A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data
Authors: Mary Chriselda Antony Oliver, Kaitlyn Hohmeier, Tuyen Tran, Alejandra Castillo, Caroline Moosmüller, Shiying Li
Organizations: Department of Applied Mathematics and Theoretical Physics, University of Cambridge · Department of Mathematics, University of North Carolina at Chapel Hill · Department of Mathematics and Statistics, Loyola University Chicago · Department of Mathematics, Pomona College · Department of Mathematics, University of Nebraska-Lincoln
High-throughput single-cell and spatial transcriptomic technologies provide high-resolution snapshots of heterogeneous cellular states, but their destructive nature prevents repeated measurements of the same cells over time. Consequently, temporal and spatial dynamics must be inferred from independently sampled, unaligned cell populations, making it challenging to reconstruct developmental trajectories. Optimal transport (OT) offers a geometric framework for aligning cell populations and inferring developmental trajectories, but many existing approaches focus on modeling the evolution of distributions of cells in gene expression space rather than the relational structure encoded by gene expression networks. To address this limitation, we introduce a geometric framework for analyzing the spatiotemporal evolution of gene expression networks through embeddings in Gromov--Wasserstein (GW) space. By representing each developmental stage as a graph combining gene expression and spatial proximity, our approach enables comparisons of network structure across time, continuous interpolation between developmental stages via GW geodesics, and quantification of network-level changes using Ollivier-Ricci curvature. We evaluate our framework on a spatiotemporal transcriptomic Drosophila dataset and show that GW geodesic interpolations reproduce main trends in curvature dynamics observed in empirical gene expression networks. Agreement with higher-order Co-Optimal Transport (COOT) distances, which jointly represent spatial and temporal information, further validates the framework and suggests that hypernetwork representations successfully record salient biological changes across time. In general, our approach provides a unified geometric approach to study dynamically evolving biological networks.
Figures & tables
Figure 1: Flow chart on the steps involved for pre-processing the Drosophila dataset.
Figure 2
Figure 4: Rank ordering classification results for the KL divergence and Wasserstein methods.
Method
Accuracy
Number Correct
Kendall τ
Spearman ρ
Wasserstein
100.00%
50
1.0000
1.0000
KL Divergence
94.00%
47
0.9735
0.9867
Scalar Mean
75.00%
37
0.7098
0.8073
Scalar Median
60.00%
30
0.3162
0.4617
Table 1: Classification results by comparing the curvature distributions with the common gene space gene expression networks for each developmental stage to the spatial slices at each developmental stage.
Figure 5: Plot of per-slice, per-reference-stage discrepancy values for the Wasserstein and KL divergence classification methods. Correctly-classified slices are marked with a star symbol and a black box, while incorrectly-classified slices are denoted with an x symbol, with a black box around the class into which it was incorrectly classified.
Figure 6: Comparison of gene expression networks with a thresholding of μ=0.2 .
Score
Chance baseline
S
0.72
0.50
Sϵ(ϵ=0.0041)
0.75
0.66
Sϵ(ϵ=0.0082)
0.82
0.73
Table 2: Sign Assignment ( S ) and Relaxed Sign Assignment ( Sϵ ) comparing the geodesic from E14-16 to L3 and the piecewise stitched geodesic for Section 4.3 with ϵ scaled to 5.0% ( ϵ=0.0041) and 10.0% of maximum variation ( ϵ=0.0082 ).
Figure 7: Average OR curvatures for the geodesics computed between E14-16 and Larvae 3 and the stitched geodesic as detailed in Section 4.3 . κGs , s=1,…,5 , denote the observed weighted graphs at the five developmental stage.
Transition
Raw δWavg
τ counts with largest δW
Centered/scaled δW,csavg
τ counts with largest δW,cs
E14–16 to E16–18
0.125
0/19
0.133
0/19
E16–18 to L1
0.158
0/19
0.193
0/19
L1 to L2
0.195
12/19
0.244
0/19
L2 to L3
0.092
7/19
0.353
19/19
Table 3: Dynamic OR curvature comparisons for consecutive developmental stages. The Wasserstein discrepancies are averaged over log10(τ) . The third and fifth columns report the number of the 19 sampled diffusion parameters at which each transition has the largest corresponding discrepancies, δW and δW,cs respectively, among the four consecutive transitions.
Figure 8: Empirical cumulative distribution functions of the edge-wise distance-scaled dynamic OR curvature values for the five observed developmental-stage networks at τ=1 and τ=10 . The common x-axis range [0.25,0.95] contains the central 98% of the curvature values for each developmental stage at both τ=1 and τ=10 .
Figure 9: Comparison of developmental-stage relationships inferred from gene correlation networks. (a) COOT-based multidimensional scaling (MDS) embedding of developmental stages (E14–16, E16–18, L1, L2, L3), illustrating the trajectory of network evolution over time. Early embryonic stages cluster closely, whereas later stages, particularly L3, show a larger shift in network structure. (b) Heatmap of pairwise CO-Optimal Transport (CO-OT) distances between developmental stages. Lower values indicate greater similarity, revealing gradual transcriptional changes from E14–16 through L2 and a more pronounced divergence at L3.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 10: Heatmap of pairwise gene correlations across all Drosophila developmental stages (E14-16, E16-18, L1, L2, L3), using a union of common genes across all five life stages. Other than large changes in correlation values at L3, the correlation values as well as the structure of the gene relationships at each time step appear visually similar.
Figure 11: Rank ordering results for the mean method.
Figure 12: Plot of per-slice, per-reference-stage discrepancy values for the mean and median classification methods. Correctly-classified slices are marked with a star symbol and a black box, while incorrectly-classified slices are denoted with an x symbol, with a black box around the class into which it was incorrectly classified.
Figure 13: Average OR curvatures for the geodesics between different stages of the fruit fly life spans as detailed in Section 4.3 . A gene network threshold of μ=0.2 was used.
s=1
s=2
s=3
s=4
s=5
κGs
0.4067
0.4038
0.4805
0.4656
0.5365
κGBw,ts
0.4067
0.4680
0.4558
0.5199
0.5365
Appendix
Table 4: Values for κGBw,ts and κGs in Section 4.3 .
Figure 14: Histograms of distance-scaled dynamic OR curvature values κx,yds(τ) with τ=0.1 for L1, L2, and L3. The kernel density estimates (KDEs) are included for visualization.
Figure 15: Histograms of distance-scaled dynamic OR curvature values κx,yds(τ) with τ=1 for L1, L2, and L3. The kernel density estimates (KDEs) are included for visualization.
Gromov--Wasserstein optimal transport (GW-OT) extends classical optimal transport by introducing structure-aware transport cost. This is particularly relevant for spatial transcriptomics, where dynamical reconstruction should preserve tissue structure in addition to matching expression patterns. While static formulations have been widely used for such structure-aware alignment, a general dynamic formulation for reconstructing continuous trajectories is still missing. We introduce Travelling Pair Dynamical Alignment and Trajectory Estimation (TP-DATE), a theoretical and computational framework to generalize GW-OT dynamically in a simulation-free manner. We formulate a broad class of static and dynamic Quadratic-form OT (QOT) through path actions and prove the static dynamic equivalence. We further develop travelling-pair flow matching, which allows interacting conditional paths and marginalizes their interactions into a single vector field. On synthetic and real spatial transcriptomics data, TP-DATE better preserves spatial structure and improves continuous 3D dynamics reconstruction.
Junda Ying, Zhiwei Zeng, Peijie Zhou +1
Beijing International Center for Mathematical Research, Peking University · Center for Quantitative Biology, Peking University · Center for Machine Learning Research, Peking University +4
Understanding cellular trajectories via time-resolved single-cell transcriptomics is vital for studying development, regeneration, and disease. A key challenge is inferring continuous trajectories from discrete snapshots. Biological complexity stems from stochastic cell fate decisions, temporal proliferation changes, and spatial environmental influences. Current methods often use deterministic interpolations treating cells in isolation, failing to capture the probabilistic branching, population shifts, and niche-dependent signaling driving real biological processes. We introduce Manifold Interpolating Optimal-Transport Flow (MIOFlow) 2.0. This framework learns biologically informed cellular trajectories by integrating manifold learning, optimal transport, and neural differential equations. It models three core processes: (1) stochasticity and branching via Neural Stochastic Differential Equations; (2) non-conservative population changes using a learned growth-rate model initialized with unbalanced optimal transport; and (3) environmental influence through a joint latent space unifying gene expression with spatial features like local cell type composition and signaling. By operating in a PHATE-distance matching autoencoder latent space, MIOFlow 2.0 ensures trajectories respect the data's intrinsic geometry. Empirical comparisons show expressive trajectory learning via neural differential equations outperforms existing generative models, including simulation-free flow matching. Validated on synthetic datasets, embryoid body differentiation, and spatially resolved axolotl brain regeneration, MIOFlow 2.0 improves trajectory accuracy and reveals hidden drivers of cellular transitions, like specific signaling niches. MIOFlow 2.0 thus bridges single-cell and spatial transcriptomics to uncover tissue-scale trajectories.
Xingzhi Sun, João Felipe Rocha, Brett Phelan +11
Yale University, USA · University of Wisconsin-Madison, USA · Université de Montréal; Mila - Quebec AI Institute, Canada +2
Spatial Transcriptomics (ST) measures gene expression while preserving spatial context, but its high cost and low throughput leave public datasets small. Inferring expression directly from widely available Hematoxylin and Eosin (H&E) stained histology offers a cost-effective alternative. However, existing approaches face several limitations: regression methods over-smooth toward the conditional mean, while generative methods are faithful but require slow multi-step inference; most methods treat genes as independent and equally important, ignoring inter-gene dependencies and heterogeneous gene informativeness; and most are tailored to a single resolution, either spot-level or cell-level. To address these issues, we propose DriftST, a unified framework for inferring spatially resolved gene expression from H&E images. DriftST builds on a Cellular Drifting generative model that learns a direct drift from a histology-conditioned source to the expression distribution, retaining generative expressiveness while enabling efficient one-step generation. To capture gene structure, we introduce the STransformer, which combines a co-expression attention module for inter-gene dependencies with a gene residual gate for differential gene importance. Operating on a generic gene-panel representation, DriftST applies directly to both spot-level and cell-level data in one framework, and extensive experiments across diverse tissues and platforms show that it achieves state-of-the-art performance at both resolutions.
Yuhang Yang, Yonggan Bu, Shengyuan Zhou +2
University of Science and Technology of China, Hefei, China · Peking University Cancer Hospital, Beijing, China