Elastic interactions compete with persistent cell motility to drive durotaxis

Abstract

The directed migration of cells toward stiffer substrate regions or durotaxis is relevant to tissue development and tumor progression. Here, we introduce a phenomenological model for single cell durotaxis that incorporates both elastic deformation-mediated cell-substrate interactions and the stochasticity of cell migration. Our model is motivated by a key observation in an early demonstration of durotaxis: a single, contractile cell at a sharp interface between a softer and a stiffer region of an elastic substrate reorients and migrates towards the stiffer region. We model migrating cells as self-propelling, persistently motile agents that exert contractile traction forces on their elastic substrate. The resulting substrate deformations induce elastic interactions with mechanical boundaries, captured by an elastic potential. Cell dynamics are governed by two critical parameters: the strength of the traction-induced boundary interaction (A) and the persistence of cell motility (Pe). The resulting elastic forces and torques align cells perpendicular (parallel) to the boundary and accumulate (deplete) them at clamped (free) boundaries. A clamped boundary induces an attractive potential, promoting durotaxis, while a free boundary generates a repulsive potential, preventing anti-durotaxis. By analyzing steady-state position and orientation probabilities, we show how accumulation and depletion depend on elastic potential strength and motility. We compare our findings with biological microswimmers and other active particles that accumulate at confining boundaries. The model defines metrics for boundary accumulation and durotaxis, presenting a phase diagram with three regimes: durotaxis, adurotaxis, and motility-induced accumulation. Our model predicts how durotaxis depends on cell contractility and motility, offering insights and testable predictions for future experiments.