Morphogen Patterning in Dynamic Tissues
Embryogenesis integrates morphogenesis--coordinated cell movements--with morphogen patterning and cell differentiation. While largely studied independently, morphogenesis and patterning often unfold simultaneously in early embryos. Yet, how cell movements affect patterning remains unclear, as most pattern formation models assume static tissues. We address this gap by developing a mathematical framework for morphogen patterning in dynamic tissues, reformulating advection-reaction-diffusion models in cells reference frames--the most natural for signal interpretation and fate decisions. This framework (i) elucidates how morphogenesis mediates morphogen transport and compartmentalization: multicellular attractors enhance cell-cell diffusive transport, while repellers act as barriers, affecting cell fate induction and bifurcations. (ii) It formalizes cell-cell signaling ranges in dynamic tissues, deconfounding morphogenetic movements and identifying which cells can communicate. (iii) It provides two nondimensional numbers--typically distinct from the Peclet number--to assess when and where morphogenesis is relevant for patterning. (iv) It elucidates the generative role of cell density dynamics in patterning. We apply this framework to classic patterning models, morphogenetic motifs, and avian gastrulation data. Broadly, our work provides a quantitative perspective to rationalize dynamic tissue patterning in natural and synthetic embryos.