A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data

Mary Chriselda Antony Oliver; Kaitlyn Hohmeier; Tuyen Tran; Alejandra Castillo; Caroline Moosmüller; Shiying Li

stat.MLAug 15, 2026

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 ·

Abstract

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.

Panel of Figure 4 — Figure 4: Rank ordering classification results for the KL divergence and Wasserstein methods.

Method	Accuracy	Number Correct	Kendall $\tau$	Spearman $\rho$
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.

Panel of Figure 5 — 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.

Panel of Figure 6 — Figure 6: Comparison of gene expression networks with a thresholding of $\mu=0.2$ .

	Score	Chance baseline
$S$	0.72	0.50
$S_{\epsilon}\,(\epsilon=0.0041)$	0.75	0.66
$S_{\epsilon}\,(\epsilon=0.0082)$	0.82	0.73

Table 2: Sign Assignment (

S

) and Relaxed Sign Assignment (

S_{\epsilon}

) comparing the geodesic from E14-16 to L3 and the piecewise stitched geodesic for Section 4.3 with

\epsilon

scaled to 5.0% (

\epsilon=0.0041)

and 10.0% of maximum variation (

\epsilon=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 .

\widetilde{\kappa}_{G_{s}}

s=1,\ldots,5

, denote the observed weighted graphs at the five developmental stage.

Transition	Raw $\delta_{W}^{\textrm{avg}}$	$\tau$ counts with largest $\delta_{W}$	Centered/scaled $\delta_{W,\mathrm{cs}}^{\mathrm{avg}}$	$\tau$ counts with largest $\delta_{W,\mathrm{cs}}$
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

\log_{10}(\tau)

. The third and fifth columns report the number of the 19 sampled diffusion parameters at which each transition has the largest corresponding discrepancies,

\delta_{W}

and

\delta_{W,\mathrm{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

\tau=1

and

\tau=10

. The common x-axis range

[0.25,0.95]

contains the central 98% of the curvature values for each developmental stage at both

\tau=1

and

\tau=10

Panel of Figure 9 — 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.

Panel of Figure 12 — 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.

Panel of Figure 13 — 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 $\mu=0.2$ was used.

	$s=1$	$s=2$	$s=3$	$s=4$	$s=5$
$\widetilde{\kappa}_{G_{s}}$	0.4067	0.4038	0.4805	0.4656	0.5365
$\widetilde{\kappa}_{G_{B}^{w,t_{s}}}$	0.4067	0.4680	0.4558	0.5199	0.5365

Appendix

Table 4: Values for

\widetilde{\kappa}_{G_{B}^{w,t_{s}}}

and

\widetilde{\kappa}_{G_{s}}

in Section 4.3 .

Figure 14: Histograms of distance-scaled dynamic OR curvature values $\kappa^{\mathrm{ds}}_{x,y}(\tau)$ with $\tau=0.1$ for L1, L2, and L3. The kernel density estimates (KDEs) are included for visualization.

Figure 15: Histograms of distance-scaled dynamic OR curvature values $\kappa^{\mathrm{ds}}_{x,y}(\tau)$ with $\tau=1$ for L1, L2, and L3. The kernel density estimates (KDEs) are included for visualization.

Ollivier-Ricci Curvature

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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.

Abstract

Ollivier-Ricci Curvature

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