IZMB, Theoretische Biologie, Universität Bonn, Bonn, Germany.
Bull Math Biol. 2010 Oct;72(7):1696-731. doi: 10.1007/s11538-009-9498-3. Epub 2010 Jan 16.
Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein-Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.
Voronoi 镶嵌已被用于模拟形态发生或癌变组织中细胞的几何排列，然而，到目前为止，仅使用平面超曲面作为细胞-细胞接触边界。为了再现实验观察到的分段球形边界形状，我们开发了一个一致的乘法加权距离函数理论框架，定义了具有不同半径的细胞体的广义有限 Voronoi 邻域，作为所得模型组织的不均匀生成器。细胞之间的相互作用由细胞接触边界上的粘附和排斥力密度表示。此外，在组织边缘的细胞边界上实现了突起运动力，随机扰动允许非确定性运动效应。对细胞中心位置和速度的随机微分方程系统的出现进行模拟表明，这种 Voronoi 方法生成真实细胞形状是可行的。在短暂接触的单个细胞对的极限情况下，对动力学非线性 Ornstein-Uhlenbeck 过程进行了分析研究。一般来说，观察到拓扑上不同的组织构象，在不同的时间尺度上表现出稳定性，并且通过适当的特征来量化组织的一致性。最后，得出一个论点，指出自然组织在细胞大小异质性和细胞薄片的扩展之间存在权衡。
