Free volume theory explains the unusual behavior of viscosity in a non-confluent tissue during morphogenesis

A recent experiment on zebrafish blastoderm morphogenesis showed that the viscosity (η) of a non-confluent embryonic tissue grows sharply until a critical cell packing fraction (ϕ). The increase in η up to ϕ is similar to the behavior observed in several glass-forming materials, which suggests that the cell dynamics is sluggish or glass-like. Surprisingly, η is a constant above ϕ. To determine the mechanism of this unusual dependence of η on ϕ, we performed extensive simulations using an agent-based model of a dense non-confluent two-dimensional tissue. We show that polydispersity in the cell size, and the propensity of the cells to deform, results in the saturation of the available free area per cell beyond a critical packing fraction. Saturation in the free space not only explains the viscosity plateau above ϕ but also provides a relationship between equilibrium geometrical packing to the dramatic increase in the relaxation dynamics.