Topological graph description of multicellular dynamics based on vertex model. (21st January 2018) - Hashimoto, Atsushi; Nagao, Atsuki; Okuda, Satoru
Highlights: We introduce graph descriptions of multicellular topological dynamics in 2D and 3D space. The descriptions are based on the 2D/3D vertex models used for physical analyses. The descriptions soundly express the entire pattern of multicellular topological dynamics. The descriptions provide the mathematical guarantee of the 2D and 3D vertex models. Graphical abstract: Abstract: Vertex models are generally powerful tools for exploring biological insights into multicellular dynamics. In these models, a multicellular structure is represented by a network, which is dynamically rearranged using topological operations. Remarkably, the topological dynamics of the network are important in guaranteeing the results from the models and their biological implications. However, it remains unclear whether the entire pattern of multicellular topological dynamics can be accurately expressed by a set of operators in the models. Surprisingly, vertex models have been empirically used for several decades without any mathematical verification. In this study, we propose a rigorous two-/three-dimensional (2D/3D) vertex model to describe multicellular topological dynamics. To do this, we classify several types of vertex models from a graph-theoretic perspective. Based on the classification, mathematical analyses reveal several conditions that enable us to apply the operators accurately without topological errors. Under these conditions, the operators can completely express the entire pattern of multicellular topological dynamics. From these results, we newly propose rigorous 2D/3D vertex models that can be applied to general multicellular dynamics, and we clarify several points to verify the results obtained from previous models.