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Topological velocity inference from spatial transcriptomic data

Nature Biotechnology (2025)Cite this article

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Abstract

Incorporating space and time into models of cell fate transition will be a key step toward characterizing how interactions among neighboring cells, local niche factors and cell migration contribute to tissue development. Here we propose Topological Velocity Inference (TopoVelo), a model for jointly inferring spatial and temporal dynamics of cell fate transition from spatial transcriptomic data. TopoVelo extends the RNA velocity framework to model single-cell gene expression dynamics of an entire tissue with spatially coupled differential equations. TopoVelo estimates cell velocity from developing mouse cerebral cortex, learns interpretable spatial cell state dependencies that correlate with the expression of ligand–receptor genes and reveals spatial signatures of mouse neural tube closure. Finally, we generate Slide-seq data from an in vitro model of human development and use TopoVelo to study the spatial patterns of early differentiation. Our work introduces a new dimension into the study of cell fate transitions and lays a foundation for modeling the collective dynamics of cells comprising an entire tissue.

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Fig. 1: Diagram of TopoVelo model.
Fig. 2: Incorporating spatial position improves velocity inference from spatial transcriptomic data.
Fig. 3: Cell velocity from TopoVelo on Slide-tags mouse brain captures spatial patterns of embryonic brain development.
Fig. 4: TopoVelo analysis of mouse neural tube closure.
Fig. 5: TopoVelo analysis of human EBs.

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Data availability

Raw and processed Curio Seeker data from human embryoid bodies are available from the Gene Expression Omnibus (GEO) under accession code GSE291200. The Slide-tags mouse brain data are from previous work by Russell et al.12, and raw data can be accessed via GEO under accession code GSE244355. The Slide-seq E9 3D mouse embryo data are from previous work by Kumar et al.49, and raw data can be accessed via GEO under accession code GSE197353. Metadata can be downloaded from CELLxGENE (https://cellxgene.cziscience.com/collections/d74b6979-efba-47cd-990a-9d80ccf29055) and figshare at https://doi.org/10.6084/m9.figshare.21695879.v1 (ref. 70). Stereo-seq data are available from the China National GeneBank: https://db.cngb.org/stomics/mosta/. Human thymus Visium data were downloaded from the European Nucleotide Archive (accession code PRJEB77091). We also uploaded all AnnData files with preprocessed count matrices (https://doi.org/10.6084/m9.figshare.28516139.v2) (ref. 71) and with TopoVelo results (https://doi.org/10.6084/m9.figshare.28516184.v1) (ref. 72) to figshare.

Code availability

The source code is available via GitHub at https://github.com/welch-lab/TopoVelo (ref. 73).

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Acknowledgements

This work was supported by the University of Michigan Rackham Predoctoral Fellowship to J.L., Ruth L. Kirschstein Predoctoral Fellowship F31AI177258 to C.L., and National Institutes of Health grants R01 HG010883 and UM1 MH130966 to J.D.W. We thank M. Karikomi and C. Gao for helpful discussions. We thank the University of Michigan Genomics Core for their help in generating the human embryoid data. We also thank N. Yayon for help with access to the mouse thymus data.

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Author notes

  1. These authors contributed equally: Yichen Gu, Jialin Liu, Kun H. Lee.

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, MI, USA

    Yichen Gu

  2. Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA

    Jialin Liu, Kun H. Lee, Chen Li, Lu Lu, Jaimee Moline, Renxiang Guan & Joshua D. Welch

  3. Department of Computer Science and Engineering, University of Michigan, Ann Arbor, MI, USA

    Joshua D. Welch

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Contributions

J.D.W. conceived the idea of Topological Velocity Inference. K.H.L., J.M. and R.G. generated the human embryoid body spatial transcriptomic data. Y.G. and J.L. implemented the method. Y.G., J.L., K.H.L. and J.D.W. performed data analyses and wrote the manuscript. J.L., C.L., K.H.L. and L.L. performed data preprocessing. All authors read and approved the final manuscript.

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Correspondence to Joshua D. Welch.

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Extended data

Extended Data Fig. 1 TopoVelo accurately models spatial dynamics in simulated datasets.

(a) Simulated datasets with layered(top) and radial(bottom) growth patterns. The true velocity stream is plotted over spatial coordinates colored by the true time values. Note that these stream plots indicate cell velocity in physical coordinates, not RNA velocity in abstract UMAP coordinates. (b) Quantitative performance. The top row shows six metrics used for benchmarking. Higher values are better for all metrics except for reconstruction error. The bottom row contains line plots showing k-CBDir using k-step neighbors with k up to five. Note that we computed two versions of k-CBDIR, one using spatial nearest neighbors and one using nearest neighbors in gene expression space. (c) Comparison of inferred cell velocity. Coordinates are colored by inferred time values. Note that all 2D stream plot visualizations in this paper, including (a) and (c), represent ‘cell velocity’ in physical coordinates, not abstract coordinates computed from UMAP as in previous RNA velocity papers.

Extended Data Fig. 2 Comparison of cell velocity.

Each velocity stream plot visualizes the cell velocity inferred from a dataset (row) by a method (column). The stream plots, from top to bottom, are generated from Slide-seq E15 mouse brain cortex, Slide-seq E13.5 mouse gut and mouse lung, and Visium postnatal 3-month mouse thymus. Each column is a separate RNA velocity method shown at the top. ‘N/A’ means a method failed to run on a certain dataset.

Extended Data Fig. 3 Comparison of inferred cell time.

Each heatmap visualizes the cell time inferred from a dataset (row) by a method (column). The stream plots, from top to bottom, are generated from Slide-seq E15 mouse brain cortex, Slide-seq E13.5 mouse gut and mouse lung, and Visium postnatal 3-month mouse thymus. Each column is a separate RNA velocity method shown at the top. ‘N/A’ means a method failed to run on a certain dataset.

Extended Data Fig. 4 TopoVelo robustly models spatial dynamics in simulated datasets with low spatial resolution.

Simulated datasets with layered growth patterns (a) and radial growth patterns (b). The true velocity stream is plotted over spatial coordinates colored by the true time values. Note that these stream plots indicate cell velocity in physical coordinates, not RNA velocity in abstract UMAP coordinates.

Extended Data Fig. 5 Sensitivity of TopoVelo performance to epsilon-ball graph radius.

(a) Bar plots comparing time correlation, cross-boundary direction correctness, spatial velocity consistency and spatial time consistency from TopoVelo with trained with different radii. (b) Line plots showing the k-step CBDir (k-CBDir) metric. Each line with distinct marker and color represents one radius choice. x-axis is the step number and y-axis is the k-CBDir value.

Extended Data Fig. 6 Sensitivity of inferred cell time to epsilon-ball graph radius.

Each panel is a heatmap showing the inferred cell time using an epsilon-ball graph with the given radius shown at the top. For the Visium thymus dataset, a different set of radius values are used due to a lower spatial resolution.

Extended Data Fig. 7 Sensitivity of cell velocity to epsilon-ball graph radius.

Each panel is a velocity stream plot showing the cell velocity on the physical coordinates. The radius is shown at the top. For the Visium thymus dataset, a different set of radius values are used due to a lower spatial resolution.

Extended Data Fig. 8 Results on Visium data (part 1).

Results from the first (a), fourth (b) and fifth (c) slices from a 10X Visium mouse embryo dataset. Each panel shows the spatial coordinates colored by cell type annotations, cell velocity stream plot and a heatmap plot showing the developmental time inferred by TopoVelo.

Extended Data Fig. 9 Quantifying mouse neural tube closure with cell velocity.

Top: 2D quiver plot showing cell velocity on the transformed anterior-posterior(A-P) axis of a E9 mouse neural tube. Bottom: 1D divergence of cell velocity on the A-P axis.

Extended Data Fig. 10 RCTD clustering and cell type compositions of spatial transcriptomic data of human embryoid bodies (EBs).

(a) Cell type assignments of each location inferred by RCTD. (b) EB identity assignments of each location. (c) Cell type composition within each whole EB.

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Gu, Y., Liu, J., Lee, K.H. et al. Topological velocity inference from spatial transcriptomic data. Nat Biotechnol (2025). https://doi.org/10.1038/s41587-025-02688-8

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