An elastic-damaging cohesive law for cell–substrate adhesion with positive and negative durotaxis

Durotaxis of cells anchored to the extracellular matrix through focal adhesions has been systematically studied through both analytical and computational approaches. However, recent experiments have revealed the attitude of certain cells to unexpectedly migrate towards comparatively softer substrates, thus suggesting the possibility for negative durotaxis to manifest. Cell migration is possible because focal adhesions grow and disrupt, thus operating like adhesive structures undergoing a chemo-physical degradation process. In the present contribution, this degradation process is described through an elastic-damaging cohesive law deduced from a convex–concave pseudo-elastic potential, which confers a variational structure to the mechanical model of the adhesion structure and makes the derivation of analytical solutions possible. Furthermore, the obtained traction-separation cohesive law is amenable to a straightforward implementation into finite element codes. Finite elasticity of the cell body is considered while durotaxis is triggered by applying a contractile pre-stretch to the cell. It is shown that displacement- or force-driven degradation processes may lead to different kinds of durotaxis. The consistency and effectiveness of the proposed approach are showcased in one- and three-dimensional examples of cell–substrate systems.