiens hsa04510) pathway as being related to reproductive development, with no statistically significant pathways related to the cell cycle (Fig 4C). GO analysis confirmed significant associations with apoptotic processes, such as the intrinsic apoptotic signaling pathway mediated by p53 (GO:0072332) and the defense response to viruses (GO:0051607); see Fig 4D and S4 Table in the Supplementary Materials. Unfortunately, KEGG analysis for both positively and negatively correlated genes did not get small enough p-value, which may be due to the relatively small number of genes selected (< 100). The results shown in the figure were exported from R, and the specific data can be found in S2–S5 Tables.

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Fig 4.

(A) KEGG enrichment analysis results of genes that were positively correlated with pseudotime, using the enrichment dataset “KEGG 2016”. (B) GO enrichment analysis results of genes that were positively correlated with pseudotime, using the enrichment dataset “GO Biological Process 2021”. (C) KEGG enrichment analysis results of genes that were negatively correlated with pseudotime, using the enrichment dataset “KEGG 2016”. (D) GO enrichment analysis results of genes that were negatively correlated with pseudotime, using the enrichment dataset “GO Biological Process 2021”.

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Validation of GRN inference methods on simulated data

We used the Gillespie algorithm to simulate the expression control of five genes in the network to simulate the up-regulation and down-regulation of different gene expressions in the process of cell differentiation (S11 Fig). The stoichiometric matrix and propensity function of the chemical reaction (Table 2) are as follows [22]. Where x_i represents the expression of Genei. The initial values we selected for x₁, x₂ and x₃ are sampled from uniform distributed between 40 and 80, and the initial values for x₄ and x₅ are 0. Based on our solution of the GRN, we found that over time, the gene regulation within the cell gradually stabilizes. Additionally, the changes in gene regulation are very drastic in the initial stage (Fig 5), which aligns with the fact that the cell’s state is relatively chaotic during the early stages of differentiation.

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Fig 5. Evolution of gene-gene regulatory strength over time, showing that the gene regulatory network underwent significant changes initially.

After a certain period, only Gene2 continued to exhibit slight regulatory influence on other genes until it eventually stabilized.

https://doi.org/10.1371/journal.pcbi.1012638.g005

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Table 2. Chemical reactions and their propensity functions.

https://doi.org/10.1371/journal.pcbi.1012638.t002