Incompatibility-driven growth and size control during development

Abstract

Size regulation in living organisms is a major unsolved problem in developmental biology. This is due to the intrinsic complexity of biological growth, which simultaneously involves genetic, biochemical, and mechanical factors. In this article, we propose a novel theoretical framework that explores the role of incompatibility, the geometric source of residual stress in a growing body, as a possible regulator of size termination during development. We explore this paradigm both at the level of a model 2D cell, and at the level of continuous tissues. After establishing a parallel between incompatibility and the shape parameter of vertex models, we show that incompatibility-driven growth leads to size control in a model 2D cell. We then extend the same paradigm to the level of continuous bodies, where incompatibility is measured by the Ricci curvature of the growth tensor. By using the model 2D cell as a template, we now derive an evolutionary law for the growth tensor with curvature fixed at a physiological value. When the analysis is specialised to radial symmetry (discs and spheres), this model captures the salient features observed in Drosophila wing discs and multicellular spheroids: these systems have a target size and build up residual stresses that cause the tissue to open in response to a radial cut, with the cut edges curling outward. The theory proposed in this work suggests that incompatibility in a growing biological tissue is potentially controllable at the cell level, and that incompatibility-driven growth provides an effective method of controlling global information (stress, size) through local geometric controls.