# Signaling gradients in surface dynamics as basis for planarian regeneration

## TL;DR

This paper presents a mathematical model for planarian regeneration that captures polarity preservation and size-invariant regeneration by incorporating long-range signaling gradients and cell differentiation dynamics.

## Contribution

It introduces a novel differential equation model that accurately reproduces experimental regeneration behaviors, including polarity preservation and size robustness, improving upon previous models.

## Key findings

- Model reproduces cut and graft experiments accurately
- Preserves polarity over large size variations
- Highlights importance of wnt-related signaling gradients

## Abstract

We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt-related signaling gradient. We motivate our model in relation to experimental data and demonstrate how it correctly reproduces cut and graft experiments. In particular, our system improves on previous models by preserving polarity in regeneration, over orders of magnitude in body size during cutting experiments and growth phases. Our model relies on tristability in cell density dynamics, between head, trunk, and tail. In addition, key to polarity preservation in regeneration, our system includes sensitivity of cell differentiation to gradients of wnt-related signals %measured relative to the tissue surface. This process is particularly relevant in a small tissue layer close to wounds during their healing, and modeled here in a robust fashion through dynamic boundary conditions.