CellBRIDGE: Learning Cellular Trajectories via Interaction-Aware Alignment

# Supplementary Material for CellBRIDGE
Source: [https://arxiv.org/html/2605.30635](https://arxiv.org/html/2605.30635)
Nicolas HuynhTennison LiuRoderik M\. KortleverGerard I\. EvanDavid L\. BentleyMihaela van der Schaar

###### Abstract

Inferring dynamics from population snapshots is a fundamental challenge in machine learning and biology\. In scRNA\-sequencing \(scRNA\-seq\), destructive measurements preclude direct tracking of individual cells across time, making trajectory inference underdetermined\. Optimal Transport \(OT\) provides a principled framework for snapshot*alignment*, but a long\-standing modeling question is which*cost functions*yield biologically meaningful couplings\. Standard OT approaches rely on gene\-expression distances, implicitly treating cells as independent points and neglecting structured cell–cell communication mediated by ligand–receptor signaling\. We introduceCellBRIDGE\(Cell\-Based Regularized Interaction\-Driven Gene Expression\), which augments feature\-based OT with a directed, typed interaction cost derived from ligand–receptor activity\. By explicitly modeling cell–cell communication,CellBRIDGEimproves cross\-snapshot couplings and downstream trajectory estimates across synthetic and real scRNA\-seq datasets relative to feature\-only baselines\. Notably,CellBRIDGEenables mechanistically interpretable*in silico*perturbations: on lung cancer data, silencing specific ligand–receptor pairs induces trajectory shifts that recapitulate expected effects of targeted pathway inhibition\.

Machine Learning, ICML

## 1Introduction

Understanding how cellular populations evolve over time is fundamental to development, disease, and therapeutic intervention\(Yeoet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib84); Qiuet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib83)\)\. Single\-cell RNA sequencing \(scRNA\-seq\) measures gene expression at unprecedented resolution, but its destructive nature precludes tracking individual cells across time, making trajectory inference from population snapshots inherently underdetermined\(Schiebingeret al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib37); Bunneet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib79)\)\. The ability to infer the trajectories of single cells has major implications for drug discovery, where experiments to probe mechanisms and interventions are costly and slow\(Sertkayaet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib95)\):in silicodynamics can guide experiment design and prioritize targets\(Yue and Dutta,[2022](https://arxiv.org/html/2605.30635#bib.bib96)\)\.

Challenges of inferring cellular dynamics\.Learning the trajectories of individual cells, i\.e\. the task of*trajectory inference*\(Bunneet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib79)\), requires reconstructing smooth dynamics from unaligned snapshots\. This presents a unique challenge: because measurements are destructive, the same cell cannot be observed at multiple time points\. These difficulties are further exacerbated by imbalanced cell populations and the noisy, sparse nature of gene expression\(Adilet al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib87); Schiebingeret al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib37)\)\.

From graph heuristics to couplings\.Classical approaches build a cell–cellkkNN graph and extract pseudotime and branches via diffusion distances or spanning\-tree heuristics\(Haghverdiet al\.,[2016](https://arxiv.org/html/2605.30635#bib.bib44); Streetet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib45)\)\. These locality\-based methods assume that proximity within a snapshot reflects temporal adjacency, which can yield biased pseudotimes and spurious lineage structure\(Weileret al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib33)\)\. To address these limitations, more recent methods recast alignment as the task of finding a coupling between*distributions*\.

Biologically meaningful cost functions\.A popular approach for distributional alignment is Optimal Transport \(OT\)\(Peyré and Cuturi,[2019](https://arxiv.org/html/2605.30635#bib.bib5)\)\. While OT makes the search for a coupling computationally tractable, the biological validity of the result hinges entirely on the choice of a cost function\. As noted byBunneet al\.\([2024](https://arxiv.org/html/2605.30635#bib.bib79)\), incorporating meaningful priors via this cost is a*central bottleneck*in single\-cell and spatial omics\. Standard OT approaches rely on gene\-expression distances, effectively enforcing a principle of least action, assuming that cells evolve smoothly via the shortest path in expression space\.

In this work, we ask:*Can we design a biologically meaningful prior for trajectory inference, which is orthogonal to the principle of least action in gene expression?*

We begin with a key observation: feature\-only OT, which relies solely on gene expression distances, implicitly treats cells as independent particles\. This discards structured*cell–cell interactions*\(CCIs\) and ignores the biological reality that trajectories are shaped by intercellular signaling\. Specifically, directed CCIs mediated by ligand–receptor \(LR\) pairs drive development and disease\(He and Xu,[2020](https://arxiv.org/html/2605.30635#bib.bib98); Liuet al\.,[2023](https://arxiv.org/html/2605.30635#bib.bib99)\)\. We posit that the*relational structure*of these interactions can also evolve smoothly over time, and hence can offer a robust signal for alignment\.

We incorporate this prior viaCellBRIDGE\(Cell\-Based Regularized Interaction\-Driven Gene Expression\)\. To avoid reliance on spatial data, we construct*proxy*communication networks within each snapshot by scoring directed LR pairs across local expression neighborhoods\. We then frame the search of a coupling as a*Fused Gromov–Wasserstein*\(FGW\) problem\. FGW simultaneously minimizes the cost of transport in gene expression space and the structural distortion of these inferred communication networks\.

Importantly, our interaction\-aware prior is orthogonal to standard priors \(such as least action in gene expressions or unbalanced transport\)\. As a consequence, this modularity enablesCellBRIDGEto be seamlessly integrated into state\-of\-the\-art pipelines for velocity field regression\(Lipmanet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib29); Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20); Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\)\. In our experiments across synthetic and real\-world datasets, we demonstrate thatCellBRIDGEleads to improved trajectory inference, with best results obtained when paired with orthogonal priors\. To summarize, our contributions are the following:

ContributionsCost\-function view\.We identify OT cost design as a key design choice for snapshot alignment, and propose the smoothness of directed, typed cell–cell communication as a biologically grounded and interpretable prior, which complements gene expression smoothness\.Typed, directed FGW\.We generalize FGW to*multi\-relation*\(vector\-valued\) directed interaction structure derived from ligand–receptor signaling, yielding interaction\-aware couplings between snapshots\.Broad applicability\.Interaction\-aware couplings improve trajectory inference across various trajectory inference frameworks, showing that CCI structure is a general\-purpose prior orthogonal to conventional assumptions\.Empirical and mechanistic evidence\.We demonstrate improved performance on synthetic and real scRNA\-seq datasets, and show interpretable*in silico*perturbations: silencing specific LR pairs induces trajectory shifts consistent with targeted pathway inhibition\.

## 2Background

Problem formulation: cell trajectory inference\.We considerkkpopulation snapshots\{𝒟i\}i=1k\\\{\\mathcal\{D\}\_\{i\}\\\}\_\{i=1\}^\{k\}, where each𝒟i⊂ℝd\\mathcal\{D\}\_\{i\}\\subset\\mathbb\{R\}^\{d\}is a set of single\-cell states measured at timetit\_\{i\}\. The goal is to learn a time\-continuous flowψ:ℝd×ℝ\+→ℝd\\psi:\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}\_\{\+\}\\rightarrow\\mathbb\{R\}^\{d\}such thatψ​\(x,t\)\\psi\(x,t\)returns the state obtained by evolving an initial statexxto timett\. Because scRNA\-seq is*destructive*, the same cell cannot be observed at two times, so there is no one\-to\-one correspondence between cells in𝒟i\\mathcal\{D\}\_\{i\}and𝒟i\+1\\mathcal\{D\}\_\{i\+1\}\. Classical time\-series and ODE\-fitting methods that require repeated observations of the same object are thus not directly applicable; trajectory inference must instead recover dynamics from*unaligned snapshots*\.

Global alignment of snapshots\.Rather than inferring trajectories from neighborhoods within a single snapshot\(Haghverdiet al\.,[2016](https://arxiv.org/html/2605.30635#bib.bib44)\), recent work aligns*multiple snapshots at the population level*\(Schiebingeret al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib37)\), treating each snapshot as a probability distribution over cell states\. This alignment is inherently underdetermined: without additional structure, many matchings between snapshots are equally compatible with the observed marginals\.

Standard OT for snapshots\.For two timepointst0<t1t\_\{0\}<t\_\{1\}with datasets𝒟0=\{xi\}i=1n0\\mathcal\{D\}\_\{0\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\_\{0\}\}and𝒟1=\{yj\}j=1n1\\mathcal\{D\}\_\{1\}=\\\{y\_\{j\}\\\}\_\{j=1\}^\{n\_\{1\}\}, wherexi,yj∈ℝdx\_\{i\},y\_\{j\}\\in\\mathbb\{R\}^\{d\}are gene\-expression vectors, we form the empirical measuresρ0=∑i=1n0ai​δxi\\rho\_\{0\}\\;=\\;\\sum\_\{i=1\}^\{n\_\{0\}\}a\_\{i\}\\,\\delta\_\{x\_\{i\}\}andρ1=∑j=1n1bj​δyj,\\rho\_\{1\}\\;=\\;\\sum\_\{j=1\}^\{n\_\{1\}\}b\_\{j\}\\,\\delta\_\{y\_\{j\}\},witha∈Σn0a\\in\\Sigma\_\{n\_\{0\}\},b∈Σn1b\\in\\Sigma\_\{n\_\{1\}\}, andΣn:=\{w∈ℝ\+n:∑k=1nwk=1\}\\Sigma\_\{n\}:=\\\{w\\in\\mathbb\{R\}\_\{\+\}^\{n\}:\\sum\_\{k=1\}^\{n\}w\_\{k\}=1\\\}\(e\.g\.,ai=1/n0a\_\{i\}=1/n\_\{0\}for uniform weights\)\. The alignment problem seeks a couplingΓ⋆∈Π​\(a,b\)\\Gamma^\{\\star\}\\in\\Pi\(a,b\)betweenρ0\\rho\_\{0\}andρ1\\rho\_\{1\}that respects the marginalsaaandbb:

Γ⋆:=\{Γ∈ℝ\+n0×n1\|Γ​1n1=a,Γ⊤​𝟏n0=b\}\.\\Gamma^\{\\star\}\\;:=\\;\\bigl\\\{\\,\\Gamma\\in\\mathbb\{R\}\_\{\+\}^\{n\_\{0\}\\times n\_\{1\}\}\\ \\big\|\\ \\Gamma\\,\\mathbf\{1\}\_\{n\_\{1\}\}=a,\\ \\Gamma^\{\\top\}\\mathbf\{1\}\_\{n\_\{0\}\}=b\\,\\bigr\\\}\.\(1\)
OT as a regularization principle\.Among all couplings inΠ​\(a,b\)\\Pi\(a,b\), how do we select biologically plausible ones? OT answers this via an optimization problem where the cost function is based on the*least\-action prior*: cellular states should evolve smoothly over time, so matchings that incur small feature\-wise changes are more likely\(Villani and others,[2008](https://arxiv.org/html/2605.30635#bib.bib30); Bunneet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib79)\)\. Given a cost matrixC∈ℝ\+n0×n1C\\in\\mathbb\{R\}\_\{\+\}^\{n\_\{0\}\\times n\_\{1\}\}, whereCi​j=c​\(xi,yj\)C\_\{ij\}=c\(x\_\{i\},y\_\{j\}\)is the cost of transporting a unit of mass fromxix\_\{i\}toyjy\_\{j\}, the discrete Kantorovich formulation solves

Γ⋆∈argminΓ∈Π​\(a,b\)⟨Γ,C⟩F,\\Gamma^\{\\star\}\\in\\arg\\min\_\{\\Gamma\\in\\Pi\(a,b\)\}\\langle\\Gamma,C\\rangle\_\{F\},\(2\)
where⟨⋅,⋅⟩F\\langle\\cdot,\\cdot\\rangle\_\{F\}denotes the Frobenius inner product\. The optimizerΓ⋆\\Gamma^\{\\star\}is a*soft*alignment that minimizes expected transport cost underc​\(⋅,⋅\)c\(\\cdot,\\cdot\)\. However, becauseCCdepends only on expression features, feature\-only OT cannot exploit intra\-snapshot structure such as cell–cell interactions \(CCIs\) that coordinate population dynamics\.

Table 1:Related work\.A comparison of inductive biases for aligning single\-cell population snapshots\.Method familyIntuitionSignal used for alignmentHowCellBRIDGEdiffersFeature\-based TI / OTDevelopment is assumed to be smooth in gene\-expression space\.Expression similarity, pseudotime graphs, or least\-action OT costs\.Uses intercellular communication structure rather than only cell\-intrinsic transcriptional similarity\.Velocity / dynamic priorsTransitions should follow a forward\-time direction and may include growth or death\.RNA velocity, fate probabilities, proliferation, apoptosis, or unbalanced dynamics\.Regularizes the coupling with directed, typed CCI rather than only temporal directionality\.Spatial / geometric OTAlignment should preserve physical proximity or relational geometry\.Spatial coordinates, neighborhood graphs, GW/FGW structural costs\.Does not require spatial measurements and uses ligand–receptor channels rather than generic geometry\.Communication\-aware modelsCell transitions may depend on signals exchanged with other cells\.Learned interaction effects or ligand–receptor\-derived features\.Uses interpretable, directed, typed CCI as a plug\-and\-play OT regularizer\.Incorporating intra\-snapshot structure\.Beyond inter\-snapshot feature distances, it is common to have access to structural information within each snapshot \(e\.g\., communication, adjacency, or interaction motifs\)\. The Kantorovich objective in[Equation2](https://arxiv.org/html/2605.30635#S2.E2)cannot exploit such information, since it depends only on cross\-snapshot costsCC\. The Gromov–Wasserstein \(GW\) problem extends OT by comparing distributions through their*pairwise relational*structure, favoring couplings that approximately preserve within\-snapshot relations across time\. This can be viewed as a*structure\-stationarity prior*: over the relevant time window, key relational patterns are assumed to evolve smoothly to constrain the otherwise underdetermined alignment\. Intuitively, if cell populations maintain consistent communication motifs across snapshots, then cells participating in similar interaction patterns att0t\_\{0\}should map to cells with similar patterns att1t\_\{1\}\.

We represent intra\-snapshot structure by relational matricesG\(0\)∈ℝn0×n0G^\{\(0\)\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times n\_\{0\}\}\(source\) andG\(1\)∈ℝn1×n1G^\{\(1\)\}\\in\\mathbb\{R\}^\{n\_\{1\}\\times n\_\{1\}\}\(target\)\. GW then seeks a couplingΓ⋆∈Π​\(a,b\)\\Gamma^\{\\star\}\\in\\Pi\(a,b\)that minimizes the distortion betweenG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}:

minΓ∈Π​\(a,b\)​∑i,k=1n0∑j,l=1n1L​\(Gi​k\(0\),Gj​l\(1\)\)​Γi​j​Γk​l\\min\_\{\\Gamma\\in\\Pi\(a,b\)\}\\sum\_\{i,k=1\}^\{n\_\{0\}\}\\sum\_\{j,l=1\}^\{n\_\{1\}\}L\(G^\{\(0\)\}\_\{ik\},G^\{\(1\)\}\_\{jl\}\)\\Gamma\_\{ij\}\\Gamma\_\{kl\}\(3\)
HereLLdenotes a pairwise distortion function\. We treatG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}as generic relational matrices \(not necessarily symmetric\), with the choice ofLLdetermining the notion of structure preservation\. Finally, the Fused Gromov–Wasserstein \(FGW\) problem combines feature matching with structure preservation via a parameterα∈\[0,1\]\\alpha\\in\[0,1\]:

minΓ∈Π​\(a,b\)⁡\(1−α\)​⟨Γ,C⟩F\+α​∑i,k=1n0∑j,l=1n1L​\(Gi​k\(0\),Gj​l\(1\)\)​Γi​j​Γk​l\\min\_\{\\Gamma\\in\\Pi\(a,b\)\}\(1\-\\alpha\)\\langle\\Gamma,C\\rangle\_\{F\}\+\\alpha\\sum\_\{i,k=1\}^\{n\_\{0\}\}\\sum\_\{j,l=1\}^\{n\_\{1\}\}L\(G^\{\(0\)\}\_\{ik\},G^\{\(1\)\}\_\{jl\}\)\\Gamma\_\{ij\}\\Gamma\_\{kl\}\(4\)
Settingα=0\\alpha=0recovers feature\-only OT, whileα=1\\alpha=1recovers GW\. In our setting,G\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}are not generic neighborhood graphs but directed, typed ligand–receptor communication networks: mechanistically interpretable structure obtainable from snapshot scRNA\-seq using curated LR catalogs, and editable for counterfactual analyses\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x1.png)Figure 1:Overview ofCellBRIDGE\.From an LR catalogue, we build directed CCI matrices\. A structure\-aware OT problem balances feature similarity with interaction structure to produce snapshot couplings used to train a vector field to recover cell trajectories\.
## 3Related Work

Inferring cellular trajectories from population snapshots is inherently underdetermined: many couplings between consecutive timepoints may be consistent with the observed marginal distributions\. Existing methods address this ambiguity by imposing differentinductive biaseson the alignment problem\. Table[1](https://arxiv.org/html/2605.30635#S2.T1)summarizes these biases and contrasts them with our communication\-aware regularizer\. We refer to[AppendixA](https://arxiv.org/html/2605.30635#A1)for a more detailed discussion, and provide a short summary in what follows\.

Classical trajectory\-inference methods infer developmental structure from transcriptomic neighborhoods, using pseudotime, branching, or graph abstractions to order cells along putative progressions\(Qiuet al\.,[2017](https://arxiv.org/html/2605.30635#bib.bib43); Wolfet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib46)\)\. Optimal\-transport approaches\(Peyré and Cuturi,[2019](https://arxiv.org/html/2605.30635#bib.bib5)\)provide a population\-level alternative by coupling distributions across timepoints under a least\-action principle in gene\-expression space, as in Waddington\-OT and related continuous\-time formulations\(Schiebingeret al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib37); Tonget al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib42)\)\. These methods are effective when transcriptional change is smooth, but their alignment signal is primarily cell\-intrinsic\.

A complementary line of work adds temporal directionality or population\-level constraints\. RNA velocity and its extensions use spliced and unspliced counts to infer local directions of motion, which can then be combined with transcriptomic similarity to estimate fate probabilities\(La Mannoet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib38); Bergenet al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib39); Langeet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib40)\)\. Other approaches relax mass conservation or explicitly model growth and death, thereby accounting for changes in population size across time\. Such dynamic priors help orient trajectories, but they typically constrain transitions through cell\-intrinsic dynamics rather than through structured interactions between cells\.

Spatial and geometric methods instead regularize alignment using relational structure\. Spatial OT methods exploit measured tissue coordinates to relate cells across modalities or timepoints\(Cang and Nie,[2020](https://arxiv.org/html/2605.30635#bib.bib41)\)\. These approaches show that relational information can be a powerful alignment signal\. However, spatial coordinates are unavailable in standard dissociated scRNA\-seq, and generic neighborhood graphs do not specify which cells signal to which others, nor through which ligand–receptor channels\.

Our work introduces a complementary communication\-based inductive bias\. From ligand–receptor expression, we construct a directed and typed representation of cell–cell communication and use it to regularize the OT coupling\. Thus, unlike feature\-based, velocity\-based, or spatial/geometric priors, our method encourages temporal alignments that preserve smoothly evolving signaling roles across time\. Related communication\-aware models also recognize that cell transitions may depend on intercellular effects\(Zhanget al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib10)\)\. However, our contribution is to inject interpretable ligand–receptor communication structure directly at the coupling level, making the regularizer plug\-and\-play with downstream trajectory learners\. More generally, our work integrates information from biological knowledge, an approach followed for other tasks \(e\.g\. cell annotation\(Wanget al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib6); Tanget al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib7); di Montesanoet al\.,[2026](https://arxiv.org/html/2605.30635#bib.bib3)\), or gene regulatory networks\(Hossainet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib4)\)\)\.

## 4CellBRIDGE: Interaction\-Aware Optimal Transport

Overview\.We study whether incorporating a structural prior on cell–cell interactions \(CCIs\), i\.e\. favoring couplings that encode smoothly evolving communication structure across snapshots, improves trajectory inference\. We introduceCellBRIDGE, which integrates gene\-expression features and interaction networks into a unified OT objective\. Given source and target snapshots𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\},CellBRIDGEfirst computes a cross\-snapshot coupling that assigns source to target cells probabilistically\. The coupling is designed to satisfy two desiderata:\(D1\) feature coherence, preserving smooth evolution in expression space, and\(D2\) communication evolution smoothness, favoring couplings that encode a smooth evolution of the directed CCI geometry induced by ligand–receptor expression\. We achieve this by solving a Fused Gromov–Wasserstein \(FGW\) problem that balances feature similarity and CCI structure, yieldingΓ⋆\\Gamma^\{\\star\}\. Because the interaction prior is encoded at the coupling level,Γ⋆\\Gamma^\{\\star\}can be reused as a plug\-and\-play input to downstream continuous\-time models, including flow matching, diffusion Schrödinger bridges \(DSB\), and unbalanced extensions\.

### 4\.1Interaction\-aware transport via multi LR\-pair FGW

Modeling cell–cell interactions from scRNA\.Given a ligand–receptor \(LR\) catalog𝒫=\{\(lp,rp\)∣p∈\[P\]\}\\mathcal\{P\}=\\\{\(l\_\{p\},r\_\{p\}\)\\mid p\\in\[P\]\\\}ofPPligand–receptor pairs and a dataset ofnncells, our aim is to construct a directed, nonnegative CCI tensorG∈ℝ≥0n×n×PG\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\\times n\\times P\}that summarizes potential signaling from any sender celliito receiver celljj\. Starting from raw expression counts, we first apply library\-size normalization, i\.e\. rescaling each cell’s counts by its total count and multiplying by a fixed scale factor, so that ligand and receptor expression levels are comparable across cells\. Rather than alog\(1\+⋅\)\\log\(1\{\+\}\\cdot\)transform, which compresses high\-expression signals logarithmically and can obscure fold\-change differences at high abundances, we map each gene to\[0,1\]\[0,1\]using a Hill saturation function\. The Hill form captures saturation/occupancy effects common in receptor systems and yields bounded activations while preserving rank ordering\. For geneggand cellcc, we definesc​g=\(xc​ghg\)/\(xc​ghg\+κghg\),s\_\{cg\}=\(x\_\{cg\}^\{h\_\{g\}\}\)/\(x\_\{cg\}^\{h\_\{g\}\}\+\\kappa\_\{g\}^\{h\_\{g\}\}\),with robust scaleκg\\kappa\_\{g\}\(e\.g\., theq=0\.9q\{=\}0\.9quantile of nonzero values in\{xc​g∣c∈\[n\]\}\\\{x\_\{cg\}\\mid c\\in\[n\]\\\}\) and exponenthgh\_\{g\}controlling sharpness \(we usehg=1h\_\{g\}=1in our experiments\)\.

This gives bounded activations where near\-saturating expression contributes strongly\. For an LR pairp=\(lp,rp\)p=\(l\_\{p\},r\_\{p\}\)and cellsii\(sender\) andjj\(receiver\), we score the interaction aszi→j\(p\)=si​lp​sj​rpz^\{\(p\)\}\_\{i\\to j\}=s\_\{il\_\{p\}\}\\,s\_\{jr\_\{p\}\}, capturing the intuitive requirement that ligand availability and receptor readiness must co\-occur\. We then setGi​j​p=zi→j\(p\)G\_\{ijp\}=z^\{\(p\)\}\_\{i\\to j\}\. The CCI tensorGGserves as the directed, multi\-channel structure we aim to preserve during cross\-snapshot alignment111The preprocessing described here is used only for constructing the CCI tensors\. Gene\-expression features used for the OT feature cost, downstream models, and baselines follow the standard preprocessing pipeline described in[SectionD\.1](https://arxiv.org/html/2605.30635#A4.SS1)\.\.

Interaction\-aware transport via multi LR\-pair FGW\.Given two snapshots𝒟0=\{xi\}i=1n0\\mathcal\{D\}\_\{0\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{n\_\{0\}\}and𝒟1=\{yj\}j=1n1\\mathcal\{D\}\_\{1\}=\\\{y\_\{j\}\\\}\_\{j=1\}^\{n\_\{1\}\}, we define a feature cost matrixC∈ℝ≥0n0×n1C\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\_\{0\}\\times n\_\{1\}\}such thatCi​j=c​\(xi,yj\)C\_\{ij\}=c\(x\_\{i\},y\_\{j\}\)\(typically squared Euclidean distance\)\. From the CCI construction above, we obtain directed, nonnegative tensorsG\(0\)∈ℝ≥0n0×n0×PG^\{\(0\)\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\_\{0\}\\times n\_\{0\}\\times P\}andG\(1\)∈ℝ≥0n1×n1×PG^\{\(1\)\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\_\{1\}\\times n\_\{1\}\\times P\}for the source and target snapshots\. Our objective is to find a couplingΓ∈ℝ≥0n0×n1\\Gamma\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\_\{0\}\\times n\_\{1\}\}that aligns cells while respecting CCI structure\. Letgi→k\(0\):=Gi​k:\(0\)∈ℝ≥0Pg^\{\(0\)\}\_\{i\\to k\}:=G^\{\(0\)\}\_\{ik:\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{P\}andgj→ℓ\(1\):=Gj​ℓ:\(1\)∈ℝ≥0Pg^\{\(1\)\}\_\{j\\to\\ell\}:=G^\{\(1\)\}\_\{j\\ell:\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{P\}denote LR\-channel vectors for directed edges\. We seek a coupling that is a solution to the following optimization problem:

minΓ∈Π​\(a,b\)\\displaystyle\\min\_\{\\Gamma\\in\\Pi\(a,b\)\}\(1−α\)​ℱ​\(Γ\)\+α​𝒮​\(Γ\),\\displaystyle\(1\-\\alpha\)\\,\\mathcal\{F\}\(\\Gamma\)\+\\alpha\\,\\mathcal\{S\}\(\\Gamma\),\(5\)ℱ​\(Γ\)\\displaystyle\\mathcal\{F\}\(\\Gamma\)=⟨Γ,C⟩,\\displaystyle=\\langle\\Gamma,C\\rangle,𝒮​\(Γ\)\\displaystyle\\mathcal\{S\}\(\\Gamma\)=∑i,k=1n0∑j,ℓ=1n1φ​\(gi→k\(0\),gj→ℓ\(1\)\)​Γi​j​Γk​ℓ\.\\displaystyle=\\sum\_\{i,k=1\}^\{n\_\{0\}\}\\sum\_\{j,\\ell=1\}^\{n\_\{1\}\}\\varphi\\big\(g^\{\(0\)\}\_\{i\\to k\},g^\{\(1\)\}\_\{j\\to\\ell\}\\big\)\\Gamma\_\{ij\}\\Gamma\_\{k\\ell\}\.The similarityφ​\(⋅,⋅\)\\varphi\(\\cdot,\\cdot\)measures how well the*interaction profile*between a sender–receiver pair in the source snapshot matches that of a pair in the target snapshot\. Intuitively, we penalize couplings that map cells in one snapshot to cells in the other snapshot when doing so would mismatch their directed, multi\-channel signaling context\. By default, we use squared Euclidean distanceφ​\(u,v\)=‖u−v‖22\\varphi\(u,v\)=\\\|u\-v\\\|\_\{2\}^\{2\}, which is natural here because interaction vectors are bounded in\[0,1\]P\[0,1\]^\{P\}and the loss decomposes across LR channels\. Unlike the classical FGW setting\(Vayeret al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib12)\),CellBRIDGEoperates on*multi\-typed*interactions, since each directed relation is a vector inℝP\\mathbb\{R\}^\{P\}rather than a scalar\.

Optimization\.[Equation5](https://arxiv.org/html/2605.30635#S4.E5)is non\-convex due to the quadratic structure term \(note that we do not use entropic regularization in our objective because it would favor more diffuse couplings\)\. We solve it using a customized conditional\-gradient routine adapted fromBraunet al\.\([2022](https://arxiv.org/html/2605.30635#bib.bib24)\), detailed in[SectionD\.3](https://arxiv.org/html/2605.30635#A4.SS3)\. Because the feature costsCCand interaction tensorsG\(m\)G^\{\(m\)\}can have different magnitudes and units, the trade\-off parameterα\\alphain[Equation5](https://arxiv.org/html/2605.30635#S4.E5)does not, by itself, guarantee a meaningful balance\. Without normalization, one term can dominate the objective, makingα\\alphadifficult to interpret and tune\. To balance the two contributions, we normalize by*endpoints*: we solve the feature\-only problem \(α=0\\alpha=0\) and the structure\-only problem \(α=1\\alpha=1\), obtaining reference objective valuesF⋆=⟨Γα=0⋆,C⟩FF^\{\\star\}=\\langle\\Gamma^\{\\star\}\_\{\\alpha=0\},C\\rangle\_\{F\}andS⋆=𝒮​\(Γα=1⋆\)S^\{\\star\}=\\mathcal\{S\}\(\\Gamma^\{\\star\}\_\{\\alpha=1\}\)\. We then rescale the objective so that intermediateα\\alphavalues interpolate comparably between these extremes \(details in[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4)\)\.

### 4\.2From couplings to continuous dynamics

Coupling\-level prior\.A key design principle ofCellBRIDGEis that interaction structure is injected*only at the level of the cross\-snapshot coupling*\. Solving the interaction\-aware FGW problem in[Equation5](https://arxiv.org/html/2605.30635#S4.E5)yields a couplingΓ⋆\\Gamma^\{\\star\}between𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\}\. Crucially,Γ⋆\\Gamma^\{\\star\}can be reused by any downstream method that learns continuous\-time dynamics from paired endpoints\. Thus,CellBRIDGEprovides a*plug\-and\-play structural prior*that is orthogonal to the choice of dynamics model\.

Coupling\-induced endpoint distribution\.From the couplingΓ⋆\\Gamma^\{\\star\}we form a joint distributionΠ=∑i,jΓ¯i​j⋆​δ\(xi,yj\),Γ¯i​j⋆:=Γi​j⋆/M,M:=∑i,jΓi​j⋆,\\Pi\\;=\\;\\sum\_\{i,j\}\\bar\{\\Gamma\}^\{\\star\}\_\{ij\}\\,\\delta\_\{\(x\_\{i\},y\_\{j\}\)\},\\qquad\\bar\{\\Gamma\}^\{\\star\}\_\{ij\}:=\\Gamma^\{\\star\}\_\{ij\}/M,\\ \\ M:=\\sum\_\{i,j\}\\Gamma^\{\\star\}\_\{ij\},whose marginals match the empirical snapshot measuresρ0\\rho\_\{0\}andρ1\\rho\_\{1\}induced by𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\}\. Downstream continuous\-time models differ in how they construct intermediate\-time states between paired endpoints\. We capture this choice through a \(possibly stochastic\)*interpolant*ItI\_\{t\}\(Albergoet al\.,[2023](https://arxiv.org/html/2605.30635#bib.bib11)\):\(X,Y\)∼Π,Zt=It​\(X,Y,ξ\),ρt:=Law⁡\(Zt\),\(X,Y\)\\sim\\Pi,\\qquad Z\_\{t\}=I\_\{t\}\(X,Y,\\xi\),\\qquad\\rho\_\{t\}:=\\operatorname\{Law\}\(Z\_\{t\}\),whereξ\\xidenotes auxiliary randomness \(absent for deterministic interpolants\)\. Intuitively,Π\\Pifixes*which*endpoints are paired \(and with what mass\), while the interpolant specifies*how*mass is distributed at intermediate times\.

Many trajectory\-learning objectives can be written as regressing one or more model fields to targets induced by the chosen interpolant\. Letfθ\(m\)​\(z,t\)f\_\{\\theta\}^\{\(m\)\}\(z,t\)denote model outputs \(e\.g\. a velocity head and/or a score head\), and letτt\(m\)​\(z∣X,Y,ξ\)\\tau\_\{t\}^\{\(m\)\}\(z\\mid X,Y,\\xi\)be the corresponding interpolant\-induced targets \(e\.g\. conditional velocity, conditional drift, or conditional score\)\. We consider the generic objective

ℒ​\(θ\)\\displaystyle\\mathcal\{L\}\(\\theta\)=𝔼Π,t,ξ​\[∑m∈ℳλm​\(t\)​‖fθ\(m\)​\(Zt,t\)−Tt\(m\)‖\(t\)2\],\\displaystyle=\\mathbb\{E\}\_\{\\Pi,t,\\xi\}\\\!\\left\[\\sum\_\{m\\in\\mathcal\{M\}\}\\lambda\_\{m\}\(t\)\\left\\\|f\_\{\\theta\}^\{\(m\)\}\(Z\_\{t\},t\)\-T\_\{t\}^\{\(m\)\}\\right\\\|\_\{\(t\)\}^\{2\}\\right\],\(6\)Tt\(m\)\\displaystyle T\_\{t\}^\{\(m\)\}=τt\(m\)​\(Zt∣X,Y,ξ\)\.\\displaystyle=\\tau\_\{t\}^\{\(m\)\}\(Z\_\{t\}\\mid X,Y,\\xi\)\.where the expectation is over\(X,Y\)∼Π\(X,Y\)\\sim\\Pi,t∼Unif​\[0,1\]t\\sim\\mathrm\{Unif\}\[0,1\],ξ\\xi, andZt=It​\(X,Y,ξ\)Z\_\{t\}=I\_\{t\}\(X,Y,\\xi\)\. Here,λm​\(t\)\\lambda\_\{m\}\(t\)are scalar weights and∥⋅∥\(t\)\\\|\\cdot\\\|\_\{\(t\)\}is either Euclidean or metric\-induced\. CFM\(Lipmanet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib29)\), SF2M\(Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\), and MFM\(Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20)\)correspond to particular choices of interpolantItI\_\{t\}and targetsτt\(m\)\\tau\_\{t\}^\{\(m\)\}, whileCellBRIDGEspecifies the endpoint distributionΠ\\Pi\.

Conditional Flow Matching \(CFM\)\.CFM\(Lipmanet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib34)\)is traditionally implemented with the deterministic affine interpolantIt​\(X,Y\)=\(1−t\)​X\+t​YI\_\{t\}\(X,Y\)=\(1\-t\)X\+tY, i\.e\.Zt=\(1−t\)​X\+t​YZ\_\{t\}=\(1\-t\)X\+tY\. With a single velocity headfθ\(v\)=vθf\_\{\\theta\}^\{\(v\)\}=v\_\{\\theta\}, the target is constantτt\(v\)​\(Zt∣X,Y\)=Y−X\\tau\_\{t\}^\{\(v\)\}\(Z\_\{t\}\\mid X,Y\)=Y\-X, yielding:

ℒCFM​\(θ\)=𝔼\(X,Y\)∼Πt∼Unif​\[0,1\]​\[‖vθ​\(Zt,t\)−\(Y−X\)‖22\]\.\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(X,Y\)\\sim\\Pi\\\\ t\\sim\\mathrm\{Unif\}\[0,1\]\\end\{subarray\}\}\\Big\[\\,\\big\\\|\\,v\_\{\\theta\}\\big\(Z\_\{t\},t\\big\)\\;\-\\;\(Y\-X\)\\,\\big\\\|\_\{2\}^\{2\}\\,\\Big\]\.\(7\)Trajectories are then obtained by integratingz˙​\(t\)=vθ​\(z​\(t\),t\)\\dot\{z\}\(t\)=v\_\{\\theta\}\(z\(t\),t\)\.

Stochastic bridges via SF2M\.SF2M\(Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\)replaces the deterministic affine interpolant by a stochastic interpolant, e\.g\. samplingZtZ\_\{t\}from a Brownian bridge between endpoints\(X,Y\)∼Π\(X,Y\)\\sim\\Pi, which provides closed\-form conditional drifts and scores at intermediate times\. We combineCellBRIDGEwith SF2M by instantiating the endpoint distribution withΠ\\Pi\(i\.e\. sampling\(X,Y\)∼Π\(X,Y\)\\sim\\Pi\), while keeping the SF2M training objective and its interpolant\-induced targets unchanged\. We provide more details in[SectionE\.4](https://arxiv.org/html/2605.30635#A5.SS4)\.

Geometric priors via Metric Flow Matching \(MFM\)\.MFM\(Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20)\)corresponds to a geometry\-aware interpolant in which intermediate states follow geodesic interpolants under a learned, data\-dependent Riemannian metric\. In[Equation6](https://arxiv.org/html/2605.30635#S4.E6), this amounts to choosingZt=Itgeo​\(X,Y\)Z\_\{t\}=I\_\{t\}^\{\\mathrm\{geo\}\}\(X,Y\)\(or its learned approximation\), using a metric\-induced norm∥⋅∥\(t\)\\\|\\cdot\\\|\_\{\(t\)\}, and setting the target to the associated geodesic velocityτt\(v\)=∂tItgeo​\(X,Y\)\\tau\_\{t\}^\{\(v\)\}=\\partial\_\{t\}I\_\{t\}^\{\\mathrm\{geo\}\}\(X,Y\)\. We combineCellBRIDGEwith MFM by reusing the same endpoint pairs sampled fromΠ\\Pi, while replacing linear interpolation with the learned geodesic interpolant \(see[SectionE\.3](https://arxiv.org/html/2605.30635#A5.SS3)\)\.

Unbalanced dynamics\.When total population mass changes between snapshots \(e\.g\. proliferation or apoptosis\), we extendCellBRIDGEwith an unbalanced OT formulation to infer non\-uniform marginals prior to solving the interaction\-aware FGW problem \(see[SectionE\.2](https://arxiv.org/html/2605.30635#A5.SS2)\)\. The resulting coupling \(and thusΠ\\Pi\) can again be passed unchanged into any downstream choice of interpolant and matching objective, including CFM, SF2M, or MFM\.

## 5Experiments

We evaluate whether incorporating cell–cell interaction \(CCI\) structure improves cross\-snapshot alignment and downstream continuous\-time trajectory inference, and whether the resulting inductive bias is biologically grounded\. Our evaluation is organized around four questions:\(Q1\) Couplings:doesCellBRIDGEproduce more faithful transport maps than feature\-only baselines?\(Q2\) Trajectories:do improved couplings translate into improved continuous\-time dynamics?\(Q3\) Grounding:are the gains driven by biologically meaningful ligand–receptor \(LR\) structure?\(Q4\) Failure modes:when does the interaction prior become uninformative or harmful?

### 5\.1Do interaction\-aware couplings improve cross\-snapshot alignment?

Throughout this section, we evaluate the quality of a cross\-snapshot coupling by*held\-out interpolation*\. Given two endpoint snapshots att0<t2t\_\{0\}<t\_\{2\}, we infer a couplingΓ\\Gammabetween𝒟0\\mathcal\{D\}\_\{0\}and𝒟2\\mathcal\{D\}\_\{2\}\. We then define an intermediate distribution att1t\_\{1\}via affine interpolation along endpoint pairs sampled from the coupling and compare the interpolated marginal to the empirical snapshot att1t\_\{1\}\. This isolates coupling quality*independently of any downstream dynamics model*\. In what follows, the coupling is computed once at the full snapshot level, i\.e\. using the full source and target snapshots rather than mini\-batches222Mini\-batching is used only when we train the downstream flow model in[Section4\.2](https://arxiv.org/html/2605.30635#S4.SS2), and we sample from the joint distribution induced by the coupling to define the batches\.\.

Synthetic setup\.We consider two 2D snapshots, each composed of three clusters\. The second snapshot is obtained by translating each cluster by a distinct vector, inducing a known one\-to\-one ground\-truth transport\. We define an interaction structure withP=2P=2types : the middle cluster points to the left \(Pathway 1\) and to the right \(Pathway 2\), mirrored in the target snapshot \(see[SectionC\.1](https://arxiv.org/html/2605.30635#A3.SS1)for more details\)\. We then obtain a coupling for eachα∈\{0,0\.1,…,1\}\\alpha\\in\\\{0,0\.1,\\ldots,1\\\}by solving the FGW problem defined in[Equation5](https://arxiv.org/html/2605.30635#S4.E5)with the ground\-truth interaction structures\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x2.png)Figure 2:Structure\-aware coupling recovers the ground\-truth transport map\.We consider within\-snapshot interactions with two channels A \(orange to blue\) and B \(orange to green\)\.Synthetic results\.Representative couplings acrossα\\alphaare shown in[Figure2](https://arxiv.org/html/2605.30635#S5.F2)\. Withα=0\\alpha=0\(feature\-only\), the interaction structure is ignored and clusters are misaligned; withα=1\\alpha=1\(structure\-only\), interaction types are satisfied but geometry is distorted\. An intermediate setting \(α≈0\.7\\alpha\\approx 0\.7\) preserves the directed relations while maintaining within\-interaction geometry\. We refer to[SectionG\.1](https://arxiv.org/html/2605.30635#A7.SS1)for a theoretical analysis of this synthetic setup\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x3.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x4.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x5.png)

Figure 3:Coupling quality via held\-out interpolation\.We plot theW1W\_\{1\}andW2W\_\{2\}distances between the interpolated and empiricalt1t\_\{1\}snapshots asα\\alphavaries\. Optimal performance occurs at dataset\-specificα∗\>0\\alpha^\{\\ast\}\>0\.Table 2:Interpolation error for continuous\-time dynamics \(lower is better\)\.We report mean±\\pmstd over55runs\.V1 LightDendritic StimulusLung tumorMethod𝜶\\boldsymbol\{\\alpha\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}TrajectoryNet—3\.022​\(0\.061\)3\.022\{\\scriptstyle\\,\(\\,0\.061\\,\)\}3\.338​\(0\.056\)3\.338\{\\scriptstyle\\,\(\\,0\.056\\,\)\}4\.410​\(0\.102\)4\.410\{\\scriptstyle\\,\(\\,0\.102\\,\)\}4\.607​\(0\.107\)4\.607\{\\scriptstyle\\,\(\\,0\.107\\,\)\}2\.712​\(0\.090\)2\.712\{\\scriptstyle\\,\(\\,0\.090\\,\)\}3\.056​\(0\.099\)3\.056\{\\scriptstyle\\,\(\\,0\.099\\,\)\}DSB—3\.819​\(0\.152\)3\.819\{\\scriptstyle\\,\(\\,0\.152\\,\)\}3\.875​\(0\.143\)3\.875\{\\scriptstyle\\,\(\\,0\.143\\,\)\}4\.099​\(0\.155\)4\.099\{\\scriptstyle\\,\(\\,0\.155\\,\)\}4\.249​\(0\.153\)4\.249\{\\scriptstyle\\,\(\\,0\.153\\,\)\}3\.700​\(0\.116\)3\.700\{\\scriptstyle\\,\(\\,0\.116\\,\)\}3\.967​\(0\.102\)3\.967\{\\scriptstyle\\,\(\\,0\.102\\,\)\}VGFM—6\.446​\(0\.114\)6\.446\{\\scriptstyle\\,\(\\,0\.114\\,\)\}6\.745​\(0\.102\)6\.745\{\\scriptstyle\\,\(\\,0\.102\\,\)\}7\.087​\(0\.022\)7\.087\{\\scriptstyle\\,\(\\,0\.022\\,\)\}7\.261​\(0\.026\)7\.261\{\\scriptstyle\\,\(\\,0\.026\\,\)\}2\.175​\(0\.017\)2\.175\{\\scriptstyle\\,\(\\,0\.017\\,\)\}2\.478​\(0\.019\)2\.478\{\\scriptstyle\\,\(\\,0\.019\\,\)\}MIOFlow—6\.360​\(0\.010\)6\.360\{\\scriptstyle\\,\(\\,0\.010\\,\)\}6\.655​\(0\.009\)6\.655\{\\scriptstyle\\,\(\\,0\.009\\,\)\}6\.970​\(0\.043\)6\.970\{\\scriptstyle\\,\(\\,0\.043\\,\)\}7\.159​\(0\.034\)7\.159\{\\scriptstyle\\,\(\\,0\.034\\,\)\}2\.001​\(0\.003\)2\.001\{\\scriptstyle\\,\(\\,0\.003\\,\)\}2\.316​\(0\.009\)2\.316\{\\scriptstyle\\,\(\\,0\.009\\,\)\}SnapMMD—2\.420​\(0\.005\)2\.420\{\\scriptstyle\\,\(\\,0\.005\\,\)\}2\.657​\(0\.005\)2\.657\{\\scriptstyle\\,\(\\,0\.005\\,\)\}3\.863​\(0\.036\)3\.863\{\\scriptstyle\\,\(\\,0\.036\\,\)\}4\.022​\(0\.048\)4\.022\{\\scriptstyle\\,\(\\,0\.048\\,\)\}2\.237​\(0\.143\)2\.237\{\\scriptstyle\\,\(\\,0\.143\\,\)\}2\.520​\(0\.115\)2\.520\{\\scriptstyle\\,\(\\,0\.115\\,\)\}Moscot—6\.242​\(0\.000\)6\.242\{\\scriptstyle\\,\(\\,0\.000\\,\)\}6\.545​\(0\.000\)6\.545\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.115​\(0\.000\)7\.115\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.331​\(0\.000\)7\.331\{\\scriptstyle\\,\(\\,0\.000\\,\)\}2\.000​\(0\.000\)2\.000\{\\scriptstyle\\,\(\\,0\.000\\,\)\}2\.335​\(0\.000\)2\.335\{\\scriptstyle\\,\(\\,0\.000\\,\)\}OT\-CFM—2\.392​\(0\.005\)2\.392\{\\scriptstyle\\,\(\\,0\.005\\,\)\}2\.625​\(0\.007\)2\.625\{\\scriptstyle\\,\(\\,0\.007\\,\)\}3\.696​\(0\.007\)3\.696\{\\scriptstyle\\,\(\\,0\.007\\,\)\}3\.857​\(0\.009\)3\.857\{\\scriptstyle\\,\(\\,0\.009\\,\)\}1\.993​\(0\.004\)1\.993\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.275​\(0\.005\)2\.275\{\\scriptstyle\\,\(\\,0\.005\\,\)\}OT\-MFM—2\.401​\(0\.003\)2\.401\{\\scriptstyle\\,\(\\,0\.003\\,\)\}2\.636​\(0\.003\)2\.636\{\\scriptstyle\\,\(\\,0\.003\\,\)\}3\.714​\(0\.008\)3\.714\{\\scriptstyle\\,\(\\,0\.008\\,\)\}3\.880​\(0\.009\)3\.880\{\\scriptstyle\\,\(\\,0\.009\\,\)\}1\.984​\(0\.004\)1\.984\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.285​\(0\.004\)2\.285\{\\scriptstyle\\,\(\\,0\.004\\,\)\}UOT\-FM—2\.411​\(0\.005\)2\.411\{\\scriptstyle\\,\(\\,0\.005\\,\)\}2\.649​\(0\.006\)2\.649\{\\scriptstyle\\,\(\\,0\.006\\,\)\}3\.701​\(0\.006\)3\.701\{\\scriptstyle\\,\(\\,0\.006\\,\)\}3\.867​\(0\.007\)3\.867\{\\scriptstyle\\,\(\\,0\.007\\,\)\}1\.998​\(0\.004\)1\.998\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.348​\(0\.004\)2\.348\{\\scriptstyle\\,\(\\,0\.004\\,\)\}SF2M—3\.254​\(0\.192\)3\.254\{\\scriptstyle\\,\(\\,0\.192\\,\)\}3\.368​\(0\.182\)3\.368\{\\scriptstyle\\,\(\\,0\.182\\,\)\}4\.333​\(0\.279\)4\.333\{\\scriptstyle\\,\(\\,0\.279\\,\)\}4\.436​\(0\.282\)4\.436\{\\scriptstyle\\,\(\\,0\.282\\,\)\}3\.826​\(0\.265\)3\.826\{\\scriptstyle\\,\(\\,0\.265\\,\)\}3\.974​\(0\.308\)3\.974\{\\scriptstyle\\,\(\\,0\.308\\,\)\}0\.53\.199​\(0\.117\)3\.199\{\\scriptstyle\\,\(\\,0\.117\\,\)\}3\.315​\(0\.110\)3\.315\{\\scriptstyle\\,\(\\,0\.110\\,\)\}4\.303​\(0\.213\)4\.303\{\\scriptstyle\\,\(\\,0\.213\\,\)\}4\.397​\(0\.205\)4\.397\{\\scriptstyle\\,\(\\,0\.205\\,\)\}3\.809​\(0\.302\)3\.809\{\\scriptstyle\\,\(\\,0\.302\\,\)\}3\.968​\(0\.374\)3\.968\{\\scriptstyle\\,\(\\,0\.374\\,\)\}CellBRIDGE\+SF2M13\.226​\(0\.075\)3\.226\{\\scriptstyle\\,\(\\,0\.075\\,\)\}3\.339​\(0\.073\)3\.339\{\\scriptstyle\\,\(\\,0\.073\\,\)\}4\.289​\(0\.110\)4\.289\{\\scriptstyle\\,\(\\,0\.110\\,\)\}4\.387​\(0\.108\)4\.387\{\\scriptstyle\\,\(\\,0\.108\\,\)\}3\.638​\(0\.308\)3\.638\{\\scriptstyle\\,\(\\,0\.308\\,\)\}3\.739​\(0\.335\)3\.739\{\\scriptstyle\\,\(\\,0\.335\\,\)\}0\.52\.393​\(0\.007\)2\.393\{\\scriptstyle\\,\(\\,0\.007\\,\)\}2\.631​\(0\.008\)2\.631\{\\scriptstyle\\,\(\\,0\.008\\,\)\}3\.679​\(0\.007\)3\.679\{\\scriptstyle\\,\(\\,0\.007\\,\)\}3\.838​\(0\.009\)3\.838\{\\scriptstyle\\,\(\\,0\.009\\,\)\}1\.978​\(0\.004\)1\.978\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.277​\(0\.003\)2\.277\{\\scriptstyle\\,\(\\,0\.003\\,\)\}CellBRIDGE\+MFM12\.363​\(0\.002\)2\.363\{\\scriptstyle\\,\(\\,0\.002\\,\)\}2\.606​\(0\.002\)2\.606\{\\scriptstyle\\,\(\\,0\.002\\,\)\}3\.668​\(0\.010\)3\.668\{\\scriptstyle\\,\(\\,0\.010\\,\)\}3\.824​\(0\.011\)3\.824\{\\scriptstyle\\,\(\\,0\.011\\,\)\}2\.013​\(0\.003\)2\.013\{\\scriptstyle\\,\(\\,0\.003\\,\)\}2\.304​\(0\.003\)2\.304\{\\scriptstyle\\,\(\\,0\.003\\,\)\}0\.52\.377​\(0\.004\)2\.377\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.619​\(0\.005\)2\.619\{\\scriptstyle\\,\(\\,0\.005\\,\)\}3\.688​\(0\.012\)3\.688\{\\scriptstyle\\,\(\\,0\.012\\,\)\}3\.854​\(0\.012\)3\.854\{\\scriptstyle\\,\(\\,0\.012\\,\)\}1\.971​\(0\.005\)1\.971\{\\scriptstyle\\,\(\\,0\.005\\,\)\}2\.322​\(0\.005\)2\.322\{\\scriptstyle\\,\(\\,0\.005\\,\)\}CellBRIDGE\+UOT\-FM12\.360​\(0\.002\)2\.360\{\\scriptstyle\\,\(\\,0\.002\\,\)\}2\.605​\(0\.001\)2\.605\{\\scriptstyle\\,\(\\,0\.001\\,\)\}3\.624​\(0\.004\)3\.624\{\\scriptstyle\\,\(\\,0\.004\\,\)\}3\.780​\(0\.002\)3\.780\{\\scriptstyle\\,\(\\,0\.002\\,\)\}1\.993​\(0\.004\)1\.993\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.335​\(0\.005\)2\.335\{\\scriptstyle\\,\(\\,0\.005\\,\)\}CellBRIDGE\+CFM0\.52\.381​\(0\.004\)2\.381\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.618​\(0\.003\)2\.618\{\\scriptstyle\\,\(\\,0\.003\\,\)\}3\.679​\(0\.009\)3\.679\{\\scriptstyle\\,\(\\,0\.009\\,\)\}3\.835​\(0\.010\)3\.835\{\\scriptstyle\\,\(\\,0\.010\\,\)\}1\.989​\(0\.004\)1\.989\{\\scriptstyle\\,\(\\,0\.004\\,\)\}2\.272​\(0\.005\)2\.272\{\\scriptstyle\\,\(\\,0\.005\\,\)\}12\.362​\(0\.003\)2\.362\{\\scriptstyle\\,\(\\,0\.003\\,\)\}2\.601​\(0\.005\)2\.601\{\\scriptstyle\\,\(\\,0\.005\\,\)\}3\.639​\(0\.021\)3\.639\{\\scriptstyle\\,\(\\,0\.021\\,\)\}3\.788​\(0\.021\)3\.788\{\\scriptstyle\\,\(\\,0\.021\\,\)\}2\.057​\(0\.005\)2\.057\{\\scriptstyle\\,\(\\,0\.005\\,\)\}2\.329​\(0\.005\)2\.329\{\\scriptstyle\\,\(\\,0\.005\\,\)\}

Real\-world datasets\.We evaluateCellBRIDGEon six real\-world scRNA\-seq datasets whose characteristics are summarized in[Table6](https://arxiv.org/html/2605.30635#A3.T6)\. We selected these datasets because their temporal coverage provides a favorable window in which ligand–receptor \(LR\) interactions are expected to remain approximately persistent\. Following standard preprocessing, we project gene\-expression profiles onto the topd=20d=20principal components \([SectionD\.1](https://arxiv.org/html/2605.30635#A4.SS1)\) and standardize them as inTonget al\.\([2024a](https://arxiv.org/html/2605.30635#bib.bib36)\)\. Additional details on dataset collection are provided in[AppendixC](https://arxiv.org/html/2605.30635#A3), and results on further datasets are in[AppendixF](https://arxiv.org/html/2605.30635#A6)\.

Setup\.We build CCI tensors by selecting dataset\-specific ligand–receptor pairs via an automated procedure that accounts for stability of expression levels across snapshots \(cf\.[SectionD\.5](https://arxiv.org/html/2605.30635#A4.SS5)for more details\)\. Given three time pointst0<t1<t2t\_\{0\}<t\_\{1\}<t\_\{2\}, we hold out the snapshot att1t\_\{1\}\. Using onlyt0t\_\{0\}andt2t\_\{2\}, and for a chosen LR catalog𝒫\\mathcal\{P\}and hyperparameterα∈\{0,0\.1,…,1\.0\}\\alpha\\in\\\{0,0\.1,\\ldots,1\.0\\\}, we obtain a couplingΓ​\(α,𝒫\)\\Gamma\(\\alpha,\\mathcal\{P\}\)by solving the OT problem defined in[Equation5](https://arxiv.org/html/2605.30635#S4.E5)\. We define the marginal att1t\_\{1\}by affine interpolation and denote it byρt1​\(α,𝒫\)\\rho\_\{t\_\{1\}\}\(\\alpha,\\mathcal\{P\}\)\. For eachα\\alphaand𝒫\\mathcal\{P\}, we compareρt1​\(α,𝒫\)\\rho\_\{t\_\{1\}\}\(\\alpha,\\mathcal\{P\}\)with the empirical distributionρt1\\rho\_\{t\_\{1\}\}observed att1t\_\{1\}, computing the Wasserstein\-1 and Wasserstein\-2 distances\.

Results\.Across datasets, incorporating CCI structure improves alignment, with optimal performance at a dataset\-specificα∗\>0\\alpha^\{\\ast\}\>0\(see[Figure3](https://arxiv.org/html/2605.30635#S5.F3)\)\. We observe two regimes: a U\-shaped curve with0<α∗<10<\\alpha^\{\\ast\}<1, indicating that combining CCI with feature\-only OT is best, and an almost monotonic decrease with a minimum atα∗=1\\alpha^\{\\ast\}=1for theDendritic Stimulusdataset\. For the latter, this may reflect the dataset’s smaller size and coherent stimulation response, which make feature\-only OT comparatively less informative than the interaction structure\. Taken together with the synthetic results, these findings support \(Q1\): encoding directed, typed CCI structure improves coupling fidelity in settings where interaction geometry is \(approximately\) persistent\.

### 5\.2Cross\-snapshot trajectory inference with learned velocity fields

Setup\.We evaluateCellBRIDGEagainst a comprehensive set of state\-of\-the\-art trajectory inference methods to assess whether improvements in coupling quality translate into improved continuous\-time dynamics\. Starting from a couplingΓ\\Gammainferred betweent0t\_\{0\}andt2t\_\{2\}, we learn a time\-dependent velocity field and integrate it to transport cells fromt0t\_\{0\}to the held\-out snapshot att1t\_\{1\}\. Performance is measured by the Wasserstein\-1 and Wasserstein\-2 distances between the transported distribution and the empirical distribution att1t\_\{1\}\. We report results forα∈\{0,0\.5,1\}\\alpha\\in\\\{0,0\.5,1\\\}\. We benchmark against representative methods spanning multiple paradigms: neural ODE models\(Tonget al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib42); Huguetet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib61)\), stochastic Schrödinger bridges\(De Bortoliet al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib25); Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\), flow\-matching approaches\(Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20); Eyringet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib14); Wanget al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib18)\), OT baselines\(Kleinet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib19)\)and kernel\-based methods\(Berlinghieriet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib9)\)\. Because performance in continuous\-time models can be sensitive to how endpoint correspondences are established, we evaluateCellBRIDGEas a*plug\-and\-play coupling prior*\. Specifically, for four representative baselines \(CFM,MFM,UOT\-FM, andSF2M\), we replace the default coupling or matching mechanism with the interaction\-aware coupling yielding “CellBRIDGE\+” variants\. In these settings, the dynamics model, architecture, and training objective are held fixed, isolating the effect of coupling quality\. Implementation details are provided in[SectionsE\.3](https://arxiv.org/html/2605.30635#A5.SS3),[E\.2](https://arxiv.org/html/2605.30635#A5.SS2)and[E\.4](https://arxiv.org/html/2605.30635#A5.SS4)\.

Results\.[Table2](https://arxiv.org/html/2605.30635#S5.T2)reports interpolation error at the held\-out timet1t\_\{1\}\. Across datasets and trajectory\-learning paradigms, we observe consistent improvements when replacing feature\-only or entropic couplings with theCellBRIDGEcoupling\. Bothα=0\.5\\alpha=0\.5andα=1\\alpha=1generally improve over the feature\-only case \(α=0\\alpha=0\), with the balanced settingα=0\.5\\alpha=0\.5providing the most consistent gains across the paired comparisons\. A paired Wilcoxon significance analysis confirms that these improvements remain significant after Holm correction, with adjustedp<0\.05p<0\.05for all model families and Cohen’sdzd\_\{z\}values ranging from0\.270\.27to0\.630\.63, corresponding to small\-to\-moderate paired effect sizes\. Notably, the same interaction\-aware coupling improves multiple downstream methods, demonstrating that gains arise from improved cross\-snapshot alignment rather than from method\-specific architectural choices\.

### 5\.3Are the gains driven by biologically meaningful ligand–receptor structure?

We useCellBRIDGEto simulate intercellular perturbations on theLung Tumordataset by ablating specific pathways from the ligand–receptor catalog, recomputing the CCI tensors, and re\-solving[Equation5](https://arxiv.org/html/2605.30635#S4.E5)\. This intervention alters only the interaction prior \(with baseline expression att=0t=0fixed\), mimicking a pharmacological blockade prior to transcriptional adaptation\(Leeet al\.,[2016](https://arxiv.org/html/2605.30635#bib.bib110)\)\. We quantify trajectory shifts relative to the unperturbed baseline using the 20 Hallmarks of Cancer gene sets \(Appendix[D\.7\.1](https://arxiv.org/html/2605.30635#A4.SS7.SSS1)\) over a 24h interpolation window\.

Results\.[Figure4\(a\)](https://arxiv.org/html/2605.30635#S5.F4.sf1)shows the relative decrease in tumour\-associated progression scores under different catalog edits\. Attenuating signaling through EGFR, ALK, or MET produces measurable reductions \(up to 15\.5%\), indicating that the inferred trajectories are sensitive to these pathways\. This aligns with their established therapeutic relevance in non–small cell lung cancer, where EGFR inhibitors \(e\.g\., gefitinib, osimertinib\), ALK inhibitors \(e\.g\., crizotinib, alectinib\), and MET inhibitors \(e\.g\., capmatinib, tepotinib\) are used clinically\(Domvriet al\.,[2013](https://arxiv.org/html/2605.30635#bib.bib90)\)\. By contrast, edits to unrelated cardio–renal pathways \(RAAS, vasopressin, natriuretic peptides\) yield negligible changes, suggesting thatCellBRIDGEresponds specifically to biologically relevant ligand–receptor structure rather than arbitrary perturbations\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x6.png)\(a\)Insilicointerventions\.
![Refer to caption](https://arxiv.org/html/2605.30635v1/x7.png)\(b\)Embryo dataset\.

Figure 4:When interaction priors help and when they do not\.Left: pathway\-specific LR edits produce measurable trajectory shifts\. Right: in rapidly remodeling development, CCI structure may be non\-persistent and provides no benefit\.Ablations\.Motivated by the observation that editing the LR catalog shifts inferred trajectories, we test whether gains are driven by coherent LR structure rather than arbitrary regularization by applying three controlled perturbations to the CCI construction atα=1\\alpha=1:*Random LR catalog*—replace the curated LR catalog with a random subset of the same size;*Shuffling*—randomly permute all entries of the CCI tensors, destroying coherent structure;*Metacells*—aggregate cells into metacells before constructing CCIs and then lift interactions back to the cell level \(see[SectionD\.2](https://arxiv.org/html/2605.30635#A4.SS2)\), thereby smoothing the signal\. We report the results in Table[3](https://arxiv.org/html/2605.30635#S5.T3), where we compute the metrics based on the interpolation setup described in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)\. Shuffling the CCI leads to a performance drop, confirming that the*structural organization*of LR interactions drives the gains\. Using a random LR catalog also degrades interpolation, highlighting the importance ofLR specificity\. Metacell\-based CCI yields intermediate performance by smoothing dropout noise, but can oversmooth, degrading performance, consistent with prior spatiotemporal analyses\(Kleinet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib19)\)\.

Table 3:Sensitivity analysis on the CCI construction\.We report results mimicking the marginal interpolation setup followed in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)\. Baselines are reported as mean±\\pmstandard deviation over five seeds, whileCellBRIDGEis deterministic\.V1 LightDendritic StimulusLung tumorMethodW1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}Shuffle2\.441​\(0\.002\)2\.441\{\\scriptstyle\\,\(\\,0\.002\\,\)\}2\.646​\(0\.002\)2\.646\{\\scriptstyle\\,\(\\,0\.002\\,\)\}3\.644​\(0\.004\)3\.644\{\\scriptstyle\\,\(\\,0\.004\\,\)\}3\.748​\(0\.007\)3\.748\{\\scriptstyle\\,\(\\,0\.007\\,\)\}2\.188​\(0\.007\)2\.188\{\\scriptstyle\\,\(\\,0\.007\\,\)\}2\.392​\(0\.007\)2\.392\{\\scriptstyle\\,\(\\,0\.007\\,\)\}Random LR2\.446​\(0\.006\)2\.446\{\\scriptstyle\\,\(\\,0\.006\\,\)\}2\.655​\(0\.008\)2\.655\{\\scriptstyle\\,\(\\,0\.008\\,\)\}3\.602​\(0\.022\)3\.602\{\\scriptstyle\\,\(\\,0\.022\\,\)\}3\.729​\(0\.019\)3\.729\{\\scriptstyle\\,\(\\,0\.019\\,\)\}2\.171​\(0\.030\)2\.171\{\\scriptstyle\\,\(\\,0\.030\\,\)\}2\.388​\(0\.033\)2\.388\{\\scriptstyle\\,\(\\,0\.033\\,\)\}Metacell2\.329​\(0\.000\)2\.329\{\\scriptstyle\\,\(\\,0\.000\\,\)\}2\.567​\(0\.002\)2\.567\{\\scriptstyle\\,\(\\,0\.002\\,\)\}3\.587​\(0\.005\)3\.587\{\\scriptstyle\\,\(\\,0\.005\\,\)\}3\.725​\(0\.000\)3\.725\{\\scriptstyle\\,\(\\,0\.000\\,\)\}2\.052​\(0\.002\)2\.052\{\\scriptstyle\\,\(\\,0\.002\\,\)\}2\.344​\(0\.002\)2\.344\{\\scriptstyle\\,\(\\,0\.002\\,\)\}CellBRIDGE2\.3502\.5873\.5853\.7322\.0282\.298
### 5\.4Does structure always help?

Setup\.Our formulation does not assume static cell–cell interaction \(CCI\) structure across time\. Instead, as in standard optimal transport, we impose a smoothness principle: among admissible couplings, we favor those that minimize feature displacement while approximately preserving directed interaction geometry\. This is appropriate when snapshots are separated by modest temporal gaps and the system evolves smoothly\. Under rapid, large\-scale remodeling this assumption can fail, a known limitation of OT\-based alignment rather than aCellBRIDGE\-specific issue\(Bunneet al\.,[2023](https://arxiv.org/html/2605.30635#bib.bib35)\)\. We illustrate this regime on a developing mouse embryo dataset\(Moonet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib47)\), where tissue composition, size, and function shift abruptly and snapshots are six days apart\(Qiuet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib106)\)\.

Results\.[Figure4\(b\)](https://arxiv.org/html/2605.30635#S5.F4.sf2)shows thatCellBRIDGEprovides no improvement over feature\-only OT \(α=0\\alpha=0\) on this dataset: interaction structure is not transferable across six\-day developmental intervals and becomes uninformative\. Accordingly, performance is essentially flat inα\\alpha, with degradation at largerα\\alpha\. Practically, when cross\-snapshot interaction geometry does not persist, the structural term should be downweighted or omitted\.

## 6Discussion

Disambiguating alignment via biological structure\.OT–based alignment is often*underdetermined*\.CellBRIDGEaddresses this bottleneck by injecting a biological inductive bias: a least action principle of ligand–receptor communication\. Crucially, LR\-derived signaling provides*complementary*information to expression similarity: cells may be transcriptionally similar yet play distinct signaling roles, or conversely exhibit different expression profiles while participating in similar communication patterns\. By formulating alignment as a multi\-channel FGW objective,CellBRIDGEproduces couplings that are simultaneously feature\-coherent and consistent with directed, typed communication structure\. A practical consequence of making structure an*explicit*prior is editability: because the prior is expressed through the LR catalog used to construct the CCI tensor,CellBRIDGEenables mechanism\-specific counterfactuals by quantifying how pathway\-level catalog edits shift inferred trajectories\.

A coupling\-level, modular prior\.A central design principle ofCellBRIDGEis the separation between*structural priors*and*trajectory parameterization*\.CellBRIDGEencodes signaling structure once at the level of the cross\-snapshot coupling and exposes this as a reusable interface to downstream models\. Empirically, we find thatCellBRIDGEimproves performance across deterministic flows \(CFM\), geometry\-aware interpolation \(MFM\), stochastic bridge dynamics \(SF2M\), and unbalanced transport \(UOT\-FM\), and that replacing baseline couplings with theCellBRIDGEcouplings improves those methods\. These results suggest that interaction\-aware couplings complement advances in generative modeling: they refine the*endpoint correspondence*problem many trajectory learners rely on, without changing architectures or objectives\. Finally, negative controls \(e\.g\., random LR assignments or permuted channels\) support that gains often depend on biologically meaningful communication structure rather than arbitrary regularization\.

Limitations, extensions, and practical guidance\.CellBRIDGEis most appropriate when cross\-snapshot communication structure is at least partially conserved\. In rapidly remodeling systems with substantial composition shifts or long temporal gaps, signaling patterns may be non\-persistent and the structural term should be downweighted\. Several extensions could broaden applicability: \(1\)*time\-varying LR catalogs*to handle non\-stationary signaling programs, and \(2\) tighter*integration with spatial transcriptomics*to validate and refine CCI proxies when spatial coordinates are available\. More broadly, scalability to human atlas\-scale datasets remains an important direction, both computationally \(large\-nncoupling optimization\) and statistically \(robust CCI estimation under extreme sparsity\)\.

Broader impact\.CellBRIDGEprovides a general mechanism for population alignment by injecting typed interaction priors into OT\. Beyond biology, the same principle applies whenever entities interact through directed, typed relations\. For example, in financial networks, directed transaction patterns could regularize market states across regime shifts by constraining correspondences\. Analogous ideas apply to social and multi\-agent systems, where preserving directed relational structure can reduce ambiguity in cross\-time alignment and improve downstream dynamics modeling\.

## Acknowledgments

We thank the anonymous ICML reviewers for their comments and suggestions\. NH thanks Illumina for their funding and support\. TL would like to thank AstraZeneca for their sponsorship and support\. The Cambridge Centre for AI in Medicine \(CCAIM\) receives funding from GSK, Boehringer\-Ingelheim, AstraZeneca, Sanofi and Quantum Black, AI by McKinsey\.

## Impact Statement

This paper presents work whose goal is to advance the field of Machine Learning\. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here\. We release the code forCellBRIDGEunder[https://github\.com/nicolashuynh/cellbridge](https://github.com/nicolashuynh/cellbridge)and at the wider lab repository[https://github\.com/vanderschaarlab/cellbridge](https://github.com/vanderschaarlab/cellbridge)\.

## References

- A\. Adil, V\. Kumar, A\. T\. Jan, and M\. Asger \(2021\)Single\-Cell Transcriptomics: Current Methods and Challenges in Data Acquisition and Analysis\.Frontiers in Neuroscience15\.External Links:ISSN 1662\-453XCited by:[§1](https://arxiv.org/html/2605.30635#S1.p2.1)\.
- M\. S\. Albergo, N\. M\. Boffi, and E\. Vanden\-Eijnden \(2023\)Stochastic interpolants: A unifying framework for flows and diffusions\.arXiv preprint arXiv:2303\.08797\.Cited by:[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p2.10)\.
- P\. S\. Arunachalam, M\. K\. D\. Scott, T\. Hagan, C\. Li, Y\. Feng, F\. Wimmers, L\. Grigoryan, M\. Trisal, V\. V\. Edara, L\. Lai, S\. E\. Chang, A\. Feng, S\. Dhingra, M\. Shah, A\. S\. Lee, S\. Chinthrajah, S\. B\. Sindher, V\. Mallajosyula, F\. Gao, N\. Sigal, S\. Kowli, S\. Gupta, K\. Pellegrini, G\. Tharp, S\. Maysel\-Auslender, S\. Hamilton, H\. Aoued, K\. Hrusovsky, M\. Roskey, S\. E\. Bosinger, H\. T\. Maecker, S\. D\. Boyd, M\. M\. Davis, P\. J\. Utz, M\. S\. Suthar, P\. Khatri, K\. C\. Nadeau, and B\. Pulendran \(2021\)Systems vaccinology of the BNT162b2 mRNA vaccine in humans\.Nature596\(7872\),pp\. 410–416\.External Links:ISSN 1476\-4687Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.2.3.1.1)\.
- G\. Arvanitidis, L\. K\. Hansen, and S\. Hauberg \(2016\)A locally adaptive normal distribution\.Advances in Neural Information Processing Systems29\.Cited by:[§E\.3](https://arxiv.org/html/2605.30635#A5.SS3.p2.4)\.
- L\. Atanackovic, X\. Zhang, B\. Amos, M\. Blanchette, L\. J\. Lee, Y\. Bengio, A\. Tong, and K\. Neklyudov \(2024\)Meta flow matching: integrating vector fields on the wasserstein manifold\.arXiv preprint arXiv:2408\.14608\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1)\.
- J\. Benamou and Y\. Brenier \(2000\)A computational fluid mechanics solution to the Monge\-Kantorovich mass transfer problem\.Numerische Mathematik84\(3\),pp\. 375–393\.Cited by:[§G\.2](https://arxiv.org/html/2605.30635#A7.SS2.p7.2)\.
- V\. Bergen, M\. Lange, S\. Peidli, F\. A\. Wolf, and F\. J\. Theis \(2020\)Generalizing RNA velocity to transient cell states through dynamical modeling\.Nature Biotechnology38\(12\),pp\. 1408–1414\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.5.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§3](https://arxiv.org/html/2605.30635#S3.p3.1)\.
- R\. Berlinghieri, Y\. Shen, J\. Jiang, and T\. Broderick \(2025\)Oh SnapMMD\! Forecasting Stochastic Dynamics Beyond the Schrödinger Bridge’s End\.arXiv preprint arXiv:2505\.16082\.Cited by:[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- G\. Braun, A\. Carderera, C\. W\. Combettes, H\. Hassani, A\. Karbasi, A\. Mokhtari, and S\. Pokutta \(2022\)Conditional gradient methods\.arXiv preprint arXiv:2211\.14103\.Cited by:[§4\.1](https://arxiv.org/html/2605.30635#S4.SS1.p4.9)\.
- C\. Bunne, G\. Schiebinger, A\. Krause, A\. Regev, and M\. Cuturi \(2024\)Optimal transport for single\-cell and spatial omics\.Nature Reviews Methods Primers4\(1\)\.External Links:ISSN 2662\-8449Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p1.1),[§1](https://arxiv.org/html/2605.30635#S1.p2.1),[§1](https://arxiv.org/html/2605.30635#S1.p4.1),[§2](https://arxiv.org/html/2605.30635#S2.p4.5)\.
- C\. Bunne, S\. G\. Stark, G\. Gut, J\. S\. Del Castillo, M\. Levesque, K\. Lehmann, L\. Pelkmans, A\. Krause, and G\. Rätsch \(2023\)Learning single\-cell perturbation responses using neural optimal transport\.Nature methods20\(11\),pp\. 1759–1768\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.15.8),[§5\.4](https://arxiv.org/html/2605.30635#S5.SS4.p1.1)\.
- J\. G\. Camp, F\. Badsha, M\. Florio, S\. Kanton, T\. Gerber, M\. Wilsch\-Bräuninger, E\. Lewitus, A\. Sykes, W\. Hevers, M\. Lancaster, J\. A\. Knoblich, R\. Lachmann, S\. Pääbo, W\. B\. Huttner, and B\. Treutlein \(2015\)Human cerebral organoids recapitulate gene expression programs of fetal neocortex development\.Proceedings of the National Academy of Sciences112\(51\),pp\. 15672–15677\.External Links:ISSN 1091\-6490Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.3.3.1.1)\.
- Z\. Cang and Q\. Nie \(2020\)Inferring spatial and signaling relationships between cells from single\-cell transcriptomic data\.Nature Communications11\(1\),pp\. 2084\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1),[§3](https://arxiv.org/html/2605.30635#S3.p4.1)\.
- G\. Chen, C\. Ren, Y\. Xiao, Y\. Wang, R\. Yao, Q\. Wang, G\. You, M\. Lu, S\. Yan, X\. Zhang, J\. Zhang, Y\. Yao, and H\. Zhou \(2023\)Time\-resolved single\-cell transcriptomics reveals the landscape and dynamics of hepatic cells in sepsis\-induced acute liver dysfunction\.JHEP Reports5\(6\),pp\. 100718\.External Links:ISSN 2589\-5559Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.7.3.1.1)\.
- V\. De Bortoli, J\. Thornton, J\. Heng, and A\. Doucet \(2021\)Diffusion Schrödinger Bridge with Applications to Score\-Based Generative Modeling\.Advances in neural information processing systems34,pp\. 17695–17709\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.13.8),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p3.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- P\. Demetci, R\. Santorella, B\. Sandstede, W\. S\. Noble, and R\. Singh \(2022\)SCOT: Single\-Cell Multi\-Omics Alignment with Optimal Transport\.Journal of Computational Biology29\(1\),pp\. 3–18\.External Links:ISSN 1557\-8666Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.8.8)\.
- S\. C\. di Montesano, D\. D’Ascenzo, S\. Raghavan, A\. P\. Amini, P\. S\. Winter, and L\. Crawford \(2026\)Improving atlas\-scale single\-cell annotation models with hierarchical cross\-entropy loss\.Nature Computational Science6\(3\),pp\. 243\.Cited by:[§3](https://arxiv.org/html/2605.30635#S3.p5.1)\.
- D\. Dimitrov, P\. S\. L\. Schäfer, E\. Farr, P\. Rodriguez\-Mier, S\. Lobentanzer, P\. Badia\-i\-Mompel, A\. Dugourd, J\. Tanevski, R\. O\. Ramirez Flores, and J\. Saez\-Rodriguez \(2024\)LIANA\+ provides an all\-in\-one framework for cell–cell communication inference\.Nature Cell Biology26\(9\),pp\. 1613–1622\.External Links:ISSN 1476\-4679Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p5.1),[§D\.5](https://arxiv.org/html/2605.30635#A4.SS5.p1.8)\.
- K\. Domvri, P\. Zarogoulidis, K\. Darwiche, R\. F\. Browning, Q\. Li, J\. F\. Turner, I\. Kioumis, D\. Spyratos, K\. Porpodis, A\. Papaiwannou, T\. Tsiouda, L\. Freitag, and K\. Zarogoulidis \(2013\)Molecular Targeted Drugs and Biomarkers in NSCLC, the Evolving Role of Individualized Therapy\.Journal of Cancer4\(9\),pp\. 736–754\.External Links:ISSN 1837\-9664Cited by:[§5\.3](https://arxiv.org/html/2605.30635#S5.SS3.p2.1)\.
- M\. Efremova, M\. Vento\-Tormo, S\. A\. Teichmann, and R\. Vento\-Tormo \(2020\)CellPhoneDB: inferring cell–cell communication from combined expression of multi\-subunit ligand–receptor complexes\.Nature Protocols15\(4\),pp\. 1484–1506\.External Links:ISSN 1750\-2799Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p5.1)\.
- L\. Eyring, D\. Klein, T\. Uscidda, G\. Palla, N\. Kilbertus, Z\. Akata, and F\. J\. Theis \(2024\)Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation\.InThe Twelfth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p9.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- N\. Farbehi, R\. Patrick, A\. Dorison, M\. Xaymardan, V\. Janbandhu, K\. Wystub\-Lis, J\. W\. Ho, R\. E\. Nordon, and R\. P\. Harvey \(2019\)Single\-cell expression profiling reveals dynamic flux of cardiac stromal, vascular and immune cells in health and injury\.eLife8\.External Links:ISSN 2050\-084XCited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.4.3.1.1)\.
- K\. Fatras, Y\. Zine, S\. Majewski, R\. Flamary, R\. Gribonval, and N\. Courty \(2021\)Minibatch optimal transport distances; analysis and applications\.arXiv preprint arXiv:2101\.01792\.Cited by:[2nd item](https://arxiv.org/html/2605.30635#A4.I2.i2.p1.1)\.
- R\. Flamary, C\. Vincent\-Cuaz, N\. Courty, A\. Gramfort, O\. Kachaiev, H\. Quang Tran, L\. David, C\. Bonet, N\. Cassereau, T\. Gnassounou, E\. Tanguy, J\. Delon, A\. Collas, S\. Mazelet, L\. Chapel, T\. Kerdoncuff, X\. Yu, M\. Feickert, P\. Krzakala, T\. Liu, and E\. Fernandes Montesuma \(2024\)POT Python Optimal Transport \(version 0\.9\.5\)\.External Links:[Link](https://github.com/PythonOT/POT)Cited by:[item 2](https://arxiv.org/html/2605.30635#A4.I1.i2.p1.1),[§D\.3](https://arxiv.org/html/2605.30635#A4.SS3.p1.5)\.
- N\. Fournier and A\. Guillin \(2015\)On the rate of convergence in Wasserstein distance of the empirical measure\.Probability theory and related fields162\(3\),pp\. 707–738\.Cited by:[§G\.1](https://arxiv.org/html/2605.30635#A7.SS1.3.p3.21)\.
- K\. C\. Goldfarbmuren, N\. D\. Jackson, S\. P\. Sajuthi, N\. Dyjack, K\. S\. Li, C\. L\. Rios, E\. G\. Plender, M\. T\. Montgomery, J\. L\. Everman, P\. E\. Bratcher, E\. K\. Vladar, and M\. A\. Seibold \(2020\)Dissecting the cellular specificity of smoking effects and reconstructing lineages in the human airway epithelium\.Nature Communications11\(1\)\.External Links:ISSN 2041\-1723Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.5.3.1.1)\.
- L\. Haghverdi, M\. Büttner, F\. A\. Wolf, F\. Buettner, and F\. J\. Theis \(2016\)Diffusion pseudotime robustly reconstructs lineage branching\.Nature Methods13\(10\),pp\. 845–848\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.4.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§1](https://arxiv.org/html/2605.30635#S1.p3.1),[§2](https://arxiv.org/html/2605.30635#S2.p2.1)\.
- D\. Hanahan and R\. A\. Weinberg \(2011\)Hallmarks of cancer: the next generation\.Cell144\(5\),pp\. 646–674\.External Links:ISSN 0092\-8674Cited by:[§D\.7\.1](https://arxiv.org/html/2605.30635#A4.SS7.SSS1.p1.1)\.
- X\. He and C\. Xu \(2020\)Immune checkpoint signaling and cancer immunotherapy\.Cell Research30\(8\),pp\. 660–669\.External Links:ISSN 1748\-7838Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p6.1)\.
- I\. Hossain, V\. Fanfani, J\. Fischer, J\. Quackenbush, and R\. Burkholz \(2024\)Biologically informed NeuralODEs for genome\-wide regulatory dynamics\.Genome Biology25\(1\),pp\. 127\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§3](https://arxiv.org/html/2605.30635#S3.p5.1)\.
- S\. Hrvatin, D\. R\. Hochbaum, M\. A\. Nagy, M\. Cicconet, K\. Robertson, L\. Cheadle, R\. Zilionis, A\. Ratner, R\. Borges\-Monroy, A\. M\. Klein,et al\.\(2018\)Single\-cell analysis of experience\-dependent transcriptomic states in the mouse visual cortex\.Nature neuroscience21\(1\),pp\. 120–129\.Cited by:[§C\.3](https://arxiv.org/html/2605.30635#A3.SS3.p1.6),[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.3.2)\.
- G\. Huguet, D\. S\. Magruder, A\. Tong, O\. Fasina, M\. Kuchroo, G\. Wolf, and S\. Krishnaswamy \(2022\)Manifold Interpolating Optimal\-Transport Flows for Trajectory Inference\.InAdvances in Neural Information Processing Systems,A\. H\. Oh, A\. Agarwal, D\. Belgrave, and K\. Cho \(Eds\.\),Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p4.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- E\. L\. Jackson, N\. Willis, K\. Mercer, R\. T\. Bronson, D\. Crowley, R\. Montoya, T\. Jacks, and D\. A\. Tuveson \(2001\)Analysis of lung tumor initiation and progression using conditional expression of oncogenic K\-ras\.Genes & Development15\(24\),pp\. 3243–3248\.External Links:ISSN 1549\-5477Cited by:[§C\.2](https://arxiv.org/html/2605.30635#A3.SS2.p1.3)\.
- K\. Kapusniak, P\. Potaptchik, T\. Reu, L\. Zhang, A\. Tong, M\. Bronstein, J\. Bose, and F\. Di Giovanni \(2024\)Metric Flow Matching for Smooth Interpolations on the Data Manifold\.Advances in Neural Information Processing Systems37,pp\. 135011–135042\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p7.1),[§E\.3](https://arxiv.org/html/2605.30635#A5.SS3.p1.1),[§1](https://arxiv.org/html/2605.30635#S1.p8.1),[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p3.11),[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p6.4),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- D\. Klein, G\. Palla, M\. Lange, M\. Klein, Z\. Piran, M\. Gander, L\. Meng\-Papaxanthos, M\. Sterr, L\. Saber, C\. Jing, A\. Bastidas\-Ponce, P\. Cota, M\. Tarquis\-Medina, S\. Parikh, I\. Gold, H\. Lickert, M\. Bakhti, M\. Nitzan, M\. Cuturi, and F\. J\. Theis \(2025\)Mapping cells through time and space with moscot\.Nature638\(8052\),pp\. 1065–1075\.External Links:ISSN 1476\-4687Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.9.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p5.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7),[§5\.3](https://arxiv.org/html/2605.30635#S5.SS3.p3.1)\.
- D\. Klein, T\. Uscidda, F\. J\. Theis, and M\. Cuturi \(2024\)GENOT: Entropic \(Gromov\) Wasserstein Flow Matching with Applications to Single\-Cell Genomics\.InThe Thirty\-eighth Annual Conference on Neural Information Processing Systems,Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1)\.
- R\. M\. Kortlever, N\. M\. Sodir, C\. H\. Wilson, D\. L\. Burkhart, L\. Pellegrinet, L\. Brown Swigart, T\. D\. Littlewood, and G\. I\. Evan \(2017\)Myc Cooperates with Ras by Programming Inflammation and Immune Suppression\.Cell171\(6\),pp\. 1301–1315\.e14\.External Links:ISSN 0092\-8674Cited by:[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.2.2)\.
- G\. La Manno, R\. Soldatov, A\. Zeisel, E\. Braun, H\. Hochgerner, V\. Petukhov, K\. Lidschreiber, M\. E\. Kastriti, P\. Lönnerberg, A\. Furlan, J\. Fan, L\. E\. Borm, Z\. Liu, D\. van Bruggen, J\. Guo, X\. He, R\. Barker, E\. Sundström, G\. Castelo\-Branco, P\. Cramer, I\. Adameyko, S\. Linnarsson, and P\. V\. Kharchenko \(2018\)RNA velocity of single cells\.Nature560\(7719\),pp\. 494–498\.External Links:ISSN 1476\-4687Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§3](https://arxiv.org/html/2605.30635#S3.p3.1)\.
- M\. Lange, V\. Bergen, M\. Klein, M\. Setty, B\. Reuter, M\. Bakhti, H\. Lickert, M\. Ansari, J\. Schniering, H\. B\. Schiller, D\. Pe’er, and F\. J\. Theis \(2022\)CellRank for directed single\-cell fate mapping\.Nature Methods19\(2\),pp\. 159–170\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§3](https://arxiv.org/html/2605.30635#S3.p3.1)\.
- J\. Y\. Lee, H\. T\. Lee, W\. Shin, J\. Chae, J\. Choi, S\. H\. Kim, H\. Lim, T\. Won Heo, K\. Y\. Park, Y\. J\. Lee, S\. E\. Ryu, J\. Y\. Son, J\. U\. Lee, and Y\. Heo \(2016\)Structural basis of checkpoint blockade by monoclonal antibodies in cancer immunotherapy\.Nature Communications7\(1\)\.External Links:ISSN 2041\-1723Cited by:[§5\.3](https://arxiv.org/html/2605.30635#S5.SS3.p1.1)\.
- A\. Liberzon, C\. Birger, H\. Thorvaldsdóttir, M\. Ghandi, J\. P\. Mesirov, and P\. Tamayo \(2015\)The Molecular Signatures Database Hallmark Gene Set Collection\.Cell Systems1\(6\),pp\. 417–425\.External Links:ISSN 2405\-4712Cited by:[§D\.7\.1](https://arxiv.org/html/2605.30635#A4.SS7.SSS1.p1.1),[§D\.7\.1](https://arxiv.org/html/2605.30635#A4.SS7.SSS1.p2.1)\.
- Y\. Lipman, R\. T\. Chen, H\. Ben\-Hamu, M\. Nickel, and M\. Le \(2022\)Flow matching for generative modeling\.InThe Eleventh International Conference on Learning Representations,Cited by:[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p4.4)\.
- Y\. Lipman, M\. Havasi, P\. Holderrieth, N\. Shaul, M\. Le, B\. Karrer, R\. T\. Chen, D\. Lopez\-Paz, H\. Ben\-Hamu, and I\. Gat \(2024\)Flow matching guide and code\.arXiv preprint arXiv:2412\.06264\.Cited by:[§E\.1](https://arxiv.org/html/2605.30635#A5.SS1.SSS0.Px2.p1.7),[§G\.3](https://arxiv.org/html/2605.30635#A7.SS3.p3.3),[§1](https://arxiv.org/html/2605.30635#S1.p8.1),[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p3.11)\.
- Z\. Liu, H\. Chen, L\. Zheng, L\. Sun, and L\. Shi \(2023\)Angiogenic signaling pathways and anti\-angiogenic therapy for cancer\.Signal Transduction and Targeted Therapy8\(1\)\.External Links:ISSN 2059\-3635Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p6.1)\.
- K\. R\. Moon, D\. Van Dijk, Z\. Wang, S\. Gigante, D\. B\. Burkhardt, W\. S\. Chen, K\. Yim, A\. v\. d\. Elzen, M\. J\. Hirn, R\. R\. Coifman,et al\.\(2019\)Visualizing structure and transitions in high\-dimensional biological data\.Nature Biotechnology37\(12\),pp\. 1482–1492\.Cited by:[§C\.6](https://arxiv.org/html/2605.30635#A3.SS6.p1.1),[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.5.2),[§5\.4](https://arxiv.org/html/2605.30635#S5.SS4.p1.1)\.
- D\. J\. Murphy, M\. R\. Junttila, L\. Pouyet, A\. Karnezis, K\. Shchors, D\. A\. Bui, L\. Brown\-Swigart, L\. Johnson, and G\. I\. Evan \(2008\)Distinct Thresholds Govern Myc’s Biological Output In Vivo\.Cancer Cell14\(6\),pp\. 447–457\.External Links:ISSN 1535\-6108Cited by:[§C\.2](https://arxiv.org/html/2605.30635#A3.SS2.p1.3)\.
- H\. Pan, C\. Xue, B\. J\. Auerbach, J\. Fan, A\. C\. Bashore, J\. Cui, D\. Y\. Yang, S\. B\. Trignano, W\. Liu, J\. Shi, C\. O\. Ihuegbu, E\. C\. Bush, J\. Worley, L\. Vlahos, P\. Laise, R\. A\. Solomon, E\. S\. Connolly, A\. Califano, P\. A\. Sims, H\. Zhang, M\. Li, and M\. P\. Reilly \(2020\)Single\-Cell Genomics Reveals a Novel Cell State During Smooth Muscle Cell Phenotypic Switching and Potential Therapeutic Targets for Atherosclerosis in Mouse and Human\.Circulation142\(21\),pp\. 2060–2075\.External Links:ISSN 1524\-4539Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.4.3.1.1)\.
- G\. Peyré and M\. Cuturi \(2019\)Computational optimal transport: with applications to data science\.Foundations and Trends in Machine Learning\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[1st item](https://arxiv.org/html/2605.30635#A4.I2.i1.p1.1),[§1](https://arxiv.org/html/2605.30635#S1.p4.1),[§3](https://arxiv.org/html/2605.30635#S3.p2.1)\.
- C\. Qiu, J\. Cao, B\. K\. Martin, T\. Li, I\. C\. Welsh, S\. Srivatsan, X\. Huang, D\. Calderon, W\. S\. Noble, C\. M\. Disteche, S\. A\. Murray, M\. Spielmann, C\. B\. Moens, C\. Trapnell, and J\. Shendure \(2022\)Systematic reconstruction of cellular trajectories across mouse embryogenesis\.Nature Genetics54\(3\),pp\. 328–341\.External Links:ISSN 1546\-1718Cited by:[§C\.7](https://arxiv.org/html/2605.30635#A3.SS7.p1.2),[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.7.2),[§F\.3](https://arxiv.org/html/2605.30635#A6.SS3.p1.1),[§1](https://arxiv.org/html/2605.30635#S1.p1.1)\.
- C\. Qiu, B\. K\. Martin, I\. C\. Welsh, R\. M\. Daza, T\. Le, X\. Huang, E\. K\. Nichols, M\. L\. Taylor, O\. Fulton, D\. R\. O’Day, A\. R\. Gomes, S\. Ilcisin, S\. Srivatsan, X\. Deng, C\. M\. Disteche, W\. S\. Noble, N\. Hamazaki, C\. B\. Moens, D\. Kimelman, J\. Cao, A\. F\. Schier, M\. Spielmann, S\. A\. Murray, C\. Trapnell, and J\. Shendure \(2024\)A single\-cell time\-lapse of mouse prenatal development from gastrula to birth\.Nature626\(8001\),pp\. 1084–1093\.External Links:ISSN 1476\-4687Cited by:[§5\.4](https://arxiv.org/html/2605.30635#S5.SS4.p1.1)\.
- X\. Qiu, Q\. Mao, Y\. Tang, L\. Wang, R\. Chawla, H\. A\. Pliner, and C\. Trapnell \(2017\)Reversed graph embedding resolves complex single\-cell trajectories\.Nature Methods14\(10\),pp\. 979–982\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§3](https://arxiv.org/html/2605.30635#S3.p2.1)\.
- K\. Sakalyan, A\. Palma, F\. Guerranti, F\. J\. Theis, and S\. Günnemann \(2025\)Modeling Microenvironment Trajectories on Spatial Transcriptomics with NicheFlow\.InThe Thirty\-ninth Annual Conference on Neural Information Processing Systems,Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.16.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1)\.
- G\. Schiebinger, J\. Shu, M\. Tabaka, B\. Cleary, V\. Subramanian, A\. Solomon, J\. Gould, S\. Liu, S\. Lin, P\. Berube, L\. Lee, J\. Chen, J\. Brumbaugh, P\. Rigollet, K\. Hochedlinger, R\. Jaenisch, A\. Regev, and E\. S\. Lander \(2019\)Optimal\-Transport Analysis of Single\-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming\.Cell176\(4\),pp\. 928–943\.e22\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.7.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§1](https://arxiv.org/html/2605.30635#S1.p1.1),[§1](https://arxiv.org/html/2605.30635#S1.p2.1),[§2](https://arxiv.org/html/2605.30635#S2.p2.1),[§3](https://arxiv.org/html/2605.30635#S3.p2.1)\.
- A\. Sertkaya, T\. Beleche, A\. Jessup, and B\. D\. Sommers \(2024\)Costs of Drug Development and Research and Development Intensity in the US, 2000\-2018\.JAMA Network Open7\(6\),pp\. e2415445\.External Links:ISSN 2574\-3805Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p1.1)\.
- A\. K\. Shalek, R\. Satija, J\. Shuga, J\. J\. Trombetta, D\. Gennert, D\. Lu, P\. Chen, R\. S\. Gertner, J\. T\. Gaublomme, N\. Yosef,et al\.\(2014\)Single\-cell RNA\-seq reveals dynamic paracrine control of cellular variation\.Nature510\(7505\),pp\. 363–369\.Cited by:[§C\.4](https://arxiv.org/html/2605.30635#A3.SS4.p1.1),[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.4.2),[§F\.11](https://arxiv.org/html/2605.30635#A6.SS11.p1.1)\.
- K\. Street, D\. Risso, R\. B\. Fletcher, D\. Das, J\. Ngai, N\. Yosef, E\. Purdom, and S\. Dudoit \(2018\)Slingshot: cell lineage and pseudotime inference for single\-cell transcriptomics\.BMC Genomics19\(1\),pp\. 477\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§1](https://arxiv.org/html/2605.30635#S1.p3.1)\.
- B\. J\. Strober, R\. Elorbany, K\. Rhodes, N\. Krishnan, K\. Tayeb, A\. Battle, and Y\. Gilad \(2019\)Dynamic genetic regulation of gene expression during cellular differentiation\.Science364\(6447\),pp\. 1287–1290\.External Links:ISSN 1095\-9203Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.6.3.1.1)\.
- Z\. Tang, G\. Chen, S\. Chen, H\. He, L\. You, and C\. Y\. Chen \(2024\)Knowledge\-based inductive bias and domain adaptation for cell type annotation\.Communications biology7\(1\),pp\. 1440\.Cited by:[§3](https://arxiv.org/html/2605.30635#S3.p5.1)\.
- A\. Tong, K\. Fatras, N\. Malkin, G\. Huguet, Y\. Zhang, J\. Rector\-Brooks, G\. Wolf, and Y\. Bengio \(2024a\)Improving and generalizing flow\-based generative models with minibatch optimal transport\.Transactions on Machine Learning Research,pp\. 1–34\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.12.8),[§G\.2](https://arxiv.org/html/2605.30635#A7.SS2.p7.2),[§5\.1](https://arxiv.org/html/2605.30635#S5.SS1.p4.1)\.
- A\. Tong, J\. Huang, G\. Wolf, D\. van Dijk, and S\. Krishnaswamy \(2020\)TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics\.InProceedings of the 37th International Conference on Machine Learning,Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.11.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p2.1),[§3](https://arxiv.org/html/2605.30635#S3.p2.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- A\. Tong, N\. Malkin, K\. Fatras, L\. Atanackovic, Y\. Zhang, G\. Huguet, G\. Wolf, and Y\. Bengio \(2024b\)Simulation\-free Schrödinger bridges via score and flow matching\.AISTATS\.Cited by:[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p8.1),[§E\.4](https://arxiv.org/html/2605.30635#A5.SS4.p1.2),[§1](https://arxiv.org/html/2605.30635#S1.p8.1),[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p3.11),[§4\.2](https://arxiv.org/html/2605.30635#S4.SS2.p5.4),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- B\. K\. Tusi, S\. L\. Wolock, C\. Weinreb, Y\. Hwang, D\. Hidalgo, R\. Zilionis, A\. Waisman, J\. R\. Huh, A\. M\. Klein, and M\. Socolovsky \(2018\)Population snapshots predict early haematopoietic and erythroid hierarchies\.Nature555\(7694\),pp\. 54–60\.External Links:ISSN 1476\-4687Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.6.3.1.1)\.
- T\. Vayer, L\. Chapel, R\. Flamary, R\. Tavenard, and N\. Courty \(2020\)Fused Gromov\-Wasserstein distance for structured objects\.Algorithms13\(9\),pp\. 212\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p4.1),[§4\.1](https://arxiv.org/html/2605.30635#S4.SS1.p3.13)\.
- C\. Villaniet al\.\(2008\)Optimal transport: old and new\.Vol\.338,Springer\.Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§2](https://arxiv.org/html/2605.30635#S2.p4.5)\.
- D\. Wang, Y\. Jiang, Z\. Zhang, X\. Gu, P\. Zhou, and J\. Sun \(2025\)Joint Velocity\-Growth Flow Matching for Single\-Cell Dynamics Modeling\.InThe Thirty\-ninth Annual Conference on Neural Information Processing Systems,Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p3.1),[§D\.6](https://arxiv.org/html/2605.30635#A4.SS6.p6.1),[§5\.2](https://arxiv.org/html/2605.30635#S5.SS2.p1.7)\.
- S\. Wang, A\. O\. Pisco, A\. McGeever, M\. Brbic, M\. Zitnik, S\. Darmanis, J\. Leskovec, J\. Karkanias, and R\. B\. Altman \(2021\)Leveraging the Cell Ontology to classify unseen cell types\.Nature communications12\(1\),pp\. 5556\.Cited by:[§3](https://arxiv.org/html/2605.30635#S3.p5.1)\.
- P\. Weiler, K\. Van den Berge, K\. Street, and S\. Tiberi \(2022\)A guide to trajectory inference and RNA velocity\.InSingle Cell Transcriptomics: Methods and Protocols,pp\. 269–292\.Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p3.1)\.
- K\. A\. Wierenga, F\. M\. Riemers, B\. Westendorp, J\. R\. Harkema, and J\. J\. Pestka \(2022\)Single cell analysis of docosahexaenoic acid suppression of sequential LPS\-induced proinflammatory and interferon\-regulated gene expression in the macrophage\.Frontiers in Immunology13,pp\. 993614\.Cited by:[§C\.5](https://arxiv.org/html/2605.30635#A3.SS5.p1.1),[Table 6](https://arxiv.org/html/2605.30635#A3.T6.9.6.2)\.
- F\. A\. Wolf, F\. K\. Hamey, M\. Plass, J\. Solana, J\. S\. Dahlin, B\. Göttgens, N\. Rajewsky, L\. Simon, and F\. J\. Theis \(2019\)PAGA: graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells\.Genome Biology20\(1\),pp\. 59\.Cited by:[Table 4](https://arxiv.org/html/2605.30635#A1.T4.9.1.3.8),[Appendix A](https://arxiv.org/html/2605.30635#A1.p2.1),[§3](https://arxiv.org/html/2605.30635#S3.p2.1)\.
- A\. T\. Yeo, S\. Rawal, B\. Delcuze, A\. Christofides, A\. Atayde, L\. Strauss, L\. Balaj, V\. A\. Rogers, E\. J\. Uhlmann, H\. Varma, B\. S\. Carter, V\. A\. Boussiotis, and A\. Charest \(2022\)Single\-cell RNA sequencing reveals evolution of immune landscape during glioblastoma progression\.Nature Immunology23\(6\),pp\. 971–984\.External Links:ISSN 1529\-2916Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p1.1)\.
- R\. Yue and A\. Dutta \(2022\)Computational systems biology in disease modeling and control, review and perspectives\.npj Systems Biology and Applications8\(1\)\.External Links:ISSN 2056\-7189Cited by:[§1](https://arxiv.org/html/2605.30635#S1.p1.1)\.
- F\. Zanini, M\. L\. Robinson, D\. Croote, M\. K\. Sahoo, A\. M\. Sanz, E\. Ortiz\-Lasso, L\. L\. Albornoz, F\. Rosso, J\. G\. Montoya, L\. Goo, B\. A\. Pinsky, S\. R\. Quake, and S\. Einav \(2018\)Virus\-inclusive single\-cell RNA sequencing reveals the molecular signature of progression to severe dengue\.Proceedings of the National Academy of Sciences115\(52\)\.External Links:ISSN 1091\-6490Cited by:[Table 5](https://arxiv.org/html/2605.30635#A2.T5.7.2.3.1.1)\.
- Z\. Zhang, Z\. Wang, Y\. Sun, T\. Li, and P\. Zhou \(2025\)Modeling cell dynamics and interactions with unbalanced mean field schrödinger bridge\.InThe Thirty\-ninth Annual Conference on Neural Information Processing Systems,Cited by:[Appendix A](https://arxiv.org/html/2605.30635#A1.p5.1),[§3](https://arxiv.org/html/2605.30635#S3.p5.1)\.

## Appendix AExtended Related Works

We situate our framework within trajectory inference and optimal transport by organizing prior methods according to the*inductive bias*they impose to resolve the intrinsic underdetermination of aligning population snapshots\. Existing approaches regularize alignment through feature smoothness, dynamical directionality, geometric/spatial structure, or conservation laws\. Our contribution introduces a complementary bias: smoothness of a*directed, typed cell–cell communication structure*, injected as a plug\-and\-play regularizer at the level of the OT coupling\.

Feature\-based priors: gene\-expression smoothness\.Classical trajectory\-inference methods reconstruct cellular progressions from neighborhood graphs using pseudotime and branching heuristics\(Qiuet al\.,[2017](https://arxiv.org/html/2605.30635#bib.bib43); Haghverdiet al\.,[2016](https://arxiv.org/html/2605.30635#bib.bib44); Streetet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib45); Wolfet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib46)\)\. When applied to time\-course data, these methods typically pool cells from all observed timepoints into a single expression\-space graph and then infer pseudotime or branching structure from local transcriptomic neighborhoods\. Optimal transport \(OT\) provides a population\-level alternative by coupling entire distributions across timepoints under a least\-action bias in gene\-expression space\(Villani and others,[2008](https://arxiv.org/html/2605.30635#bib.bib30); Peyré and Cuturi,[2019](https://arxiv.org/html/2605.30635#bib.bib5)\)\. Waddington\-OT \(WOT\) extends this idea to sequences of snapshots via adjacent\-time couplings\(Schiebingeret al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib37)\), while continuous\-time models such as TrajectoryNet learn neural ODE flows constrained by transport\(Tonget al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib42)\)\. More recent work connects OT couplings to continuous\-time generative dynamics using flow matching or diffusion\-based formulations\(Kleinet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib77)\)\. These approaches resolve underdetermination by favoring*feature\-smooth*matchings, with alignment costs defined in expression space\.

Dynamic priors: directionality and population\-level constraints\.A separate line of work encodes*directional*priors or population\-scale constraints\. RNA velocity and its extensions infer directionality from spliced/unspliced counts and propagate it overkkNN graphs\(La Mannoet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib38); Bergenet al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib39)\), while CellRank combines velocity with transcriptomic similarity to estimate fate probabilities\(Langeet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib40)\)\. Other approaches modify the mass\-conservation assumption: unbalanced OT and flow\-matching methods such as UOT\-FM relax exact mass preservation to account for proliferation or apoptosis\(Eyringet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib14)\), while VGFM explicitly models cellular growth rates within generative flows\(Wanget al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib18)\)\. Related biologically informed NeuralODE frameworks incorporate mechanistic gene\-regulatory priors into continuous\-time expression dynamics\. For example, PHOENIX\(Hossainet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib4)\)uses Hill–Langmuir\-inspired neural dynamics and prior GRN structure to learn sparse, interpretable genome\-scale regulatory ODEs\. However, such models regularize intracellular gene–gene regulatory dynamics rather than the intercellular, directed ligand–receptor communication structure used in our coupling\-level prior\. Meta Flow Matching\(Atanackovicet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib16)\)learns amortized vector fields using graph neural networks but requires multiple datasets for training, limiting applicability when only a single time\-series experiment is available\.

Geometric priors: spatial and structural alignment\.Spatial optimal transport methods leverage physical proximity to infer communication or alignment, using spatial transcriptomics measurements\(Cang and Nie,[2020](https://arxiv.org/html/2605.30635#bib.bib41)\)\. For example, NicheFlow models microenvironment\-mediated effects but assumes access to spatial coordinates\(Sakalyanet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib109)\)\. This requirement is a practical barrier in the most common setting of dissociated scRNA\-seq, where spatial coordinates are unavailable; in contrast, our approach enables communication\-aware alignment*from dissociated scRNA\-seq alone*by using curated ligand–receptor knowledge as a mechanistically grounded prior\. More generally, Gromov–Wasserstein \(GW\) and Fused GW formulations compare samples via relational structure rather than raw features\(Vayeret al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib12)\)\. In single\-cell applications, the relational structure is often instantiated as scalar similarity graphs capturing generic topology; such graphs are useful geometric summaries, but they do not encode*who signals to whom via which pathway*\. Recent OT toolkits such as MosCOT provide scalable solvers for linear and fused OT, yet standard pipelines largely rely on feature\-space or spatial distances, leaving the rich landscape of interaction\-driven constraints unexplored\(Kleinet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib19)\)\. Related flow\-based approaches incorporate geometric priors by restricting dynamics to the data manifold or a learned Riemannian metric\(Huguetet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib61); Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20)\)\.

Communication priors \(ours\): directed interaction structure\.A large literature infers putative cell–cell communication from dissociated scRNA\-seq using curated ligand–receptor catalogs and expression\-based heuristics, providing biologically grounded*priors*on plausible signaling even without spatial coordinates\. Tools such as CellPhoneDB systematically enumerate LR co\-expression across cell types\(Efremovaet al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib73)\)\. Meta\-frameworks like LIANA\+ unify and standardize CCI scoring across multiple LR resources and methods, facilitating method\-agnostic comparisons and consensus analyses\(Dimitrovet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib91)\)\. In our work, we derive a directed, typed interaction representation from ligand–receptor expression and inject it into an FGW objective, favoring couplings that encode smoothly evolving channel\-specific signaling context across time\. Relatedly, CytoBridge\(Zhanget al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib10)\)formulates an unbalanced mean\-field Schrödinger bridge that learns interaction effects via a neural interaction potential alongside growth and transition dynamics, but it does not leverageinterpretable, directed, typed ligand–receptor channels as an explicit coupling regularizer\. Furthermore, because our contribution operates at the coupling level, it is agnostic to the choice of downstream continuous\-time learner, unlike CytoBridge\. In particular,CellBRIDGEcan be used as a plug\-and\-play regularizer on top of the priors discussed above\.

##### Comparison with Related Works

Table[4](https://arxiv.org/html/2605.30635#A1.T4)provides a non\-exhaustive comparison betweenCellBRIDGEand different methods across five key capabilities essential for modeling complex cellular dynamics\. We define these criteria as follows:

- •Dynamicindicates whether the method explicitly models temporal evolution across multiple experimental timepoints, as opposed to inferring dynamics or trajectories from a single static snapshot\.
- •Trajectoriesdistinguishes methods that recover a continuous smooth path enabling predictions at unobserved intermediate timepoints from those that solely compute discrete couplings or transport maps between timepoints\.
- •In\-silico Perturbationrefers to the capability to perform principled interventions, allowing users to simulate and predict the system’s response to specific stimuli or perturbations\.
- •Structure\-Awareassesses whether the optimization objective explicitly models interactions between cells \(e\.g\., via cell\-cell communication or topological constraints\) rather than treating cells as independent, isolated entities\.
- •scRNA Data Sufficientconfirms whether the method can operate effectively using standard single\-cell RNA sequencing inputs alone, without requiring auxiliary spatial transcriptomics data or multi\-modal integration that are often unavailable\.

Table 4:Capability matrix for trajectory inference methods\.Comparison of graph\-based, optimal transport, and continuous\-time generative approaches for learning cellular dynamics from population snapshots\.CellBRIDGEuniquely integrates typed interaction structure into transport\-based trajectory inference using scRNA\-seq data only\.MethodDynamicTrajectoriesIn\-silico PerturbationStructure\-AwarescRNA OnlyMechanismExample ReferenceGraph\-Based and Pseudotime MethodsPAGA \(Scanpy\)✗✗✗✗✓Graph heuristicsWolfet al\.\([2019](https://arxiv.org/html/2605.30635#bib.bib46)\)Monocle / DPT✗✗✗✗✓Pseudotime graphsHaghverdiet al\.\([2016](https://arxiv.org/html/2605.30635#bib.bib44)\)scVelo✓✗✗✗✓RNA velocityBergenet al\.\([2020](https://arxiv.org/html/2605.30635#bib.bib39)\)Optimal Transport AlignmentWaddington\-OT✓~✗✗✓Feature OTSchiebingeret al\.\([2019](https://arxiv.org/html/2605.30635#bib.bib37)\)SCOT✗✗✗~✗GW alignment\(Demetciet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib103)\)MOSCOT✗✗✗~✓GW / FGW OT\(Kleinet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib19)\)OT\-Based Continuous\-Time DynamicsTrajectoryNet✓✓✗✗✓Neural ODE \+ OT\(Tonget al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib42)\)OT\-CFM✓✓✗✗✓Flow matching\(Tonget al\.,[2024a](https://arxiv.org/html/2605.30635#bib.bib36)\)Diffusion SB \(DSB\)✓✓✗✗✓Schrödinger bridge\(De Bortoliet al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib25)\)Perturbation and Conditional TransportCellOT✗✗✓✗✗Conditional OT\(Bunneet al\.,[2023](https://arxiv.org/html/2605.30635#bib.bib35)\)NicheFlow / Spatial OT✗✗~~✗Spatial structure\(Sakalyanet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib109)\)\\rowcolorblue\!5CellBRIDGE\(Ours\)✓✓✓✓✓Interaction\-aware FGW \+ FM–

## Appendix BPotential Applications ofCellBRIDGE

Snapshots of cellular systems using single\-cell RNA sequencing are now pervasive across diverse areas of biology and medicine\. A few representative longitudinal datasets are summarized in Table[5](https://arxiv.org/html/2605.30635#A2.T5)\.CellBRIDGEprovides a principled framework to analyze such data by combining snapshot measurements with biologically typed ligand–receptor structure\. This enables the reconstruction of coherent cell\-state trajectories through optimal transport couplings and a learned continuous flow, as well as the exploration of counterfactual scenarios by selectively re\-weighting interaction channels\. The resulting outputs \(shifts in lineage fate, changes in pathway usage, and differences in progression timing\) offer interpretable readouts that can guide mechanistic hypotheses and help prioritize therapeutic strategies before experimental validation\.

Table 5:Examples of public longitudinal single\-cell datasets\.Each row summarizes a biomedical area, the available longitudinal single\-cell setting, and representative references\. The list is illustrative rather than exhaustive\.AreaDataset descriptionRepresentative referencesVirologyLongitudinal PBMC or tissue scRNA\-seq during viral infection, vaccination, or challenge studies, capturing early immune activation, peak response, and recovery\.Dengue virus:\(Zaniniet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib63)\); vaccine response:\(Arunachalamet al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib64)\)NeurologyBrain single\-cell time courses, including organoid and tissue systems that profile neuronal, glial, and immune\-state changes over development or disease progression\.Brain organoids:\(Campet al\.,[2015](https://arxiv.org/html/2605.30635#bib.bib69)\)CardiologyCardiac and vascular single\-cell time courses following injury or during disease progression, capturing inflammation, remodeling, and repair\.Post\-MI heart:\(Farbehiet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib66)\); atherosclerosis:\(Panet al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib65)\)ImmunologyTissue and immune\-cell scRNA\-seq across baseline, inflammatory activation, and resolution or recovery in experimental model systems\.Lung inflammation:\(Goldfarbmurenet al\.,[2020](https://arxiv.org/html/2605.30635#bib.bib70)\)DevelopmentHuman iPSC or hPSC differentiation series that track lineage commitment, maturation, and cell\-state transitions over multiple sampling stages\.Cardiomyocytes:\(Stroberet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib67)\); blood cells:\(Tusiet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib68)\)RegenerationInjury\-response time courses in organs such as liver, kidney, or muscle, capturing damage response, repair, and cellular remodeling\.Liver injury:\(Chenet al\.,[2023](https://arxiv.org/html/2605.30635#bib.bib71)\)
## Appendix CDatasets

In addition to the synthetic dataset, we used 6 real\-world scRNA datasets to showcase the effectiveness and limitations of our method\. Details on the number of genes and the number of cells in each dataset can be found in Table[6](https://arxiv.org/html/2605.30635#A3.T6)\.

Table 6:Datasets used in our experiments\.Counts reflect the preprocessed objects used byCellBRIDGE\. Time points indicate the observed sampling times for each dataset\. The mouse cell atlas spans embryonic day E3\.5 to E13\.5\.DatasetReferenceTime points\#Cells\#GenesTumour\(Kortleveret al\.,[2017](https://arxiv.org/html/2605.30635#bib.bib54)\)0, 8, 24, 168 \(h\)31,53622,681V1 Cortex\(Hrvatinet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib26)\)0, 1, 4 \(h\)6,50517,008Dendritic Stimulus\(Shaleket al\.,[2014](https://arxiv.org/html/2605.30635#bib.bib28)\)0, 1, 2, 4, 6 \(h\)2,38210,972Mouse embryo\(Moonet al\.,[2019](https://arxiv.org/html/2605.30635#bib.bib47)\)0, 6, 12, 18, 24 \(d\)18,20317,789Macrophage Stimulus\(Wierengaet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib27)\)0, 3, 5 \(h\)223478Mouse Cell Atlas\(Qiuet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib83)\)E3\.5 \- E13\.51\.7 M29,452### C\.1Synthetic example

In this section we detail the synthetic setup used in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)\. We construct𝒟0\\mathcal\{D\}\_\{0\}as three 2D Gaussian clusters,

𝒟0=⋃k=02𝒮k,𝒮k=\{Xi\(k\)\}i=135,Xi\(k\)∼i\.i\.d\.𝒩​\(μk,0\.1​I2\),\\mathcal\{D\}\_\{0\}=\\bigcup\_\{k=0\}^\{2\}\\mathcal\{S\}\_\{k\},\\qquad\\mathcal\{S\}\_\{k\}=\\\{X\_\{i\}^\{\(k\)\}\\\}\_\{i=1\}^\{35\},\\qquad X\_\{i\}^\{\(k\)\}\\stackrel\{\{\\scriptstyle\\text\{i\.i\.d\.\}\}\}\{\{\\sim\}\}\\mathcal\{N\}\(\\mu\_\{k\},\\,0\.1\\,I\_\{2\}\),with centersμ0=\(−2,2\)\\mu\_\{0\}=\(\-2,2\),μ1=\(0,2\)\\mu\_\{1\}=\(0,2\), andμ2=\(2,2\)\\mu\_\{2\}=\(2,2\)\. The target snapshot𝒟1=⋃k=02𝒮k′\\mathcal\{D\}\_\{1\}=\\bigcup\_\{k=0\}^\{2\}\\mathcal\{S\}^\{\\prime\}\_\{k\}is obtained by translating each cluster via

T0​\(x\)=x\+\(4,−4\),T1​\(x\)=x\+\(0,−4\),T2​\(x\)=x\+\(−4,−4\),T\_\{0\}\(x\)=x\+\(4,\-4\),\\quad T\_\{1\}\(x\)=x\+\(0,\-4\),\\quad T\_\{2\}\(x\)=x\+\(\-4,\-4\),so that𝒮k′=\{Tk​\(X\):X∈𝒮k\}\\mathcal\{S\}^\{\\prime\}\_\{k\}=\\\{T\_\{k\}\(X\):\\,X\\in\\mathcal\{S\}\_\{k\}\\\}\.

For structure, we define two\-channel, directed relation tensorsG,G′∈\{0,1\}105×105×2G,G^\{\\prime\}\\in\\\{0,1\\\}^\{105\\times 105\\times 2\}over𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\}, respectively\. WritingG\(c\)G^\{\(c\)\}for channelcc, we set

Gi​j\(1\)=𝟏​\{Xi∈𝒮1,Xj∈𝒮0\},Gi​j\(2\)=𝟏​\{Xi∈𝒮1,Xj∈𝒮2\},G^\{\(1\)\}\_\{ij\}=\\mathbf\{1\}\\\{X\_\{i\}\\in\\mathcal\{S\}\_\{1\},\\;X\_\{j\}\\in\\mathcal\{S\}\_\{0\}\\\},\\qquad G^\{\(2\)\}\_\{ij\}=\\mathbf\{1\}\\\{X\_\{i\}\\in\\mathcal\{S\}\_\{1\},\\;X\_\{j\}\\in\\mathcal\{S\}\_\{2\}\\\},withG′G^\{\\prime\}defined analogously on𝒟1\\mathcal\{D\}\_\{1\}\. Thus, channel 1 encodes𝒮1→𝒮0\\mathcal\{S\}\_\{1\}\\\!\\to\\\!\\mathcal\{S\}\_\{0\}and channel 2 encodes𝒮1→𝒮2\\mathcal\{S\}\_\{1\}\\\!\\to\\\!\\mathcal\{S\}\_\{2\}\.

### C\.2Lung Tumor

We use a private scRNA\-seq dataset to study rapid tumour progression driven by RAS–MYC signalling using aKrasG12Dlung tumour model with tamoxifen\-inducible MycER\. Samples were collected at 0 h \(vehicle\), 8 h, 24 h \(n=8n=8biological replicates per condition; 0 h is time zero\)\. Lungs from LSL\-KrasG12D\(Jacksonet al\.,[2001](https://arxiv.org/html/2605.30635#bib.bib100)\)and LSL\-Rosa26MIE/MIE\(MycERT2\) mice\(Murphyet al\.,[2008](https://arxiv.org/html/2605.30635#bib.bib57)\)were dissociated to single cells, red blood cells removed, filtered \(70μ\\mum\), and 6,000 cells per sample were loaded for 10x Chromium3′3^\{\\prime\}v3 libraries\. Libraries were sequenced on a NovaSeq 6000 and processed with Cell Ranger v6\.1\.1 againstmm10\. All animal work complied with institutional ethical regulations of the Francis Crick Institute\.

### C\.3V1 Cortex \- Light stimulation

Adult \(6–8 week\) mice were dark\-adapted for 7 days, then either euthanized in darkness \(0h, control\) or exposed to ambient light for11h or44h\(Hrvatinet al\.,[2018](https://arxiv.org/html/2605.30635#bib.bib26)\)\. The visual cortex was profiled by scRNA\-seq to capture early transcriptional responses to sensory input\. We treat0h as the source snapshot,44h as the target snapshot, and use11h as an intermediate time point for interpolation/validation\.

### C\.4Dendritic\-cell stimulus

We use the Shalek et al\.\(Shaleket al\.,[2014](https://arxiv.org/html/2605.30635#bib.bib28)\)dendritic\-cell stimulus\-response dataset, which profiles primary mouse bone\-marrow\-derived dendritic cells under innate immune stimulation\. The dataset includes wild\-type cells stimulated with LPS across early response time points, as well as matched knockout conditions used to dissect paracrine signalling\. In our interpolation experiments, we use wild\-type unstimulated cells as the source snapshot, wild\-type LPS\-stimulated cells at 4 h as the target snapshot, and intermediate LPS\-stimulated time points for validation where applicable\. For the perturbation analysis in[SectionF\.11](https://arxiv.org/html/2605.30635#A6.SS11), we additionally use the experimentally observed 4 h knockout populations forIfnar1,Stat1, andTnfras held\-out perturbed targets\.

### C\.5Macrophage stimulus

We use the single\-cell RNA\-seq dataset of Wierenga et al\.\(Wierengaet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib27)\), which profiles murine fetal liver\-derived macrophages exposed to LPS with or without 24 h pre\-treatment with docosahexaenoic acid \(DHA, 25μ\\muM\)\. Cells were treated with LPS \(20 ng/mL\) and collected at 0 h, 1 h, and 4 h, then sequenced using the 10x Chromium platform\. In our interpolation experiments, we use 0 h as the source snapshot, 4 h as the target snapshot, and 1 h as the held\-out intermediate snapshot\. Where condition\-specific analyses are performed, cells are stratified by vehicle versus DHA pre\-treatment before subsampling\.

### C\.6Embryo development

In[Section5\.4](https://arxiv.org/html/2605.30635#S5.SS4), we analyze a mouse embryoid body \(EB\) differentiation time course used inMoonet al\.\([2019](https://arxiv.org/html/2605.30635#bib.bib47)\), which profiles embryonic stem cells differentiating toward germ layers over 27 days by scRNA\-seq\. We use the first \(Day 0\) and third \(Day 12\) snapshot to infer the cellular dynamics, reserving data at Day 6 for interpolation/validation\.

### C\.7Embryo cell atlas

To evaluate scalability on atlas\-scale data and assess performance under challenging developmental dynamics, we additionally considered the mouse embryo cell atlas of\(Qiuet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib83)\)\. We constructed two held\-out interpolation tasks: E7\.5→\\rightarrowE8, holding out E7\.75, and E7\.75→\\rightarrowE8\.25, holding out E8\. These transitions span rapid embryonic cell\-state diversification and tissue remodeling, providing a challenging benchmark for trajectory inference methods\.

## Appendix DExperimental Details

In what follows, we provide details about our experiments presented in[Section5](https://arxiv.org/html/2605.30635#S5)\.

### D\.1Data pre\-processing

Raw scRNA\-seq files for all datasets were converted to AnnData to standardize processing\. We applied basic QC, removing cells with<300<300detected genes and genes expressed in<3<3cells\. Counts were library\-size normalized per cell \(fixed total\), then log\-normalized\. We then selected the20002000highly variable genes and computed a2020\-component PCA on these features\. Finally, we performed Harmony batch correction in PCA space \(retaining both corrected and uncorrected embeddings for downstream analyses\)\.

### D\.2Constructing CCIs using metacells

We detail how we construct CCIs using metacells in the ablation presented in[Section5\.3](https://arxiv.org/html/2605.30635#S5.SS3)\. Without loss of generality and to keep the presentation simple \(with matrix multiplications\), we assumeK=1K=1\(i\.e\., one LR pair\) reducing the CCI tensors to matrices\. Before constructing the CCI matrices, we cluster the cells in each snapshot using Leiden community detection on akk\-nearest\-neighbour \(kNN\) graph built from the PCA representations with Euclidean distances andk=10k=10\. An example of the Leiden clustering with subsequent cell annotations is provided in Figure[5](https://arxiv.org/html/2605.30635#A4.F5)\. We select the resolutionρ⋆\\rho^\{\\star\}by scanning a small grid of resolutions and choosing the value whose*median*cluster size is closest to a target ofn⋆=40n^\{\\star\}=40cells\.

LetS∈ℝ≥0n×gS\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\\times g\}be the membership matrix of the resultingggclusters \(rows sum to 1 and correspond to one\-hot assignments\)\. We obtain metacell\-level activations by averaging thesc​gs\_\{cg\}within clusters and form the metacell CCI inℝg×g\\mathbb\{R\}^\{g\\times g\}similarly as in the setting with individual cells\.

Having constructed the metacell CCI matrixG¯\\bar\{G\}, we lift it back to the cell level via

G~=S​\(S⊤​S\)−1​G¯​\(S⊤​S\)−1​S⊤,\\tilde\{G\}\\;=\\;S\\,\(S^\{\\top\}S\)^\{\-1\}\\,\\bar\{G\}\\,\(S^\{\\top\}S\)^\{\-1\}\\,S^\{\\top\},This lifting operation ensuresS⊤​G~​S=G¯S^\{\\top\}\\tilde\{G\}S=\\bar\{G\}\. In contrast toGG, the matrixG~\\tilde\{G\}is constrained to lie in the subspace\{S​M​S⊤∣M∈ℝg×g\}\\\{SMS^\{\\top\}\\mid M\\in\\mathbb\{R\}^\{g\\times g\}\\\}, i\.e\., cell–cell interactions inG~\\tilde\{G\}are entirely mediated by metacell–metacell interactions\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/images/umap_cell_type.png)Figure 5:Metacell construction example\.UMAP visualization of the single\-cell RNA\-seq data of the lung cancer dataset after Leiden clustering\. Each point corresponds to an individual cell, colored by its assigned cluster and annotated with the corresponding cell type based on marker genes\.
### D\.3Optimal transport solver

We extend POT’s\(Flamaryet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib32)\)conditional\-gradient \(Frank–Wolfe\) solver to handle multi\-channel interactions\. Given structure tensorsC1∈ℝns×ns×dC\_\{1\}\\\!\\in\\\!\\mathbb\{R\}^\{n\_\{s\}\\times n\_\{s\}\\times d\},C2∈ℝnt×nt×dC\_\{2\}\\\!\\in\\\!\\mathbb\{R\}^\{n\_\{t\}\\times n\_\{t\}\\times d\}, marginalsp,qp,q\(uniform by default\), and a matrixΣ⪰0∈ℝd×d\\Sigma\\succeq 0\\in\\mathbb\{R\}^\{d\\times d\}, we measure discrepancies with the Mahalanobis norm‖x‖Σ=x⊤​Σ−1​x\\\|x\\\|\_\{\\Sigma\}=\\sqrt\{x^\{\\top\}\\Sigma^\{\-1\}x\}\.

Let⟨A,B⟩=∑i,jAi​j​Bi​j\\langle A,B\\rangle=\\sum\_\{i,j\}A\_\{ij\}B\_\{ij\}and writeC\(r\)C^\{\(r\)\}for therr\-th channel of a structure tensorCC\. The GW quadratic term is

𝒬​\(Γ\)=∑i,k,j,l‖C1​\[i,k\]−C2​\[j,l\]‖Σ2​Γi​j​Γk​l=⟨constC,Γ⟩−⟨ℬ​\(Γ\),Γ⟩,\\mathcal\{Q\}\(\\Gamma\)=\\sum\_\{i,k,j,l\}\\\!\\big\\\|C\_\{1\}\[i,k\]\-C\_\{2\}\[j,l\]\\big\\\|\_\{\\Sigma\}^\{2\}\\,\\Gamma\_\{ij\}\\Gamma\_\{kl\}=\\langle\\mathrm\{constC\},\\Gamma\\rangle\-\\langle\\mathcal\{B\}\(\\Gamma\),\\Gamma\\rangle,with

constCi​j=∑k‖C1​\[i,k\]‖Σ2​pk\+∑l‖C2​\[j,l\]‖Σ2​ql,ℬ​\(Γ\)=∑r=1dC1\(r\)​Γ​\(C2\(r\)\)⊤\.\\mathrm\{constC\}\_\{ij\}=\\sum\_\{k\}\\\!\\\|C\_\{1\}\[i,k\]\\\|\_\{\\Sigma\}^\{2\}p\_\{k\}\+\\sum\_\{l\}\\\!\\\|C\_\{2\}\[j,l\]\\\|\_\{\\Sigma\}^\{2\}q\_\{l\},\\qquad\\mathcal\{B\}\(\\Gamma\)=\\sum\_\{r=1\}^\{d\}C\_\{1\}^\{\(r\)\}\\,\\Gamma\\,\\big\(C\_\{2\}^\{\(r\)\}\\big\)^\{\\\!\\top\}\.The gradient computed by the solver is

∇𝒬​\(Γ\)=2​\(constC−ℬ​\(Γ\)\)\\nabla\\mathcal\{Q\}\(\\Gamma\)=2\\big\(\\mathrm\{constC\}\-\\mathcal\{B\}\(\\Gamma\)\\big\)\(8\)
We keep POT’s CG loop, stopping criteria, and line\-search options unchanged\.

We minimize

minΓ∈Π​\(p,q\)⁡\(1−α\)​⟨M,Γ⟩\+α​𝒬​\(Γ\),\\min\_\{\\Gamma\\in\\Pi\(p,q\)\}\\;\\,\(1\-\\alpha\)\\,\\langle M,\\Gamma\\rangle\+\\alpha\\mathcal\{Q\}\(\\Gamma\),with the same CG loop, where this objective is linearized using the gradient in[Equation8](https://arxiv.org/html/2605.30635#A4.E8)\.

Whend=1d=1\(scalar edges\), the method reduces to the original POT solver\.

### D\.4Normalization

To balance the contributions of the feature term and the structure term in the objective described at[Equation5](https://arxiv.org/html/2605.30635#S4.E5), we rescale the feature cost matrixCCand the CCI tensorsG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}\. We first compute the two endpoint couplings by solving the feature\-only \(α=0\\alpha=0\) and structure\-only \(α=1\\alpha=1\) problems, yieldingΓα=0⋆\\Gamma^\{\\star\}\_\{\\alpha=0\}andΓα=1⋆\\Gamma^\{\\star\}\_\{\\alpha=1\}\. We then define the scaling factors as follows:

Δ​ℱ\\displaystyle\\Delta\\mathcal\{F\}:=ℱ​\(Γα=0⋆\)−ℱ​\(Γα=1⋆\),\\displaystyle:=\\mathcal\{F\}\\\!\\left\(\\Gamma^\{\\star\}\_\{\\alpha=0\}\\right\)\-\\mathcal\{F\}\\\!\\left\(\\Gamma^\{\\star\}\_\{\\alpha=1\}\\right\),\(9\)Δ​𝒮\\displaystyle\\Delta\\mathcal\{S\}:=𝒮​\(Γα=1⋆\)−𝒮​\(Γα=0⋆\),\\displaystyle:=\\mathcal\{S\}\\\!\\left\(\\Gamma^\{\\star\}\_\{\\alpha=1\}\\right\)\-\\mathcal\{S\}\\\!\\left\(\\Gamma^\{\\star\}\_\{\\alpha=0\}\\right\),\(10\)and rescale the feature cost matrix and the CCI tensors:

C\\displaystyle C←C\|Δ​ℱ\|,\\displaystyle\\leftarrow\\frac\{C\}\{\\,\\lvert\\Delta\\mathcal\{F\}\\rvert\\,\},\(12\)G\(0\)\\displaystyle G^\{\(0\)\}←G\(0\)Δ​𝒮,G\(1\)←G\(1\)Δ​𝒮\.\\displaystyle\\leftarrow\\frac\{G^\{\(0\)\}\}\{\\sqrt\{\\Delta\\mathcal\{S\}\}\},\\qquad G^\{\(1\)\}\\leftarrow\\frac\{G^\{\(1\)\}\}\{\\sqrt\{\\Delta\\mathcal\{S\}\}\}\.\(13\)This places the terms on comparable scales so thatα\\alphameaningfully reflects the feature/structure trade\-off, and increasingα\\alphafrom0to11smoothly interpolates between the Kantorovich and the Gromov–Wasserstein problems\.

### D\.5Selection of ligand / receptor pairs

We apply LIANA’s\(Dimitrovet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib91)\)consensus rank aggregation withexpr\_prop= 0\.1 to obtain per–cell\-type interaction scores\. We retain interactions withcellphone\_pvals≤0\.05\\text\{\{cellphone\\\_pvals\}\}\\leq 0\.05andlr\_logfc≥0\\text\{\{lr\\\_logfc\}\}\\geq 0, then keep ligand–receptor pairs whoseexpr\_prodexceeds the median within that significant set\. We require the same significance criteria in each snapshot\. For every surviving pair, we aggregate LIANA results across significant edges to compute the mean expression product, average specificity ranks, counts of significant source→\\rightarrowtarget edges, and the numbers of unique source and target cell types\. We define coverage ascoverage=n\_edges/Nsig edges\\text\{\{coverage\}\}=\\text\{\{n\\\_edges\}\}/N\_\{\\text\{sig edges\}\}and retain only pairs with0\.10≤coverage≤0\.400\.10\\leq\\text\{\{coverage\}\}\\leq 0\.40and at least two sources and two targets\. We compute a standardized scores=0\.6​z​\(mean\_expr\)\+0\.4​z​\(−spec\_rank\)s=0\.6\\,z\(\\text\{\{mean\\\_expr\}\}\)\+0\.4\\,z\(\-\\text\{\{spec\\\_rank\}\}\)and greedily select pairs in descendingsswhile preventing repeated ligands or receptors\. We keep the top1010pairs for each dataset\.

### D\.6Baselines

Table 7:Flow\-based baseline hyperparameters\.Baselines based on flow matching, diffusion bridges, or unbalanced optimal transport\.MethodCategoryHyperparameterSetting / NotesDiffusion Schrödinger Bridge \(DSB\)DSBModelScore networkEncoder \[16, 32\], Decoder \[64, 64, 64\], latent dim 16TrainingIPF iterations10 outer IPF roundsOptimisation steps10,000 gradient updatesLangevin steps12 per bridge trajectoryBatch size128Learning rate1×10−41\\times 10^\{\-4\}Regularizationγ\\gammascheduleγmin=γmax=10−3\\gamma\_\{\\min\}=\\gamma\_\{\\max\}=10^\{\-3\}Mean matchingEnabledEMADisabledMFM \(Flow Matching \+ Riemannian Correction\)MFMVelocity netArchitectureMLP \(hidden dim 64, depth 3\)Time embeddingSinusoidal \(dim 16\)TrainingEpochs500Batch size128 \(train\), 2048 \(val\)OptimizerAdamW \(lr=10−3\\text\{lr\}=10^\{\-3\}, wd10−410^\{\-4\}\)Grad clipping1\.0GeoPathArchitectureMLP \(hidden dim 128, depth 3\)ActivationSELUOptimizerAdam \(lr=10−4\\text\{lr\}=10^\{\-4\}\)MetricLANDγ=0\.2\\gamma=0\.2,ρ=10−3\\rho=10^\{\-3\},α=1\.0\\alpha=1\.0Max samples4096SF2MSF2MModelVelocity \+ score MLPHidden dim 64, depth 3Time embeddingSinusoidal \(dim 16\)Distributionσbridge\\sigma\_\{\\text\{bridge\}\}1\.0σsample\\sigma\_\{\\text\{sample\}\}1\.0TrainingEpochs500Batch size128 \(train\), 2048 \(val\)OptimizerAdamW \(lr=10−3\\text\{lr\}=10^\{\-3\}, wd10−410^\{\-4\}\)Grad clipping1\.0UOT\-FMUOT\-FMModelVelocity MLPHidden dim 64, depth 3Time embeddingSinusoidal \(dim 16\)TrainingEpochs500Batch size128 \(train\), 2048 \(val\)OptimizerAdamW \(lr=10−3\\text\{lr\}=10^\{\-3\}, wd10−410^\{\-4\}\)OTConvergence tol\.10−910^\{\-9\}\(rel\./abs\.\)Marginal reg\.1\.0Flow MatchingFlow MatchingModelVelocity MLPHidden dim 64, depth 3Time embeddingSinusoidal \(dim 16\)TrainingEpochs500Batch size128 \(train\), 2048 \(val\)OptimizerAdamW \(lr=10−3\\text\{lr\}=10^\{\-3\}, wd10−410^\{\-4\}\)Table 8:Non–flow\-based baseline hyperparameters\.Baselines not relying on flow matching or diffusion bridges\.MethodCategoryHyperparameterSetting / NotesMioFlowModelNetwork layers\[64, 64, 64\]TrainingLearning rate1×10−41\\times 10^\{\-4\}Total epochs20Local epochs5Post\-local epochs5Batch size256Batches / epoch100MoscotOTEntropic reg\. \(ϵ\\epsilon\)0\.001Source reg\. \(τa\\tau\_\{a\}\)1\.0Target reg\. \(τb\\tau\_\{b\}\)1\.0VGFMModelHidden dimension64Hidden layers3ActivationTanhTrainingPre\-train epochs300Training epochs50Batch size256Learning rate \(init\)1×10−31\\times 10^\{\-3\}Learning rate \(second\)1×10−41\\times 10^\{\-4\}SolverStep size0\.01TrajectoryNetTrainingIterations1000Batch size1000Learning rate1×10−31\\times 10^\{\-3\}Weight decay1×10−51\\times 10^\{\-5\}ModelArchitecture1 block, concatsquash layers \(64–64–64\)RegularizationsL2ints\_\{\\mathrm\{L2int\}\}1×10−31\\times 10^\{\-3\}ktopk\_\{\\text\{top\}\}1×10−21\\times 10^\{\-2\}Training noise0\.1ODE solverSolverdopri5Time scale0\.4 \(5 integration points\)Tolerancesrtol=atol=10−5\\text\{rtol\}=\\text\{atol\}=10^\{\-5\}Conditional Flow Matching hyperparametersWe detail the hyperparameters used for downstream in[Table7](https://arxiv.org/html/2605.30635#A4.T7), which we kept fixed across the datasets\. Given a0\.9/0\.10\.9/0\.1train/val split, we keep the checkpoint that minimizes the validation loss over the run\.

MoscotWe use the implementation from the authors\(Kleinet al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib19)\)available at[https://github\.com/theislab/moscot](https://github.com/theislab/moscot)\. We summarize the hyperparameters used in[Table8](https://arxiv.org/html/2605.30635#A4.T8)\.

VGFMWe use the implementation from the authors\(Wanget al\.,[2025](https://arxiv.org/html/2605.30635#bib.bib18)\)available at[https://github\.com/DongyiWang\-66/VGFM](https://github.com/DongyiWang-66/VGFM)\. We summarize the hyperparameters used in[Table8](https://arxiv.org/html/2605.30635#A4.T8)\.

SF2MWe developed a custom implementation of the SF2M framework\(Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\)to enable the integration of theCellBRIDGEstructural prior into the simulation\-free training objective\. We summarize the hyperparameters used in[Table7](https://arxiv.org/html/2605.30635#A4.T7)\.

UOT\-FMTo incorporate the interaction\-aware coupling in an unbalanced setting, we utilized a custom implementation of UOT\-FM based on the original formulation\(Eyringet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib14)\)\. We summarize the hyperparameters used in[Table7](https://arxiv.org/html/2605.30635#A4.T7)\.

### D\.7Lung cancer data experiment

For the experiment described in[Section5\.3](https://arxiv.org/html/2605.30635#S5.SS3), we annotated the lung cancer dataset using canonical lineage and state markers \(Table[9](https://arxiv.org/html/2605.30635#A4.T9)\); an overview of the full dataset is shown in Fig\.[5](https://arxiv.org/html/2605.30635#A4.F5)\. Because whole\-lung profiling dilutes treatment effects \(the tumour comprises only a small fraction of total cells\), we constructed a focused*tumour\-niche*subset to increase sensitivity and interpretability\. Concretely, we retained all tumour cells and subsampled an equal number of T cells, B cells, fibroblasts, and endothelial cells from the same specimens to form a minimal viable tumour microenvironment\. We then reused the analysis pipeline described earlier with matched timepoints at0h,88h, and2424h\. The only modification was to the ligand–receptor \(LR\) library: for pathway\-specific probes, we toggled custom LR pairs to mimic the presence or absence of a given ligand \(e\.g\., EGFR\) and quantified the resulting changes in inferred communication and downstream dynamics\. Marker definitions are provided in Table[9](https://arxiv.org/html/2605.30635#A4.T9), and a dot\-plot confirming marker specificity and minimal cross\-lineage leakage is shown in Fig\.[6](https://arxiv.org/html/2605.30635#A4.F6)\.

Table 9:Curated panel of positive marker genes used for per\-cell scoring and assignment in the lung cancer dataset\.Cell TypePositive MarkersDifferentiated AT1RTKN2, AGERAT1CLDN18Tumour \(AT2\)SFTPD, LAMP3, SCGB3A2Mucous EpithelialDNAH12, AZGP1EndothelialSEMA3GLow IEG EndothelialCDH5Alveolar Capillary EndothelialEDNRB, RPRMLLymphatic Vein EndothelialLYVE1, SELE, VWFFibroblastsCOL1A2, PDGFRASmooth Muscle FibroblastsACTA2, LGR6Fibroblast SubsetDCNPericytesCSPG4MegakaryocytesPPBP, PF4ErythrocytesALAS2LymphocytesCCL21ACyclingTOP2ANeutrophilsS100A9, RETNLGBasophils & Mast cellsMCPT8, MS4A2MacrophagesMARCOMonocytesLY6IDC 1 and 2CLEC9A, XCR1, C1QA, SIGLECHDC 3FSCN1, IL12BNK cell likeNCR1, EOMES, TBX21ILCRORA, RORC, IL2RAAdaptive T cellsFOXP3, CD4, CD8AB cellsCD79A![Refer to caption](https://arxiv.org/html/2605.30635v1/images/marker_dotplot.png)Figure 6:Dot‐plot validation of curated marker genes across annotated cell types\. Each column corresponds to a marker gene and each row to a cell‐type label\. Dot size encodes the fraction of cells expressing that gene, while color intensity represents its standardized expression level\.#### D\.7\.1Tumour progression quantification using Hallmark gene sets

There is no single, universally accepted definition of tumour progression\. Clinical assessments typically use lesion size, extent of metastasis, and histopathology\. While we observe distinct cellular changes and invasion over our 24 h window, these measures are not applicable at single\-cell resolution\. Instead, we construct an approximate*tumour differentiation*score based on the Hallmarks of Cancer\(Hanahan and Weinberg,[2011](https://arxiv.org/html/2605.30635#bib.bib104)\), using the MSigDB*Hallmark*gene sets\(Liberzonet al\.,[2015](https://arxiv.org/html/2605.30635#bib.bib105)\)\.

For each hallmark, we compute a per\-cell score as the*median*expression across its member genes \(chosen over the mean for robustness to sparsity and outliers\)\. The overall progression score is then the mean across the 20 retained hallmarks\. The full hallmark definitions are available in MSigDB\(Liberzonet al\.,[2015](https://arxiv.org/html/2605.30635#bib.bib105)\); the selected hallmarks, their gene counts, and five example genes each are listed in Table[10](https://arxiv.org/html/2605.30635#A4.T10)\. Hallmarks not applicable to our tumour context \(e\.g\., hormonal signalling for breast/prostate, long\-term metabolic programs\) were excluded\.

As a baseline check, we verify that tumour cells exhibit coherent changes along the selected hallmarks over0h→24\\rightarrow 24h; see Fig\.[7](https://arxiv.org/html/2605.30635#A4.F7)\.

Table 10:Hallmark gene sets used for trajectory summarization\.We list each set’s size and five randomly sampled member genes\.Gene set\# GenesRandom gene examples \(5\)Angiogenesis36TIMP1, POSTN, VTN, THBD, NRP1Apoptosis161ERBB2, IL1B, DPYD, NEDD9, MADDDNA Repair150GTF2B, RAE1, ADCY6, POLA2, TAF1CE2F Targets200MCM7, PCNA, MCM4, RFC2, GINS1Epithelial–Mesenchymal Transition200SPP1, GPX7, LOX, THBS1, SLC6A8G2M Checkpoint200RBM14, AMD1, CDC27, UCK2, NDC80Glycolysis200SPAG4, PKP2, SLC25A13, PRPS1, ZNF292Hypoxia200S100A4, CSRP2, DTNA, PIM1, TPST2KRAS Signaling v1200FSHB, YPEL1, BARD1, SLC6A3, ATP6V1B1KRAS Signaling v2200CIDEA, KIF5C, LAT2, PDCD1LG2, PIGRMYC Targets v1200RAD23B, USP1, NAP1L1, NDUFAB1, SNRPA1MYC Targets v258PRMT3, AIMP2, SRM, EXOSC5, SUPV3L1Myogenesis200EIF4A2, PDE4DIP, ANKRD2, EPHB3, ATP6AP1Notch Signaling32SKP1, MAML2, HES1, FBXW11, DTX1Oxidative Phosphorylation200NDUFS8, VDAC1, UQCRQ, NDUFB3, NDUFB2p53 Pathway200TNNI1, SLC35D1, BTG1, FDXR, JAG2Peroxisome104IDH2, FIS1, EPHX2, SLC23A2, SLC25A4Reactive Oxygen Species Pathway49PRNP, OXSR1, SOD1, PDLIM1, TXNTNFα\\alphaSignaling via NFκ\\kappaB200DUSP2, CEBPB, OLR1, CCL20, IL1AXenobiotic Metabolism200SSR3, HACL1, ARPP19, AHCY, GSR![Refer to caption](https://arxiv.org/html/2605.30635v1/images/hallmark_median_barplot_t0_t24.png)Figure 7:Hallmark changes\.Changes in our dataset over 24 h following combined KRAS and MYC signalling across the 20 selected Hallmark gene sets\.

### D\.8Computational and memory costs

##### Complexity of the full OT solver\.

We solve[Equation5](https://arxiv.org/html/2605.30635#S4.E5)with a custom conditional\-gradient \(Frank–Wolfe\) solver detailed in[SectionD\.3](https://arxiv.org/html/2605.30635#A4.SS3)\. Letn0n\_\{0\}andn1n\_\{1\}be the numbers of cells in the two snapshots andKKthe number of ligand–receptor \(LR\) pairs \(interaction channels\)\.

Each Frank–Wolfe iteration consists of two main steps:

1. 1\.Gradient computation\.This yields a per\-iteration cost 𝒪​\(K​n0​n1​\(n0\+n1\)\)\\mathcal\{O\}\\bigl\(K\\,n\_\{0\}n\_\{1\}\(n\_\{0\}\+n\_\{1\}\)\\bigr\)because it requires performing the matrix multiplication ofC1\(r\)​ΓC^\{\(r\)\}\_\{1\}\\Gammaand\(C1\(r\)​Γ\)​\(C2\(r\)\)⊤\\bigl\(C^\{\(r\)\}\_\{1\}\\Gamma\\bigr\)\\bigl\(C^\{\(r\)\}\_\{2\}\\bigr\)^\{\\top\}for each channelrr\.
2. 2\.Linear OT subproblem\.Given the linearized objective, we solve a linear OT problem overΠ​\(a,b\)\\Pi\(a,b\)using POT’s\(Flamaryet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib32)\)existing OT routine\. Its complexity is 𝒪​\(costOT​\(n0,n1\)\),\\mathcal\{O\}\\bigl\(\\mathrm\{cost\}\_\{\\mathrm\{OT\}\}\(n\_\{0\},n\_\{1\}\)\\bigr\),\(e\.g\. cubic innnfor a network\-simplex LP, or𝒪​\(TSinkhorn​n0​n1\)\\mathcal\{O\}\(T\_\{\\mathrm\{Sinkhorn\}\}\\,n\_\{0\}n\_\{1\}\)for entropic OT\)\.

IfTCGT\_\{\\mathrm\{CG\}\}denotes the number of Frank–Wolfe iterations required to reach the desired tolerance, the total complexity of theCellBRIDGEOT stage is

𝒪​\(TCG​\[K​n0​n1​\(n0\+n1\)\+costOT​\(n0,n1\)\]\)\.\\mathcal\{O\}\\Bigl\(T\_\{\\mathrm\{CG\}\}\\bigl\[K\\,n\_\{0\}n\_\{1\}\(n\_\{0\}\+n\_\{1\}\)\+\\mathrm\{cost\}\_\{\\mathrm\{OT\}\}\(n\_\{0\},n\_\{1\}\)\\bigr\]\\Bigr\)\.
Wall\-clock runtimes\.We report the wall\-clock runtimes \(seconds\) in[Table11](https://arxiv.org/html/2605.30635#A4.T11), decomposing it into the OT part \(finding the couplingΓ∗\\Gamma^\{\*\}\) and the Flow matching part \(fitting the velocity model\)\.

Table 11:Wall\-clock runtime \(in seconds\) ofCellBRIDGE, decomposing into the OT part and the Flow matching part\.Lung TumourV1 LightDendritic StimulusMouse Cell AtlasOT \[s\]211\.3107\.04\.32656FM \[s\]189\.1186\.024\.24739
##### Memory footprint of the OT stage\.

The dominant memory costs come from: \(i\) the couplingΓ∈ℝn0×n1\\Gamma\\in\\mathbb\{R\}^\{n\_\{0\}\\times n\_\{1\}\}, \(ii\) the feature cost matrixC∈ℝn0×n1C\\in\\mathbb\{R\}^\{n\_\{0\}\\times n\_\{1\}\}, \(iii\) the multi\-channel structure tensorsC1∈ℝn0×n0×KC\_\{1\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times n\_\{0\}\\times K\}andC2∈ℝn1×n1×KC\_\{2\}\\in\\mathbb\{R\}^\{n\_\{1\}\\times n\_\{1\}\\times K\}\(corresponding to the CCI tensorsG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}\), and \(iv\) a small number of auxiliary matrices of sizen0×n1n\_\{0\}\\times n\_\{1\}\(e\.g\.constC\\mathrm\{constC\},B​\(Γ\)B\(\\Gamma\), and the gradient\)\. Crucially, we never construct the full tensor of pairwise structure discrepancies in[Equation5](https://arxiv.org/html/2605.30635#S4.E5)\. Instead, the structure term is implemented through the matrix productsC1\(r\)​Γ​\(C2\(r\)\)⊤C^\{\(r\)\}\_\{1\}\\Gamma\\bigl\(C^\{\(r\)\}\_\{2\}\\bigr\)^\{\\top\}\. As a result, the memory complexity of the OT solver scales as

𝒪​\(K​\(n02\+n12\)\+n0​n1\),\\mathcal\{O\}\\bigl\(K\(n\_\{0\}^\{2\}\+n\_\{1\}^\{2\}\)\+n\_\{0\}n\_\{1\}\\bigr\),Hence it is quadratic in the number of cells per snapshot and linear in the number of LR pairsKK\. In comparison, a feature\-only OT solver \(α=0\\alpha=0\) needsCCandΓ\\Gamma, with memory𝒪​\(n0​n1\)\\mathcal\{O\}\(n\_\{0\}n\_\{1\}\)\.

UsingCellBRIDGEwith large\-scale datasets\.While the computational and memory cost remained reasonable across the datasets we used, for very large datasets it can be mitigated using standard scalable techniques that are orthogonal to our formulation:

- •adding entropic regularization\(Peyré and Cuturi,[2019](https://arxiv.org/html/2605.30635#bib.bib5)\)and using Sinkhorn\-type solvers, which make the problem easier to optimize and reduce memory at the price of a small, controllable bias
- •employing mini\-batch optimization\(Fatraset al\.,[2021](https://arxiv.org/html/2605.30635#bib.bib15)\), where the CCI prior is estimated from couplings computed on minibatches instead of the whole dataset
- •constructing metacells, with details provided in[SectionD\.2](https://arxiv.org/html/2605.30635#A4.SS2)\. Using metacells reduces the effective sample size\. We evaluate this variant in[Section5\.3](https://arxiv.org/html/2605.30635#S5.SS3)\.

These strategies preserve the form of theCellBRIDGEprior while substantially improving scalability for large\-scale datasets\.

## Appendix EDownstream Dynamics withCellBRIDGE

Overview\.This appendix details how the interaction\-aware coupling produced byCellBRIDGEcan be combined with different continuous\-time trajectory models\. Across all variants, the interaction prior is encoded exclusively in the cross\-snapshot couplingΓ⋆\\Gamma^\{\\star\}\(or its induced joint distributionΠ\\Pi\), while the downstream dynamics model determines how trajectories are parameterized between coupled endpoints\. We consider four settings: deterministic flows via Conditional Flow Matching \(CFM\), geometry\-aware flows via Metric Flow Matching \(MFM\), stochastic Schrödinger bridges via SF2M, and population\-changing dynamics via unbalanced OT\. Importantly, no modification of the interaction\-aware FGW objective is required when switching between these mechanisms\.

### E\.1Conditional Flow Matching \(CFM\)

We begin with Conditional Flow Matching \(CFM\), which serves as the simplest and default downstream instantiation ofCellBRIDGE\. CFM learns a deterministic, time\-dependent velocity field whose induced flow matches a prescribed probability path between two endpoint distributions\.

##### Coupling\-induced probability path\.

Letρ0\\rho\_\{0\}andρ1\\rho\_\{1\}denote the empirical distributions associated with𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\}\. LetΓ⋆∈ℝ\+n0×n1\\Gamma^\{\\star\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times n\_\{1\}\}\_\{\+\}be the optimal coupling obtained from the interaction\-aware FGW problem[Equation5](https://arxiv.org/html/2605.30635#S4.E5), and defineM=∑i,jΓi​j⋆M=\\sum\_\{i,j\}\\Gamma^\{\\star\}\_\{ij\}\. We introduce the joint distribution

Π=∑i=1n0∑j=1n1Γi​j⋆M​δ\(xi,yj\),\\Pi\\;=\\;\\sum\_\{i=1\}^\{n\_\{0\}\}\\sum\_\{j=1\}^\{n\_\{1\}\}\\frac\{\\Gamma^\{\\star\}\_\{ij\}\}\{M\}\\,\\delta\_\{\(x\_\{i\},y\_\{j\}\)\},whose marginals areρ0\\rho\_\{0\}andρ1\\rho\_\{1\}\. Fort∈\[0,1\]t\\in\[0,1\], we define the affine interpolation

Zt=\(1−t\)​X\+t​Y,\(X,Y\)∼Π,Z\_\{t\}=\(1\-t\)X\+tY,\\qquad\(X,Y\)\\sim\\Pi,and letρt=ℒ​\(Zt\)\\rho\_\{t\}=\\mathcal\{L\}\(Z\_\{t\}\)\. By construction,\{ρt\}t∈\[0,1\]\\\{\\rho\_\{t\}\\\}\_\{t\\in\[0,1\]\}forms a probability path connectingρ0\\rho\_\{0\}andρ1\\rho\_\{1\}\.

##### Learning the velocity field\.

We learn a time\-dependent velocity fieldvθ:ℝd×\[0,1\]→ℝdv\_\{\\theta\}:\\mathbb\{R\}^\{d\}\\times\[0,1\]\\to\\mathbb\{R\}^\{d\}that generates the path\{ρt\}\\\{\\rho\_\{t\}\\\}\. For\(X,Y\)∼Π\(X,Y\)\\sim\\PiandZt=\(1−t\)​X\+t​YZ\_\{t\}=\(1\-t\)X\+tY, the interpolation implies a constant conditional drift

ut​\(Zt∣X,Y\)=Y−X\.u\_\{t\}\(Z\_\{t\}\\mid X,Y\)=Y\-X\.We trainvθv\_\{\\theta\}by minimizing the Conditional Flow Matching objective

ℒCFM​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=𝔼\(X,Y\)∼Πt∼Unif​\[0,1\]\[∥vθ\(Zt,t\)−ut\(Zt∣X,Y\)∥22\]\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(X,Y\)\\sim\\Pi\\\\ t\\sim\\mathrm\{Unif\}\[0,1\]\\end\{subarray\}\}\\Big\[\\big\\\|v\_\{\\theta\}\(Z\_\{t\},t\)\-u\_\{t\}\(Z\_\{t\}\\mid X,Y\)\\big\\\|\_\{2\}^\{2\}\\Big\]\(14\)=𝔼\(X,Y\)∼Πt∼Unif​\[0,1\]​\[‖vθ​\(Zt,t\)−\(Y−X\)‖22\]\.\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(X,Y\)\\sim\\Pi\\\\ t\\sim\\mathrm\{Unif\}\[0,1\]\\end\{subarray\}\}\\Big\[\\big\\\|v\_\{\\theta\}\(Z\_\{t\},t\)\-\(Y\-X\)\\big\\\|\_\{2\}^\{2\}\\Big\]\.\(15\)As shown inLipmanet al\.\([2024](https://arxiv.org/html/2605.30635#bib.bib29)\), the minimizer of this objective generates the target probability path\. After training, trajectories are obtained by integrating the ODEz˙​\(t\)=vθ​\(z​\(t\),t\)\\dot\{z\}\(t\)=v\_\{\\theta\}\(z\(t\),t\)with initial conditionz​\(0\)=xz\(0\)=x\.

### E\.2Unbalanced interaction\-aware dynamics

We next describe howCellBRIDGEcan be extended to settings where the total population mass changes between snapshots, e\.g\. due to cell proliferation or apoptosis\. Rather than enforcing exact marginal constraints, we adopt an unbalanced OT formulation that relaxes mass conservation\.

##### Step 1: inferring non\-uniform marginals\.

We first ignore interaction structure and solve an unbalanced feature\-only OT problem

Γu∈arg​minΓ∈ℝ≥0n0×n1⁡\{⟨Γ,C⟩\+λ0​KL​\(Γ​𝟏∥a\)\+λ1​KL​\(Γ⊤​𝟏∥b\)\},\\Gamma^\{\\mathrm\{u\}\}\\;\\in\\;\\operatorname\*\{arg\\,min\}\_\{\\Gamma\\in\\mathbb\{R\}\_\{\\geq 0\}^\{n\_\{0\}\\times n\_\{1\}\}\}\\Big\\\{\\langle\\Gamma,C\\rangle\+\\lambda\_\{0\}\\,\\mathrm\{KL\}\(\\Gamma\\mathbf\{1\}\\,\\\|\\,a\)\+\\lambda\_\{1\}\\,\\mathrm\{KL\}\(\\Gamma^\{\\top\}\\mathbf\{1\}\\,\\\|\\,b\)\\Big\\\},\(16\)where𝟏\\mathbf\{1\}denotes the all\-ones vector\. From the optimal solution we extract the reweighted marginals

a~=Γu​𝟏,b~=\(Γu\)⊤​𝟏,\\tilde\{a\}=\\Gamma^\{\\mathrm\{u\}\}\\mathbf\{1\},\\qquad\\tilde\{b\}=\(\\Gamma^\{\\mathrm\{u\}\}\)^\{\\top\}\\mathbf\{1\},which are renormalized to sum to one\. This step is independent of the interaction weightα\\alpha, allowing the same marginals to be reused across different FGW trade\-offs\.

##### Step 2: interaction\-aware FGW with frozen marginals\.

In a second step, we fix\(a~,b~\)\(\\tilde\{a\},\\tilde\{b\}\)and solve the interaction\-aware FGW problem

minΓ∈Π​\(a~,b~\)⁡\(1−α\)​ℱ​\(Γ\)\+α​𝒮​\(Γ\),\\min\_\{\\Gamma\\in\\Pi\(\\tilde\{a\},\\tilde\{b\}\)\}\(1\-\\alpha\)\\,\\mathcal\{F\}\(\\Gamma\)\+\\alpha\\,\\mathcal\{S\}\(\\Gamma\),\(17\)whereℱ\\mathcal\{F\}and𝒮\\mathcal\{S\}are defined as in[Equation5](https://arxiv.org/html/2605.30635#S4.E5)\. The resulting coupling preserves multi\-LR\-pair interaction structure while allowing unequal total mass between snapshots\. This coupling can be passed unchanged to any downstream dynamics model, including CFM, MFM, or SF2M\.

### E\.3CombiningCellBRIDGEwith Metric Flow Matching \(MFM\)

We now describe how the interaction\-aware coupling produced byCellBRIDGEcan be combined with Metric Flow Matching \(MFM\)\(Kapusniaket al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib20)\)\. MFM generalizes CFM by encouraging trajectories to follow geodesics of a data\-dependent Riemannian metricgg\.

Given a couplingqqbetween endpoint distributions, MFM first learns interpolants

xt,η=\(1−t\)​x0\+t​x1\+t​\(1−t\)​ϕt,η​\(x0,x1\),x\_\{t,\\eta\}=\(1\-t\)x\_\{0\}\+tx\_\{1\}\+t\(1\-t\)\\phi\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\),by minimizing the geodesic energy

Lg​\(η\)=𝔼\(x0,x1\)∼q,t​\[x˙t,η⊤​G​\(xt,η;𝒟\)​x˙t,η\],L\_\{g\}\(\\eta\)=\\mathbb\{E\}\_\{\(x\_\{0\},x\_\{1\}\)\\sim q,\\,t\}\\big\[\\dot\{x\}\_\{t,\\eta\}^\{\\top\}G\(x\_\{t,\\eta\};\\mathcal\{D\}\)\\dot\{x\}\_\{t,\\eta\}\\big\],whereG​\(⋅;𝒟\)G\(\\cdot;\\mathcal\{D\}\)denotes the coordinate representation ofgg\. In our experiments we use the LAND metricgLANDg\_\{\\mathrm\{LAND\}\}\(Arvanitidiset al\.,[2016](https://arxiv.org/html/2605.30635#bib.bib22)\)\.

After fittingη⋆\\eta^\{\\star\}, we train a velocity field using the interaction\-aware couplingΠ\\Pivia

ℒMFM​\(θ,η\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{MFM\}\}\(\\theta,\\eta\)=𝔼\(X,Y\)∼Πt∼Unif​\[0,1\]​\[‖vθ​\(Zt,η,t\)−x˙t,η​\(X,Y\)‖g​\(Zt,η\)2\],\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(X,Y\)\\sim\\Pi\\\\ t\\sim\\mathrm\{Unif\}\[0,1\]\\end\{subarray\}\}\\Big\[\\big\\\|v\_\{\\theta\}\(Z\_\{t,\\eta\},t\)\-\\dot\{x\}\_\{t,\\eta\}\(X,Y\)\\big\\\|\_\{g\(Z\_\{t,\\eta\}\)\}^\{2\}\\Big\],\(18\)where∥⋅∥g​\(Zt,η\)\\\|\\cdot\\\|\_\{g\(Z\_\{t,\\eta\}\)\}denotes the norm induced by the metric atZt,ηZ\_\{t,\\eta\}\.

### E\.4CombiningCellBRIDGEwith SF2M

Finally, we describe howCellBRIDGEcan be combined with SF2M\(Tonget al\.,[2024b](https://arxiv.org/html/2605.30635#bib.bib21)\)to obtain interaction\-aware Schrödinger\-bridge dynamics\. SF2M learns both a driftvθv\_\{\\theta\}and a scoresθs\_\{\\theta\}by regressing to the conditional drift and score of a mixture of Brownian bridges between coupled endpoints\.

For a single bridge with diffusionσ\\sigma, the conditional marginal at timet∈\(0,1\)t\\in\(0,1\)is

pt​\(x∣x0,x1\)=𝒩​\(x;\(1−t\)​x0\+t​x1,σ2​t​\(1−t\)​Id\),p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\mathcal\{N\}\\\!\\left\(x;\\,\(1\-t\)x\_\{0\}\+tx\_\{1\},\\,\\sigma^\{2\}t\(1\-t\)I\_\{d\}\\right\),with closed\-form drift and score\. To combine SF2M withCellBRIDGE, we simply replace the entropic OT endpoint coupling with the interaction\-aware couplingΠ\\Pi\. Training samples are generated by

t∼Unif\[0,1\],\(X,Y\)∼Π,Zt∼pt\(⋅∣X,Y\)\.t\\sim\\mathrm\{Unif\}\[0,1\],\\qquad\(X,Y\)\\sim\\Pi,\\qquad Z\_\{t\}\\sim p\_\{t\}\(\\cdot\\mid X,Y\)\.The resulting SF2M objective is

ℒSF2​M​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{SF\}^\{2\}\\mathrm\{M\}\}\(\\theta\)=𝔼\[∥vθ\(t,Zt\)−ut∘\(Zt∣X,Y\)∥22\\displaystyle=\\mathbb\{E\}\\Big\[\\\|v\_\{\\theta\}\(t,Z\_\{t\}\)\-u\_\{t\}^\{\\circ\}\(Z\_\{t\}\\mid X,Y\)\\\|\_\{2\}^\{2\}\(19\)\+λ\(t\)2∥sθ\(t,Zt\)−∇zlogpt\(Zt∣X,Y\)∥22\],\\displaystyle\\qquad\+\\lambda\(t\)^\{2\}\\\|s\_\{\\theta\}\(t,Z\_\{t\}\)\-\\nabla\_\{z\}\\log p\_\{t\}\(Z\_\{t\}\\mid X,Y\)\\\|\_\{2\}^\{2\}\\Big\],\(20\)whereλ​\(t\)=2​t​\(1−t\)/σ\\lambda\(t\)=2\\sqrt\{t\(1\-t\)\}/\\sigma\.

## Appendix FAdditional Results

### F\.1Synthetic dataset

On synthetic data, where the ground\-truth is known, we also evaluate direct matching metrics \(Hits@1 and Transport Rank Error \(TRE\)\)\. The results are reported in[Figure8](https://arxiv.org/html/2605.30635#A6.F8)\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x8.png)Figure 8:Structure\-aware coupling recovers the ground\-truth transport map on the synthetic dataset\.
### F\.2Stimulus datasets

We reproduce the experimental setup described in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)and[Section5\.2](https://arxiv.org/html/2605.30635#S5.SS2)with the macrophage stimulus\-response dataset\. We report the results in[Figure9](https://arxiv.org/html/2605.30635#A6.F9)and[Table12](https://arxiv.org/html/2605.30635#A6.T12), which are consistent with the findings on the other datasets\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x9.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x10.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x11.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x12.png)

Figure 9:Interpolationresults for the macrophage stimulus datasets\.Table 12:Interpolation error for continuous time dynamics \(lower is better\)\.CellBRIDGEwith varying structure weightα\\alphavs\. baselines for the macrophage stimulus datasets\. We report mean±\\pmstd over55runs\. MIOFlow failed to converge on stimulus CPG \(numerical instability\)\.Stimulus PICStimulus CPGStimulus LPSStimulus PCSK3Method𝜶\\boldsymbol\{\\alpha\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}TrajectoryNet—5\.628​\(0\.055\)5\.628\{\\scriptstyle\\,\(\\,0\.055\\,\)\}5\.930​\(0\.049\)5\.930\{\\scriptstyle\\,\(\\,0\.049\\,\)\}5\.361​\(0\.085\)5\.361\{\\scriptstyle\\,\(\\,0\.085\\,\)\}5\.826​\(0\.080\)5\.826\{\\scriptstyle\\,\(\\,0\.080\\,\)\}5\.087​\(0\.109\)5\.087\{\\scriptstyle\\,\(\\,0\.109\\,\)\}5\.589​\(0\.078\)5\.589\{\\scriptstyle\\,\(\\,0\.078\\,\)\}5\.033​\(0\.051\)5\.033\{\\scriptstyle\\,\(\\,0\.051\\,\)\}5\.434​\(0\.051\)5\.434\{\\scriptstyle\\,\(\\,0\.051\\,\)\}DSB—5\.796​\(0\.574\)5\.796\{\\scriptstyle\\,\(\\,0\.574\\,\)\}5\.833​\(0\.571\)5\.833\{\\scriptstyle\\,\(\\,0\.571\\,\)\}4\.500​\(0\.128\)4\.500\{\\scriptstyle\\,\(\\,0\.128\\,\)\}4\.594​\(0\.114\)4\.594\{\\scriptstyle\\,\(\\,0\.114\\,\)\}4\.685​\(0\.533\)4\.685\{\\scriptstyle\\,\(\\,0\.533\\,\)\}4\.815​\(0\.528\)4\.815\{\\scriptstyle\\,\(\\,0\.528\\,\)\}4\.648​\(0\.328\)4\.648\{\\scriptstyle\\,\(\\,0\.328\\,\)\}4\.749​\(0\.324\)4\.749\{\\scriptstyle\\,\(\\,0\.324\\,\)\}OT\-CFM—5\.544​\(0\.038\)5\.544\{\\scriptstyle\\,\(\\,0\.038\\,\)\}5\.614​\(0\.036\)5\.614\{\\scriptstyle\\,\(\\,0\.036\\,\)\}4\.430​\(0\.019\)4\.430\{\\scriptstyle\\,\(\\,0\.019\\,\)\}4\.512​\(0\.020\)4\.512\{\\scriptstyle\\,\(\\,0\.020\\,\)\}4\.434​\(0\.052\)4\.434\{\\scriptstyle\\,\(\\,0\.052\\,\)\}4\.582​\(0\.058\)4\.582\{\\scriptstyle\\,\(\\,0\.058\\,\)\}4\.423​\(0\.020\)4\.423\{\\scriptstyle\\,\(\\,0\.020\\,\)\}4\.551​\(0\.018\)4\.551\{\\scriptstyle\\,\(\\,0\.018\\,\)\}OT\-MFM—5\.501​\(0\.020\)5\.501\{\\scriptstyle\\,\(\\,0\.020\\,\)\}5\.566​\(0\.022\)5\.566\{\\scriptstyle\\,\(\\,0\.022\\,\)\}4\.464​\(0\.013\)4\.464\{\\scriptstyle\\,\(\\,0\.013\\,\)\}4\.556​\(0\.015\)4\.556\{\\scriptstyle\\,\(\\,0\.015\\,\)\}4\.440​\(0\.033\)4\.440\{\\scriptstyle\\,\(\\,0\.033\\,\)\}4\.592​\(0\.038\)4\.592\{\\scriptstyle\\,\(\\,0\.038\\,\)\}4\.384​\(0\.008\)4\.384\{\\scriptstyle\\,\(\\,0\.008\\,\)\}4\.505​\(0\.008\)4\.505\{\\scriptstyle\\,\(\\,0\.008\\,\)\}UOT\-FM—5\.414​\(0\.027\)5\.414\{\\scriptstyle\\,\(\\,0\.027\\,\)\}5\.492​\(0\.024\)5\.492\{\\scriptstyle\\,\(\\,0\.024\\,\)\}4\.572​\(0\.033\)4\.572\{\\scriptstyle\\,\(\\,0\.033\\,\)\}4\.733​\(0\.045\)4\.733\{\\scriptstyle\\,\(\\,0\.045\\,\)\}4\.570​\(0\.122\)4\.570\{\\scriptstyle\\,\(\\,0\.122\\,\)\}4\.785​\(0\.143\)4\.785\{\\scriptstyle\\,\(\\,0\.143\\,\)\}4\.332​\(0\.007\)4\.332\{\\scriptstyle\\,\(\\,0\.007\\,\)\}4\.483​\(0\.008\)4\.483\{\\scriptstyle\\,\(\\,0\.008\\,\)\}SF2M—6\.132​\(0\.059\)6\.132\{\\scriptstyle\\,\(\\,0\.059\\,\)\}6\.212​\(0\.058\)6\.212\{\\scriptstyle\\,\(\\,0\.058\\,\)\}4\.959​\(0\.044\)4\.959\{\\scriptstyle\\,\(\\,0\.044\\,\)\}5\.053​\(0\.043\)5\.053\{\\scriptstyle\\,\(\\,0\.043\\,\)\}4\.930​\(0\.051\)4\.930\{\\scriptstyle\\,\(\\,0\.051\\,\)\}5\.057​\(0\.054\)5\.057\{\\scriptstyle\\,\(\\,0\.054\\,\)\}5\.048​\(0\.031\)5\.048\{\\scriptstyle\\,\(\\,0\.031\\,\)\}5\.155​\(0\.032\)5\.155\{\\scriptstyle\\,\(\\,0\.032\\,\)\}VGFM—9\.796​\(0\.658\)9\.796\{\\scriptstyle\\,\(\\,0\.658\\,\)\}9\.863​\(0\.683\)9\.863\{\\scriptstyle\\,\(\\,0\.683\\,\)\}8\.242​\(0\.192\)8\.242\{\\scriptstyle\\,\(\\,0\.192\\,\)\}8\.362​\(0\.205\)8\.362\{\\scriptstyle\\,\(\\,0\.205\\,\)\}8\.026​\(0\.135\)8\.026\{\\scriptstyle\\,\(\\,0\.135\\,\)\}8\.158​\(0\.1469\)8\.158\{\\scriptstyle\\,\(\\,0\.1469\\,\)\}8\.205​\(0\.1101\)8\.205\{\\scriptstyle\\,\(\\,0\.1101\\,\)\}8\.297​\(0\.110\)8\.297\{\\scriptstyle\\,\(\\,0\.110\\,\)\}MIOFlow—9\.365​\(0\.382\)9\.365\{\\scriptstyle\\,\(\\,0\.382\\,\)\}9\.440​\(0\.404\)9\.440\{\\scriptstyle\\,\(\\,0\.404\\,\)\}16899​\(22805\)16899\{\\scriptstyle\\,\(\\,22805\\,\)\}22007​\(31264\)22007\{\\scriptstyle\\,\(\\,31264\\,\)\}7\.807​\(0\.038\)7\.807\{\\scriptstyle\\,\(\\,0\.038\\,\)\}7\.927​\(0\.034\)7\.927\{\\scriptstyle\\,\(\\,0\.034\\,\)\}8\.179​\(0\.150\)8\.179\{\\scriptstyle\\,\(\\,0\.150\\,\)\}8\.281​\(0\.166\)8\.281\{\\scriptstyle\\,\(\\,0\.166\\,\)\}Moscot—7\.213​\(0\.000\)7\.213\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.244​\(0\.0000\)7\.244\{\\scriptstyle\\,\(\\,0\.0000\\,\)\}6\.983​\(0\.000\)6\.983\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.087​\(0\.000\)7\.087\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.663​\(0\.000\)7\.663\{\\scriptstyle\\,\(\\,0\.000\\,\)\}7\.786​\(0\.000\)7\.786\{\\scriptstyle\\,\(\\,0\.000\\,\)\}6\.734​\(0\.000\)6\.734\{\\scriptstyle\\,\(\\,0\.000\\,\)\}6\.832​\(0\.000\)6\.832\{\\scriptstyle\\,\(\\,0\.000\\,\)\}0\.56\.109​\(0\.081\)6\.109\{\\scriptstyle\\,\(\\,0\.081\\,\)\}6\.193​\(0\.086\)6\.193\{\\scriptstyle\\,\(\\,0\.086\\,\)\}4\.948​\(0\.066\)4\.948\{\\scriptstyle\\,\(\\,0\.066\\,\)\}5\.027​\(0\.063\)5\.027\{\\scriptstyle\\,\(\\,0\.063\\,\)\}4\.846​\(0\.046\)4\.846\{\\scriptstyle\\,\(\\,0\.046\\,\)\}4\.973​\(0\.046\)4\.973\{\\scriptstyle\\,\(\\,0\.046\\,\)\}4\.960​\(0\.058\)4\.960\{\\scriptstyle\\,\(\\,0\.058\\,\)\}5\.084​\(0\.058\)5\.084\{\\scriptstyle\\,\(\\,0\.058\\,\)\}CellBRIDGE\+SF2M16\.110​\(0\.062\)6\.110\{\\scriptstyle\\,\(\\,0\.062\\,\)\}6\.197​\(0\.063\)6\.197\{\\scriptstyle\\,\(\\,0\.063\\,\)\}4\.951​\(0\.056\)4\.951\{\\scriptstyle\\,\(\\,0\.056\\,\)\}5\.037​\(0\.058\)5\.037\{\\scriptstyle\\,\(\\,0\.058\\,\)\}4\.874​\(0\.068\)4\.874\{\\scriptstyle\\,\(\\,0\.068\\,\)\}5\.016​\(0\.065\)5\.016\{\\scriptstyle\\,\(\\,0\.065\\,\)\}5\.016​\(0\.029\)5\.016\{\\scriptstyle\\,\(\\,0\.029\\,\)\}5\.173​\(0\.039\)5\.173\{\\scriptstyle\\,\(\\,0\.039\\,\)\}0\.55\.483​\(0\.012\)5\.483\{\\scriptstyle\\,\(\\,0\.012\\,\)\}5\.544​\(0\.014\)5\.544\{\\scriptstyle\\,\(\\,0\.014\\,\)\}4\.448​\(0\.007\)4\.448\{\\scriptstyle\\,\(\\,0\.007\\,\)\}4\.527​\(0\.006\)4\.527\{\\scriptstyle\\,\(\\,0\.006\\,\)\}4\.442​\(0\.035\)4\.442\{\\scriptstyle\\,\(\\,0\.035\\,\)\}4\.589​\(0\.041\)4\.589\{\\scriptstyle\\,\(\\,0\.041\\,\)\}4\.386​\(0\.016\)4\.386\{\\scriptstyle\\,\(\\,0\.016\\,\)\}4\.520​\(0\.013\)4\.520\{\\scriptstyle\\,\(\\,0\.013\\,\)\}CellBRIDGE\+MFM15\.376​\(0\.021\)5\.376\{\\scriptstyle\\,\(\\,0\.021\\,\)\}5\.440​\(0\.022\)5\.440\{\\scriptstyle\\,\(\\,0\.022\\,\)\}4\.460​\(0\.079\)4\.460\{\\scriptstyle\\,\(\\,0\.079\\,\)\}4\.543​\(0\.082\)4\.543\{\\scriptstyle\\,\(\\,0\.082\\,\)\}4\.477​\(0\.050\)4\.477\{\\scriptstyle\\,\(\\,0\.050\\,\)\}4\.641​\(0\.062\)4\.641\{\\scriptstyle\\,\(\\,0\.062\\,\)\}4\.329​\(0\.023\)4\.329\{\\scriptstyle\\,\(\\,0\.023\\,\)\}4\.507​\(0\.029\)4\.507\{\\scriptstyle\\,\(\\,0\.029\\,\)\}0\.55\.355​\(0\.021\)5\.355\{\\scriptstyle\\,\(\\,0\.021\\,\)\}5\.451​\(0\.018\)5\.451\{\\scriptstyle\\,\(\\,0\.018\\,\)\}4\.544​\(0\.033\)4\.544\{\\scriptstyle\\,\(\\,0\.033\\,\)\}4\.700​\(0\.035\)4\.700\{\\scriptstyle\\,\(\\,0\.035\\,\)\}4\.489​\(0\.054\)4\.489\{\\scriptstyle\\,\(\\,0\.054\\,\)\}4\.692​\(0\.061\)4\.692\{\\scriptstyle\\,\(\\,0\.061\\,\)\}4\.343​\(0\.025\)4\.343\{\\scriptstyle\\,\(\\,0\.025\\,\)\}4\.522​\(0\.033\)4\.522\{\\scriptstyle\\,\(\\,0\.033\\,\)\}CellBRIDGE\+UOT\-FM15\.346​\(0\.024\)5\.346\{\\scriptstyle\\,\(\\,0\.024\\,\)\}5\.487​\(0\.038\)5\.487\{\\scriptstyle\\,\(\\,0\.038\\,\)\}4\.547​\(0\.035\)4\.547\{\\scriptstyle\\,\(\\,0\.035\\,\)\}4\.755​\(0\.040\)4\.755\{\\scriptstyle\\,\(\\,0\.040\\,\)\}4\.489​\(0\.038\)4\.489\{\\scriptstyle\\,\(\\,0\.038\\,\)\}4\.716​\(0\.041\)4\.716\{\\scriptstyle\\,\(\\,0\.041\\,\)\}4\.325​\(0\.038\)4\.325\{\\scriptstyle\\,\(\\,0\.038\\,\)\}4\.531​\(0\.038\)4\.531\{\\scriptstyle\\,\(\\,0\.038\\,\)\}CellBRIDGE\+CFM0\.55\.490​\(0\.018\)5\.490\{\\scriptstyle\\,\(\\,0\.018\\,\)\}5\.555​\(0\.019\)5\.555\{\\scriptstyle\\,\(\\,0\.019\\,\)\}4\.427​\(0\.021\)4\.427\{\\scriptstyle\\,\(\\,0\.021\\,\)\}4\.502​\(0\.025\)4\.502\{\\scriptstyle\\,\(\\,0\.025\\,\)\}4\.380​\(0\.021\)4\.380\{\\scriptstyle\\,\(\\,0\.021\\,\)\}4\.517​\(0\.021\)4\.517\{\\scriptstyle\\,\(\\,0\.021\\,\)\}4\.396​\(0\.023\)4\.396\{\\scriptstyle\\,\(\\,0\.023\\,\)\}4\.531​\(0\.019\)4\.531\{\\scriptstyle\\,\(\\,0\.019\\,\)\}15\.446​\(0\.018\)5\.446\{\\scriptstyle\\,\(\\,0\.018\\,\)\}5\.512​\(0\.016\)5\.512\{\\scriptstyle\\,\(\\,0\.016\\,\)\}4\.440​\(0\.041\)4\.440\{\\scriptstyle\\,\(\\,0\.041\\,\)\}4\.518​\(0\.044\)4\.518\{\\scriptstyle\\,\(\\,0\.044\\,\)\}4\.431​\(0\.126\)4\.431\{\\scriptstyle\\,\(\\,0\.126\\,\)\}4\.577​\(0\.141\)4\.577\{\\scriptstyle\\,\(\\,0\.141\\,\)\}4\.352​\(0\.020\)4\.352\{\\scriptstyle\\,\(\\,0\.020\\,\)\}4\.530​\(0\.033\)4\.530\{\\scriptstyle\\,\(\\,0\.033\\,\)\}

### F\.3ScalingCellBRIDGEto cell atlas level datasets

We extended the evaluation to substantially larger datasets with two additional interpolation tasks on a mouse development cell atlas\(Qiuet al\.,[2022](https://arxiv.org/html/2605.30635#bib.bib83)\)\. These results show thatCellBRIDGEremains feasible in this larger\-scale regime \(see Table[13](https://arxiv.org/html/2605.30635#A6.T13)\)\. We did not include UOT\-FM on this benchmark because the unbalanced OT solver did not scale to datasets of this size in our experiments\.

Table 13:Interpolation error for embryo developmental transitions \(lower is better\)\.We report mean±\\pmstd over runs forW1/W2W\_\{1\}/W\_\{2\}\.E7\.5→\\rightarrowE8E7\.75→\\rightarrowE8\.25MethodW1W\_\{1\}W2W\_\{2\}W1W\_\{1\}W2W\_\{2\}OT\-CFM3\.266​\(0\.011\)3\.266\{\\scriptstyle\\,\(\\,0\.011\\,\)\}3\.575​\(0\.023\)3\.575\{\\scriptstyle\\,\(\\,0\.023\\,\)\}3\.229​\(0\.018\)3\.229\{\\scriptstyle\\,\(\\,0\.018\\,\)\}3\.249​\(0\.010\)3\.249\{\\scriptstyle\\,\(\\,0\.010\\,\)\}OT\-MFM3\.249​\(0\.007\)3\.249\{\\scriptstyle\\,\(\\,0\.007\\,\)\}3\.586​\(0\.025\)3\.586\{\\scriptstyle\\,\(\\,0\.025\\,\)\}3\.235​\(0\.015\)3\.235\{\\scriptstyle\\,\(\\,0\.015\\,\)\}3\.250​\(0\.007\)3\.250\{\\scriptstyle\\,\(\\,0\.007\\,\)\}SF2M4\.967​\(0\.078\)4\.967\{\\scriptstyle\\,\(\\,0\.078\\,\)\}5\.382​\(0\.078\)5\.382\{\\scriptstyle\\,\(\\,0\.078\\,\)\}4\.748​\(0\.108\)4\.748\{\\scriptstyle\\,\(\\,0\.108\\,\)\}4\.884​\(0\.125\)4\.884\{\\scriptstyle\\,\(\\,0\.125\\,\)\}CellBRIDGE\+CFM3\.068​\(0\.007\)3\.068\{\\scriptstyle\\,\(\\,0\.007\\,\)\}3\.555​\(0\.005\)3\.555\{\\scriptstyle\\,\(\\,0\.005\\,\)\}3\.131​\(0\.022\)3\.131\{\\scriptstyle\\,\(\\,0\.022\\,\)\}3\.323​\(0\.037\)3\.323\{\\scriptstyle\\,\(\\,0\.037\\,\)\}CellBRIDGE\+MFM3\.054​\(0\.009\)3\.054\{\\scriptstyle\\,\(\\,0\.009\\,\)\}3\.546​\(0\.014\)3\.546\{\\scriptstyle\\,\(\\,0\.014\\,\)\}3\.140​\(0\.023\)3\.140\{\\scriptstyle\\,\(\\,0\.023\\,\)\}3\.384​\(0\.020\)3\.384\{\\scriptstyle\\,\(\\,0\.020\\,\)\}CellBRIDGE\+SF2M4\.585​\(0\.110\)4\.585\{\\scriptstyle\\,\(\\,0\.110\\,\)\}4\.872​\(0\.132\)4\.872\{\\scriptstyle\\,\(\\,0\.132\\,\)\}4\.767​\(0\.238\)4\.767\{\\scriptstyle\\,\(\\,0\.238\\,\)\}4\.947​\(0\.225\)4\.947\{\\scriptstyle\\,\(\\,0\.225\\,\)\}
### F\.4Sensitivity of couplings to catalog edits

The experiment presented in[Section5\.3](https://arxiv.org/html/2605.30635#S5.SS3)involved perturbing the LR catalog by removing specific LR pairs\. In[Table14](https://arxiv.org/html/2605.30635#A6.T14), we show how the coupling changes, by computing the fraction of source cells whose target argmax differs between ”active” vs\. ”inactive” LR libraries for each pathway\.

Table 14:Coupling changes \(argmax\) atα=1\.0\\alpha=1\.0\.Fraction of source cells whose target argmax differs between “active” vs\. “inactive” LR libraries for each pathway;N=2195N\{=\}2195source cells\. Targeted pathways \(EGFR/ALK/MET\) show large shifts, while cardio–renal controls \(RAAS, Vasopressin, Natriuretic\) show little or moderate effect, as expected\.Pathway / SystemCoupling changed \(count /NN\)PercentEGFR \(targeted\)2071/21952071/219594\.35%94\.35\\%ALK \(targeted\)2164/21952164/219598\.59%98\.59\\%MET \(targeted\)2154/21952154/219598\.13%98\.13\\%RAAS \(control\)0/21950/21950\.00%0\.00\\%Vasopressin \(control\)0/21950/21950\.00%0\.00\\%Natriuretic \(control\)1582/21951582/219572\.07%72\.07\\%
### F\.5Comparing cell interaction types

We further examined how different classes of molecular interactions influence the resulting transport couplings\. Using our automated selection procedure \([SectionD\.5](https://arxiv.org/html/2605.30635#A4.SS5)\), we identified a top\-ranking set of1010ligand\-receptor pairs for each of the datasets\. We contrasted this against a matched set of1010canonical long\-range soluble cytokines and growth factors: \(CXCL12\-CXCR4, VEGFA\-KDR, CCL5\-CCR5, TGFB1\-TGFBR2, IL6\-IL6R, EGF\-EGFR, TNF\-TNFRSF1A, IGF1\-IGF1R, CSF1\-CSF1R, IFNG\-IFNGR1\)\. As shown in[Table15](https://arxiv.org/html/2605.30635#A6.T15), the cytokine pairs exhibit slightly higher Wasserstein \(W1W\_\{1\}andW2W\_\{2\}\) distances compared to the results obtained previously with our selection procedure\. This suggests that the specific interaction modes we keep have more informative topological constraints on the transport map than generic diffusive signaling, effectively recovering structure\-aware couplings that reflect the physical tissue architecture\.

Table 15:Wasserstein distances for pure structural alignment \(α=1\\alpha=1\)\.Comparison of interpolation performance using generic Long Range priors versus our automated selection procedure\. Lower values indicate better alignment\.DatasetInteraction Prior𝒲1↓\\mathcal\{W\}\_\{1\}\\downarrow𝒲2↓\\mathcal\{W\}\_\{2\}\\downarrowV1 LightLong range2\.422\.63Dataset\-specific2\.352\.59ImmuneLong range3\.583\.73Dataset\-specific3\.593\.73Lung CancerLong range2\.102\.33Dataset\-specific2\.022\.30
### F\.6CombiningCellBRIDGEwith other priors

A key advantage ofCellBRIDGEis its modularity: the CCI\-derived prior only depends on the CCI tensors\(G\(0\),G\(1\)\)\(G^\{\(0\)\},G^\{\(1\)\}\)and on a couplingΓ\\Gamma, and is therefore largely orthogonal to howΓ\\Gammais obtained\. As a consequence, the CCI prior can be combined with a wide range of existing priors or architectural choices for trajectory inference\. Here, we illustrate this flexibility by extendingCellBRIDGEto two settings: \(i\) unbalanced OT, which explicitly accounts for cell birth and death between snapshots, and \(ii\) metric flow matching, which replaces the standard Euclidean flow\-matching objective with a geometry\-aware variant\. Details on both of these implementations can be found in[SectionE\.2](https://arxiv.org/html/2605.30635#A5.SS2)and[SectionE\.3](https://arxiv.org/html/2605.30635#A5.SS3)\.

We reproduce the experiment in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)with theseCellBRIDGEvariants, and report the results in[Figure10](https://arxiv.org/html/2605.30635#A6.F10)and[Figure11](https://arxiv.org/html/2605.30635#A6.F11)\. We notice the following:

- •α\>0\\alpha\>0remains optimal\.For all datasets and the twoCellBRIDGEvariants, the bestW1W\_\{1\}/W2W\_\{2\}values occur at a non\-zero structure weightα\\alpha, mirroring the behavior observed in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)\.
- •Complementary to other priors\.The fact thatα\>0\\alpha\>0remains optimal shows that adding the CCI prior on top of MFM or UOT\-FM yields consistent improvements over the corresponding feature\-only baselines, highlighting thatCellBRIDGE’s gains are not tied to a specific OT or flow\-matching objective, but rather come from the biological prior\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x13.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x14.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x15.png)

Figure 10:Incorporating the CCI prior with MFM\.We plot theW1W\_\{1\}andW2W\_\{2\}distances between the interpolated and empiricalt1t\_\{1\}snapshots asα\\alphavaries\.![Refer to caption](https://arxiv.org/html/2605.30635v1/x16.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x17.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x18.png)

Figure 11:Incorporating the CCI prior with UOT\-CFM\.We plot theW1W\_\{1\}andW2W\_\{2\}distances between the interpolated and empiricalt1t\_\{1\}snapshots asα\\alphavaries\.
### F\.7Sensitivity analysis on the LR expressions

In this section, we study the sensitivity ofCellBRIDGEto measurement noise in the LR expressions\. We inject this noise in LR genes expression by adding zero\-mean Gaussian noise to the gene expressions before applying the Hill transform and clipping below by0, i\.e\. we definex~c​g=max⁡\(0,xc​g\+ϵc​g\)\\tilde\{x\}\_\{cg\}=\\max\(0,x\_\{cg\}\+\\epsilon\_\{cg\}\)whereϵc​g∼𝒩​\(0,σg2\)\\epsilon\_\{cg\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{g\}^\{2\}\)\.σg2\\sigma\_\{g\}^\{2\}denotes a gene\-specific noise variance, defined asσg=β​σ^g\\sigma\_\{g\}=\\beta\\hat\{\\sigma\}\_\{g\}, withσ^g\\hat\{\\sigma\}\_\{g\}the empirical standard deviation of\{xc​g∣c∈\[n\]\}\\\{x\_\{cg\}\\mid c\\in\[n\]\\\}\(to take into account per\-gene variance\) andβ\\betaa scaling factor\. From these perturbed expressions, we compute the activationss~c​g\\tilde\{s\}\_\{cg\}and we construct the CCI tensors with the entriesq~i→j\(pk\)=s~i​ℓk​s~j​rk\\tilde\{q\}^\{\(p\_\{k\}\)\}\_\{i\\to j\}=\\tilde\{s\}\_\{i\\ell\_\{k\}\}\\,\\tilde\{s\}\_\{jr\_\{k\}\}and obtain the couplings by solving[Equation5](https://arxiv.org/html/2605.30635#S4.E5)\.

We report the results in[Figure12](https://arxiv.org/html/2605.30635#A6.F12), where we sweepβ\\betafor different values across the interval\[0,2\]\[0,2\], withα=1\\alpha=1\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x19.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x20.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x21.png)

Figure 12:Perturbation of the LR expressions with Gaussian noise\.We plot theW1W\_\{1\}andW2W\_\{2\}distances between the interpolated and empiricalt1t\_\{1\}snapshots as the scaling factor of the noiseβ\\betaincreases\.As the noise scaleβ\\betaincreases, bothW1W\_\{1\}andW2W\_\{2\}gradually deteriorate across all three datasets\. This non\-zero sensitivity is expected and desirable: if the CCI prior was irrelevant, corrupting the LR expressions would leave the interpolation error unchanged\. Instead, adding noise worsens alignment, showing the benefits of the prior\. Furthermore, the performance is relatively robust to small levels of noise for the Lung tumour and V1 Light datasets\. Interestingly, for the V1 Light dataset, we see thatβ∈\{0\.1,0\.2\}\\beta\\in\\\{0\.1,0\.2\\\}improves the results uponβ=0\\beta=0, which we attribute to a small regularization / denoising effect\. Adding a small amount of centered Gaussian noise before the Hill transform and clipping makes low\-intensity ligand or receptor expressions become zero while leaving strongly expressed pairs essentially unchanged\. The results are noisier for the Dendritic Stimulus dataset, which we attribute to the smaller size of the dataset\.

### F\.8Sensitivity with respect toKgK\_\{g\}andhgh\_\{g\}

In this section, we conduct a sensitivity analysis on the hyperparametersKgK\_\{g\}andhgh\_\{g\}, used to define the interaction scores in[Section4\.1](https://arxiv.org/html/2605.30635#S4.SS1)assc​g=xc​ghg/\(xc​ghg\+Kghg\)s\_\{cg\}=x\_\{cg\}^\{h\_\{g\}\}/\(x\_\{cg\}^\{h\_\{g\}\}\+K\_\{g\}^\{h\_\{g\}\}\)\. We consider different values of the percentile levelp∈\{80,90,99\}p\\in\\\{80,90,99\\\}\(withKg=Qg​\(p\)K\_\{g\}=Q\_\{g\}\(p\)denoting thepp\-th percentile of\{xc​g∣c∈\[n\]\}\\\{x\_\{cg\}\\mid c\\in\[n\]\\\}\) andhg∈\{1,2,4\}h\_\{g\}\\in\\\{1,2,4\\\}, forα=0\.5\\alpha=0\.5\. We report the interpolation results in[Figure13](https://arxiv.org/html/2605.30635#A6.F13)\. We observe that the performance is largely insensitive to the specific choice of these parameters\. This stability justifies the use of standard default values \(9090th percentile andhg=1h\_\{g\}=1\) across our experiments without the need for extensive per\-dataset tuning\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x22.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x23.png)
![Refer to caption](https://arxiv.org/html/2605.30635v1/x24.png)

Figure 13:Sensitivity analysis on the hyperparameters of the Hill transform\.We plot theW1W\_\{1\}distances between the interpolated and empiricalt1t\_\{1\}snapshots for different values ofKgK\_\{g\}andhgh\_\{g\}\.
### F\.9Path curvature analysis

To empirically demonstrate thatCellBRIDGElearns non\-linear interaction effects, we measure the average path length ratio \(displacement divided by path length\) of the inferred trajectories:

S​\(z0,vθ\)=‖z1−z0‖2∫01‖vθ​\(zt,t\)‖2​𝑑tS\(z\_\{0\},v\_\{\\theta\}\)=\\frac\{\\\|z\_\{1\}\-z\_\{0\}\\\|\_\{2\}\}\{\\int\_\{0\}^\{1\}\\\|v\_\{\\theta\}\(z\_\{t\},t\)\\\|\_\{2\}\\,dt\}\(21\)
wherezt=z0\+∫0tvθ​\(zt,t\)​𝑑tz\_\{t\}=z\_\{0\}\+\\int\_\{0\}^\{t\}v\_\{\\theta\}\(z\_\{t\},t\)dtfort∈\[0,1\]t\\in\[0,1\]andz0z\_\{0\}denotes the initial point\.

A ratio of1\.01\.0indicates a straight line, while values<1\.0<1\.0indicate curvature\.

As shown in[Table16](https://arxiv.org/html/2605.30635#A6.T16), increasing the interaction weightα\\alphaleads to significantly higher curvature \(lower ratios\), confirming that incorporating interactions prevents the model from simply learning independent straight lines\.

Table 16:Path length ratio comparison across different datasets\.α\\alphaLung TumourDendritic StimulusV1 Light00\.974±0\.0010\.974\\pm 0\.0010\.982±0\.0020\.982\\pm 0\.0020\.954±0\.0040\.954\\pm 0\.0040\.50\.952±0\.0020\.952\\pm 0\.0020\.979±0\.0030\.979\\pm 0\.0030\.907±0\.0110\.907\\pm 0\.01110\.862±0\.0110\.862\\pm 0\.0110\.969±0\.0050\.969\\pm 0\.0050\.635±0\.0090\.635\\pm 0\.009
### F\.10Normalization procedure ablation

Our normalization scheme described in[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4)requires solving two OT problems \(forα=0\\alpha=0andα=1\\alpha=1\) to calibrate the relative scales of the cost and structural terms\. The motivation is thatα\\alphashould smoothly control the balance between the two terms\. We test a simpler normalization strategy on the tumour dataset that avoids these endpoint OT solves, by scalingCC,G\(0\)G^\{\(0\)\}, andG\(1\)G^\{\(1\)\}by their respective medians\. The results in Figure[14](https://arxiv.org/html/2605.30635#A6.F14)show that our normalization provides better calibration between the feature and structural terms\.

![Refer to caption](https://arxiv.org/html/2605.30635v1/x25.png)Figure 14:Comparison between our proposed normalization schemes and a median\-based normalization for the Lung Tumour dataset, evaluated via held\-out interpolation as in[Section5\.1](https://arxiv.org/html/2605.30635#S5.SS1)\.
### F\.11Perturbation analysis on the dendritic\-cell stimulus dataset

To test whetherCellBRIDGEcan model signalling\-level perturbations beyond the lung tumour setting, we performed an additional analysis on the dendritic\-cell stimulus\-response dataset\(Shaleket al\.,[2014](https://arxiv.org/html/2605.30635#bib.bib28)\)\. The dataset contains mouse bone\-marrow\-derived dendritic cells under LPS stimulation, including wild\-type cells at baseline and 4 h after stimulation, together with experimentally observed 4 h knockout conditions forIfnar1,Stat1, andTnfr\.

We learned couplings from wild\-type baseline cells to wild\-type 4 h LPS\-stimulated cells and compared two in silico perturbation strategies for each pathway axis\. In the gene\-edit baseline, we zeroed out the perturbed gene in the wild\-type endpoint snapshots before computing the predicted 4 h distribution\. In the pathway\-edit setting, we left the wild\-type expression snapshots unchanged but modified the ligand–receptor catalogue so that the perturbation acted through the signalling\-structure term in the coupling\. We then evaluated each perturbation by comparing the predicted 4 h distribution against the experimentally observed 4 h knockout population\.

Table 17:In silico perturbation analysis on the dendritic\-cell stimulus dataset\.We compare gene\-level edits \(α=0\\alpha=0\) and pathway\-level edits \(α=1\\alpha=1\) against experimentally observed knockout populations\. Lower is better\.ConditionMethod𝐖𝟏\\mathbf\{W\_\{1\}\}𝐖𝟐\\mathbf\{W\_\{2\}\}IFNAR1 KOα=0\\alpha=0\(gene edit\)4\.5064\.648IFNAR1 KOα=1\\alpha=1\(pathway edit\)4\.3764\.490STAT1 KOα=0\\alpha=0\(gene edit\)5\.6065\.686STAT1 KOα=1\\alpha=1\(pathway edit\)5\.5025\.567TNFR KOα=0\\alpha=0\(gene edit\)4\.4874\.572TNFR KOα=1\\alpha=1\(pathway edit\)4\.3614\.437

## Appendix GTheoretical Analysis

### G\.1Synthetic setup

In this section, we provide a theoretical guarantee for the synthetic setup of Section[C\.1](https://arxiv.org/html/2605.30635#A3.SS1)\.

###### Theorem 1\.

Let𝒟0\\mathcal\{D\}\_\{0\}and𝒟1\\mathcal\{D\}\_\{1\}be the source and target datasets defined by the synthetic clusters, and letG\(0\),G\(1\)G^\{\(0\)\},G^\{\(1\)\}be the associated directed interaction tensors\.

Consider the two candidate couplings:

1. 1\.ΓG​T\\Gamma\_\{GT\}:The transport plan corresponding to the true translation vectors \(preserving cluster identity\)\.
2. 2\.ΓF​O\\Gamma\_\{FO\}:The transport plan corresponding to the feature\-only map\.

As in[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4), define the \(unnormalized\) feature and structure gaps between these two couplings as

Δ​ℱ:=ℱ​\(ΓF​O\)−ℱ​\(ΓG​T\),Δ​𝒮:=𝒮​\(ΓF​O\)−𝒮​\(ΓG​T\),\\Delta\\mathcal\{F\}:=\\mathcal\{F\}\(\\Gamma\_\{FO\}\)\-\\mathcal\{F\}\(\\Gamma\_\{GT\}\),\\qquad\\Delta\\mathcal\{S\}:=\\mathcal\{S\}\(\\Gamma\_\{FO\}\)\-\\mathcal\{S\}\(\\Gamma\_\{GT\}\),and the corresponding normalized feature and structure terms

ℱ~​\(Γ\):=ℱ​\(Γ\)\|Δ​ℱ\|,𝒮~​\(Γ\):=𝒮​\(Γ\)Δ​𝒮\.\\tilde\{\\mathcal\{F\}\}\(\\Gamma\):=\\frac\{\\mathcal\{F\}\(\\Gamma\)\}\{\|\\Delta\\mathcal\{F\}\|\},\\qquad\\tilde\{\\mathcal\{S\}\}\(\\Gamma\):=\\frac\{\\mathcal\{S\}\(\\Gamma\)\}\{\\Delta\\mathcal\{S\}\}\.Let

J​\(Γ,α\)=\(1−α\)​ℱ~​\(Γ\)\+α​𝒮~​\(Γ\)J\(\\Gamma,\\alpha\)=\(1\-\\alpha\)\\,\\tilde\{\\mathcal\{F\}\}\(\\Gamma\)\+\\alpha\\,\\tilde\{\\mathcal\{S\}\}\(\\Gamma\)be the normalized FGW objective function\.

LetNNdenote the size of each cluster\. Then, for sufficiently largeNN, there exists a critical thresholdα∗∈\(0,1\)\\alpha^\{\*\}\\in\(0,1\)such that for allα\>α∗\\alpha\>\\alpha^\{\*\}, the ground truth coupling strictly minimizes the objective relative to the feature\-only alternative:J​\(ΓG​T,α\)<J​\(ΓF​O,α\)J\(\\Gamma\_\{GT\},\\alpha\)<J\(\\Gamma\_\{FO\},\\alpha\)\.

###### Proof\.

Let the source measure beμ\\muand the target measure beν\\nu, with the variance of the Normal distributions set toσ2=0\.1\\sigma^\{2\}=0\.1\. The centroids are located atμ0=\(−2,2\),μ1=\(0,2\),μ2=\(2,2\)\\mu\_\{0\}=\(\-2,2\),\\mu\_\{1\}=\(0,2\),\\mu\_\{2\}=\(2,2\)andμ0′=\(2,−2\),μ1′=\(0,−2\),μ2′=\(−2,−2\)\\mu^\{\\prime\}\_\{0\}=\(2,\-2\),\\mu^\{\\prime\}\_\{1\}=\(0,\-2\),\\mu^\{\\prime\}\_\{2\}=\(\-2,\-2\)\. The interaction tensorsG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}encode directed edges from the middle cluster \(k=1k=1\) to the left \(k=0k=0\) via Channel 1, and to the right \(k=2k=2\) via Channel 2\. In what follows, we first compute the*unnormalized*feature and structure costs\. We then incorporate the normalization scheme of[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4)in the final threshold derivation\.

1\. Analysis of the feature costℱ\\mathcal\{F\}

For the ground truth couplingΓG​T\\Gamma\_\{GT\}, each clusterkkmaps to its true imageμk′\\mu^\{\\prime\}\_\{k\}\. The cost is the mean squared norm of the translation vectorsv0=\(4,−4\)v\_\{0\}=\(4,\-4\),v1=\(0,−4\)v\_\{1\}=\(0,\-4\), andv2=\(−4,−4\)v\_\{2\}=\(\-4,\-4\):

ℱ​\(ΓG​T\)=13​∑k=02‖vk‖2=13​\(32\+16\+32\)=803\.\\mathcal\{F\}\(\\Gamma\_\{GT\}\)=\\frac\{1\}\{3\}\\sum\_\{k=0\}^\{2\}\|\|v\_\{k\}\|\|^\{2\}=\\frac\{1\}\{3\}\(32\+16\+32\)=\\frac\{80\}\{3\}\.\(22\)For the feature\-only couplingΓF​O\\Gamma\_\{FO\},𝒮0\\mathcal\{S\}\_\{0\}maps to𝒮2′\\mathcal\{S\}^\{\\prime\}\_\{2\},𝒮1\\mathcal\{S\}\_\{1\}to𝒮1′\\mathcal\{S\}^\{\\prime\}\_\{1\}, and𝒮2\\mathcal\{S\}\_\{2\}to𝒮0′\\mathcal\{S\}^\{\\prime\}\_\{0\}\. In the finite sample regime withNNpoints, the optimal transport cost between two empirical Gaussian distributions with identical covariance matrices converges to the squared Euclidean distance between their means\. We denote the finite\-sample deviation byδN\\delta\_\{N\}:

ℱ​\(ΓF​O\)=13​\(‖μ0−μ2′‖2\+‖μ1−μ1′‖2\+‖μ2−μ0′‖2\)\+δN\.\\mathcal\{F\}\(\\Gamma\_\{FO\}\)=\\frac\{1\}\{3\}\\left\(\|\|\\mu\_\{0\}\-\\mu^\{\\prime\}\_\{2\}\|\|^\{2\}\+\|\|\\mu\_\{1\}\-\\mu^\{\\prime\}\_\{1\}\|\|^\{2\}\+\|\|\\mu\_\{2\}\-\\mu^\{\\prime\}\_\{0\}\|\|^\{2\}\\right\)\+\\delta\_\{N\}\.\(23\)Hence:

ℱ​\(ΓF​O\)=16\+δN\.\\mathcal\{F\}\(\\Gamma\_\{FO\}\)=16\+\\delta\_\{N\}\.\(24\)The termδN\\delta\_\{N\}represents the error between the empirical measures and their population counterparts\. For distributions in dimensiond=2d=2, this error decays at a rate ofδN=O​\(N−1/2\)\\delta\_\{N\}=O\(N^\{\-1/2\}\)\(Fournier and Guillin,[2015](https://arxiv.org/html/2605.30635#bib.bib17)\)\. ProvidedNNis sufficiently large, standard OT \(α=0\\alpha=0\) prefers the incorrect mapping since16<80316<\\frac\{80\}\{3\}\.

2\. Analysis of structure cost𝒮\\mathcal\{S\}

The structure cost is the Gromov\-Wasserstein cost:

𝒮​\(Γ\)=∑i,k∑j,l‖Gi​k\(0\)−Gj​l\(1\)‖2​Γi​j​Γk​l\.\\mathcal\{S\}\(\\Gamma\)=\\sum\_\{i,k\}\\sum\_\{j,l\}\|\|G^\{\(0\)\}\_\{ik\}\-G^\{\(1\)\}\_\{jl\}\|\|^\{2\}\\Gamma\_\{ij\}\\Gamma\_\{kl\}\.\(25\)SinceΓG​T\\Gamma\_\{GT\}maps every source clusterkkto the target cluster with the same indexkk, andG\(1\)G^\{\(1\)\}is defined to preserve the index\-based structure ofG\(0\)G^\{\(0\)\}, we have:

𝒮​\(ΓG​T\)=0\.\\mathcal\{S\}\(\\Gamma\_\{GT\}\)=0\.\(26\)ForΓF​O\\Gamma\_\{FO\}, the mapping permutes indices asπ​\(0\)=2,π​\(1\)=1,π​\(2\)=0\\pi\(0\)=2,\\pi\(1\)=1,\\pi\(2\)=0\. We evaluate the cost for the two active interactions inG\(0\)G^\{\(0\)\}:

- •Edge1→01\\to 0\(Channel 1\):The source relation is\[1,0\]⊤\[1,0\]^\{\\top\}and the target relation \(fromπ​\(1\)=1\\pi\(1\)=1toπ​\(0\)=2\\pi\(0\)=2\) is Channel 2 \(\[0,1\]⊤\[0,1\]^\{\\top\}\), which yields a squared difference of22with mass weight1/91/9\.
- •Edge1→21\\to 2\(Channel 2\):The source relation is\[0,1\]⊤\[0,1\]^\{\\top\}and the target relation \(fromπ​\(1\)=1\\pi\(1\)=1toπ​\(2\)=0\\pi\(2\)=0\) is Channel 1 \(\[1,0\]⊤\[1,0\]^\{\\top\}\), which yields a squared difference of22with mass weight1/91/9\.

The total structure cost is:

𝒮​\(ΓF​O\)=19×2\+19×2=49\.\\mathcal\{S\}\(\\Gamma\_\{FO\}\)=\\frac\{1\}\{9\}\\times 2\+\\frac\{1\}\{9\}\\times 2=\\frac\{4\}\{9\}\.\(27\)
3\. Threshold derivation with normalization

We now incorporate the normalization scheme of[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4)\. Define the \(unnormalized\) feature and structure gaps between the two couplings as

Δ​ℱ:=ℱ​\(ΓF​O\)−ℱ​\(ΓG​T\),Δ​𝒮:=𝒮​\(ΓF​O\)−𝒮​\(ΓG​T\)\.\\Delta\\mathcal\{F\}:=\\mathcal\{F\}\(\\Gamma\_\{FO\}\)\-\\mathcal\{F\}\(\\Gamma\_\{GT\}\),\\qquad\\Delta\\mathcal\{S\}:=\\mathcal\{S\}\(\\Gamma\_\{FO\}\)\-\\mathcal\{S\}\(\\Gamma\_\{GT\}\)\.\(28\)From the computations above,

Δ​ℱ=16\+δN−803,Δ​𝒮=49\.\\Delta\\mathcal\{F\}=16\+\\delta\_\{N\}\-\\frac\{80\}\{3\},\\qquad\\Delta\\mathcal\{S\}=\\frac\{4\}\{9\}\.\(29\)For sufficiently largeNN, we haveℱ​\(ΓG​T\)\>ℱ​\(ΓF​O\)\\mathcal\{F\}\(\\Gamma\_\{GT\}\)\>\\mathcal\{F\}\(\\Gamma\_\{FO\}\), so\|Δ​ℱ\|=ℱ​\(ΓG​T\)−ℱ​\(ΓF​O\)\>0\|\\Delta\\mathcal\{F\}\|=\\mathcal\{F\}\(\\Gamma\_\{GT\}\)\-\\mathcal\{F\}\(\\Gamma\_\{FO\}\)\>0and the normalization is well\-defined\. Normalizing, we get:

ℱ~​\(Γ\)=ℱ​\(Γ\)\|Δ​ℱ\|,𝒮~​\(Γ\)=𝒮​\(Γ\)Δ​𝒮\.\\tilde\{\\mathcal\{F\}\}\(\\Gamma\)=\\frac\{\\mathcal\{F\}\(\\Gamma\)\}\{\|\\Delta\\mathcal\{F\}\|\},\\qquad\\tilde\{\\mathcal\{S\}\}\(\\Gamma\)=\\frac\{\\mathcal\{S\}\(\\Gamma\)\}\{\\Delta\\mathcal\{S\}\}\.\(30\)The normalized FGW objective can therefore be written as

J​\(Γ,α\)=\(1−α\)​ℱ~​\(Γ\)\+α​𝒮~​\(Γ\)\.J\(\\Gamma,\\alpha\)=\(1\-\\alpha\)\\tilde\{\\mathcal\{F\}\}\(\\Gamma\)\+\\alpha\\tilde\{\\mathcal\{S\}\}\(\\Gamma\)\.\(31\)For the two couplings of interest, we obtain

𝒮~​\(ΓG​T\)=𝒮​\(ΓG​T\)Δ​𝒮=0,𝒮~​\(ΓF​O\)=𝒮​\(ΓF​O\)Δ​𝒮=1,\\tilde\{\\mathcal\{S\}\}\(\\Gamma\_\{GT\}\)=\\frac\{\\mathcal\{S\}\(\\Gamma\_\{GT\}\)\}\{\\Delta\\mathcal\{S\}\}=0,\\qquad\\tilde\{\\mathcal\{S\}\}\(\\Gamma\_\{FO\}\)=\\frac\{\\mathcal\{S\}\(\\Gamma\_\{FO\}\)\}\{\\Delta\\mathcal\{S\}\}=1,\(32\)and

ℱ~​\(ΓG​T\)−ℱ~​\(ΓF​O\)=ℱ​\(ΓG​T\)−ℱ​\(ΓF​O\)\|Δ​ℱ\|=\|Δ​ℱ\|\|Δ​ℱ\|=1\.\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{GT\}\)\-\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{FO\}\)=\\frac\{\\mathcal\{F\}\(\\Gamma\_\{GT\}\)\-\\mathcal\{F\}\(\\Gamma\_\{FO\}\)\}\{\|\\Delta\\mathcal\{F\}\|\}=\\frac\{\|\\Delta\\mathcal\{F\}\|\}\{\|\\Delta\\mathcal\{F\}\|\}=1\.\(33\)We seekα\\alphasuch thatJ​\(ΓG​T,α\)<J​\(ΓF​O,α\)J\(\\Gamma\_\{GT\},\\alpha\)<J\(\\Gamma\_\{FO\},\\alpha\)under this normalization, i\.e\.

\(1−α\)​ℱ~​\(ΓG​T\)<\(1−α\)​ℱ~​\(ΓF​O\)\+α\.\(1\-\\alpha\)\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{GT\}\)<\(1\-\\alpha\)\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{FO\}\)\+\\alpha\.\(34\)Usingℱ~​\(ΓG​T\)−ℱ~​\(ΓF​O\)=1\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{GT\}\)\-\\tilde\{\\mathcal\{F\}\}\(\\Gamma\_\{FO\}\)=1, this inequality becomes

\(1−α\)<α⟺α\>12\.\(1\-\\alpha\)<\\alpha\\quad\\Longleftrightarrow\\quad\\alpha\>\\frac\{1\}\{2\}\.\(35\)Hence, under the normalization of[SectionD\.4](https://arxiv.org/html/2605.30635#A4.SS4)and in the asymptotic regime, a critical thresholdα∗=1/2\\alpha^\{\*\}=1/2exists above which the ground truth coupling strictly improves the normalized objective relative to the feature\-only alternative:J​\(ΓG​T,α\)<J​\(ΓF​O,α\)J\(\\Gamma\_\{GT\},\\alpha\)<J\(\\Gamma\_\{FO\},\\alpha\)for allα\>1/2\\alpha\>1/2\. ∎

Remarks\.In theory,α∗=0\.5\\alpha^\{\*\}=0\.5comes from an idealized analysis of the normalized objective that only compares the feature\-only and structure\-only couplings in the*population limit*\. Finite\-sample effects, approximate normalization, and the existence of many ’almost\-correct’ couplings break the symmetry of the idealized setting and make the optimalα\\alphaslightly bigger than0\.50\.5\.

Second, Theorem 1 shows that the ground truth couplingΓG​T\\Gamma\_\{GT\}is better thanΓF​O\\Gamma\_\{FO\}atα=1\\alpha=1\. However, it is not theonlyone\. The interaction tensorsG\(0\)G^\{\(0\)\}andG\(1\)G^\{\(1\)\}are constant for all points within a cluster\. Therefore, the structure cost𝒮​\(Γ\)\\mathcal\{S\}\(\\Gamma\)depends only on which clusters are matched, not on how individual points are mapped within them\.

Any coupling that correctly maps source clusters to their corresponding target clusters yields a structure cost of0\. This includes the ground truth couplingΓG​T\\Gamma\_\{GT\}, but also any coupling that correctly matches clusters while randomly permuting points inside them\. This explains the results observed in[Figure2](https://arxiv.org/html/2605.30635#S5.F2): atα=1\\alpha=1, the solver returns a solution that is structurally perfect but fails to recover the exact point\-to\-point correspondence\.

### G\.2Dynamic interpretation ofCellBRIDGE

We provide a dynamic viewpoint onCellBRIDGE, showing that it can be seen as the solution of a joint static\-dynamic energy minimization problem combining kinetic energy in expression space and a structure\-preserving term\.

As before, let

Π​\(a,b\):=\{Γ∈ℝ\+n0×n1:Γ​𝟏n1=a,Γ⊤​𝟏n0=b\}\.\\Pi\(a,b\):=\\left\\\{\\Gamma\\in\\mathbb\{R\}\_\{\+\}^\{n\_\{0\}\\times n\_\{1\}\}:\\Gamma\\mathbf\{1\}\_\{n\_\{1\}\}=a,\\ \\Gamma^\{\\top\}\\mathbf\{1\}\_\{n\_\{0\}\}=b\\right\\\}\.We further consider the common choice of feature cost

Ci​j=‖xi−yj‖2,1≤i≤n0,1≤j≤n1\.C\_\{ij\}\\;=\\;\\,\\\|x\_\{i\}\-y\_\{j\}\\\|^\{2\},\\qquad 1\\leq i\\leq n\_\{0\},\\;1\\leq j\\leq n\_\{1\}\.\(36\)
Admissible processes for a fixed coupling\.LetΓ∈Π​\(a,b\)\\Gamma\\in\\Pi\(a,b\)and define the associated joint law on endpoints

ΠΓ:=∑i=1n0∑j=1n1Γi​j​δ\(xi,yj\)\.\\Pi\_\{\\Gamma\}\\;:=\\;\\sum\_\{i=1\}^\{n\_\{0\}\}\\sum\_\{j=1\}^\{n\_\{1\}\}\\Gamma\_\{ij\}\\,\\delta\_\{\(x\_\{i\},y\_\{j\}\)\}\.\(37\)In the balanced case∑i,jΓi​j=1\\sum\_\{i,j\}\\Gamma\_\{ij\}=1, soΠΓ\\Pi\_\{\\Gamma\}is a probability measure with marginalsρ0,ρ1\\rho\_\{0\},\\rho\_\{1\}\.

We consider continuous\-time processes\(Xt\)t∈\[0,1\]\(X\_\{t\}\)\_\{t\\in\[0,1\]\}taking values inℝd\\mathbb\{R\}^\{d\}and satisfying:

- •X⋅X\_\{\\cdot\}has almost surely absolutely continuous paths
- •the joint law of its endpoints is\(X0,X1\)∼ΠΓ\(X\_\{0\},X\_\{1\}\)\\sim\\Pi\_\{\\Gamma\}

We write𝒜​\(ΠΓ\)\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)for the class of all such processes\. For anyX⋅∈𝒜​\(ΠΓ\)X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\), define the kinetic energy as:

𝒦​\(X⋅\):=𝔼​\[∫01‖X˙t‖2​𝑑t\]\.\\mathcal\{K\}\(X\_\{\\cdot\}\)\\;:=\\;\\mathbb\{E\}\\Bigg\[\\int\_\{0\}^\{1\}\\,\\big\\\|\\dot\{X\}\_\{t\}\\big\\\|^\{2\}\\,dt\\Bigg\]\.\(38\)
The following lemma is standard but we include it for completeness\.

###### Lemma 1\.

Letx,y∈ℝdx,y\\in\\mathbb\{R\}^\{d\}and letγ:\[0,1\]→ℝd\\gamma:\[0,1\]\\to\\mathbb\{R\}^\{d\}be absolutely continuous withγ​\(0\)=x\\gamma\(0\)=x,γ​\(1\)=y\\gamma\(1\)=y\. Then

∫01‖γ˙​\(t\)‖2​𝑑t≥‖y−x‖2,\\int\_\{0\}^\{1\}\\,\\big\\\|\\dot\{\\gamma\}\(t\)\\big\\\|^\{2\}\\,dt\\;\\geq\\;\\,\\\|y\-x\\\|^\{2\},\(39\)with equality if and only ifγ​\(t\)=\(1−t\)​x\+t​y\\gamma\(t\)=\(1\-t\)x\+tyfor allt∈\[0,1\]t\\in\[0,1\]\.

###### Proof\.

By Cauchy–Schwarz inequality,

‖∫01γ˙​\(t\)​𝑑t‖2≤∫01‖γ˙​\(t\)‖2​𝑑t,\\left\\\|\\int\_\{0\}^\{1\}\\dot\{\\gamma\}\(t\)\\,dt\\right\\\|^\{2\}\\;\\leq\\;\\int\_\{0\}^\{1\}\\big\\\|\\dot\{\\gamma\}\(t\)\\big\\\|^\{2\}\\,dt,\(40\)with equality if and only ifγ˙​\(t\)\\dot\{\\gamma\}\(t\)is constant intt\. Sinceγ​\(1\)−γ​\(0\)=y−x\\gamma\(1\)\-\\gamma\(0\)=y\-x, this yields

‖y−x‖2=‖∫01γ˙​\(t\)​𝑑t‖2≤∫01‖γ˙​\(t\)‖2​𝑑t,\\\|y\-x\\\|^\{2\}=\\left\\\|\\int\_\{0\}^\{1\}\\dot\{\\gamma\}\(t\)\\,dt\\right\\\|^\{2\}\\leq\\int\_\{0\}^\{1\}\\big\\\|\\dot\{\\gamma\}\(t\)\\big\\\|^\{2\}\\,dt,\(41\)Equality holds if and only ifγ˙​\(t\)=y−x\\dot\{\\gamma\}\(t\)=y\-x, i\.e\.γ​\(t\)=\(1−t\)​x\+t​y\\gamma\(t\)=\(1\-t\)x\+ty\. ∎

###### Proposition 1\.

LetΓ∈Π​\(a,b\)\\Gamma\\in\\Pi\(a,b\)andΠΓ\\Pi\_\{\\Gamma\}be as above\. Consider the admissible class𝒜​\(ΠΓ\)\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)and the kinetic energy𝒦\\mathcal\{K\}\. Then:

1. 1\.The energy𝒦​\(X⋅\)\\mathcal\{K\}\(X\_\{\\cdot\}\)is minimized over𝒜​\(ΠΓ\)\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)by the process Xtlin:=\(1−t\)​X\+t​Y,\(X,Y\)∼ΠΓ\.X^\{\\mathrm\{lin\}\}\_\{t\}\\;:=\\;\(1\-t\)X\+tY,\\qquad\(X,Y\)\\sim\\Pi\_\{\\Gamma\}\.\(42\)
2. 2\.The minimal value of the kinetic energy is infX⋅∈𝒜​\(ΠΓ\)𝒦​\(X⋅\)=𝔼\(X,Y\)∼ΠΓ​\[‖X−Y‖2\]=∑i,jΓi​j​Ci​j\.\\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{K\}\(X\_\{\\cdot\}\)\\;=\\;\\,\\mathbb\{E\}\_\{\(X,Y\)\\sim\\Pi\_\{\\Gamma\}\}\\big\[\\\|X\-Y\\\|^\{2\}\\big\]\\;=\\;\\sum\_\{i,j\}\\Gamma\_\{ij\}\\,C\_\{ij\}\.\(43\)

###### Proof\.

AnyX⋅∈𝒜​\(ΠΓ\)X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)satisfies\(X0,X1\)∼ΠΓ\(X\_\{0\},X\_\{1\}\)\\sim\\Pi\_\{\\Gamma\}\. Condition on the endpoints:

𝒦​\(X⋅\)=𝔼\(X,Y\)∼ΠΓ​\[𝔼​\[∫01‖X˙t‖2​𝑑t\|\(X0,X1\)=\(X,Y\)\]\]\.\\mathcal\{K\}\(X\_\{\\cdot\}\)=\\mathbb\{E\}\_\{\(X,Y\)\\sim\\Pi\_\{\\Gamma\}\}\\Bigg\[\\mathbb\{E\}\\Big\[\\int\_\{0\}^\{1\}\\big\\\|\\dot\{X\}\_\{t\}\\big\\\|^\{2\}\\,dt\\,\\Big\|\\,\(X\_\{0\},X\_\{1\}\)=\(X,Y\)\\Big\]\\Bigg\]\.\(44\)For each fixed pair\(X,Y\)=\(x,y\)\(X,Y\)=\(x,y\), Lemma[1](https://arxiv.org/html/2605.30635#Thmlemma1)shows that the conditional energy is minimized by the straight\-line patht↦\(1−t\)​x\+t​yt\\mapsto\(1\-t\)x\+ty, with minimal value‖x−y‖2\\\|x\-y\\\|^\{2\}\. Thus the global minimizer over𝒜​\(ΠΓ\)\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)is the straight\-line processXtlinX^\{\\mathrm\{lin\}\}\_\{t\}, and

infX⋅∈𝒜​\(ΠΓ\)𝒦​\(X⋅\)=𝔼\(X,Y\)∼ΠΓ​\[‖X−Y‖2\]=∑i,jΓi​j​‖xi−yj‖2=∑i,jΓi​j​Ci​j\.\\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{K\}\(X\_\{\\cdot\}\)=\\mathbb\{E\}\_\{\(X,Y\)\\sim\\Pi\_\{\\Gamma\}\}\\Big\[\\\|X\-Y\\\|^\{2\}\\Big\]=\\sum\_\{i,j\}\\Gamma\_\{ij\}\\,\\\|x\_\{i\}\-y\_\{j\}\\\|^\{2\}=\\sum\_\{i,j\}\\Gamma\_\{ij\}C\_\{ij\}\.\(45\)∎

Joint static–dynamic energy and reduction to FGW\.We can viewCellBRIDGEas minimizing over both couplings and dynamics the joint energy functional

ℰα​\(Γ,X⋅\):=\(1−α\)​𝒦​\(X⋅\)\+α​S​\(Γ\),\\mathcal\{E\}\_\{\\alpha\}\(\\Gamma,X\_\{\\cdot\}\):=\(1\-\\alpha\)\\,\\mathcal\{K\}\(X\_\{\\cdot\}\)\\;\+\\;\\alpha\\,S\(\\Gamma\),\(46\)subject toΓ∈Π​\(a,b\)\\Gamma\\in\\Pi\(a,b\)andX⋅∈𝒜​\(ΠΓ\)X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\.

###### Proposition 2\.

Fixα∈\[0,1\]\\alpha\\in\[0,1\]\. Consider the optimization problem

infΓ∈Π​\(a,b\)infX⋅∈𝒜​\(ΠΓ\)ℰα​\(Γ,X⋅\)=infΓ∈Π​\(a,b\)infX⋅∈𝒜​\(ΠΓ\)\[\(1−α\)​𝒦​\(X⋅\)\+α​S​\(Γ\)\]\.\\inf\_\{\\Gamma\\in\\Pi\(a,b\)\}\\ \\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\ \\mathcal\{E\}\_\{\\alpha\}\(\\Gamma,X\_\{\\cdot\}\)\\;=\\;\\inf\_\{\\Gamma\\in\\Pi\(a,b\)\}\\ \\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\ \\Big\[\(1\-\\alpha\)\\mathcal\{K\}\(X\_\{\\cdot\}\)\+\\alpha S\(\\Gamma\)\\Big\]\.\(47\)Then:

1. 1\.For any fixedΓ\\Gamma, the inner infimum overX⋅∈𝒜​\(ΠΓ\)X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)is attained by the straight\-line processXtlin=\(1−t\)​X\+t​YX^\{\\mathrm\{lin\}\}\_\{t\}=\(1\-t\)X\+tY,\(X,Y\)∼ΠΓ\(X,Y\)\\sim\\Pi\_\{\\Gamma\}, and infX⋅∈𝒜​\(ΠΓ\)ℰα​\(Γ,X⋅\)=\(1−α\)​∑i,jΓi​j​Ci​j\+α​S​\(Γ\)\.\\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{E\}\_\{\\alpha\}\(\\Gamma,X\_\{\\cdot\}\)=\(1\-\\alpha\)\\sum\_\{i,j\}\\Gamma\_\{ij\}C\_\{ij\}\+\\alpha S\(\\Gamma\)\.\(48\)
2. 2\.Consequently, the joint static–dynamic problem reduces to the purely static FGW problem infΓ∈Π​\(a,b\)infX⋅∈𝒜​\(ΠΓ\)ℰα​\(Γ,X⋅\)=infΓ∈Π​\(a,b\)\[\(1−α\)​⟨Γ,C⟩F\+α​S​\(Γ\)\],\\inf\_\{\\Gamma\\in\\Pi\(a,b\)\}\\ \\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{E\}\_\{\\alpha\}\(\\Gamma,X\_\{\\cdot\}\)\\;=\\;\\inf\_\{\\Gamma\\in\\Pi\(a,b\)\}\\Big\[\(1\-\\alpha\)\\,\\langle\\Gamma,C\\rangle\_\{F\}\+\\alpha S\(\\Gamma\)\\Big\],\(49\)whose minimizers are exactly the FGW\-optimal couplings used byCellBRIDGE\.

###### Proof\.

Point \(1\) follows directly from Proposition[1](https://arxiv.org/html/2605.30635#Thmproposition1): for anyΓ\\Gamma,

infX⋅∈𝒜​\(ΠΓ\)ℰα​\(Γ,X⋅\)=\(1−α\)​infX⋅∈𝒜​\(ΠΓ\)𝒦​\(X⋅\)\+α​S​\(Γ\)=\(1−α\)​∑i,jΓi​j​Ci​j\+α​S​\(Γ\)\.\\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{E\}\_\{\\alpha\}\(\\Gamma,X\_\{\\cdot\}\)=\(1\-\\alpha\)\\inf\_\{X\_\{\\cdot\}\\in\\mathcal\{A\}\(\\Pi\_\{\\Gamma\}\)\}\\mathcal\{K\}\(X\_\{\\cdot\}\)\+\\alpha S\(\\Gamma\)=\(1\-\\alpha\)\\sum\_\{i,j\}\\Gamma\_\{ij\}C\_\{ij\}\+\\alpha S\(\\Gamma\)\.\(50\)Taking the infimum overΓ∈Π​\(a,b\)\\Gamma\\in\\Pi\(a,b\)yields \(2\), which coincides with the static FGW objective\. ∎

Intuitively, Proposition[2](https://arxiv.org/html/2605.30635#Thmproposition2)shows thatCellBRIDGEdoes not use linear interpolations between matched cells as a heuristic, but as the*unique*minimal\-action choice once the couplingΓ⋆\\Gamma^\{\\star\}is fixed\. The static FGW step therefore selects an interaction\-aware coupling that trades off feature displacement and CCI preservation, and the subsequent dynamic step realizes this coupling by approximating the lowest\-kinetic\-energy flow in expression space\. Whenα=0\\alpha=0, this recovers the classical OT–CFM\(Tonget al\.,[2024a](https://arxiv.org/html/2605.30635#bib.bib36)\)/ Benamou\-Brenier\(Benamou and Brenier,[2000](https://arxiv.org/html/2605.30635#bib.bib13)\)interpolation\.

### G\.3Connection to the velocity field learned with CFM\.

In practice,CellBRIDGEdoes not explicitly construct the processXtlinX^\{\\mathrm\{lin\}\}\_\{t\}but instead uses CFM to learn a time–dependent vector fieldvθv\_\{\\theta\}that generates the same probability path\.

In the infinite–capacity and optimization limit, the minimizerv⋆v^\{\\star\}ofℒCFM\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}coincides with the velocity field ofXtlinX^\{\\mathrm\{lin\}\}\_\{t\}constructed in[SectionG\.2](https://arxiv.org/html/2605.30635#A7.SS2), in the sense that

v⋆​\(z,t\)=𝔼​\[Y−X∣Zt=z\],v^\{\\star\}\(z,t\)=\\mathbb\{E\}\[\\,Y\-X\\mid Z\_\{t\}=z\\,\],and the ODE

z˙t=v⋆​\(zt,t\)\\dot\{z\}\_\{t\}=v^\{\\star\}\(z\_\{t\},t\)generates exactly the probability path\{ρt\}t∈\[0,1\]\\\{\\rho\_\{t\}\\\}\_\{t\\in\[0,1\]\}induced byΓ⋆\\Gamma^\{\\star\}\.

We can relatev⋆v^\{\\star\}to the kinetic energy of the straight–line process, following a similar technique as in\(Lipmanet al\.,[2024](https://arxiv.org/html/2605.30635#bib.bib29)\)\. For a time–dependent vector fieldw:ℝd×\[0,1\]→ℝdw:\\mathbb\{R\}^\{d\}\\times\[0,1\]\\to\\mathbb\{R\}^\{d\}that generates\{ρt\}\\\{\\rho\_\{t\}\\\}, define its kinetic energy along this path by

𝒦Eul​\(w\):=∫01𝔼Zt∼ρt​\[‖w​\(Zt,t\)‖22\]​𝑑t\.\\mathcal\{K\}\_\{\\mathrm\{Eul\}\}\(w\):=\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{Z\_\{t\}\\sim\\rho\_\{t\}\}\\Big\[\\,\\\|w\(Z\_\{t\},t\)\\\|\_\{2\}^\{2\}\\Big\]\\,dt\.Using the formula ofv⋆v^\{\\star\}above and Jensen’s inequality, we obtain

𝒦Eul​\(v⋆\)\\displaystyle\\mathcal\{K\}\_\{\\mathrm\{Eul\}\}\(v^\{\\star\}\)=∫01𝔼\[∥𝔼\[Y−X∣Zt\]∥22\]dt\\displaystyle=\\int\_\{0\}^\{1\}\\mathbb\{E\}\\Big\[\\;\\big\\\|\\mathbb\{E\}\[\\,Y\-X\\mid Z\_\{t\}\\,\]\\big\\\|\_\{2\}^\{2\}\\Big\]\\,dt≤∫01𝔼​\[𝔼​\[‖Y−X‖22∣Zt\]\]​𝑑t\\displaystyle\\leq\\int\_\{0\}^\{1\}\\mathbb\{E\}\\Big\[\\;\\mathbb\{E\}\\big\[\\\|Y\-X\\\|\_\{2\}^\{2\}\\mid Z\_\{t\}\\big\]\\Big\]\\,dt=∫01𝔼​\[‖Y−X‖22\]​𝑑t\\displaystyle=\\int\_\{0\}^\{1\}\\mathbb\{E\}\\Big\[\\;\\\|Y\-X\\\|\_\{2\}^\{2\}\\Big\]\\,dt=𝔼\(X,Y\)∼Π​\[‖Y−X‖22\]\.\\displaystyle=\\mathbb\{E\}\_\{\(X,Y\)\\sim\\Pi\}\\Big\[\\;\\\|Y\-X\\\|\_\{2\}^\{2\}\\Big\]\.By[Proposition1](https://arxiv.org/html/2605.30635#Thmproposition1)we have

𝔼\(X,Y\)∼ΠΓ⋆​\[‖Y−X‖22\]=∑i,jΓi​j⋆​Ci​j=K​\(X⋅lin\),\\mathbb\{E\}\_\{\(X,Y\)\\sim\\Pi\_\{\\Gamma^\{\\star\}\}\}\\Big\[\\;\\\|Y\-X\\\|\_\{2\}^\{2\}\\Big\]=\\sum\_\{i,j\}\\Gamma^\{\\star\}\_\{ij\}C\_\{ij\}=K\\\!\\big\(X^\{\\mathrm\{lin\}\}\_\{\\cdot\}\\big\),Hence

𝒦Eul​\(v⋆\)≤∑i,jΓi​j⋆​Ci​j=K​\(X⋅lin\)\.\\mathcal\{K\}\_\{\\mathrm\{Eul\}\}\(v^\{\\star\}\)\\;\\leq\\;\\sum\_\{i,j\}\\Gamma^\{\\star\}\_\{ij\}C\_\{ij\}=K\\\!\\big\(X^\{\\mathrm\{lin\}\}\_\{\\cdot\}\\big\)\.
In other words, for a fixed couplingΓ\\Gamma, the feature term

F​\(Γ\)=⟨Γ,C⟩F=∑i,jΓi​j​Ci​jF\(\\Gamma\)=\\langle\\Gamma,C\\rangle\_\{F\}=\\sum\_\{i,j\}\\Gamma\_\{ij\}C\_\{ij\}provides an explicit upper bound on the kinetic energy of the velocity field recovered by CFM from the corresponding straight–line dynamics\. Combined with the joint static–dynamic formulation in[Equation46](https://arxiv.org/html/2605.30635#A7.E46), this shows that theCellBRIDGEobjective

\(1−α\)​⟨Γ,C⟩F\+α​S​\(Γ\)\(1\-\\alpha\)\\,\\langle\\Gamma,C\\rangle\_\{F\}\+\\alpha\\,S\(\\Gamma\)can be viewed as selecting a coupling that balances CCI preservation with a surrogate upper bound on the kinetic energy of the flow that CFM learns from that coupling\.