Spatiotemporal modeling of focal adhesion growth on viscoelastic substrate

Focal adhesions (FAs) serve as key structures that mediate the transmission of mechanical forces between cells and the extracellular matrix (ECM). Recent in vitro studies have indicated that ECM viscoelasticity significantly regulates FA growth. However, the regulatory role of viscoelasticity in FA growth and its underlying spatiotemporal mechanisms remain incompletely understood. Herein, by integrating integrin internalization into the classical molecular clutch model, we systematically investigated how ECM viscoelastic parameters (additional stiffness k_a, long-term stiffness k_l, and viscosity η) affect FA growth. Theoretical results demonstrate that FA length increases sigmoidally with increasing k_a and k_l, rising from 0.2 μm to 2.9 μm as k_a increases from 0.01 to 100 pN/nm, and from 0.5 μm to 2.6 μm as k_l increases over the same range. This trend is consistent with existing experimental observations. In addition, the influence of η on FA exhibits stiffness dependence. On soft substrate, FA length increases from 0.5 μm to 2.5 μm as η decreases from 100 pN·s/nm to 0.1 pN·s/nm. Conversely, FA length becomes insensitive to η on stiff substrate. The macroscopic trends of cell adhesion experiments performed on four specific substrates are consistent with the model predictions, providing qualitative support for these model predictions. These results deepen our understanding of the effects of substrate viscoelasticity on FA growth and provides a certain theoretical foundation for the rational design of biomaterials with tailored cell-mechanical crosstalk.