Computational modeling of eukaryotic cells moving on substrates is an extraordinarily complex task: many physical processes, such as actin polymerization, action of motors, formation of adhesive contacts concomitant with both substrate deformation and recruitment of actin etc., as well as regulatory pathways are intertwined. Moreover, highly nontrivial cell responses emerge when the substrate becomes deformable and/or heterogeneous. Here we extended a computational model for motile cell fragments, based on an earlier developed phase field approach, to account for explicit dynamics of adhesion site formation, as well as for substrate compliance via an effective elastic spring. Our model displays steady motion vs. stick-slip transitions with concomitant shape oscillations as a function of the actin protrusion rate, the substrate stiffness, and the rates of adhesion. Implementing a step in the substrate’s elastic modulus, as well as periodic patterned surfaces exemplified by alternating stripes of high and low adhesiveness, we were able to reproduce the correct motility modes and shape phenomenology found experimentally. We also predict the following nontrivial behavior: the direction of motion of cells can switch from parallel to perpendicular to the stripes as a function of both the adhesion strength and the width ratio of adhesive to non-adhesive stripes.
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