Mechanotropism of single cells adhering to elastic substrates subject to exogenous forces

Adherent cells are able to actively generate internal forces, channelled by cytoskeletal protein filaments and transmitted through transmembrane receptors to the surrounding environment by means of focal adhesions. Cells also dynamically interact with extracellular matrix by sensing external chemo-mechanical stimuli and then inducing formation of stress fibres mediated by polymerization/de-polymerization processes which continuously redesign the interplay between structural organization and contractility activities in the cytoskeleton, so orchestrating selected signal pathways at the basis of many important cell's physiological functions like adhesion, migration and division. Despite chemo- and duro-taxis have been intensively studied in the last years to understand how cells move on a substrate, their polarization and reorientation – observed during gastrulation, wound healing and morphogenesis – are only partially understood. By starting from the evidence highlighted in some recent studies that cells reorient in response to substrata deformation by essentially obeying a pure mechanical interaction, we propose to interpret the seeming overall cells’ rotation resulting from the reconfiguration of their cytoskeleton – here named mechanotropism – as a nonlinear optimization problem in which the adherent cell aims to align along directions leading it to minimize a physically coherent measure of work spent to deform the elastic substrate while retaining a prescribed level of homeostatic contractile force. To do this, the single-cell is simply modelled as a finite size contractile force dipole that acts on the boundary of a half-space perturbed by applied point loads. The effects of fences of forces acting orthogonally and tangentially on the boundary of the adhesion medium and encircling the cell are then investigated, so obtaining solutions that predict multiple non-trivial cell polarizations as functions of number, direction, relative magnitude and distance from the cell-dipole of the forces, as well as of the substrate's Poisson ratio, with unprecedented outcomes under the hypothesis of auxetic (i.e. negative Poisson ratio) materials. The results lead to envisage that the proposed theoretical model might contribute to unveil the mechanobiological principles ruling in vivo cells orientation processes by guiding the design of novel experimental strategies and to conceive new mechanics-based markers for guessing cells’ pathological conditions from their mechanotropism.