Traversing Geodesics to Grow Biological Structures

Biological tissues reliably grow into precise, functional structures from simple starting states during development. Throughout the developmental process, the energy of a tissue changes depending on its natural resistance to deformations such as stretching, bending, shearing, and torsion. In this paper, we represent tissue structures as shapes and develop a mathematical framework to discover paths on the tissue shape manifold to minimize the total energy during development. We find that paths discovered by gradient descent and the geodesic algorithm outperform naive shape interpolation in energetic terms and resemble strategies observed in development. Broadly, these tools can be used to understand and compare shape transformations in biology and propose optimal strategies for synthetic tissue engineering.