Viscoelastic dynamics of cell-matrix adhesions: Insights from a one-dimensional perspective

We develop a mathematical treatment of viscoelastic cell-matrix interactions via a one- dimensional continuum-mechanical model within a thermodynamically consistent framework. The model explicitly incorporates the time-dependent behaviour of both the adhesion interface and substrate, with the former modelled as a Kelvin-Voigt material and the latter as a standard linear solid (i.e., a parallel arrangement of an elastic spring and a Maxwell element). We de- rive a system of coupled partial differential equations and show that they can be solved analyti- cally in the Laplace domain, yielding a concise expression for the time-dependent internal length scale that mediates force transmission. We also perform numerical simulations to highlight the model’s main features in prototypical cases. Our theoretical and numerical predictions demon- strate that the coupling between interface and substrate viscoelasticity modulates the system’s time-dependent response to a mechanical stimulus applied to the substrate and critically deter- mines how much deformation is transmitted to the adhesion plaque. These findings align with experimental observations of frequency-dependent cellular responses and provide mechanistic insights into phenomena that purely elastic models cannot capture.