Morphogenesis of active shells

We consider the active shell as a single-cell or epithelial sheet surface that, sharing basic properties of stretched elastic shells, is capable of active planar movement owing to recruiting of the new surface elements. As model examples of their morphogenesis, we consider the growth and differentiation of single-cell hairs (trichomes) in plants of the genus Draba, and the epiboly and formation of the dorsoventral polarity in loach. The essential feature of the active shell behavior at both cellular and supracellular levels is regular deviating from the spatially homogeneous form, which is a primary cause of originating of the active mechanical stresses inside the shell in addition to its passive stretching by the intrinsic forces. Analyzing the quantitative morphological data, we derive the equations in which the temporal self-oscillations and spatial differentiation are distinguishable only at the parametric level depending on the proportion of active to passive stresses. In contrast to the ordinary activator-inhibitor systems, the self-oscillation dynamics is principally non-local and, consequently, one-parametric, the shell surface curvature being an analog of the inhibitor, while its spatial variance being an analog of the activator of shaping. Analyzing variability and evolution of the hair cell branching, we argue that the linear ontogeny (succession of the developmental stages) is a secondary evolutionary phenomenon originating from cyclic self-organizing algorithms of the active shell shaping.