Generating and navigating single cell dynamics via a geodesic bridge between nonlinear transcriptional and linear latent manifolds

Technical Summary: GeoBridge

Problem Statement
Time-series single-cell RNA sequencing (scRNA-seq) captures cellular processes as sparse, unpaired snapshots. This limitation hinders the reconstruction of continuous cell state transitions and the ability to navigate between states in a controlled, interpretable manner. While existing methods like pseudotime ordering, RNA velocity, and Dynamic Optimal Transport (DOT) attempt to approximate continuous transitions from static data, they often rely on Euclidean metrics in high-dimensional gene expression space. Under the manifold hypothesis, biological data resides on a low-dimensional, nonlinear differentiable manifold where the "distance" between cells is a nonlinear function of gene expression. Assuming a Euclidean metric in this curved space leads to geometric distortion, resulting in inaccurate estimations of developmental speed, fate probability, and state distances. Furthermore, directly optimizing arbitrary geodesic distances in the original high-dimensional space is computationally intractable.

Methodology
The authors propose GeoBridge, a data-driven framework that formulates biological-time interpolation as a Riemannian metric-learning problem. The core innovation is the construction of a "geodesic bridge" between the nonlinear transcriptional manifold and a linear latent manifold.

1. Invertible Neural Network (INN) Mapping: GeoBridge employs an INN to learn a bijective mapping, $\phi: \mathcal{M} \to \mathcal{N}$, from the original high-dimensional gene-expression manifold ($\mathcal{M}$) to a flat, latent Euclidean manifold ($\mathcal{N}$).
2. Pullback Metric Construction: Instead of explicitly defining a complex metric tensor in the original space, GeoBridge defines a simple Euclidean metric ($g_\mathcal{N} = I$) in the latent space. The Riemannian metric on the original manifold ($g_\mathcal{M}$) is induced via the pullback construction: $g_\mathcal{M}(x) = J_\phi(x)^\top J_\phi(x)$, where $J_\phi$ is the Jacobian of the mapping.
3. Geodesic Linearization: A key theoretical result (Theorem 2 in the Supplemental Information) establishes that under this pullback metric, geodesics on $\mathcal{M}$ correspond one-to-one with geodesics on $\mathcal{N}$. Since geodesics in the flat latent space are constant-velocity straight lines, the complex nonlinear transitions in the original space can be represented as straight lines in the latent space.
4. Training Objectives:
   • Constant-Velocity Linear Loss ($\mathcal{L}_{CVL}$): Enforces that time-series data, when mapped to the latent space, follow constant-speed straight-line trajectories. This is achieved by interpolating between source and target latent vectors based on biological time ratios and minimizing the Maximum Mean Discrepancy (MMD) between generated and true intermediate distributions.
   • Local Proportional Regularization ($\mathcal{L}_{LPR}$): Prevents excessive local distortion (stretching/compression) of the mapping. It encourages the ratio of local neighborhood distances in the original space to the latent space to remain stable within a time slice, ensuring biologically plausible interpolations.
5. Downstream Applications:
   • Interpolation: Intermediate states are generated via closed-form linear interpolation in the latent space and mapped back to the original space.
   • Fate Prediction: Cell fate probabilities are estimated using optimal transport plans with Euclidean costs in the latent space.
   • Pseudotime Inference: The Kantorovich potential (KP) derived from the optimal transport in the latent Euclidean space is linearly related to real time, serving as a robust pseudotime indicator.
   • Driver Gene Identification: Dynamic driver indices are calculated by computing the dot product of trajectory expression gradients and velocity components.
   • Fate Navigation: The framework allows for "virtual navigation," generating geodesic paths between any arbitrary starting cell and a target cell state, even across different lineages.

Key Results
GeoBridge was validated across diverse biological systems:

• EMT–MET Progression in Cancer Stem Cells: GeoBridge successfully reconstructed continuous trajectories for Epithelial-Mesenchymal Transition (EMT) and Mesenchymal-Epithelial Transition (MET). It identified stage-specific regulatory modules and driver genes consistent with known biology (e.g., RHO GTPase cycles in early MET, extracellular matrix organization in late EMT).
• Human Pluripotent Stem Cell (hPSC) Differentiation: The method resolved branching dynamics toward stem cell-derived $\beta$-cells and enterochromaffin-like cells. It accurately captured time-resolved marker gene expression and identified distinct early and late regulatory programs for each lineage.
• Multi-Target Hematopoietic Lineages: In a unified multi-target model, GeoBridge navigated hematopoietic stem cells (HSCs) toward neutrophil, mast-cell, and megakaryocyte fates. Crucially, it demonstrated the ability to virtually guide neutrophil-biased cells toward mast-cell fates along biologically plausible geodesic paths, validating the interpretability of the learned geometry.
• Benchmarking: GeoBridge outperformed state-of-the-art methods (MioFlow, TIGON, DeepRUOT) in held-out interpolation accuracy (measured by Wasserstein-1 distance) and pseudotime inference (measured by Pearson correlation with true time labels). Notably, it achieved these results without requiring preliminary dimensionality reduction, operating directly on high-dimensional inputs (top 2,000 highly variable genes).

Significance and Claims
The paper claims that GeoBridge establishes a principled, pullback-geometric method that transforms sparse single-cell measurements into a continuous, controllable landscape. Its primary significance lies in:

1. Geometric Rigor: It avoids the geometric distortions of Euclidean assumptions by learning a Riemannian metric where biological time corresponds to geodesic motion.
2. Computational Efficiency: By leveraging the flatness of the latent space, it enables explicit, closed-form interpolation, eliminating the need for computationally intensive numerical integration (e.g., Neural ODEs) required by other DOT-based methods.
3. Controllability and Interpretability: It provides a framework for "navigating" cell states, allowing researchers to simulate transitions between arbitrary phenotypes and identify the specific genes driving these transitions.
4. Robustness: The method is robust to noise and snapshot sparsity, recovering smooth transcriptional programs even when trained on data without explicit temporal annotations (via iterative pseudotime refinement).

The authors position GeoBridge not merely as an interpolation tool but as a generative model grounded in explicit geometric construction, offering a new paradigm for reconstructing, interpreting, and manipulating cellular state transitions.