  Many animal cells crawling on elastic substrates exhibit durotaxis and have
implications in several biological processes including tissue development, and
tumor progression. Here, we introduce a phenomenological model for durotactic
migration incorporating both elastic deformation-mediated cell-substrate
interactions and the stochasticity of cell migration. Our model is motivated by
the key observation in one of the first demonstrations of durotaxis: a single
contractile cell at an 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, dipolar traction forces on the underlying elastic substrate. The
resulting substrate deformations induce elastic interactions with mechanical
boundaries, captured by an elastic potential that depends on cell position and
orientation relative to the boundary. The potential is attractive or repulsive
depending on whether the mechanical boundary condition is clamped or free,
which represents the cell being on the softer or stiffer side, respectively, of
a confining boundary. The forces and torques from the interactions drive cells
to orient perpendicular (parallel) to the boundary and accumulate (deplete) at
the clamped (free) boundary, extent of which is determined by elastic potential
(A) and motility (Pe). While the elastic interaction drives durotaxis, cell
migratory movements such as random reorientation and self-propulsion enable the
cell from the attractive potential thereby reducing durotaxis. We define
metrics quantifying boundary accumulation and durotaxis and present a phase
diagram that identifies three possible regimes: durotaxis, adurotaxis without
accumulation and adurotaxis with motility-induced accumulation at a confining
boundary.
