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  • Open Access

Learning Geometric Models for Developmental Dynamics

Addison Howe* and Madhav Mani

  • *Contact author: addi.howe@u.northwestern.edu
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Phys. Rev. X 15, 031070 – Published 15 September, 2025

DOI: https://doi.org/10.1103/8vpj-bj7d

Abstract

Successful development from a single cell to a complex, multicellular organism requires that cells differentiate in a coordinated and organized manner in response to a variety of signaling modalities. While the molecular underpinnings may be complex, the resulting phenomenon, in which a cell decides between one fate or another, is relatively simple. A body of work—rooted in dynamical systems theory—has formalized this notion of cellular decision making as flow in a Waddington-like landscape, in which cells evolve according to gradientlike dynamics within a potential that changes shape in response to a number of signals. We present a framework leveraging neural networks as universal function approximators to infer such a parametrized landscape from gene expression data. Inspired by the success of physics-informed machine learning in data-limited contexts, we enforce principled constraints motivated not by physical laws but by this phenomenological understanding of differentiation. Our data-driven approach infers a governing landscape atop a manifold situated within expression space, thereby describing the dynamics of interest in a biologically meaningful context. The resulting system provides an intuitive, visualizable, and interpretable model of cellular differentiation dynamics.

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Popular Summary

A central mystery in biology is how a single cell develops into a complex organism through a series of robust and reliable transitions between different cell states. Even though the molecular details inside cells are extremely complex, the outcome often looks surprisingly simple: Cells tend to move between only a few distinct states in a highly predictable way. This reliability motivates what is known as the Waddington perspective: Development may follow a kind of landscape where cells move along paths shaped by genetic and signaling constraints. To address this, we design a neural network model that learns the shape of a developmental landscape in a low-dimensional space linked to the underlying genes.

In our framework, the neural network captures the gradientlike dynamics of how cells move across the landscape. In this space, external chemical signals act like forces that push the cells in particular directions. The result is a stochastic differential equation that provides both predictive power and intuitive interpretation. We apply this model to an experimental system where cells make a simple binary decision, allowing us to test the framework in a controlled setting.

Our findings show that neural networks can be used to model the rules of development, an approach that opens the door to mapping more complex landscapes underlying tissue formation and differentiation.

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Phys. Rev. X 15, 031070– Published 15 September, 2025
  • Received 12 December 2024
  • Accepted 12 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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