Does cellular adaptation to force loading rate determine the biphasic vs monotonic response of actin retrograde flow with substrate rigidity?

The transmission of cytoskeletal forces to the extracellular matrix through focal adhesion complexes is essential for a multitude of biological processes such as cell migration, differentiation, tissue development, cancer progression, among others. During migration, focal adhesions arrest the actin retrograde flow towards the cell interior, allowing the cell front to move forward. Here, we address a puzzling observation of the existence of two distinct phenomena: a biphasic relationship of the retrograde flow and cell traction force with increasing substrate rigidity, with maximum traction force and minimum retrograde flow velocity being present at an optimal substrate stiffness; in contrast, a monotonic relationship between them where the retrograde flow decreases and traction force increases with substrate stiffness. We propose a theoretical model for cell-matrix adhesions at the leading edge of a migrating cell, incorporating a novel approach in force loading rate sensitive binding and reinforcement of focal adhesions assembly and the subsequent force-induced slowing down of actin flow. Our model unravels both biphasic and monotonic responses of the retrograde flow and cell traction force with increasing substrate rigidity, owing to the cells ability to sense and adapt to the fast-growing forces. Moreover, we also elucidate how the viscoelastic properties of the substrate regulate these nonlinear responses and alter cellular behaviours.