Does cellular ability to adapt fast varying forces determine the biphasic vs monotonic behaviour of actin retrograde flow with substrate stiffness?

Cell migration plays an important role in diverse biological processes, such as embryonic development, morphogenesis, wound healing, and tissue regeneration. During migration, focal adhesions arrest actin retrograde flow towards the cell interior, allowing actin polymerization and cell protrusion to advance at the cell front. Here, we present a theoretical model for cell-matrix adhesions at the leading edge of a crawling cell and address a puzzling observation of the biphasic versus monotonic relationship of the retrograde flow with increasing substrate rigidity. Two distinct phenomena have been observed - a biphasic behaviour of the retrograde flow and cell traction force with increasing substrate rigidity, with maximum traction force and minimum retrograde flow velocity present at an optimal substrate stiffness. In contrast, a monotonic relationship between them where the retrograde flow decreases and traction force increases with the substrate stiffness. Our study shows that the difference in the cellular behaviours could arise due to the cell's ability to sense the fast varying force and adapt to the growing force loading rate through adhesion reinforcement mechanisms. Our theory further elucidates how the substrate viscosity along with substrate elasticity alter these nonlinear cellular responses.