Anisotropic Young-Laplace-equation provides insight into tissue growth

Growing tissues are highly dynamic, flowing on sufficiently long time-scales due to cell proliferation, migration and tissue remodeling. As a consequence, living tissues can be approximated as liquids. This means the shape of microtissues is governed by a surface stress state as in fluid droplets. Recent work showed that cells in the near-surface region of fibroblastic or osteoblastic microtissues contract with highly oriented actin filaments, thus making the surface stress state anisotropic, in contrast to what is expected for an isotropic fluid. Here, we extend the Young-Laplace law to include mechanical anisotropy of the surface. We then take this into account to determine equilibrium shapes of rotationally symmetric bodies subjected to anisotropic surface stress states and derive a family of surfaces of revolution in analogy to the Delaunay surfaces. A comparison with recently published shapes of microtissues shows that this theory accurately predicts both the surface shape and the direction of the actin filaments in the surface. The anisotropic version of the Young-Laplace law might help describing the growth of microtissues and also predicts the shape of fluid bodies with highly anisotropic surface properties.