🧬 genomics

Trajectory Uncertainty Framework (TUF): A Modular Framework for Identifying Transitional and Branch-Point Cell States in Single-Cell Trajectory Analysis

This paper introduces the Trajectory Uncertainty Framework (TUF), a modular, open-source tool that quantifies single-cell trajectory uncertainty through Temporal Entropy and Trajectory Divergence scores to identify transitional states and reveal context-specific transcriptional drivers across diverse biological systems.

Original authors: Mahdavifar, M., Mohammadifar, Z., Iranpourtari, T.

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mahdavifar, M., Mohammadifar, Z., Iranpourtari, T.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to map a bustling city where people are constantly moving from their homes to work, school, or parties. In the world of biology, scientists use a technology called single-cell RNA sequencing to take a "snapshot" of millions of individual cells, capturing a moment in their lives. To understand how these cells grow and change, researchers use a method called trajectory inference. Think of this like drawing a road map based on a single photo of traffic; they try to figure out the path each car (cell) took from a starting point to a destination. They assign every car a "pseudotime" score, which is just a number telling you how far along the journey it is.

However, real life is messy. Sometimes cars are stuck in traffic jams, sometimes two roads cross over each other, and sometimes a driver is unsure whether to turn left or right. In biology, these are the "transitional states" where a cell is changing its identity, or "branch points" where it must decide to become one type of cell or another. The problem is that the old maps were too simple. They gave every cell a single number, which made it hard to tell the difference between a cell that is just confused (uncertain) and a cell that is at a real fork in the road. If you can't tell the difference, you might miss the most exciting parts of the story: the exact moment a cell decides its future.

This is where a new tool called the Trajectory Uncertainty Framework (TUF) comes in. Instead of giving a cell just one number, this paper introduces a way to give it two numbers, creating a 2D map of "uncertainty." The authors, Masoud Mahdavifar, Zahra Mohammadifar, and Tara Iranpourtari, built this system to act like a detective that can spot the difference between a cell that is just mixing with others and a cell that is actually splitting into a new path.

The framework uses two clever scores. The first is the Temporal Entropy Score (TES). Imagine a group of friends waiting for a bus. If everyone is waiting at the exact same time, the group is "ordered." But if some friends arrived an hour ago and others just walked up, the group is "mixed up." TES measures this kind of time-mixing. It tells us if a cell is surrounded by neighbors that are at very different stages of their journey. The second score is the Trajectory Divergence Score (TDS). This is like looking at which way the friends are facing. If everyone is looking straight ahead down the road, they are "coherent." But if some are looking left, some right, and some are spinning in circles, they are "diverging." TDS measures this directional confusion.

The paper's main finding is that you need both scores to make sense of the chaos. The authors tested this on fake (synthetic) data where they knew exactly what the roads looked like. They found that TDS alone is a bit of a trickster; it gets excited at real branch points, but it also gets excited when two roads simply cross over each other on a map, even if the cars aren't actually changing direction. However, when you look at TES and TDS together, the picture becomes clear. A true "fork in the road" (where a cell decides to become two different things) shows up as high divergence (looking in many directions) but low time-mixing (everyone is at the same stage). A "crossing road" (where two different groups just happen to pass each other) shows up as high scores on both metrics.

The researchers then applied this 2D map to real biological data, including cells from the pancreas, the gut, and two types of cancer: glioblastoma and breast cancer. In the pancreas and gut, the system successfully found the known "branching" cells where development splits into different types. But the most interesting part came from the cancer data. By using a special math trick called "residual analysis" to strip away the usual noise of cancer (like cells just dividing or reacting to low oxygen), they discovered that the two scores were pointing to completely different biological secrets.

In glioblastoma (a brain cancer), cells with high "time-mixing" (high TES) were found to be involved in inflammatory remodeling—basically, the cells were in a state of chaotic change, reacting to immune signals. Meanwhile, the cells with high "directional divergence" (high TDS) were the ones struggling with metabolic stress and trying to divide rapidly. In breast cancer, the high "time-mixing" cells were linked to cells changing their shape and structure (EMT), while the "directional divergence" cells were linked to specific family-lineage programs. The paper notes that the genes driving these two types of uncertainty barely overlap at all (only about 2.8% to 7.9% shared genes), suggesting that the two scores are capturing two totally different dimensions of how cells behave.

The authors are careful to say that this tool doesn't predict the future or prove that a cell will become something specific. Instead, it acts as a spotlight. It highlights the cells that are most interesting to study further. They suggest that by looking at these "high uncertainty" cells, scientists can generate new hypotheses about how diseases progress or how tissues repair themselves. The tool is available as open-source software called SiCell.jl, written in the Julia programming language, allowing other scientists to use this 2D map to explore their own cellular roadmaps.

In short, this paper suggests that by stopping the habit of squeezing cell uncertainty into a single number and instead looking at it from two angles—time and direction—we can finally distinguish between a cell that is just passing through a crowd and a cell that is standing at a crossroads, deciding its fate. It's a way to turn a blurry, confusing map into a clear guide for the next generation of biological discovery.

Technical Summary: Trajectory Uncertainty Framework (TUF)

Problem Statement

Single-cell RNA sequencing (scRNA-seq) trajectory inference methods successfully reconstruct coarse-grained developmental topologies but often fail to resolve fine-grained transitional states and ambiguous branching regions. Existing approaches typically assign cells a scalar pseudotime value, providing a deterministic mapping from high-dimensional transcriptomic space to a one-dimensional continuum. This representation inherently obscures local ambiguity, particularly at trajectory bifurcations where multiple lineage fates are plausible. Furthermore, current uncertainty estimation methods often rely on heuristic thresholds or single metrics that cannot distinguish between different sources of ambiguity, such as true fate bifurcations versus spatial crossings or asynchronous differentiation along a shared path.

Methodology

The authors introduce the Trajectory Uncertainty Framework (TUF), a modular, downstream approach that operates on outputs from standard trajectory inference workflows (e.g., Diffusion Pseudotime, k-nearest neighbor graphs). Instead of a single uncertainty score, TUF defines a 2D uncertainty coordinate system based on two complementary local metrics:

Temporal Entropy Score (TES):
- Definition: Quantifies local temporal heterogeneity by measuring the pairwise pseudotime variability among a cell's $k$ nearest neighbors.
- Calculation: Computed as the normalized average pairwise pseudotime difference (equivalent to a scaled Gini mean difference) over sorted neighbor pseudotime values.
- Interpretation: High TES indicates that neighboring cells occupy a broad range of developmental stages (temporal mixing), while low TES indicates cells are at similar developmental stages.
Trajectory Divergence Score (TDS):
- Definition: Quantifies the directional dispersion of the local neighborhood structure in embedding space.
- Calculation: Computed by calculating unit displacement vectors from a cell to its neighbors, determining the mean resultant vector length ( $R$ ), and defining TDS as $1 - R$ .
- Interpretation: High TDS indicates directional incoherence (neighbors point in divergent directions), while low TDS indicates consistent directional flow.
- Constraint: Requires geometrically meaningful embeddings (e.g., diffusion components, PCA) and is unsuitable for non-metric visualizations like UMAP or t-SNE.

The 2D Coordinate System:
The joint interpretation of TES and TDS partitions cells into distinct regimes:

High TDS / Low TES: Consistent with true fate bifurcations (directional divergence without temporal mixing).
High TES / Low TDS: Consistent with temporal transitions or asynchronous differentiation along a shared lineage (temporal mixing without directional divergence).
High TES / High TDS: Consistent with spatial crossings, convergence funnels, or artifacts.
Low TES / Low TDS: Consistent with committed, coherent lineages.

To identify transcriptional drivers of uncertainty beyond baseline biology, the authors employed a fractional logit residual analysis. This approach regressed TES and TDS against known confounders (hypoxia, stemness, cell cycle, EMT) to isolate residual uncertainty associated with context-specific transitional programs.

Key Results

Synthetic Benchmarks

Differentiation of Artifacts: In synthetic topologies, TDS alone responded strongly to both true branch points and spatial crossings. However, the joint TES/TDS system successfully distinguished them: true bifurcations showed high TDS with low TES, whereas spatial crossings showed high values in both metrics.
Noise Sensitivity: TES remained near zero along true Y-branches but rose sharply at spatial crossings. TDS scaled with the number of diverging lineages at branch points.
Comparison to Baselines: TES showed weak correlation with neighborhood entropy (based on discrete cluster labels), demonstrating its ability to capture continuous temporal heterogeneity that discrete baselines miss.

Biological Applications

The framework was applied to pancreatic endocrinogenesis, intestinal epithelium, glioblastoma (GBM), and breast cancer datasets.

Pancreatic & Intestinal: TUF successfully identified known transitional populations and branch points (e.g., Ngn3-low progenitors in pancreas; lineage divergence in intestinal epithelium).
Glioblastoma (GBM):
- Residual TES: Enriched for inflammatory remodeling (TNF- $\alpha$ , TGF- $\beta$ , IL17) and loss of mature glial identity, consistent with active temporal state transitions.
- Residual TDS: Enriched for proliferative and metabolic stress programs (G2M, mTORC1, Glycolysis), consistent with cycling progenitor states.
- Redundancy: The gene sets associated with residual TES and TDS showed minimal overlap (Jaccard index = 0.079).
Breast Cancer:
- Residual TES: Enriched for EMT-associated and extracellular matrix remodeling programs.
- Residual TDS: Enriched for distinct lineage-associated programs.
- Redundancy: Minimal overlap (Jaccard index = 0.028).

Robustness and Implementation

Cross-Platform: Results were replicated in independent Julia (SiCell.jl) and Python (Scanpy) implementations with nearly identical gene sets and preserved low redundancy between metrics.
Parameter Stability: TES and TDS remained robust across varying neighborhood sizes ( $k=10$ to $30$) and subsampling (80% of cells), with high Jaccard similarity for top uncertain cell types.
Scalability: Computation scales near-linearly with cell number.

Significance and Claims

The paper claims that TUF provides a modular, fate-agnostic uncertainty layer that complements existing trajectory inference methods. Unlike fate-conditional methods (e.g., Palantir, CellRank) that quantify uncertainty relative to predefined terminal states, TUF quantifies local geometric and pseudotemporal ambiguity directly from the neighborhood graph.

The authors assert that:

Context-Specificity: While the low redundancy between TES and TDS is universal, the biological programs they capture are context-dependent (e.g., inflammatory remodeling in GBM vs. EMT in breast cancer).
Hypothesis Generation: TUF serves as a descriptive framework for prioritizing dynamically heterogeneous cell populations for downstream analysis (e.g., differential expression, pathway enrichment) rather than predicting therapeutic response or causal cell fates.
Geometric Insight: The framework highlights that local transcriptomic uncertainty can be decomposed into distinct dimensions: temporal mixing (TES) and directional instability (TDS), which are not reducible to conventional cancer-state signatures.

The framework is implemented in the open-source package SiCell.jl (MIT license) and is available via the Julia General Registry and GitHub, with Python scripts provided for reproducibility.