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.

中文翻译：

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开发过程中不兼容驱动的增长和尺寸控制

生物体的大小调节是发育生物学中尚未解决的主要问题。这是由于生物生长的内在复杂性，同时涉及遗传、生化和机械因素。在本文中，我们提出了一个新颖的理论框架，探讨了不相容性（生长体中残余应力的几何来源）作为发育过程中尺寸终止的可能调节因素的作用。我们在模型 2D 细胞水平和连续组织水平上探索这种范例。在建立不相容性和顶点模型的形状参数之间的相似性之后，我们表明不相容性驱动的增长导致模型二维单元中的尺寸控制。然后，我们将相同的范式扩展到连续体的水平，其中不相容性通过生长张量的里奇曲率来测量。通过使用模型二维细胞作为模板，我们现在推导出曲率固定为生理值的生长张量的进化定律。当分析专门针对径向对称（圆盘和球体）时，该模型捕获了在果蝇翼盘和多细胞球体中观察到的显着特征：这些系统具有目标尺寸并积累残余应力，导致组织响应于径向切割，切割边缘向外卷曲。这项工作提出的理论表明，生长的生物组织中的不相容性可能在细胞水平上可控，并且不相容性驱动的生长提供了一种通过局部几何控制来控制全局信息（应力、尺寸）的有效方法。