  A mathematical model is proposed for shape evolution and locomotion of fish
epidermal keratocytes on elastic substrates. The model is based on
mechanosensing concepts: cells apply contractile forces onto the elastic
substrate, while cell shape evolution depends locally on the substrate stress
generated by themselves or external mechanical stimuli acting on the substrate.
We use the level set method to study the behavior of the model numerically, and
predict a number of distinct phenomena observed in experiments, such as (i)
symmetry breaking from the stationary centrosymmetric to the well-known
steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and
traveling waves in the lamellipodium leading edge (iii) response to mechanical
stress externally applied to the substrate (tensotaxis), (iv) changing
direction of motion towards an interface with a rigid substrate (durotaxis) and
(v) the configuration of substrate wrinkles induced by contractile forces
applied by the keratocyte.
