Emergence of biphasic versus monotonic response of actin retrograde flow and cell traction force with varying 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, cell differentiation, tissue development, and 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 vs a monotonic relationship of the retrograde flow and cell traction force with substrate rigidity. In the former, maximum traction force and minimum retrograde flow velocity are observed at an intermediate optimal substrate stiffness; while in the latter, the actin retrograde flow decreases and traction force increases with increasing 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 exhibits both biphasic and monotonic responses of the retrograde flow and cell traction force with increasing substrate rigidity, owing to the cell's ability to sense and adapt to the fast-growing forces. Furthermore, our analysis shows how competition between different timescales regulated by loading rate sensitivity influences the biphasic versus monotonic behavior and the emergence of optimal substrate rigidity in the biphasic scenario. We also elucidate how the viscoelastic properties of the substrate regulate these nonlinear responses and predict the loss of cell sensitivity to variation in substrate rigidity when adhesions are subjected to high forces.