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

• A mathematical model for viscoelastic cell-matrix adhesion mechanics. • Elastic adhesion plaque, Kelvin-Voigt interface, standard linear solid substrate. • Equations solved analytically in the Laplace domain with internal length analysis. • Analytical formula for frequency-dependent internal length, possibly non-monotonic. • Simulations highlight the relevance of viscoelasticity in adhesion dynamics. 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 derive a system of coupled partial differential equations and show that they can be solved analytically 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 demonstrate 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 determines 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.