An Elementary Model of Focal Adhesion Detachment and Reattachment During Cell Reorientation Using Ideas from the Kinetics of Wiggly Energies

Abstract

AbstractA simple, transparent, two-dimensional, nonlinear model of cell reorientation is constructed in this paper. The cells are attached to a substrate by “focal adhesions” that transmit the deformation of the substrate to the “stress fibers” in the cell. When the substrate is subjected to a deformation, say an in-plane bi-axial deformation with stretches $\lambda _{1}$ λ 1 and $\lambda _{2}$ λ 2 , the stress fibers deform with it and change their length and orientation. In addition, the focal adhesions can detach from the substrate and reattach to it at new nearby locations, and this process of detachment and reattachment can happen many times. In this scenario the (varying) fiber angle $\Theta $ Θ in the reference configuration plays the role of an internal variable. In addition to the elastic energy of the stress fibers, the energy associated with the focal adhesions is accounted for by a wiggly energy $\epsilon a \cos \Theta /\epsilon $ ϵ a cos Θ / ϵ , $0 < \epsilon \ll 1$ 0 < ϵ ≪ 1 . Each local minimum of this energy corresponds to a particular configuration of the focal adhesions. The small amplitude $\epsilon a$ ϵ a indicates that the energy barrier between two neighboring configurations is relatively small, and the small distance $2 \pi \epsilon $ 2 π ϵ between the local minima indicates that a focal adhesion does not have to move very far before it reattaches. The evolution of this system is studied using a gradient flow kinetic law, which is homogenized for $\epsilon \to 0$ ϵ → 0 using results from weak convergence. The results determine $(a)$ ( a ) a region of the $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 -plane in which the (referential) fiber orientation remains stuck at the angle $\Theta $ Θ and does not evolve, and $(b)$ ( b ) the evolution of the orientation when the stretches move out of this region as the fibers seek to minimize energy.