Shear-lag mechanics of adhesive bias in durotaxis and viscotaxis

We propose a variational continuum model for mechanically biased adhesion of cell plaques, describing their interactions with a substrate exhibiting spatial gradients in stiffness (durotaxis), viscosity (viscotaxis), or both. The adhesion plaque is modelled as an elasto-active bar whose internal remodelling strain evolves in response to mechanical cues, while the substrate is represented as a spatially inhomogeneous viscoelastic solid of the Standard Linear Solid type. The mechanosensitive resistance to remodelling is assumed to depend nonlinearly on the interface force between the plaque and the substrate, governed by a linear elastic law. This coupling drives an inhomogeneous spatial response to mechanical gradients, thereby inducing a bias in cell active remodelling towards mechanically favourable regions of the substrate, as shown by numerical simulations. The model captures adhesive bias towards stiffer regions in the durotaxis case, and towards more viscous regions in the viscotaxis case. We also consider scenarios in which these gradients compete, showing the interplay of characteristic parameters in determining the direction of active bias. Comparisons with experiments show qualitative agreement, suggesting the proposed framework offers a plausible continuum alternative to clutch-based models.