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首页> 美国卫生研究院文献>Biophysical Journal >From Discrete to Continuum Models of Three-Dimensional Deformations in Epithelial Sheets
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From Discrete to Continuum Models of Three-Dimensional Deformations in Epithelial Sheets

机译：从离散到上皮片三维变形的连续体模型

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Epithelial tissue, in which cells adhere tightly to each other and to the underlying substrate, is one of the four major tissue types in adult organisms. In embryos, epithelial sheets serve as versatile substrates during the formation of developing organs. Some aspects of epithelial morphogenesis can be adequately described using vertex models, in which the two-dimensional arrangement of epithelial cells is approximated by a polygonal lattice with an energy that has contributions reflecting the properties of individual cells and their interactions. Previous studies with such models have largely focused on dynamics confined to two spatial dimensions and analyzed them numerically. We show how these models can be extended to account for three-dimensional deformations and studied analytically. Starting from the extended model, we derive a continuum plate description of cell sheets, in which the effective tissue properties, such as bending rigidity, are related explicitly to the parameters of the vertex model. To derive the continuum plate model, we duly take into account a microscopic shift between the two sublattices of the hexagonal network, which has been ignored in previous work. As an application of the continuum model, we analyze tissue buckling by a line tension applied along a circular contour, a simplified set-up relevant to several situations in the developmental contexts. The buckling thresholds predicted by the continuum description are in good agreement with the results of stability calculations based on the vertex model. Our results establish a direct connection between discrete and continuum descriptions of cell sheets and can be used to probe a wide range of morphogenetic processes in epithelial tissues.
机译：上皮组织是成年生物中四种主要的组织类型之一，在上皮组织中，细胞彼此之间以及与基底的紧密粘附。在胚胎中，上皮层在发育器官形成过程中充当多功能底物。上皮形态发生的某些方面可以使用顶点模型来充分描述，其中上皮细胞的二维排列由多边形晶格近似，该多边形晶格的能量具有反映单个细胞及其相互作用的特性的作用。以前使用此类模型进行的研究主要集中在限于两个空间维度的动力学上，并对其进行了数值分析。我们展示了如何扩展这些模型以解决三维变形并进行分析研究。从扩展模型开始，我们获得了细胞片的连续板描述，其中有效的组织特性（例如弯曲刚度）与顶点模型的参数明确相关。为了导出连续板模型，我们充分考虑了六边形网络的两个子晶格之间的微观位移，这在先前的工作中已被忽略。作为连续模型的应用，我们通过沿圆形轮廓施加线张力来分析组织屈曲，这是一种与发展环境中的几种情况相关的简化设置。连续描述所预测的屈曲阈值与基于顶点模型的稳定性计算结果非常吻合。我们的结果在细胞片的离散和连续描述之间建立了直接的联系，可用于探测上皮组织中广泛的形态发生过程。

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