# Inverse modeling unveils governing law of mechano-chemical dynamics of epithelial migration

## TL;DR

A machine-learning method learns how cells collectively move based on chemical and mechanical signals, revealing new insights into tissue healing and cancer spread.

## Contribution

A data-driven framework that infers governing equations for cell migration directly from live-cell imaging data.

## Key findings

- Cells use spatiotemporal derivatives of signals to guide their movement.
- Cell-cell heterogeneity exists in how migratory rules are applied.
- Front cells respond to spatial gradients, while interior cells react to temporal changes.

## Abstract

Collective cell migration is fundamental to tissue homeostasis and underlies biological processes such as wound healing and cancer invasion. Previous work has proposed governing equations to describe how chemical and mechanical inputs regulate these movements, but the quantitative validity of such models remains to be thoroughly assessed. Here, we developed a machine-learning framework that infers the governing equation from live-cell imaging data. Applied to epithelial sheet migration driven by MAPK/ERK, our approach quantitatively predicted single-cell movement from local chemical and mechanical cues. Examination of the learned equations further indicated that cells process environmental signals by computing their spatiotemporal derivatives. Moreover, when applied to individual cells, our framework revealed cell-cell heterogeneity in the underlying migratory rules. Our framework offers a powerful tool for predictive modeling of multicellular dynamics in both physiological and pathological settings.

Tissues heal and cancers invade when cells move together, yet we still lack a testable account of how chemical and mechanical cues combine to steer this motion. We created a data-driven framework that learns the governing rule directly from live cell movies. Applied to epithelial sheets where MAPK/ERK waves drive collective migration, the learned rule predicts each cell’s next movement from local chemical and mechanical information. Interrogating the rule shows that cells rely on absolute signal levels and on how those signals change across space and time: gradients and rates of change. The framework also reveals cell to cell differences in these migratory rules and a clear front versus interior distinction within the same sheet. Cells at the wound edge respond mainly to the instantaneous spatial gradient, whereas interior cells are more sensitive to recent temporal changes. Finally, we used the learned rule to refine a forward model, improving its ability to reproduce observed behavior. By translating raw imaging data into predictive rules, our approach provides a general tool to connect data with mechanism in multicellular dynamics and to inform quantitative studies of tissue repair and disease.