Towards a formal expression of morphogenesis: a mechanics based integration of cell growth at tissue scale

In a recent paper, Coen et al. [2004] proposed a qualitative framework to study morphogenesis and express the link between genetics and shape development. This paper is an attempt to formalise this approach. It explains, from a mathematical point of view, the decomposition of local growth into the elemental transformations (scaling, anisotropy and direction change) proposed by Coen et al. [2004]. We then show how these geometrical transformations can be integrated at tissue scale using mechanics to find a compromise between linked regions with different growth. To explicit the link between genetics and local growth, the local strain of each region is computed as an elastic deformation of cells under turgor pressure. Growth is then expressed as a material synthesis by cells in response to cell walls strain. This approach proposes a comprehensive link between local genetic information and global tissue shape. To illustrate this work, we applied the proposed mechanical formalism on a 2D tissue to model the growth of a particular flower structure, the carpel.