Trajectory-informed gene feature selection in single-cell analysis with SEEK-VFI

Summary

The prioritization of highly variable genes is an important step in single-cell trajectory inference. However, when variability arises from a continuous latent cell development trajectory, standard methods may fail to differentiate trajectory-relevant genes from uninformative genes. SEEK-VFI (Spectral Ensembling of topic models with Eigenscore for K-agnostic Variable Feature Identification) is an ensemble topic modeling machine learning algorithm for trajectory inference preprocessing that prioritizes trajectory-relevant genes. It outperforms existing methods and identifies key genes that improve trajectory topology reconstruction, enhance visualization, and augment downstream trajectory analyses.

Graphical abstract

Highlights

• •Feature selection method for single-cell RNA expression analysis
• •Topic modeling and ensemble learning to estimate low-dimensional structure
• •Prioritizes trajectory-relevant genes for downstream analysis
• •Improves pseudotime inference and cellular embedding over existing methods

Motivation

Understanding cellular developmental trajectories is a key goal in the analysis of single-cell RNA sequencing (scRNA-seq) data. Trajectory inference methods typically require a feature selection preprocessing step that ideally filters to the genes with relevance to the underlying trajectory. However, standard feature selection methods are designed to identify genes that vary across discrete cell clusters. As such, these methods may fail to capture genes relevant to the underlying continuous trajectory. We introduce a method for scRNA feature selection that captures trajectory-relevant genes without needing to explicitly model the underlying cellular trajectory.

Danning et al. introduce a feature selection method for single-cell expression analysis that combines statistical and ensemble learning techniques to identify genes that vary across the latent cellular trajectory. This preprocessing method improves downstream tasks including trajectory reconstruction, low-dimensional embedding, and pseudotime analysis.

Introduction

Single-cell trajectory inference aims to understand the developmental structure underlying cell differentiation based on cell-level gene expression profiles.¹ Trajectory inference methods typically represent the trajectory as a graph comprising developmental milestones (nodes) and transitions (edges) between adjacent milestones.² Feature selection is a critical preprocessing step in trajectory inference and is typically done by choosing genes with the greatest variability in expression across cells.³ A fundamental assumption of trajectory inference is that cells lie along continuous, latent transitions among cellular states that give rise to variation in gene expression.⁴ However, standard feature selection methods^5,6,7 do not exploit the underlying low-dimensional, smooth structure of the expression data. Instead, they operate directly on the high-dimensional normalized expression matrices. These approaches are primarily designed to select genes that are differentially expressed across discrete cell clusters, such as distinct cell types.^3,8 Consequently, when applied to trajectory inference, existing feature selection methods often fail to capture genes that play key roles in developmental dynamics, particularly those whose expression changes gradually along continuous differentiation trajectories rather than sharply between clusters.

To address this limitation, here we introduce SEEK-VFI (Spectral Ensembling of topic models with Eigenscore for K-agnostic Variable Feature Identification), an ensemble machine learning method for prioritizing genes that are highly variable with respect to latent topological structure. Building on topic modeling, SEEK-VFI enables efficient detection of trajectory-associated features through its flexible representation of mixed cellular memberships along continuous developmental trajectories. SEEK-VFI is a preprocessing method and does not assume the availability of a cell embedding representing the complete trajectory; rather, it leverages the latent structure underlying the trajectory indirectly to inform feature prioritization. An overview of topic modeling for trajectory inference and the SEEK-VFI workflow is shown in Figure 1.

Figure 1.

SEEK-VFI workflow for prioritizing trajectory-relevant genes

(A) Topic modeling decomposes the gene × cell expression matrix into two low-rank matrices defined by the latent trajectory structure.

(B) The milestone × cell matrix describes the position of each cell with respect to its trajectory segment. In the diagram, cell A lies halfway between milestones 1 and 2, whereas cell B lies closer to milestone 4 than to milestone 2.

(C) The gene × milestone matrix describes gene expression patterns with respect to location on the trajectory. Gene A has high expression across the entire trajectory, whereas gene B has high expression at milestone 4 and low expression otherwise.

(D) The gene × milestone matrix can be normalized by dividing each row by its ${\mathcal{l}}^1$-norm. The normalized row corresponding to gene A is uniform, whereas the normalized row corresponding to gene B is spikier; gene B is therefore the gene that is more relevant to trajectory structure. A candidate trajectory relevance score measuring spikiness is calculated for each gene by taking the squared ${\mathcal{l}}^2$-norm of each normalized row (see STAR Methods).

(E) Topic modeling requires the dimension of the latent trajectory space K. Since the underlying trajectory structure is unknown, we require a method that ensembles the spikiness vectors from a variety of candidate trajectories.

(F) SEEK-VFI combines the candidate trajectory relevance vectors from a range of topic models, using a spectral ensembling method that yields a robust and trajectory-agnostic estimator of gene relevance.

(G) Feature selection with SEEK-VFI enables downstream tasks such as trajectory reconstruction, cell embedding, and pseudotime analysis.

Results

Topic modeling assumes a low-rank structure underlying the expression matrix⁹ and seeks to decompose the gene × cell matrix into gene × milestone and milestone × cell matrix (Figure 1A). The milestone × cell matrix captures the position of each cell along the trajectory (Figure 1B), whereas the gene × milestone matrix captures the expected expression level of each gene at each milestone (Figure 1C). Intuitively, if gene A shows relatively uniform expression across all milestones along the trajectory, while gene B is highly expressed only at a specific milestone, the variation in gene B’s expression across cells (“spikiness”) would be more biologically meaningful and informative of the underlying developmental process (Figure 1C).

To quantify the biological relevance of each gene with respect to the underlying trajectory, we compute the spikiness of each gene from the estimated gene × milestone matrix. The spikiness metric is designed to highlight genes whose expression profiles across milestones contain one or two dominant loadings and much smaller values in the remaining components (STAR Methods). Genes with high spikiness vary more significantly along the trajectory and are prioritized for downstream analysis, while genes with a low spikiness exhibit relatively uniform expression along the trajectory and are less informative. These measures are collected into a trajectory relevance vector (TRV) (Figure 1D).

SEEK-VFI uses Topic-SCORE,⁹ an efficient method based in singular value decomposition (SVD), to derive the gene × milestone matrices. Topic modeling algorithms require the specification of K, the number of milestones; however, this is often unknown. Thus, the first step of SEEK-VFI is to run Topic-SCORE for a range of plausible values of K (Figure 1E). For each candidate value of K, we can compute a TRV that summarizes each gene’s importance with respect to that milestone model. These vectors are then aggregated using an optimal spectral ensemble method^10,11 (Figure 1F). The output of SEEK-VFI is a consensus trajectory relevance score for each gene. These scores can be used to rank the genes and guide feature selection in downstream analysis (Figure 1G).

To benchmark SEEK-VFI’s ability to distinguish trajectory-relevant genes from uninformative ones, we first test it on simulated trajectory data and compare it with the three existing methods available in Seurat¹² and the recent method DELVE.¹³ The dispersion method (“Disp”) directly calculates the dispersion of each gene by computing the ratio of the sample variance to the sample mean with respect to the log-transformed expression counts.³ The scaled dispersion method (“ScDisp”) normalizes the dispersion scores with respect to bins of cells with similar average expression.⁶ The variance-stabilizing method (VST) transforms the counts into Z scores using a variance-stabilizing transformation and then calculates the variance with respect to these (potentially truncated) Z scores to assess the dispersion of the expression counts, while also controlling for the average expression of each gene.⁷ DELVE is a recently developed method for unsupervised feature selection in molecular data that is designed to capture features most relevant to the underlying cellular trajectory through the identification of modules of features that vary significantly across inferred cell states.¹³ Both SEEK-VFI and DELVE model the underlying trajectory structure prior to feature ranking; while SEEK-VFI uses SVD-based methods to model the latent space, DELVE models the cellular trajectory via a k-nearest neighbor affinity graph where the edge weight between cells is a function of similarity across all observed features.

The simulated data are generated such that 10% of the genes have expression levels related to the milestone structure (trajectory-relevant) and 90% of genes have constant expression levels across the trajectory (uninformative). We vary the signal strength parameter α, which dictates how much the expression levels of the trajectory-relevant genes vary across milestones; further details are provided in STAR Methods. Figure 2A shows the area under the curve (AUC) for SEEK-VFI and the alternative methods in identifying the trajectory-relevant genes across the settings. We found that SEEK-VFI consistently outperformed the alternative methods, particularly with an increased sample size (Figure S2A). While DELVE performs comparably to the Seurat methods, analysis of a single dataset containing only thousands of cells takes hours (Figure S3B). Notably, SEEK-VFI is the most robust to different trajectory topological structures, which can be seen clearly in Figure S2B. The SEEK-VFI results are robust to the choice of the upper limit for ensembling (Figure S2C).

Figure 2.

Comparison with existing methods in simulated and real datasets

(A) Performance of SEEK-VFI and alternative methods on simulated scRNA-seq data with 5,000 cells and 10,000 genes. The x axis varies the value of α, which controls how similar the trajectory-relevant genes are to the uninformative genes. The y axis shows the median AUC for differentiating between informative and uninformative genes across the simulations. Error bars represent the 25^th and 75^th percentiles. Results from expanded settings are shown in Figure S2.

(B) Bar plots showing the mean squared error comparing the true cell-to-cell trajectory distance matrices (see STAR Methods) with the distance matrices derived from the normalized expression of the top X% genes as selected by each of the five methods. Results for additional datasets and values of X are shown in Figure S4A.

(C) Low-dimensional embeddings using the top 500 genes as selected by the given method. The top row shows the ground-truth trajectory. Left: bifurcation of T cells in the thymus, comparing the uniform manifold approximation and projection (UMAP) embeddings from SEEK-VFI and VST. Middle: linear trajectory of oligodendrocytes, comparing the t-SNE embeddings from SEEK-VFI and Disp. Right: reprogramming of fibroblast cells, comparing the principal component analysis (PCA) embeddings from SEEK-VFI and ScDisp. Quantitative metrics assessing the visualization quality are shown in Table S1. Results from an expanded set of visualization settings can be found in Figure S6.

(D) Performance of downstream trajectory analysis tools using the top 500 genes from SEEK-VFI and the Seurat methods, and Venn diagrams showing the overlap of the gene sets. An example of the relationship between the low-dimensional embeddings and the downstream analysis is shown in Figure S5.

(E) Runtimes of the non-DELVE methods. SEEK-X ensembles the spikiness vectors from topic models with K = {3, …,X} milestones. Runtimes for the Seurat methods do not include normalization preprocessing. Runtimes for DELVE in the 5,000-cell setting are shown in Figure S3B.

We test these methods on five real single-cell RNA sequencing (scRNA-seq) datasets covering diverse developmental processes, comparing their performance in prioritizing the genes that best capture the underlying true trajectory (Figures 2B and S4A). The results for DELVE here are shown using the default hyperparameters; varying the hyperparameters did not significantly change the performance (Figure S4B). As in the simulated data, the analysis of the real data took between 20 min and 3 h, whereas the remaining methods all took less than 5 min. Thus, we chose to exclude DELVE from further analysis. Across all datasets and various gene selection cutoffs, SEEK-VFI robustly achieves the overall lowest reconstruction error of the latent trajectory. This superiority is further reflected in the low-dimensional embeddings of the cells, using the top 500 genes selected by each method (Figure 2C). These embeddings more clearly reflect the ground-truth trajectory structure and lead to improved performance in downstream tasks such as pseudotime inference and cell state identification. An example of the relationship between the embeddings and pseudotime analysis is demonstrated in Figure S5. For each of three popular trajectory inference methods (Monocle3,¹⁴ Slingshot,¹⁵ and Totem¹⁶), we restrict the input to the expression profiles of the top 500 genes selected by each feature selection method and provide a low-dimensional visualization as input where possible. We then evaluate (1) the correlation of the derived pseudotime with the ground-truth pseudotime and (2) the concordance between the inferred milestone structure and the true cell states (see STAR Methods). As shown in Figure 2D, SEEK-VFI achieved the best overall performance across all datasets and trajectory inference algorithms. It performed comparably to the existing feature selection methods on datasets with already strong baseline performance (e.g., kidney), while showing substantial improvement on more challenging datasets (e.g., fibroblast and thymus). Notably, there is smaller overlap between the genes prioritized by SEEK-VFI and the existing methods, as compared with the overlap among the existing methods themselves, suggesting the unique strength of SEEK-VFI in detecting distinct type of expression patterns commonly overlooked by conventional approaches (Figure 2D).

The runtime of SEEK-VFI depends on dataset size and the number of candidate milestone models included in the ensemble (Figure 2E). Although the conventional methods that do not require model estimation are fastest, the additional computational cost of SEEK-VFI is modest. For comparison, the other trajectory-aware algorithm DELVE took nearly 3 h to analyze datasets with 5K cells and 10K genes (Figure S3B). In our experiments, even for datasets containing 50K cells and over 20K genes, with as many as 10 candidate models and up to 12 milestones, SEEK-VFI completes within minutes on a standard laptop (MacBook Pro, 6-core Intel i7).

Discussion

In sum, SEEK-VFI is a powerful method for detecting trajectory-associated genes from single-cell transcriptomics data. We show through simulations that SEEK-VFI outperforms conventional feature selection methods at differentiating between trajectory-relevant and uninformative genes across a range of dataset sizes and cell trajectory structures. SEEK-VFI prioritizes trajectory-relevant genes in real scRNA-seq datasets across a variety of biological processes and trajectory structures, improving both trajectory embedding and pseudotime estimation.

Limitations of the study

While SEEK-VFI shows promise for improving trajectory inference, there are some limitations to consider. First, the computational cost of the underlying Topic-SCORE algorithm increases exponentially with K (Figure S3A). Second, the underlying Topic-SCORE algorithm utilizes k means, which may occasionally suffer from instability in convergence. Third, the researcher must independently determine the range of candidate values of K. Finally, as with traditional feature selection methods, SEEK-VFI does not provide a guaranteed cutoff for the number of genes to select.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Rong Ma (rongma@hsph.harvard.edu).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •This paper analyzes existing, publicly available data, accessible at Zenodo: zenodo.org/records/1443566¹⁷ as of the date of publication.
• •All original code has been deposited at Zenodo and is publicly available at Zenodo: https://doi.org/10.5281/zenodo.20475377 as of the date of publication. SEEK-VFI is also available to download as an R package at github.com/rdanning/seekvfi.
• •Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

R.D.’s work was supported by NIH grant T32-GM135117 while at the Harvard TH Chan School of Public Health. Z.T.K.’s work is supported by the Sloan Research grant FG-2023-19970. X.L.’s work is supported by NIH grants R35-CA197449, R01-HL163560, and U01-HG012064.

Author contributions

R.D., Z.T.K., X.L., and R.M. designed the methodology; R.D. and R.M. wrote the software code; R.D. designed and performed all computational analyses; R.D. generated all manuscript figures; R.D. and R.M. wrote the manuscript; X.L. and R.M. supervised the study; R.D., Z.T.K., X.L., and R.M. revised the manuscript.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Fibroblast dataset	Treutlein et al.18	GSE67310; Zenodo: zenodo.org/records/1443566
Kidney dataset	Park et al.19	GSE107585; Zenodo: zenodo.org/records/1443566
Oligodendrocyte dataset	Marques et al.20	GSE75330; Zenodo: zenodo.org/records/1443566
Thymus dataset	Han et al.21	GSE108097; Zenodo: zenodo.org/records/1443566
Software and algorithms
SEEK-VFI	This paper	Zenodo: https://doi.org/10.5281/zenodo.20475377; https://github.com/rdanning/seekvfi
Seurat (v4)	Hao et al.12	https://satijalab.org/seurat/
DELVE	Ranek et al.13	https://github.com/jranek/delve

Method details

Trajectory inference takes as its input a matrix $\mathbf{X}\in {\mathbb{R}}^{p\times n}$ of n cells and p genes, where X_ij is cell j’s expression of gene i at a particular point in time. In the following section, we will describe SEEK-VFI in more detail, including how we apply the topic modeling framework to the trajectory inference context.

Topic modeling and Topic-SCORE

Topic modeling is a statistical method from natural language processing that aims to find low-rank latent structure (“topics”) that explain the counts of vocabulary words in a collection of documents: vocabulary words have varying likelihoods of appearing across topics, and documents comprise a mixture of topics.²² The observed matrix $\mathbf{X}\in {\mathbb{R}}^{p\times n}$ summarizes the counts of p vocabulary words across n documents. Given a document i containing n_i words, the i^th column of X is distributed as $X_i\sim Multinomial\left(n_i,d_i^{\ast }\right)$, where $d_i^{\ast }$ is the i^th column of D∗.⁹ The matrix D∗ satisfies $\mathbb{E}\left[{\mathbf{D}}^{\ast }\right]=\mathbf{\text{AW}}$, where $\mathbf{A}\in {\mathbb{R}}^{p\times K}$ is a non-negative topic matrix, and $\mathbf{W}\in {\mathbb{R}}^{K\times n}$ is a non-negative topic weight matrix. The number of topics, K, is the dimension of the latent space. Each of the K columns of A is a probability mass function describing the relative likelihood of each word in the vocabulary with respect to that topic. Each of the n columns of W describes the mixture of topics within a particular document. By definition, $d_i^{\ast }$ is a convex combination of the columns of A weighted and also a probability mass function. The observed corpus matrix D is the normalized version of X such that each of the columns of D sum to 1.

In the context of trajectory inference, we treat the genes as the vocabulary, the cells as the documents, and the expression levels in the corpus matrix as the word counts. The topic matrix, then, is the matrix describing the relationship between the genes and the topics, which we posit correspond to the latent milestones.

Several methods have been proposed to recover the topic matrix from the corpus matrix, including latent Dirichlet allocation,²³ non-negative matrix factorization,²⁴ and expectation–maximization algorithm-based methods.²⁵ Topic-SCORE⁹ is a state-of-the-art and computationally efficient method that uses singular value decomposition and simplex geometry to estimate the low-rank A from the observed X. There are five main steps to Topic-SCORE.

• 1.Pre-SVD normalization. Let D′ = M^−1/2D, where M is a diagonal matrix such that M_jj is proportional to the j^th vocabulary word’s frequency in the corpus matrix.
• 2.Singular value decomposition. Calculate Ξ = [ξ₁ … ξ_K], the K left singular vectors of D′. The rows of Ξ are contained in a simplicial cone with K supporting rays. Anchor words (words with nonzero probability on only one topic) lie on the corresponding supporting ray.
• 3.Post-SVD normalization. Perform SCORE normalization²⁶ on Ξ: normalize each row of Ξ by its first component, yielding $\mathbf{R}\in {\mathbb{R}}^{p\times K-1}$ and projecting the K-dimensional simplicial cone onto a (K-1)-dimensional simplex. Row j of R is the embedding of word j into ${\mathbb{R}}^{K-1}$.
• 4.Vertex hunting with SVS. Use Sketched Vertex Search (SVS)²⁷ to find the vertices of the (K-1)-dimensional simplex. Let ${\hat{z}}_1,\dots ,{\hat{z}}_K$ be the vertices found by SVS: these represent the locations of each of the K topics in the simplex space. Points within the simplex represent a mixture of topics, and any points outside the simplex (due to noise in the estimation) can be normalized to their nearest point within the bounds of the simplex.
• 5.Topic matrix estimation. Let ${\hat{r}}_j$ be the j^th row of R, which corresponds to the location of word j in the (K-1)-dimensional simplex. For each vocabulary word, let ${\hat{\pi }}_j$ be the coefficients of the convex combination of ${\hat{z}}_1,\dots ,{\hat{z}}_K$ that yields ${\hat{r}}_j$. Write $\hat{\mathrm{\Pi }}={\left[{\hat{\pi }}_1,\dots ,{\hat{\pi }}_p\right]}^{\prime }$. The estimate of A is ${\mathbf{M}}^{-1/2}\left[\text{diag}\left({\xi }_1\right)\right]\hat{\mathrm{\Pi }}$.

The topic matrix describes the relationship between the words (genes) and topics (milestones). However, words that are high in frequency across all topics are not helpful for topic interpretation or detection. Therefore, the topic matrix is next converted into a loadings matrix by dividing each row by its ${\mathcal{l}}^1$-norm. Words corresponding to spiky rows in the loadings matrix—those with one large loading and much smaller values in the remaining components—are more strongly indicative of the topic structure; similarly, genes corresponding to spiky rows in the loadings matrix are more strongly indicative of trajectory- or milestone-relevance. To quantify such trajectory-relevance for each gene, we define a spikiness metric by taking the squared ${\mathcal{l}}^2$-norm of each row of the loadings matrix. A justification of this metric follows in the next subsection. The spikiness measures are collected into the trajectory relevance vector (TRV).

Crucially, topic modeling requires specifying the number of topics K, which cannot be reliably learned from the data. In the trajectory inference, this is equivalent to needing to specify the number of milestones, which is usually unknown. Therefore, to avoid having to select a single number of milestones, we ensemble the TRVs from a range of candidate milestone models using a spectral ensembling procedure. This spectral ensembling method is adapted from the spectral meta-learner of Parisi et al.¹⁰ and the eigenscore method of Ma et al.¹¹

Let $\mathcal{K}=\left\{K_1,\dots K_{\left|\mathcal{K}\right|}\right\}$ be the candidate values of K, the number of milestones and the number of topics for the topic model. Let L_i be the ${\mathbb{R}}^{p\times 1}$ candidate TRV resulting from running the topic model with K=K_i. We compute a $\left|\mathcal{K}\right|\times \left|\mathcal{K}\right|$ similarity matrix G where G_i,j = cov(L_i,L_j). Let $u\in {\mathbb{R}}^{\left|\mathcal{K}\right|\times 1}$ be the leading eigenvector of G. The output of SEEK-VFI is the vector

$$\widehat{\mathbf{\text{TRV}}}=\sum _{i=1}^{\mathcal{K}}\frac{\left|u_i\right|}{u}{L}_i\text{.}$$

This vector contains the ensembled trajectory relevance score of each gene, which can be used to rank the genes for feature selection and downstream analysis. Figures S1C and S1D compare the spikiness metrics from component topic models and the TRV scores to principal component analysis and demonstrates the advantage of this approach compared to standard PCA-based methods.

Spikiness metric

Here we will provide a formal argument justifying the proposed spikiness metric. Let the $\mathbf{B}\in {\mathbb{R}}^{p\times K}$ be the row-normalized gene × milestone matrix, or loadings matrix. By construction, any row b_i = [b_i,1, …,b_i,K] is such that b_i,j≥0 and ${{\mathbf{b}}_i}_1=1$, or ${\sum }_{j=1}^K{b}_{i,j}=1$. The proposed spikiness measure is ${{\mathbf{b}}_i}_2^2={\sum }_{j=1}^K{b}_{i,j}^2$. We can set up a Lagrangian with $f\left({\mathbf{b}}_i\right)={\sum }_{j=1}^K{b}_{i,j}^2$ and $g\left({\mathbf{b}}_i\right)={\sum }_{j=1}^K{b}_{i,j}-1=0$:

$$\begin{array}{cc}\mathcal{L}\left({\mathbf{b}}_i,\lambda \right)& =f\left({\mathbf{b}}_i\right)+\lambda ·g\left({\mathbf{b}}_i\right)\\ & =\sum _{j=1}^K{b}_{i,j}^2+\sum _{j=1}^K\lambda b_{i,j}-\lambda \end{array}$$

Next, we calculate the gradient:

$$\begin{array}{cc}{\nabla }_{{\mathbf{b}}_i,\lambda }\mathcal{L}\left({\mathbf{b}}_i,\lambda \right)& =\left(\frac{\partial \mathcal{L}}{\partial b_{i,1}},\dots ,\frac{\partial \mathcal{L}}{\partial b_{i,K}},\frac{\partial \mathcal{L}}{\partial \lambda }\right)\\ & =\left(2b_{i,1}+\lambda ,\dots ,2b_{i,K}+\lambda ,\sum _{j=1}^K{b}_{i,j}-1\right)\end{array}$$

Therefore, we have:

$${\nabla }_{{\mathbf{b}}_i,\lambda }\mathcal{L}\left({\mathbf{b}}_i,\lambda \right)=0\iff \{\begin{array}{l}2b_{i,1}+\lambda =0\\ \dots \\ 2b_{i,K}+\lambda =0\\ \sum _{j=1}^K{b}_{i,j}=1\end{array}$$

These equations are satisfied (and therefore maxima/minima are achieved) when either of the following is true.

• 1.λ = 0, b_i,j = 1 for some 1 ≤ j ≤ K, and b_i,l = 0 for all l≠j
• 2.$\lambda =-\frac{2}{K}$ and $b_{i,j}=\frac{1}{K}$ for all 1 ≤ j ≤ K

Under case 1, which corresponds to the highest possible spikiness setting where a gene is only expressed at one milestone, f evaluates to 1. Under case 2, which corresponds to the lowest spikiness setting where a gene is uniformly expressed across milestones, f evaluates to $\frac{1}{K}$. Thus, as desired, our spikiness metric is maximized in the highest-spikiness setting and minimized in the lowest-spikiness setting.

Figures S1A and S1B provides an intuitive visual explanation of the metric. These plots show the relationship between the spikiness of a vector v_i, with ${{\mathbf{v}}_i}_1=1$, and its squared ${\mathcal{l}}^2$-norm ${{\mathbf{v}}_i}_2^2$ in the K = 2 (Figure S1A) and K = 3 (Figure S1B) cases.

Quantification and statistical analysis

Simulation details

We tested the performance of SEEK-VFI, along with DELVE and the three comparator methods from Seurat, on simulated single cell data following six of the basic trajectories described by Saelens et al.² and shown in Figure 2A. The nodes of each trajectory graph represent milestones of cellular development, with individual cells lying on the edges representing transitions between the milestones. We consider two types of genes: trajectory-relevant genes have average expression levels related to the cell’s position in the trajectory (e.g., gene B in Figure 1C), and uninformative genes have constant average expression levels across all locations in the trajectory (e.g., gene A in Figure 1C). Each dataset simulates the expression of 10,000 genes total, with 1,000 trajectory-relevant genes and 9,000 uninformative genes. Each setting comprises 50 simulated datasets.

Let G=(V,E) denote the trajectory graph, where v₁, …,v_K are the milestones of the graph and edge e_i,j connects v_i and v_j. For a gene g and milestone i∈{1, …,K}, p_g(i) is the gene expression probability for a cell at that milestone.

For trajectory-relevant genes, the expression probabilities for a gene g are simulated as follows.

• 1.Let C be the distance matrix corresponding to the underlying trajectory: C_ij is the length of the shortest path between milestone i and milestone j.
• 2.Define $\mathbf{\Sigma }={\left(\frac{1}{\mathbf{C}+1}\right)}^{\alpha }$.
• 3.Draw g₀ ∼ MVN(0,Σ).
• 4.Transform the multivariate normal random vector to a Uniform[0,1] vector via g₁ = Φ(g₀), where Φ is the normal CDF.
• 5.Linearly rescale g₁ to g ∼ Uniform[0.2,0.8]; this ensures that the resulting cell counts are reasonable.
• 6.The expression probability for gene g with respect to milestone i is p_g(i) = g_i, the i^th entry of g. The vector g is analogous to the g^th row of the topic matrix in the standatd topic modeling framework.

In other words, for trajectory-relevant genes, the expression probability is indexed by the milestone, and the probabilities are correlated with respect to the distance between nodes in the underlying trajectory.

For uninformative genes, the expression probability is constant across nodes: p_g(i)∼Uniform[0.2,0.8] and p_g(i) = p_g(j) for all milestones i and j.

Suppose a cell c is d% along the transition from milestone v_i to milestone v_j: c lies on e_i,j, its scaled distance from v_i is d, and its scaled distance from v_j is 1-d. For a gene g, the gene expression probability for that cell is p_c,g = d·p_g(i)+(1-d)·p_g(j). Draw a value r ∼ N(2,0.5), and let r_g = max(r,0.1) to yield a gene-specific baseline expression level. Then the expression of gene g for cell c is drawn from NegBinomial(r_g,p_c,g).

For each setting (trajectory type, number of cells, and α), we generate 50 datasets representing simulated gene expression from the given number of cells.

Real-data analysis details

The real datasets analyzed in this paper are.

• •Fibroblast: bifurcation trajectory representing the reprogramming of mouse embryonic fibroblasts into myocytes and neurons¹⁸
• •Kidney: linear trajectory of kidney duct cells through a transitional cell type¹⁹
• •Oligodendrocytes: linear progression of oligodendrocyte precursor cells to mature oligodendrocytes²⁰
• •Thymus: bifurcation trajectory of t-cells in the thymus²¹

For the analysis in Figure 1B, the ground-truth distance matrices were derived using the cell labels and milestone networks provided in the data.¹⁷ The ground-truth distance between two cells is defined as the length of shortest path between the two cell states in the milestone network.

We used the default settings for UMAP and t-SNE to generate the low-dimensional visualizations in Figure 1C.

For analysis with Monocle3, Slingshot, and Totem, we provided as input data the counts matrix restricted to the top 500 genes from each method and the UMAP embedding derived from the top 500 genes as the reduced dimensions object. We ran Slingshot via dynverse.² For all methods, we provided a start cell by selecting the median point within the appropriate cell state with respect to the low-dimensional embedding.

To evaluate the performance of the downstream analysis methods, we consider a) the estimated pseudotime for each cell and b) the estimated milestone network. To evaluate the milestone network, we assign each cell to a cluster based on its nearest milestone and compare it to the ground-truth cell state using the normalized mutual information (Figures 2D, X-axis). To evaluate the pseudotime, we calculate its correlation to the pseudotime derived from the ground-truth milestone network (Figure 2D, y axis). The ground-truth milestone network pseudotime is calculated as follows:

Let G = {V,E} be the directed graph corresponding to the ground truth milestone network. Let v₀ ∈ V be the root of the graph and let V∗ ⊂ V be the set of leaf nodes: indegree(v₀) = 0 and outdegree(v∗) = 0 for v∗ ∈ V∗. For each v∗ ∈ V∗, let G[v∗] = {V[v∗],E[v∗]} be the subgraph of G corresponding to the shortest path between v₀ and v∗. For v ∈ V[v∗], define $p{t}_{v^{\ast }}\left(v\right)=dis{t}_{\mathbf{G}\left[v^{\ast }\right]}\left(v_0,v\right)$, the distance between the root node and the node of interest with respect to the subgraph. If v is not in G[v∗], we set the value as null and remove it prior to the final pseudotime calculation. The ground-truth milestone network pseudotime for a node v ∈ V is $\frac{1}{\left|{\mathbf{V}}^{\ast }\right|}{\sum }_{v^{\ast }\in {\mathbf{V}}^{\ast }}p{t}_{v^{\ast }}\left(v\right)$, or the average pseudotime across all subgraphs.

Hyperparameter selection and computational cost

A key hyperparameter for SEEK-VFI is the choice of candidate milestone counts K to ensemble. Underspecified models (chosen K is smaller than the truth) capture a subset of the signal, while overspecified models (chosen K is greater than the truth) result in signal + noise.⁹ The ensembling method is highly robust to the inclusion of noise,¹¹ and in simulations experiences minimal degradation in quality when additional overspecified models are included. However, there is a tradeoff regarding computational time: while Topic-SCORE is much faster than other topic modeling methods,⁹ its most computationally intensive step (Sketched Vertex Search for vertex hunting) requires iterating through $\left(\begin{array}{c}1.5\times K\\ K\end{array}\right)$ combinations of candidate vertices, which accumulates exponentially as K increases (Figure S3A). While the simulations and real-data results in this paper were run using SEEK-12, since the computational time is very reasonable given the data size, the results in Figures S2C and S4C show that the results for SEEK-6 are quite similar to those in the main figure, and that reduced upper ensembling limits can be chosen if faster computational time is desired.

DELVE requires three hyperparameters: k (number of nearest neighbors for the between-cell affinity graph), m (number of subsampled neighborhoods), and c (number of clusters/modules of features). In our simulations and real-data results in the main figure, we used the default DELVE hyperparameters indicated in the DELVE benchmarking analysis (k = 10, m = 1000, c = 5; github.com/jranek/delve_benchmark).

Published: July 27, 2026