A "morphogenetic action" principle for 3D shape formation by the growth of thin sheets

TL;DR

This paper introduces a mathematical framework using quasiconformal transformations to predict 3D shape formation by growing thin sheets, suggesting that nature favors growth patterns that minimize variation and optimize deformation.

Contribution

It formulates growth pattern selection as an optimization problem, providing a predictive model for shape development and cell displacement in biological tissues.

Findings

Growth patterns can be predicted by minimizing spatiotemporal variation in growth rates.

Uniformized growth rates can be achieved by introducing anisotropy.

The model explains the prevalence of anisotropic growth in development.

Abstract

How does growth encode form in developing organisms? Many different spatiotemporal growth profiles may sculpt tissues into the same target 3D shapes, but only specific growth patterns are observed in animal and plant development. In particular, growth profiles may differ in their degree of spatial variation and growth anisotropy, however, the criteria that distinguish observed patterns of growth from other possible alternatives are not understood. Here we exploit the mathematical formalism of quasiconformal transformations to formulate the problem of "growth pattern selection" quantitatively in the context of 3D shape formation by growing 2D epithelial sheets. We propose that nature settles on growth patterns that are the 'simplest' in a certain way. Specifically, 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 cell displacement along those surfaces in the process of growth. Optimization of growth trajectories for both idealized surfaces and those observed in nature show that relative growth rates can be uniformized at the cost of introducing anisotropy. Minimizing the variation of programmed growth rates can therefore be viewed as a generic mechanism for growth pattern selection and may help to understand the prevalence of anisotropy in developmental programs.