Geometry, Analysis and Morphogenesis: Problems and Prospects

TL;DR

This paper reviews the interdisciplinary study of biological shape formation, integrating geometry, analysis, and mechanics to understand and predict morphogenesis of various biological structures.

Contribution

It synthesizes current mathematical and physical theories on morphogenesis, linking classical embedding problems with biological shape development, and discusses open problems for future research.

Findings

Analysis of curvature-driven patterning in elastic films.

Asymptotic behavior of solutions as film thickness approaches zero.

Connections between Nash embedding problem and biological morphogenesis.

Abstract

The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.