Trajectory Inference: Single-Cell Dynamics

• Single-cell trajectory inference is a computational framework that maps high-dimensional omics data into pseudotemporal or graph-structured paths to reveal developmental and branching processes.
• It integrates mathematical tools like optimal transport, stochastic differential equations, and deep neural networks to model cellular differentiation, proliferation, and fate decisions.
• Recent innovations enhance scalability, noise robustness, and multi-omics integration by leveraging advanced deep learning, dynamic OT, and improved visualization techniques.

Single-cell trajectory inference encompasses a broad class of computational methods for reconstructing the dynamic, often branching, processes of cellular differentiation and state transitions from static or sparsely-sampled single-cell omics data. The fundamental aim is to map high-dimensional cell-by-feature profiles into pseudotemporal or graph-structured paths that reflect underlying biological progressions, despite the destructive nature of assays (e.g., scRNA-seq, scATAC-seq) and the inherent stochasticity, proliferation, and death events shaping cellular populations.

1. Mathematical Foundations of Single-Cell Trajectory Inference

Trajectory inference builds on the premise that a complex system's evolution can be reconstructed from temporally or pseudo-temporally ordered snapshots. Mathematically, the observed data is an $n\times p$ matrix $X$ of $n$ cells by $p$ features, with each row $x_i$ representing a cell's high-dimensional state. The central task is to recover a function $\tau: \{1,\ldots,n\}\to[0,1]$—the "pseudotime"—that assigns each cell a value representing its position along a biological progression, or more generally, to recover a topology (e.g., tree, graph, or manifold) encoding developmental lineages and transitions (Hutton et al., 13 Feb 2025).

Optimal transport (OT) and its stochastic generalizations have become core mathematical frameworks for trajectory inference. The Schrödinger bridge problem, dynamic OT, and unbalanced OT offer rigorous tools to interpolate distributions over time, reconstructing likely stochastic paths between populations observed at discrete time points (Ying et al., 1 May 2026, Hong et al., 1 Mar 2025, Lavenant et al., 2021). Extensions, such as Wasserstein–Fisher–Rao geometry, enable explicit modeling of non-conservative mass dynamics (cell proliferation, apoptosis) (Ling et al., 16 Nov 2025).

In modern implementations, trajectory inference may be posed as solving for $X_t$ evolving stochastically according to SDEs,

$dX_t = v(t,X_t)\,dt + \sigma\,dB_t$

or with additional birth–death dynamics, subject to observed marginal distributions at discrete $t$. The identification of cell-level transition maps, pseudotime orderings, and fate probabilities is then achieved via combinatorial, variational, or neural parameterizations of these underlying dynamics (Sun et al., 23 Mar 2026, Ling et al., 16 Nov 2025, Ying et al., 1 May 2026).

2. Classical and Contemporary Methodological Classes

Single-cell trajectory inference methods can be categorized as follows (Hutton et al., 13 Feb 2025):

Graph-Based & Manifold Approaches:

• Reconstruct an underlying cell-state graph (e.g., k-NN, minimum spanning tree, principal graph) whose paths reflect biological trajectories.
• Representative algorithms: Monocle (principal graph fitting), Slingshot (cluster-based MST and principal curves), PAGA (graph abstraction) (Hutton et al., 13 Feb 2025).
• Pseudotime is determined by distances along graph edges from a designated root, and complex topologies (branching, loops) are explicitly modeled.

Diffusion and Density-Based Methods:

• Compute a diffusion affinity or Markov transition matrix in the cell-cell similarity space (typically via adaptive Gaussian kernels), then embed cells into lower dimensions (diffusion maps, PHATE) (Hutton et al., 13 Feb 2025).
• Diffusion pseudotime (DPT) computes geodesic distances along this manifold, often robustly capturing continuous transitions including branchings.

Optimal Transport (OT)-Based & Dynamic Modeling Approaches:

• Classical and entropic OT, Schrödinger bridges, and their unbalanced stochastic variants provide principled solutions for reconstructing transport maps, couplings, and dynamic trajectories among cell populations at disparate time points or even from single timepoint snapshots (Ying et al., 1 May 2026, Hong et al., 1 Mar 2025, Ling et al., 16 Nov 2025, Tronstad et al., 7 Feb 2025).
• Mass-creation/destruction (proliferation, death) is modeled via unbalanced OT or explicitly via branching Schrödinger bridges.

Deep Learning with Neural ODE/SDE and OT:

• Joint autoencoder-manifold learning and dynamic modeling (e.g., CellStream, MIOFlow 2.0) embed cells in a low-dimensional geometry and directly learn latent dynamical flows using neural ODEs/SDEs and unbalanced OT constraints (Ling et al., 16 Nov 2025, Sun et al., 23 Mar 2026).
• Stochastic branching, population shifts, and niche influences are increasingly incorporated (Sun et al., 23 Mar 2026).

RNA Velocity and Velocity Field Integration:

• RNA velocity augments transcriptomic measurements with estimated per-cell change vectors, enabling directional inference of near-future cell states. Methods such as VeloTree integrate the global velocity field via kernel regression, trajectory integration, and varifold-based distance measures to robustly reconstruct differentiation trees (Maignant et al., 1 Apr 2026).

3. Recent Algorithmic Innovations

Recent work has focused on scaling, robustness to noise, modeling stochastic and non-conservative dynamics, and leveraging deep representations:

• Unbalanced Schrödinger Bridge (USB): Simulation-free learning of stochastic branching with discrete birth–death events at single-cell resolution. Efficient training via conditional Poisson–Brownian bridges and unbalanced score-matching regression (Ying et al., 1 May 2026).
• CellStream: Simultaneous learning of temporally coherent embeddings and continuous flows using neural ODEs coupled to WFR-based unbalanced dynamical OT. Empirical superiority in temporal coherence and trajectory recovery (Ling et al., 16 Nov 2025).
• MIOFlow 2.0: Manifold-regularized latent encoding (PHATE) with neural SDEs for stochastic dynamics, proliferation/death (unbalanced OT), and gene–spatial data fusion. Demonstrates reduced trajectory error and superior manifold adherence compared to baseline generative flow models (Sun et al., 23 Mar 2026).
• MultistageOT: Chains together multiple entropic OT stages within a single snapshot, enabling high-resolution modeling of intermediate trajectories, fate probabilities, and outlier detection in cell populations (Tronstad et al., 7 Feb 2025).
• Robust Local Fréchet Regression (UOT-FR): Uses unbalanced neural OT to interpolate distributions at unobserved times, reconstruct branches, and reveal trajectory-regulator genes. Achieves superior interpolation and trajectory reconstruction on classic data sets (Yan et al., 13 Jun 2025).
• PCS-Guided Neighbor Embedding (NESS): Applies predictability-computability-stability criteria to neighbor embedding methods (e.g., UMAP, t-SNE, PHATE), optimally tuning hyperparameters to reveal smooth structures and quantify embedding stability for robust trajectory visualization (Ma et al., 27 Jun 2025).
• GNN-augmented cross-sample trajectory inference (TrajLens): Graph neural network modeling of population-level transitions and robust, expert-validated visual analytics for multi-sample trajectory comparisons and gene-function integration (Wang et al., 21 Jul 2025).
• Layerwise Embedding Extraction in Foundation Models: Systematic evaluation reveals that intermediate transformer layers encode more informative features for trajectory inference than the final embedding layers—task-dependent and context-dependent optimality is observed (Civale et al., 16 Apr 2026).

4. Benchmarking, Metrics, and Biological Interpretation

Trajectory inference methods employ several evaluation and interpretability criteria:

• Trajectory Recovery: Concordance with known temporal orderings (e.g., time series, lineage tracing), typically via correlation metrics (Spearman's $\rho$) between inferred pseudotime and ground-truth, or trajectory reconstruction error (mean distance to reference path) (Civale et al., 16 Apr 2026, Sun et al., 23 Mar 2026).
• Benchmarking Against Simulations: Recovery of branching structures, fate probabilities, change points, and temporal consistency are assessed on well-characterized synthetic systems (e.g., dyngen, toggle-switch), with accuracy up to $X$0–$X$1 for transition recovery in state-of-the-art methods (Zhu et al., 7 Jan 2025, Tronstad et al., 7 Feb 2025).
• Biological Validation: Alignment with known marker gene expression along lineages, gene set enrichment in inferred branches, and recapitulation of established trajectories (e.g., hematopoiesis, neuronal differentiation) (Hutton et al., 13 Feb 2025, Yan et al., 13 Jun 2025).
• Growth and Branching Modeling: Novel metrics such as Relative Mass Error (RME), fate entropy, and branch-assignment accuracy directly quantify model fit to proliferation/death rates and fate distributions (Ying et al., 1 May 2026, Ling et al., 16 Nov 2025).

5. Implementation Practices and Best-Practice Recommendations

Efficient and robust trajectory reconstruction requires careful choices and pipeline design:

• Preprocessing: Standard best practices are applied—quality filtering of cells/genes, normalization (e.g., log1p, CPM/TPM, library size), and highly variable gene selection to reduce noise and scale features (Hutton et al., 13 Feb 2025, Ling et al., 16 Nov 2025).
• Dimensionality Reduction: Principal component analysis or advanced neighbor embedding methods (PHATE, UMAP, t-SNE), with PCS-guided or data-driven selection of neighbor/connectivity hyperparameters, ensure faithful manifold discovery and avoid fragmentation or artificial clustering (Ma et al., 27 Jun 2025).
• Algorithmic Layer Selection: When using pretrained deep models (e.g., transformer-based foundation models for scRNA-seq), optimal layer selection should be data-driven (e.g., via DPT correlation sweep), not defaulted to the final embedding—a routine layer scan can yield 13–31% higher recovery of trajectory structure (Civale et al., 16 Apr 2026).
• Unbalanced OT and Growth Modeling: Algorithms should robustly accommodate non-conservative dynamics and batch-specific effects. Hybrid regularizations (KL, WFR actions), auxiliary states for outlier detection, and cell-type–level or cell-level mass constraints are commonly implemented for enhanced interpretability (Ling et al., 16 Nov 2025, Tronstad et al., 7 Feb 2025, Zhu et al., 7 Jan 2025).
• Pipeline Ensemble and Versioning: Save layer index, embedding dimension, and parameter states in metadata; consider concatenating embeddings from adjacent layers or timepoints for ensemble analyses (Civale et al., 16 Apr 2026).
• Empirical Validation: Root selection, marker validation, and, where possible, cross-validation or permutation testing of inferred structures should be routine (Hutton et al., 13 Feb 2025).

6. Limitations, Open Challenges, and Future Directions

Several key challenges persist in single-cell trajectory inference:

• Scalability: Most methods scale $X$2 in the number of cells (e.g., full pairwise OT, varifold distances), which motivates development of GPU-accelerated, mini-batch, or factorizable algorithms and low-rank approximations (Tronstad et al., 7 Feb 2025, Maignant et al., 1 Apr 2026).
• Curse of Dimensionality: High-dimensional latent and sequence spaces can challenge basis-expansion and kernel methods (e.g., truncated orthonormal bases in phase-space SB) (Hong et al., 1 Mar 2025).
• Uncertainty Quantification: Bayesian formulations and uncertainty propagation, as well as stability/instability quantification in embeddings and trajectory assignments, are important areas for practical robustness (Ma et al., 27 Jun 2025, Lavenant et al., 2021).
• Integration of Multi-omics and Spatial Data: Robust integration of transcriptional, chromatin, proteomic, and spatial information into unified trajectory models is an ongoing frontier; manifold learning, joint autoencoders, and graph-based neighborhood modeling (e.g., in spatial transcriptomics) are increasingly adopted (Sun et al., 23 Mar 2026, Wang et al., 21 Jul 2025).
• Complex Dynamics: Accurate modeling of cyclic, reversible, or lineage barcoded dynamics, as well as explicit SDEs or Schrödinger bridge models capturing true biological stochasticity and jump processes, remain active research areas (Ying et al., 1 May 2026, Hong et al., 1 Mar 2025).
• Dependence on Annotations/Clusters: Type-level trajectory methods (e.g., TIMO) and cluster-rooted tree methods depend critically on prior annotations, which may obscure sub-lineage heterogeneity; joint inference approaches are being explored (Zhu et al., 7 Jan 2025, Hutton et al., 13 Feb 2025).

7. Summary Table: Selected Recent Methods and Key Features

Method	Principle	Handles Unbalanced Mass	Branching/Stochasticity	Embedding/Manifold
MultistageOT (Tronstad et al., 7 Feb 2025)	Multimarginal OT	Yes (auxiliary states)	Implicitly via OT	Snapshot PCA/UMAP
CellStream (Ling et al., 16 Nov 2025)	AE + Dyn. OT (WFR)	Yes	Continuous ODE (latent)	Joint AE manifold
MIOFlow 2.0 (Sun et al., 23 Mar 2026)	Neural SDE + OT	Yes	Explicit (Neural SDEs)	PHATE AE + spatial
USB (Ying et al., 1 May 2026)	Unbal. Schröd. Br.	Yes	Birth-death SDE	None/Latent AE
VeloTree (Maignant et al., 1 Apr 2026)	RNA velocity field	N/A (mass conserved)	Deterministic tree	PCA (velocity)
TIMO (Zhu et al., 7 Jan 2025)	Discrete unbalanced	Yes (cell type level)	Branching at type-level	Cluster centroids
NESS (Ma et al., 27 Jun 2025)	PCS NE framework	N/A	N/A	t-SNE/UMAP/PHATE
gWOT (Lavenant et al., 2021)	Global Entropic OT	With splitting	Some branching	PCA/Manifold

The field of single-cell trajectory inference is now defined by a convergence of optimal transport, stochastic process theory, neural generative modeling, and interpretable machine learning, with each method carefully balancing computational feasibility, robustness to noise, and biological plausibility. Progress continues toward unified frameworks modeling both global population dynamics and discrete, cell-level fate decisions in increasingly high-dimensional and multimodal settings.