A key challenge in biology is to understand how spatio-temporal patterns and structures arise during the development of an organism. An initial aggregate of spatially uniform cells develops and forms the differentiated structures of a fully developed organism. On the one hand, contact-dependent cell–cell signalling is responsible for generating a large number of complex, self-organized, spatial patterns in the distribution of the signalling molecules. On the other hand, the motility of cells coupled with their polarity can independently lead to collective motion patterns that depend on mechanical parameters influencing tissue deformation, such as cellular elasticity, cell–cell adhesion and active forces generated by actin and myosin dynamics. Although modelling efforts have, thus far, treated cell motility and cell–cell signalling separately, experiments in recent years suggest that these processes could be tightly coupled. Hence, in this paper, we study how the dynamics of cell polarity and migration influence the spatiotemporal patterning of signalling molecules. Such signalling interactions can occur only between cells that are in physical contact, either directly at the junctions of adjacent cells or through cellular protrusional contacts. We present a vertex model which accounts for contact-dependent signalling between adjacent cells and between non-adjacent neighbours through long protrusional contacts that occur along the orientation of cell polarization. We observe a rich variety of spatiotemporal patterns of signalling molecules that is influenced by polarity dynamics of the cells, relative strengths of adjacent and non-adjacent signalling interactions, range of polarized interaction, signalling activation threshold, relative time scales of signalling and polarity orientation, and cell motility. Though our results are developed in the context of Delta–Notch signalling, they are sufficiently general and can be extended to other contact dependent morpho-mechanical dynamics.

生物学的一个关键挑战是了解时空模式和结构是如何在有机体发育过程中出现的。空间上均匀的细胞的初始聚集体发育并形成完全发育的生物体的分化结构。一方面，依赖于接触的细胞间信号传导负责在信号分子的分布中产生大量复杂的、自组织的空间模式。另一方面，细胞的运动及其极性可以独立地导致集体运动模式，该模式取决于影响组织变形的机械参数，例如细胞弹性、细胞-细胞粘附以及肌动蛋白和肌球蛋白动力学产生的主动力。尽管到目前为止，建模工作已经分别处理了细胞运动和细胞间信号传导，近年来的实验表明，这些过程可以紧密耦合。因此，在本文中，我们研究了细胞极性和迁移的动力学如何影响信号分子的时空模式。这种信号相互作用只能发生在物理接触的细胞之间，直接在相邻细胞的连接处或通过细胞突起接触。我们提出了一个顶点模型，该模型通过沿细胞极化方向发生的长突起接触来解释相邻细胞之间和非相邻相邻细胞之间的依赖于接触的信号传导。我们观察到信号分子的多种时空模式，这些模式受细胞极性动力学、相邻和非相邻信号相互作用的相对强度、极化相互作用的范围、信号激活阈值、信号和极性方向的相对时间尺度以及细胞运动。虽然我们的结果是在 Delta-Notch 信号的背景下开发的，但它们足够通用，可以扩展到其他依赖于接触的形态力学动力学。