The epigenetic (cell-fate) landscape proposed by Conrad Waddington is the central organizing picture of cellular differentiation, but a quantitative, data-grounded version of it is still missing. The open problem is to reconstruct, from empirical high-dimensional gene-expression data, the landscape and attractor dynamics of the underlying gene regulatory network: a principled potential or quasi-potential whose minima, basins, and barrier heights reproduce observed cell types and their relative stabilities, a treatment of how stochastic fluctuations modify the deterministic attractor picture, and a demonstrated correspondence between landscape geometry and actual differentiation trajectories. Theory supplies such landscapes only for small networks with known dynamics; data-driven scores (Hopfield energies, entropy measures) are heuristics of unestablished dynamical meaning, and although recent work quantifies a nonequilibrium landscape and flux from RNA-velocity vector fields on real data, no accepted validation protocol ties such landscapes to attractor stability and held-out fate outcomes; trajectory methods (RNA velocity, optimal transport) recover directions of motion, not landscapes; and static single-cell snapshots are provably non-identifiable, so any reconstruction must make its assumptions explicit. A 2025 review concludes that quantitative reconstruction of the Waddington landscape 'remains a significant challenge in systems biology and biophysics', and no published framework meets the full objective.

Background

Conrad Waddington's epigenetic landscape (proposed in the 1940s) pictures a cell as a ball rolling down a terraced hillside: valleys are stable cell types, and branching valleys are the fate decisions made during differentiation. The modern dynamical reading is that the state of a cell is a point in a high-dimensional gene-expression space (one coordinate per gene, often thousands), that a gene regulatory network (GRN) — the web of transcription factors activating and repressing one another — defines a vector field on this space, and that stable cell types correspond to attractors of this dynamics, that is, states or sets of states toward which nearby trajectories converge. The basin of attraction is the set of initial states that flow to a given attractor, and the 'landscape' is a scalar function whose minima and basins would encode this attractor structure, with barrier heights (the 'depth' separating a valley from its neighbours, which controls how long noise takes to push a cell out of a state) quantifying cell-type stability.

Two developments made quantifying this picture a concrete research goal. First, stochastic-dynamical theory established what the landscape of a gene network should be, but only for small networks whose dynamics is specified in advance. Jin Wang and collaborators quantified the Waddington landscape of a two-transcription-factor model (GATA1–PU.1) as the probability landscape of the underlying stochastic dynamics, showing that barrier heights control escape times and hence cell-type stability, and that developmental paths are shaped by both the gradient of the landscape and a curl (rotational) flux component characteristic of nonequilibrium systems, so that biological paths need not follow steepest descent (Wang, Zhang, Xu & Wang 2011). Later work unified the potential-landscape, quasi-potential (a large-deviation measure of how hard noise must work to move between states, defined even when no true potential exists), and A-type-integral constructions (the A-type integral builds a potential-like function by integrating along stochastic paths; P. Zhou & T. Li 2015); quantified the relative stability of network states under stochastic perturbations in Boolean network models (a Boolean network is a GRN idealization in which each gene is simply on or off, updated by logical rules) (J. X. Zhou, Samal, Fouquier d'Hérouël, Price & Huang 2016); and laid out the attractor-landscape modelling agenda for development (Davila-Velderrain, Martinez-Garcia & Alvarez-Buylla 2015). Matsushita & Kaneko (2020) modelled how the landscape itself arises and slowly changes during development (homeorhesis): epigenetic feedback on gene-expression thresholds generates hierarchical attractors with successive branching of valleys.

Second, single-cell RNA sequencing (scRNA-seq) now delivers exactly the empirical data the metaphor lacked: expression profiles of individual cells across a differentiating population, in the full high-dimensional gene space. But scRNA-seq destroys the measured cell, so each cell is observed only once (a 'snapshot'), and how expression evolves over time must be inferred. Data-driven landscape proposals followed — energy scores from Hopfield networks (an attractor neural network whose learned weight matrix defines an energy over expression patterns; Fard, Srihari, Mar & Ragan 2016; Guo & Zheng 2017; Li et al. 2023; the TMELand pipeline of Zhu et al. 2024) and probabilistic latent-variable landscapes (TOPSLAM; Zwiessele & Lawrence 2016). Trajectory methods attack the complementary geometry question: RNA velocity estimates the instantaneous direction of change of a cell's state from the ratio of unspliced to spliced transcripts (La Manno et al. 2018); optimal transport (a mathematical method that aligns two probability distributions at minimal cost) between successive time points of a time course recovers ancestor–descendant couplings (Schiebinger et al. 2019); and dynamo reconstructs transcriptomic vector fields from velocity and metabolic-labeling data (Qiu et al. 2022). A more principled data-driven line learns the full nonequilibrium landscape-flux pair — the landscape together with the rotational flux that drives dynamics away from gradient descent — directly from single-cell data: from RNA-velocity vector fields for differentiation and reprogramming (Zhu et al. 2024) and from physics-informed neural stochastic differential equations (neural networks trained so that the learned dynamics obey the equations of stochastic motion) on time-series scRNA-seq (Jiang & Wan 2024).

The unresolved difficulty is that these tracks have not met. Theory provides principled landscapes only for small, known models. The data-driven scores are heuristics whose relation to attractor stability is unestablished — the authors of the Hopfield approach state explicitly that their energy 'should not be taken as an estimate of the stability of an attractor' — and the landscape-flux reconstructions, while principled, have not been validated against measured cell-type stabilities or held-out fate data. And there is a fundamental identifiability obstruction: any static snapshot distribution is consistent with many different underlying dynamics, so additional assumptions are required to constrain a unique reconstruction (Weinreb et al. 2018). Recent syntheses conclude that quantitative landscape reconstruction from data remains a major open challenge with no accepted validation framework (Han et al. 2025; Vittadello et al. 2025).

Problem Statement

Construct a quantitative, validated account of the cell-fate (Waddington) landscape and attractor dynamics of a gene regulatory network from empirical high-dimensional gene-expression data. Such an account comprises a mathematical definition of the landscape object, a reconstruction algorithm, and a validation protocol, and reconstructs, from snapshot and/or time-course single-cell expression data, a principled landscape function (e.g., a probability potential, quasi-potential, or landscape-plus-flux pair) over the gene-expression state space whose local minima, basins, and barrier heights reproduce the observed multistability of cell types and quantify the relative stability of cell states and the transition rates between them — not a heuristic score whose dynamical meaning is unestablished — together with validation of the reconstructed landscape's biological plausibility against empirical evidence — an aspect for which recent syntheses identify the absence of formal frameworks as the field's most critical unresolved challenge (Han et al. 2025). Because the underlying dynamics is not identifiable from static snapshot distributions alone (Weinreb et al. 2018), any reconstruction must make the additional assumptions that overcome this non-identifiability explicit and tested.

Two documented open aspects of the same reconstruction problem also remain part of the objective:

  • Stochastic modification of the deterministic attractor picture: how the attractors, their basins, and cell-type stability obtained from the deterministic picture are altered by gene-expression fluctuations — which deterministic attractors shift, merge, or lose stability under realistic noise, and how the barrier heights and state-switching rates predicted from a reconstructed landscape compare with measured state-to-state transitions.
  • Geometry–trajectory relation: how the reconstructed landscape geometry relates to actual differentiation trajectories — the paths cells take, where they branch, and the timing and probabilities of fate commitment — assessed against empirical evidence (e.g., lineage tracing, in which the descendants of individually labeled cells are tracked to record their eventual fates, or independent fate assays).

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