  Stick-slip motion, a common phenomenon observed during crawling of cells, is
found to be strongly sensitive to the substrate stiffness. Stick-slip
behaviours have previously been investigated typically using purely elastic
substrates. For a more realistic understanding of this phenomenon, we propose a
theoretical model to study the dynamics on a viscoelastic substrate. Our model
based on a reaction-diffusion framework, incorporates known important
interactions such as retrograde flow of actin, myosin contractility, force
dependent assembly and disassembly of focal adhesions coupled with
cell-substrate interaction. We show that consideration of a viscoelastic
substrate not only captures the usually observed stick-slip jumps, but also
predicts the existence of an optimal substrate viscosity corresponding to
maximum traction force and minimum retrograde flow which was hitherto
unexplored. Moreover, our theory predicts the time evolution of individual bond
force that characterizes the stick-slip patterns on soft versus stiff
substrates. Our analysis also elucidates how the duration of the stick-slip
cycles are affected by various cellular parameters.
