Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.

中文翻译：

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三维顶点模型的能量泛函的单细胞和两细胞形状分析


顶点模型已广泛用于模拟多细胞系统的演化，并产生了有关其宏观流变学或干扰转变的重要全局特性。这些模型基于能量泛函的定义，它完全决定了细胞的反应和结论。虽然二维顶点模型已被广泛使用，但三维模型却要稀缺得多，这主要是因为它们可能采用大量的配置以及它们经历的复杂的几何过渡。我们在这里研究单细胞和两细胞构型的形状作为能量项的函数，并研究最终形状对模型参数的依赖性：即惩罚细胞-细胞粘附和表面收缩性项的指数。在单细胞分析中，我们分析推导了 2D 和 3D 中线性和二次表面能项的收缩性半径和极限值。在对称和不对称的双细胞系统中，我们推导出长宽比和相对半径的演变。虽然在具有线性曲面项的泛函中，在 2D 和 3D 中产生相同的纵横比，但使用二次曲面项时的配置是不同的。我们将我们的结果与毛细管理论的著名解决方案联系起来，并用三维顶点模型验证我们的分析结果。