An action principle for the morphogenesis of thin sheets

How does growth encode form in developing organisms? Many different spatiotemporal growth profiles may sculpt 2D epithelial sheets into the same target 3D shapes, but only specific growth patterns are observed in animal and plant development. The criteria that select for these stereotypic growth patterns and the ubiquity of anisotropic growth remain poorly understood. We propose that nature settles on the 'simplest' growth patterns. Using the geometric formalism of quasiconformal transformations, we demonstrate that growth pattern selection can be formulated as an optimization problem and solved for the trajectories that minimize spatiotemporal variation in areal growth rates and deformation anisotropy. The result is a complete prediction for the growth of the surface, including not only a set of intermediate shapes, but also a prediction for how cells flow along those surfaces. Optimization of growth trajectories for both idealized surfaces and experimentally acquired data show that relative growth rates can be uniformized at the cost of introducing anisotropy. Minimizing complexity can therefore be viewed as a generic mechanism for growth pattern selection and may help to understand the prevalence of anisotropy in developmental programs. Application to appendage outgrowth in the crustacean Parhyale hawaiensis generates dynamic developmental trajectories that are consistent with experimentally observed growth patterns.