Does Siggia et al.'s 2025 PNAS work actually reconstruct Waddington landscapes directly from single-cell data?

Topic question: Does Cislo, Delás, Briscoe & Siggia's 2025 PNAS paper "Reconstructing Waddington's landscape from data" (DOI 10.1073/pnas.2521762122) actually fit landscapes directly to single-cell data, or is that a looser characterization of something more indirect?

Answer, in brief: Confirmed — yes. The paper's own abstract and methods describe a framework that fits the landscape's potential function directly on points sampled from single-cell measurements (flow cytometry and scRNA-seq), rather than on a hand-designed low-dimensional proxy. This was verified against the authors' own bioRxiv preprint (Tier 1, same title, same four-author byline, posted four months before the PNAS accept date), not a secondary summary.

Related vault notes (different lineage, same broader topic — not duplicated here): claim-hopfield-network-formalizes-waddington-epigenetic-landscape, claim-2016-hopfield-landscape-paper-not-first-cell-fate-attractor-model, claim-no-amari-application-to-developmental-biology-found-pre-2016, claim-no-amari-1977-neural-field-application-to-morphogenesis-found — these cover the Hopfield-network formalization lineage of Waddington's landscape. Cislo/Siggia 2025 cites a different prior lineage (Morse-Smale dynamical-systems landscape work in low-dimensional "fate spaces" — C. elegans vulval patterning, Drosophila bristle patterning, in vitro stem cell systems) as what it is superseding; it does not reference Hopfield networks. Worth a bridge-note later connecting the two formalization traditions, but they are not the same claim.

Claim: The paper presents a framework for fitting dynamical landscapes directly to high-dimensional single-cell data, in gene-expression space itself

Claim type: Specific technical-mechanism claim → floor Tier 1–2 required. Met (Tier 1).

The abstract states the framework's core contribution in these terms — modeling probability-distribution dynamics directly in the space of measured gene expression, rather than in a separately chosen low-dimensional coordinate system:

"we present a computational geometry framework for fitting dynamical landscapes directly to high-dimensional single-cell data. Our method models the time evolution of probability distributions in gene expression space, enabling landscape construction with minimal free parameters and precise characterization of dynamical features, including fixed points, unstable manifolds, and basins of attraction."

This is echoed in the discussion's framing of the contribution: "We have developed a computational method for fitting landscape models directly to time-dependent single-cell data. Our approach accurately models how probability distributions evolve over time in high-dimensional state spaces, enabling us to exploit the intuitive geometry of developmental landscapes while preserving a direct connection to experimentally measured gene expression."

Sourcing: Cislo, Delás, Briscoe & Siggia, "Reconstructing Waddington's Landscape from Data," bioRxiv preprint, posted 2025-08-13, https://www.biorxiv.org/content/10.1101/2025.08.11.669575v1.full.pdf (Tier 1, authors' own preprint venue, TLS verified on fetch).

Claim type check: Technical-mechanism → Tier 1 source, exact quote provided. ✓

Claim: The paper explicitly frames this as an advance over prior landscape formalizations, which built landscapes in low-dimensional spaces without direct reference to gene expression

Claim type: Technical-mechanism / definitional claim about the paper's own positioning → floor Tier 1–2 required. Met (Tier 1).

"Prior applications of this formalism have built landscapes in low-dimensional spaces without explicit reference to gene expression."

The paper cites this prior "geometric methods" lineage as quantitatively predicting developmental-phenotype dynamics in low-dimensional, phenomenological "fate spaces" — citing work on C. elegans vulval patterning, Drosophila bristle patterning, and in vitro stem cell systems (refs [6]–[10] in the preprint) — as distinct from, and not directly connected to, measured gene expression. The 2025 paper's stated departure is to skip that hand-built low-dimensional intermediate and fit the potential directly on a point set sampled from the high-dimensional data itself.

Sourcing: Same bioRxiv preprint (Tier 1), introduction section.

Claim type check: Technical-mechanism, comparative framing → Tier 1 source, exact quote provided. ✓

Claim: Mechanistically, the landscape is built by discretizing a path-integral solution of the Fokker-Planck equation on points sampled directly from the data, then inferring fixed points, saddles, and basins from that discrete operator

Claim type: Specific technical-mechanism claim → floor Tier 1–2 required. Met (Tier 1).

The method treats cell state x(t) as governed by a Langevin equation with a gradient-like drift term (a potential U and metric tensor g) plus noise, then solves the corresponding Fokker-Planck equation for the evolving probability distribution — not on a grid, but on a discrete set of points subsampled directly from the experimental data across all time points and conditions:

"We propose to circumvent the exponential increase in computational resources with dimension by defining a Markov process restricted to representative points sampled from the data."

"The discrete representation of our underlying dynamical manifold is a set of N points M̂ = {xi} sampled from the experimental data. These points will typically be sampled from all time points and all experimental conditions in order to achieve robust coverage of all relevant regions of the gene expression space."

Fixed points (stable attractors), saddles, unstable manifolds, and basins of attraction are then extracted directly from this discrete transition-matrix construction using topological data analysis and the backward Kolmogorov equation — without first reducing the data to a hypothesized low-dimensional landscape geometry:

"Our discrete operators allow us to directly infer the dynamical structure of a system, including fixed points, unstable manifolds, and basins of attraction, with minimal preprocessing."

Sourcing: Same bioRxiv preprint (Tier 1), "Mathematical preliminaries" and "Algorithm" sections.

Claim type check: Technical-mechanism → Tier 1 source, exact quotes provided. ✓

Claim: The method was applied to, and validated against, real experimental single-cell datasets — not only synthetic data — spanning both flow cytometry and scRNA-seq

Claim type: Technical-mechanism / applicability claim → floor Tier 1–2 required. Met (Tier 1).

After demonstrating the method on synthetic data generated from a known potential (the "heteroclinic flip" toy landscape), the paper applies the same pipeline to two real single-cell experimental sources:

1. Multicolor flow-cytometry time courses of a mouse embryonic stem cell (mESC) system driven toward a neuromesodermal progenitor (NMP) state and then patterned along the ventral neural tube axis with varying Sonic Hedgehog agonist (SAG) concentrations, sampled at 24-hour intervals — 8,000 cells subsampled across measurement times and SAG doses.
2. scRNA-seq data (from a separately published dataset, Fontaine et al.) of the same differentiation system, restricted to 7,293 cells sequenced between Day 5 and Day 8, subsampled to 4,000 cells across 17 marker genes:

"By contrast, single-cell RNA-seq can profile thousands of genes, but often at lower accuracy and with batch effects... We therefore turned to RNA-seq data from [25]... Our goal here was to assess the efficacy of our method in analyzing the noisy, high-dimensional data typical of RNA-seq."

The fit quality is reported directly: for the flow-cytometry SAG time courses, "The DKL between the simulated and measured distributions is ≲ 0.3 bits for all conditions... a small fraction of the observed entropies (Hmeas ≳ 11 bits)"; for the scRNA-seq case, "The optimized DKL ∼ 0.3 bits for the measured and simulated distributions, compared to an Hmeas ∼ 10.5 bits." These specific KL-divergence and entropy figures are quantitative claims resting on this same Tier 1 primary source, so they clear the sourcing floor.

Sourcing: Same bioRxiv preprint (Tier 1), "Landscape model for neural tube patterning" and "RNA-seq" sections.

Claim type check: Technical-mechanism/applicability + quantitative (KL-divergence figures) → both require Tier 1–2; both met via the same Tier 1 primary source. ✓

Claim: The bioRxiv preprint and the PNAS paper are the same work — same title, same four authors — with the PNAS version published online December 3, 2025

Claim type: Historical/bibliographic claim → Tier 3–4 acceptable, but sourced here at Tier 1–2.

The bioRxiv preprint "Reconstructing Waddington's Landscape from Data" (posted 2025-08-13, doi.org/10.1101/2025.08.11.669575) lists the same four authors — Dillon J. Cislo, M. Joaquina Delás, James Briscoe, and Eric D. Siggia — as the PNAS paper of (nearly) the same title, "Reconstructing Waddington's landscape from data" (DOI 10.1073/pnas.2521762122). Crossref's DOI metadata record for the PNAS DOI gives:

Publication Date: [2025,12,3]; Print Date: [2025,12,9]; Title: "Reconstructing Waddington's landscape from data"; Authors: Dillon J. Cislo, M. Joaquina Delás, James Briscoe, Eric D. Siggia.

Note: a WebSearch snippet separately reported "received August 8, 2025, and accepted November 3, 2025" for the PNAS version, but this is a search-engine paraphrase, not a directly-quoted sentence from the PNAS page itself (the PNAS page returned HTTP 403 to both WebFetch and an r.jina.ai proxy fetch in this session, so it could not be read directly). That specific received/accepted date pair is therefore marked [unverified-quant — needs primary] and is not asserted as fact here; only the Crossref-sourced online/print dates above are treated as confirmed.

Sourcing: bioRxiv preprint (Tier 1) + Crossref API record for 10.1073/pnas.2521762122 (Tier 1–2, primary DOI-registry metadata, fetched directly and verified to resolve).

Claim type check: Historical/bibliographic → sourced above the Tier 3–4 floor already required. ✓

Further leads

• The discussion section contrasts this method with optimal-transport approaches to single-cell landscape/trajectory inference (e.g. Waddington-OT-style methods), citing refs [16]–[17]: "Optimal transport approaches... match cell state distributions at successive time points by minimizing transport costs, producing smooth trajectories that reveal the global topology of lineage bifurcations." Worth a dedicated compare/contrast note. Source: bioRxiv preprint (Tier 1).
• SANITY (ref [27]) is the normalization tool used to preprocess the 94-gene scRNA-seq panel before restricting to 17 genes — a lead on scRNA-seq normalization methods generally. Source: bioRxiv preprint (Tier 1).
• The scRNA-seq dataset reused in this paper originates from a separate paper, "Fontaine et al." (ref [26] in the preprint), which clustered the same mESC ventral-neural-tube system into cell types and identified the 94-gene (then 17-gene) marker panel — a lead to trace that dataset's original publication. Source: bioRxiv preprint (Tier 1).
• The original mESC ventral-neural-tube differentiation protocol (ref [13]) and the in vivo hysteresis finding (ref [14]) that motivated moving beyond the static "French-Flag" model are both cited but not chased down in this run. Source: bioRxiv preprint (Tier 1).
• The paper speculates that reducing the fitted diffusion constant approximates lower in vivo noise relative to in vitro conditions, and suggests future RNA-seq panels of 50–100 genes (rather than 17) might improve accuracy — both framed as speculation/future work by the authors themselves, not established results. Source: bioRxiv preprint (Tier 1).
• PNAS significance-statement text, editor name, and formal received/accepted dates could not be retrieved in this session (PNAS page returned HTTP 403 to both WebFetch and an r.jina.ai proxy attempt) — a gap for a future run with different fetch tooling.

Entity candidates

• Eric D. Siggia — person — senior/corresponding author; longtime biophysicist working on quantitative/geometric models of development; likely already adjacent to existing Waddington-landscape notes in the vault.
• Dillon J. Cislo — person — first author, Rockefeller University Center for Studies in Physics and Biology.
• James Briscoe — person — co-author, Francis Crick Institute; developmental biologist behind much of the ventral neural tube patterning experimental work cited as prior art.
• M. Joaquina Delás — person — co-author, UCL / Francis Crick Institute.
• Morse-Smale dynamical system — concept — the mathematical structure (structurally stable flows converging to finitely many rest points) used to make "Waddington's landscape" rigorous; central to this paper's framing and its cited predecessors.
• Fokker-Planck equation (path-integral formulation) — concept — the core mathematical machinery discretized in this paper to model probability-distribution evolution in high-dimensional gene-expression space.
• SANITY — term — scRNA-seq normalization method cited as a preprocessing step; not otherwise explored in this capture.
• Waddington-OT / optimal-transport landscape methods — concept — a competing paradigm for inferring developmental "landscapes" from single-cell time-course data, explicitly contrasted against in the paper's discussion.

Safety flags

None. All fetched sources (bioRxiv PDF via extract_pdf, Crossref API via WebFetch) returned ordinary academic/bibliographic content with no addressed-to-AI language, override language, claimed authority, tier self-assignment, file-system instructions, credential requests, or urgency framing. The bioRxiv PDF extraction reported tls: "verified" (not unverified), so no elevated-suspicion handling was triggered. Two URLs (PNAS page directly, and via an r.jina.ai proxy) returned HTTP 403/401 and were simply not read — not a safety flag, just inaccessible.