Biological processes, such as embryonic development, wound repair and cancer invasion, or bacterial swarming and fruiting body formation, involve collective motion of cells as a coordinated group. Collective cell motion of eukaryotic cells often includes interactions that result in polar alignment of cell velocities, while bacterial patterns typically show features of apolar velocity alignment. For analysing the population-scale effects of these different alignment mechanisms, various on- and off-lattice agent-based models have been introduced. However, discriminating model-specific artefacts from general features of collective cell motion is challenging. In this work, we focus on equivalence criteria at the population level to compare on- and off-lattice models. In particular, we define prototypic off- and on-lattice models of polar and apolar alignment, and show how to obtain an on-lattice from an off-lattice model of velocity alignment. By characterizing the behaviour and dynamical description of collective migration models at the macroscopic level, we suggest the type of phase transitions and possible patterns in the approximative macroscopic partial differential equation descriptions as informative equivalence criteria between on- and off-lattice models.

This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

生物过程，如胚胎发育、伤口修复和癌症侵袭，或细菌聚集和子实体形成，涉及细胞作为一个协调群体的集体运动。真核细胞的集体细胞运动通常包括导致细胞速度极性对齐的相互作用，而细菌模式通常显示非极性速度对齐的特征。为了分析这些不同对齐机制的人口规模效应，已经引入了各种基于晶格和非晶格代理的模型。然而，从集体细胞运动的一般特征中区分模型特定的人工制品具有挑战性。在这项工作中，我们专注于总体水平的等效标准，以比较格内和格外模型。特别是，我们定义了极坐标和非极坐标对齐的原型非晶格和非晶格模型，并展示了如何从速度对齐的非晶格模型中获得晶格上的模型。通过在宏观层面表征集体迁移模型的行为和动态描述，我们建议将近似宏观偏微分方程描述中的相变类型和可能的模式作为网格上和网格外模型之间的信息等价标准。

本文是主题问题“生物系统中集体迁移的多尺度分析和建模”的一部分。