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 ({\eta}) of a non-confluent embryonic tissue grows sharply until a critical cell packing fraction ({\phi}S). The increase in {\eta} up to {\phi}S is similar to the behavior observed in several glass-forming materials, which suggests that the cell dynamics is sluggish or glass-like. Surprisingly, {\eta} is a constant above {\phi}S. To determine the mechanism of this unusual dependence of {\eta} on {\phi}, 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 {\phi}S but also provides a relationship between equilibrium geometrical packing to the dramatic increase in the relaxation dynamics.