Modeling of viscous frictional force in multicellular morphogenesis

Abstract

In morphogenesis, mechanical forces are the fundamental drivers of tissue deformation. To elucidate the principles governing this process, a thorough understanding of these forces is indispensable, and mathematical models are invaluable tools for this purpose. Various models have been developed, including vertex models suitable for representing epithelial cell mechanics, cell particle models for simulating the dynamics of large cell populations, and deformable cell models for describing complicated cell shapes. In these models, the motion of the objects is usually assumed to be overdamped by viscous friction. The viscous frictional forces are likely provided by liquid medium or body fluid, and they are also exerted in cell–cell and cell–extracellular matrix contacts. Furthermore, if these components, including the ECM, are in motion, the viscous frictional forces are affected by their motions. Because there seems to be little consensus on how to model the viscous frictional forces, we organize and review them in this article where we exemplified the vertex model and the particle model.

Caption of Graphical Abstract

Top: A cell on a moving ECM region (green mesh) is displaced from its initial (magenta) to final (orange) position. Total displacement (“Cell observation”) is the sum of ECM motion and intrinsic cellular migration. Bottom: Viscous friction occurs between cells, ECM, and fluid. While friction in static environments depends on absolute velocity, in dynamic environments, it depends on the cell’s velocity relative to the moving surroundings.