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MultistageOT: Multistage optimal transport infers trajectories from a snapshot of single-cell data

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Multistage optimal transport infers cell differentiation trajectories and fate potential from a single snapshot by modeling transitions over multiple stages, and predicts hematopoietic progenitor fates better than single-stage optimal…

desk verdict Multistage OT is a genuinely useful extension, but the headline fate-prediction claim rests on a benchmark that is not a single snapshot. read the letter →

arxiv 2502.05241 v1 pith:4AMBDSRK submitted 2025-02-07 q-bio.QM q-bio.CB

Magnus Tronstad , Johan Karlsson , Joakim S. Dahlin This is my paper · ORCID
classification q-bio.QMq-bio.CB MSC 49Q2292C3762P10
keywords single-cellRNA-sequencingtrajectoryinferenceoptimaltransportmulti-marginalcellfatepredictionpseudotimehematopoiesisoutlierdetection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

MultistageOT claims that cell differentiation inside a single RNA-sequencing snapshot can be modeled as mass transported in several discrete stages from known initial cell states through intermediate states to terminal states, and that this temporal structure extracts information a single-stage transport map misses. If this claim holds, a snapshot of a steady-state process such as blood formation contains enough signal to predict which lineage a progenitor will enter and to order cells by maturity without destructive time-course data. The method solves one global entropy-regularized multistage transport problem, derives per-cell transport-stage distributions and an aggregated cell-cell affinity, and converts the affinity into fate probabilities through an absorbing Markov chain. On 795 day-2 progenitors whose clonal sisters were observed at later time points, the paper reports that MultistageOT's predicted fate distributions sit closer to the observed sister-fate distributions than those of StationaryOT and a distance-weighted baseline, with higher dominant-fate and total-fate accuracy. The same framework, extended with auxiliary states, is used to flag cells that do not belong to the process and to pinpoint a rare basophil-mast cell progenitor population.

What carries the argument

The load-bearing object is the multistage optimal transport program (MultistageOT): a multi-marginal generalization in which mass leaves known initial states, moves through intermediate states across $T$ differentiation stages, and exits into terminal states, with squared Euclidean distances as the transition cost and entropy regularization for tractability. In each stage $t$, mass splits into a part staying in the system ($\tilde{\mu}_t$, sent to $\mathcal{X}$) and a part exiting ($\hat{\mu}_t$, sent to $\mathcal{X}_F$), with conservation constraints linking consecutive stages; the paper derives a generalized Sinkhorn block-coordinate ascent on the dual problem and wraps it in a proximal-point scheme to lower the effective regularization. The same optimization produces all downstream quantities: per-cell stage distributions (pseudotime), summed couplings over stages (cell-cell affinities), and an absorbing Markov chain whose absorption probabilities are the predicted fates. An extension adds an auxiliary initial, intermediate, and terminal state at fixed cost $Q$, so that off-process cells can exit through an 'unknown fate' without destabilizing the main transport.

What would settle it

Re-run the lineage-tracing benchmark with StationaryOT's growth rates estimated from data or swept over a wide range, and evaluate the comparison with clone-level resampling of the 795 day-2 cells; if the mean total variation advantage of MultistageOT over StationaryOT collapses inside the resampling uncertainty, the paper's central comparative claim would be an artifact of the baseline settings. Independently, on simulated branching data with known ground-truth trajectories, the pseudotime-stage rankings should separate known branches; failure to do so would falsify the temporal-ordering claim.

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is that a snapshot trajectory-inference problem is better posed as a multistage optimal transport problem than as the classical bimarginal one. MultistageOT transports unit mass from $\mathcal{X}_0$ to $\mathcal{X}_F$ over $T$ stages through intermediate states $\mathcal{X}$, minimizing the summed squared-Euclidean transport cost subject to flow-conservation constraints; the entropy-regularized version is solved by block-coordinate ascent on the dual. Each intermediate cell gets a probability distribution over transport stages, whose mean defines pseudotime, and summing the stage-wise transport plans yields a transition matrix whose absorption probabilities in fate classes define each cell's fate potential. In the lineage-tracing benchmark, the model's mean total variation distance to clonal sister fates is lower than StationaryOT and IDW, and it scores higher on dominant-fate and total-fate accuracy; extension with three auxiliary states gives a single-cell 'unknown fate' commitment that the authors use to identify out-of-process cells and to recover basophil/mast cell bipotent progenitors matching known markers.

Load-bearing premise

The benchmark conclusion rests on the heuristic growth-rate values supplied to StationaryOT and on treating 795 day-2 cells as independent samples even though cells of the same clone share the same ground-truth sister-fate distribution.

Editorial extensions

If this is right

  • If the benchmark result holds, lineage potential of individual progenitors is recoverable from a single snapshot, and modeling intermediate stages explicitly is what unlocks the added accuracy.
  • MultistageOT-derived pseudotime orders cells along maturation and increases with experimental time in the lineage-tracing data, giving a temporal axis without time-resolved sampling.
  • Outlier and rare-progenitor analyses suggest the framework can isolate cells outside the modeled process and nominate specific marker genes for prospective isolation, as it does for basophil-mast cell progenitors.
  • Scaling to roughly $10^4$ cells per partition on a laptop, with decision variables growing linearly in the number of stages rather than quadratically in cell number, indicates the approach is practical for large snapshots when combined with partitioning and averaging.
  • The method generalizes to any developmental process with identifiable beginning and end states, not only hematopoiesis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The reported advantage over StationaryOT may be sensitive to the baseline's growth-rate heuristic; an ablation that fits or sweeps those rates would reveal whether the gain is specifically from multistage structure.
  • Statistical significance is assessed by treating individual day-2 cells as independent despite cells within a clone sharing the same ground-truth sister-fate distribution; a clone-level bootstrap or mixed model is a natural reanalysis that could shrink the reported p-values.
  • Because the formulation is a structured optimal transport problem, the same machinery could chain across multiple time points or transfer to other steady-state systems such as intestinal crypts or reprogramming cultures where initial and terminal states are known.
  • The auxiliary-state outlier score with fixed cost $Q$ acts as an implicit contamination threshold; it could be recast as a generative model of off-process cells and validated on synthetic mixtures with known outlier fractions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper introduces MultistageOT, a multistage optimal transport framework for trajectory inference from a single-cell RNA-sequencing snapshot. The method extends bimarginal optimal transport by modeling differentiation as a sequence of transport stages from known initial states to known terminal states, with intermediate cells acting as transit hubs. The authors derive an entropy-regularized formulation and a block-coordinate Sinkhorn-type algorithm in the Supplementary Note, and apply the method to synthetic data, the Paul et al. hematopoiesis snapshot, the Weinreb et al. lineage-tracing data, and a Dahlin et al. in vivo snapshot. The central quantitative claim is that MultistageOT significantly improves cell-fate prediction over StationaryOT and an inverse-distance-weighted baseline on day-2 hematopoietic progenitors in the Weinreb benchmark (Fig. 6j).

Significance. If validated, MultistageOT would be a useful new tool: the mathematical derivation in the Supplementary Note is detailed and appears self-consistent, the code is publicly available, and the use of external lineage-tracing data as a ground truth is an appropriate validation strategy. The synthetic experiments and marker-gene correlations on the in vivo snapshot provide supporting evidence that the method produces biologically plausible pseudotime, fate probabilities, outlier calls, and bipotent progenitor predictions. However, the headline claim rests on a benchmark that is not actually a single snapshot, and the baseline comparison and statistical tests have weaknesses that currently make the strength of the reported improvement uncertain.

major comments (4)
  1. [Section 2.3, Methods 6.1.3, Methods 6.3.3] The central validation is not performed on a single snapshot. Methods 6.1.3 pools all 130,887 cells from Weinreb et al. across days 2, 4, and 6 before splitting them into 12 subsets, and Section 2.3 itself describes the result as a "unified snapshot-like landscape." The transport problem therefore has access to cells from later time points as intermediate or terminal hubs, and for at least some test clones the day-4/day-6 clonal sisters whose fates are used as ground truth may be among the cells in the same fitted partition. The Fig. 6j advantage could thus reflect temporal or clonal leakage rather than an ability to extract fate information from a genuine single snapshot. The abstract's "from a snapshot" claim requires either a day-2-only re-run, or explicit evidence that no test clone's day-4/day-6 sisters occur in any fitted partition and that results are unchanged when later-time-point cells are excluded from the fitting set.
  2. [Methods 6.3.4, Fig. 6j] The StationaryOT baseline may be artificially disadvantaged by the heuristic growth-rate specification. Equation (16) sets sink growth rates to zero, source growth rates to (nF/n0 + 1)^(1/Δt), intermediate growth rates to 1, and Δt = 0.25 is taken from a different data set; no sensitivity analysis over these choices is reported. Because StationaryOT's cell-fate predictions depend strongly on the growth rates, the lower total variation distance for MultistageOT in Fig. 6j could reflect a poorly calibrated baseline rather than an intrinsic methodological advantage. Please estimate growth rates from the lineage-tracing data or systematically vary Δt and the growth-rate model and show that the qualitative conclusion is unchanged.
  3. [Methods 6.3.3, Fig. 6j] The Mann-Whitney U-tests treat the 795 day-2 cells as independent observations, but cells belonging to the same clone share the same ground-truth sister-fate distribution (Eq. 12), and cells within each of the 12 partitions share the same fitted transport plan. The effective sample size is therefore smaller than 795, and the reported p-values in Fig. 6j overstate the significance of MultistageOT's advantage. Please provide clone-level aggregated analyses or block-bootstrap p-values, together with effect sizes and confidence intervals.
  4. [Methods 6.2.1, 6.2.5, Table S.1] The model has several free parameters (the number of transport stages, the regularization parameter ε, and the auxiliary-state cost Q), and the benchmark results in Fig. 6 use fixed chosen values without a sensitivity analysis. Because the paper's central quantitative claim is a comparative improvement over StationaryOT, the authors should show that the conclusion is robust to reasonable variations in T and ε, particularly for the Weinreb benchmark where the number of transport stages is fixed to T − 1 = 20.
minor comments (6)
  1. [Methods 6.2.1] The sentence "T − 1 = 20 intermediate transport steps" is confusing because Box 2 defines T − 1 as the number of transport stages; please state explicitly that T = 21 and clarify whether the count includes the first transport step from X0.
  2. [Methods 6.3.3, Eq. (13)] The phrase "for two probably vectors" should read "for two probability vectors."
  3. [Methods 6.1.4] The text contains a duplicated word: "keeping only genes with with 0.2 or lower Pearson's correlation coefficient."
  4. [Fig. 6g] The figure reports only "*** p < 0.001" from Mann-Whitney U-tests; please report the exact p-values and the number of cells in each time-point group.
  5. [Fig. 6j] Please clarify whether the Mann-Whitney U-tests are one-sided or two-sided and whether any multiple-testing correction was applied across the several performance metrics.
  6. [Section 2.3, Fig. 6l] The area under the threshold-dependent accuracy curve is described in the caption but not formally defined in Methods; please add a definition of the AUC computation.

Circularity Check

2 steps flagged · score 6.0 of 10

Fate-prediction benchmark pools future time points into the model input, so the reported 'snapshot' prediction is self-definitional; the IDW baseline is also fit to the ground truth.

  1. self definitional [Section 2.3; Methods 6.1.3 and 6.3.3]
    "To evaluate the accuracy of MultistageOT, we utilized the data set of Weinreb et al. [3], which provides longitudinal single-cell RNA-sequencing data of in vitro hematopoiesis, integrated into a unified snapshot-like landscape of cell states."

    The 'snapshot-like landscape' is the full 130,887-cell Weinreb set pooled across days 2, 4, and 6 (Methods 6.1.3: 'the data frame contained 25289 genes and 130887 cells ... created a partition of 12 disjoint subsets of the data, each of size 10907. MultistageOT was applied to each partition independently'). The evaluation target for each day-2 cell is 'its clonal sister fate distribution in days 4 or 6' (Methods 6.3.3), i.e., the fates of day-4/6 cells. Those day-4/6 cells are part of the same pooled fitting input, with no exclusion of benchmark clones from the data used to fit the model. Hence the 'prediction from a snapshot' is a function fitted on the very cells whose annotations define the ground truth; the temporal and clonal leakage is built into the benchmark construction.

  2. fitted input called prediction [Fig. S.3 caption; Methods 6.3.5]
    "We used the ground truth fates to optimize the parameter with respect to 'dominant fate accuracy'."

    The IDW baseline's parameter γ is selected by maximizing dominant fate accuracy (Eq. 14) against the same clonal-sister ground truth that is later used to report IDW's performance in Fig. 6j and Fig. 6l. The baseline is therefore fit to the evaluation target, and its reported accuracy is not an independent benchmark. This is a secondary circular step and does not by itself affect the MultistageOT-versus-StationaryOT comparison, but it is a fitted-input-called-prediction in the paper's own evaluation.

full rationale

The core mathematical derivation of MultistageOT is self-contained: the entropy-regularized multistage transport problem and its block-coordinate ascent algorithm are derived in the Supplementary Note, so the self-citations to the authors' prior optimal-transport work (refs [10,12,14]) are not load-bearing. The significant circularity is in the validation design. The paper claims to predict cell fates from a single snapshot, but the Weinreb benchmark is constructed by pooling all 130,887 cells from days 2, 4, and 6 into one 'snapshot-like landscape' (Section 2.3; Methods 6.1.3). The ground truth for each day-2 progenitor is the empirical fate distribution of its clonal sisters in days 4 or 6 (Methods 6.3.3), and those sister cells belong to the same pooled dataset used to fit the model. Thus the evaluation target is a function of cells in the fitting input, making the headline 'prediction from a snapshot' a self-definitional benchmark rather than an out-of-sample forecast. A second, minor circular step is the IDW baseline, whose parameter is optimized against the ground-truth fates before its accuracy is reported. The StationaryOT growth-rate heuristic is a benchmark-fairness concern rather than a circularity, and the pseudotime-versus-real-time comparison is weakened by the same pooled-input issue but is not itself definitionally forced. Because the central fate-prediction claim rests on a benchmark whose target is inside the model input, the circularity score is 6.

Assumptions & free parameters 5 free parameters · 7 assumptions · 1 invented entities

The central claim rests on several biological assumptions (snapshot as feasible states of a dynamical system, known initial and terminal states, squared-distance transition costs, all intermediate cells transporting mass) and on model hyperparameters (number of stages T, regularization epsilon, auxiliary cost Q) that are chosen per dataset for numerical stability. The invented auxiliary states are the main new model entity and they do have an external handle through predicted outlier identity. The benchmark additionally assumes clonal-sister fates are ground truth and that StationaryOT's heuristic growth rates are adequate.

free parameters (5)
  • Number of transport stages (T-1) = 20 intermediate stages for scRNA-seq; 10 stages for synthetic data (T=11)
    User-chosen hyperparameter; determines pseudotime resolution and computational cost. Methods 6.2.1.
  • Entropy regularization parameter epsilon = Initial 0.013-0.055; effective about 0.0038-0.0066 per dataset (Table S.1)
    Controls diffusion in transport plans; selected per dataset for numerical stability, not from benchmark ground truth. Methods 6.2.5.
  • Auxiliary transport cost Q = 3.25 for Dahlin data; 0.9 for synthetic; tested 1, 2, 3.25 on a Weinreb partition
    Chosen as the highest value that prevents division-by-zero in Sinkhorn iterations; affects outlier absorption strength. Methods 6.2.3.
  • StationaryOT growth-rate heuristic and time step = Delta t = 0.25; g_i = (n_F/n_0 + 1)^(1/delta t) for sources; epsilon = 0.12
    Baseline comparison parameters; chosen heuristically in absence of growth-rate data, possibly disadvantaging StationaryOT. Methods 6.3.4.
  • IDW distance weight gamma = gamma* = 20
    Baseline parameter optimized on the evaluation ground truth, which inflates IDW performance; central claim does not depend on it. Methods 6.3.5.
assumptions (7)
  • domain assumption A single-cell snapshot is generated by sampling a continuous time-invariant dynamical system; measured gene-expression states are feasible states along differentiation trajectories.
    Foundation of the model in Supplementary Note §3; if false, transport between expression states as a proxy for development is invalid.
  • domain assumption Initial states X0 and terminal states XF are known and correctly identified from markers and embeddings.
    Input to the optimization problem (P); mis-specification changes inferred trajectories. Methods 6.2.2.
  • domain assumption Squared Euclidean distance in PCA space is a valid transition cost, corresponding to a Gaussian transition kernel.
    Cost definition (3.1); standard in OT-based single-cell analysis but an assumption about the geometry of differentiation.
  • ad hoc to paper Every intermediate state must transport at least unit mass and every terminal state must receive at least unit mass (constraints 3.9-3.11).
    Ensures all cells participate in the process; forces outliers to be modeled via auxiliary states rather than excluded naturally. Supplementary Note, problem (P).
  • domain assumption Aggregated transport over stages defines transition probabilities of a stationary absorbing Markov chain, and absorption probabilities give cell-fate potentials.
    Methods 6.3.2; follows prior literature (Weinreb et al. 2018, Lange et al. 2022) but remains a modeling choice.
  • domain assumption Clonal sister fates at days 4 or 6 are a valid proxy for a day-2 cell's ground-truth fate distribution.
    Benchmark basis; ignores undifferentiated sisters and requires at least 7 sisters. Methods 6.3.3.
  • ad hoc to paper The entropy-regularized solution with proximal-point iterations approximates the unregularized LP sufficiently for biological conclusions.
    The authors solve regularized problem (3.20) rather than the original LP, with effective epsilon about 3-8 times smaller than the initial value, not zero. Supplementary Note §4.3.
invented entities (1)
  • Auxiliary initial/intermediate/terminal states (A0, A, AF) and the associated 'unknown fate' absorbing state independent evidence
    purpose: Absorb cells that are too far from all other cells (outliers) to prevent numerical instability and spurious trajectories; the terminal auxiliary state defines an unknown-fate category.
    The paper provides a falsifiable handle: cells with predicted unknown-fate probability above 0.5 were extracted and validated as late-stage neutrophils and erythroid cells, or B, T, and ILC-like cells, using marker expression in independent UMAP and Leiden analyses (Fig. 7g-i, Figs. S.8-S.11).

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Cite this review

Pith. "Pith review of MultistageOT: Multistage optimal transport infers trajectories from a snapshot of single-cell data." pith.science (2026). https://pith.science/paper/4AMBDSRK

@misc{pith2026250205241,
  author       = {Pith},
  title        = {Pith review of: MultistageOT: Multistage optimal transport infers trajectories from a snapshot of single-cell data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4AMBDSRK}},
  note         = {Machine review of arXiv:2502.05241}
}
read the original abstract

Single-cell RNA-sequencing captures a temporal slice, or a snapshot, of a cell differentiation process. A major bioinformatical challenge is the inference of differentiation trajectories from a single snapshot, and methods that account for outlier cells that are unrelated to the differentiation process have yet to be established. We present MultistageOT: a Multistage Optimal Transport-based framework for trajectory inference in a snapshot (https://github.com/dahlinlab/MultistageOT). Application of optimal transport has proven successful for many single-cell tasks, but classical bimarginal optimal transport for trajectory inference fails to model temporal progression in a snapshot. Representing a novel generalization of optimal transport, MultistageOT addresses this major limitation by introducing a temporal dimension, allowing for high resolution modeling of intermediate differentiation stages. We challenge MultistageOT with snapshot data of cell differentiation, demonstrating effectiveness in pseudotime ordering, detection of outliers, and significantly improved fate prediction accuracy over state-of-the-art.

Figures

Figures reproduced from arXiv: 2502.05241 by the authors.

Figure 1
Figure 1. Optimal transport finds an optimal assignment between points in two point cloud distri￾butions µ1 ∈ R 2 + (blue dots) and µ2 ∈ R 4 + (orange dots). The two distributions µ1 and µ2 could for example represent two cell populations. The solution is a matrix, M ∈ R 2×4 + , encoding the optimal way to redistribute the mass in µ1 to form µ2. The thickness of the black lines in the left plot represents the amount of mass s… view at source ↗
Figure 2
Figure 2. Conceptual figure of a scRNA-seq data set partitioned into three subsets: X0 (initial states), X (intermediate states), and XF (terminal states) We model the transitions of cells from X0 to XF as a multistage mass transport process, represented below in terms of marginals ˜µt (mass that continues in the system) and ˆµt (mass that exits the system) in the t:th stage of transport ( [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 3
Figure 3. Graph representation of our general multistage optimal transport framework to model cell differentiation. In each time step t = 0, 1, . . . , T − 1 mass can be sent either to intermediate states in X or to terminal states in XF . Mass sent to X stays within the system, whereas any mass sent to XF permanently leaves the system. We compute the optimal transitions between the cells by solving the following multistage o… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: MultistageOT orders simulated snapshot data along a pseudotemporal axis and induces state transition probabilities between data points (Methods). (a) Two-dimensional data (N = 105) was gener￾ated in the unit square to mimic samples from an developmental process with kn…
Figure 5
Figure 5. Figure 5: MultistageOT-based inference of fate potential and pseudotemporal ordering of single cells in a publicly available single-cell RNA-sequencing snapshot of hematopoiesis. (a) Layered gene expression level (log(x+1)-transformed) plot of known marker genes for early progen…
Figure 6
Figure 6. Figure 6: Validation of the MultistageOT framework with in vitro lineage tracing data from mouse hematopoiesis. (a) SPRING-plot of the data from Weinreb et al. [3] highlighted with initial and terminal cell 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: MultistageOT identifies outliers and predicts bipotent progenitors in a snapshot of bone marrow hematopoiesis. (a) The 2D synthetic data set with the addition of a small population of outliers; auxiliary states enables MultistageOT to model the development of outliers …

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.