Regeneration in planaria: boundary dynamics and signaling gradients

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, central body parts, 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. The mechanism proposed here resolves a conundrum in modeling efforts. In fact, many models of spontaneous formation of finite-size structure in unstructured tissue allude to a activator-inhibitor mechanism to select a finite wavelength (e.g. models for regeneration in hydra). Such Turing type mechanisms however do not scale across several orders of magnitude as seen in the patterning of planarians and hydra, nor do they incorporate robust selection of polarity. (Joint work with Arnd Scheel and Christoph Tenbrock)